#### Sample records for discontinuous galerkin methods

1. Hybridized Multiscale Discontinuous Galerkin Methods for Multiphysics

Science.gov (United States)

2015-09-14

local approximation spaces of the hybridizable discontinuous Galerkin methods with precomputed phases which are solutions of the eikonal equation in...geometrical optics. Second, we propose a systematic procedure for computing multiple solutions of the eikonal equation. Third, we utilize the eigenvalue

2. Discontinuous Galerkin Method for Hyperbolic Conservation Laws

KAUST Repository

Mousikou, Ioanna

2016-11-11

Hyperbolic conservation laws form a special class of partial differential equations. They describe phenomena that involve conserved quantities and their solutions show discontinuities which reflect the formation of shock waves. We consider one-dimensional systems of hyperbolic conservation laws and produce approximations using finite difference, finite volume and finite element methods. Due to stability issues of classical finite element methods for hyperbolic conservation laws, we study the discontinuous Galerkin method, which was recently introduced. The method involves completely discontinuous basis functions across each element and it can be considered as a combination of finite volume and finite element methods. We illustrate the implementation of discontinuous Galerkin method using Legendre polynomials, in case of scalar equations and in case of quasi-linear systems, and we review important theoretical results about stability and convergence of the method. The applications of finite volume and discontinuous Galerkin methods to linear and non-linear scalar equations, as well as to the system of elastodynamics, are exhibited.

3. Discontinuous Galerkin Method for Hyperbolic Conservation Laws

KAUST Repository

Mousikou, Ioanna

2016-01-01

Hyperbolic conservation laws form a special class of partial differential equations. They describe phenomena that involve conserved quantities and their solutions show discontinuities which reflect the formation of shock waves. We consider one-dimensional systems of hyperbolic conservation laws and produce approximations using finite difference, finite volume and finite element methods. Due to stability issues of classical finite element methods for hyperbolic conservation laws, we study the discontinuous Galerkin method, which was recently introduced. The method involves completely discontinuous basis functions across each element and it can be considered as a combination of finite volume and finite element methods. We illustrate the implementation of discontinuous Galerkin method using Legendre polynomials, in case of scalar equations and in case of quasi-linear systems, and we review important theoretical results about stability and convergence of the method. The applications of finite volume and discontinuous Galerkin methods to linear and non-linear scalar equations, as well as to the system of elastodynamics, are exhibited.

4. Discontinuous Galerkin finite element methods for hyperbolic differential equations

NARCIS (Netherlands)

van der Vegt, Jacobus J.W.; van der Ven, H.; Boelens, O.J.; Boelens, O.J.; Toro, E.F.

2002-01-01

In this paper a suryey is given of the important steps in the development of discontinuous Galerkin finite element methods for hyperbolic partial differential equations. Special attention is paid to the application of the discontinuous Galerkin method to the solution of the Euler equations of gas

5. Class of reconstructed discontinuous Galerkin methods in computational fluid dynamics

International Nuclear Information System (INIS)

Luo, Hong; Xia, Yidong; Nourgaliev, Robert

2011-01-01

A class of reconstructed discontinuous Galerkin (DG) methods is presented to solve compressible flow problems on arbitrary grids. The idea is to combine the efficiency of the reconstruction methods in finite volume methods and the accuracy of the DG methods to obtain a better numerical algorithm in computational fluid dynamics. The beauty of the resulting reconstructed discontinuous Galerkin (RDG) methods is that they provide a unified formulation for both finite volume and DG methods, and contain both classical finite volume and standard DG methods as two special cases of the RDG methods, and thus allow for a direct efficiency comparison. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via a so-called in-cell reconstruction process. The devised in-cell reconstruction is aimed to augment the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. These three reconstructed discontinuous Galerkin methods are used to compute a variety of compressible flow problems on arbitrary meshes to assess their accuracy. The numerical experiments demonstrate that all three reconstructed discontinuous Galerkin methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstructed DG method provides the best performance in terms of both accuracy, efficiency, and robustness. (author)

6. On cell entropy inequality for discontinuous Galerkin methods

Science.gov (United States)

Jiang, Guangshan; Shu, Chi-Wang

1993-01-01

We prove a cell entropy inequality for a class of high order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one dimensional scalar convex case.

DEFF Research Database (Denmark)

Grooss, Jesper; Hesthaven, Jan

2006-01-01

We present a discontinuous Galerkin method on a fully unstructured grid for the modeling of unsteady incompressible fluid flows with free surfaces. The surface is modeled by embedding and represented by a levelset. We discuss the discretization of the flow equations and the level set equation...

8. Clearance gap flow: Simulations by discontinuous Galerkin method and experiments

Czech Academy of Sciences Publication Activity Database

Hála, Jindřich; Luxa, Martin; Bublík, O.; Prausová, H.; Vimmr, J.

2016-01-01

Roč. 92, May (2016), 02073-02073 ISSN 2100-014X. [EFM14 – Experimental Fluid Mechanics 2014. Český Krumlov, 18.11.2014-21.11.2014] Institutional support: RVO:61388998 Keywords : compressible fluid flow * narrow channel flow * discontinuous Galerkin finite element method Subject RIV: BK - Fluid Dynamics

9. Modeling shallow water flows using the discontinuous Galerkin method

CERN Document Server

Khan, Abdul A

2014-01-01

Replacing the Traditional Physical Model Approach Computational models offer promise in improving the modeling of shallow water flows. As new techniques are considered, the process continues to change and evolve. Modeling Shallow Water Flows Using the Discontinuous Galerkin Method examines a technique that focuses on hyperbolic conservation laws and includes one-dimensional and two-dimensional shallow water flows and pollutant transports. Combines the Advantages of Finite Volume and Finite Element Methods This book explores the discontinuous Galerkin (DG) method, also known as the discontinuous finite element method, in depth. It introduces the DG method and its application to shallow water flows, as well as background information for implementing and applying this method for natural rivers. It considers dam-break problems, shock wave problems, and flows in different regimes (subcritical, supercritical, and transcritical). Readily Adaptable to the Real World While the DG method has been widely used in the fie...

10. A Streaming Language Implementation of the Discontinuous Galerkin Method

Science.gov (United States)

Barth, Timothy; Knight, Timothy

2005-01-01

We present a Brook streaming language implementation of the 3-D discontinuous Galerkin method for compressible fluid flow on tetrahedral meshes. Efficient implementation of the discontinuous Galerkin method using the streaming model of computation introduces several algorithmic design challenges. Using a cycle-accurate simulator, performance characteristics have been obtained for the Stanford Merrimac stream processor. The current Merrimac design achieves 128 Gflops per chip and the desktop board is populated with 16 chips yielding a peak performance of 2 Teraflops. Total parts cost for the desktop board is less than $20K. Current cycle-accurate simulations for discretizations of the 3-D compressible flow equations yield approximately 40-50% of the peak performance of the Merrimac streaming processor chip. Ongoing work includes the assessment of the performance of the same algorithm on the 2 Teraflop desktop board with a target goal of achieving 1 Teraflop performance. 11. A discontinuous Galerkin method on kinetic flocking models OpenAIRE Tan, Changhui 2014-01-01 We study kinetic representations of flocking models. They arise from agent-based models for self-organized dynamics, such as Cucker-Smale and Motsch-Tadmor models. We prove flocking behavior for the kinetic descriptions of flocking systems, which indicates a concentration in velocity variable in infinite time. We propose a discontinuous Galerkin method to treat the asymptotic$\\delta$-singularity, and construct high order positive preserving scheme to solve kinetic flocking systems. 12. Finite element and discontinuous Galerkin methods for transient wave equations CERN Document Server Cohen, Gary 2017-01-01 This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem ... 13. Planet-disc interactions with Discontinuous Galerkin Methods using GPUs Science.gov (United States) Velasco Romero, David A.; Veiga, Maria Han; Teyssier, Romain; Masset, Frédéric S. 2018-05-01 We present a two-dimensional Cartesian code based on high order discontinuous Galerkin methods, implemented to run in parallel over multiple GPUs. A simple planet-disc setup is used to compare the behaviour of our code against the behaviour found using the FARGO3D code with a polar mesh. We make use of the time dependence of the torque exerted by the disc on the planet as a mean to quantify the numerical viscosity of the code. We find that the numerical viscosity of the Keplerian flow can be as low as a few 10-8r2Ω, r and Ω being respectively the local orbital radius and frequency, for fifth order schemes and resolution of ˜10-2r. Although for a single disc problem a solution of low numerical viscosity can be obtained at lower computational cost with FARGO3D (which is nearly an order of magnitude faster than a fifth order method), discontinuous Galerkin methods appear promising to obtain solutions of low numerical viscosity in more complex situations where the flow cannot be captured on a polar or spherical mesh concentric with the disc. 14. Discontinuous Galerkin finite element method with anisotropic local grid refinement for inviscid compressible flows NARCIS (Netherlands) van der Vegt, Jacobus J.W.; van der Ven, H. 1998-01-01 A new discretization method for the three-dimensional Euler equations of gas dynamics is presented, which is based on the discontinuous Galerkin finite element method. Special attention is paid to an efficient implementation of the discontinuous Galerkin method that minimizes the number of flux 15. Adaptive discontinuous Galerkin methods for non-linear reactive flows CERN Document Server Uzunca, Murat 2016-01-01 The focus of this monograph is the development of space-time adaptive methods to solve the convection/reaction dominated non-stationary semi-linear advection diffusion reaction (ADR) equations with internal/boundary layers in an accurate and efficient way. After introducing the ADR equations and discontinuous Galerkin discretization, robust residual-based a posteriori error estimators in space and time are derived. The elliptic reconstruction technique is then utilized to derive the a posteriori error bounds for the fully discrete system and to obtain optimal orders of convergence. As coupled surface and subsurface flow over large space and time scales is described by (ADR) equation the methods described in this book are of high importance in many areas of Geosciences including oil and gas recovery, groundwater contamination and sustainable use of groundwater resources, storing greenhouse gases or radioactive waste in the subsurface. 16. Implementation of the entropy viscosity method with the discontinuous Galerkin method KAUST Repository Zingan, Valentin 2013-01-01 The notion of entropy viscosity method introduced in Guermond and Pasquetti [21] is extended to the discontinuous Galerkin framework for scalar conservation laws and the compressible Euler equations. © 2012 Elsevier B.V. 17. Interior penalty discontinuous Galerkin method for coupled elasto-acoustic media OpenAIRE Dudouit , Yohann; Giraud , Luc; Millot , Florence; Pernet , Sébastien 2016-01-01 We introduce a high order interior penalty discontinuous Galerkin scheme for the nu- merical solution of wave propagation in coupled elasto-acoustic media. A displacement formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same framework. Weakly imposing the correct transmission condition is achieved by the derivation of adapted numerical fluxes. This generalization does not weaken the discontinuous Galerkin method, thus hp-non-conforming m... 18. hpGEM -- A software framework for discontinuous Galerkin finite element methods NARCIS (Netherlands) Pesch, L.; Bell, A.; Sollie, W.E.H.; Ambati, V.R.; Bokhove, Onno; van der Vegt, Jacobus J.W. 2006-01-01 hpGEM, a novel framework for the implementation of discontinuous Galerkin finite element methods, is described. We present structures and methods that are common for many (discontinuous) finite element methods and show how we have implemented the components as an object-oriented framework. This 19. A discontinuous Galerkin method for solving transient Maxwell equations with nonlinear material properties KAUST Repository Sirenko, Kostyantyn; Asirim, Ozum Emre; Bagci, Hakan 2014-01-01 Discontinuous Galerkin time-domain method (DGTD) has been used extensively in computational electromagnetics for analyzing transient electromagnetic wave interactions on structures described with linear constitutive relations. DGTD expands unknown fields independently on disconnected mesh elements and uses numerical flux to realize information exchange between fields on different elements (J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Method, 2008). The numerical flux of choice for 'linear' Maxwell equations is the upwind flux, which mimics accurately the physical behavior of electromagnetic waves on discontinuous boundaries. It is obtained from the analytical solution of the Riemann problem defined on the boundary of two neighboring mesh elements. 20. A discontinuous Galerkin method for solving transient Maxwell equations with nonlinear material properties KAUST Repository Sirenko, Kostyantyn 2014-07-01 Discontinuous Galerkin time-domain method (DGTD) has been used extensively in computational electromagnetics for analyzing transient electromagnetic wave interactions on structures described with linear constitutive relations. DGTD expands unknown fields independently on disconnected mesh elements and uses numerical flux to realize information exchange between fields on different elements (J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Method, 2008). The numerical flux of choice for \\'linear\\' Maxwell equations is the upwind flux, which mimics accurately the physical behavior of electromagnetic waves on discontinuous boundaries. It is obtained from the analytical solution of the Riemann problem defined on the boundary of two neighboring mesh elements. 1. A second order discontinuous Galerkin method for advection on unstructured triangular meshes NARCIS (Netherlands) Geijselaers, Hubertus J.M.; Huetink, Han 2003-01-01 In this paper the advection of element data which are linearly distributed inside the elements is addressed. Across element boundaries the data are assumed discontinuous. The equations are discretized by the Discontinuous Galerkin method. For stability and accuracy at large step sizes (large values 2. Super-convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations Science.gov (United States) Atkins, Harold L. 2009-01-01 The practical benefits of the hyper-accuracy properties of the discontinuous Galerkin method are examined. In particular, we demonstrate that some flow attributes exhibit super-convergence even in the absence of any post-processing technique. Theoretical analysis suggest that flow features that are dominated by global propagation speeds and decay or growth rates should be super-convergent. Several discrete forms of the discontinuous Galerkin method are applied to the simulation of unsteady viscous flow over a two-dimensional cylinder. Convergence of the period of the naturally occurring oscillation is examined and shown to converge at 2p+1, where p is the polynomial degree of the discontinuous Galerkin basis. Comparisons are made between the different discretizations and with theoretical analysis. 3. A study on discontinuous Galerkin finite element methods for elliptic problems NARCIS (Netherlands) Janivita Joto Sudirham, J.J.S.; Sudirham, J.J.; van der Vegt, Jacobus J.W.; van Damme, Rudolf M.J. 2003-01-01 In this report we study several approaches of the discontinuous Galerkin finite element methods for elliptic problems. An important aspect in these formulations is the use of a lifting operator, for which we present an efficient numerical approximation technique. Numerical experiments for two 4. Discontinuous Galerkin finite element methods for radiative transfer in spherical symmetry Science.gov (United States) Kitzmann, D.; Bolte, J.; Patzer, A. B. C. 2016-11-01 The discontinuous Galerkin finite element method (DG-FEM) is successfully applied to treat a broad variety of transport problems numerically. In this work, we use the full capacity of the DG-FEM to solve the radiative transfer equation in spherical symmetry. We present a discontinuous Galerkin method to directly solve the spherically symmetric radiative transfer equation as a two-dimensional problem. The transport equation in spherical atmospheres is more complicated than in the plane-parallel case owing to the appearance of an additional derivative with respect to the polar angle. The DG-FEM formalism allows for the exact integration of arbitrarily complex scattering phase functions, independent of the angular mesh resolution. We show that the discontinuous Galerkin method is able to describe accurately the radiative transfer in extended atmospheres and to capture discontinuities or complex scattering behaviour which might be present in the solution of certain radiative transfer tasks and can, therefore, cause severe numerical problems for other radiative transfer solution methods. 5. A second order discontinuous Galerkin fast sweeping method for Eikonal equations Science.gov (United States) Li, Fengyan; Shu, Chi-Wang; Zhang, Yong-Tao; Zhao, Hongkai 2008-09-01 In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin (DG) local solver for computing viscosity solutions of a class of static Hamilton-Jacobi equations, namely the Eikonal equations. Our piecewise linear DG local solver is built on a DG method developed recently [Y. Cheng, C.-W. Shu, A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics 223 (2007) 398-415] for the time-dependent Hamilton-Jacobi equations. The causality property of Eikonal equations is incorporated into the design of this solver. The resulting local nonlinear system in the Gauss-Seidel iterations is a simple quadratic system and can be solved explicitly. The compactness of the DG method and the fast sweeping strategy lead to fast convergence of the new scheme for Eikonal equations. Extensive numerical examples verify efficiency, convergence and second order accuracy of the proposed method. 6. Tensor-product preconditioners for higher-order space-time discontinuous Galerkin methods Science.gov (United States) Diosady, Laslo T.; Murman, Scott M. 2017-02-01 A space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equations. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high-order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows. 7. Tensor-Product Preconditioners for Higher-Order Space-Time Discontinuous Galerkin Methods Science.gov (United States) Diosady, Laslo T.; Murman, Scott M. 2016-01-01 space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equat ions. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows. 8. Discontinuous Galerkin for the Radiative Transport Equation KAUST Repository Guermond, Jean-Luc; Kanschat, Guido; Ragusa, Jean C. 2013-01-01 This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed. 9. Discontinuous Galerkin for the Radiative Transport Equation KAUST Repository Guermond, Jean-Luc 2013-10-11 This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed. 10. Optimal convergence of discontinuous Galerkin methods for continuum modeling of supply chain networks KAUST Repository Zhang, Shuhua; Sun, Shuyu; Yang, Hongtao 2014-01-01 A discontinuous Galerkin method is considered to simulate materials flow in a supply chain network problem which is governed by a system of conservation laws. By means of a novel interpolation and superclose analysis technique, the optimal and superconvergence error estimates are established under two physically meaningful assumptions on the connectivity matrix. Numerical examples are presented to validate the theoretical results. © 2014 Elsevier Ltd. All rights reserved. 11. Optimal convergence of discontinuous Galerkin methods for continuum modeling of supply chain networks KAUST Repository Zhang, Shuhua 2014-09-01 A discontinuous Galerkin method is considered to simulate materials flow in a supply chain network problem which is governed by a system of conservation laws. By means of a novel interpolation and superclose analysis technique, the optimal and superconvergence error estimates are established under two physically meaningful assumptions on the connectivity matrix. Numerical examples are presented to validate the theoretical results. © 2014 Elsevier Ltd. All rights reserved. 12. Penyelesaian Numerik Persamaan Advection Dengan Radial Point Interpolation Method dan Integrasi Waktu Dengan Discontinuous Galerkin Method Directory of Open Access Journals (Sweden) Kresno Wikan Sadono 2016-12-01 Full Text Available Persamaan differensial banyak digunakan untuk menggambarkan berbagai fenomena dalam bidang sains dan rekayasa. Berbagai masalah komplek dalam kehidupan sehari-hari dapat dimodelkan dengan persamaan differensial dan diselesaikan dengan metode numerik. Salah satu metode numerik, yaitu metode meshfree atau meshless berkembang akhir-akhir ini, tanpa proses pembuatan elemen pada domain. Penelitian ini menggabungkan metode meshless yaitu radial basis point interpolation method (RPIM dengan integrasi waktu discontinuous Galerkin method (DGM, metode ini disebut RPIM-DGM. Metode RPIM-DGM diaplikasikan pada advection equation pada satu dimensi. RPIM menggunakan basis function multiquadratic function (MQ dan integrasi waktu diturunkan untuk linear-DGM maupun quadratic-DGM. Hasil simulasi menunjukkan, metode ini mendekati hasil analitis dengan baik. Hasil simulasi numerik dengan RPIM DGM menunjukkan semakin banyak node dan semakin kecil time increment menunjukkan hasil numerik semakin akurat. Hasil lain menunjukkan, integrasi numerik dengan quadratic-DGM untuk suatu time increment dan jumlah node tertentu semakin meningkatkan akurasi dibandingkan dengan linear-DGM. [Title: Numerical solution of advection equation with radial basis interpolation method and discontinuous Galerkin method for time integration] Differential equation is widely used to describe a variety of phenomena in science and engineering. A variety of complex issues in everyday life can be modeled with differential equations and solved by numerical method. One of the numerical methods, the method meshfree or meshless developing lately, without making use of the elements in the domain. The research combines methods meshless, i.e. radial basis point interpolation method with discontinuous Galerkin method as time integration method. This method is called RPIM-DGM. The RPIM-DGM applied to one dimension advection equation. The RPIM using basis function multiquadratic function and time 13. Divergence-Conforming Discontinuous Galerkin Methods and$C^0$Interior Penalty Methods KAUST Repository Kanschat, Guido 2014-01-01 © 2014 Society for Industrial and Applied Mathematics. In this paper, we show that recently developed divergence-conforming methods for the Stokes problem have discrete stream functions. These stream functions in turn solve a continuous interior penalty problem for biharmonic equations. The equivalence is established for the most common methods in two dimensions based on interior penalty terms. Then, extensions of the concept to discontinuous Galerkin methods defined through lifting operators, for different weak formulations of the Stokes problem, and to three dimensions are discussed. Application of the equivalence result yields an optimal error estimate for the Stokes velocity without involving the pressure. Conversely, combined with a recent multigrid method for Stokes flow, we obtain a simple and uniform preconditioner for harmonic problems with simply supported and clamped boundary. 14. An efficient discontinuous Galerkin finite element method for highly accurate solution of maxwell equations KAUST Repository Liu, Meilin 2012-08-01 A discontinuous Galerkin finite element method (DG-FEM) with a highly accurate time integration scheme for solving Maxwell equations is presented. The new time integration scheme is in the form of traditional predictor-corrector algorithms, PE CE m, but it uses coefficients that are obtained using a numerical scheme with fully controllable accuracy. Numerical results demonstrate that the proposed DG-FEM uses larger time steps than DG-FEM with classical PE CE) m schemes when high accuracy, which could be obtained using high-order spatial discretization, is required. © 1963-2012 IEEE. 15. An efficient discontinuous Galerkin finite element method for highly accurate solution of maxwell equations KAUST Repository Liu, Meilin; Sirenko, Kostyantyn; Bagci, Hakan 2012-01-01 A discontinuous Galerkin finite element method (DG-FEM) with a highly accurate time integration scheme for solving Maxwell equations is presented. The new time integration scheme is in the form of traditional predictor-corrector algorithms, PE CE m, but it uses coefficients that are obtained using a numerical scheme with fully controllable accuracy. Numerical results demonstrate that the proposed DG-FEM uses larger time steps than DG-FEM with classical PE CE) m schemes when high accuracy, which could be obtained using high-order spatial discretization, is required. © 1963-2012 IEEE. 16. A Gas-kinetic Discontinuous Galerkin Method for Viscous Flow Equations International Nuclear Information System (INIS) Liu, Hongwei; Xu, Kun 2007-01-01 This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The construction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at the cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gaskinetic BGK model and the Navier-Stokes equations, the Navier-Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (10) and two dimensional(20) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method 17. Discontinuous Galerkin method for computing gravitational waveforms from extreme mass ratio binaries International Nuclear Information System (INIS) Field, Scott E; Hesthaven, Jan S; Lau, Stephen R 2009-01-01 Gravitational wave emission from extreme mass ratio binaries (EMRBs) should be detectable by the joint NASA-ESA LISA project, spurring interest in analytical and numerical methods for investigating EMRBs. We describe a discontinuous Galerkin (dG) method for solving the distributionally forced 1+1 wave equations which arise when modeling EMRBs via the perturbation theory of Schwarzschild black holes. Despite the presence of jump discontinuities in the relevant polar and axial gravitational 'master functions', our dG method achieves global spectral accuracy, provided that we know the instantaneous position, velocity and acceleration of the small particle. Here these variables are known, since we assume that the particle follows a timelike geodesic of the Schwarzschild geometry. We document the results of several numerical experiments testing our method, and in our concluding section discuss the possible inclusion of gravitational self-force effects. 18. Hybridizable discontinuous Galerkin method for the 2-D frequency-domain elastic wave equations Science.gov (United States) Bonnasse-Gahot, Marie; Calandra, Henri; Diaz, Julien; Lanteri, Stéphane 2018-04-01 Discontinuous Galerkin (DG) methods are nowadays actively studied and increasingly exploited for the simulation of large-scale time-domain (i.e. unsteady) seismic wave propagation problems. Although theoretically applicable to frequency-domain problems as well, their use in this context has been hampered by the potentially large number of coupled unknowns they incur, especially in the 3-D case, as compared to classical continuous finite element methods. In this paper, we address this issue in the framework of the so-called hybridizable discontinuous Galerkin (HDG) formulations. As a first step, we study an HDG method for the resolution of the frequency-domain elastic wave equations in the 2-D case. We describe the weak formulation of the method and provide some implementation details. The proposed HDG method is assessed numerically including a comparison with a classical upwind flux-based DG method, showing better overall computational efficiency as a result of the drastic reduction of the number of globally coupled unknowns in the resulting discrete HDG system. 19. Study of flow over object problems by a nodal discontinuous Galerkin-lattice Boltzmann method Science.gov (United States) Wu, Jie; Shen, Meng; Liu, Chen 2018-04-01 The flow over object problems are studied by a nodal discontinuous Galerkin-lattice Boltzmann method (NDG-LBM) in this work. Different from the standard lattice Boltzmann method, the current method applies the nodal discontinuous Galerkin method into the streaming process in LBM to solve the resultant pure convection equation, in which the spatial discretization is completed on unstructured grids and the low-storage explicit Runge-Kutta scheme is used for time marching. The present method then overcomes the disadvantage of standard LBM for depending on the uniform meshes. Moreover, the collision process in the LBM is completed by using the multiple-relaxation-time scheme. After the validation of the NDG-LBM by simulating the lid-driven cavity flow, the simulations of flows over a fixed circular cylinder, a stationary airfoil and rotating-stationary cylinders are performed. Good agreement of present results with previous results is achieved, which indicates that the current NDG-LBM is accurate and effective for flow over object problems. 20. Numerical solution of the Navier-Stokes equations by discontinuous Galerkin method Science.gov (United States) Krasnov, M. M.; Kuchugov, P. A.; E Ladonkina, M.; E Lutsky, A.; Tishkin, V. F. 2017-02-01 Detailed unstructured grids and numerical methods of high accuracy are frequently used in the numerical simulation of gasdynamic flows in areas with complex geometry. Galerkin method with discontinuous basis functions or Discontinuous Galerkin Method (DGM) works well in dealing with such problems. This approach offers a number of advantages inherent to both finite-element and finite-difference approximations. Moreover, the present paper shows that DGM schemes can be viewed as Godunov method extension to piecewise-polynomial functions. As is known, DGM involves significant computational complexity, and this brings up the question of ensuring the most effective use of all the computational capacity available. In order to speed up the calculations, operator programming method has been applied while creating the computational module. This approach makes possible compact encoding of mathematical formulas and facilitates the porting of programs to parallel architectures, such as NVidia CUDA and Intel Xeon Phi. With the software package, based on DGM, numerical simulations of supersonic flow past solid bodies has been carried out. The numerical results are in good agreement with the experimental ones. 1. galerkin's methods African Journals Online (AJOL) user The assumed deflection shapes used in the approximate methods such as in the Galerkin's method were normally ... to direct compressive forces Nx, was derived by Navier. [3]. ..... tend to give higher frequency and stiffness, as well as. 2. The direct Discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids Science.gov (United States) Yang, Xiaoquan; Cheng, Jian; Liu, Tiegang; Luo, Hong 2015-11-01 The direct discontinuous Galerkin (DDG) method based on a traditional discontinuous Galerkin (DG) formulation is extended and implemented for solving the compressible Navier-Stokes equations on arbitrary grids. Compared to the widely used second Bassi-Rebay (BR2) scheme for the discretization of diffusive fluxes, the DDG method has two attractive features: first, it is simple to implement as it is directly based on the weak form, and therefore there is no need for any local or global lifting operator; second, it can deliver comparable results, if not better than BR2 scheme, in a more efficient way with much less CPU time. Two approaches to perform the DDG flux for the Navier- Stokes equations are presented in this work, one is based on conservative variables, the other is based on primitive variables. In the implementation of the DDG method for arbitrary grid, the definition of mesh size plays a critical role as the formation of viscous flux explicitly depends on the geometry. A variety of test cases are presented to demonstrate the accuracy and efficiency of the DDG method for discretizing the viscous fluxes in the compressible Navier-Stokes equations on arbitrary grids. 3. Convergence of Discontinuous Galerkin Methods for Incompressible Two-Phase Flow in Heterogeneous Media KAUST Repository Kou, Jisheng; Sun, Shuyu 2013-01-01 A class of discontinuous Galerkin methods with interior penalties is presented for incompressible two-phase flow in heterogeneous porous media with capillary pressures. The semidiscrete approximate schemes for fully coupled system of two-phase flow are formulated. In highly heterogeneous permeable media, the saturation is discontinuous due to different capillary pressures, and therefore, the proposed methods incorporate the capillary pressures in the pressure equation instead of saturation equation. By introducing a coupling approach for stability and error estimates instead of the conventional separate analysis for pressure and saturation, the stability of the schemes in space and time and a priori hp error estimates are presented in the L2(H 1) for pressure and in the L∞(L2) and L2(H1) for saturation. Two time discretization schemes are introduced for effectively computing the discrete solutions. © 2013 Societ y for Industrial and Applied Mathematics. 4. Lagrangian Particle Tracking in a Discontinuous Galerkin Method for Hypersonic Reentry Flows in Dusty Environments Science.gov (United States) Ching, Eric; Lv, Yu; Ihme, Matthias 2017-11-01 Recent interest in human-scale missions to Mars has sparked active research into high-fidelity simulations of reentry flows. A key feature of the Mars atmosphere is the high levels of suspended dust particles, which can not only enhance erosion of thermal protection systems but also transfer energy and momentum to the shock layer, increasing surface heat fluxes. Second-order finite-volume schemes are typically employed for hypersonic flow simulations, but such schemes suffer from a number of limitations. An attractive alternative is discontinuous Galerkin methods, which benefit from arbitrarily high spatial order of accuracy, geometric flexibility, and other advantages. As such, a Lagrangian particle method is developed in a discontinuous Galerkin framework to enable the computation of particle-laden hypersonic flows. Two-way coupling between the carrier and disperse phases is considered, and an efficient particle search algorithm compatible with unstructured curved meshes is proposed. In addition, variable thermodynamic properties are considered to accommodate high-temperature gases. The performance of the particle method is demonstrated in several test cases, with focus on the accurate prediction of particle trajectories and heating augmentation. Financial support from a Stanford Graduate Fellowship and the NASA Early Career Faculty program are gratefully acknowledged. 5. Discontinuous Galerkin methods and a posteriori error analysis for heterogenous diffusion problems International Nuclear Information System (INIS) Stephansen, A.F. 2007-12-01 In this thesis we analyse a discontinuous Galerkin (DG) method and two computable a posteriori error estimators for the linear and stationary advection-diffusion-reaction equation with heterogeneous diffusion. The DG method considered, the SWIP method, is a variation of the Symmetric Interior Penalty Galerkin method. The difference is that the SWIP method uses weighted averages with weights that depend on the diffusion. The a priori analysis shows optimal convergence with respect to mesh-size and robustness with respect to heterogeneous diffusion, which is confirmed by numerical tests. Both a posteriori error estimators are of the residual type and control the energy (semi-)norm of the error. Local lower bounds are obtained showing that almost all indicators are independent of heterogeneities. The exception is for the non-conforming part of the error, which has been evaluated using the Oswald interpolator. The second error estimator is sharper in its estimate with respect to the first one, but it is slightly more costly. This estimator is based on the construction of an H(div)-conforming Raviart-Thomas-Nedelec flux using the conservativeness of DG methods. Numerical results show that both estimators can be used for mesh-adaptation. (author) 6. A discontinuous galerkin time domain-boundary integral method for analyzing transient electromagnetic scattering KAUST Repository Li, Ping 2014-07-01 This paper presents an algorithm hybridizing discontinuous Galerkin time domain (DGTD) method and time domain boundary integral (BI) algorithm for 3-D open region electromagnetic scattering analysis. The computational domain of DGTD is rigorously truncated by analytically evaluating the incoming numerical flux from the outside of the truncation boundary through BI method based on the Huygens\\' principle. The advantages of the proposed method are that it allows the truncation boundary to be conformal to arbitrary (convex/ concave) scattering objects, well-separated scatters can be truncated by their local meshes without losing the physics (such as coupling/multiple scattering) of the problem, thus reducing the total mesh elements. Furthermore, low frequency waves can be efficiently absorbed, and the field outside the truncation domain can be conveniently calculated using the same BI formulation. Numerical examples are benchmarked to demonstrate the accuracy and versatility of the proposed method. 7. A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems KAUST Repository Efendiev, Yalchin R. 2015-08-01 We design a multiscale model reduction framework within the hybridizable discontinuous Galerkin finite element method. Our approach uses local snapshot spaces and local spectral decomposition following the concept of Generalized Multiscale Finite Element Methods. We propose several multiscale finite element spaces on the coarse edges that provide a reduced dimensional approximation for numerical traces within the HDG framework. We provide a general framework for systematic construction of multiscale trace spaces. Using local snapshots, we avoid high dimensional representation of trace spaces and use some local features of the solution space in constructing a low dimensional trace space. We investigate the solvability and numerically study the performance of the proposed method on a representative number of numerical examples. 8. Discontinuous Galerkin finite element method for solving population density functions of cortical pyramidal and thalamic neuronal populations. Science.gov (United States) Huang, Chih-Hsu; Lin, Chou-Ching K; Ju, Ming-Shaung 2015-02-01 Compared with the Monte Carlo method, the population density method is efficient for modeling collective dynamics of neuronal populations in human brain. In this method, a population density function describes the probabilistic distribution of states of all neurons in the population and it is governed by a hyperbolic partial differential equation. In the past, the problem was mainly solved by using the finite difference method. In a previous study, a continuous Galerkin finite element method was found better than the finite difference method for solving the hyperbolic partial differential equation; however, the population density function often has discontinuity and both methods suffer from a numerical stability problem. The goal of this study is to improve the numerical stability of the solution using discontinuous Galerkin finite element method. To test the performance of the new approach, interaction of a population of cortical pyramidal neurons and a population of thalamic neurons was simulated. The numerical results showed good agreement between results of discontinuous Galerkin finite element and Monte Carlo methods. The convergence and accuracy of the solutions are excellent. The numerical stability problem could be resolved using the discontinuous Galerkin finite element method which has total-variation-diminishing property. The efficient approach will be employed to simulate the electroencephalogram or dynamics of thalamocortical network which involves three populations, namely, thalamic reticular neurons, thalamocortical neurons and cortical pyramidal neurons. Copyright © 2014 Elsevier Ltd. All rights reserved. 9. An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media KAUST Repository Chung, Eric T. 2017-02-07 Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method. 10. A hybrid time-domain discontinuous galerkin-boundary integral method for electromagnetic scattering analysis KAUST Repository Li, Ping; Shi, Yifei; Jiang, Lijun; Bagci, Hakan 2014-01-01 A scheme hybridizing discontinuous Galerkin time-domain (DGTD) and time-domain boundary integral (TDBI) methods for accurately analyzing transient electromagnetic scattering is proposed. Radiation condition is enforced using the numerical flux on the truncation boundary. The fields required by the flux are computed using the TDBI from equivalent currents introduced on a Huygens' surface enclosing the scatterer. The hybrid DGTDBI ensures that the radiation condition is mathematically exact and the resulting computation domain is as small as possible since the truncation boundary conforms to scatterer's shape and is located very close to its surface. Locally truncated domains can also be defined around each disconnected scatterer additionally reducing the size of the overall computation domain. Numerical examples demonstrating the accuracy and versatility of the proposed method are presented. © 2014 IEEE. 11. Discontinuous Galerkin Subgrid Finite Element Method for Heterogeneous Brinkman’s Equations KAUST Repository Iliev, Oleg P. 2010-01-01 We present a two-scale finite element method for solving Brinkman\\'s equations with piece-wise constant coefficients. This system of equations model fluid flows in highly porous, heterogeneous media with complex topology of the heterogeneities. We make use of the recently proposed discontinuous Galerkin FEM for Stokes equations by Wang and Ye in [12] and the concept of subgrid approximation developed for Darcy\\'s equations by Arbogast in [4]. In order to reduce the error along the coarse-grid interfaces we have added a alternating Schwarz iteration using patches around the coarse-grid boundaries. We have implemented the subgrid method using Deal.II FEM library, [7], and we present the computational results for a number of model problems. © 2010 Springer-Verlag Berlin Heidelberg. 12. hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes CERN Document Server Cangiani, Andrea; Georgoulis, Emmanuil H; Houston, Paul 2017-01-01 Over the last few decades discontinuous Galerkin finite element methods (DGFEMs) have been witnessed tremendous interest as a computational framework for the numerical solution of partial differential equations. Their success is due to their extreme versatility in the design of the underlying meshes and local basis functions, while retaining key features of both (classical) finite element and finite volume methods. Somewhat surprisingly, DGFEMs on general tessellations consisting of polygonal (in 2D) or polyhedral (in 3D) element shapes have received little attention within the literature, despite the potential computational advantages. This volume introduces the basic principles of hp-version (i.e., locally varying mesh-size and polynomial order) DGFEMs over meshes consisting of polygonal or polyhedral element shapes, presents their error analysis, and includes an extensive collection of numerical experiments. The extreme flexibility provided by the locally variable elemen t-shapes, element-sizes, and elemen... 13. A hybrid time-domain discontinuous galerkin-boundary integral method for electromagnetic scattering analysis KAUST Repository Li, Ping 2014-05-01 A scheme hybridizing discontinuous Galerkin time-domain (DGTD) and time-domain boundary integral (TDBI) methods for accurately analyzing transient electromagnetic scattering is proposed. Radiation condition is enforced using the numerical flux on the truncation boundary. The fields required by the flux are computed using the TDBI from equivalent currents introduced on a Huygens\\' surface enclosing the scatterer. The hybrid DGTDBI ensures that the radiation condition is mathematically exact and the resulting computation domain is as small as possible since the truncation boundary conforms to scatterer\\'s shape and is located very close to its surface. Locally truncated domains can also be defined around each disconnected scatterer additionally reducing the size of the overall computation domain. Numerical examples demonstrating the accuracy and versatility of the proposed method are presented. © 2014 IEEE. 14. Analysis of the discontinuous Petrov-Galerkin method with optimal test functions for the Reissner-Mindlin plate bending model KAUST Repository Calo, Victor M.; Collier, Nathan; Niemi, Antti H. 2014-01-01 We analyze the discontinuous Petrov-Galerkin (DPG) method with optimal test functions when applied to solve the Reissner-Mindlin model of plate bending. We prove that the hybrid variational formulation underlying the DPG method is well-posed (stable 15. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations NARCIS (Netherlands) Rhebergen, Sander; Bokhove, Onno; van der Vegt, Jacobus J.W. We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the formulation is that if the system of nonconservative partial 16. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations NARCIS (Netherlands) Rhebergen, Sander; Bokhove, Onno; van der Vegt, Jacobus J.W. 2008-01-01 We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial 17. An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations KAUST Repository Pani, Amiya K. 2010-06-06 In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L 2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains. © 2010 Springer Science+Business Media, LLC. 18. A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity KAUST Repository Demkowicz, Leszek 2012-04-01 We continue our theoretical and numerical study on the Discontinuous Petrov-Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: ε=10 -11 for 1D and ε=10 -7 for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only. © 2011 IMACS. Published by Elsevier B.V. All rights reserved. 19. A discontinuous Galerkin method for numerical pricing of European options under Heston stochastic volatility Science.gov (United States) Hozman, J.; Tichý, T. 2016-12-01 The paper is based on the results from our recent research on multidimensional option pricing problems. We focus on European option valuation when the price movement of the underlying asset is driven by a stochastic volatility following a square root process proposed by Heston. The stochastic approach incorporates a new additional spatial variable into this model and makes it very robust, i.e. it provides a framework to price a variety of options that is closer to reality. The main topic is to present the numerical scheme arising from the concept of discontinuous Galerkin methods and applicable to the Heston option pricing model. The numerical results are presented on artificial benchmarks as well as on reference market data. 20. An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations KAUST Repository Pani, Amiya K.; Yadav, Sangita 2010-01-01 In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L 2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains. © 2010 Springer Science+Business Media, LLC. 1. Sharp Penalty Term and Time Step Bounds for the Interior Penalty Discontinuous Galerkin Method for Linear Hyperbolic Problems NARCIS (Netherlands) Geevers, Sjoerd; van der Vegt, J.J.W. 2017-01-01 We present sharp and sucient bounds for the interior penalty term and time step size to ensure stability of the symmetric interior penalty discontinuous Galerkin (SIPDG) method combined with an explicit time-stepping scheme. These conditions hold for generic meshes, including unstructured 2. Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods KAUST Repository Kubatko, Ethan J.; Yeager, Benjamin A.; Ketcheson, David I. 2013-01-01 Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge–Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases. 3. Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods KAUST Repository Kubatko, Ethan J. 2013-10-29 Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge–Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases. 4. High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations Science.gov (United States) Vaziri Astaneh, Ali; Fuentes, Federico; Mora, Jaime; Demkowicz, Leszek 2018-04-01 This work represents the first endeavor in using ultraweak formulations to implement high-order polygonal finite element methods via the discontinuous Petrov-Galerkin (DPG) methodology. Ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space, while most of the trial space can be chosen as copies of$L^2$-discretizations that have no need to be continuous across adjacent elements. Additionally, the test spaces are broken along the mesh interfaces. This allows one to construct conforming polygonal finite element methods, termed here as PolyDPG methods, by defining most spaces by restriction of a bounding triangle or box to the polygonal element. The only variables that require nontrivial compatibility across elements are the so-called interface or skeleton variables, which can be defined directly on the element boundaries. Unlike other high-order polygonal methods, PolyDPG methods do not require ad hoc stabilization terms thanks to the crafted stability of the DPG methodology. A proof of convergence of the form$h^p$is provided and corroborated through several illustrative numerical examples. These include polygonal meshes with$n$-sided convex elements and with highly distorted concave elements, as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid. Since PolyDPG methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes. This work is also accompanied by an open-source$\\texttt{PolyDPG}$software supporting polygonal and conventional elements. 5. Bound-Preserving Discontinuous Galerkin Methods for Conservative Phase Space Advection in Curvilinear Coordinates Energy Technology Data Exchange (ETDEWEB) Mezzacappa, Anthony [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Endeve, Eirik [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Hauck, Cory D. [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Xing, Yulong [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States) 2015-02-01 We extend the positivity-preserving method of Zhang & Shu [49] to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stabilitypreserving, Runge-Kutta (SSP-RK) time integration. Special care in taken to ensure that the method preserves strict bounds for the phase space distribution function f; i.e., f ϵ [0, 1]. The combination of suitable CFL conditions and the use of the high-order limiter proposed in [49] is su cient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergencefree property of the phase space ow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments - including one example in spherical symmetry adopting the Schwarzschild metric - demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle. 6. Analysis of a combined mixed finite element and discontinuous Galerkin method for incompressible two-phase flow in porous media KAUST Repository Kou, Jisheng; Sun, Shuyu 2013-01-01 We analyze a combined method consisting of the mixed finite element method for pressure equation and the discontinuous Galerkin method for saturation equation for the coupled system of incompressible two-phase flow in porous media. The existence and uniqueness of numerical solutions are established under proper conditions by using a constructive approach. Optimal error estimates in L2(H1) for saturation and in L∞(H(div)) for velocity are derived. Copyright © 2013 John Wiley & Sons, Ltd. 7. Analysis of a combined mixed finite element and discontinuous Galerkin method for incompressible two-phase flow in porous media KAUST Repository Kou, Jisheng 2013-06-20 We analyze a combined method consisting of the mixed finite element method for pressure equation and the discontinuous Galerkin method for saturation equation for the coupled system of incompressible two-phase flow in porous media. The existence and uniqueness of numerical solutions are established under proper conditions by using a constructive approach. Optimal error estimates in L2(H1) for saturation and in L∞(H(div)) for velocity are derived. Copyright © 2013 John Wiley & Sons, Ltd. 8. Diffusion and dispersion characteristics of hybridized discontinuous Galerkin methods for under-resolved turbulence simulations Science.gov (United States) Moura, Rodrigo; Fernandez, Pablo; Mengaldo, Gianmarco 2017-11-01 We investigate the dispersion and diffusion characteristics of hybridized discontinuous Galerkin (DG) methods. This provides us with insights to develop robust and accurate high-order DG discretizations for under-resolved flow simulations. Using the eigenanalysis technique introduced in (Moura et al., JCP, 2015 and Mengaldo et al., Computers & Fluids, 2017), we present a dispersion-diffusion analysis for the linear advection-diffusion equation. The effect of the accuracy order, the Riemann flux and the viscous stabilization are investigated. Next, we examine the diffusion characteristics of hybridized DG methods for under-resolved turbulent flows. The implicit large-eddy simulation (iLES) of the inviscid and viscous Taylor-Green vortex (TGV) problems are considered to this end. The inviscid case is relevant in the limit of high Reynolds numbers Re , i.e. negligible molecular viscosity, while the viscous case explores the effect of Re on the accuracy and robustness of the simulations. The TGV cases considered here are particularly crucial to under-resolved turbulent free flows away from walls. We conclude the talk with a discussion on the connections between hybridized and standard DG methods for under-resolved flow simulations. 9. Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations Science.gov (United States) Jiang, Zhen-Hua; Yan, Chao; Yu, Jian 2013-08-01 Two types of implicit algorithms have been improved for high order discontinuous Galerkin (DG) method to solve compressible Navier-Stokes (NS) equations on triangular grids. A block lower-upper symmetric Gauss-Seidel (BLU-SGS) approach is implemented as a nonlinear iterative scheme. And a modified LU-SGS (LLU-SGS) approach is suggested to reduce the memory requirements while retain the good convergence performance of the original LU-SGS approach. Both implicit schemes have the significant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory (HWENO) limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock transition and the designed high-order accuracy simultaneously. 10. The hybridized Discontinuous Galerkin method for Implicit Large-Eddy Simulation of transitional turbulent flows Science.gov (United States) Fernandez, P.; Nguyen, N. C.; Peraire, J. 2017-05-01 We present a high-order Implicit Large-Eddy Simulation (ILES) approach for transitional aerodynamic flows. The approach encompasses a hybridized Discontinuous Galerkin (DG) method for the discretization of the Navier-Stokes (NS) equations, and a parallel preconditioned Newton-GMRES solver for the resulting nonlinear system of equations. The combination of hybridized DG methods with an efficient solution procedure leads to a high-order accurate NS solver that is competitive to alternative approaches, such as finite volume and finite difference codes, in terms of computational cost. The proposed approach is applied to transitional flows over the NACA 65-(18)10 compressor cascade and the Eppler 387 wing at Reynolds numbers up to 460,000. Grid convergence studies are presented and the required resolution to capture transition at different Reynolds numbers is investigated. Numerical results show rapid convergence and excellent agreement with experimental data. In short, this work aims to demonstrate the potential of high-order ILES for simulating transitional aerodynamic flows. This is illustrated through numerical results and supported by theoretical considerations. 11. Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods Science.gov (United States) Pazner, Will; Persson, Per-Olof 2018-02-01 In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O (p2d) storage and O (p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O (p d + 1) storage, O (p d + 1) work in two spatial dimensions, and O (p d + 2) work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O (p9) to O (p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier-Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p. 12. Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics KAUST Repository Niemi, Antti H.; Bramwell, Jamie A.; Demkowicz, Leszek F. 2011-01-01 We study the applicability of the discontinuous Petrov-Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial 13. A spectral hybridizable discontinuous Galerkin method for elastic-acoustic wave propagation Science.gov (United States) Terrana, S.; Vilotte, J. P.; Guillot, L. 2018-04-01 We introduce a time-domain, high-order in space, hybridizable discontinuous Galerkin (DG) spectral element method (HDG-SEM) for wave equations in coupled elastic-acoustic media. The method is based on a first-order hyperbolic velocity-strain formulation of the wave equations written in conservative form. This method follows the HDG approach by introducing a hybrid unknown, which is the approximation of the velocity on the elements boundaries, as the only globally (i.e. interelement) coupled degrees of freedom. In this paper, we first present a hybridized formulation of the exact Riemann solver at the element boundaries, taking into account elastic-elastic, acoustic-acoustic and elastic-acoustic interfaces. We then use this Riemann solver to derive an explicit construction of the HDG stabilization function τ for all the above-mentioned interfaces. We thus obtain an HDG scheme for coupled elastic-acoustic problems. This scheme is then discretized in space on quadrangular/hexahedral meshes using arbitrary high-order polynomial basis for both volumetric and hybrid fields, using an approach similar to the spectral element methods. This leads to a semi-discrete system of algebraic differential equations (ADEs), which thanks to the structure of the global conservativity condition can be reformulated easily as a classical system of first-order ordinary differential equations in time, allowing the use of classical explicit or implicit time integration schemes. When an explicit time scheme is used, the HDG method can be seen as a reformulation of a DG with upwind fluxes. The introduction of the velocity hybrid unknown leads to relatively simple computations at the element boundaries which, in turn, makes the HDG approach competitive with the DG-upwind methods. Extensive numerical results are provided to illustrate and assess the accuracy and convergence properties of this HDG-SEM. The approximate velocity is shown to converge with the optimal order of k + 1 in the L2-norm 14. Adaptive mixed-hybrid and penalty discontinuous Galerkin method for two-phase flow in heterogeneous media KAUST Repository Hou, Jiangyong 2016-02-05 In this paper, we present a hybrid method, which consists of a mixed-hybrid finite element method and a penalty discontinuous Galerkin method, for the approximation of a fractional flow formulation of a two-phase flow problem in heterogeneous media with discontinuous capillary pressure. The fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients. For the wetting phase pressure equation, the continuous mixed-hybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous. While it can reduce the computational cost by using less degrees of freedom and avoiding the post-processing of velocity reconstruction, this method can also keep several good properties of the discontinuous Galerkin method, which are important to the fractional flow formulation, such as the local mass balance, continuous normal flux and capability of handling the discontinuous capillary pressure. For the wetting phase saturation equation, the penalty discontinuous Galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation. Furthermore, an adaptive algorithm for the hybrid method together with the centroidal Voronoi Delaunay triangulation technique is proposed. Five numerical examples are presented to illustrate the features of proposed numerical method, such as the optimal convergence order, the accurate and efficient velocity approximation, and the applicability to the simulation of water flooding in oil field and the oil-trapping or barrier effect phenomena. 15. Adaptive mixed-hybrid and penalty discontinuous Galerkin method for two-phase flow in heterogeneous media KAUST Repository Hou, Jiangyong; Chen, Jie; Sun, Shuyu; Chen, Zhangxin 2016-01-01 In this paper, we present a hybrid method, which consists of a mixed-hybrid finite element method and a penalty discontinuous Galerkin method, for the approximation of a fractional flow formulation of a two-phase flow problem in heterogeneous media with discontinuous capillary pressure. The fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients. For the wetting phase pressure equation, the continuous mixed-hybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous. While it can reduce the computational cost by using less degrees of freedom and avoiding the post-processing of velocity reconstruction, this method can also keep several good properties of the discontinuous Galerkin method, which are important to the fractional flow formulation, such as the local mass balance, continuous normal flux and capability of handling the discontinuous capillary pressure. For the wetting phase saturation equation, the penalty discontinuous Galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation. Furthermore, an adaptive algorithm for the hybrid method together with the centroidal Voronoi Delaunay triangulation technique is proposed. Five numerical examples are presented to illustrate the features of proposed numerical method, such as the optimal convergence order, the accurate and efficient velocity approximation, and the applicability to the simulation of water flooding in oil field and the oil-trapping or barrier effect phenomena. 16. A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media International Nuclear Information System (INIS) Wilcox, Lucas C.; Stadler, Georg; Burstedde, Carsten; Ghattas, Omar 2010-01-01 We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic-acoustic media. A velocity-strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic-acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic-acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer. 17. Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes Science.gov (United States) Zhu, Jun; Zhong, Xinghui; Shu, Chi-Wang; Qiu, Jianxian 2013-09-01 In this paper we generalize a new type of limiters based on the weighted essentially non-oscillatory (WENO) finite volume methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving nonlinear hyperbolic conservation laws, which were recently developed in [32] for structured meshes, to two-dimensional unstructured triangular meshes. The key idea of such limiters is to use the entire polynomials of the DG solutions from the troubled cell and its immediate neighboring cells, and then apply the classical WENO procedure to form a convex combination of these polynomials based on smoothness indicators and nonlinear weights, with suitable adjustments to guarantee conservation. The main advantage of this new limiter is its simplicity in implementation, especially for the unstructured meshes considered in this paper, as only information from immediate neighbors is needed and the usage of complicated geometric information of the meshes is largely avoided. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good performance of this procedure. 18. A discontinuous Galerkin method for P-wave modeling in tilted TI media KAUST Repository Amler, Thomas; Alkhalifah, Tariq Ali; Hoteit, Ibrahim 2014-01-01 The acoustic approximation is an efficient alternative to the equations of elastodynamics for modeling Pwave propagation in weakly anisotropic media. We present a stable discontinuous Galerkin (DG) method for solving the acoustic approximation in tilted TI media (acoustic TI approximation). The acoustic TI approximation is considered as a modification of the equations of elastodynamics from which a modified energy is derived. The modified energy is obtained by eliminating the shear stress in the coordinates determined by the tilt angle and finding an energy for the remaining unknowns. This construction is valid if the medium is not elliptically anisotropic, a requirement frequently found in the literature. In the fully discrete setting, the modified energy is also conserved in time the presence of sharp contrasts in material parameters. By construction, the scheme can be coupled to the (fully) acoustic wave equation in the same way as the equations of elastodynamics. Hence, the number of unknowns can be reduced in acoustic regions. Our numerical examples confirm the conservation of energy in the discrete setting and the stability of the scheme. 19. Long-time stability effects of quadrature and artificial viscosity on nodal discontinuous Galerkin methods for gas dynamics Science.gov (United States) Durant, Bradford; Hackl, Jason; Balachandar, Sivaramakrishnan 2017-11-01 Nodal discontinuous Galerkin schemes present an attractive approach to robust high-order solution of the equations of fluid mechanics, but remain accompanied by subtle challenges in their consistent stabilization. The effect of quadrature choices (full mass matrix vs spectral elements), over-integration to manage aliasing errors, and explicit artificial viscosity on the numerical solution of a steady homentropic vortex are assessed over a wide range of resolutions and polynomial orders using quadrilateral elements. In both stagnant and advected vortices in periodic and non-periodic domains the need arises for explicit stabilization beyond the numerical surface fluxes of discontinuous Galerkin spectral elements. Artificial viscosity via the entropy viscosity method is assessed as a stabilizing mechanism. It is shown that the regularity of the artificial viscosity field is essential to its use for long-time stabilization of small-scale features in nodal discontinuous Galerkin solutions of the Euler equations of gas dynamics. Supported by the Department of Energy Predictive Science Academic Alliance Program Contract DE-NA0002378. 20. Hybrid Fourier pseudospectral/discontinuous Galerkin time-domain method for wave propagation Science.gov (United States) Pagán Muñoz, Raúl; Hornikx, Maarten 2017-11-01 The Fourier Pseudospectral time-domain (Fourier PSTD) method was shown to be an efficient way of modelling acoustic propagation problems as described by the linearized Euler equations (LEE), but is limited to real-valued frequency independent boundary conditions and predominantly staircase-like boundary shapes. This paper presents a hybrid approach to solve the LEE, coupling Fourier PSTD with a nodal Discontinuous Galerkin (DG) method. DG exhibits almost no restrictions with respect to geometrical complexity or boundary conditions. The aim of this novel method is to allow the computation of complex geometries and to be a step towards the implementation of frequency dependent boundary conditions by using the benefits of DG at the boundaries, while keeping the efficient Fourier PSTD in the bulk of the domain. The hybridization approach is based on conformal meshes to avoid spatial interpolation of the DG solutions when transferring values from DG to Fourier PSTD, while the data transfer from Fourier PSTD to DG is done utilizing spectral interpolation of the Fourier PSTD solutions. The accuracy of the hybrid approach is presented for one- and two-dimensional acoustic problems and the main sources of error are investigated. It is concluded that the hybrid methodology does not introduce significant errors compared to the Fourier PSTD stand-alone solver. An example of a cylinder scattering problem is presented and accurate results have been obtained when using the proposed approach. Finally, no instabilities were found during long-time calculation using the current hybrid methodology on a two-dimensional domain. 1. The Application of Discontinuous Galerkin Methods in Conjugate Heat Transfer Simulations of Gas Turbines Directory of Open Access Journals (Sweden) Zeng-Rong Hao 2014-11-01 Full Text Available The performance of modern heavy-duty gas turbines is greatly determined by the accurate numerical predictions of thermal loading on the hot-end components. The purpose of this paper is: (1 to present an approach applying a novel numerical technique—the discontinuous Galerkin (DG method—to conjugate heat transfer (CHT simulations, develop the engineering-oriented numerical platform, and validate the feasibility of the methodology and tool preliminarily; and (2 to utilize the constructed platform to investigate the aerothermodynamic features of a typical transonic turbine vane with convection cooling. Fluid dynamic and solid heat conductive equations are discretized into explicit DG formulations. A centroid-expanded Taylor basis is adopted for various types of elements. The Bassi-Rebay method is used in the computation of gradients. A coupled strategy based on a data exchange process via numerical flux on interface quadrature points is simply devised. Additionally, various turbulence Reynolds-Averaged-Navier-Stokes (RANS models and the local-variable-based transition model γ-Reθ are assimilated into the integral framework, combining sophisticated modelling with the innovative algorithm. Numerical tests exhibit good consistency between computational and analytical or experimental results, demonstrating that the presented approach and tool can handle well general CHT simulations. Application and analysis in the turbine vane, focusing on features around where there in cluster exist shock, separation and transition, illustrate the effects of Bradshaw’s shear stress limitation and separation-induced-transition modelling. The general overestimation of heat transfer intensity behind shock is conjectured to be associated with compressibility effects on transition modeling. This work presents an unconventional formulation in CHT problems and achieves its engineering applications in gas turbines. 2. Spacetime Discontinuous Galerkin FEM: Spectral Response International Nuclear Information System (INIS) Abedi, R; Omidi, O; Clarke, P L 2014-01-01 Materials in nature demonstrate certain spectral shapes in terms of their material properties. Since successful experimental demonstrations in 2000, metamaterials have provided a means to engineer materials with desired spectral shapes for their material properties. Computational tools are employed in two different aspects for metamaterial modeling: 1. Mircoscale unit cell analysis to derive and possibly optimize material's spectral response; 2. macroscale to analyze their interaction with conventional material. We compare two different approaches of Time-Domain (TD) and Frequency Domain (FD) methods for metamaterial applications. Finally, we discuss advantages of the TD method of Spacetime Discontinuous Galerkin finite element method (FEM) for spectral analysis of metamaterials 3. Discontinuous Petrov-Galerkin method based on the optimal test space norm for one-dimensional transport problems KAUST Repository Niemi, Antti; Collier, Nathan; Calo, Victor M. 2011-01-01 We revisit the finite element analysis of convection dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can 4. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations International Nuclear Information System (INIS) Rhebergen, S.; Bokhove, O.; Vegt, J.J.W. van der 2008-01-01 We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form, then the formulation must reduce to that for conservative systems. Standard DGFEM formulations cannot be applied to nonconservative systems of partial differential equations. We therefore introduce the theory of weak solutions for nonconservative products into the DGFEM formulation leading to the new question how to define the path connecting left and right states across a discontinuity. The effect of different paths on the numerical solution is investigated and found to be small. We also introduce a new numerical flux that is able to deal with nonconservative products. Our scheme is applied to two different systems of partial differential equations. First, we consider the shallow water equations, where topography leads to nonconservative products, in which the known, possibly discontinuous, topography is formally taken as an unknown in the system. Second, we consider a simplification of a depth-averaged two-phase flow model which contains more intrinsic nonconservative products 5. Resolution of the Vlasov-Maxwell system by PIC discontinuous Galerkin method on GPU with OpenCL Directory of Open Access Journals (Sweden) Crestetto Anaïs 2013-01-01 Full Text Available We present an implementation of a Vlasov-Maxwell solver for multicore processors. The Vlasov equation describes the evolution of charged particles in an electromagnetic field, solution of the Maxwell equations. The Vlasov equation is solved by a Particle-In-Cell method (PIC, while the Maxwell system is computed by a Discontinuous Galerkin method. We use the OpenCL framework, which allows our code to run on multicore processors or recent Graphic Processing Units (GPU. We present several numerical applications to two-dimensional test cases. 6. Application of discontinuous Galerkin method for solving a compressible five-equation two-phase flow model Science.gov (United States) Saleem, M. Rehan; Ali, Ishtiaq; Qamar, Shamsul 2018-03-01 In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme. 7. Analysis and development of adjoint-based h-adaptive direct discontinuous Galerkin method for the compressible Navier-Stokes equations Science.gov (United States) Cheng, Jian; Yue, Huiqiang; Yu, Shengjiao; Liu, Tiegang 2018-06-01 In this paper, an adjoint-based high-order h-adaptive direct discontinuous Galerkin method is developed and analyzed for the two dimensional steady state compressible Navier-Stokes equations. Particular emphasis is devoted to the analysis of the adjoint consistency for three different direct discontinuous Galerkin discretizations: including the original direct discontinuous Galerkin method (DDG), the direct discontinuous Galerkin method with interface correction (DDG(IC)) and the symmetric direct discontinuous Galerkin method (SDDG). Theoretical analysis shows the extra interface correction term adopted in the DDG(IC) method and the SDDG method plays a key role in preserving the adjoint consistency. To be specific, for the model problem considered in this work, we prove that the original DDG method is not adjoint consistent, while the DDG(IC) method and the SDDG method can be adjoint consistent with appropriate treatment of boundary conditions and correct modifications towards the underlying output functionals. The performance of those three DDG methods is carefully investigated and evaluated through typical test cases. Based on the theoretical analysis, an adjoint-based h-adaptive DDG(IC) method is further developed and evaluated, numerical experiment shows its potential in the applications of adjoint-based adaptation for simulating compressible flows. 8. Accurate characterization of 3D diffraction gratings using time domain discontinuous Galerkin method with exact absorbing boundary conditions KAUST Repository Sirenko, Kostyantyn 2013-07-01 Exact absorbing and periodic boundary conditions allow to truncate grating problems\\' infinite physical domains without introducing any errors. This work presents exact absorbing boundary conditions for 3D diffraction gratings and describes their discretization within a high-order time-domain discontinuous Galerkin finite element method (TD-DG-FEM). The error introduced by the boundary condition discretization matches that of the TD-DG-FEM; this results in an optimal solver in terms of accuracy and computation time. Numerical results demonstrate the superiority of this solver over TD-DG-FEM with perfectly matched layers (PML)-based domain truncation. © 2013 IEEE. 9. Discontinuous Petrov-Galerkin method based on the optimal test space norm for one-dimensional transport problems KAUST Repository Niemi, Antti 2011-05-14 We revisit the finite element analysis of convection dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the so called optimal test space norm by using an element subgrid discretization. This should make the DPG method not only stable but also robust, that is, uniformly stable with respect to the Ṕeclet number in the current application. The e_ectiveness of the algorithm is demonstrated on two problems for the linear advection-di_usion equation. 10. Numerical simulation of the interaction between a nonlinear elastic structure and compressible flow by the discontinuous Galerkin method Czech Academy of Sciences Publication Activity Database Kosík, Adam; Feistauer, M.; Hadrava, Martin; Horáček, Jaromír 2015-01-01 Roč. 267, September (2015), s. 382-396 ISSN 0096-3003 R&D Projects: GA ČR(CZ) GAP101/11/0207 Institutional support: RVO:61388998 Keywords : discontinuous Galerkin method * nonlinear elasticity * compressible viscous flow * fluid–structure interaction Subject RIV: BI - Acoustics Impact factor: 1.345, year: 2015 http://www.sciencedirect.com/science/article/pii/S0096300315002453/pdfft?md5=02d46bc730e3a7fb8a5008aaab1da786&pid=1-s2.0-S0096300315002453-main.pdf 11. Modeling of Multicomponent Diffusions and Natural Convection in Unfractured and Fractured Media by Discontinuous Galerkin and Mixed Methods KAUST Repository Hoteit, Hussein 2017-12-29 Computation of the distribution of species in hydrocarbon reservoirs from diffusions (thermal, molecular, and pressure) and natural convection is an important step in reservoir initialization. Current methods, which are mainly based on the conventional finite difference approach, may not be numerically efficient in fractured and other media with complex heterogeneities. In this work, the discontinuous Galerkin (DG) method combined with the mixed finite element (MFE) method is used for the calculation of compositional variation in fractured hydrocarbon reservoirs. The use of unstructured gridding allows efficient computations for fractured media when the crossflow equilibrium concept is invoked. The DG method has less numerical dispersion than the upwind finite difference (FD) methods. The MFE method ensures continuity of fluxes at the interface of the grid elements. We also use the local discontinuous Galerkin (LDG) method instead of the MFE calculate the diffusion fluxes. Results from several numerical examples are presented to demonstrate the efficiency, robustness, and accuracy of the model. Various features of convection and diffusion in homogeneous, layered, and fractured media are also discussed. 12. Modeling of Multicomponent Diffusions and Natural Convection in Unfractured and Fractured Media by Discontinuous Galerkin and Mixed Methods KAUST Repository Hoteit, Hussein; Firoozabadi, Abbas 2017-01-01 Computation of the distribution of species in hydrocarbon reservoirs from diffusions (thermal, molecular, and pressure) and natural convection is an important step in reservoir initialization. Current methods, which are mainly based on the conventional finite difference approach, may not be numerically efficient in fractured and other media with complex heterogeneities. In this work, the discontinuous Galerkin (DG) method combined with the mixed finite element (MFE) method is used for the calculation of compositional variation in fractured hydrocarbon reservoirs. The use of unstructured gridding allows efficient computations for fractured media when the crossflow equilibrium concept is invoked. The DG method has less numerical dispersion than the upwind finite difference (FD) methods. The MFE method ensures continuity of fluxes at the interface of the grid elements. We also use the local discontinuous Galerkin (LDG) method instead of the MFE calculate the diffusion fluxes. Results from several numerical examples are presented to demonstrate the efficiency, robustness, and accuracy of the model. Various features of convection and diffusion in homogeneous, layered, and fractured media are also discussed. 13. The Reverse Time Migration technique coupled with Interior Penalty Discontinuous Galerkin method. Science.gov (United States) Baldassari, C.; Barucq, H.; Calandra, H.; Denel, B.; Diaz, J. 2009-04-01 Seismic imaging is based on the seismic reflection method which produces an image of the subsurface from reflected waves recordings by using a tomography process and seismic migration is the industrial standard to improve the quality of the images. The migration process consists in replacing the recorded wavefields at their actual place by using various mathematical and numerical methods but each of them follows the same schedule, according to the pioneering idea of Claerbout: numerical propagation of the source function (propagation) and of the recorded wavefields (retropropagation) and next, construction of the image by applying an imaging condition. The retropropagation step can be realized accouting for the time reversibility of the wave equation and the resulting algorithm is currently called Reverse Time Migration (RTM). To be efficient, especially in three dimensional domain, the RTM requires the solution of the full wave equation by fast numerical methods. Finite element methods are considered as the best discretization method for solving the wave equation, even if they lead to the solution of huge systems with several millions of degrees of freedom, since they use meshes adapted to the domain topography and the boundary conditions are naturally taken into account in the variational formulation. Among the different finite element families, the spectral element one (SEM) is very interesting because it leads to a diagonal mass matrix which dramatically reduces the cost of the numerical computation. Moreover this method is very accurate since it allows the use of high order finite elements. However, SEM uses meshes of the domain made of quadrangles in 2D or hexaedra in 3D which are difficult to compute and not always suitable for complex topographies. Recently, Grote et al. applied the IPDG (Interior Penalty Discontinuous Galerkin) method to the wave equation. This approach is very interesting since it relies on meshes with triangles in 2D or tetrahedra in 3D 14. Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations Science.gov (United States) Atkins, H. L.; Helenbrook, B. T. 2005-01-01 This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of di usion. Gauss-Seidel relaxation converges 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel. 15. Discontinuous Petrov-Galerkin method based on the optimal test space norm for steady transport problems in one space dimension KAUST Repository Niemi, Antti; Collier, Nathan; Calo, Victor M. 2013-01-01 We revisit the finite element analysis of convection-dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the optimal test space norm. This makes the DPG method not only stable but also robust, that is, uniformly stable with respect to the Péclet number in the current application. We employ discontinuous piecewise Bernstein polynomials as trial functions and construct a subgrid discretization that accounts for the singular perturbation character of the problem to resolve the corresponding optimal test functions. We also show that a smooth B-spline basis has certain computational advantages in the subgrid discretization. The overall effectiveness of the algorithm is demonstrated on two problems for the linear advection-diffusion equation. © 2011 Elsevier B.V. 16. Discontinuous Petrov-Galerkin method based on the optimal test space norm for steady transport problems in one space dimension KAUST Repository Niemi, Antti 2013-05-01 We revisit the finite element analysis of convection-dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the optimal test space norm. This makes the DPG method not only stable but also robust, that is, uniformly stable with respect to the Péclet number in the current application. We employ discontinuous piecewise Bernstein polynomials as trial functions and construct a subgrid discretization that accounts for the singular perturbation character of the problem to resolve the corresponding optimal test functions. We also show that a smooth B-spline basis has certain computational advantages in the subgrid discretization. The overall effectiveness of the algorithm is demonstrated on two problems for the linear advection-diffusion equation. © 2011 Elsevier B.V. 17. The development of high performance numerical simulation code for transient groundwater flow and reactive solute transport problems based on local discontinuous Galerkin method International Nuclear Information System (INIS) Suzuki, Shunichi; Motoshima, Takayuki; Naemura, Yumi; Kubo, Shin; Kanie, Shunji 2009-01-01 The authors develop a numerical code based on Local Discontinuous Galerkin Method for transient groundwater flow and reactive solute transport problems in order to make it possible to do three dimensional performance assessment on radioactive waste repositories at the earliest stage possible. Local discontinuous Galerkin Method is one of mixed finite element methods which are more accurate ones than standard finite element methods. In this paper, the developed numerical code is applied to several problems which are provided analytical solutions in order to examine its accuracy and flexibility. The results of the simulations show the new code gives highly accurate numeric solutions. (author) 18. Discontinuous Galerkin methods for plasma physics in the scrape-off layer of tokamaks International Nuclear Information System (INIS) Michoski, C.; Meyerson, D.; Isaac, T.; Waelbroeck, F. 2014-01-01 A new parallel discontinuous Galerkin solver, called ArcOn, is developed to describe the intermittent turbulent transport of filamentary blobs in the scrape-off layer (SOL) of fusion plasma. The model is comprised of an elliptic subsystem coupled to two convection-dominated reaction–diffusion–convection equations. Upwinding is used for a class of numerical fluxes developed to accommodate cross product driven convection, and the elliptic solver uses SIPG, NIPG, IIPG, Brezzi, and Bassi–Rebay fluxes to formulate the stiffness matrix. A novel entropy sensor is developed for this system, designed for a space–time varying artificial diffusion/viscosity regularization algorithm. Some numerical experiments are performed to show convergence order on manufactured solutions, regularization of blob/streamer dynamics in the SOL given unstable parameterizations, long-time stability of modon (or dipole drift vortex) solutions arising in simulations of drift-wave turbulence, and finally the formation of edge mode turbulence in the scrape-off layer under turbulent saturation conditions 19. Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes Science.gov (United States) Liu, Yong; Shu, Chi-Wang; Zhang, Mengping 2018-02-01 We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules [15] for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov [32], the entropy stable DG framework with suitable quadrature rules [15], the entropy conservative flux in [14] inside each cell and the entropy dissipative approximate Godunov type numerical flux at cell interfaces to make the scheme entropy stable. The main difficulty in the generalization of the results in [15] is the appearance of the non-conservative "source terms" added in the modified MHD model introduced by Godunov [32], which do not exist in the general hyperbolic system studied in [15]. Special care must be taken to discretize these "source terms" adequately so that the resulting DG scheme satisfies entropy stability. Total variation diminishing / bounded (TVD/TVB) limiters and bound-preserving limiters are applied to control spurious oscillations. We demonstrate the accuracy and robustness of this new scheme on standard MHD examples. 20. Discontinuous Galerkin methods and a posteriori error analysis for heterogenous diffusion problems; Methodes de Galerkine discontinues et analyse d'erreur a posteriori pour les problemes de diffusion heterogene Energy Technology Data Exchange (ETDEWEB) Stephansen, A.F 2007-12-15 In this thesis we analyse a discontinuous Galerkin (DG) method and two computable a posteriori error estimators for the linear and stationary advection-diffusion-reaction equation with heterogeneous diffusion. The DG method considered, the SWIP method, is a variation of the Symmetric Interior Penalty Galerkin method. The difference is that the SWIP method uses weighted averages with weights that depend on the diffusion. The a priori analysis shows optimal convergence with respect to mesh-size and robustness with respect to heterogeneous diffusion, which is confirmed by numerical tests. Both a posteriori error estimators are of the residual type and control the energy (semi-)norm of the error. Local lower bounds are obtained showing that almost all indicators are independent of heterogeneities. The exception is for the non-conforming part of the error, which has been evaluated using the Oswald interpolator. The second error estimator is sharper in its estimate with respect to the first one, but it is slightly more costly. This estimator is based on the construction of an H(div)-conforming Raviart-Thomas-Nedelec flux using the conservativeness of DG methods. Numerical results show that both estimators can be used for mesh-adaptation. (author) 1. Advances in the discontinuous Galerkin method: Hybrid schemes and applications to the reactive infiltration instability in an upwelling compacting mantle Science.gov (United States) Schiemenz, Alan R. High-order methods are emerging in the scientific computing community as superior alternatives to the classical finite difference, finite volume, and continuous finite element methods. The discontinuous Galerkin (DG) method in particular combines many of the positive features of all of these methods. This thesis presents two projects involving the DG method. First, a Hybrid scheme is presented, which implements DG areas where the solution is considered smooth, while dropping the order of the scheme elsewhere and implementing a finite volume scheme with high-order, non-oscillatory solution reconstructions suitable for unstructured mesh. Two such reconstructions from the ENO class are considered in the Hybrid. Successful numerical results are presented for nonlinear systems of conservation laws in one dimension. Second, the high-order discontinuous Galerkin and Fourier spectral methods are applied to an application modeling three-phase fluid flow through a porous medium, undergoing solid-fluid reaction due to the reactive infiltration instability (RII). This model incorporates a solid upwelling term and an equation to track the abundance of the reacting mineral orthopyroxene (opx). After validating the numerical discretization, results are given that provide new insight into the formation of melt channels in the Earth's mantle. Mantle heterogeneities are observed to be one catalyst for the development of melt channels, and the dissolution of opx produces interesting bifurcations in the melt channels. An alternative formulation is considered where the mass transfer rate relative to velocity is taken to be infinitely large. In this setting, the stiffest terms are removed, greatly reducing the cost of time integration. 2. Automatically stable discontinuous Petrov-Galerkin methods for stationary transport problems: Quasi-optimal test space norm KAUST Repository Niemi, Antti H.; Collier, Nathan; Calo, Victor M. 2013-01-01 We investigate the application of the discontinuous Petrov-Galerkin (DPG) finite element framework to stationary convection-diffusion problems. In particular, we demonstrate how the quasi-optimal test space norm improves the robustness of the DPG method with respect to vanishing diffusion. We numerically compare coarse-mesh accuracy of the approximation when using the quasi-optimal norm, the standard norm, and the weighted norm. Our results show that the quasi-optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions. We address the problems associated to the resolution of the optimal test functions with respect to the quasi-optimal norm by studying their convergence numerically. In order to facilitate understanding of the method, we also include a detailed explanation of the methodology from the algorithmic point of view. © 2013 Elsevier Ltd. All rights reserved. 3. Automatically stable discontinuous Petrov-Galerkin methods for stationary transport problems: Quasi-optimal test space norm KAUST Repository Niemi, Antti H. 2013-12-01 We investigate the application of the discontinuous Petrov-Galerkin (DPG) finite element framework to stationary convection-diffusion problems. In particular, we demonstrate how the quasi-optimal test space norm improves the robustness of the DPG method with respect to vanishing diffusion. We numerically compare coarse-mesh accuracy of the approximation when using the quasi-optimal norm, the standard norm, and the weighted norm. Our results show that the quasi-optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions. We address the problems associated to the resolution of the optimal test functions with respect to the quasi-optimal norm by studying their convergence numerically. In order to facilitate understanding of the method, we also include a detailed explanation of the methodology from the algorithmic point of view. © 2013 Elsevier Ltd. All rights reserved. 4. Analysis of the discontinuous Petrov-Galerkin method with optimal test functions for the Reissner-Mindlin plate bending model KAUST Repository Calo, Victor M. 2014-01-01 We analyze the discontinuous Petrov-Galerkin (DPG) method with optimal test functions when applied to solve the Reissner-Mindlin model of plate bending. We prove that the hybrid variational formulation underlying the DPG method is well-posed (stable) with a thickness-dependent constant in a norm encompassing the L2-norms of the bending moment, the shear force, the transverse deflection and the rotation vector. We then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p. We show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p+3. We prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non-affine mesh sequences. © 2013 Elsevier Ltd. All rights reserved. 5. Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics KAUST Repository Niemi, Antti H. 2011-02-01 We study the applicability of the discontinuous Petrov-Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate, local computation of the corresponding \\'optimal\\' test functions. In the Timoshenko beam problem, the proposed method is shown to provide the best approximation in an energy-type norm which is equivalent to the L2-norm for all the unknowns, uniformly with respect to the thickness parameter. The same formulation remains valid also for the asymptotic Euler-Bernoulli solution. As another one-dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations. In particular, we derive a special norm for the test space which leads to a robust method in terms of the shell thickness. Finally, we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp-mesh for resolving the boundary layer. © 2010 Elsevier B.V. 6. Upwind discontinuous Galerkin methods with mass conservation of both phases for incompressible two-phase flow in porous media KAUST Repository Kou, Jisheng; Sun, Shuyu 2014-01-01 Discontinuous Galerkin methods with interior penalties and upwind schemes are applied to the original formulation modeling incompressible two-phase flow in porous media with the capillary pressure. The pressure equation is obtained by summing the discretized conservation equations of two phases. This treatment is very different from the conventional approaches, and its great merit is that the mass conservations hold for both phases instead of only one phase in the conventional schemes. By constructing a new continuous map and using the fixed-point theorem, we prove the global existence of discrete solutions under the proper conditions, and furthermore, we obtain a priori hp error estimates of the pressures in L 2 (H 1) and the saturations in L ∞(L 2) and L 2 (H 1). © 2014 Wiley Periodicals, Inc. 7. Upwind discontinuous Galerkin methods with mass conservation of both phases for incompressible two-phase flow in porous media KAUST Repository Kou, Jisheng 2014-03-22 Discontinuous Galerkin methods with interior penalties and upwind schemes are applied to the original formulation modeling incompressible two-phase flow in porous media with the capillary pressure. The pressure equation is obtained by summing the discretized conservation equations of two phases. This treatment is very different from the conventional approaches, and its great merit is that the mass conservations hold for both phases instead of only one phase in the conventional schemes. By constructing a new continuous map and using the fixed-point theorem, we prove the global existence of discrete solutions under the proper conditions, and furthermore, we obtain a priori hp error estimates of the pressures in L 2 (H 1) and the saturations in L ∞(L 2) and L 2 (H 1). © 2014 Wiley Periodicals, Inc. 8. Incorporation of exact boundary conditions into a discontinuous galerkin finite element method for accurately solving 2d time-dependent maxwell equations KAUST Repository Sirenko, Kostyantyn; Liu, Meilin; Bagci, Hakan 2013-01-01 A scheme that discretizes exact absorbing boundary conditions (EACs) to incorporate them into a time-domain discontinuous Galerkin finite element method (TD-DG-FEM) is described. The proposed TD-DG-FEM with EACs is used for accurately characterizing 9. PHYSICAL-CONSTRAINT-PRESERVING CENTRAL DISCONTINUOUS GALERKIN METHODS FOR SPECIAL RELATIVISTIC HYDRODYNAMICS WITH A GENERAL EQUATION OF STATE Energy Technology Data Exchange (ETDEWEB) Wu, Kailiang [School of Mathematical Sciences, Peking University, Beijing 100871 (China); Tang, Huazhong, E-mail: wukl@pku.edu.cn, E-mail: hztang@math.pku.edu.cn [HEDPS, CAPT and LMAM, School of Mathematical Sciences, Peking University, Beijing 100871 (China) 2017-01-01 The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics (RHD). This paper develops high-order accurate, physical-constraints-preserving (PCP), central, discontinuous Galerkin (DG) methods for the one- and two-dimensional special RHD equations with a general EOS. It is built on our theoretical analysis of the admissible states for RHD and the PCP limiting procedure that enforce the admissibility of central DG solutions. The convexity, scaling invariance, orthogonal invariance, and Lax–Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form. Then, the high-order central DG methods with the PCP limiting procedure and strong stability-preserving time discretization are proved, to preserve the positivity of the density, pressure, specific internal energy, and the bound of the fluid velocity, maintain high-order accuracy, and be L {sup 1}-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density/pressure, etc. 10. An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media Science.gov (United States) Le Bouteiller, P.; Benjemaa, M.; Métivier, L.; Virieux, J. 2018-03-01 Accurate numerical computation of wave traveltimes in heterogeneous media is of major interest for a large range of applications in seismics, such as phase identification, data windowing, traveltime tomography and seismic imaging. A high level of precision is needed for traveltimes and their derivatives in applications which require quantities such as amplitude or take-off angle. Even more challenging is the anisotropic case, where the general Eikonal equation is a quartic in the derivatives of traveltimes. Despite their efficiency on Cartesian meshes, finite-difference solvers are inappropriate when dealing with unstructured meshes and irregular topographies. Moreover, reaching high orders of accuracy generally requires wide stencils and high additional computational load. To go beyond these limitations, we propose a discontinuous-finite-element-based strategy which has the following advantages: (1) the Hamiltonian formalism is general enough for handling the full anisotropic Eikonal equations; (2) the scheme is suitable for any desired high-order formulation or mixing of orders (p-adaptivity); (3) the solver is explicit whatever Hamiltonian is used (no need to find the roots of the quartic); (4) the use of unstructured meshes provides the flexibility for handling complex boundary geometries such as topographies (h-adaptivity) and radiation boundary conditions for mimicking an infinite medium. The point-source factorization principles are extended to this discontinuous Galerkin formulation. Extensive tests in smooth analytical media demonstrate the high accuracy of the method. Simulations in strongly heterogeneous media illustrate the solver robustness to realistic Earth-sciences-oriented applications. 11. Divergence-Conforming Discontinuous Galerkin Methods and$C^0$Interior Penalty Methods KAUST Repository Kanschat, Guido; Sharma, Natasha 2014-01-01 © 2014 Society for Industrial and Applied Mathematics. In this paper, we show that recently developed divergence-conforming methods for the Stokes problem have discrete stream functions. These stream functions in turn solve a continuous interior 12. -Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations Directory of Open Access Journals (Sweden) Lee HyunYoung 2010-01-01 Full Text Available We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction. 13. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry Energy Technology Data Exchange (ETDEWEB) Wintermeyer, Niklas [Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln (Germany); Winters, Andrew R., E-mail: awinters@math.uni-koeln.de [Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln (Germany); Gassner, Gregor J. [Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln (Germany); Kopriva, David A. [Department of Mathematics, The Florida State University, Tallahassee, FL 32306 (United States) 2017-07-01 We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem. 14. A fully-coupled discontinuous Galerkin spectral element method for two-phase flow in petroleum reservoirs Science.gov (United States) Taneja, Ankur; Higdon, Jonathan 2018-01-01 A high-order spectral element discontinuous Galerkin method is presented for simulating immiscible two-phase flow in petroleum reservoirs. The governing equations involve a coupled system of strongly nonlinear partial differential equations for the pressure and fluid saturation in the reservoir. A fully implicit method is used with a high-order accurate time integration using an implicit Rosenbrock method. Numerical tests give the first demonstration of high order hp spatial convergence results for multiphase flow in petroleum reservoirs with industry standard relative permeability models. High order convergence is shown formally for spectral elements with up to 8th order polynomials for both homogeneous and heterogeneous permeability fields. Numerical results are presented for multiphase fluid flow in heterogeneous reservoirs with complex geometric or geologic features using up to 11th order polynomials. Robust, stable simulations are presented for heterogeneous geologic features, including globally heterogeneous permeability fields, anisotropic permeability tensors, broad regions of low-permeability, high-permeability channels, thin shale barriers and thin high-permeability fractures. A major result of this paper is the demonstration that the resolution of the high order spectral element method may be exploited to achieve accurate results utilizing a simple cartesian mesh for non-conforming geological features. Eliminating the need to mesh to the boundaries of geological features greatly simplifies the workflow for petroleum engineers testing multiple scenarios in the face of uncertainty in the subsurface geology. 15. A non-linear optimal Discontinuous Petrov-Galerkin method for stabilising the solution of the transport equation International Nuclear Information System (INIS) Merton, S. R.; Smedley-Stevenson, R. P.; Pain, C. C.; Buchan, A. G.; Eaton, M. D. 2009-01-01 This paper describes a new Non-Linear Discontinuous Petrov-Galerkin (NDPG) method and application to the one-speed Boltzmann Transport Equation (BTE) for space-time problems. The purpose of the method is to remove unwanted oscillations in the transport solution which occur in the vicinity of sharp flux gradients, while improving computational efficiency and numerical accuracy. This is achieved by applying artificial dissipation in the solution gradient direction, internal to an element using a novel finite element (FE) Riemann approach. The amount of dissipation added acts internal to each element. This is done using a gradient-informed scaling of the advection velocities in the stabilisation term. This makes the method in its most general form non-linear. The method is designed to be independent of angular expansion framework. This is demonstrated for the both discrete ordinates (S N ) and spherical harmonics (P N ) descriptions of the angular variable. Results show the scheme performs consistently well in demanding time dependent and multi-dimensional radiation transport problems. (authors) 16. A conservative local discontinuous Galerkin method for the solution of nonlinear Schr(o)dinger equation in two dimensions Institute of Scientific and Technical Information of China (English) ZHANG RongPei; YU XiJun; LI MingJun; LI XiangGui 2017-01-01 In this study,we present a conservative local discontinuous Galerkin (LDG) method for numerically solving the two-dimensional nonlinear Schr(o)dinger (NLS) equation.The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux.The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central,alternative and upwind-based flux.We will propose two kinds of time discretization methods for the semi-discrete formulation.One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation.The other one is Krylov implicit integration factor (ⅡF) method which demands much less computational effort.Various numerical experiments are presented to demonstrate the conservation law of mass and energy,the optimal rates of convergence,and the blow-up phenomenon. 17. A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier-Stokes equations Science.gov (United States) Piatkowski, Marian; Müthing, Steffen; Bastian, Peter 2018-03-01 In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on H (div) reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability. 18. An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property KAUST Repository Friedrich, Lucas 2017-12-29 This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between nonconforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h/p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples. 19. Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes KAUST Repository Pelties, Christian 2012-02-18 Accurate and efficient numerical methods to simulate dynamic earthquake rupture and wave propagation in complex media and complex fault geometries are needed to address fundamental questions in earthquake dynamics, to integrate seismic and geodetic data into emerging approaches for dynamic source inversion, and to generate realistic physics-based earthquake scenarios for hazard assessment. Modeling of spontaneous earthquake rupture and seismic wave propagation by a high-order discontinuous Galerkin (DG) method combined with an arbitrarily high-order derivatives (ADER) time integration method was introduced in two dimensions by de la Puente et al. (2009). The ADER-DG method enables high accuracy in space and time and discretization by unstructured meshes. Here we extend this method to three-dimensional dynamic rupture problems. The high geometrical flexibility provided by the usage of tetrahedral elements and the lack of spurious mesh reflections in the ADER-DG method allows the refinement of the mesh close to the fault to model the rupture dynamics adequately while concentrating computational resources only where needed. Moreover, ADER-DG does not generate spurious high-frequency perturbations on the fault and hence does not require artificial Kelvin-Voigt damping. We verify our three-dimensional implementation by comparing results of the SCEC TPV3 test problem with two well-established numerical methods, finite differences, and spectral boundary integral. Furthermore, a convergence study is presented to demonstrate the systematic consistency of the method. To illustrate the capabilities of the high-order accurate ADER-DG scheme on unstructured meshes, we simulate an earthquake scenario, inspired by the 1992 Landers earthquake, that includes curved faults, fault branches, and surface topography. Copyright 2012 by the American Geophysical Union. 20. Regionally Implicit Discontinuous Galerkin Methods for Solving the Relativistic Vlasov-Maxwell System Submitted to Iowa State University Science.gov (United States) Guthrey, Pierson Tyler The relativistic Vlasov-Maxwell system (RVM) models the behavior of collisionless plasma, where electrons and ions interact via the electromagnetic fields they generate. In the RVM system, electrons could accelerate to significant fractions of the speed of light. An idea that is actively being pursued by several research groups around the globe is to accelerate electrons to relativistic speeds by hitting a plasma with an intense laser beam. As the laser beam passes through the plasma it creates plasma wakes, much like a ship passing through water, which can trap electrons and push them to relativistic speeds. Such setups are known as laser wakefield accelerators, and have the potential to yield particle accelerators that are significantly smaller than those currently in use. Ultimately, the goal of such research is to harness the resulting electron beams to generate electromagnetic waves that can be used in medical imaging applications. High-order accurate numerical discretizations of kinetic Vlasov plasma models are very effective at yielding low-noise plasma simulations, but are computationally expensive to solve because of the high dimensionality. In addition to the general difficulties inherent to numerically simulating Vlasov models, the relativistic Vlasov-Maxwell system has unique challenges not present in the non-relativistic case. One such issue is that operator splitting of the phase gradient leads to potential instabilities, thus we require an alternative to operator splitting of the phase. The goal of the current work is to develop a new class of high-order accurate numerical methods for solving kinetic Vlasov models of plasma. The main discretization in configuration space is handled via a high-order finite element method called the discontinuous Galerkin method (DG). One difficulty is that standard explicit time-stepping methods for DG suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL 1. The Discontinuous Galerkin Finite Element Method for Solving the MEG and the Combined MEG/EEG Forward Problem Directory of Open Access Journals (Sweden) Maria Carla Piastra 2018-02-01 Full Text Available In Electro- (EEG and Magnetoencephalography (MEG, one important requirement of source reconstruction is the forward model. The continuous Galerkin finite element method (CG-FEM has become one of the dominant approaches for solving the forward problem over the last decades. Recently, a discontinuous Galerkin FEM (DG-FEM EEG forward approach has been proposed as an alternative to CG-FEM (Engwer et al., 2017. It was shown that DG-FEM preserves the property of conservation of charge and that it can, in certain situations such as the so-called skull leakages, be superior to the standard CG-FEM approach. In this paper, we developed, implemented, and evaluated two DG-FEM approaches for the MEG forward problem, namely a conservative and a non-conservative one. The subtraction approach was used as source model. The validation and evaluation work was done in statistical investigations in multi-layer homogeneous sphere models, where an analytic solution exists, and in a six-compartment realistically shaped head volume conductor model. In agreement with the theory, the conservative DG-FEM approach was found to be superior to the non-conservative DG-FEM implementation. This approach also showed convergence with increasing resolution of the hexahedral meshes. While in the EEG case, in presence of skull leakages, DG-FEM outperformed CG-FEM, in MEG, DG-FEM achieved similar numerical errors as the CG-FEM approach, i.e., skull leakages do not play a role for the MEG modality. In particular, for the finest mesh resolution of 1 mm sources with a distance of 1.59 mm from the brain-CSF surface, DG-FEM yielded mean topographical errors (relative difference measure, RDM% of 1.5% and mean magnitude errors (MAG% of 0.1% for the magnetic field. However, if the goal is a combined source analysis of EEG and MEG data, then it is highly desirable to employ the same forward model for both EEG and MEG data. Based on these results, we conclude that the newly presented 2. The Discontinuous Galerkin Finite Element Method for Solving the MEG and the Combined MEG/EEG Forward Problem. Science.gov (United States) Piastra, Maria Carla; Nüßing, Andreas; Vorwerk, Johannes; Bornfleth, Harald; Oostenveld, Robert; Engwer, Christian; Wolters, Carsten H 2018-01-01 In Electro- (EEG) and Magnetoencephalography (MEG), one important requirement of source reconstruction is the forward model. The continuous Galerkin finite element method (CG-FEM) has become one of the dominant approaches for solving the forward problem over the last decades. Recently, a discontinuous Galerkin FEM (DG-FEM) EEG forward approach has been proposed as an alternative to CG-FEM (Engwer et al., 2017). It was shown that DG-FEM preserves the property of conservation of charge and that it can, in certain situations such as the so-called skull leakages , be superior to the standard CG-FEM approach. In this paper, we developed, implemented, and evaluated two DG-FEM approaches for the MEG forward problem, namely a conservative and a non-conservative one. The subtraction approach was used as source model. The validation and evaluation work was done in statistical investigations in multi-layer homogeneous sphere models, where an analytic solution exists, and in a six-compartment realistically shaped head volume conductor model. In agreement with the theory, the conservative DG-FEM approach was found to be superior to the non-conservative DG-FEM implementation. This approach also showed convergence with increasing resolution of the hexahedral meshes. While in the EEG case, in presence of skull leakages, DG-FEM outperformed CG-FEM, in MEG, DG-FEM achieved similar numerical errors as the CG-FEM approach, i.e., skull leakages do not play a role for the MEG modality. In particular, for the finest mesh resolution of 1 mm sources with a distance of 1.59 mm from the brain-CSF surface, DG-FEM yielded mean topographical errors (relative difference measure, RDM%) of 1.5% and mean magnitude errors (MAG%) of 0.1% for the magnetic field. However, if the goal is a combined source analysis of EEG and MEG data, then it is highly desirable to employ the same forward model for both EEG and MEG data. Based on these results, we conclude that the newly presented conservative DG 3. An Unstructured Grid Morphodynamic Model with a Discontinuous Galerkin Method for Bed Evolution National Research Council Canada - National Science Library Kubatko, Ethan J; Westerink, Joannes J; Dawson, Clint 2005-01-01 ...) method for the solution of the sediment continuity equation. The DG method is a robust finite element method that is particularly well suited for this type of advection dominated transport equation... 4. Discontinuous Galerkin Subgrid Finite Element Method for Heterogeneous Brinkman’s Equations KAUST Repository Iliev, Oleg P.; Lazarov, Raytcho D.; Willems, Joerg 2010-01-01 We present a two-scale finite element method for solving Brinkman's equations with piece-wise constant coefficients. This system of equations model fluid flows in highly porous, heterogeneous media with complex topology of the heterogeneities. We 5. Hybrid Fourier pseudospectral/discontinuous Galerkin time-domain method for arbitrary boundary conditions NARCIS (Netherlands) Pagan Munoz, R.; Hornikx, M.C.J. The wave-based Fourier Pseudospectral time-domain (Fourier-PSTD) method was shown to be an effective way of modeling outdoor acoustic propagation problems as described by the linearized Euler equations (LEE), but is limited to real-valued frequency independent boundary conditions and predominantly 6. Exploiting Superconvergence in Discontinuous Galerkin Methods for Improved Time-Stepping and Visualization Science.gov (United States) 2016-09-08 Filters for Multi-dimensional data”, International Conference on Spectral and Higher Order Methods (ICOSAHOM). Rio de Janeiro, Brazil. June 27 - July...Accuracy Conserving (SIAC) filter when applied to nonuniform meshes; 2) Theoretically and numerical demonstration of the 2k+1 order accuracy of the SIAC...filter to reduce the kernel footprint; 4) Establishing the theoretical viability of the SIAC filter for nonlinear scalar hyperbolic conservation laws 7. Discontinuous Galerkin Time-Domain Analysis of Power-Ground Planes Taking Into Account Decoupling Capacitors KAUST Repository Li, Ping; Jiang, Li Jun; Bagci, Hakan 2017-01-01 In this paper, a discontinuous Galerkin time-domain (DGTD) method is developed to analyze the power-ground planes taking into account the decoupling capacitors. In the presence of decoupling capacitors, the whole physical system can be split 8. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms International Nuclear Information System (INIS) Xing Yulong; Shu Chiwang 2006-01-01 Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. In our earlier work [J. Comput. Phys. 208 (2005) 206-227; J. Sci. Comput., accepted], we designed a well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which at the same time maintains genuine high order accuracy for general solutions, to a class of hyperbolic systems with separable source terms including the shallow water equations, the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. In this paper, we generalize high order finite volume WENO schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property. Finite volume and discontinuous Galerkin finite element schemes are more flexible than finite difference schemes to treat complicated geometry and adaptivity. However, because of a different computational framework, the maintenance of the well-balanced property requires different technical approaches. After the description of our well-balanced high order finite volume WENO and RKDG schemes, we perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions 9. On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations Science.gov (United States) Fehn, Niklas; Wall, Wolfgang A.; Kronbichler, Martin 2017-12-01 The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for small time step sizes. Since the critical time step size depends on the viscosity and the spatial resolution, these instabilities limit the robustness of the Navier-Stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly. 10. L2-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations Directory of Open Access Journals (Sweden) Hyun Young Lee 2010-01-01 Full Text Available We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal ℓ∞(L2 error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction. 11. Incorporation of exact boundary conditions into a discontinuous galerkin finite element method for accurately solving 2d time-dependent maxwell equations KAUST Repository Sirenko, Kostyantyn 2013-01-01 A scheme that discretizes exact absorbing boundary conditions (EACs) to incorporate them into a time-domain discontinuous Galerkin finite element method (TD-DG-FEM) is described. The proposed TD-DG-FEM with EACs is used for accurately characterizing transient electromagnetic wave interactions on two-dimensional waveguides. Numerical results demonstrate the proposed method\\'s superiority over the TD-DG-FEM that employs approximate boundary conditions and perfectly matched layers. Additionally, it is shown that the proposed method can produce the solution with ten-eleven digit accuracy when high-order spatial basis functions are used to discretize the Maxwell equations as well as the EACs. © 1963-2012 IEEE. 12. An H1(Ph)-Coercive Discontinuous Galerkin Formulation for the Poisson Problem : 1-D Analysis NARCIS (Netherlands) Van der Zee, K.G.; Van Brummelen, E.H. 2005-01-01 Discontinuous Galerkin (DG) methods are finite element techniques for the solution of partial differential equations. They allow shape functions which are discontinuous across inter-element edges. In principle, DG methods are ideally suited for hp-adaptivity, as they handle nonconforming meshes and 13. A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D KAUST Repository Zitelli, J.; Muga, Ignacio; Demkowicz, Leszek F.; Gopalakrishnan, Jayadeep; Pardo, David; Calo, Victor M. 2011-01-01 The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed. © 2010 Elsevier Inc. 14. A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D KAUST Repository Zitelli, J. 2011-04-01 The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed. © 2010 Elsevier Inc. 15. ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics Science.gov (United States) Fambri, F.; Dumbser, M.; Köppel, S.; Rezzolla, L.; Zanotti, O. 2018-03-01 We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes. In this paper we assume the background spacetime to be given and static, i.e. we make use of the Cowling approximation. The governing partial differential equations are solved via a new family of fully-discrete and arbitrary high-order accurate path-conservative discontinuous Galerkin (DG) finite-element methods combined with adaptive mesh refinement and time accurate local timestepping. In order to deal with shock waves and other discontinuities, the high-order DG schemes are supplemented with a novel a-posteriori subcell finite-volume limiter, which makes the new algorithms as robust as classical second-order total-variation diminishing finite-volume methods at shocks and discontinuities, but also as accurate as unlimited high-order DG schemes in smooth regions of the flow. We show the advantages of this new approach by means of various classical two- and three-dimensional benchmark problems on fixed spacetimes. Finally, we present a performance and accuracy comparisons between Runge-Kutta DG schemes and ADER high-order finite-volume schemes, showing the higher efficiency of DG schemes. 16. Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation NARCIS (Netherlands) P.W. Hemker (Piet); M.H. van Raalte (Marc) 2002-01-01 textabstractIn this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann-Oden and for the symmetric DG method, we give a detailed analysis of the 17. Fourier two-level analysis for discontinuous Galerkin discretization with linear elements NARCIS (Netherlands) P.W. Hemker (Piet); W. Hoffmann; M.H. van Raalte (Marc) 2002-01-01 textabstractIn this paper we study the convergence of a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence fordifferent block-relaxation strategies. In addition to an 18. Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretisation NARCIS (Netherlands) P.W. Hemker (Piet); W. Hoffmann; M.H. van Raalte (Marc) 2002-01-01 textabstractIn this paper we study a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, andwe give a detailed analysis of the convergence for different block-relaxation strategies.We find that point-wise 19. A discontinuous Galerkin method with a bound preserving limiter for the advection of non-diffusive fields in solid Earth geodynamics Science.gov (United States) He, Ying; Puckett, Elbridge Gerry; Billen, Magali I. 2017-02-01 Mineral composition has a strong effect on the properties of rocks and is an essentially non-diffusive property in the context of large-scale mantle convection. Due to the non-diffusive nature and the origin of compositionally distinct regions in the Earth the boundaries between distinct regions can be nearly discontinuous. While there are different methods for tracking rock composition in numerical simulations of mantle convection, one must consider trade-offs between computational cost, accuracy or ease of implementation when choosing an appropriate method. Existing methods can be computationally expensive, cause over-/undershoots, smear sharp boundaries, or are not easily adapted to tracking multiple compositional fields. Here we present a Discontinuous Galerkin method with a bound preserving limiter (abbreviated as DG-BP) using a second order Runge-Kutta, strong stability-preserving time discretization method for the advection of non-diffusive fields. First, we show that the method is bound-preserving for a point-wise divergence free flow (e.g., a prescribed circular flow in a box). However, using standard adaptive mesh refinement (AMR) there is an over-shoot error (2%) because the cell average is not preserved during mesh coarsening. The effectiveness of the algorithm for convection-dominated flows is demonstrated using the falling box problem. We find that the DG-BP method maintains sharper compositional boundaries (3-5 elements) as compared to an artificial entropy-viscosity method (6-15 elements), although the over-/undershoot errors are similar. When used with AMR the DG-BP method results in fewer degrees of freedom due to smaller regions of mesh refinement in the neighborhood of the discontinuity. However, using Taylor-Hood elements and a uniform mesh there is an over-/undershoot error on the order of 0.0001%, but this error increases to 0.01-0.10% when using AMR. Therefore, for research problems in which a continuous field method is desired the DG 20. An adaptive simplex cut-cell method for high-order discontinuous Galerkin discretizations of elliptic interface problems and conjugate heat transfer problems Science.gov (United States) Sun, Huafei; Darmofal, David L. 2014-12-01 In this paper we propose a new high-order solution framework for interface problems on non-interface-conforming meshes. The framework consists of a discontinuous Galerkin (DG) discretization, a simplex cut-cell technique, and an output-based adaptive scheme. We first present a DG discretization with a dual-consistent output evaluation for elliptic interface problems on interface-conforming meshes, and then extend the method to handle multi-physics interface problems, in particular conjugate heat transfer (CHT) problems. The method is then applied to non-interface-conforming meshes using a cut-cell technique, where the interface definition is completely separate from the mesh generation process. No assumption is made on the interface shape (other than Lipschitz continuity). We then equip our strategy with an output-based adaptive scheme for an accurate output prediction. Through numerical examples, we demonstrate high-order convergence for elliptic interface problems and CHT problems with both smooth and non-smooth interface shapes. 1. Stability Analysis of Discontinuous Galerkin Approximations to the Elastodynamics Problem KAUST Repository Antonietti, Paola F. 2015-11-21 We consider semi-discrete discontinuous Galerkin approximations of both displacement and displacement-stress formulations of the elastodynamics problem. We prove the stability analysis in the natural energy norm and derive optimal a-priori error estimates. For the displacement-stress formulation, schemes preserving the total energy of the system are introduced and discussed. We verify our theoretical estimates on two and three dimensions test problems. 2. Stability Analysis of Discontinuous Galerkin Approximations to the Elastodynamics Problem KAUST Repository Antonietti, Paola F.; Ayuso de Dios, Blanca; Mazzieri, Ilario; Quarteroni, Alfio 2015-01-01 We consider semi-discrete discontinuous Galerkin approximations of both displacement and displacement-stress formulations of the elastodynamics problem. We prove the stability analysis in the natural energy norm and derive optimal a-priori error estimates. For the displacement-stress formulation, schemes preserving the total energy of the system are introduced and discussed. We verify our theoretical estimates on two and three dimensions test problems. 3. DNS of Low-Pressure Turbine Cascade Flows with Elevated Inflow Turbulence Using a Discontinuous-Galerkin Spectral-Element Method Science.gov (United States) Garai, Anirban; Diosady, Laslo T.; Murman, Scott M.; Madavan, Nateri K. 2016-01-01 Recent progress towards developing a new computational capability for accurate and efficient high-fidelity direct numerical simulation (DNS) and large-eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy- stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in a computationally efficient manner on a modern high performance computer architecture. An inflow turbulence generation procedure based on a linear forcing approach has been incorporated in this framework and DNS conducted to study the effect of inflow turbulence on the suction- side separation bubble in low-pressure turbine (LPT) cascades. The T106 series of airfoil cascades in both lightly (T106A) and highly loaded (T106C) configurations at exit isentropic Reynolds numbers of 60,000 and 80,000, respectively, are considered. The numerical simulations are performed using 8th-order accurate spatial and 4th-order accurate temporal discretization. The changes in separation bubble topology due to elevated inflow turbulence is captured by the present method and the physical mechanisms leading to the changes are explained. The present results are in good agreement with prior numerical simulations but some expected discrepancies with the experimental data for the T106C case are noted and discussed. 4. Discontinuous Galerkin Approaches for Stokes Flow and Flow in Porous Media Science.gov (United States) Lehmann, Ragnar; Kaus, Boris; Lukacova, Maria 2014-05-01 Firstly, we present results of a study comparing two different numerical approaches for solving the Stokes equations with strongly varying viscosity: the continuous Galerkin (i.e., FEM) and the discontinuous Galerkin (DG) method. Secondly, we show how the latter method can be extended and applied to flow in porous media governed by Darcy's law. Nonlinearities in the viscosity or other material parameters can lead to discontinuities in the velocity-pressure solution that may not be approximated well with continuous elements. The DG method allows for discontinuities across interior edges of the underlying mesh. Furthermore, depending on the chosen basis functions, it naturally enforces local mass conservation, i.e., in every mesh cell. Computationally, it provides the capability to locally adapt the polynomial degree and needs communication only between directly adjacent mesh cells making it highly flexible and easy to parallelize. The methods are compared for several geophysically relevant benchmarking setups and discussed with respect to speed, accuracy, computational efficiency. 5. Off-fault plasticity in three-dimensional dynamic rupture simulations using a modal Discontinuous Galerkin method on unstructured meshes: Implementation, verification, and application Science.gov (United States) Wollherr, Stephanie; Gabriel, Alice-Agnes; Uphoff, Carsten 2018-05-01 The dynamics and potential size of earthquakes depend crucially on rupture transfers between adjacent fault segments. To accurately describe earthquake source dynamics, numerical models can account for realistic fault geometries and rheologies such as nonlinear inelastic processes off the slip interface. We present implementation, verification, and application of off-fault Drucker-Prager plasticity in the open source software SeisSol (www.seissol.org). SeisSol is based on an arbitrary high-order derivative modal Discontinuous Galerkin (ADER-DG) method using unstructured, tetrahedral meshes specifically suited for complex geometries. Two implementation approaches are detailed, modelling plastic failure either employing sub-elemental quadrature points or switching to nodal basis coefficients. At fine fault discretizations the nodal basis approach is up to 6 times more efficient in terms of computational costs while yielding comparable accuracy. Both methods are verified in community benchmark problems and by three dimensional numerical h- and p-refinement studies with heterogeneous initial stresses. We observe no spectral convergence for on-fault quantities with respect to a given reference solution, but rather discuss a limitation to low-order convergence for heterogeneous 3D dynamic rupture problems. For simulations including plasticity, a high fault resolution may be less crucial than commonly assumed, due to the regularization of peak slip rate and an increase of the minimum cohesive zone width. In large-scale dynamic rupture simulations based on the 1992 Landers earthquake, we observe high rupture complexity including reverse slip, direct branching, and dynamic triggering. The spatio-temporal distribution of rupture transfers are altered distinctively by plastic energy absorption, correlated with locations of geometrical fault complexity. Computational cost increases by 7% when accounting for off-fault plasticity in the demonstrating application. Our results 6. An efficient Adaptive Mesh Refinement (AMR) algorithm for the Discontinuous Galerkin method: Applications for the computation of compressible two-phase flows Science.gov (United States) Papoutsakis, Andreas; Sazhin, Sergei S.; Begg, Steven; Danaila, Ionut; Luddens, Francky 2018-06-01 We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices. A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces. The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian 7. Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes KAUST Repository Pelties, Christian; de la Puente, Josep; Ampuero, Jean-Paul; Brietzke, Gilbert B.; Kä ser, Martin 2012-01-01 Accurate and efficient numerical methods to simulate dynamic earthquake rupture and wave propagation in complex media and complex fault geometries are needed to address fundamental questions in earthquake dynamics, to integrate seismic and geodetic 8. An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property KAUST Repository Friedrich, Lucas; Winters, Andrew R.; Ferná ndez, David C. Del Rey; Gassner, Gregor J.; Parsani, Matteo; Carpenter, Mark H. 2017-01-01 analysis are discretely mimicked. Special attention is given to the coupling between nonconforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead 9. Seismic wave propagation in fractured media: A discontinuous Galerkin approach KAUST Repository De Basabe, Jonás D. 2011-01-01 We formulate and implement a discontinuous Galekin method for elastic wave propagation that allows for discontinuities in the displacement field to simulate fractures or faults using the linear- slip model. We show numerical results using a 2D model with one linear- slip discontinuity and different frequencies. The results show a good agreement with analytic solutions. © 2011 Society of Exploration Geophysicists. 10. Topology Optimization of Nano-Mechanical Cantilever Sensors Using a C0 Discontinuous Galerkin-Type Approach DEFF Research Database (Denmark) Marhadi, Kun Saptohartyadi; Evgrafov, Anton; Sørensen, Mads Peter 2011-01-01 We demonstrate the use of a C0 discontinuous Galerkin method for topology optimization of nano-mechanical sensors, namely temperature, surface stress, and mass sensors. The sensors are modeled using classical thin plate theory, which requires C1 basis functions in the standard finite element method... 11. Discontinuous Galerkin time-domain analysis of power/ground plate pairs with wave port excitation KAUST Repository Li, Ping; Jiang, Li Jun; Bagci, Hakan 2018-01-01 In this work, a discontinuous Galerkin time-domain method is developed to analyze the power/ground plate pairs taking into account arbitrarily shaped antipads. To implement proper source excitations over the antipads, the magnetic surface current expanded by the electric eigen-modes supported by the corresponding antipad is employed as the excitation. For irregularly shaped antipads, the eigen-modes are obtained by numerical approach. Accordingly, the methodology for the S-parameter extraction is derived based on the orthogonal properties of the different modes. Based on the approach, the transformation between different modes can be readily evaluated. 12. Discontinuous Galerkin time-domain analysis of power/ground plate pairs with wave port excitation KAUST Repository Li, Ping 2018-04-06 In this work, a discontinuous Galerkin time-domain method is developed to analyze the power/ground plate pairs taking into account arbitrarily shaped antipads. To implement proper source excitations over the antipads, the magnetic surface current expanded by the electric eigen-modes supported by the corresponding antipad is employed as the excitation. For irregularly shaped antipads, the eigen-modes are obtained by numerical approach. Accordingly, the methodology for the S-parameter extraction is derived based on the orthogonal properties of the different modes. Based on the approach, the transformation between different modes can be readily evaluated. 13. A discontinous Galerkin finite element method with an efficient time integration scheme for accurate simulations KAUST Repository Liu, Meilin; Bagci, Hakan 2011-01-01 A discontinuous Galerkin finite element method (DG-FEM) with a highly-accurate time integration scheme is presented. The scheme achieves its high accuracy using numerically constructed predictor-corrector integration coefficients. Numerical results 14. h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems Science.gov (United States) Botti, L.; Colombo, A.; Bassi, F. 2017-10-01 In this work we exploit agglomeration based h-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature h-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element L2 projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented. 15. A Discontinuous Galerkin Model for Fluorescence Loss in Photobleaching DEFF Research Database (Denmark) Hansen, Christian Valdemar; Schroll, Achim; Wüstner, Daniel 2018-01-01 Fluorescence loss in photobleaching (FLIP) is a modern microscopy method for visualization of transport processes in living cells. This paper presents the simulation of FLIP sequences based on a calibrated reaction–di usion system de ned on segmented cell images. By the use of a discontinuous...... of the nuclear membrane for GFP passage, directly from the FLIP image series. Thus, we present for the rst time, to our knowledge, a quantitative computational FLIP method for inferring several molecular transport parameters in parallel from FLIP image data acquired at commercial microscope systems.... 16. Parallel discontinuous Galerkin FEM for computing hyperbolic conservation law on unstructured grids Science.gov (United States) Ma, Xinrong; Duan, Zhijian 2018-04-01 High-order resolution Discontinuous Galerkin finite element methods (DGFEM) has been known as a good method for solving Euler equations and Navier-Stokes equations on unstructured grid, but it costs too much computational resources. An efficient parallel algorithm was presented for solving the compressible Euler equations. Moreover, the multigrid strategy based on three-stage three-order TVD Runge-Kutta scheme was used in order to improve the computational efficiency of DGFEM and accelerate the convergence of the solution of unsteady compressible Euler equations. In order to make each processor maintain load balancing, the domain decomposition method was employed. Numerical experiment performed for the inviscid transonic flow fluid problems around NACA0012 airfoil and M6 wing. The results indicated that our parallel algorithm can improve acceleration and efficiency significantly, which is suitable for calculating the complex flow fluid. 17. A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion Science.gov (United States) Huynh, H. T. 2009-01-01 We introduce a new approach to high-order accuracy for the numerical solution of diffusion problems by solving the equations in differential form using a reconstruction technique. The approach has the advantages of simplicity and economy. It results in several new high-order methods including a simplified version of discontinuous Galerkin (DG). It also leads to new definitions of common value and common gradient quantities at each interface shared by the two adjacent cells. In addition, the new approach clarifies the relations among the various choices of new and existing common quantities. Fourier stability and accuracy analyses are carried out for the resulting schemes. Extensions to the case of quadrilateral meshes are obtained via tensor products. For the two-point boundary value problem (steady state), it is shown that these schemes, which include most popular DG methods, yield exact common interface quantities as well as exact cell average solutions for nearly all cases. 18. Discontinuous Galerkin methodology for Large-Eddy Simulations of wind turbine airfoils DEFF Research Database (Denmark) Frére, A.; Sørensen, Niels N.; Hillewaert, K. 2016-01-01 This paper aims at evaluating the potential of the Discontinuous Galerkin (DG) methodology for Large-Eddy Simulation (LES) of wind turbine airfoils. The DG method has shown high accuracy, excellent scalability and capacity to handle unstructured meshes. It is however not used in the wind energy...... sector yet. The present study aims at evaluating this methodology on an application which is relevant for that sector and focuses on blade section aerodynamics characterization. To be pertinent for large wind turbines, the simulations would need to be at low Mach numbers (M ≤ 0.3) where compressible...... at low and high Reynolds numbers and compares the results to state-of-the-art models used in industry, namely the panel method (XFOIL with boundary layer modeling) and Reynolds Averaged Navier-Stokes (RANS). At low Reynolds number (Re = 6 × 104), involving laminar boundary layer separation and transition... 19. Discontinuous Galerkin discretization and hp-refinement for the resolution of the neutron transport equation International Nuclear Information System (INIS) Fournier, Damien; Le-Tellier, Romain; Herbin, Raphaele 2013-01-01 This paper presents an hp-refinement method for a first order scalar transport reaction equation discretized by a discontinuous Galerkin method. First, the theoretical rates of convergence of h- and p-refinement are recalled and numerically tested. Then, in order to design some meshes, we propose two different estimators of the local error on the spatial domain. These quantities are analyzed and compared depending on the regularity of the solution so as to find the best way to lead the refinement process and the best strategy to choose between h- and p-refinement. Finally, the different possible refinement strategies are compared first on analytical examples and then on realistic applications for neutron transport in a nuclear reactor core. (authors) 20. Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals NARCIS (Netherlands) Lu, Zhongjie; Cesmelioglu, A.; van der Vegt, Jacobus J.W.; Xu, Yan We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with 1. Propel: A Discontinuous-Galerkin Finite Element Code for Solving the Reacting Navier-Stokes Equations Science.gov (United States) Johnson, Ryan; Kercher, Andrew; Schwer, Douglas; Corrigan, Andrew; Kailasanath, Kazhikathra 2017-11-01 This presentation focuses on the development of a Discontinuous Galerkin (DG) method for application to chemically reacting flows. The in-house code, called Propel, was developed by the Laboratory of Computational Physics and Fluid Dynamics at the Naval Research Laboratory. It was designed specifically for developing advanced multi-dimensional algorithms to run efficiently on new and innovative architectures such as GPUs. For these results, Propel solves for convection and diffusion simultaneously with detailed transport and thermodynamics. Chemistry is currently solved in a time-split approach using Strang-splitting with finite element DG time integration of chemical source terms. Results presented here show canonical unsteady reacting flow cases, such as co-flow and splitter plate, and we report performance for higher order DG on CPU and GPUs. 2. On discontinuous Galerkin and discrete ordinates approximations for neutron transport equation and the critical eigenvalue International Nuclear Information System (INIS) Asadzadeh, M.; Thevenot, L. 2010-01-01 The objective of this paper is to give a mathematical framework for a fully discrete numerical approach for the study of the neutron transport equation in a cylindrical domain (container model,). More specifically, we consider the discontinuous Galerkin (D G) finite element method for spatial approximation of the mono-energetic, critical neutron transport equation in an infinite cylindrical domain ??in R3 with a polygonal convex cross-section ? The velocity discretization relies on a special quadrature rule developed to give optimal estimates in discrete ordinate parameters compatible with the quasi-uniform spatial mesh. We use interpolation spaces and derive optimal error estimates, up to maximal available regularity, for the fully discrete scalar flux. Finally we employ a duality argument and prove superconvergence estimates for the critical eigenvalue. 3. Dual-scale Galerkin methods for Darcy flow Science.gov (United States) Wang, Guoyin; Scovazzi, Guglielmo; Nouveau, Léo; Kees, Christopher E.; Rossi, Simone; Colomés, Oriol; Main, Alex 2018-02-01 The discontinuous Galerkin (DG) method has found widespread application in elliptic problems with rough coefficients, of which the Darcy flow equations are a prototypical example. One of the long-standing issues of DG approximations is the overall computational cost, and many different strategies have been proposed, such as the variational multiscale DG method, the hybridizable DG method, the multiscale DG method, the embedded DG method, and the Enriched Galerkin method. In this work, we propose a mixed dual-scale Galerkin method, in which the degrees-of-freedom of a less computationally expensive coarse-scale approximation are linked to the degrees-of-freedom of a base DG approximation. We show that the proposed approach has always similar or improved accuracy with respect to the base DG method, with a considerable reduction in computational cost. For the specific definition of the coarse-scale space, we consider Raviart-Thomas finite elements for the mass flux and piecewise-linear continuous finite elements for the pressure. We provide a complete analysis of stability and convergence of the proposed method, in addition to a study on its conservation and consistency properties. We also present a battery of numerical tests to verify the results of the analysis, and evaluate a number of possible variations, such as using piecewise-linear continuous finite elements for the coarse-scale mass fluxes. 4. On the time-stepping stability of continuous mass-lumped and discontinuous Galerkin finite elements for the 3D acoustic wave equation NARCIS (Netherlands) Zhebel, E.; Minisini, S.; Mulder, W.A. 2012-01-01 We solve the three-dimensional acoustic wave equation, discretized on tetrahedral meshes. Two methods are considered: mass-lumped continuous finite elements and the symmetric interior-penalty discontinuous Galerkin method (SIP-DG). Combining the spatial discretization with the leap-frog 5. Dynamic mesh adaptation for front evolution using discontinuous Galerkin based weighted condition number relaxation International Nuclear Information System (INIS) Greene, Patrick T.; Schofield, Samuel P.; Nourgaliev, Robert 2017-01-01 A new mesh smoothing method designed to cluster cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered fields, such as a volume fraction or index function, is provided. Results show that the low-order level set works equally well as the actual level set for mesh smoothing. Meshes generated for a number of interface geometries are presented, including cases with multiple level sets. Lastly, dynamic cases with moving interfaces show the new method is capable of maintaining a desired resolution near the interface with an acceptable number of relaxation iterations per time step, which demonstrates the method's potential to be used as a mesh relaxer for arbitrary Lagrangian Eulerian (ALE) methods. 6. Dynamic Mesh Adaptation for Front Evolution Using Discontinuous Galerkin Based Weighted Condition Number Mesh Relaxation Energy Technology Data Exchange (ETDEWEB) Greene, Patrick T. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Schofield, Samuel P. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Nourgaliev, Robert [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States) 2016-06-21 A new mesh smoothing method designed to cluster mesh cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function being computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered elds, such as a volume fraction or index function, is provided. Results show that the low-order level set works equally well for the weight function as the actual level set. Meshes generated for a number of interface geometries are presented, including cases with multiple level sets. Dynamic cases for moving interfaces are presented to demonstrate the method's potential usefulness to arbitrary Lagrangian Eulerian (ALE) methods. 7. A nodal discontinuous Galerkin approach to 3-D viscoelastic wave propagation in complex geological media Science.gov (United States) Lambrecht, L.; Lamert, A.; Friederich, W.; Möller, T.; Boxberg, M. S. 2018-03-01 A nodal discontinuous Galerkin (NDG) approach is developed and implemented for the computation of viscoelastic wavefields in complex geological media. The NDG approach combines unstructured tetrahedral meshes with an element-wise, high-order spatial interpolation of the wavefield based on Lagrange polynomials. Numerical fluxes are computed from an exact solution of the heterogeneous Riemann problem. Our implementation offers capabilities for modelling viscoelastic wave propagation in 1-D, 2-D and 3-D settings of very different spatial scale with little logistical overhead. It allows the import of external tetrahedral meshes provided by independent meshing software and can be run in a parallel computing environment. Computation of adjoint wavefields and an interface for the computation of waveform sensitivity kernels are offered. The method is validated in 2-D and 3-D by comparison to analytical solutions and results from a spectral element method. The capabilities of the NDG method are demonstrated through a 3-D example case taken from tunnel seismics which considers high-frequency elastic wave propagation around a curved underground tunnel cutting through inclined and faulted sedimentary strata. The NDG method was coded into the open-source software package NEXD and is available from GitHub. 8. Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients KAUST Repository Ayuso Dios, Blanca 2013-10-30 We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. © 2013 American Mathematical Society. 9. Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients KAUST Repository Ayuso Dios, Blanca; Holst, Michael; Zhu, Yunrong; Zikatanov, Ludmil 2013-01-01 We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. © 2013 American Mathematical Society. 10. Analysis of an a posteriori error estimator for the transport equation with SN and discontinuous Galerkin discretizations International Nuclear Information System (INIS) Fournier, D.; Le Tellier, R.; Suteau, C. 2011-01-01 We present an error estimator for the S N neutron transport equation discretized with an arbitrary high-order discontinuous Galerkin method. As a starting point, the estimator is obtained for conforming Cartesian meshes with a uniform polynomial order for the trial space then adapted to deal with non-conforming meshes and a variable polynomial order. Some numerical tests illustrate the properties of the estimator and its limitations. Finally, a simple shielding benchmark is analyzed in order to show the relevance of the estimator in an adaptive process. 11. A high order compact least-squares reconstructed discontinuous Galerkin method for the steady-state compressible flows on hybrid grids Science.gov (United States) Cheng, Jian; Zhang, Fan; Liu, Tiegang 2018-06-01 In this paper, a class of new high order reconstructed DG (rDG) methods based on the compact least-squares (CLS) reconstruction [23,24] is developed for simulating the two dimensional steady-state compressible flows on hybrid grids. The proposed method combines the advantages of the DG discretization with the flexibility of the compact least-squares reconstruction, which exhibits its superior potential in enhancing the level of accuracy and reducing the computational cost compared to the underlying DG methods with respect to the same number of degrees of freedom. To be specific, a third-order compact least-squares rDG(p1p2) method and a fourth-order compact least-squares rDG(p2p3) method are developed and investigated in this work. In this compact least-squares rDG method, the low order degrees of freedom are evolved through the underlying DG(p1) method and DG(p2) method, respectively, while the high order degrees of freedom are reconstructed through the compact least-squares reconstruction, in which the constitutive relations are built by requiring the reconstructed polynomial and its spatial derivatives on the target cell to conserve the cell averages and the corresponding spatial derivatives on the face-neighboring cells. The large sparse linear system resulted by the compact least-squares reconstruction can be solved relatively efficient when it is coupled with the temporal discretization in the steady-state simulations. A number of test cases are presented to assess the performance of the high order compact least-squares rDG methods, which demonstrates their potential to be an alternative approach for the high order numerical simulations of steady-state compressible flows. 12. Galerkin method for solving diffusion equations International Nuclear Information System (INIS) Tsapelkin, E.S. 1975-01-01 A programme for the solution of the three-dimensional two-group multizone neutron diffusion problem in (x, y, z)-geometry is described. The programme XYZ-5 gives the currents of both groups, the effective neutron multiplication coefficient and several integral properties of the reactor. The solution was found with the Galerkin method using speciallly constructed and chosen coordinate functions. The programme is written in ALGOL-60 and consists of 5 parts. Its text is given 13. OpenACC directive-based GPU acceleration of an implicit reconstructed discontinuous Galerkin method for compressible flows on 3D unstructured grids Energy Technology Data Exchange (ETDEWEB) Lou, Jialin [North Carolina State Univ., Raleigh, NC (United States); Xia, Yidong [Idaho National Lab. (INL), Idaho Falls, ID (United States); Luo, Lixiang [North Carolina State Univ., Raleigh, NC (United States); Luo, Hong [North Carolina State Univ., Raleigh, NC (United States); Edwards, Jack [North Carolina State Univ., Raleigh, NC (United States); Mueller, Frank [North Carolina State Univ., Raleigh, NC (United States) 2016-09-01 In this study, we use a combination of modeling techniques to describe the relationship between fracture radius that might be accomplished in a hypothetical enhanced geothermal system (EGS) and drilling distance required to create and access those fractures. We use a combination of commonly applied analytical solutions for heat transport in parallel fractures and 3D finite-element method models of more realistic heat extraction geometries. For a conceptual model involving multiple parallel fractures developed perpendicular to an inclined or horizontal borehole, calculations demonstrate that EGS will likely require very large fractures, of greater than 300 m radius, to keep interfracture drilling distances to ~10 km or less. As drilling distances are generally inversely proportional to the square of fracture radius, drilling costs quickly escalate as the fracture radius decreases. It is important to know, however, whether fracture spacing will be dictated by thermal or mechanical considerations, as the relationship between drilling distance and number of fractures is quite different in each case. Information about the likelihood of hydraulically creating very large fractures comes primarily from petroleum recovery industry data describing hydraulic fractures in shale. Those data suggest that fractures with radii on the order of several hundred meters may, indeed, be possible. The results of this study demonstrate that relatively simple calculations can be used to estimate primary design constraints on a system, particularly regarding the relationship between generated fracture radius and the total length of drilling needed in the fracture creation zone. Comparison of the numerical simulations of more realistic geometries than addressed in the analytical solutions suggest that simple proportionalities can readily be derived to relate a particular flow field. 14. Discontinuous Galerkin Time-Domain Modeling of Graphene Nano-Ribbon Incorporating the Spatial Dispersion Effects KAUST Repository Li, Ping; Jiang, Li Jun; Bagci, Hakan 2018-01-01 It is well known that graphene demonstrates spatial dispersion properties, i.e., its conductivity is nonlocal and a function of spectral wave number (momentum operator) q. In this paper, to account for effects of spatial dispersion on transmission of high speed signals along graphene nano-ribbon (GNR) interconnects, a discontinuous Galerkin time-domain (DGTD) algorithm is proposed. The atomically-thick GNR is modeled using a nonlocal transparent surface impedance boundary condition (SIBC) incorporated into the DGTD scheme. Since the conductivity is a complicated function of q (and one cannot find an analytical Fourier transform pair between q and spatial differential operators), an exact time domain SIBC model cannot be derived. To overcome this problem, the conductivity is approximated by its Taylor series in spectral domain under low-q assumption. This approach permits expressing the time domain SIBC in the form of a second-order partial differential equation (PDE) in current density and electric field intensity. To permit easy incorporation of this PDE with the DGTD algorithm, three auxiliary variables, which degenerate the second-order (temporal and spatial) differential operators to first-order ones, are introduced. Regarding to the temporal dispersion effects, the auxiliary differential equation (ADE) method is utilized to eliminates the expensive temporal convolutions. To demonstrate the applicability of the proposed scheme, numerical results, which involve characterization of spatial dispersion effects on the transfer impedance matrix of GNR interconnects, are presented. 15. Discontinuous Galerkin Time-Domain Modeling of Graphene Nano-Ribbon Incorporating the Spatial Dispersion Effects KAUST Repository Li, Ping 2018-04-13 It is well known that graphene demonstrates spatial dispersion properties, i.e., its conductivity is nonlocal and a function of spectral wave number (momentum operator) q. In this paper, to account for effects of spatial dispersion on transmission of high speed signals along graphene nano-ribbon (GNR) interconnects, a discontinuous Galerkin time-domain (DGTD) algorithm is proposed. The atomically-thick GNR is modeled using a nonlocal transparent surface impedance boundary condition (SIBC) incorporated into the DGTD scheme. Since the conductivity is a complicated function of q (and one cannot find an analytical Fourier transform pair between q and spatial differential operators), an exact time domain SIBC model cannot be derived. To overcome this problem, the conductivity is approximated by its Taylor series in spectral domain under low-q assumption. This approach permits expressing the time domain SIBC in the form of a second-order partial differential equation (PDE) in current density and electric field intensity. To permit easy incorporation of this PDE with the DGTD algorithm, three auxiliary variables, which degenerate the second-order (temporal and spatial) differential operators to first-order ones, are introduced. Regarding to the temporal dispersion effects, the auxiliary differential equation (ADE) method is utilized to eliminates the expensive temporal convolutions. To demonstrate the applicability of the proposed scheme, numerical results, which involve characterization of spatial dispersion effects on the transfer impedance matrix of GNR interconnects, are presented. 16. CosmosDG: An hp-adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD Science.gov (United States) Anninos, Peter; Bryant, Colton; Fragile, P. Chris; Holgado, A. Miguel; Lau, Cheuk; Nemergut, Daniel 2017-08-01 We have extended Cosmos++, a multidimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high-order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for the regularization of shocks. High-order multistage forward Euler and strong-stability preserving Runge-Kutta time integration options complement high-order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performed separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, although we note that an equivalent capability currently also exists in CosmosDG for Newtonian systems. 17. A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations Science.gov (United States) Sharifian, Mohammad Kazem; Kesserwani, Georges; Hassanzadeh, Yousef 2018-05-01 This work extends a robust second-order Runge-Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green-Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modeling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model. 18. CosmosDG: An hp -adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD Energy Technology Data Exchange (ETDEWEB) Anninos, Peter; Lau, Cheuk [Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 (United States); Bryant, Colton [Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois, 60208 (United States); Fragile, P. Chris [Department of Physics and Astronomy, College of Charleston, 66 George Street, Charleston, SC 29424 (United States); Holgado, A. Miguel [Department of Astronomy and National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, Illinois, 61801 (United States); Nemergut, Daniel [Operations and Engineering Division, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 (United States) 2017-08-01 We have extended Cosmos++, a multidimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high-order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for the regularization of shocks. High-order multistage forward Euler and strong-stability preserving Runge–Kutta time integration options complement high-order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performed separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, although we note that an equivalent capability currently also exists in CosmosDG for Newtonian systems. 19. CosmosDG: An hp -adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD International Nuclear Information System (INIS) Anninos, Peter; Lau, Cheuk; Bryant, Colton; Fragile, P. Chris; Holgado, A. Miguel; Nemergut, Daniel 2017-01-01 We have extended Cosmos++, a multidimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high-order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for the regularization of shocks. High-order multistage forward Euler and strong-stability preserving Runge–Kutta time integration options complement high-order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performed separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, although we note that an equivalent capability currently also exists in CosmosDG for Newtonian systems. 20. Modeling Storm Surges Using Discontinuous Galerkin Methods Science.gov (United States) 2016-06-01 layer non-reflecting boundary condition (NRBC) on the right wall of the model. A NRBC is when an artificial boundary , B, is created, which truncates the... applications ,” Journal of Computational Physics, 2004. [30] P. L. Butzer and R. Weis, “On the lax equivalence theorem equipped with orders,” Journal of...closer to the shoreline. In our simulation, we also learned of the effects spurious waves can have on the results. Due to boundary conditions, a 1. A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers Science.gov (United States) Tavelli, Maurizio; Dumbser, Michael 2017-07-01 We propose a new arbitrary high order accurate semi-implicit space-time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier-Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows. The new DG scheme extends the seminal ideas outlined in [1], where a second order semi-implicit finite volume method for the solution of the compressible Navier-Stokes equations with a general equation of state was introduced on staggered Cartesian grids. Regarding the high order extension we follow [2], where a staggered space-time DG scheme for the incompressible Navier-Stokes equations was presented. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. Then, the mass conservation equation, as well as the nonlinear convective terms in the momentum equation and the transport of kinetic energy in the energy equation are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Formal substitution of the discrete momentum equation into the total energy conservation equation yields a linear system for only one unknown, namely the scalar pressure. Here the equation of state is assumed linear with respect to the pressure. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems and it is a block four-point system in the two dimensional case. Furthermore, for high order in space and piecewise constant polynomials in time, the system is observed to be symmetric and positive definite. This allows to use fast linear solvers such as the conjugate gradient (CG) method. In 2. Discontinuous Galerkin Time-Domain Analysis of Power-Ground Planes Taking Into Account Decoupling Capacitors KAUST Repository Li, Ping 2017-03-22 In this paper, a discontinuous Galerkin time-domain (DGTD) method is developed to analyze the power-ground planes taking into account the decoupling capacitors. In the presence of decoupling capacitors, the whole physical system can be split into two subsystems: 1) the field subsystem that is governed by Maxwell\\'s equations that will be solved by the DGTD method, and 2) the circuit subsystem including the capacitor and its parasitic inductor and resistor, which is going to be characterized by the modified nodal analysis algorithm constructed circuit equations. With the aim to couple the two subsystems together, a lumped port is defined over a coaxial surface between the via barrel and the ground plane. To reach the coupling from the field to the circuit subsystem, a lumped voltage source calculated by the integration of electric field along the radial direction is introduced. On the other hand, to facilitate the coupling from the circuit to field subsystem, a lumped port current source calculated from the circuit equation is introduced, which serves as an impressed current source for the field subsystem. With these two auxiliary terms, a hybrid field-circuit matrix equation is established, which enables the field and circuit subsystems are solved in a synchronous scheme. Furthermore, the arbitrarily shaped antipads are considered by enforcing the proper wave port excitation using the magnetic surface current source derived from the antipads supported electric eigenmodes. In this way, the S-parameters corresponding to different modes can be conveniently extracted. To further improve the efficiency of the proposed algorithm in handling multiscale meshes, the local time-stepping marching scheme is applied. The proposed algorithm is verified by several representative examples. 3. Extending the Riemann-Solver-Free High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme (DG-CVS) to Solve Compressible Magnetohydrodynamics Equations Science.gov (United States) 2016-06-08 Ideal Magnetohydrodynamics,” J. Com- put. Phys., Vol. 153, No. 2, 1999, pp. 334–352. [14] Tang, H.-Z. and Xu, K., “A high-order gas -kinetic method for...notwithstanding any other provision of law , no person shall be subject to any penalty for failing to comply with a collection of information if it does...Riemann-solver-free spacetime discontinuous Galerkin method for general conservation laws to solve compressible magnetohydrodynamics (MHD) equations. The 4. Effective implementation of wavelet Galerkin method Science.gov (United States) Finěk, Václav; Šimunková, Martina 2012-11-01 It was proved by W. Dahmen et al. that an adaptive wavelet scheme is asymptotically optimal for a wide class of elliptic equations. This scheme approximates the solution u by a linear combination of N wavelets and a benchmark for its performance is the best N-term approximation, which is obtained by retaining the N largest wavelet coefficients of the unknown solution. Moreover, the number of arithmetic operations needed to compute the approximate solution is proportional to N. The most time consuming part of this scheme is the approximate matrix-vector multiplication. In this contribution, we will introduce our implementation of wavelet Galerkin method for Poisson equation -Δu = f on hypercube with homogeneous Dirichlet boundary conditions. In our implementation, we identified nonzero elements of stiffness matrix corresponding to the above problem and we perform matrix-vector multiplication only with these nonzero elements. 5. Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation International Nuclear Information System (INIS) Lin Lin; Lu Jianfeng; Ying Lexing; Weinan, E 2012-01-01 Kohn–Sham density functional theory is one of the most widely used electronic structure theories. In the pseudopotential framework, uniform discretization of the Kohn–Sham Hamiltonian generally results in a large number of basis functions per atom in order to resolve the rapid oscillations of the Kohn–Sham orbitals around the nuclei. Previous attempts to reduce the number of basis functions per atom include the usage of atomic orbitals and similar objects, but the atomic orbitals generally require fine tuning in order to reach high accuracy. We present a novel discretization scheme that adaptively and systematically builds the rapid oscillations of the Kohn–Sham orbitals around the nuclei as well as environmental effects into the basis functions. The resulting basis functions are localized in the real space, and are discontinuous in the global domain. The continuous Kohn–Sham orbitals and the electron density are evaluated from the discontinuous basis functions using the discontinuous Galerkin (DG) framework. Our method is implemented in parallel and the current implementation is able to handle systems with at least thousands of atoms. Numerical examples indicate that our method can reach very high accuracy (less than 1 meV) with a very small number (4–40) of basis functions per atom. 6. HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis NARCIS (Netherlands) van der Vegt, Jacobus J.W.; Rhebergen, Sander 2011-01-01 The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the 7. Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit KAUST Repository Guermond, Jean-Luc; Kanschat, Guido 2010-01-01 We revisit some results from M. L. Adams [Nu cl. Sci. Engrg., 137 (2001), pp. 298- 333]. Using functional analytic tools we prove that a necessary and sufficient condition for the standard upwind discontinuous Galerkin approximation to converge to the correct limit solution in the diffusive regime is that the approximation space contains a linear space of continuous functions, and the restrictions of the functions of this space to each mesh cell contain the linear polynomials. Furthermore, the discrete diffusion limit converges in the Sobolev space H1 to the continuous one if the boundary data is isotropic. With anisotropic boundary data, a boundary layer occurs, and convergence holds in the broken Sobolev space H with s < 1/2 only © 2010 Society for Industrial and Applied Mathematics. 8. On discontinuous Galerkin approach for atmospheric flow in the mesoscale with and without moisture Directory of Open Access Journals (Sweden) Dieter Schuster 2014-09-01 Full Text Available We present and discuss discontinuous Galerkin (DG schemes for dry and moist atmospheric flows in the mesoscale. We derive terrain-following coordinates on the sphere in strong-conservation form, which makes it possible to perform the computation on a Cartesian grid and yet conserves the momentum density on an f$f$-plane. A new DG model, i.e. DG-COSMO, is compared to the operational model COSMO of the Deutscher Wetterdienst (DWD. A simplified version of the suggested terrain-following coordinates is implemented in DG-COSMO and is compared against the DG dynamical core implemented within the DUNE framework, which uses unstructured grids to capture orography. Finally, a few idealised test cases, including 3d and moisture, are used for validation. In addition an estimate of efficiency for locally adaptive grids is derived for locally and non-locally occurring phenomena. 9. Comparison of two Galerkin quadrature methods International Nuclear Information System (INIS) Morel, J. E.; Warsa, J. S.; Franke, B. C.; Prinja, A. K. 2013-01-01 We compare two methods for generating Galerkin quadrature for problems with highly forward-peaked scattering. In Method 1, the standard Sn method is used to generate the moment-to-discrete matrix and the discrete-to-moment is generated by inverting the moment-to-discrete matrix. In Method 2, which we introduce here, the standard Sn method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. Method 1 has the advantage that it preserves both N eigenvalues and N eigenvectors (in a pointwise sense) of the scattering operator with an N-point quadrature. Method 2 has the advantage that it generates consistent angular moment equations from the corresponding S N equations while preserving N eigenvalues of the scattering operator with an N-point quadrature. Our computational results indicate that these two methods are quite comparable for the test problem considered. (authors) 10. A discontinous Galerkin finite element method with an efficient time integration scheme for accurate simulations KAUST Repository Liu, Meilin 2011-07-01 A discontinuous Galerkin finite element method (DG-FEM) with a highly-accurate time integration scheme is presented. The scheme achieves its high accuracy using numerically constructed predictor-corrector integration coefficients. Numerical results show that this new time integration scheme uses considerably larger time steps than the fourth-order Runge-Kutta method when combined with a DG-FEM using higher-order spatial discretization/basis functions for high accuracy. © 2011 IEEE. 11. A Discontinuous Galerkin Finite Element Method for Dynamic of Fully Saturated Soil / Rzwiazanie Zadania Dynamiki Całkowicie Nawodnionego Gruntu Przy Zastosowaniu Mes Z Nieciagłym Sformułowaniem Galerkina W Czasie Directory of Open Access Journals (Sweden) Wrana B. 2011-03-01 Full Text Available Artykuł podejmuje zagadnienie analizy rozchodzenia sie fal naprezeniowych w gruncie w ujeciu metody elementów skonczonych bazujac na sformułowaniu rozwiazania ciagłego w przestrzeni i nieciagłego w dziedzinie czasu Galerkina (space and time-discontinous Galerkin TDG finite element method. W tym sformułowaniu zarówno przemieszczenia jak i predkosci sa wielkosciami nieznanymi wzajemnie niezaleznymi aproksymowanymi ciagłymi funkcjami kształtu w przestrzeni i nieciagłymi funkcjami kształtu w czasie. Do opisu zachowania sie gruntu w pełni nasyconego woda zastosowano sformułowanie u-p w ujeciu metody elementów skonczonych. Grunt traktowany jest, jako osrodek dwufazowy składajacy sie ze szkieletu i wody w porach. Zastosowane sformułowanie uwzglednia tłumienie osrodka przez uwzglednienie dyssypacji energii proporcjonalnej do predkosci wody wzgledem szkieletu. W artykule przedstawiono porównanie proponowanej metody rozwiazania numerycznego w dziedzinie czasu do metod obecnie stosowanych, takich jak: metoda róznicy centralnej, metoda Houbolta, Wilsona θ, HHT-α oraz najczesciej stosowanej metody Newmarka. Z porównania wynika, ze proponowana metoda jest metoda stabilna o małym błedzie numerycznego rozwiazania. 12. Mollified birth in natural-age-grid Galerkin methods for age-structured biological systems International Nuclear Information System (INIS) Ayati, Bruce P; Dupont, Todd F 2009-01-01 We present natural-age-grid Galerkin methods for a model of a biological population undergoing aging. We use a mollified birth term in the method and analysis. The error due to mollification is of arbitrary order, depending on the choice of mollifier. The methods in this paper generalize the methods presented in [1], where the approximation space in age was taken to be a discontinuous piecewise polynomial subspace of L 2 . We refer to these methods as 'natural-age-grid' Galerkin methods since transport in the age variable is computed through the smooth movement of the age grid at the natural dimensionless velocity of one. The time variable has been left continuous to emphasize this smooth motion, as well as the independence of the time and age discretizations. The methods are shown to be superconvergent in the age variable 13. Edge reconstruction in armchair phosphorene nanoribbons revealed by discontinuous Galerkin density functional theory. Science.gov (United States) Hu, Wei; Lin, Lin; Yang, Chao 2015-12-21 With the help of our recently developed massively parallel DGDFT (Discontinuous Galerkin Density Functional Theory) methodology, we perform large-scale Kohn-Sham density functional theory calculations on phosphorene nanoribbons with armchair edges (ACPNRs) containing a few thousands to ten thousand atoms. The use of DGDFT allows us to systematically achieve a conventional plane wave basis set type of accuracy, but with a much smaller number (about 15) of adaptive local basis (ALB) functions per atom for this system. The relatively small number of degrees of freedom required to represent the Kohn-Sham Hamiltonian, together with the use of the pole expansion the selected inversion (PEXSI) technique that circumvents the need to diagonalize the Hamiltonian, results in a highly efficient and scalable computational scheme for analyzing the electronic structures of ACPNRs as well as their dynamics. The total wall clock time for calculating the electronic structures of large-scale ACPNRs containing 1080-10,800 atoms is only 10-25 s per self-consistent field (SCF) iteration, with accuracy fully comparable to that obtained from conventional planewave DFT calculations. For the ACPNR system, we observe that the DGDFT methodology can scale to 5000-50,000 processors. We use DGDFT based ab initio molecular dynamics (AIMD) calculations to study the thermodynamic stability of ACPNRs. Our calculations reveal that a 2 × 1 edge reconstruction appears in ACPNRs at room temperature. 14. Equivalence between the Energy Stable Flux Reconstruction and Filtered Discontinuous Galerkin Schemes Science.gov (United States) Zwanenburg, Philip; Nadarajah, Siva 2016-02-01 The aim of this paper is to demonstrate the equivalence between filtered Discontinuous Galerkin (DG) schemes and the Energy Stable Flux Reconstruction (ESFR) schemes, expanding on previous demonstrations in 1D [1] and for straight-sided elements in 3D [2]. We first derive the DG and ESFR schemes in strong form and compare the respective flux penalization terms while highlighting the implications of the fundamental assumptions for stability in the ESFR formulations, notably that all ESFR scheme correction fields can be interpreted as modally filtered DG correction fields. We present the result in the general context of all higher dimensional curvilinear element formulations. Through a demonstration that there exists a weak form of the ESFR schemes which is both discretely and analytically equivalent to the strong form, we then extend the results obtained for the strong formulations to demonstrate that ESFR schemes can be interpreted as a DG scheme in weak form where discontinuous edge flux is substituted for numerical edge flux correction. Theoretical derivations are then verified with numerical results obtained from a 2D Euler testcase with curved boundaries. Given the current choice of high-order DG-type schemes and the question as to which might be best to use for a specific application, the main significance of this work is the bridge that it provides between them. Clearly outlining the similarities between the schemes results in the important conclusion that it is always less efficient to use ESFR schemes, as opposed to the weak DG scheme, when solving problems implicitly. 15. Discontinuous Galerkin modeling of the Columbia River's coupled estuary-plume dynamics Science.gov (United States) Vallaeys, Valentin; Kärnä, Tuomas; Delandmeter, Philippe; Lambrechts, Jonathan; Baptista, António M.; Deleersnijder, Eric; Hanert, Emmanuel 2018-04-01 The Columbia River (CR) estuary is characterized by high river discharge and strong tides that generate high velocity flows and sharp density gradients. Its dynamics strongly affects the coastal ocean circulation. Tidal straining in turn modulates the stratification in the estuary. Simulating the hydrodynamics of the CR estuary and plume therefore requires a multi-scale model as both shelf and estuarine circulations are coupled. Such a model has to keep numerical dissipation as low as possible in order to correctly represent the plume propagation and the salinity intrusion in the estuary. Here, we show that the 3D baroclinic discontinuous Galerkin finite element model SLIM 3D is able to reproduce the main features of the CR estuary-to-ocean continuum. We introduce new vertical discretization and mode splitting that allow us to model a region characterized by complex bathymetry and sharp density and velocity gradients. Our model takes into account the major forcings, i.e. tides, surface wind stress and river discharge, on a single multi-scale grid. The simulation period covers the end of spring-early summer of 2006, a period of high river flow and strong changes in the wind regime. SLIM 3D is validated with in-situ data on the shelf and at multiple locations in the estuary and compared with an operational implementation of SELFE. The model skill in the estuary and on the shelf indicate that SLIM 3D is able to reproduce the key processes driving the river plume dynamics, such as the occurrence of bidirectional plumes or reversals of the inner shelf coastal currents. 16. Feed back Petrov-Galerkin methods for convection dominated problems International Nuclear Information System (INIS) Carmo, E.G.D. do; Galeao, A.C. 1988-09-01 The Petrov-Galerkin method is adaptively applied to convection dominated problems. To this end a feedback function is created which increases the control of derivatives in the direction of he gradient of the approximate solution. This leads to a method with good stability properties close to boundary layers and high accuracy in those regions where regular solutions do occur. (author) [pt 17. Multilevel Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems with Jump Coefficients Science.gov (United States) 2010-12-01 discontinuous coefficients on geometrically nonconforming substructures. Technical Report Serie A 634, Instituto de Matematica Pura e Aplicada, Brazil, 2009...Instituto de Matematica Pura e Aplicada, Brazil, 2010. submitted. [41] M. Dryja, M. V. Sarkis, and O. B. Widlund. Multilevel Schwarz methods for 18. Multicomponent gas flow computations by a discontinuous Galerkin scheme using L2-projection of perfect gas EOS Science.gov (United States) Franchina, N.; Savini, M.; Bassi, F. 2016-06-01 A new formulation of multicomponent gas flow computation, suited to a discontinuous Galerkin discretization, is here presented and discussed. The original key feature is the use of L2-projection form of the (perfect gas) equation of state that allows all thermodynamic variables to span the same functional space. This choice greatly mitigates problems encountered by the front-capturing schemes in computing discontinuous flow field, retaining at the same time their conservation properties at the discrete level and ease of use. This new approach, combined with an original residual-based artificial dissipation technique, shows itself capable, through a series of tests illustrated in the paper, to both control the spurious oscillations of flow variables occurring in high-order accurate computations and reduce them increasing the degree of the polynomial representation of the solution. This result is of great importance in computing reacting gaseous flows, where the local accuracy of temperature and species mass fractions is crucial to the correct evaluation of the chemical source terms contained in the equations, even if the presence of the physical diffusivities somewhat brings relief to these problems. The present work can therefore also be considered, among many others already presented in the literature, as the authors' first step toward the construction of a new discontinuous Galerkin scheme for reacting gas mixture flows. 19. Nonlinear dynamic analysis using Petrov-Galerkin natural element method International Nuclear Information System (INIS) Lee, Hong Woo; Cho, Jin Rae 2004-01-01 According to our previous study, it is confirmed that the Petrov-Galerkin Natural Element Method (PG-NEM) completely resolves the numerical integration inaccuracy in the conventional Bubnov-Galerkin Natural Element Method (BG-NEM). This paper is an extension of PG-NEM to two-dimensional nonlinear dynamic problem. For the analysis, a constant average acceleration method and a linearized total Lagrangian formulation is introduced with the PG-NEM. At every time step, the grid points are updated and the shape functions are reproduced from the relocated nodal distribution. This process enables the PG-NEM to provide more accurate and robust approximations. The representative numerical experiments performed by the test Fortran program, and the numerical results confirmed that the PG-NEM effectively and accurately approximates the nonlinear dynamic problem 20. Arbitrary-Lagrangian-Eulerian Discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes Science.gov (United States) Boscheri, Walter; Dumbser, Michael 2017-10-01 We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes, like molecular viscosity or heat conduction. High order piecewise polynomials of degree N are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor. A novel nodal solver algorithm based on the HLL flux is derived to compute the velocity for each nodal degree of freedom that describes the current mesh geometry. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Each technique generates a corresponding number of geometrical degrees of freedom needed to describe the current mesh configuration and which must be considered by the nodal solver for determining the grid velocity. The connection of the old mesh configuration at time tn with the new one at time t n + 1 provides the space-time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our numerical method belongs to the category of so-called direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry (including a possible rezoning step) directly during the computation of the numerical fluxes. We emphasize that our method is a moving mesh method, as opposed to total 1. Reactivity perturbation formulation for a discontinuous Galerkin-based transport solver and its use with adaptive mesh refinement International Nuclear Information System (INIS) Le Tellier, R.; Fournier, D.; Suteau, C. 2011-01-01 Within the framework of a Discontinuous Galerkin spatial approximation of the multigroup discrete ordinates transport equation, we present a generalization of the exact standard perturbation formula that takes into account spatial discretization-induced reactivity changes. It encompasses in two separate contributions the nuclear data-induced reactivity change and the reactivity modification induced by two different spatial discretizations. The two potential uses of such a formulation when considering adaptive mesh refinement are discussed, and numerical results on a simple two-group Cartesian two-dimensional benchmark are provided. In particular, such a formulation is shown to be useful to filter out a more accurate estimate of nuclear data-related reactivity effects from initial and perturbed calculations based on independent adaptation processes. (authors) 2. Error Analysis of Galerkin's Method for Semilinear Equations Directory of Open Access Journals (Sweden) Tadashi Kawanago 2012-01-01 Full Text Available We establish a general existence result for Galerkin's approximate solutions of abstract semilinear equations and conduct an error analysis. Our results may be regarded as some extension of a precedent work (Schultz 1969. The derivation of our results is, however, different from the discussion in his paper and is essentially based on the convergence theorem of Newton’s method and some techniques for deriving it. Some of our results may be applicable for investigating the quality of numerical verification methods for solutions of ordinary and partial differential equations. 3. Galerkin finite element methods for wave problems Indian Academy of Sciences (India) basis functions (called G1FEM here) and quadratic basis functions (called G2FEM) ... mulation of Brookes & Hughes (1982) that implicitly incorporates numerical ..... functions and (c) SUPG method in the (kh − ω t)-plane for explicit Euler. 4. Applications of mixed Petrov-Galerkin finite element methods to transient and steady state creep analysis International Nuclear Information System (INIS) Guerreiro, J.N.C.; Loula, A.F.D. 1988-12-01 The mixed Petrov-Galerkin finite element formulation is applied to transiente and steady state creep problems. Numerical analysis has shown additional stability of this method compared to classical Galerkin formulations. The accuracy of the new formulation is confirmed in some representative examples of two dimensional and axisymmetric problems. (author) [pt 5. A hybridized discontinuous Galerkin framework for high-order particle-mesh operator splitting of the incompressible Navier-Stokes equations Science.gov (United States) Maljaars, Jakob M.; Labeur, Robert Jan; Möller, Matthias 2018-04-01 A generic particle-mesh method using a hybridized discontinuous Galerkin (HDG) framework is presented and validated for the solution of the incompressible Navier-Stokes equations. Building upon particle-in-cell concepts, the method is formulated in terms of an operator splitting technique in which Lagrangian particles are used to discretize an advection operator, and an Eulerian mesh-based HDG method is employed for the constitutive modeling to account for the inter-particle interactions. Key to the method is the variational framework provided by the HDG method. This allows to formulate the projections between the Lagrangian particle space and the Eulerian finite element space in terms of local (i.e. cellwise) ℓ2-projections efficiently. Furthermore, exploiting the HDG framework for solving the constitutive equations results in velocity fields which excellently approach the incompressibility constraint in a local sense. By advecting the particles through these velocity fields, the particle distribution remains uniform over time, obviating the need for additional quality control. The presented methodology allows for a straightforward extension to arbitrary-order spatial accuracy on general meshes. A range of numerical examples shows that optimal convergence rates are obtained in space and, given the particular time stepping strategy, second-order accuracy is obtained in time. The model capabilities are further demonstrated by presenting results for the flow over a backward facing step and for the flow around a cylinder. 6. Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: A numerical comparison KAUST Repository Bä ck, Joakim; Nobile, Fabio; Tamellini, Lorenzo; Tempone, Raul 2010-01-01 Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncertainty quantification. An open and relevant research topic is the comparison of these two methods 7. application of the galerkin-vlasov method to the flexural analysis African Journals Online (AJOL) user In this research, the Galerkin-Vlasov variational method was used to present a general formulation of the Kirchhoff plate problem with simply supported edges and under distributed ..... analysed for elastic, dynamic and stability behaviour,. 8. A blended continuous–discontinuous finite element method for solving the multi-fluid plasma model Energy Technology Data Exchange (ETDEWEB) Sousa, E.M., E-mail: sousae@uw.edu; Shumlak, U., E-mail: shumlak@uw.edu 2016-12-01 The multi-fluid plasma model represents electrons, multiple ion species, and multiple neutral species as separate fluids that interact through short-range collisions and long-range electromagnetic fields. The model spans a large range of temporal and spatial scales, which renders the model stiff and presents numerical challenges. To address the large range of timescales, a blended continuous and discontinuous Galerkin method is proposed, where the massive ion and neutral species are modeled using an explicit discontinuous Galerkin method while the electrons and electromagnetic fields are modeled using an implicit continuous Galerkin method. This approach is able to capture large-gradient ion and neutral physics like shock formation, while resolving high-frequency electron dynamics in a computationally efficient manner. The details of the Blended Finite Element Method (BFEM) are presented. The numerical method is benchmarked for accuracy and tested using two-fluid one-dimensional soliton problem and electromagnetic shock problem. The results are compared to conventional finite volume and finite element methods, and demonstrate that the BFEM is particularly effective in resolving physics in stiff problems involving realistic physical parameters, including realistic electron mass and speed of light. The benefit is illustrated by computing a three-fluid plasma application that demonstrates species separation in multi-component plasmas. 9. Galerkin projection methods for solving multiple related linear systems Energy Technology Data Exchange (ETDEWEB) Chan, T.F.; Ng, M.; Wan, W.L. 1996-12-31 We consider using Galerkin projection methods for solving multiple related linear systems A{sup (i)}x{sup (i)} = b{sup (i)} for 1 {le} i {le} s, where A{sup (i)} and b{sup (i)} are different in general. We start with the special case where A{sup (i)} = A and A is symmetric positive definite. The method generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems orthogonally onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated with another unsolved system as a seed until all the systems are solved. We observe in practice a super-convergence behaviour of the CG process of the seed system when compared with the usual CG process. We also observe that only a small number of restarts is required to solve all the systems if the right-hand sides are close to each other. These two features together make the method particularly effective. In this talk, we give theoretical proof to justify these observations. Furthermore, we combine the advantages of this method and the block CG method and propose a block extension of this single seed method. The above procedure can actually be modified for solving multiple linear systems A{sup (i)}x{sup (i)} = b{sup (i)}, where A{sup (i)} are now different. We can also extend the previous analytical results to this more general case. Applications of this method to multiple related linear systems arising from image restoration and recursive least squares computations are considered as examples. 10. A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids NARCIS (Netherlands) Pesch, L.; van der Vegt, Jacobus J.W. 2008-01-01 Using the generalized variable formulation of the Euler equations of fluid dynamics, we develop a numerical method that is capable of simulating the flow of fluids with widely differing thermodynamic behavior: ideal and real gases can be treated with the same method as an incompressible fluid. The 11. A stochastic Galerkin method for the Euler equations with Roe variable transformation KAUST Repository Pettersson, Per; Iaccarino, Gianluca; Nordströ m, Jan 2014-01-01 The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy. © 2013 Elsevier Inc. 12. Applicability of the Galerkin method to the approximate solution of the multigroup diffusion equation International Nuclear Information System (INIS) Obradovic, D. 1970-04-01 In the study of the nuclear reactors space-time behaviour the modal analysis is very often used though some basic mathematical problems connected with application of this methods are still unsolved. In this paper the modal analysis is identified as a set of the methods in the mathematical literature known as the Galerkin methods (or projection methods, or sometimes direct methods). Using the results of the mathematical investigations of these methods the applicability of the Galerkin type methods to the calculations of the eigenvalue and eigenvectors of the stationary and non-stationary diffusion operator, as well as for the solutions of the corresponding functional equations, is established (author) 13. Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes Science.gov (United States) Dumbser, Michael; Guercilena, Federico; Köppel, Sven; Rezzolla, Luciano; Zanotti, Olindo 2018-04-01 We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential equations system is written in nonconservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations. 14. Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method Science.gov (United States) Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye 2018-04-01 The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective. 15. Analysis of circular fibers with an arbitrary index profile by the Galerkin method. Science.gov (United States) Guo, Shangping; Wu, Feng; Ikram, Khalid; Albin, Sacharia 2004-01-01 We propose a full-vectorial Galerkin method for the analysis of circular symmetric fibers with arbitrary index profiles. A set of orthogonal Laguerre-Gauss functions is used to calculate the dispersion relation and mode fields of TE and TM modes. Examples are given for both standard step-index fibers and Bragg fibers. For standard step-index fiber with low or high index contrast, the Galerkin method agrees well with the analytical results. In the case of the TE mode of a Bragg fiber it agrees well with the asymptotic results. 16. Error analysis of some Galerkin - least squares methods for the elasticity equations International Nuclear Information System (INIS) Franca, L.P.; Stenberg, R. 1989-05-01 We consider the recent technique of stabilizing mixed finite element methods by augmenting the Galerkin formulation with least squares terms calculated separately on each element. The error analysis is performed in a unified manner yielding improved results for some methods introduced earlier. In addition, a new formulation is introduced and analyzed [pt 17. Stochastic Galerkin methods for the steady-state Navier–Stokes equations Energy Technology Data Exchange (ETDEWEB) Sousedík, Bedřich, E-mail: sousedik@umbc.edu [Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 (United States); Elman, Howard C., E-mail: elman@cs.umd.edu [Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742 (United States) 2016-07-01 We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems. 18. Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical (PEC) solar cells Science.gov (United States) Harmon, Michael; Gamba, Irene M.; Ren, Kui 2016-12-01 This work concerns the numerical solution of a coupled system of self-consistent reaction-drift-diffusion-Poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical (PEC) solar cells with reactive semiconductor and electrolyte interfaces. We present three numerical algorithms, mainly based on a mixed finite element and a local discontinuous Galerkin method for spatial discretization, with carefully chosen numerical fluxes, and implicit-explicit time stepping techniques, for solving the time-dependent nonlinear systems of partial differential equations. We perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model. 19. A new finite element formulation for CFD:VIII. The Galerkin/least-squares method for advective-diffusive equations International Nuclear Information System (INIS) Hughes, T.J.R.; Hulbert, G.M.; Franca, L.P. 1988-10-01 Galerkin/least-squares finite element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of SUPG, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented. (author) [pt 20. The Stochastic Galerkin Method for Darcy Flow Problem with Log-Normal Random Czech Academy of Sciences Publication Activity Database Beres, Michal; Domesová, Simona 2017-01-01 Roč. 15, č. 2 (2017), s. 267-279 ISSN 1336-1376 R&D Projects: GA MŠk LQ1602 Institutional support: RVO:68145535 Keywords : Darcy flow * Gaussian random field * Karhunen-Loeve decomposition * polynomial chaos * Stochastic Galerkin method Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics http://advances.utc.sk/index.php/AEEE/article/view/2280 1. Topology optimization using the improved element-free Galerkin method for elasticity* International Nuclear Information System (INIS) Wu Yi; Ma Yong-Qi; Feng Wei; Cheng Yu-Min 2017-01-01 The improved element-free Galerkin (IEFG) method of elasticity is used to solve the topology optimization problems. In this method, the improved moving least-squares approximation is used to form the shape function. In a topology optimization process, the entire structure volume is considered as the constraint. From the solid isotropic microstructures with penalization, we select relative node density as a design variable. Then we choose the minimization of compliance to be an objective function, and compute its sensitivity with the adjoint method. The IEFG method in this paper can overcome the disadvantages of the singular matrices that sometimes appear in conventional element-free Galerkin (EFG) method. The central processing unit (CPU) time of each example is given to show that the IEFG method is more efficient than the EFG method under the same precision, and the advantage that the IEFG method does not form singular matrices is also shown. (paper) 2. An element-free Galerkin (EFG) method for generalized Fisher equations (GFE) International Nuclear Information System (INIS) Shi Ting-Yu; Ge Hong-Xia; Cheng Rong-Jun 2013-01-01 A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The exact mathematical result of the GFE has been widely used in population dynamics and genetics, where it originated. Many researchers have studied the numerical solutions of the GFE, up to now. In this paper, we introduce an element-free Galerkin (EFG) method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics. Compared with other numerical methods, the EFG method for the GFE needs only scattered nodes instead of meshing the domain of the problem. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. In comparison with the traditional method, numerical solutions show that the new method has higher accuracy and better convergence. Several numerical examples are presented to demonstrate the effectiveness of the method 3. A weak Galerkin least-squares finite element method for div-curl systems Science.gov (United States) Li, Jichun; Ye, Xiu; Zhang, Shangyou 2018-06-01 In this paper, we introduce a weak Galerkin least-squares method for solving div-curl problem. This finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh, polytopal mesh and mesh with hanging nodes. Error estimates of the finite element solution are derived. The numerical examples demonstrate the robustness and flexibility of the proposed method. 4. Numerical and experimental validation of a particle Galerkin method for metal grinding simulation Science.gov (United States) Wu, C. T.; Bui, Tinh Quoc; Wu, Youcai; Luo, Tzui-Liang; Wang, Morris; Liao, Chien-Chih; Chen, Pei-Yin; Lai, Yu-Sheng 2018-03-01 In this paper, a numerical approach with an experimental validation is introduced for modelling high-speed metal grinding processes in 6061-T6 aluminum alloys. The derivation of the present numerical method starts with an establishment of a stabilized particle Galerkin approximation. A non-residual penalty term from strain smoothing is introduced as a means of stabilizing the particle Galerkin method. Additionally, second-order strain gradients are introduced to the penalized functional for the regularization of damage-induced strain localization problem. To handle the severe deformation in metal grinding simulation, an adaptive anisotropic Lagrangian kernel is employed. Finally, the formulation incorporates a bond-based failure criterion to bypass the prospective spurious damage growth issues in material failure and cutting debris simulation. A three-dimensional metal grinding problem is analyzed and compared with the experimental results to demonstrate the effectiveness and accuracy of the proposed numerical approach. 5. An h-p Taylor-Galerkin finite element method for compressible Euler equations Science.gov (United States) Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O. 1991-01-01 An extension of the familiar Taylor-Galerkin method to arbitrary h-p spatial approximations is proposed. Boundary conditions are analyzed, and a linear stability result for arbitrary meshes is given, showing the unconditional stability for the parameter of implicitness alpha not less than 0.5. The wedge and blunt body problems are solved with both linear, quadratic, and cubic elements and h-adaptivity, showing the feasibility of higher orders of approximation for problems with shocks. 6. The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation OpenAIRE Jin, Bangti; Lazarov, Raytcho; Liu, Yikan; Zhou, Zhi 2014-01-01 We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite... 7. Galerkin methods for Boltzmann-Poisson transport with reflection conditions on rough boundaries Science.gov (United States) Morales Escalante, José A.; Gamba, Irene M. 2018-06-01 We consider in this paper the mathematical and numerical modeling of reflective boundary conditions (BC) associated to Boltzmann-Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modeling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently, a pointwise zero flux mathematical condition for the electron transport equation. Such condition balances the incident and reflective momentum flux at the microscopic level, pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability p (k →). We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and observe that the diffusive condition influences the kinetic moments over the whole domain in position space. 8. Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients KAUST Repository Beck, Joakim; Nobile, Fabio; Tamellini, Lorenzo; Tempone, Raul 2014-01-01 In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. © 2013 Elsevier Ltd. All rights reserved. 9. Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients KAUST Repository Beck, Joakim 2014-03-01 In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. © 2013 Elsevier Ltd. All rights reserved. 10. And still, a new beginning: the Galerkin least-squares gradient method International Nuclear Information System (INIS) Franca, L.P.; Carmo, E.G.D. do 1988-08-01 A finite element method is proposed to solve a scalar singular diffusion problem. The method is constructed by adding to the standard Galerkin a mesh-dependent term obtained by taking the gradient of the Euler-lagrange equation and multiplying it by its least-squares. For the one-dimensional homogeneous problem the method is designed to develop nodal exact solution. An error estimate shows that the method converges optimaly for any value of the singular parameter. Numerical results demonstrate the good stability and accuracy properties of the method. (author) [pt 11. Robust Discontinuity Preserving Optical Flow Methods Directory of Open Access Journals (Sweden) Nelson Monzón 2016-11-01 Full Text Available In this work, we present an implementation of discontinuity-preserving strategies in TV-L1 optical flow methods. These are based on exponential functions that mitigate the regularization at image edges, which usually provide precise flow boundaries. Nevertheless, if the smoothing is not well controlled, it may produce instabilities in the computed motion fields. We present an algorithm that allows three regularization strategies: the first one uses an exponential function together with a TV process; the second one combines this strategy with a small constant that ensures a minimum isotropic smoothing; the third one is a fully automatic approach that adapts the diffusion depending on the histogram of the image gradients. The last two alternatives are aimed at reducing the effect of instabilities. In the experiments, we observe that the pure exponential function is highly unstable while the other strategies preserve accurate motion contours for a large range of parameters. 12. A Galerkin Finite Element Method for Numerical Solutions of the Modified Regularized Long Wave Equation Directory of Open Access Journals (Sweden) Liquan Mei 2014-01-01 Full Text Available A Galerkin method for a modified regularized long wave equation is studied using finite elements in space, the Crank-Nicolson scheme, and the Runge-Kutta scheme in time. In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. A Fourier stability analysis for the method is shown to be marginally stable. Three invariants of motion are investigated. Numerical experiments are presented to check the theoretical study of this method. 13. Multigrid for the Galerkin least squares method in linear elasticity: The pure displacement problem Energy Technology Data Exchange (ETDEWEB) Yoo, Jaechil [Univ. of Wisconsin, Madison, WI (United States) 1996-12-31 Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we prove the convergence of a multigrid (W-cycle) method. This multigrid is robust in that the convergence is uniform as the parameter, v, goes to 1/2 Computational experiments are included. 14. Individualized drug dosing using RBF-Galerkin method: Case of anemia management in chronic kidney disease. Science.gov (United States) Mirinejad, Hossein; Gaweda, Adam E; Brier, Michael E; Zurada, Jacek M; Inanc, Tamer 2017-09-01 Anemia is a common comorbidity in patients with chronic kidney disease (CKD) and is frequently associated with decreased physical component of quality of life, as well as adverse cardiovascular events. Current treatment methods for renal anemia are mostly population-based approaches treating individual patients with a one-size-fits-all model. However, FDA recommendations stipulate individualized anemia treatment with precise control of the hemoglobin concentration and minimal drug utilization. In accordance with these recommendations, this work presents an individualized drug dosing approach to anemia management by leveraging the theory of optimal control. A Multiple Receding Horizon Control (MRHC) approach based on the RBF-Galerkin optimization method is proposed for individualized anemia management in CKD patients. Recently developed by the authors, the RBF-Galerkin method uses the radial basis function approximation along with the Galerkin error projection to solve constrained optimal control problems numerically. The proposed approach is applied to generate optimal dosing recommendations for individual patients. Performance of the proposed approach (MRHC) is compared in silico to that of a population-based anemia management protocol and an individualized multiple model predictive control method for two case scenarios: hemoglobin measurement with and without observational errors. In silico comparison indicates that hemoglobin concentration with MRHC method has less variation among the methods, especially in presence of measurement errors. In addition, the average achieved hemoglobin level from the MRHC is significantly closer to the target hemoglobin than that of the other two methods, according to the analysis of variance (ANOVA) statistical test. Furthermore, drug dosages recommended by the MRHC are more stable and accurate and reach the steady-state value notably faster than those generated by the other two methods. The proposed method is highly efficient for 15. A fractional spline collocation-Galerkin method for the time-fractional diffusion equation Directory of Open Access Journals (Sweden) Pezza L. 2018-03-01 Full Text Available The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented. 16. Coupling multipoint flux mixed finite element methodswith continuous Galerkin methods for poroelasticity KAUST Repository Wheeler, Mary 2013-11-16 We study the numerical approximation on irregular domains with general grids of the system of poroelasticity, which describes fluid flow in deformable porous media. The flow equation is discretized by a multipoint flux mixed finite element method and the displacements are approximated by a continuous Galerkin finite element method. First-order convergence in space and time is established in appropriate norms for the pressure, velocity, and displacement. Numerical results are presented that illustrate the behavior of the method. © Springer Science+Business Media Dordrecht 2013. 17. An Element Free Galerkin method for an elastoplastic coupled to damage analysis Directory of Open Access Journals (Sweden) Sendi Zohra 2016-01-01 Full Text Available In this work, a Meshless approach for nonlinear solid mechanics is developed based on the Element Free Galerkin method. Furthermore, Meshless is combined with an elastoplastic model coupled to ductile damage. The efficiency of the proposed methodology is evaluated through various numerical examples. Besides these, two-dimensional tensile tests under several boundary conditions were studied and solved by a Dynamic-Explicit resolution scheme. Finally, the results obtained from the numerical simulations are analyzed and critically compared with Finite Element Method results. 18. The dimension split element-free Galerkin method for three-dimensional potential problems Science.gov (United States) Meng, Z. J.; Cheng, H.; Ma, L. D.; Cheng, Y. M. 2018-02-01 This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method. 19. kantorovich-euler lagrange-galerkin's method for bending analysis African Journals Online (AJOL) user OF CIVIL ENGINEERING, ENUGU STATE UNIVERSITY OF SCIENCE & TECHNOLOGY, ... In this work, the Kantorovich method is applied to solve the bending problem of thin ... Lagrange differential equation is determined for this functional. 20. Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems Energy Technology Data Exchange (ETDEWEB) Lee, Kookjin [Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science; Carlberg, Kevin [Sandia National Lab. (SNL-CA), Livermore, CA (United States); Elman, Howard C. [Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science and Inst. for Advanced Computer Studies 2018-03-29 Here, we consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite-dimensional subspace. It has been shown that the stochastic Galerkin projection technique fails to minimize any measure of the solution error. As a remedy for this, we propose a novel stochatic least-squares Petrov--Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted$\\ell^2$-norm of the residual over all solutions in a given finite-dimensional subspace. Moreover, the method can be adapted to minimize the solution error in different weighted$\\ell^2$-norms by simply applying a weighting function within the least-squares formulation. In addition, a goal-oriented seminorm induced by an output quantity of interest can be minimized by defining a weighting function as a linear functional of the solution. We establish optimality and error bounds for the proposed method, and extensive numerical experiments show that the weighted LSPG method outperforms other spectral methods in minimizing corresponding target weighted norms. 1. A high-order Petrov-Galerkin method for the Boltzmann transport equation International Nuclear Information System (INIS) Pain, C.C.; Candy, A.S.; Piggott, M.D.; Buchan, A.; Eaton, M.D.; Goddard, A.J.H.; Oliveira, C.R.E. de 2005-01-01 We describe a new Petrov-Galerkin method using high-order terms to introduce dissipation in a residual-free formulation. The method is developed following both a Taylor series analysis and a variational principle, and the result has much in common with traditional Petrov-Galerkin, Self Adjoint Angular Flux (SAAF) and Even Parity forms of the Boltzmann transport equation. In addition, we consider the subtleties in constructing appropriate boundary conditions. In sub-grid scale (SGS) modelling of fluids the advantages of high-order dissipation are well known. Fourth-order terms, for example, are commonly used as a turbulence model with uniform dissipation. They have been shown to have superior properties to SGS models based upon second-order dissipation or viscosity. Even higher-order forms of dissipation (e.g. 16.-order) can offer further advantages, but are only easily realised by spectral methods because of the solution continuity requirements that these higher-order operators demand. Higher-order operators are more effective, bringing a higher degree of representation to the solution locally. Second-order operators, for example, tend to relax the solution to a linear variation locally, whereas a high-order operator will tend to relax the solution to a second-order polynomial locally. The form of the dissipation is also important. For example, the dissipation may only be applied (as it is in this work) in the streamline direction. While for many problems, for example Large Eddy Simulation (LES), simply adding a second or fourth-order dissipation term is a perfectly satisfactory SGS model, it is well known that a consistent residual-free formulation is required for radiation transport problems. This motivated the consideration of a new Petrov-Galerkin method that is residual-free, but also benefits from the advantageous features that SGS modelling introduces. We close with a demonstration of the advantages of this new discretization method over standard Petrov-Galerkin 2. Approximate solution of the transport equation by methods of Galerkin type International Nuclear Information System (INIS) Pitkaranta, J. 1977-01-01 Questions of the existence, uniqueness, and convergence of approximate solutions of transport equations by methods of the Galerkin type (where trial and weighting functions are the same) are discussed. The results presented do not exclude the infinite-dimensional case. Two strategies can be followed in the variational approximation of the transport operator: one proceeds from the original form of the transport equation, while the other is based on the partially symmetrized equation. Both principles are discussed in this paper. The transport equation is assumed in a discretized multigroup form 3. The Galerkin finite element method for a multi-term time-fractional diffusion equation KAUST Repository Jin, Bangti 2015-01-01 © 2014 The Authors. We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one- and two-dimensional problems confirm the theoretical convergence rates. 4. Steady and transient analyses of natural convection in a horizontal porous annulus with Galerkin method International Nuclear Information System (INIS) Rao, Y.F.; Fukuda, K.; Hasegawa, S. 1986-01-01 Steady and transient analytical investigation with the Galerkin method has been performed on natural convection in a horizontal porous annulus heated from the inner surface. Three families of convergent solutions, appearing one after another with increasing RaDa numbers, were obtained corresponding to different initial conditions. Despite the fact that the flow structures of two branching solutions are quite different, there exists a critical RaDa number at which their overall heat transfer rates have the same value. The bifurcation point was determined numerically, which coincided very well with that from experimental observation. The solutions in which higher wavenumber modes are dominant agree better with experimental data of overall heat transfer 5. Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels Directory of Open Access Journals (Sweden) Haotao Cai 2017-01-01 Full Text Available We develop a generalized Jacobi-Galerkin method for second kind Volterra integral equations with weakly singular kernels. In this method, we first introduce some known singular nonpolynomial functions in the approximation space of the conventional Jacobi-Galerkin method. Secondly, we use the Gauss-Jacobi quadrature rules to approximate the integral term in the resulting equation so as to obtain high-order accuracy for the approximation. Then, we establish that the approximate equation has a unique solution and the approximate solution arrives at an optimal convergence order. One numerical example is presented to demonstrate the effectiveness of the proposed method. 6. Imposition of Dirichlet Boundary Conditions in Element Free Galerkin Method through an Object-Oriented Implementation Directory of Open Access Journals (Sweden) Samira Hosseini Full Text Available Abstract One of the main drawbacks of Element Free Galerkin (EFG method is its dependence on moving least square shape functions which don’t satisfy the Kronecker Delta property, so in this method it’s not possible to apply Dirichlet boundary conditions directly. The aim of the present paper is to discuss different aspects of three widely used methods of applying Dirichlet boundary conditions in EFG method, called Lagrange multipliers, penalty method, and coupling with finite element method. Numerical simulations are presented to compare the results of these methods form the perspective of accuracy, convergence and computational expense. These methods have been implemented in an object oriented programing environment, called INSANE, and the results are presented and compared with the analytical solutions. 7. Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review Directory of Open Access Journals (Sweden) Bermejo Rodolfo 2016-09-01 Full Text Available We review in this paper the development of Lagrange-Galerkin (LG methods to integrate the incompressible Navier-Stokes equations (NSEs for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers. 8. Constant Jacobian Matrix-Based Stochastic Galerkin Method for Probabilistic Load Flow Directory of Open Access Journals (Sweden) Yingyun Sun 2016-03-01 Full Text Available An intrusive spectral method of probabilistic load flow (PLF is proposed in the paper, which can handle the uncertainties arising from renewable energy integration. Generalized polynomial chaos (gPC expansions of dependent random variables are utilized to build a spectral stochastic representation of PLF model. Instead of solving the coupled PLF model with a traditional, cumbersome method, a modified stochastic Galerkin (SG method is proposed based on the P-Q decoupling properties of load flow in power system. By introducing two pre-calculated constant sparse Jacobian matrices, the computational burden of the SG method is significantly reduced. Two cases, IEEE 14-bus and IEEE 118-bus systems, are used to verify the computation speed and efficiency of the proposed method. 9. Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: A numerical comparison KAUST Repository Bäck, Joakim 2010-09-17 Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncertainty quantification. An open and relevant research topic is the comparison of these two methods. By introducing a suitable generalization of the classical sparse grid SC method, we are able to compare SG and SC on the same underlying multivariate polynomial space in terms of accuracy vs. computational work. The approximation spaces considered here include isotropic and anisotropic versions of Tensor Product (TP), Total Degree (TD), Hyperbolic Cross (HC) and Smolyak (SM) polynomials. Numerical results for linear elliptic SPDEs indicate a slight computational work advantage of isotropic SC over SG, with SC-SM and SG-TD being the best choices of approximation spaces for each method. Finally, numerical results corroborate the optimality of the theoretical estimate of anisotropy ratios introduced by the authors in a previous work for the construction of anisotropic approximation spaces. © 2011 Springer. 10. New implementation method for essential boundary condition to extended element-free Galerkin method. Application to nonlinear problem International Nuclear Information System (INIS) Saitoh, Ayumu; Matsui, Nobuyuki; Itoh, Taku; Kamitani, Atsushi; Nakamura, Hiroaki 2011-01-01 A new method has been proposed for implementing essential boundary conditions to the Element-Free Galerkin Method (EFGM) without using the Lagrange multiplier. Furthermore, the performance of the proposed method has been investigated for a nonlinear Poisson problem. The results of computations show that, as interpolation functions become closer to delta functions, the accuracy of the solution is improved on the boundary. In addition, the accuracy of the proposed method is higher than that of the conventional EFGM. Therefore, it might be concluded that the proposed method is useful for solving the nonlinear Poisson problem. (author) 11. The streamline upwind Petrov-Galerkin stabilising method for the numerical solution of highly advective problems Directory of Open Access Journals (Sweden) Carlos Humberto Galeano Urueña 2009-05-01 Full Text Available This article describes the streamline upwind Petrov-Galerkin (SUPG method as being a stabilisation technique for resolving the diffusion-advection-reaction equation by finite elements. The first part of this article has a short analysis of the importance of this type of differential equation in modelling physical phenomena in multiple fields. A one-dimensional description of the SUPG me- thod is then given to extend this basis to two and three dimensions. The outcome of a strongly advective and a high numerical complexity experiment is presented. The results show how the version of the implemented SUPG technique allowed stabilised approaches in space, even for high Peclet numbers. Additional graphs of the numerical experiments presented here can be downloaded from www.gnum.unal.edu.co. 12. Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients KAUST Repository Bonito, Andrea; DeVore, Ronald A.; Nochetto, Ricardo H. 2013-01-01 Elliptic PDEs with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electromagnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces. The standard approach to numerically treating such problems using finite element methods is to assume that the discontinuities lie on the boundaries of the cells in the initial triangulation. However, this does not match applications where discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the L∞ norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated. We present a new approach based on distortion of the coefficients in an Lq norm with q < ∞ which therefore does not require the exact matching of the discontinuities. We then use this new distortion theory to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in the sense of distortion versus number of computations, and report insightful numerical results supporting our analysis. © 2013 Societ y for Industrial and Applied Mathematics. 13. ON THE APPLICATION OF THE METHOD OF B.G. GALERKIN TO LINEAR PROBLEMS ARISING FROM DYNAMICAL SYSTEMS WITH DISTRIBUTED PARAMETERS Energy Technology Data Exchange (ETDEWEB) Gurevich, S. G. 1955-07-01 Galerkin's method is applied to the solution of a linear partial differential equation of arbitrary order under specified initial and boundary conditions. An example is carried through in complete detail to illustrate the method. (auth) 14. Minimizers with discontinuous velocities for the electromagnetic variational method International Nuclear Information System (INIS) De Luca, Jayme 2010-01-01 The electromagnetic two-body problem has neutral differential delay equations of motion that, for generic boundary data, can have solutions with discontinuous derivatives. If one wants to use these neutral differential delay equations with arbitrary boundary data, solutions with discontinuous derivatives must be expected and allowed. Surprisingly, Wheeler-Feynman electrodynamics has a boundary value variational method for which minimizer trajectories with discontinuous derivatives are also expected, as we show here. The variational method defines continuous trajectories with piecewise defined velocities and accelerations, and electromagnetic fields defined by the Euler-Lagrange equations on trajectory points. Here we use the piecewise defined minimizers with the Lienard-Wierchert formulas to define generalized electromagnetic fields almost everywhere (but on sets of points of zero measure where the advanced/retarded velocities and/or accelerations are discontinuous). Along with this generalization we formulate the generalized absorber hypothesis that the far fields vanish asymptotically almost everywhere and show that localized orbits with far fields vanishing almost everywhere must have discontinuous velocities on sewing chains of breaking points. We give the general solution for localized orbits with vanishing far fields by solving a (linear) neutral differential delay equation for these far fields. We discuss the physics of orbits with discontinuous derivatives stressing the differences to the variational methods of classical mechanics and the existence of a spinorial four-current associated with the generalized variational electrodynamics. 15. An improved wavelet-Galerkin method for dynamic response reconstruction and parameter identification of shear-type frames Science.gov (United States) Bu, Haifeng; Wang, Dansheng; Zhou, Pin; Zhu, Hongping 2018-04-01 An improved wavelet-Galerkin (IWG) method based on the Daubechies wavelet is proposed for reconstructing the dynamic responses of shear structures. The proposed method flexibly manages wavelet resolution level according to excitation, thereby avoiding the weakness of the wavelet-Galerkin multiresolution analysis (WGMA) method in terms of resolution and the requirement of external excitation. IWG is implemented by this work in certain case studies, involving single- and n-degree-of-freedom frame structures subjected to a determined discrete excitation. Results demonstrate that IWG performs better than WGMA in terms of accuracy and computation efficiency. Furthermore, a new method for parameter identification based on IWG and an optimization algorithm are also developed for shear frame structures, and a simultaneous identification of structural parameters and excitation is implemented. Numerical results demonstrate that the proposed identification method is effective for shear frame structures. 16. A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation NARCIS (Netherlands) Hoitinga, W.; Brummelen, van E.H. 2011-01-01 To develop and analyze new computational techniques for the Boltzmann equation based on model or approximation adaptivity, it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the Boltzmann equation and that admits the extraction of highly 17. A Study into Discontinuous Galerkin Methods for the Second Order Wave Equation Science.gov (United States) 2015-06-01 solution directly at a set of points in a domain. In terms of the calculus of finite differences, we are looking to approximate the derivatives by...example, (∇p)∗ is the coupled numerical flux computation for the gradient of the pressure at the boundaries (∂Ω j) for neighboring elements within the...the last variational equation, we are going to multiply the gradient of ( ∂p ∂t −ω ) with the gradient of a test function (∇ψ): ∫ Ω j ∇ψ∇ ( ∂p ∂t −w 18. Application of Parallel Time-Implicit Discontinuous Galerkin Finite Element Methods to Hypersonic Nonequilibrium Flow Problems Science.gov (United States) 2014-05-01 we however focus on the continuum regime, where we can use the governing equations like Euler equations or Navier Stokes equations. The flow chemistry can...assumption. Instead the flow is considered to be a mixture of ideal gases, and the flow chemistry accounts for production and destruction of all the species. A 19. A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity KAUST Repository Demkowicz, Leszek; Gopalakrishnan, Jay; Niemi, Antti H. 2012-01-01 space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish 20. A 3-D discontinuous Galerkin Method for jet engine buzz-saw noise propagation International Nuclear Information System (INIS) Remaki, M.; Habashi, W.G.; Ait-Ali-Yahia, D.; Jay, A. 2002-01-01 This paper presents a 3-D methodology for solving jet engine aero-acoustics problems in the presence of strong shocks and rarefactions. For example, turbofan engines suffer from Multiple Pure Tone noise, also called Buzz-saw noise, generated by the fan when the blade rotational tip speed is supersonic. These waves are composed of a series of shocks and rarefactions produced by a coalescence of shocks due to non-uniformities in the blade spacing and in the blade stagger angles, arising from manufacturing tolerances 1. An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media KAUST Repository Chung, Eric T.; Efendiev, Yalchin R.; Leung, Wing Tat 2017-01-01 . Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples 2. Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media NARCIS (Netherlands) Minisini, S.; Zhebel, E.; Kononov, A.; Mulder, W.A. 2013-01-01 Modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources. Seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings. Finite elements enable accurate simulation of time-dependent wave propagation 3. A Moving Discontinuous Galerkin Finite Element Method for Flows with Interfaces Science.gov (United States) 2017-12-07 compressible (Euler) flow , both in steady and spacetime form. The Euler flow state variable is given by y = (r,rv1, . . . ,rvn,rE) 2 Rm, (18) where m = n+2...analogous flow was computed using a larger spatial velocity vx = 32 using MDG-ICE(p = 0), as shown in Figure 3. In this case, the upstream state ...spacetime domainW= (0,1)⇥ 0, 12 with six quadrilateral cells is shown in Figure 4. The flow state variable y was discretized using Q2 elements, while the 4. A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems KAUST Repository Efendiev, Yalchin R.; Lazarov, Raytcho D.; Moon, Minam; Shi, Ke 2015-01-01 of multiscale trace spaces. Using local snapshots, we avoid high dimensional representation of trace spaces and use some local features of the solution space in constructing a low dimensional trace space. We investigate the solvability and numerically study 5. Comparisons of Particle Tracking Techniques and Galerkin Finite Element Methods in Flow Simulations on Watershed Scales Science.gov (United States) Shih, D.; Yeh, G. 2009-12-01 This paper applies two numerical approximations, the particle tracking technique and Galerkin finite element method, to solve the diffusive wave equation in both one-dimensional and two-dimensional flow simulations. The finite element method is one of most commonly approaches in numerical problems. It can obtain accurate solutions, but calculation times may be rather extensive. The particle tracking technique, using either single-velocity or average-velocity tracks to efficiently perform advective transport, could use larger time-step sizes than the finite element method to significantly save computational time. Comparisons of the alternative approximations are examined in this poster. We adapt the model WASH123D to examine the work. WASH123D is an integrated multimedia, multi-processes, physics-based computational model suitable for various spatial-temporal scales, was first developed by Yeh et al., at 1998. The model has evolved in design capability and flexibility, and has been used for model calibrations and validations over the course of many years. In order to deliver a locally hydrological model in Taiwan, the Taiwan Typhoon and Flood Research Institute (TTFRI) is working with Prof. Yeh to develop next version of WASH123D. So, the work of our preliminary cooperationx is also sketched in this poster. 6. Numerical Analysis of an H1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation Directory of Open Access Journals (Sweden) Jinfeng Wang 2014-01-01 Full Text Available We discuss and analyze an H1-Galerkin mixed finite element (H1-GMFE method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H1-GMFE method. Based on the discussion on the theoretical error analysis in L2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H1-norm. Moreover, we derive and analyze the stability of H1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure. 7. Application of a mixed Galerkin/least-squares method to axisymetric shell problems subjected to arbitrary loading International Nuclear Information System (INIS) Loula, A.F.D.; Toledo, E.M.; Franca, L.P.; Garcia, E.L.M. 1989-08-01 A variationaly consistent finite element formulation for constrained problems free from shear or membrane locking is applied to axisymetric shells subjected to arbitrary loading. The governing equations are writen according to Love's classical theory for a problem of bending of axisymetric thin and moderately thick shells accounting for shear deformation. The mixed variational formulation, in terms of stresses and displacements here presented consists of classical Galerkin method plus mesh-dependent least-square type terms employed with equal-order finite element polynomials. The additional terms enhance stability and accuracy of the original Galerkin method, as already proven theoretically and confirmed trough numerical experiments. Numerical results of some examples are presented to demonstrate the good stability and accuracy of the formulation. (author) [pt 8. An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings Energy Technology Data Exchange (ETDEWEB) Jin, Shi, E-mail: sjin@wisc.edu [Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 (United States); Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240 (China); Lu, Hanqing, E-mail: hanqing@math.wisc.edu [Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 (United States) 2017-04-01 In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro–macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (in the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime. 9. On the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods KAUST Repository Beck, Joakim; Tempone, Raul; Nobile, Fabio; Tamellini, Lorenzo 2012-01-01 In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company. 10. On the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods KAUST Repository Beck, Joakim 2012-09-01 In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company. 11. Convergence Improvement of Response Matrix Method with Large Discontinuity Factors International Nuclear Information System (INIS) Yamamoto, Akio 2003-01-01 In the response matrix method, a numerical divergence problem has been reported when extremely small or large discontinuity factors are utilized in the calculations. In this paper, an alternative response matrix formulation to solve the divergence problem is discussed, and properties of iteration matrixes are investigated through eigenvalue analyses. In the conventional response matrix formulation, partial currents between adjacent nodes are assumed to be discontinuous, and outgoing partial currents are converted into incoming partial currents by the discontinuity factor matrix. Namely, the partial currents of the homogeneous system (i.e., homogeneous partial currents) are treated in the conventional response matrix formulation. In this approach, the spectral radius of an iteration matrix for the partial currents may exceed unity when an extremely small or large discontinuity factor is used. Contrary to this, an alternative response matrix formulation using heterogeneous partial currents is discussed in this paper. In the latter approach, partial currents are assumed to be continuous between adjacent nodes, and discontinuity factors are directly considered in the coefficients of a response matrix. From the eigenvalue analysis of the iteration matrix for the one-group, one-dimensional problem, the spectral radius for the heterogeneous partial current formulation does not exceed unity even if an extremely small or large discontinuity factor is used in the calculation; numerical stability of the alternative formulation is superior to the conventional one. The numerical stability of the heterogeneous partial current formulation is also confirmed by the two-dimensional light water reactor core analysis. Since the heterogeneous partial current formulation does not require any approximation, the converged solution exactly reproduces the reference solution when the discontinuity factors are directly derived from the reference calculation 12. Energy dependent mesh adaptivity of discontinuous isogeometric discrete ordinate methods with dual weighted residual error estimators Science.gov (United States) Owens, A. R.; Kópházi, J.; Welch, J. A.; Eaton, M. D. 2017-04-01 In this paper a hanging-node, discontinuous Galerkin, isogeometric discretisation of the multigroup, discrete ordinates (SN) equations is presented in which each energy group has its own mesh. The equations are discretised using Non-Uniform Rational B-Splines (NURBS), which allows the coarsest mesh to exactly represent the geometry for a wide range of engineering problems of interest; this would not be the case using straight-sided finite elements. Information is transferred between meshes via the construction of a supermesh. This is a non-trivial task for two arbitrary meshes, but is significantly simplified here by deriving every mesh from a common coarsest initial mesh. In order to take full advantage of this flexible discretisation, goal-based error estimators are derived for the multigroup, discrete ordinates equations with both fixed (extraneous) and fission sources, and these estimators are used to drive an adaptive mesh refinement (AMR) procedure. The method is applied to a variety of test cases for both fixed and fission source problems. The error estimators are found to be extremely accurate for linear NURBS discretisations, with degraded performance for quadratic discretisations owing to a reduction in relative accuracy of the "exact" adjoint solution required to calculate the estimators. Nevertheless, the method seems to produce optimal meshes in the AMR process for both linear and quadratic discretisations, and is ≈×100 more accurate than uniform refinement for the same amount of computational effort for a 67 group deep penetration shielding problem. 13. On the Preconditioning of a Newton-Krylov Solver for a High-Order reconstructed Discontinuous Galerkin Discretization of All-Speed Compressible Flow with Phase Change for Application in Laser-Based Additive Manufacturing Energy Technology Data Exchange (ETDEWEB) Weston, Brian T. [Univ. of California, Davis, CA (United States) 2017-05-17 This dissertation focuses on the development of a fully-implicit, high-order compressible ow solver with phase change. The work is motivated by laser-induced phase change applications, particularly by the need to develop large-scale multi-physics simulations of the selective laser melting (SLM) process in metal additive manufacturing (3D printing). Simulations of the SLM process require precise tracking of multi-material solid-liquid-gas interfaces, due to laser-induced melting/ solidi cation and evaporation/condensation of metal powder in an ambient gas. These rapid density variations and phase change processes tightly couple the governing equations, requiring a fully compressible framework to robustly capture the rapid density variations of the ambient gas and the melting/evaporation of the metal powder. For non-isothermal phase change, the velocity is gradually suppressed through the mushy region by a variable viscosity and Darcy source term model. The governing equations are discretized up to 4th-order accuracy with our reconstructed Discontinuous Galerkin spatial discretization scheme and up to 5th-order accuracy with L-stable fully implicit time discretization schemes (BDF2 and ESDIRK3-5). The resulting set of non-linear equations is solved using a robust Newton-Krylov method, with the Jacobian-free version of the GMRES solver for linear iterations. Due to the sti nes associated with the acoustic waves and thermal and viscous/material strength e ects, preconditioning the GMRES solver is essential. A robust and scalable approximate block factorization preconditioner was developed, which utilizes the velocity-pressure (vP) and velocity-temperature (vT) Schur complement systems. This multigrid block reduction preconditioning technique converges for high CFL/Fourier numbers and exhibits excellent parallel and algorithmic scalability on classic benchmark problems in uid dynamics (lid-driven cavity ow and natural convection heat transfer) as well as for laser 14. Numerical solution of the helmholtz equation for the superellipsoid via the galerkin method Directory of Open Access Journals (Sweden) Hy Dinh 2013-01-01 Full Text Available The objective of this work was to find the numerical solution of the Dirichlet problem for the Helmholtz equation for a smooth superellipsoid. The superellipsoid is a shape that is controlled by two parameters. There are some numerical issues in this type of an analysis; any integration method is affected by the wave number k, because of the oscillatory behavior of the fundamental solution. In this case we could only obtain good numerical results for super ellipsoids that were more shaped like super cones, which is a narrow range of super ellipsoids. The formula for these shapes was:$x=cos(xsin(y^{n},y=sin(xsin(y^{n},z=cos(y$where$nvaried from 0.5 to 4. The Helmholtz equation, which is the modified wave equation, is used in many scattering problems. This project was funded by NASA RI Space Grant for testing of the Dirichlet boundary condition for the shape of the superellipsoid. One practical value of all these computations can be getting a shape for the engine nacelles in a ray tracing the space shuttle. We are researching the feasibility of obtaining good convergence results for the superellipsoid surface. It was our view that smaller and lighter wave numbers would reduce computational costs associated with obtaining Galerkin coefficients. In addition, we hoped to significantly reduce the number of terms in the infinite series needed to modify the original integral equation, all of which were achieved in the analysis of the superellipsoid in a finite range. We used the Green's theorem to solve the integral equation for the boundary of the surface. Previously, multiple surfaces were used to test this method, such as the sphere, ellipsoid, and perturbation of the sphere, pseudosphere and the oval of Cassini Lin and Warnapala , Warnapala and Morgan . 15. Galerkin method for unsplit 3-D Dirac equation using atomically/kinetically balanced B-spline basis International Nuclear Information System (INIS) Fillion-Gourdeau, F.; Lorin, E.; Bandrauk, A.D. 2016-01-01 A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron–molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field. 16. Comparison between results of solution of Burgers' equation and Laplace's equation by Galerkin and least-square finite element methods Science.gov (United States) Adib, Arash; Poorveis, Davood; Mehraban, Farid 2018-03-01 In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers' equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace's equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace's equation in an earth dam is considered. By solution of Laplace's equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers' equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers' equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace's equation, Galerkin and least square methods can show water table correctly in earth dam. 17. Dual and primal mixed Petrov-Galerkin finite element methods in heat transfer problems International Nuclear Information System (INIS) Loula, A.F.D.; Toledo, E.M. 1988-12-01 New mixed finite element formulations for the steady state heat transfer problem are presented with no limitation in the choice of conforming finite element spaces. Adding least square residual forms of the governing equations of the classical Galerkin formulation the original saddle point problem is transformed into a minimization problem. Stability analysis, error estimates and numerical results are presented, confirming the error estimates and the good performance of this new formulation. (author) [pt 18. OPTIMAL ESTIMATES FOR THE SEMIDISCRETE GALERKIN METHOD APPLIED TO PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONSMOOTH DATA KAUST Repository GOSWAMI, DEEPJYOTI; PANI, AMIYA K.; YADAV, SANGITA 2014-01-01 AWe propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal L2-error estimate is derived for the semidiscrete approximation when the initial data is in L2. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain. © 2014 Australian Mathematical Society. 19. Comparison of a Material Point Method and a Galerkin Meshfree Method for the Simulation of Cohesive-Frictional Materials Directory of Open Access Journals (Sweden) Ilaria Iaconeta 2017-09-01 Full Text Available The simulation of large deformation problems, involving complex history-dependent constitutive laws, is of paramount importance in several engineering fields. Particular attention has to be paid to the choice of a suitable numerical technique such that reliable results can be obtained. In this paper, a Material Point Method (MPM and a Galerkin Meshfree Method (GMM are presented and verified against classical benchmarks in solid mechanics. The aim is to demonstrate the good behavior of the methods in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. The vast majority of MPM techniques in the literatrue are based on some sort of explicit time integration. The techniques proposed in the current work, on the contrary, are based on implicit approaches, which can also be easily adapted to the simulation of static cases. The two methods are presented so as to highlight the similarities to rather than the differences from “standard” Updated Lagrangian (UL approaches commonly employed by the Finite Elements (FE community. Although both methods are able to give a good prediction, it is observed that, under very large deformation of the medium, GMM lacks robustness due to its meshfree natrue, which makes the definition of the meshless shape functions more difficult and expensive than in MPM. On the other hand, the mesh-based MPM is demonstrated to be more robust and reliable for extremely large deformation cases. 20. Sensitivity analysis of the Galerkin finite element method neutron diffusion solver to the shape of the elements Energy Technology Data Exchange (ETDEWEB) Hosseini, Seyed Abolfaz [Dept. of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of) 2017-02-15 The purpose of the present study is the presentation of the appropriate element and shape function in the solution of the neutron diffusion equation in two-dimensional (2D) geometries. To this end, the multigroup neutron diffusion equation is solved using the Galerkin finite element method in both rectangular and hexagonal reactor cores. The spatial discretization of the equation is performed using unstructured triangular and quadrilateral finite elements. Calculations are performed using both linear and quadratic approximations of shape function in the Galerkin finite element method, based on which results are compared. Using the power iteration method, the neutron flux distributions with the corresponding eigenvalue are obtained. The results are then validated against the valid results for IAEA-2D and BIBLIS-2D benchmark problems. To investigate the dependency of the results to the type and number of the elements, and shape function order, a sensitivity analysis of the calculations to the mentioned parameters is performed. It is shown that the triangular elements and second order of the shape function in each element give the best results in comparison to the other states. 1. The nonlinear Galerkin method: A multi-scale method applied to the simulation of homogeneous turbulent flows Science.gov (United States) Debussche, A.; Dubois, T.; Temam, R. 1993-01-01 Using results of Direct Numerical Simulation (DNS) in the case of two-dimensional homogeneous isotropic flows, the behavior of the small and large scales of Kolmogorov like flows at moderate Reynolds numbers are first analyzed in detail. Several estimates on the time variations of the small eddies and the nonlinear interaction terms were derived; those terms play the role of the Reynolds stress tensor in the case of LES. Since the time step of a numerical scheme is determined as a function of the energy-containing eddies of the flow, the variations of the small scales and of the nonlinear interaction terms over one iteration can become negligible by comparison with the accuracy of the computation. Based on this remark, a multilevel scheme which treats differently the small and the large eddies was proposed. Using mathematical developments, estimates of all the parameters involved in the algorithm, which then becomes a completely self-adaptive procedure were derived. Finally, realistic simulations of (Kolmorov like) flows over several eddy-turnover times were performed. The results are analyzed in detail and a parametric study of the nonlinear Galerkin method is performed. 2. A locally conservative stabilized continuous Galerkin finite element method for two-phase flow in poroelastic subsurfaces Science.gov (United States) Deng, Q.; Ginting, V.; McCaskill, B.; Torsu, P. 2017-10-01 We study the application of a stabilized continuous Galerkin finite element method (CGFEM) in the simulation of multiphase flow in poroelastic subsurfaces. The system involves a nonlinear coupling between the fluid pressure, subsurface's deformation, and the fluid phase saturation, and as such, we represent this coupling through an iterative procedure. Spatial discretization of the poroelastic system employs the standard linear finite element in combination with a numerical diffusion term to maintain stability of the algebraic system. Furthermore, direct calculation of the normal velocities from pressure and deformation does not entail a locally conservative field. To alleviate this drawback, we propose an element based post-processing technique through which local conservation can be established. The performance of the method is validated through several examples illustrating the convergence of the method, the effectivity of the stabilization term, and the ability to achieve locally conservative normal velocities. Finally, the efficacy of the method is demonstrated through simulations of realistic multiphase flow in poroelastic subsurfaces. 3. Adaptive Meshless Local Petrov-Galerkin Method with Variable Domain of Influence in 2D Elastostatic Problems Directory of Open Access Journals (Sweden) Pamuda Pudjisuryadi 2008-01-01 Full Text Available A meshless local Petrov-Galerkin (MLPG method that employs polygonal sub-domains constructed from several triangular patches rather than the typically used circular sub-domains is presented. Moving least-squares approximation is used to construct the trial displacements and linear, Lagrange interpolation functions are used to construct the test functions. An adaptive technique to improve the accuracy of approximate solutions is developed to minimize the computational cost. Variable domain of influence (VDOI and effective stress gradient indicator (EK for local error assessment are the focus of this study. Several numerical examples are presented to verify the efficiency and accuracy of the proposed adaptive MLPG method. The results show that the proposed adaptive technique performs as expected that is refining the problem domain in area with high stress concentration in which higher accuracy is commonly required. 4. Numerical Solution and Simulation of Second-Order Parabolic PDEs with Sinc-Galerkin Method Using Maple Directory of Open Access Journals (Sweden) Aydin Secer 2013-01-01 Full Text Available An efficient solution algorithm for sinc-Galerkin method has been presented for obtaining numerical solution of PDEs with Dirichlet-type boundary conditions by using Maple Computer Algebra System. The method is based on Whittaker cardinal function and uses approximating basis functions and their appropriate derivatives. In this work, PDEs have been converted to algebraic equation systems with new accurate explicit approximations of inner products without the need to calculate any numeric integrals. The solution of this system of algebraic equations has been reduced to the solution of a matrix equation system via Maple. The accuracy of the solutions has been compared with the exact solutions of the test problem. Computational results indicate that the technique presented in this study is valid for linear partial differential equations with various types of boundary conditions. 5. Alternating direction transport sweeps for linear discontinuous SN method International Nuclear Information System (INIS) Yavuz, M.; Aykanat, C. 1993-01-01 The performance of Alternating Direction Transport Sweep (ADTS) method is investigated for spatially differenced Linear Discontinuous S N (LD-S N ) problems on a MIMD multicomputer, Intel IPSC/2. The method consists of dividing a transport problem spatially into sub-problems, assigning each sub-problem to a separate processor. Then, the problem is solved by performing transport sweeps iterating on the scattering source and interface fluxes between the sub-problems. In each processor, the order of transport sweeps is scheduled such that a processor completing its computation in a quadrant of a transport sweep is able to use the most recent information (exiting fluxes of neighboring processor) as its incoming fluxes to start the next quadrant calculation. Implementation of this method on the Intel IPSC/2 multicomputer displays significant speedups over the one-processor method. Also, the performance of the method is compared with those reported previously for the Diamond Differenced S N (DD-S N ) method. Our experimental experience illustrates that the parallel performance of both the ADTS LD- and DD-S N methods is the same. (orig.) 6. Hydrodynamic analysis of wave interactions with a moored floating breakwater using the element-free Galerkin method International Nuclear Information System (INIS) Lee, J.; Cho, W. 2003-01-01 This paper deals with a numerical investigation of incident wave interactions with a moored pontoon-type floating breakwater. The element-free Galerkin method, in which only nodal data are required to analyze the problem, is employed to solve the diffraction and radiation boundary value problems addressed by the modified Helmholtz equation. The numerical model includes the hydrodynamic and mooring analyses, and it is validated by previous numerical and experimental results. Using the numerical model, we are able to assess the hydrodynamic performance of a moored pontoon-type floating breakwater in regular waves. Numerical results are presented to show the effects of wave conditions and mooring system configuration. This paper also presents the simple forms of stiffness coefficients of a slack mooring line. The influence of mooring line condition on the performance of a floating breakwater is highlighted. (author) 7. Stable Galerkin versus equal-order Galerkin least-squares elements for the stokes flow problem International Nuclear Information System (INIS) Franca, L.P.; Frey, S.L.; Sampaio, R. 1989-11-01 Numerical experiments are performed for the stokes flow problem employing a stable Galerkin method and a Galerkin/Least-squares method with equal-order elements. Error estimates for the methods tested herein are reviewed. The numerical results presented attest the good stability properties of all methods examined herein. (A.C.A.S.) [pt 8. Optimal trace inequality constants for interior penalty discontinuous Galerkin discretisations of elliptic operators using arbitrary elements with non-constant Jacobians Science.gov (United States) Owens, A. R.; Kópházi, J.; Eaton, M. D. 2017-12-01 In this paper, a new method to numerically calculate the trace inequality constants, which arise in the calculation of penalty parameters for interior penalty discretisations of elliptic operators, is presented. These constants are provably optimal for the inequality of interest. As their calculation is based on the solution of a generalised eigenvalue problem involving the volumetric and face stiffness matrices, the method is applicable to any element type for which these matrices can be calculated, including standard finite elements and the non-uniform rational B-splines of isogeometric analysis. In particular, the presented method does not require the Jacobian of the element to be constant, and so can be applied to a much wider variety of element shapes than are currently available in the literature. Numerical results are presented for a variety of finite element and isogeometric cases. When the Jacobian is constant, it is demonstrated that the new method produces lower penalty parameters than existing methods in the literature in all cases, which translates directly into savings in the solution time of the resulting linear system. When the Jacobian is not constant, it is shown that the naive application of existing approaches can result in penalty parameters that do not guarantee coercivity of the bilinear form, and by extension, the stability of the solution. The method of manufactured solutions is applied to a model reaction-diffusion equation with a range of parameters, and it is found that using penalty parameters based on the new trace inequality constants result in better conditioned linear systems, which can be solved approximately 11% faster than those produced by the methods from the literature. 9. An Equal-Order DG Method for the Incompressible Navier-Stokes Equations KAUST Repository Cockburn, Bernardo; Kanschat, Guido; Schö tzau, Dominik 2008-01-01 We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability 10. Numerical analysis on interactions between fluid flow and structure deformation in plate-fin heat exchanger by Galerkin method Science.gov (United States) Liu, Jing-cheng; Wei, Xiu-ting; Zhou, Zhi-yong; Wei, Zhen-wen 2018-03-01 The fluid-structure interaction performance of plate-fin heat exchanger (PFHE) with serrated fins in large scale air-separation equipment was investigated in this paper. The stress and deformation of fins were analyzed, besides, the interaction equations were deduced by Galerkin method. The governing equations of fluid flow and heat transfer in PFHE were deduced by finite volume method (FVM). The distribution of strain and stress were calculated in large scale air separation equipment and the coupling situation of serrated fins under laminar situation was analyzed. The results indicated that the interactions between fins and fluid flow in the exchanger have significant impacts on heat transfer enhancement, meanwhile, the strain and stress of fins includes dynamic pressure of the sealing head and flow impact with the increase of flow velocity. The impacts are especially significant at the conjunction of two fins because of the non-alignment fins. It can be concluded that the soldering process and channel width led to structure deformation of fins in the exchanger, and degraded heat transfer efficiency. 11. A high-order method for the integration of the Galerkin semi-discretized nuclear reactor kinetics equations International Nuclear Information System (INIS) Vargas, L. 1988-01-01 The numerical approximate solution of the space-time nuclear reactor kinetics equation is investigated using a finite-element discretization of the space variable and a high order integration scheme for the resulting semi-discretized parabolic equation. The Galerkin method with spatial piecewise polynomial Lagrange basis functions are used to obtained a continuous time semi-discretized form of the space-time reactor kinetics equation. A temporal discretization is then carried out with a numerical scheme based on the Iterated Defect Correction (IDC) method using piecewise quadratic polynomials or exponential functions. The kinetics equations are thus solved with in a general finite element framework with respect to space as well as time variables in which the order of convergence of the spatial and temporal discretizations is consistently high. A computer code GALFEM/IDC is developed, to implement the numerical schemes described above. This issued to solve a one space dimensional benchmark problem. The results of the numerical experiments confirm the theoretical arguments and show that the convergence is very fast and the overall procedure is quite efficient. This is due to the good asymptotic properties of the numerical scheme which is of third order in the time interval 12. Discontinuous Galerkin Dynamical Core in HOMME Energy Technology Data Exchange (ETDEWEB) Nair, R. D. [Univ. of Colorado, Boulder, CO (United States); Tufo, Henry [Univ. of Colorado, Boulder, CO (United States) 2012-08-14 Atmospheric numerical modeling has been going through radical changes over the past decade. One major reason for this trend is due to the recent paradigm change in scientific computing , triggered by the arrival of petascale computing resources with core counts in the tens of thousands to hundreds of thousands range. Modern atmospheric modelers must adapt grid systems and numerical algorithms to facilitate an unprecedented levels of scalability on these modern highly parallel computer architectures. The numerical algorithms which can address these challenges should have the local properties such as high on-processor floating-point operation count to bytes moved and minimum parallel communication overhead. 13. A Floating Node Method for the Modelling of Discontinuities Within a Finite Element Science.gov (United States) Pinho, Silvestre T.; Chen, B. Y.; DeCarvalho, Nelson V.; Baiz, P. M.; Tay, T. E. 2013-01-01 This paper focuses on the accurate numerical representation of complex networks of evolving discontinuities in solids, with particular emphasis on cracks. The limitation of the standard finite element method (FEM) in approximating discontinuous solutions has motivated the development of re-meshing, smeared crack models, the eXtended Finite Element Method (XFEM) and the Phantom Node Method (PNM). We propose a new method which has some similarities to the PNM, but crucially: (i) does not introduce an error on the crack geometry when mapping to natural coordinates; (ii) does not require numerical integration over only part of a domain; (iii) can incorporate weak discontinuities and cohesive cracks more readily; (iv) is ideally suited for the representation of multiple and complex networks of (weak, strong and cohesive) discontinuities; (v) leads to the same solution as a finite element mesh where the discontinuity is represented explicitly; and (vi) is conceptually simpler than the PNM. 14. An implementation analysis of the linear discontinuous finite element method International Nuclear Information System (INIS) Becker, T. L. 2013-01-01 This paper provides an implementation analysis of the linear discontinuous finite element method (LD-FEM) that spans the space of (l, x, y, z). A practical implementation of LD includes 1) selecting a computationally efficient algorithm to solve the 4 x 4 matrix system Ax = b that describes the angular flux in a mesh element, and 2) choosing how to store the data used to construct the matrix A and the vector b to either reduce memory consumption or increase computational speed. To analyze the first of these, three algorithms were selected to solve the 4 x 4 matrix equation: Cramer's rule, a streamlined implementation of Gaussian elimination, and LAPACK's Gaussian elimination subroutine dgesv. The results indicate that Cramer's rule and the streamlined Gaussian elimination algorithm perform nearly equivalently and outperform LAPACK's implementation of Gaussian elimination by a factor of 2. To analyze the second implementation detail, three formulations of the discretized LD-FEM equations were provided for implementation in a transport solver: 1) a low-memory formulation, which relies heavily on 'on-the-fly' calculations and less on the storage of pre-computed data, 2) a high-memory formulation, which pre-computes much of the data used to construct A and b, and 3) a reduced-memory formulation, which lies between the low - and high-memory formulations. These three formulations were assessed in the Jaguar transport solver based on relative memory footprint and computational speed for increasing mesh size and quadrature order. The results indicated that the memory savings of the low-memory formulation were not sufficient to warrant its implementation. The high-memory formulation resulted in a significant speed advantage over the reduced-memory option (10-50%), but also resulted in a proportional increase in memory consumption (5-45%) for increasing quadrature order and mesh count; therefore, the practitioner should weigh the system memory constraints against any 15. An implementation analysis of the linear discontinuous finite element method Energy Technology Data Exchange (ETDEWEB) Becker, T. L. [Bechtel Marine Propulsion Corporation, Knolls Atomic Power Laboratory, P.O. Box 1072, Schenectady, NY 12301-1072 (United States) 2013-07-01 This paper provides an implementation analysis of the linear discontinuous finite element method (LD-FEM) that spans the space of (l, x, y, z). A practical implementation of LD includes 1) selecting a computationally efficient algorithm to solve the 4 x 4 matrix system Ax = b that describes the angular flux in a mesh element, and 2) choosing how to store the data used to construct the matrix A and the vector b to either reduce memory consumption or increase computational speed. To analyze the first of these, three algorithms were selected to solve the 4 x 4 matrix equation: Cramer's rule, a streamlined implementation of Gaussian elimination, and LAPACK's Gaussian elimination subroutine dgesv. The results indicate that Cramer's rule and the streamlined Gaussian elimination algorithm perform nearly equivalently and outperform LAPACK's implementation of Gaussian elimination by a factor of 2. To analyze the second implementation detail, three formulations of the discretized LD-FEM equations were provided for implementation in a transport solver: 1) a low-memory formulation, which relies heavily on 'on-the-fly' calculations and less on the storage of pre-computed data, 2) a high-memory formulation, which pre-computes much of the data used to construct A and b, and 3) a reduced-memory formulation, which lies between the low - and high-memory formulations. These three formulations were assessed in the Jaguar transport solver based on relative memory footprint and computational speed for increasing mesh size and quadrature order. The results indicated that the memory savings of the low-memory formulation were not sufficient to warrant its implementation. The high-memory formulation resulted in a significant speed advantage over the reduced-memory option (10-50%), but also resulted in a proportional increase in memory consumption (5-45%) for increasing quadrature order and mesh count; therefore, the practitioner should weigh the system memory 16. Transient Analysis of Dispersive Power-Ground Plate Pairs With Arbitrarily Shaped Antipads by the DGTD Method With Wave Port Excitation KAUST Repository Li, Ping; Jiang, Li Jun; Bagci, Hakan 2016-01-01 A discontinuous Galerkin time-domain (DGTD) method analyzing signal/power integrity on multilayered power-ground parallel plate pairs is proposed. The excitation is realized by introducing wave ports on the antipads where electric/magnetic current 17. Crack edge collocation for the direct computation of stress intensity factors using the displacement discontinuity method CSIR Research Space (South Africa) Napier, JAL 2002-09-01 Full Text Available The numerical solution of problems relating to crack fracture and failure can be accomplished using the displacement discontinuity boundary element method. This paper presents an extension to the normal formulation of this method to enable stress... 18. An Analysis of Contraceptive Discontinuation among Female, Reversible Method Users in Urban Honduras Science.gov (United States) Barden-O’Fallon, Janine; Speizer, Ilene S.; Cálix, Javier; Rodriguez, Francisco 2013-01-01 A panel study examining the effects of individual characteristics, side effects experienced, and service quality on contraceptive discontinuation was undertaken in four urban areas of Honduras. Data were collected from October 2006 to December 2007. The baseline population included 800 women aged 15–44 who were new or continuing users of the injectable, IUD, or oral contraceptive pill. A total of 671 women (84%) were re-interviewed after one year. Life tables and Cox proportional hazards models are used to present discontinuation rates and factors associated with contraceptive discontinuation. Among new users, discontinuation of the baseline method at 12 months was high (45%); especially for users of the injectable (50%). In the hazards model, service quality had little effect on discontinuation, while individual characteristics and the experience of specific side effects showed significant effects. The results suggest that programs should emphasize continuous contraceptive coverage rather than continuous use of a particular method. PMID:21500697 19. One-Sided Smoothness-Increasing Accuracy-Conserving Filtering for Enhanced Streamline Integration through Discontinuous Fields NARCIS (Netherlands) Walfisch, D.; Ryan, J.K.; Kirby, R.M.; Haimes, R. 2008-01-01 The discontinuous Galerkin (DG) method continues to maintain heightened levels of interest within the simulation community because of the discretization flexibility it provides. One of the fundamental properties of the DG methodology and arguably its most powerful property is the ability to combine 20. Coupling multipoint flux mixed finite element methodswith continuous Galerkin methods for poroelasticity KAUST Repository Wheeler, Mary; Xue, Guangri; Yotov, Ivan 2013-01-01 We study the numerical approximation on irregular domains with general grids of the system of poroelasticity, which describes fluid flow in deformable porous media. The flow equation is discretized by a multipoint flux mixed finite element method 1. Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns Science.gov (United States) Zhou, Qiuling; Tang, Chen; Li, Biyuan; Wang, Linlin; Lei, Zhenkun; Tang, Shuwei 2018-01-01 The filtering of discontinuous optical fringe patterns is a challenging problem faced in this area. This paper is concerned with oriented partial differential equations (OPDEs)-based image filtering methods for discontinuous optical fringe patterns. We redefine a new controlling speed function to depend on the orientation coherence. The orientation coherence can be used to distinguish the continuous regions and the discontinuous regions, and can be calculated by utilizing fringe orientation. We introduce the new controlling speed function to the previous OPDEs and propose adaptive OPDEs filtering models. According to our proposed adaptive OPDEs filtering models, the filtering in the continuous and discontinuous regions can be selectively carried out. We demonstrate the performance of the proposed adaptive OPDEs via application to the simulated and experimental fringe patterns, and compare our methods with the previous OPDEs. 2. A new method for automatic discontinuity traces sampling on rock mass 3D model Science.gov (United States) Umili, G.; Ferrero, A.; Einstein, H. H. 2013-02-01 A new automatic method for discontinuity traces mapping and sampling on a rock mass digital model is described in this work. The implemented procedure allows one to automatically identify discontinuity traces on a Digital Surface Model: traces are detected directly as surface breaklines, by means of maximum and minimum principal curvature values of the vertices that constitute the model surface. Color influence and user errors, that usually characterize the trace mapping on images, are eliminated. Also trace sampling procedures based on circular windows and circular scanlines have been implemented: they are used to infer trace data and to calculate values of mean trace length, expected discontinuity diameter and intensity of rock discontinuities. The method is tested on a case study: results obtained applying the automatic procedure on the DSM of a rock face are compared to those obtained performing a manual sampling on the orthophotograph of the same rock face. 3. A Review of Element-Based Galerkin Methods for Numerical Weather Prediction Science.gov (United States) 2015-04-01 weighted residuals because a linear system of algebraic equations in the unknowns q̂ is built by imposing that∫ Ω wRdΩ = 0, (30) where R = L−S is the (non...space; this spectral transform is evaluated using a combination of Fourier and 10 Simone Marras1 et al. Legendre transforms. We perform an elementary ...For the first algebraic system, the GMRES method with a simple diagonal preconditioning is efficient in most of the cases, and few iterations are 4. A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics KAUST Repository Schick, M.; Heuveline, V.; Le Ma, O. P. 2014-01-01 The determination of stable limit-cycles plays an important role in quantifying the characteristics of dynamical systems. In practice, exact knowledge of model parameters is rarely available leading to parameter uncertainties, which can be modeled as an input of random variables. This has the effect that the limit-cycles become stochastic themselves, resulting in almost surely time-periodic solutions with a stochastic period. In this paper we introduce a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method using polynomial chaos (PC). We are able to overcome the difficulties of PC regarding its well-known convergence breakdown for long term integration. To this end, we introduce a stochastic time scaling which treats the stochastic period as an additional random variable and controls the phase-drift of the stochastic trajectories, keeping the necessary PC order low. Based on the rescaled governing equations, we aim at determining an initial condition and a period such that the trajectories close after completion of one stochastic cycle. Furthermore, we verify the numerical method by computation of a vortex shedding of a flow around a circular domain with stochastic inflow boundary conditions as a benchmark problem. The results are verified by comparison to purely deterministic reference problems and demonstrate high accuracy up to machine precision in capturing the stochastic variations of the limit-cycle. 5. A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation Directory of Open Access Journals (Sweden) Yunying Zheng 2011-01-01 Full Text Available The spatial transport process in fractal media is generally anomalous. The space-fractional advection-diffusion equation can be used to characterize such a process. In this paper, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In the spatial direction, we use the finite element method, and in the temporal direction, we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical examples are also included which are in line with the theoretical analysis. 6. Adaptive algorithms for a self-shielding wavelet-based Galerkin method International Nuclear Information System (INIS) Fournier, D.; Le Tellier, R. 2009-01-01 The treatment of the energy variable in deterministic neutron transport methods is based on a multigroup discretization, considering the flux and cross-sections to be constant within a group. In this case, a self-shielding calculation is mandatory to correct sections of resonant isotopes. In this paper, a different approach based on a finite element discretization on a wavelet basis is used. We propose adaptive algorithms constructed from error estimates. Such an approach is applied to within-group scattering source iterations. A first implementation is presented in the special case of the fine structure equation for an infinite homogeneous medium. Extension to spatially-dependent cases is discussed. (authors) 7. A Galerkin-Parameterization Method for the Optimal Control of Smart Microbeams Directory of Open Access Journals (Sweden) Marwan Abukhaled 2009-01-01 Full Text Available A proposed computational method is applied to damp out the excess vibrations in smart microbeams, where the control action is implemented using piezoceramic actuators. From a mathematical point of view, we wish to determine the optimal boundary actuators that minimize a given energy-based performance measure. The minimization of the performance measure over the actuators is subjected to the full motion of the structural vibrations of the micro-beams. A direct state-control parametrization approach is proposed where the shifted Legendre polynomials are employed to solve the optimization problem. Legendre operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the lumped parameter systems with respect to the quadratic cost function by solving linear algebraic equations. Numerical examples are provided to demonstrate the applicability and efficiency of the proposed approach. 8. Utilization and discontinuation of contraceptive methods: the University of Calabar Teaching Hospital (UCTH experience Directory of Open Access Journals (Sweden) Njoku CO 2014-09-01 Full Text Available Background: Contraception has an important role to play in reducing the high rate of maternal morbidity and mortality in developing countries. Objective: The objective is to determine the prevalence rate, methods and reasons for discontinuation of contraceptive methods at UCTH, Calabar. Method: This was a retrospective study of all clients that utilised different forms of contraceptives at UCTH, Calabar from 1st January, 2009 to 31st December, 2013. Results: A total of 5,381 clients used various methods of contraception while 13,492 live births were recorded giving the prevalence rate of 39.9% of total live birth. Common methods were intrauterine contraceptive device (IUCD 1,745(32.8% and injectable contraceptives 1,268(23.8%. Most clients 1,876(35.2% were graduates while 81(1.5% had no formal education. A total of 535(10.1% clients discontinued different family planning method commonly due to desire for pregnancy and side effects. IUCD had the highest discontinuation rate. Conclusion: The study revealed low prevalence rate of contraceptive use which was more among teenagers and illiterate women. The main reasons for discontinuation of different methods were desire for pregnancy, side effects and menopause. Creating more contraceptive awareness, improvement in contraceptive counselling and female education will help to improve contraceptive utilisation rate and reduce discontinuation rate. 9. Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces Science.gov (United States) Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J.; Frankel, Steven H. 2013-01-01 Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations. 10. A new method for automated discontinuity trace mapping on rock mass 3D surface model Science.gov (United States) Li, Xiaojun; Chen, Jianqin; Zhu, Hehua 2016-04-01 This paper presents an automated discontinuity trace mapping method on a 3D surface model of rock mass. Feature points of discontinuity traces are first detected using the Normal Tensor Voting Theory, which is robust to noisy point cloud data. Discontinuity traces are then extracted from feature points in four steps: (1) trace feature point grouping, (2) trace segment growth, (3) trace segment connection, and (4) redundant trace segment removal. A sensitivity analysis is conducted to identify optimal values for the parameters used in the proposed method. The optimal triangular mesh element size is between 5 cm and 6 cm; the angle threshold in the trace segment growth step is between 70° and 90°; the angle threshold in the trace segment connection step is between 50° and 70°, and the distance threshold should be at least 15 times the mean triangular mesh element size. The method is applied to the excavation face trace mapping of a drill-and-blast tunnel. The results show that the proposed discontinuity trace mapping method is fast and effective and could be used as a supplement to traditional direct measurement of discontinuity traces. 11. Application of Galerkin meshfree methods to nonlinear thermo-mechanical simulation of solids under extremely high pulsed loading International Nuclear Information System (INIS) Ibáñez, Daniel Iglesias; García Orden, Juan C.; Brañas, B.; Carmona, J.M.; Molla, J. 2013-01-01 Highlights: • The paper presents a novel application of meshfree methods, valid for its implementation on a multibody framework. • Coupled nonlinear thermo-mechanical formulation is detailed and described in the reference configuration, as this allows to compute the shape functions only once. • We show the conditions in which future information induces inefficiency. • Beam parameters are the only information needed to apply the thermal load. • The solution procedure takes charge of updating the volumetric heat rate as the body moves and deforms. -- Abstract: Beam facing elements of the International Fusion Materials Irradiation Facility (IFMIF) Linear Particle Accelerator prototype (LIPAc) must stop 5–40 MeV D + ions with a peak current of 125 mA. The duty cycle of the beam loading varies from 0.1% to 100% (CW), depending on the device, with the ions being stopped in the first hundreds microns of the beam facing material. For intermediate duty cycles up to CW, the thermal load can be considered a heat flux load on the boundary, but this approximation gets too conservative as the duty cycle is reduced because the thermal diffusion becomes more important. Instant heat flux produced by the beam can reach up to 3 GW/m 2 in elements such as the beam dump and slits during short times of hundredths of microseconds. In these cases, the accuracy of the volumetric heat generation is critical for obtaining realistic results. Meshfree Galerkin methods discretize a continuum using scattered nodes. As opposed to FEM, no predefined connectivity is needed between the nodes, so C ∞ (infinitely differentiable) locally supported shape functions can be used to approximate both the trial and the test functions. This feature makes these type of methods well suited for those problems where the domain experiences very large deformations or has high gradients of the state variables. Radial basis (RBF) and moving least squares (MLS) functions have been applied to the 12. Seismic detection method for small-scale discontinuities based on dictionary learning and sparse representation Science.gov (United States) Yu, Caixia; Zhao, Jingtao; Wang, Yanfei 2017-02-01 Studying small-scale geologic discontinuities, such as faults, cavities and fractures, plays a vital role in analyzing the inner conditions of reservoirs, as these geologic structures and elements can provide storage spaces and migration pathways for petroleum. However, these geologic discontinuities have weak energy and are easily contaminated with noises, and therefore effectively extracting them from seismic data becomes a challenging problem. In this paper, a method for detecting small-scale discontinuities using dictionary learning and sparse representation is proposed that can dig up high-resolution information by sparse coding. A K-SVD (K-means clustering via Singular Value Decomposition) sparse representation model that contains two stage of iteration procedure: sparse coding and dictionary updating, is suggested for mathematically expressing these seismic small-scale discontinuities. Generally, the orthogonal matching pursuit (OMP) algorithm is employed for sparse coding. However, the method can only update one dictionary atom at one time. In order to improve calculation efficiency, a regularized version of OMP algorithm is presented for simultaneously updating a number of atoms at one time. Two numerical experiments demonstrate the validity of the developed method for clarifying and enhancing small-scale discontinuities. The field example of carbonate reservoirs further demonstrates its effectiveness in revealing masked tiny faults and small-scale cavities. 13. Validating the Galerkin least-squares finite element methods in predicting mixing flows in stirred tank reactors International Nuclear Information System (INIS) Johnson, K.; Bittorf, K.J. 2002-01-01 A novel approach for computer aided modeling and optimizing mixing process has been developed using Galerkin least-squares finite element technology. Computer aided mixing modeling and analysis involves Lagrangian and Eulerian analysis for relative fluid stretching, and energy dissipation concepts for laminar and turbulent flows. High quality, conservative, accurate, fluid velocity, and continuity solutions are required for determining mixing quality. The ORCA Computational Fluid Dynamics (CFD) package, based on a finite element formulation, solves the incompressible Reynolds Averaged Navier Stokes (RANS) equations. Although finite element technology has been well used in areas of heat transfer, solid mechanics, and aerodynamics for years, it has only recently been applied to the area of fluid mixing. ORCA, developed using the Galerkin Least-Squares (GLS) finite element technology, provides another formulation for numerically solving the RANS based and LES based fluid mechanics equations. The ORCA CFD package is validated against two case studies. The first, a free round jet, demonstrates that the CFD code predicts the theoretical velocity decay rate, linear expansion rate, and similarity profile. From proper prediction of fundamental free jet characteristics, confidence can be derived when predicting flows in a stirred tank, as a stirred tank reactor can be considered a series of free jets and wall jets. (author) 14. The discontinuous finite element method for solving Eigenvalue problems of transport equations International Nuclear Information System (INIS) Yang, Shulin; Wang, Ruihong 2011-01-01 In this paper, the multigroup transport equations for solving the eigenvalues λ and K_e_f_f under two dimensional cylindrical coordinate are discussed. Aimed at the equations, the discretizing way combining discontinuous finite element method (DFE) with discrete ordinate method (SN) is developed, and the iterative algorithms and steps are studied. The numerical results show that the algorithms are efficient. (author) 15. Multigrid technique and Optimized Schwarz method on block-structured grids with discontinuous interfaces DEFF Research Database (Denmark) Kolmogorov, Dmitry; Sørensen, Niels N.; Shen, Wen Zhong 2013-01-01 An Optimized Schwarz method using Robin boundary conditions for relaxation scheme is presented in the frame of Multigrid method on discontinuous grids. At each iteration the relaxation scheme is performed in two steps: one step with Dirichlet and another step with Robin boundary conditions at inn... 16. Numerical Methods for Forward and Inverse Problems in Discontinuous Media Energy Technology Data Exchange (ETDEWEB) Chartier, Timothy P. 2011-03-08 The research emphasis under this grant's funding is in the area of algebraic multigrid methods. The research has two main branches: 1) exploring interdisciplinary applications in which algebraic multigrid can make an impact and 2) extending the scope of algebraic multigrid methods with algorithmic improvements that are based in strong analysis.The work in interdisciplinary applications falls primarily in the field of biomedical imaging. Work under this grant demonstrated the effectiveness and robustness of multigrid for solving linear systems that result from highly heterogeneous finite element method models of the human head. The results in this work also give promise to medical advances possible with software that may be developed. Research to extend the scope of algebraic multigrid has been focused in several areas. In collaboration with researchers at the University of Colorado, Lawrence Livermore National Laboratory, and Los Alamos National Laboratory, the PI developed an adaptive multigrid with subcycling via complementary grids. This method has very cheap computing costs per iterate and is showing promise as a preconditioner for conjugate gradient. Recent work with Los Alamos National Laboratory concentrates on developing algorithms that take advantage of the recent advances in adaptive multigrid research. The results of the various efforts in this research could ultimately have direct use and impact to researchers for a wide variety of applications, including, astrophysics, neuroscience, contaminant transport in porous media, bi-domain heart modeling, modeling of tumor growth, and flow in heterogeneous porous media. This work has already led to basic advances in computational mathematics and numerical linear algebra and will continue to do so into the future. 17. Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier--Stokes Equations KAUST Repository Parsani, Matteo 2016-10-04 Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier--Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, M. Parsani, M. H. Carpenter, and E. J. Nielsen, J. Comput. Phys., 292 (2015), pp. 88--113], extends the applicable set of points from tensor product, Legendre--Gauss--Lobatto (LGL), to a combination of tensor product Legendre--Gauss (LG) and LGL points. The new semidiscrete operators discretely conserve mass, momentum, energy, and satisfy a mathematical entropy inequality for the compressible Navier--Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly from a theoretical point of view. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinear stability proof for the compressible Navier--Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators). 18. Extension of meshless Galerkin/Petrov-Galerkin approach without using Lagrange multipliers International Nuclear Information System (INIS) Kamitani, Atsushi; Takayama, Teruou; Itoh, Taku; Nakamura, Hiroaki 2011-01-01 By directly discretizing the weak form used in the finite element method, meshless methods have been derived. Neither the Lagrange multiplier method nor the penalty method is employed in the derivation of the methods. The resulting methods are divided into two groups, depending on whether the discretization is based on the Galerkin or the Petrov-Galerkin approach. Each group is further subdivided into two groups, according to the method for imposing the essential boundary condition. Hence, four types of the meshless methods have been formulated. The accuracy of these methods is illustrated for two-dimensional Poisson problems. (author) 19. Stable and high order accurate difference methods for the elastic wave equation in discontinuous media KAUST Repository Duru, Kenneth 2014-12-01 © 2014 Elsevier Inc. In this paper, we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media. We consider both planar and curved interfaces where media parameters are allowed to be discontinuous. The key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise. We discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule. Conditions at layer interfaces are imposed weakly using penalties. By deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability. We present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials. The numerical experiments verify high order accuracy and time stability. 20. Thermo-mechanical nonlinear vibration analysis of fluid-conveying structures subjected to different boundary conditions using Galerkin-Newton-Harmonic balancing method Directory of Open Access Journals (Sweden) Gbeminiyi Sobamowo 2017-04-01 Full Text Available The development of mathematical models for describing the dynamic behaviours of fluid conveying pipes, micro-pipes and nanotubes under the influence of some thermo-mechanical parameters results into nonlinear equations that are very difficult to solve analytically. In cases where the exact analytical solutions are presented either in implicit or explicit forms, high skills and rigorous mathematical analyses were employed. It is noted that such solutions do not provide general exact solutions. Inevitably, comparatively simple, flexible yet accurate and practicable solutions are required for the analyses of these structures. Therefore, in this study, approximate analytical solutions are provided to the nonlinear equations arising in flow-induced vibration of pipes, micro-pipes and nanotubes using Galerkin-Newton-Harmonic Method (GNHM. The developed approximate analytical solutions are shown to be valid for both small and large amplitude oscillations. The accuracies and explicitness of these solutions were examined in limiting cases to establish the suitability of the method. 1. Linear Discontinuous Expansion Method using the Subcell Balances for Unstructured Geometry SN Transport International Nuclear Information System (INIS) Hong, Ser Gi; Kim, Jong Woon; Lee, Young Ouk; Kim, Kyo Youn 2010-01-01 The subcell balance methods have been developed for one- and two-dimensional SN transport calculations. In this paper, a linear discontinuous expansion method using sub-cell balances (LDEM-SCB) is developed for neutral particle S N transport calculations in 3D unstructured geometrical problems. At present, this method is applied to the tetrahedral meshes. As the name means, this method assumes the linear distribution of the particle flux in each tetrahedral mesh and uses the balance equations for four sub-cells of each tetrahedral mesh to obtain the equations for the four sub-cell average fluxes which are unknowns. This method was implemented in the computer code MUST (Multi-group Unstructured geometry S N Transport). The numerical tests show that this method gives more robust solution than DFEM (Discontinuous Finite Element Method) 2. 2D deterministic radiation transport with the discontinuous finite element method International Nuclear Information System (INIS) Kershaw, D.; Harte, J. 1993-01-01 This report provides a complete description of the analytic and discretized equations for 2D deterministic radiation transport. This computational model has been checked against a wide variety of analytic test problems and found to give excellent results. We make extensive use of the discontinuous finite element method 3. Geotechnical applications of a two-dimensional elastodynamic displacement discontinuity method CSIR Research Space (South Africa) Siebrits, E 1993-12-01 Full Text Available A general two-dimensional elastodynamic displacement discontinuity method is used to model a variety of application problems. The plane strain problems are: the elastodynamic motions induced on a cavity by shear slip on a nearby crack; the dynamic... 4. An Alternate Approach to Optimal L 2 -Error Analysis of Semidiscrete Galerkin Methods for Linear Parabolic Problems with Nonsmooth Initial Data KAUST Repository Goswami, Deepjyoti 2011-09-01 In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic problems. In both cases, optimal L2-error estimates are derived, when the initial data is in L2. A superconvergence phenomenon is also observed, which is then used to prove L∞-estimates for linear parabolic problems defined on two-dimensional spatial domain again with rough initial data. Copyright © Taylor & Francis Group, LLC. 5. Numerical study of the stress-strain state of reinforced plate on an elastic foundation by the Bubnov-Galerkin method Science.gov (United States) Beskopylny, Alexey; Kadomtseva, Elena; Strelnikov, Grigory 2017-10-01 The stress-strain state of a rectangular slab resting on an elastic foundation is considered. The slab material is isotropic. The slab has stiffening ribs that directed parallel to both sides of the plate. Solving equations are obtained for determining the deflection for various mechanical and geometric characteristics of the stiffening ribs which are parallel to different sides of the plate, having different rigidity for bending and torsion. The calculation scheme assumes an orthotropic slab having different cylindrical stiffness in two mutually perpendicular directions parallel to the reinforcing ribs. An elastic foundation is adopted by Winkler model. To determine the deflection the Bubnov-Galerkin method is used. The deflection is taken in the form of an expansion in a series with unknown coefficients by special polynomials, which are a combination of Legendre polynomials. 6. Early discontinuation DEFF Research Database (Denmark) Hansen, Dorte Gilså; Felde, Lina; Gichangi, Anthony 2007-01-01 prevalence and rate of early discontinuation of different drugs consisting of, in this study, lipid-lowering drugs, antihypertensive drugs, antidepressants, antidiabetics and drugs against osteoporosis. Material and methods This was a register study based on prescription data covering a 4-year period...... and consisting of 470,000 citizens. For each practice and group of drug, a 1-year prevalence for 2002 and the rate of early discontinuation among new users in 2002-2003 were estimated. Early discontinuation was defined as no prescriptions during the second half-year following the first prescription....... There was a positive association between the prevalence of prescribing for the specific drugs studied (antidepressants, antidiabetics, drugs against osteoporosis and lipid-lowering drugs) and early discontinuation (r = 0.29 -0.44), but not for anti-hypertensive drugs. The analysis of the association between prevalence... 7. Determinants of Method Switching among Social Franchise Clients Who Discontinued the Use of Intrauterine Contraceptive Device. Science.gov (United States) Hameed, Waqas; Azmat, Syed Khurram; Ali, Moazzam; Hussain, Wajahat; Mustafa, Ghulam; Ishaque, Muhammad; Ali, Safdar; Ahmed, Aftab; Temmerman, Marleen 2015-01-01 Introduction. Women who do not switch to alternate methods after contraceptive discontinuation, for reasons other than the desire to get pregnant or not needing it, are at obvious risk for unplanned pregnancies or unwanted births. This paper examines the factors that influence women to switch from Intrauterine Contraceptive Device (IUCD) to other methods instead of terminating contraceptive usage altogether. Methods. The data used for this study comes from a larger cross-sectional survey conducted in nine (9) randomly selected districts of Sindh and Punjab provinces of Pakistan, during January 2011. Using Stata 11.2, we analyzed data on 333 women, who reported the removal of IUCDs due to reasons other than the desire to get pregnant. Results. We found that 39.9% of the women do not switch to another method of contraception within one month after IUCD discontinuation. Use of contraception before IUCD insertion increases the odds for method switching by 2.26 times after removal. Similarly, postremoval follow-up by community health worker doubles (OR = 2.0) the chances of method switching. Compared with women who received free IUCD service (via voucher scheme), the method switching is 2.01 times higher among women who had paid for IUCD insertion. Conclusion. To increase the likelihood of method switching among IUCD discontinuers this study emphasizes the need for postremoval client counseling, follow-up by healthcare provider, improved choices to a wider range of contraceptives for poor clients, and user satisfaction. 8. Determinants of Method Switching among Social Franchise Clients Who Discontinued the Use of Intrauterine Contraceptive Device Directory of Open Access Journals (Sweden) Waqas Hameed 2015-01-01 Full Text Available Introduction. Women who do not switch to alternate methods after contraceptive discontinuation, for reasons other than the desire to get pregnant or not needing it, are at obvious risk for unplanned pregnancies or unwanted births. This paper examines the factors that influence women to switch from Intrauterine Contraceptive Device (IUCD to other methods instead of terminating contraceptive usage altogether. Methods. The data used for this study comes from a larger cross-sectional survey conducted in nine (9 randomly selected districts of Sindh and Punjab provinces of Pakistan, during January 2011. Using Stata 11.2, we analyzed data on 333 women, who reported the removal of IUCDs due to reasons other than the desire to get pregnant. Results. We found that 39.9% of the women do not switch to another method of contraception within one month after IUCD discontinuation. Use of contraception before IUCD insertion increases the odds for method switching by 2.26 times after removal. Similarly, postremoval follow-up by community health worker doubles (OR = 2.0 the chances of method switching. Compared with women who received free IUCD service (via voucher scheme, the method switching is 2.01 times higher among women who had paid for IUCD insertion. Conclusion. To increase the likelihood of method switching among IUCD discontinuers this study emphasizes the need for postremoval client counseling, follow-up by healthcare provider, improved choices to a wider range of contraceptives for poor clients, and user satisfaction. 9. Numerical solution of the Neutron Transport Equation using discontinuous nodal methods at X-Y geometry International Nuclear Information System (INIS) Delfin L, A. 1996-01-01 The purpose of this work is to solve the neutron transport equation in discrete-ordinates and X-Y geometry by developing and using the strong discontinuous and strong modified discontinuous nodal finite element schemes. The strong discontinuous and modified strong discontinuous nodal finite element schemes go from two to ten interpolation parameters per cell. They are describing giving a set D c and polynomial space S c corresponding for each scheme BDMO, RTO, BL, BDM1, HdV, BDFM1, RT1, BQ and BDM2. The solution is obtained solving the neutron transport equation moments for each nodal scheme by developing the basis functions defined by Pascal triangle and the Legendre moments giving in the polynomial space S c and, finally, looking for the non singularity of the resulting linear system. The linear system is numerically solved using a computer program for each scheme mentioned . It uses the LU method and forward and backward substitution and makes a partition of the domain in cells. The source terms and angular flux are calculated, using the directions and weights associated to the S N approximation and solving the angular flux moments to find the effective multiplication constant. The programs are written in Fortran language, using the dynamic allocation of memory to increase efficiently the available memory of the computing equipment. (Author) 10. Upwind methods for the Baer–Nunziato equations and higher-order reconstruction using artificial viscosity International Nuclear Information System (INIS) Fraysse, F.; Redondo, C.; Rubio, G.; Valero, E. 2016-01-01 This article is devoted to the numerical discretisation of the hyperbolic two-phase flow model of Baer and Nunziato. A special attention is paid on the discretisation of intercell flux functions in the framework of Finite Volume and Discontinuous Galerkin approaches, where care has to be taken to efficiently approximate the non-conservative products inherent to the model equations. Various upwind approximate Riemann solvers have been tested on a bench of discontinuous test cases. New discretisation schemes are proposed in a Discontinuous Galerkin framework following the criterion of Abgrall and the path-conservative formalism. A stabilisation technique based on artificial viscosity is applied to the high-order Discontinuous Galerkin method and compared against classical TVD-MUSCL Finite Volume flux reconstruction. 11. Upwind methods for the Baer–Nunziato equations and higher-order reconstruction using artificial viscosity Energy Technology Data Exchange (ETDEWEB) Fraysse, F., E-mail: francois.fraysse@rs2n.eu [RS2N, St. Zacharie (France); E. T. S. de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Madrid (Spain); Redondo, C.; Rubio, G.; Valero, E. [E. T. S. de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Madrid (Spain) 2016-12-01 This article is devoted to the numerical discretisation of the hyperbolic two-phase flow model of Baer and Nunziato. A special attention is paid on the discretisation of intercell flux functions in the framework of Finite Volume and Discontinuous Galerkin approaches, where care has to be taken to efficiently approximate the non-conservative products inherent to the model equations. Various upwind approximate Riemann solvers have been tested on a bench of discontinuous test cases. New discretisation schemes are proposed in a Discontinuous Galerkin framework following the criterion of Abgrall and the path-conservative formalism. A stabilisation technique based on artificial viscosity is applied to the high-order Discontinuous Galerkin method and compared against classical TVD-MUSCL Finite Volume flux reconstruction. 12. An Alternate Approach to Optimal L 2 -Error Analysis of Semidiscrete Galerkin Methods for Linear Parabolic Problems with Nonsmooth Initial Data KAUST Repository Goswami, Deepjyoti; Pani, Amiya K. 2011-01-01 In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis 13. New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology International Nuclear Information System (INIS) Khader, M. M.; Kumar, Sunil; Abbasbandy, S. 2013-01-01 We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential—difference equations. The proposed method is based on the Laplace transform with the homotopy analysis method (HAM). This method is a powerful tool for solving a large amount of problems. This technique provides a series of functions which may converge to the exact solution of the problem. A good agreement between the obtained solution and some well-known results is obtained 14. Reasons for discontinuation of contraceptive methods among couples with different family size and educational status. Science.gov (United States) Rizvi, Farwa; Irfan, Ghazia 2012-01-01 High rates of contraceptive discontinuation for reasons other than the desire for pregnancy are a public health concern because of their association with negative reproductive health outcomes. The objective of this study was to determine reasons for discontinuation of contraceptive methods among couples with different family size and educational status. This cross-sectional study was carried out at the Obstetrics/Gynaecology Out-Patient Department of Pakistan Institute of Medical Sciences, Islamabad from April-September 2012. Patients (241) were selected by consecutive sampling after informed written consent and acquiring approval of Ethical Committee. The survey interview tool was a semi-structured questionnaire. Majority (68%) of women belonged to urban, and the rest were from rural areas. Mean age of these women was 29.43 +/- 5.384 year. Reasons for discontinuation of contraceptives included fear of injectable contraceptives (2.9%), contraceptive failure/pregnancy (7.46%), desire to become pregnant (63.48%), husband away at job (2.49%), health concerns/side effects (16.18%), affordability (0.83%), inconvenient to use (1.24%), acceptability (0.83%) and accessibility/lack of information (4.56%). Association of different reasons of discontinuation (chi square test) with the family size (actual number of children) was significant (p = 0.019) but was not significant with husband's or wife's educational status (p = 0.33 and 0.285 respectively). Keeping in mind the complex socioeconomic conditions in our country, Family planning programmers and stake holders need to identify women who strongly want to avoid a pregnancy and finding ways to help the couples successfully initiate and maintain appropriate contraceptive use. 15. The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations. Science.gov (United States) Thamareerat, N; Luadsong, A; Aschariyaphotha, N 2016-01-01 In this paper, we present a numerical scheme used to solve the nonlinear time fractional Navier-Stokes equations in two dimensions. We first employ the meshless local Petrov-Galerkin (MLPG) method based on a local weak formulation to form the system of discretized equations and then we will approximate the time fractional derivative interpreted in the sense of Caputo by a simple quadrature formula. The moving Kriging interpolation which possesses the Kronecker delta property is applied to construct shape functions. This research aims to extend and develop further the applicability of the truly MLPG method to the generalized incompressible Navier-Stokes equations. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed algorithm. Very good agreement between the numerically and analytically computed solutions can be observed in the verification. The present MLPG method has proved its efficiency and reliability for solving the two-dimensional time fractional Navier-Stokes equations arising in fluid dynamics as well as several other problems in science and engineering. 16. A constrained Delaunay discretization method for adaptively meshing highly discontinuous geological media Science.gov (United States) Wang, Yang; Ma, Guowei; Ren, Feng; Li, Tuo 2017-12-01 A constrained Delaunay discretization method is developed to generate high-quality doubly adaptive meshes of highly discontinuous geological media. Complex features such as three-dimensional discrete fracture networks (DFNs), tunnels, shafts, slopes, boreholes, water curtains, and drainage systems are taken into account in the mesh generation. The constrained Delaunay triangulation method is used to create adaptive triangular elements on planar fractures. Persson's algorithm (Persson, 2005), based on an analogy between triangular elements and spring networks, is enriched to automatically discretize a planar fracture into mesh points with varying density and smooth-quality gradient. The triangulated planar fractures are treated as planar straight-line graphs (PSLGs) to construct piecewise-linear complex (PLC) for constrained Delaunay tetrahedralization. This guarantees the doubly adaptive characteristic of the resulted mesh: the mesh is adaptive not only along fractures but also in space. The quality of elements is compared with the results from an existing method. It is verified that the present method can generate smoother elements and a better distribution of element aspect ratios. Two numerical simulations are implemented to demonstrate that the present method can be applied to various simulations of complex geological media that contain a large number of discontinuities. 17. Regularization of the Fourier series of discontinuous functions by various summation methods Energy Technology Data Exchange (ETDEWEB) Ahmad, S.S.; Beghi, L. (Padua Univ. (Italy). Seminario Matematico) 1983-07-01 In this paper the regularization by various summation methods of the Fourier series of functions containing discontinuities of the first and second kind are studied and the results of the numerical analyses referring to some typical periodic functions are presented. In addition to the Cesaro and Lanczos weightings, a new (i.e. cosine) weighting for accelerating the convergence rate is proposed. A comparison with the results obtained by Garibotti and Massaro with the punctual Pade approximants (PPA) technique in case of a periodic step function is also carried out. 18. Stable and high order accurate difference methods for the elastic wave equation in discontinuous media KAUST Repository Duru, Kenneth; Virta, Kristoffer 2014-01-01 to be discontinuous. The key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise. We discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule. Conditions 19. Vibration analysis of orthotropic circular and elliptical nano-plates embedded in elastic medium based on nonlocal Mindlin plate theory and using Galerkin method International Nuclear Information System (INIS) Anjomshoa, Amin; Tahani, Masoud 2016-01-01 In the present study a continuum model based on the nonlocal elasticity theory is developed for free vibration analysis of embedded ortho tropic thick circular and elliptical nano-plates rested on an elastic foundation. The elastic foundation is considered to behave like a Pasternak type of foundations. Governing equations for vibrating nano-plate are derived according to the Mindlin plate theory in which the effects of shear deformations of nano-plate are also included. The Galerkin method is then employed to obtain the size dependent natural frequencies of nano-plate. The solution procedure considers the entire nano-plate as a single super-continuum element. Effect of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties, thickness and foundation on circular frequencies are investigated. It is seen that the nonlocal frequencies of the nano-plate are smaller in comparison to those from the classical theory and this is more pronounced for small lengths and higher vibration modes. It is also found that as the aspect ratio increases or the nanoplate becomes more elliptical, the small scale effect on natural frequencies increases. Further, it is observed that the elastic foundation decreases the influence of nonlocal parameter on the results. Since the effect of shear deformations plays an important role in vibration analysis and design of nano-plates, by predicting smaller values for fundamental frequencies, the study of these nano-structures using thick plate theories such as Mindlin plate theory is essential. 20. Large deformation of uniaxially loaded slender microbeams on the basis of modified couple stress theory: Analytical solution and Galerkin-based method Science.gov (United States) Kiani, Keivan 2017-09-01 Large deformation regime of micro-scale slender beam-like structures subjected to axially pointed loads is of high interest to nanotechnologists and applied mechanics community. Herein, size-dependent nonlinear governing equations are derived by employing modified couple stress theory. Under various boundary conditions, analytical relations between axially applied loads and deformations are presented. Additionally, a novel Galerkin-based assumed mode method (AMM) is established to solve the highly nonlinear equations. In some particular cases, the predicted results by the analytical approach are also checked with those of AMM and a reasonably good agreement is reported. Subsequently, the key role of the material length scale on the load-deformation of microbeams is discussed and the deficiencies of the classical elasticity theory in predicting such a crucial mechanical behavior are explained in some detail. The influences of slenderness ratio and thickness of the microbeam on the obtained results are also examined. The present work could be considered as a pivotal step in better realizing the postbuckling behavior of nano-/micro- electro-mechanical systems consist of microbeams. 1. Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method International Nuclear Information System (INIS) Belendez, A.; Hernandez, A.; Belendez, T.; Neipp, C.; Marquez, A. 2008-01-01 He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient 2. A two-dimensional discontinuous heterogeneous finite element method for neutron transport calculations International Nuclear Information System (INIS) Masiello, E.; Sanchez, R. 2007-01-01 A discontinuous heterogeneous finite element method is presented and discussed. The method is intended for realistic numerical pin-by-pin lattice calculations when an exact representation of the geometric shape of the pins is made without need for homogenization. The method keeps the advantages of conventional discrete ordinate methods, such as fast execution together with the possibility to deal with a large number of spatial meshes, while minimizing the need for geometric modeling. It also provides a complete factorization in space, angle, and energy for the discretized matrices and minimizes, thus, storage requirements. An angular multigrid acceleration technique has also been developed to speed up the rate of convergence of the inner iterations. A particular aspect of this acceleration is the introduction of boundary restriction and prolongation operators that minimize oscillatory behavior and enhance positivity. Numerical tests are presented that show the high precision of the method and the efficiency of the angular multigrid acceleration. (authors) 3. Acceleration of step and linear discontinuous schemes for the method of characteristics in DRAGON5 Directory of Open Access Journals (Sweden) Alain Hébert 2017-09-01 Full Text Available The applicability of the algebraic collapsing acceleration (ACA technique to the method of characteristics (MOC in cases with scattering anisotropy and/or linear sources was investigated. Previously, the ACA was proven successful in cases with isotropic scattering and uniform (step sources. A presentation is first made of the MOC implementation, available in the DRAGON5 code. Two categories of schemes are available for integrating the propagation equations: (1 the first category is based on exact integration and leads to the classical step characteristics (SC and linear discontinuous characteristics (LDC schemes and (2 the second category leads to diamond differencing schemes of various orders in space. The acceleration of these MOC schemes using a combination of the generalized minimal residual [GMRES(m] method preconditioned with the ACA technique was focused on. Numerical results are provided for a two-dimensional (2D eight-symmetry pressurized water reactor (PWR assembly mockup in the context of the DRAGON5 code. 4. Generalized multiscale finite element methods. nonlinear elliptic equations KAUST Repository Efendiev, Yalchin R.; Galvis, Juan; Li, Guanglian; Presho, Michael 2013-01-01 In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples. © 2014 Global-Science Press. 5. Effect of flux discontinuity on spatial approximations for discrete ordinates methods International Nuclear Information System (INIS) Duo, J.I.; Azmy, Y.Y. 2005-01-01 This work presents advances on error analysis of the spatial approximation of the discrete ordinates method for solving the neutron transport equation. Error norms for different non-collided flux problems over a two dimensional pure absorber medium are evaluated using three numerical methods. The problems are characterized by the incoming flux boundary conditions to obtain solutions with different level of differentiability. The three methods considered are the Diamond Difference (DD) method, the Arbitrarily High Order Transport method of the Nodal type (AHOT-N), and of the Characteristic type (AHOT-C). The last two methods are employed in constant, linear and quadratic orders of spatial approximation. The cell-wise error is computed as the difference between the cell-averaged flux computed by each method and the exact value, then the L 1 , L 2 , and L ∞ error norms are calculated. The results of this study demonstrate that the level of differentiability of the exact solution profoundly affects the rate of convergence of the numerical methods' solutions. Furthermore, in the case of discontinuous exact flux the methods fail to converge in the maximum error norm, or in the pointwise sense, in accordance with previous local error analysis. (authors) 6. Comparative study of finite element method, isogeometric analysis, and finite volume method in elastic wave propagation of stress discontinuities Czech Academy of Sciences Publication Activity Database Berezovski, A.; Kolman, Radek; Blažek, Jiří; Kopačka, Ján; Gabriel, Dušan; Plešek, Jiří 2014-01-01 Roč. 19, č. 12 (2014) ISSN 1435-4934. [European Conference on Non-Destructive Testing (ECNDT 2014) /11./. Praha, 06.10.2014-10.10.2014] R&D Projects: GA ČR(CZ) GAP101/11/0288; GA ČR(CZ) GAP101/12/2315 Institutional support: RVO:61388998 Keywords : elastic wave propagation * finite element method * isogeometric analysis * finite volume method * stress discontinuities * spurious oscillations Subject RIV: JR - Other Machinery http://www.ndt.net/events/ECNDT2014/app/content/Paper/25_Berezovski_Rev1.pdf 7. A (Dis)continuous finite element model for generalized 2D vorticity dynamics NARCIS (Netherlands) Bernsen, E.; Bokhove, Onno; van der Vegt, Jacobus J.W. 2005-01-01 A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) 8. Discontinuous finite element and characteristics methods for neutrons transport equation solution in heterogeneous grids International Nuclear Information System (INIS) Masiello, E. 2006-01-01 The principal goal of this manuscript is devoted to the investigation of a new type of heterogeneous mesh adapted to the shape of the fuel pins (fuel-clad-moderator). The new heterogeneous mesh guarantees the spatial modelling of the pin-cell with a minimum of regions. Two methods are investigated for the spatial discretization of the transport equation: the discontinuous finite element method and the method of characteristics for structured cells. These methods together with the new representation of the pin-cell result in an appreciable reduction of calculation points. They allow an exact modelling of the fuel pin-cell without spatial homogenization. A new synthetic acceleration technique based on an angular multigrid is also presented for the speed up of the inner iterations. These methods are good candidates for transport calculations for a nuclear reactor core. A second objective of this work is the application of method of characteristics for non-structured geometries to the study of double heterogeneity problem. The letters is characterized by fuel material with a stochastic dispersion of heterogeneous grains, and until now was solved with a model based on collision probabilities. We propose a new statistical model based on renewal-Markovian theory, which makes possible to take into account the stochastic nature of the problem and to avoid the approximations of the collision probability model. The numerical solution of this model is guaranteed by the method of characteristics. (author) 9. B-spline based finite element method in one-dimensional discontinuous elastic wave propagation Czech Academy of Sciences Publication Activity Database Kolman, Radek; Okrouhlík, Miloslav; Berezovski, A.; Gabriel, Dušan; Kopačka, Ján; Plešek, Jiří 2017-01-01 Roč. 46, June (2017), s. 382-395 ISSN 0307-904X R&D Projects: GA ČR(CZ) GAP101/12/2315; GA MŠk(CZ) EF15_003/0000493 Grant - others:AV ČR(CZ) DAAD-16-12; AV ČR(CZ) ETA-15-03 Program:Bilaterální spolupráce; Bilaterální spolupráce Institutional support: RVO:61388998 Keywords : discontinuous elastic wave propagation * B-spline finite element method * isogeometric analysis * implicit and explicit time integration * dispersion * spurious oscillations Subject RIV: BI - Acoustics OBOR OECD: Acoustics Impact factor: 2.350, year: 2016 http://www.sciencedirect.com/science/article/pii/S0307904X17300835 10. Variational Methods for Discontinuous Structures : Applications to Image Segmentation, Continuum Mechanics CERN Document Server Tomarelli, Franco 1996-01-01 In recent years many researchers in material science have focused their attention on the study of composite materials, equilibrium of crystals and crack distribution in continua subject to loads. At the same time several new issues in computer vision and image processing have been studied in depth. The understanding of many of these problems has made significant progress thanks to new methods developed in calculus of variations, geometric measure theory and partial differential equations. In particular, new technical tools have been introduced and successfully applied. For example, in order to describe the geometrical complexity of unknown patterns, a new class of problems in calculus of variations has been introduced together with a suitable functional setting: the free-discontinuity problems and the special BV and BH functions. The conference held at Villa Olmo on Lake Como in September 1994 spawned successful discussion of these topics among mathematicians, experts in computer science and material scientis... 11. Simplified microstrip discontinuity modeling using the transmission line matrix method interfaced to microwave CAD Science.gov (United States) Thompson, James H.; Apel, Thomas R. 1990-07-01 A technique for modeling microstrip discontinuities is presented which is derived from the transmission line matrix method of solving three-dimensional electromagnetic problems. In this technique the microstrip patch under investigation is divided into an integer number of square and half-square (triangle) subsections. An equivalent lumped-element model is calculated for each subsection. These individual models are then interconnected as dictated by the geometry of the patch. The matrix of lumped elements is then solved using either of two microwave CAD software interfaces with each port properly defined. Closed-form expressions for the lumped-element representation of the individual subsections is presented and experimentally verified through the X-band frequency range. A model demonstrating the use of symmetry and block construction of a circuit element is discussed, along with computer program development and CAD software interface. 12. Three-Dimensional Numerical Simulation of a 30-GHz Gyrotron Resonator With an Explicit High-Order Discontinuous-Galerkin-Based Parallel Particle-In-Cell Method DEFF Research Database (Denmark) Stock, Andreas; Neudorfer, Jonathan; Riedlinger, Marc 2012-01-01 Fast design codes for the simulation of the particle–field interaction in the interior of gyrotron resonators are available. They procure their rapidity by making strong physical simplifications and approximations, which are not known to be valid for many variations of the geometry and the operat... 13. A Convergence Result for the Euler-Maruyama Method for a Simple Stochastic Differential Equation with Discontinuous Drift DEFF Research Database (Denmark) Simonsen, Maria; Schiøler, Henrik; Leth, John-Josef 2014-01-01 The Euler-Maruyama method is applied to a simple stochastic differential equation (SDE) with discontinuous drift. Convergence aspects are investigated in the case, where the Euler-Maruyama method is simulated in dyadic points. A strong rate of convergence is presented for the numerical simulations... 14. A one-level FETI method for the drift–diffusion-Poisson system with discontinuities at an interface KAUST Repository Baumgartner, Stefan; Heitzinger, Clemens 2013-01-01 A 3d feti method for the drift-diffusion-Poisson system including discontinuities at a 2d interface is developed. The motivation for this work is to provide a parallel numerical algorithm for a system of PDEs that are the basic model equations 15. New formulations on the finite element method for boundary value problems with internal/external boundary layers International Nuclear Information System (INIS) Pereira, Luis Carlos Martins 1998-06-01 New Petrov-Galerkin formulations on the finite element methods for convection-diffusion problems with boundary layers are presented. Such formulations are based on a consistent new theory on discontinuous finite element methods. Existence and uniqueness of solutions for these problems in the new finite element spaces are demonstrated. Some numerical experiments shows how the new formulation operate and also their efficacy. (author) 16. A one-level FETI method for the drift–diffusion-Poisson system with discontinuities at an interface KAUST Repository Baumgartner, Stefan 2013-06-01 A 3d feti method for the drift-diffusion-Poisson system including discontinuities at a 2d interface is developed. The motivation for this work is to provide a parallel numerical algorithm for a system of PDEs that are the basic model equations for the simulation of semiconductor devices such as transistors and sensors. Moreover, discontinuities or jumps in the potential and its normal derivative at a 2d surface are included for the simulation of nanowire sensors based on a homogenized model. Using the feti method, these jump conditions can be included with the usual numerical properties and the original Farhat-Roux feti method is extended to the drift-diffusion-Poisson equations including discontinuities. We show two numerical examples. The first example verifies the correct implementation including the discontinuities on a 2d grid divided into eight subdomains. The second example is 3d and shows the application of the algorithm to the simulation of nanowire sensors with high aspect ratios. The Poisson-Boltzmann equation and the drift-diffusion-Poisson system with jump conditions are solved on a 3d grid with real-world boundary conditions. © 2013 Elsevier Inc.. 17. Time-integration methods for finite element discretisations of the second-order Maxwell equation NARCIS (Netherlands) Sarmany, D.; Bochev, Mikhail A.; van der Vegt, Jacobus J.W. This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and theH(\\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic 18. Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction International Nuclear Information System (INIS) Carlberg, Kevin Thomas; Barone, Matthew F.; Antil, Harbir 2016-01-01 Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. Furthermore, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude. 19. Krylov Iterative Methods and the Degraded Effectiveness of Diffusion Synthetic Acceleration for Multidimensional SN Calculations in Problems with Material Discontinuities International Nuclear Information System (INIS) Warsa, James S.; Wareing, Todd A.; Morel, Jim E. 2004-01-01 A loss in the effectiveness of diffusion synthetic acceleration (DSA) schemes has been observed with certain S N discretizations on two-dimensional Cartesian grids in the presence of material discontinuities. We will present more evidence supporting the conjecture that DSA effectiveness will degrade for multidimensional problems with discontinuous total cross sections, regardless of the particular physical configuration or spatial discretization. Fourier analysis and numerical experiments help us identify a set of representative problems for which established DSA schemes are ineffective, focusing on diffusive problems for which DSA is most needed. We consider a lumped, linear discontinuous spatial discretization of the S N transport equation on three-dimensional, unstructured tetrahedral meshes and look at a fully consistent and a 'partially consistent' DSA method for this discretization. The effectiveness of both methods is shown to degrade significantly. A Fourier analysis of the fully consistent DSA scheme in the limit of decreasing cell optical thickness supports the view that the DSA itself is failing when material discontinuities are present in a problem. We show that a Krylov iterative method, preconditioned with DSA, is an effective remedy that can be used to efficiently compute solutions for this class of problems. We show that as a preconditioner to the Krylov method, a partially consistent DSA method is more than adequate. In fact, it is preferable to a fully consistent method because the partially consistent method is based on a continuous finite element discretization of the diffusion equation that can be solved relatively easily. The Krylov method can be implemented in terms of the original S N source iteration coding with only slight modification. Results from numerical experiments show that replacing source iteration with a preconditioned Krylov method can efficiently solve problems that are virtually intractable with accelerated source iteration 20. Organising medication discontinuation DEFF Research Database (Denmark) Nixon, Michael; Kousgaard, Marius Brostrøm 2016-01-01 medication? Methods: Twenty four GPs were interviewed using a maximum variation sample strategy. Participant observations were done in three general practices, for one day each, totalling approximately 30 consultations. Results: The results show that different discontinuation cues (related to the type...... a medication, in agreement with the patients, from a professional perspective. Three research questions were examined in this study: when does medication discontinuation occur in general practice, how is discontinuing medication handled in the GP’s practice and how do GPs make decisions about discontinuing... 1. New formulations on the finite element method for boundary value problems with internal/external boundary layers; Novas formulacoes de elementos finitos para problemas de valor de contorno com camadas limite interna/externa Energy Technology Data Exchange (ETDEWEB) Pereira, Luis Carlos Martins 1998-06-15 New Petrov-Galerkin formulations on the finite element methods for convection-diffusion problems with boundary layers are presented. Such formulations are based on a consistent new theory on discontinuous finite element methods. Existence and uniqueness of solutions for these problems in the new finite element spaces are demonstrated. Some numerical experiments shows how the new formulation operate and also their efficacy. (author) 2. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media KAUST Repository Wang, Yi; Yu, Bo; Sun, Shuyu 2017-01-01 Fast prediction modeling via proper orthogonal decomposition method combined with Galerkin projection is applied to incompressible single-phase fluid flow in porous media. Cases for different configurations of porous media, boundary conditions 3. Non-Galerkin Coarse Grids for Algebraic Multigrid Energy Technology Data Exchange (ETDEWEB) Falgout, Robert D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Schroder, Jacob B. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States) 2014-06-26 Algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. And while AMG has been effectively implemented on large scale parallel machines, challenges remain, especially when moving to exascale. Particularly, stencil sizes (the number of nonzeros in a row) tend to increase further down in the coarse grid hierarchy, and this growth leads to more communication. Therefore, as problem size increases and the number of levels in the hierarchy grows, the overall efficiency of the parallel AMG method decreases, sometimes dramatically. This growth in stencil size is due to the standard Galerkin coarse grid operator,$P^T A P$, where$Pis the prolongation (i.e., interpolation) operator. For example, the coarse grid stencil size for a simple three-dimensional (3D) seven-point finite differencing approximation to diffusion can increase into the thousands on present day machines, causing an associated increase in communication costs. We therefore consider algebraically truncating coarse grid stencils to obtain a non-Galerkin coarse grid. First, the sparsity pattern of the non-Galerkin coarse grid is determined by employing a heuristic minimal “safe” pattern together with strength-of-connection ideas. Second, the nonzero entries are determined by collapsing the stencils in the Galerkin operator using traditional AMG techniques. The result is a reduction in coarse grid stencil size, overall operator complexity, and parallel AMG solve phase times. 4. Study of Continuance Rate and Related Causes of Discontinuance of Pregnancy Prevention Methods among Women in Yazd Directory of Open Access Journals (Sweden) H Fallahzadeh 2008-04-01 Full Text Available Introduction: From maturity to menopause, women are worried about pregnancy. Abstinence from sex or use of pregnancy prevention methods are choices for them. As abstinence is impossible, the only remaining choice is use of pregnancy prevention methods. Effective control of pregnancy is really essential for the health of mother and infant and also control of unplanned increase in population. Regarding the importance of continuance rate of pregnancy prevention methods (OCP, IUD, Condom &DMPA & the reasons for their disruption, this study was carried out with the aim of determining the continuance rate and reasons for discontinuance of pregnancy prevention methods in Yazd women. Methods: This was a cross-sectional study. Six urban health care centers of Yazd were selected as study clusters and information of 15-49 year old women using the pregnancy prevention methods (OCP, IUD, Condom& injection was collected via a questionnaire. The data collected was analyzed by Coplan- Mayer statistic method and variance analysis test. Results: Pregnancy prevention methods were most prevalent in the 25-34 years old age group (57%. Mean duration of pregnancy prevention method usage was 27.98 months using Caplan-mayer method with a median of 24 months. 86.3% for 6 months, 72.8% for 12 months, 62.5% for 18 months, 47.9% for 24 months, 39.9% for 30 months and 37% for 37 months had used four certain methods of pregnancy prevention (OCP, IUD, Condom and Injection. The reasons of discontinuance were disease (15.6% for OCPS, bleeding (27% for IUD, unwanted pregnancy (21% for Condoms and also disease (75% for Injection method. Discussion: According to the results, not only education programs regarding family planning before starting each pregnancy prevention method to women is recommended, but a complete incentive consultation about these methods is essential. This educational & consultation programs should be implemented initially for women using OCP method. 5. Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities. Science.gov (United States) Chremmos, Ioannis 2010-01-01 The scattering of a surface plasmon polariton (SPP) by a rectangular dielectric channel discontinuity is analyzed through a rigorous magnetic field integral equation method. The scattering phenomenon is formulated by means of the magnetic-type scalar integral equation, which is subsequently treated through an entire-domain Galerkin method of moments (MoM), based on a Fourier-series plane wave expansion of the magnetic field inside the discontinuity. The use of Green's function Fourier transform allows all integrations over the area and along the boundary of the discontinuity to be performed analytically, resulting in a MoM matrix with entries that are expressed as spectral integrals of closed-form expressions. Complex analysis techniques, such as Cauchy's residue theorem and the saddle-point method, are applied to obtain the amplitudes of the transmitted and reflected SPP modes and the radiated field pattern. Through numerical results, we examine the wavelength selectivity of transmission and reflection against the channel dimensions as well as the sensitivity to changes in the refractive index of the discontinuity, which is useful for sensing applications. 6. Generalized multiscale finite element method. Symmetric interior penalty coupling KAUST Repository Efendiev, Yalchin R.; Galvis, Juan; Lazarov, Raytcho D.; Moon, M.; Sarkis, Marcus V. 2013-01-01 Motivated by applications to numerical simulations of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the framework of the discontinuous Galerkin finite element method. We propose two different finite element spaces on the coarse mesh. The first space is based on a local eigenvalue problem that uses an interior weighted L2-norm and a boundary weighted L2-norm for computing the "mass" matrix. The second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension. The approximation with these multiscale spaces is based on the discontinuous Galerkin finite element method framework. We investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples. © 2013 Elsevier Inc. 7. Generalized multiscale finite element method. Symmetric interior penalty coupling KAUST Repository Efendiev, Yalchin R. 2013-12-01 Motivated by applications to numerical simulations of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the framework of the discontinuous Galerkin finite element method. We propose two different finite element spaces on the coarse mesh. The first space is based on a local eigenvalue problem that uses an interior weighted L2-norm and a boundary weighted L2-norm for computing the "mass" matrix. The second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension. The approximation with these multiscale spaces is based on the discontinuous Galerkin finite element method framework. We investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples. © 2013 Elsevier Inc. 8. A hybrid perturbation-Galerkin technique for partial differential equations Science.gov (United States) Geer, James F.; Anderson, Carl M. 1990-01-01 A two-step hybrid perturbation-Galerkin technique for improving the usefulness of perturbation solutions to partial differential equations which contain a parameter is presented and discussed. In the first step of the method, the leading terms in the asymptotic expansion(s) of the solution about one or more values of the perturbation parameter are obtained using standard perturbation methods. In the second step, the perturbation functions obtained in the first step are used as trial functions in a Bubnov-Galerkin approximation. This semi-analytical, semi-numerical hybrid technique appears to overcome some of the drawbacks of the perturbation and Galerkin methods when they are applied by themselves, while combining some of the good features of each. The technique is illustrated first by a simple example. It is then applied to the problem of determining the flow of a slightly compressible fluid past a circular cylinder and to the problem of determining the shape of a free surface due to a sink above the surface. Solutions obtained by the hybrid method are compared with other approximate solutions, and its possible application to certain problems associated with domain decomposition is discussed. 9. The discrete maximum principle for Galerkin solutions of elliptic problems Czech Academy of Sciences Publication Activity Database Vejchodský, Tomáš 2012-01-01 Roč. 10, č. 1 (2012), s. 25-43 ISSN 1895-1074 R&D Projects: GA AV ČR IAA100760702 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete maximum principle * monotone methods * Galerkin solution Subject RIV: BA - General Mathematics Impact factor: 0.405, year: 2012 http://www.springerlink.com/content/x73624wm23x4wj26 10. A high-order particle-in-cell method for low density plasma flow and the simulation of gyrotron resonator devices International Nuclear Information System (INIS) Stock, Andreas 2013-01-01 Within this thesis a parallelized, transient, three-dimensional, high-order discontinuous Galerkin Particle-in-Cell solver is developed and used to simulate the resonant cavity of a gyrotron. The high-order discontinuous Galerkin approach - a Finite-Element type method - provides a fast and efficient algorithm to numerically solve Maxwell's equations used within this thesis. Besides its outstanding dissipation and dispersion properties, the discontinuous Galerkin approach easily allows for using unstructured grids, as required to simulate complex-shaped engineering devices. The discontinuous Galerkin approach approximates a wavelength with significantly less degrees of freedom compared to other methods, e.g. Finite Difference methods. Furthermore, the parallelization capabilities of the discontinuous Galerkin framework are excellent due to the very local dependencies between the elements. These properties are essential for the efficient numerical treatment of the Vlasov-Maxwell system with the Particle-in-Cell method. This system describes the self-consistent interaction of charged particles and the electromagnetic field. As central application within this thesis gyrotron resonators are simulated with the discontinuous Galerkin Particle-in-Cell method on high-performance-computers. The gyrotron is a high-power millimeter wave source, used for the electron cyclotron resonance heating of magnetically confined fusion plasma, e.g. in the Wendelstein 7-X experimental fusion-reactor. Compared to state-of-the-art simulation tools used for the design of gyrotron resonators the Particle-in-Cell method does not use any significant physically simplifications w.r.t. the modelling of the particle-field-interaction, the geometry and the wave-spectrum. Hence, it is the method of choice for validation of current simulation tools being restricted by these simplifications. So far, the Particle-in-Cell method was restricted to be used for demonstration calculations only, because 11. A high-order particle-in-cell method for low density plasma flow and the simulation of gyrotron resonator devices Energy Technology Data Exchange (ETDEWEB) Stock, Andreas 2013-04-26 Within this thesis a parallelized, transient, three-dimensional, high-order discontinuous Galerkin Particle-in-Cell solver is developed and used to simulate the resonant cavity of a gyrotron. The high-order discontinuous Galerkin approach - a Finite-Element type method - provides a fast and efficient algorithm to numerically solve Maxwell's equations used within this thesis. Besides its outstanding dissipation and dispersion properties, the discontinuous Galerkin approach easily allows for using unstructured grids, as required to simulate complex-shaped engineering devices. The discontinuous Galerkin approach approximates a wavelength with significantly less degrees of freedom compared to other methods, e.g. Finite Difference methods. Furthermore, the parallelization capabilities of the discontinuous Galerkin framework are excellent due to the very local dependencies between the elements. These properties are essential for the efficient numerical treatment of the Vlasov-Maxwell system with the Particle-in-Cell method. This system describes the self-consistent interaction of charged particles and the electromagnetic field. As central application within this thesis gyrotron resonators are simulated with the discontinuous Galerkin Particle-in-Cell method on high-performance-computers. The gyrotron is a high-power millimeter wave source, used for the electron cyclotron resonance heating of magnetically confined fusion plasma, e.g. in the Wendelstein 7-X experimental fusion-reactor. Compared to state-of-the-art simulation tools used for the design of gyrotron resonators the Particle-in-Cell method does not use any significant physically simplifications w.r.t. the modelling of the particle-field-interaction, the geometry and the wave-spectrum. Hence, it is the method of choice for validation of current simulation tools being restricted by these simplifications. So far, the Particle-in-Cell method was restricted to be used for demonstration calculations only, because 12. Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface Science.gov (United States) Coco, Armando; Russo, Giovanni 2018-05-01 In this paper we propose a second-order accurate numerical method to solve elliptic problems with discontinuous coefficients (with general non-homogeneous jumps in the solution and its gradient) in 2D and 3D. The method consists of a finite-difference method on a Cartesian grid in which complex geometries (boundaries and interfaces) are embedded, and is second order accurate in the solution and the gradient itself. In order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface, two numerical values are assigned on grid points that are close to the interface: a real value, that represents the numerical solution on that grid point, and a ghost value, that represents the numerical solution extrapolated from the other side of the interface, obtained by enforcing the assigned non-homogeneous jump conditions on the solution and its flux. The method is also extended to the case of matrix coefficient. The linear system arising from the discretization is solved by an efficient multigrid approach. Unlike the 1D case, grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient. A proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor, comparable with the one obtained by Local Fourier Analysis for rectangular domains. The method is robust enough to handle large jump in the coefficients: order of accuracy, monotonicity of the errors and good convergence factor are maintained by the scheme. 13. Generalized multiscale finite element method for elasticity equations KAUST Repository Chung, Eric T. 2014-10-05 In this paper, we discuss the application of generalized multiscale finite element method (GMsFEM) to elasticity equation in heterogeneous media. We consider steady state elasticity equations though some of our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and offline spaces. The latter is constructed using local spectral decomposition in the snapshot space. The spectral decomposition is based on the analysis which is provided in the paper. We consider both continuous Galerkin and discontinuous Galerkin coupling of basis functions. Both approaches have their cons and pros. Continuous Galerkin methods allow avoiding penalty parameters though they involve partition of unity functions which can alter the properties of multiscale basis functions. On the other hand, discontinuous Galerkin techniques allow gluing multiscale basis functions without any modifications. Because basis functions are constructed independently from each other, this approach provides an advantage. We discuss the use of oversampling techniques that use snapshots in larger regions to construct the offline space. We provide numerical results to show that one can accurately approximate the solution using reduced number of degrees of freedom. 14. An acceleration of the characteristics by a space-angle two-level method using surface discontinuity factors Energy Technology Data Exchange (ETDEWEB) Grassi, G. [Commissariat a l' Energie Atomique, CEA de Saclay, DM2S/SERMA/LENR, 91191, Gif-sur-Yvette (France) 2006-07-01 We present a non-linear space-angle two-level acceleration scheme for the method of the characteristics (MOC). To the fine level on which the MOC transport calculation is performed, we associate a more coarsely discretized phase space in which a low-order problem is solved as an acceleration step. Cross sections on the coarse level are obtained by a flux-volume homogenisation technique, which entails the non-linearity of the acceleration. Discontinuity factors per surface are introduced as additional degrees of freedom on the coarse level in order to ensure the equivalence of the heterogeneous and the homogenised problem. After each fine transport iteration, a low-order transport problem is iteratively solved on the homogenised grid. The solution of this problem is then used to correct the angular moments of the flux resulting from the previous free transport sweep. Numerical tests for a given benchmark have been performed. Results are discussed. (authors) 15. An acceleration of the characteristics by a space-angle two-level method using surface discontinuity factors International Nuclear Information System (INIS) Grassi, G. 2006-01-01 We present a non-linear space-angle two-level acceleration scheme for the method of the characteristics (MOC). To the fine level on which the MOC transport calculation is performed, we associate a more coarsely discretized phase space in which a low-order problem is solved as an acceleration step. Cross sections on the coarse level are obtained by a flux-volume homogenisation technique, which entails the non-linearity of the acceleration. Discontinuity factors per surface are introduced as additional degrees of freedom on the coarse level in order to ensure the equivalence of the heterogeneous and the homogenised problem. After each fine transport iteration, a low-order transport problem is iteratively solved on the homogenised grid. The solution of this problem is then used to correct the angular moments of the flux resulting from the previous free transport sweep. Numerical tests for a given benchmark have been performed. Results are discussed. (authors) 16. The AFEN Method in Cylindrical (r,θ ,z) Geometry for Pebble Bed Reactors -Incorporation of Acceleration and Discontinuity Factor International Nuclear Information System (INIS) Lee, Jaejun; Cho, Namzin 2007-01-01 Most existing methods of nuclear design analysis for pebble bed reactors (PBRs) are based on old finite difference solvers or on statistical methods. These methods require very long computer times. Therefore, there is strong desire of making available high fidelity coarse-mesh nodal computer codes. Recently, we extended the analytic function expansion nodal (AFEN) method developed quite extensively in Cartesian (x,y,z) geometry and in hexagonal-z geometry to the treatment of the full three dimensional cylindrical (r,θ,z) geometry for pebble bed reactors(PBRs). The AFEN methodology in this geometry as in hexagonal geometry is 'robust', due to the unique feature of the AFEN method that it does not use the transverse integration. This paper presents an acceleration scheme based on the coarse-group rebalance (CGR) concept and provides test results verifying the method and its implementation in the TOPS code. Also, we implemented discontinuity factors in the TOPS code and tested on benchmark problems. The TOPS results are in excellent agreement with those of the VENTURE code, using significantly less computer time 17. Strength Reduction Method for Stability Analysis of Local Discontinuous Rock Mass with Iterative Method of Partitioned Finite Element and Interface Boundary Element Directory of Open Access Journals (Sweden) Tongchun Li 2015-01-01 element is proposed to solve the safety factor of local discontinuous rock mass. Slope system is divided into several continuous bodies and local discontinuous interface boundaries. Each block is treated as a partition of the system and contacted by discontinuous joints. The displacements of blocks are chosen as basic variables and the rigid displacements in the centroid of blocks are chosen as motion variables. The contact forces on interface boundaries and the rigid displacements to the centroid of each body are chosen as mixed variables and solved iteratively using the interface boundary equations. Flexibility matrix is formed through PFE according to the contact states of nodal pairs and spring flexibility is used to reflect the influence of weak structural plane so that nonlinear iteration is only limited to the possible contact region. With cohesion and friction coefficient reduced gradually, the states of all nodal pairs at the open or slip state for the first time are regarded as failure criterion, which can decrease the effect of subjectivity in determining safety factor. Examples are used to verify the validity of the proposed method. 18. Explicit evaluation of discontinuities in 2-D unsteady flows solved by the method of characteristics Science.gov (United States) Osnaghi, C. When shock waves appear in the numerical solution of flows, a choice is necessary between shock capturing techniques, possible when equations are written in conservative form, and shock fitting techniques. If the second one is preferred, e.g. in order to obtain better definition and more physical description of the shock evolution in time, the method of characteristics is advantageous in the vicinity of the shock and it seems natural to use this method everywhere. This choice requires to improve the efficiency of the numerical scheme in order to produce competitive codes, preserving accuracy and flexibility, which are intrinsic features of the method: this is the goal of the present work. 19. On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems International Nuclear Information System (INIS) De Cezaro, A; Leitão, A; Tai, X-C 2013-01-01 We investigate level-set-type methods for solving ill-posed problems with discontinuous (piecewise constant) coefficients. The goal is to identify the level sets as well as the level values of an unknown parameter function on a model described by a nonlinear ill-posed operator equation. The PCLS approach is used here to parametrize the solution of a given operator equation in terms of a L 2 level-set function, i.e. the level-set function itself is assumed to be a piecewise constant function. Two distinct methods are proposed for computing stable solutions of the resulting ill-posed problem: the first is based on Tikhonov regularization, while the second is based on the augmented Lagrangian approach with total variation penalization. Classical regularization results (Engl H W et al 1996 Mathematics and its Applications (Dordrecht: Kluwer)) are derived for the Tikhonov method. On the other hand, for the augmented Lagrangian method, we succeed in proving the existence of (generalized) Lagrangian multipliers in the sense of (Rockafellar R T and Wets R J-B 1998 Grundlehren der Mathematischen Wissenschaften (Berlin: Springer)). Numerical experiments are performed for a 2D inverse potential problem (Hettlich F and Rundell W 1996 Inverse Problems 12 251–66), demonstrating the capabilities of both methods for solving this ill-posed problem in a stable way (complicated inclusions are recovered without any a priori geometrical information on the unknown parameter). (paper) 20. A stochastic collocation method for the second order wave equation with a discontinuous random speed KAUST Repository Motamed, Mohammad; Nobile, Fabio; Tempone, Raul 2012-01-01 In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical 1. A stochastic collocation method for the second order wave equation with a discontinuous random speed KAUST Repository Motamed, Mohammad 2012-08-31 In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems. © 2012 Springer-Verlag. 2. An autonomous surface discontinuity detection and quantification method by digital image correlation and phase congruency Science.gov (United States) Cinar, A. F.; Barhli, S. M.; Hollis, D.; Flansbjer, M.; Tomlinson, R. A.; Marrow, T. J.; Mostafavi, M. 2017-09-01 Digital image correlation has been routinely used to measure full-field displacements in many areas of solid mechanics, including fracture mechanics. Accurate segmentation of the crack path is needed to study its interaction with the microstructure and stress fields, and studies of crack behaviour, such as the effect of closure or residual stress in fatigue, require data on its opening displacement. Such information can be obtained from any digital image correlation analysis of cracked components, but it collection by manual methods is quite onerous, particularly for massive amounts of data. We introduce the novel application of Phase Congruency to detect and quantify cracks and their opening. Unlike other crack detection techniques, Phase Congruency does not rely on adjustable threshold values that require user interaction, and so allows large datasets to be treated autonomously. The accuracy of the Phase Congruency based algorithm in detecting cracks is evaluated and compared with conventional methods such as Heaviside function fitting. As Phase Congruency is a displacement-based method, it does not suffer from the noise intensification to which gradient-based methods (e.g. strain thresholding) are susceptible. Its application is demonstrated to experimental data for cracks in quasi-brittle (Granitic rock) and ductile (Aluminium alloy) materials. 3. Numerical sensitivity computation for discontinuous gradient-only optimization problems using the complex-step method CSIR Research Space (South Africa) Wilke, DN 2012-07-01 Full Text Available problems that utilise remeshing (i.e. the mesh topology is allowed to change) between design updates. Here, changes in mesh topology result in abrupt changes in the discretization error of the computed response. These abrupt changes in turn manifests... in shape optimization but may be present whenever (partial) differential equations are ap- proximated numerically with non-constant discretization methods e.g. remeshing of spatial domains or automatic time stepping in temporal domains. Keywords: Complex... 4. Timing, method and discontinuation of hydrocortisone administration for septic shock patients OpenAIRE Ibarra-Estrada, Miguel A; Ch?vez-Pe?a, Quetzalc?atl; Reynoso-Estrella, Claudia I; Rios-Zerme?o, Jorge; Aguilera-Gonz?lez, P?vel E; Garc?a-Soto, Miguel A; Aguirre-Avalos, Guadalupe 2017-01-01 AIM To characterize the prescribing patterns for hydrocortisone for patients with septic shock and perform an exploratory analysis in order to identify the variables associated with better outcomes. METHODS This prospective cohort study included 59 patients with septic shock who received stress-dose hydrocortisone. It was performed at 2 critical care units in academic hospitals from June 1st, 2015, to July 31st, 2016. Demographic data, comorbidities, medical management details, adverse effect... 5. Implementation of LDG method for 3D unstructured meshes Directory of Open Access Journals (Sweden) Filander A. Sequeira Chavarría 2012-07-01 Full Text Available This paper describes an implementation of the Local Discontinuous Galerkin method (LDG applied to elliptic problems in 3D. The implementation of the major operators is discussed. In particular the use of higher-order approximations and unstructured meshes. Efficient data structures that allow fast assembly of the linear system in the mixed formulation are described in detail. Keywords: Discontinuous finite element methods, high-order approximations, unstructured meshes, object-oriented programming. Mathematics Subject Classification: 65K05, 65N30, 65N55. 6. Global Scale Exploration Seismics: Mapping Mantle Discontinuities with Inverse Scattering Methods and Millions of Seismograms Science.gov (United States) van der Hilst, R. D.; de Hoop, M. V.; Shim, S. H.; Shang, X.; Wang, P.; Cao, Q. 2012-04-01 Over the past three decades, tremendous progress has been made with the mapping of mantle heterogeneity and with the understanding of these structures in terms of, for instance, the evolution of Earth's crust, continental lithosphere, and thermo-chemical mantle convection. Converted wave imaging (e.g., receiver functions) and reflection seismology (e.g. SS stacks) have helped constrain interfaces in crust and mantle; surface wave dispersion (from earthquake or ambient noise signals) characterizes wavespeed variations in continental and oceanic lithosphere, and body wave and multi-mode surface wave data have been used to map trajectories of mantle convection and delineate mantle regions of anomalous elastic properties. Collectively, these studies have revealed substantial ocean-continent differences and suggest that convective flow is strongly influenced by but permitted to cross the upper mantle transition zone. Many questions have remained unanswered, however, and further advances in understanding require more accurate depictions of Earth's heterogeneity at a wider range of length scales. To meet this challenge we need new observations—more, better, and different types of data—and methods that help us extract and interpret more information from the rapidly growing volumes of broadband data. The huge data volumes and the desire to extract more signal from them means that we have to go beyond 'business as usual' (that is, simplified theory, manual inspection of seismograms, …). Indeed, it inspires the development of automated full wave methods, both for tomographic delineation of smooth wavespeed variations and the imaging (for instance through inverse scattering) of medium contrasts. Adjoint tomography and reverse time migration, which are closely related wave equation methods, have begun to revolutionize seismic inversion of global and regional waveform data. In this presentation we will illustrate this development - and its promise - drawing from our work 7. Symmetric-Galerkin BEM simulation of fracture with frictional contact CSIR Research Space (South Africa) Phan, AV 2003-06-14 Full Text Available FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2003; 57:835?851 (DOI: 10.1002/nme.707) Symmetric-Galerkin BEM simulation of fracture with frictional contact A.-V. Phan1;asteriskmath;?, J. A. L. Napier2, L. J. Gray3 and T. Kaplan3 1Department... Methods in Engineering 1975; 9:495?507. 35. Barsoum RS. On the use of isoparametric FFnite elements in linear fracture mechanics. International Journal for Numerical Methods in Engineering 1976; 10:25?37. 36. Gray LJ, Phan A-V, Paulino GH, Kaplan T... 8. Application of the Galerkin's method to the solution of the one-dimensional integral transport equation: generalized collision probabilities taken in account the flux gradient and the linearly anisotropic scattering International Nuclear Information System (INIS) Sanchez, Richard. 1975-04-01 For the one-dimensional geometries, the transport equation with linearly anisotropic scattering can be reduced to a single integral equation; this is a singular-kernel FREDHOLM equation of the second kind. When applying a conventional projective method that of GALERKIN, to the solution of this equation the well-known collision probability algorithm is obtained. Piecewise polynomial expansions are used to represent the flux. In the ANILINE code, the flux is supposed to be linear in plane geometry and parabolic in both cylindrical and spherical geometries. An integral relationship was found between the one-dimensional isotropic and anisotropic kernels; this allows to reduce the new matrix elements (issuing from the anisotropic kernel) to classic collision probabilities of the isotropic scattering equation. For cylindrical and spherical geometries used an approximate representation of the current was used to avoid an additional numerical integration. Reflective boundary conditions were considered; in plane geometry the reflection is supposed specular, for the other geometries the isotropic reflection hypothesis has been adopted. Further, the ANILINE code enables to deal with an incoming isotropic current. Numerous checks were performed in monokinetic theory. Critical radii and albedos were calculated for homogeneous slabs, cylinders and spheres. For heterogeneous media, the thermal utilization factor obtained by this method was compared with the theoretical result based upon a formula by BENOIST. Finally, ANILINE was incorporated into the multigroup APOLLO code, which enabled to analyse the MINERVA experimental reactor in transport theory with 99 groups. The ANILINE method is particularly suited to the treatment of strongly anisotropic media with considerable flux gradients. It is also well adapted to the calculation of reflectors, and in general, to the exact analysis of anisotropic effects in large-sized media [fr 9. Assembly Discontinuity Factors for the Neutron Diffusion Equation discretized with the Finite Volume Method. Application to BWR International Nuclear Information System (INIS) Bernal, A.; Roman, J.E.; Miró, R.; Verdú, G. 2016-01-01 Highlights: • A method is proposed to solve the eigenvalue problem of the Neutron Diffusion Equation in BWR. • The Neutron Diffusion Equation is discretized with the Finite Volume Method. • The currents are calculated by using a polynomial expansion of the neutron flux. • The current continuity and boundary conditions are defined implicitly to reduce the size of the matrices. • Different structured and unstructured meshes were used to discretize the BWR. - Abstract: The neutron flux spatial distribution in Boiling Water Reactors (BWRs) can be calculated by means of the Neutron Diffusion Equation (NDE), which is a space- and time-dependent differential equation. In steady state conditions, the time derivative terms are zero and this equation is rewritten as an eigenvalue problem. In addition, the spatial partial derivatives terms are transformed into algebraic terms by discretizing the geometry and using numerical methods. As regards the geometrical discretization, BWRs are complex systems containing different components of different geometries and materials, but they are usually modelled as parallelepiped nodes each one containing only one homogenized material to simplify the solution of the NDE. There are several techniques to correct the homogenization in the node, but the most commonly used in BWRs is that based on Assembly Discontinuity Factors (ADFs). As regards numerical methods, the Finite Volume Method (FVM) is feasible and suitable to be applied to the NDE. In this paper, a FVM based on a polynomial expansion method has been used to obtain the matrices of the eigenvalue problem, assuring the accomplishment of the ADFs for a BWR. This eigenvalue problem has been solved by means of the SLEPc library. 10. Testing discontinuities in nonparametric regression KAUST Repository Dai, Wenlin 2017-01-19 In nonparametric regression, it is often needed to detect whether there are jump discontinuities in the mean function. In this paper, we revisit the difference-based method in [13 H.-G. Müller and U. Stadtmüller, Discontinuous versus smooth regression, Ann. Stat. 27 (1999), pp. 299–337. doi: 10.1214/aos/1018031100 11. Testing discontinuities in nonparametric regression KAUST Repository Dai, Wenlin; Zhou, Yuejin; Tong, Tiejun 2017-01-01 In nonparametric regression, it is often needed to detect whether there are jump discontinuities in the mean function. In this paper, we revisit the difference-based method in [13 H.-G. Müller and U. Stadtmüller, Discontinuous versus smooth regression, Ann. Stat. 27 (1999), pp. 299–337. doi: 10.1214/aos/1018031100 12. A multiscale method for compressible liquid-vapor flow with surface tension* Directory of Open Access Journals (Sweden) Jaegle Felix 2013-01-01 Full Text Available Discontinuous Galerkin methods have become a powerful tool for approximating the solution of compressible flow problems. Their direct use for two-phase flow problems with phase transformation is not straightforward because this type of flows requires a detailed tracking of the phase front. We consider the fronts in this contribution as sharp interfaces and propose a novel multiscale approach. It combines an efficient high-order Discontinuous Galerkin solver for the computation in the bulk phases on the macro-scale with the use of a generalized Riemann solver on the micro-scale. The Riemann solver takes into account the effects of moderate surface tension via the curvature of the sharp interface as well as phase transformation. First numerical experiments in three space dimensions underline the overall performance of the method. 13. Velocity structure around the 410 km discontinuity beneath the East China Sea based on the waveform modeling method Science.gov (United States) Li, W.; Cui, Q.; Gao, Y.; Wei, R.; Zhou, Y.; Yu, J. 2017-12-01 The 410 km discontinuity is the upper boundary of the mantle transition zone. Seismic detections on the structure and morphology of the 410 km discontinuity are helpful to understand the compositions of the Earth's interior and the relevant geodynamics. In this paper, we select the broadband P waveforms of an intermediate earthquake that occurred in the Ryukyu subduction zone and retrieved from the China Digital Seismograph Network, and study the fine velocity structure around the 410 km discontinuity by matching the observed triplicated waveforms with the theoretical ones. Our results reveal that (1) the 410 km discontinuity beneath the East China Sea is mostly a sharp boundary with a small-scale uplift of 8-15 km and a gradient boundary up to 20 km in the most southern part, and (2) there exist a low velocity layer atop the 410 km discontinuity with the thickness of 50-62 km and P-wave velocity decrease of 0.5%-1.5%, and (3) a high velocity anomaly with P-wave decrease of 1.0%-3.0% below 440 km. Combining with the previous topographic results in this area, we speculate that the high velocity anomaly is relevant to the stagnancy of the western Pacific slab in the mantle transition zone, the decomposition of phase E in the slab results in the increase of water content, which would cause the uplift of the 410 km discontinuity, and the low velocity layer atop the discontinuity should be related to the partial melting of the mantle peridotite induced by the dehydration of the hydrous minerals. 14. Strategies for discontinuation of proton pump inhibitors DEFF Research Database (Denmark) Haastrup, Peter; Paulsen, Maja S; Begtrup, Luise M 2014-01-01 PURPOSE: Proton pump inhibitors (PPIs) are considered to be overprescribed. Consensus on how to attempt discontinuation is, however, lacking. We therefore conducted a systematic review of clinical studies on discontinuation of PPIs. METHODS: Systematic review based on clinical studies investigating... 15. The combined Petrov-Galerkin method with auto-adapting schemes and its applications in numerical resolution of problems with limit layer International Nuclear Information System (INIS) Silva, R.S.; Galeao, A.C.; Carmo, E.G.D. do 1989-07-01 In this paper a new finite element model is constructed combining an r- refinement scheme with the CCAU method. The new formulation gives better approximation for boundary and internal layers compared to the standard CCAU, without increasing computer codes. (author) [pt 16. Intergenerational continuity and discontinuity in Mexican-origin youths' participation in organized activities: insights from mixed-methods. Science.gov (United States) Simpkins, Sandra D; Vest, Andrea E; Price, Chara D 2011-12-01 Motivation theories suggest that parents are an integral support for adolescents' participation in organized activities. Despite the importance of parents, the field knows very little about how parents' own experiences in activities influence the participation of their adolescent children. The goals of this study were to examine (a) the patterns of intergenerational continuity and discontinuity in parents' activity participation during adolescence and their adolescents' activity participation, and (b) the processes underlying each of these patterns within Mexican-origin families. Qualitative and quantitative data were collected through three in-depth interviews conducted with 31 seventh-grade adolescents and their parents at three time points over a year. The quantitative data suggested there was modest intergenerational continuity in activity participation. There were three distinct patterns: nine families were continuous participants, seven families were continuous nonparticipants, and 15 families were discontinuous, where the parent did not participate but the youth did participate in activities. The continuous participant families included families in which parents valued how organized activities contributed to their own lives and actively encouraged their adolescents' participation. The continuous nonparticipant families reported less knowledge and experience with activities along with numerous barriers to participation. There were three central reasons for the change in the discontinuous families. For a third of these families, parents felt strongly about providing a different childhood for their adolescents than what they experienced. The intergenerational discontinuity in participation was also likely to be sparked by someone else in the family or an external influence (i.e., friends, schools). 17. Discretization methods for evolving discontinuities : proceedings of the IUTAM symposium held in Lyon, France, 4-7 September 2006 NARCIS (Netherlands) Combescure, Alain; Borst, de R.; Belytschko, Ted 2007-01-01 Discontinuities arean important domain in the mechanics of solids and fluids. With mechanics focusing on smaller and smaller length scales in order to understand the physics that underly many phenomena that hitherto were modelled in a phenomenological manner, the need to properly model d- 18. Galerkin v. discrete-optimal projection in nonlinear model reduction Energy Technology Data Exchange (ETDEWEB) Carlberg, Kevin Thomas [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Barone, Matthew Franklin [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Antil, Harbir [George Mason Univ., Fairfax, VA (United States) 2015-04-01 Discrete-optimal model-reduction techniques such as the Gauss{Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible ow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform projection at the time-continuous level, while discrete-optimal techniques do so at the time-discrete level. This work provides a detailed theoretical and experimental comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge{Kutta schemes. We present a number of new ndings, including conditions under which the discrete-optimal ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and experimentally that decreasing the time step does not necessarily decrease the error for the discrete-optimal ROM; instead, the time step should be `matched' to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible- ow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the discrete-optimal reduced-order model by an order of magnitude. 19. Element free Galerkin formulation of composite beam with longitudinal slip Energy Technology Data Exchange (ETDEWEB) Ahmad, Dzulkarnain; Mokhtaram, Mokhtazul Haizad [Department of Civil Engineering, Universiti Selangor, Bestari Jaya, Selangor (Malaysia); Badli, Mohd Iqbal; Yassin, Airil Y. Mohd [Faculty of Civil Engineering, Universiti Teknologi Malaysia, Skudai, Johor (Malaysia) 2015-05-15 Behaviour between two materials in composite beam is assumed partially interact when longitudinal slip at its interfacial surfaces is considered. Commonly analysed by the mesh-based formulation, this study used meshless formulation known as Element Free Galerkin (EFG) method in the beam partial interaction analysis, numerically. As meshless formulation implies that the problem domain is discretised only by nodes, the EFG method is based on Moving Least Square (MLS) approach for shape functions formulation with its weak form is developed using variational method. The essential boundary conditions are enforced by Langrange multipliers. The proposed EFG formulation gives comparable results, after been verified by analytical solution, thus signify its application in partial interaction problems. Based on numerical test results, the Cubic Spline and Quartic Spline weight functions yield better accuracy for the EFG formulation, compares to other proposed weight functions. 20. Depuration Study of Heavy Metal Lead (Pb) and Copper (Cu) in Green Mussels Perna viridis through Continues-discontinues and Acid Extraction Methods Science.gov (United States) Budiawan; Bakri, Ridla; Cahaya Dani, Intan; Handayani, Sri; Ade Kurnia Putri, Rizki; Tamala, Riska 2018-01-01 Green mussel or Perna viridis is filter feeder, which is very susceptible to heavy metals. It takes an effort to release heavy metal contents on the green shell, one of method that can be used to release heavy metal from green shell is depuration proccess. In this research, the depuration process was conducted by continues method of depuration, discontinues method by using various kind of water and acid extraction. The optimum time of continues depuration method is 1.5 hours, with circulation speed 250 L/h and result of Pb metal content decreased is equal to 30.048% and 29.748% for Cu. In the discontinues method, the optimum result was reached at 100oC by using PAM water as the media at 3 h immersion period with decrease of Pb metal content 35.001% and Cu metal content 39.015%. In the acid extraction method, the optimum condition was achieved by 11% acetic acid solvent with decreasing of Pb and Cu levels are 88.224% and 76.298%. For the determination of protein content, the decrease of protein content obtained by treatment with 11% acetic acid extract showed decrease of protein content 36.656% with Kjeldahl method. 1. A Galerkin least squares approach to viscoelastic flow. Energy Technology Data Exchange (ETDEWEB) Rao, Rekha R. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Schunk, Peter Randall [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States) 2015-10-01 A Galerkin/least-squares stabilization technique is applied to a discrete Elastic Viscous Stress Splitting formulation of for viscoelastic flow. From this, a possible viscoelastic stabilization method is proposed. This method is tested with the flow of an Oldroyd-B fluid past a rigid cylinder, where it is found to produce inaccurate drag coefficients. Furthermore, it fails for relatively low Weissenberg number indicating it is not suited for use as a general algorithm. In addition, a decoupled approach is used as a way separating the constitutive equation from the rest of the system. A Pressure Poisson equation is used when the velocity and pressure are sought to be decoupled, but this fails to produce a solution when inflow/outflow boundaries are considered. However, a coupled pressure-velocity equation with a decoupled constitutive equation is successful for the flow past a rigid cylinder and seems to be suitable as a general-use algorithm. 2. A zonal Galerkin-free POD model for incompressible flows Science.gov (United States) Bergmann, Michel; Ferrero, Andrea; Iollo, Angelo; Lombardi, Edoardo; Scardigli, Angela; Telib, Haysam 2018-01-01 A domain decomposition method which couples a high and a low-fidelity model is proposed to reduce the computational cost of a flow simulation. This approach requires to solve the high-fidelity model in a small portion of the computational domain while the external field is described by a Galerkin-free Proper Orthogonal Decomposition (POD) model. We propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models. This zonal approach can be used to study multi-body configurations or to perform detailed local analyses in the framework of shape optimisation problems. The efficiency of the method to perform predictive low-cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem. 3. The influence of method-related partner violence on covert pill use and pill discontinuation among women living in La Paz, El Alto and Santa Cruz, Bolivia. Science.gov (United States) McCarraher, Donna R; Martin, Sandra L; Bailey, Patricia E 2006-03-01 Intimate partner violence is widespread worldwide. While assumed to impact women's ability to use contraceptive methods, few data are available to support this claim. In this study, eight focus group discussions were conducted to guide questionnaire development and to provide contextual information. Participants were women who were currently using the pill and women who had used the pill previously. In addition, 300 women were interviewed who initiated oral contraceptive pill use between December 1995 and April 1996. Participants were interviewed 3-6 months later to investigate the role intimate partner violence played in covert pill use and pill discontinuation. Special study procedures for asking women questions about violence were employed. Nineteen per cent of the women interviewed were using the pill covertly. The odds of covert pill use were four times higher in El Alto and La Paz than in Santa Cruz. Women who used the pill covertly were more likely to have experienced method-related partner violence (OR = 21.27) than women whose partners knew of their pill use. One-third of the women had discontinued pill use at the time of the interview. In the final multivariate analysis, having experienced side-effects (OR = 2.37) was a significant predictor of pill discontinuation and method-related partner violence was marginally predictive (OR = 1.91; 95% CI 1.0-3.66). While efforts are ongoing to incorporate men into family planning programmes, some male partners oppose, and in some situations violently oppose, contraceptive use. The needs of women with these types of partners must not be overlooked. 4. Space-time least-squares Petrov-Galerkin projection in nonlinear model reduction. Energy Technology Data Exchange (ETDEWEB) Choi, Youngsoo [Sandia National Laboratories (SNL-CA), Livermore, CA (United States). Extreme-scale Data Science and Analytics Dept.; Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Carlberg, Kevin Thomas [Sandia National Laboratories (SNL-CA), Livermore, CA (United States). Extreme-scale Data Science and Analytics Dept. 2017-09-01 Our work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply Petrov-Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted ℓ2-norm. This norm can be de ned to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time GNAT variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include: (1) a reduction of both the spatial and temporal dimensions of the dynamical system, (2) the removal of spurious temporal modes (e.g., unstable growth) from the state space, and (3) error bounds that exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy. 5. Research on advancement of method for evaluating aseismatic ability of rock discontinuity plane in ground and surrounding slopes of nuclear power facilities International Nuclear Information System (INIS) Kusunose, Kinichiro; Cho, Akio; Takahashi, Manabu; Kamai, Toshitaka 1997-01-01 The purpose of this research is to carry out the technical development required for exploring with high accuracy the distribution and shapes of the discontinuity planes in rocks in the ground and surrounding cut-off slopes of nuclear power facilities, and to advance the techniques of interpreting and evaluating quantitatively the stability against earthquakes of the discontinuity planes. This research consists of two themes: the research on the method of investigating the three-dimensional distribution of the crevices in the ground and the research on the method of evaluating the aseismatic ability in the slopes. As for the first theme, one of the techniques for exploring underground structure with elastic waves, tomography, is explained, and the development of the 12 channel receiver and the program for the multi-channel analysis and processing of waveform are reported. As for the second theme, the stability analysis was carried out on three actual cases of landslide. The equation for stability analysis is shown, and the results are reported. The strength at the time of forming separation plane gives the most proper result. (K.I.) 6. Dynamic Rupture Benchmarking of the ADER-DG Method Science.gov (United States) Gabriel, Alice; Pelties, Christian 2013-04-01 We will verify the arbitrary high-order derivative Discontinuous Galerkin (ADER-DG) method in various test cases of the 'SCEC/USGS Dynamic Earthquake Rupture Code Verification Exercise' benchmark suite (Harris et al. 2009). The ADER-DG scheme is able to solve the spontaneous rupture problem with high-order accuracy in space and time on three-dimensional unstructured tetrahedral meshes. Strong mesh coarsening or refinement at areas of interest can be applied to keep the computational costs feasible. Moreover, the method does not generate spurious high-frequency contributions in the slip rate spectra and therefore does not require any artificial damping as demonstrated in previous presentations and publications (Pelties et al. 2010 and 2012). We will show that the mentioned features hold also for more advanced setups as e.g. a branching fault system, heterogeneous background stresses and bimaterial faults. The advanced geometrical flexibility combined with an enhanced accuracy will make the ADER-DG method a useful tool to study earthquake dynamics on complex fault systems in realistic rheologies. References: Harris, R.A., M. Barall, R. Archuleta, B. Aagaard, J.-P. Ampuero, H. Bhat, V. Cruz-Atienza, L. Dalguer, P. Dawson, S. Day, B. Duan, E. Dunham, G. Ely, Y. Kaneko, Y. Kase, N. Lapusta, Y. Liu, S. Ma, D. Oglesby, K. Olsen, A. Pitarka, S. Song, and E. Templeton, The SCEC/USGS Dynamic Earthquake Rupture Code Verification Exercise, Seismological Research Letters, vol. 80, no. 1, pages 119-126, 2009 Pelties, C., J. de la Puente, and M. Kaeser, Dynamic Rupture Modeling in Three Dimensions on Unstructured Meshes Using a Discontinuous Galerkin Method, AGU 2010 Fall Meeting, abstract #S21C-2068 Pelties, C., J. de la Puente, J.-P. Ampuero, G. Brietzke, and M. Kaeser, Three-Dimensional Dynamic Rupture Simulation with a High-order Discontinuous Galerkin Method on Unstructured Tetrahedral Meshes, JGR. - Solid Earth, VOL. 117, B02309, 2012 7. Discrete maximum principle for the P1 - P0 weak Galerkin finite element approximations Science.gov (United States) Wang, Junping; Ye, Xiu; Zhai, Qilong; Zhang, Ran 2018-06-01 This paper presents two discrete maximum principles (DMP) for the numerical solution of second order elliptic equations arising from the weak Galerkin finite element method. The results are established by assuming an h-acute angle condition for the underlying finite element triangulations. The mathematical theory is based on the well-known De Giorgi technique adapted in the finite element context. Some numerical results are reported to validate the theory of DMP. 8. Proteus-MOC: A 3D deterministic solver incorporating 2D method of characteristics International Nuclear Information System (INIS) Marin-Lafleche, A.; Smith, M. A.; Lee, C. 2013-01-01 A new transport solution methodology was developed by combining the two-dimensional method of characteristics with the discontinuous Galerkin method for the treatment of the axial variable. The method, which can be applied to arbitrary extruded geometries, was implemented in PROTEUS-MOC and includes parallelization in group, angle, plane, and space using a top level GMRES linear algebra solver. Verification tests were performed to show accuracy and stability of the method with the increased number of angular directions and mesh elements. Good scalability with parallelism in angle and axial planes is displayed. (authors) 9. Proteus-MOC: A 3D deterministic solver incorporating 2D method of characteristics Energy Technology Data Exchange (ETDEWEB) Marin-Lafleche, A.; Smith, M. A.; Lee, C. [Argonne National Laboratory, 9700 S. Cass Avenue, Lemont, IL 60439 (United States) 2013-07-01 A new transport solution methodology was developed by combining the two-dimensional method of characteristics with the discontinuous Galerkin method for the treatment of the axial variable. The method, which can be applied to arbitrary extruded geometries, was implemented in PROTEUS-MOC and includes parallelization in group, angle, plane, and space using a top level GMRES linear algebra solver. Verification tests were performed to show accuracy and stability of the method with the increased number of angular directions and mesh elements. Good scalability with parallelism in angle and axial planes is displayed. (authors) 10. A hybrid Pade-Galerkin technique for differential equations Science.gov (United States) Geer, James F.; Andersen, Carl M. 1993-01-01 A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter epsilon associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade approximation in the form of a rational function in the parameter epsilon. In the third step, the various powers of epsilon which appear in the Pade approximation are replaced by new (unknown) parameters (delta(sub j)). These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade approximations fail to do so. The method is discussed and topics for future investigations are indicated. 11. An H1(Ph)-coercive discontinuous Galerkin formulation for the Poisson problem: 1D analysis NARCIS (Netherlands) Zee, van der K.G.; Brummelen, van E.H.; Borst, de R. 2006-01-01 Coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite-element discretizations: By the classical Lax–Milgram theorem, any conforming discretization of a coercive variational problem is stable; i.e., discrete approximations are well-posed and possess 12. An H1(Ph)-Coercive Discontinuous Galerkin Formulation for The Poisson Problem : 1-D Analysis NARCIS (Netherlands) Van der Zee, K.G.; Van Brummelen, E.H.; De Borst, R. 2006-01-01 Coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite-element discretizations: By the classical Lax–Milgram theorem, any conforming discretization of a coercive variational problem is stable; i.e., discrete approximations are well-posed and possess 13. High-order discontinuous Galerkin nonlocal transport and energy equations scheme for radiation hydrodynamics Czech Academy of Sciences Publication Activity Database Holec, M.; Limpouch, J.; Liska, R.; Weber, Stefan A. 2017-01-01 Roč. 83, č. 10 (2017), s. 779-797 ISSN 0271-2091 R&D Projects: GA MŠk EF15_008/0000162; GA MŠk LQ1606 Grant - others:ELI Beamlines(XE) CZ.02.1.01/0.0/0.0/15_008/0000162 Institutional support: RVO:68378271 Keywords : radiation hydrodynamics * nonlocal transport * Knudsen number * multigroup diffusion * radiation coupling Subject RIV: BG - Nuclear, Atomic and Molecular Physics, Colliders OBOR OECD: Nuclear physics Impact factor: 1.652, year: 2016 14. High-Order Approximation of Chromatographic Models using a Nodal Discontinuous Galerkin Approach DEFF Research Database (Denmark) Meyer, Kristian; Huusom, Jakob Kjøbsted; Abildskov, Jens 2018-01-01 by Javeed et al. (2011a,b, 2013) with an efficient quadrature-free implementation. The framework is used to simulate linear and non-linear multicomponent chromatographic systems. The results confirm arbitrary high-order accuracy and demonstrate the potential for accuracy and speed-up gains obtainable... 15. Transient Thermal Analysis of 3-D Integrated Circuits Packages by the DGTD Method KAUST Repository Li, Ping; Dong, Yilin; Tang, Min; Mao, Junfa; Jiang, Li Jun; Bagci, Hakan 2017-01-01 Since accurate thermal analysis plays a critical role in the thermal design and management of the 3-D system-level integration, in this paper, a discontinuous Galerkin time-domain (DGTD) algorithm is proposed to achieve this purpose 16. Green's function approach to neutron flux discontinuities International Nuclear Information System (INIS) Saad, E.A.; El-Wakil, S.A. 1980-01-01 The present work is devoted to the presentation of analytical method for the calculation of elastically and inelastically slowed down neutrons in an infinite non-absorbing medium. On the basis of the central limit theory (CLT) and the integral transform technique the slowing down equation including inelastic scattering, in terms of the Green function of elastic scattering, is solved. The Green function is decomposed according to the number of collisions. Placzec discontinuity associated with elastic scattering in addition to two discontinuities due to inelastic scattering are investigated. Numerical calculations for Fe 56 show that the elastic discontinuity produces about 41.8% change in the collision density whilst the ratio of the inelastic collision density discontinuity at qsub(o)sup(+) to the Placzec discontinuity at usub(o) + 1n 1/oc gives 55.7 percent change. (author) 17. AxiSEM3D: a new fast method for global wave propagation in 3-D Earth models with undulating discontinuities Science.gov (United States) Leng, K.; Nissen-Meyer, T.; van Driel, M.; Al-Attar, D. 2016-12-01 We present a new, computationally efficient numerical method to simulate global seismic wave propagation in realistic 3-D Earth models with laterally heterogeneous media and finite boundary perturbations. Our method is a hybrid of pseudo-spectral and spectral element methods (SEM). We characterize the azimuthal dependence of 3-D wavefields in terms of Fourier series, such that the 3-D equations of motion reduce to an algebraic system of coupled 2-D meridional equations, which can be solved by a 2-D spectral element method (based on www.axisem.info). Computational efficiency of our method stems from lateral smoothness of global Earth models (with respect to wavelength) as well as axial singularity of seismic point sources, which jointly confine the Fourier modes of wavefields to a few lower orders. All boundary perturbations that violate geometric spherical symmetry, including Earth's ellipticity, topography and bathymetry, undulations of internal discontinuities such as Moho and CMB, are uniformly considered by means of a Particle Relabeling Transformation.The MPI-based high performance C++ code AxiSEM3D, is now available for forward simulations upon 3-D Earth models with fluid outer core, ellipticity, and both mantle and crustal structures. We show novel benchmarks for global wave solutions in 3-D mantle structures between our method and an independent, fully discretized 3-D SEM with remarkable agreement. Performance comparisons are carried out on three state-of-the-art tomography models, with seismic period going down to 5s. It is shown that our method runs up to two orders of magnitude faster than the 3-D SEM for such settings, and such computational advantage scales favourably with seismic frequency. By examining wavefields passing through hypothetical Gaussian plumes of varying sharpness, we identify in model-wavelength space the limits where our method may lose its advantage. 18. High order spectral difference lattice Boltzmann method for incompressible hydrodynamics Science.gov (United States) Li, Weidong 2017-09-01 This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners-Burggraf flow; and (c) the unsteady Taylor-Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified. 19. A Legendre-pseudospectral method for computing travelling waves with corners (slope discontinuities) in one space dimension with application to Whitham's equation family International Nuclear Information System (INIS) Boyd, John P. 2003-01-01 If the dispersion in a nonlinear hyperbolic wave equation is weak in the sense that the frequency ω(k) of cos(kx) is bounded as k→∞, it is common that (i) travelling waves exist up to a limiting amplitude with wave-breaking for higher amplitudes, and (ii) the limiting wave has a corner, that is, a discontinuity in slope. Because 'corner' waves are not smooth, standard numerical methods converge poorly as the number of grid points is increased. However, the corner wave is important because, at least in some systems, it is the attractor for all large amplitude initial conditions. Here we devise a Legendre-pseudospectral method which is uncorrupted by the singularity. The symmetric (u(X)=u(-X)) wave can be computed on an interval spanning only half the spatial period; since u is smooth on this domain which does not include the corner except as an endpoint, all numerical difficulties are avoided. A key step is to derive an extra boundary condition which uniquely identifies the corner wave. With both the grid point values of u(x) and phase speed c as unknowns, the discretized equations, imposing three boundary conditions on a second order differential equation, are solved by a Newton-Raphson iteration. Although our method is illustrated by the so-called 'Whitham's equation', u t +uu x =∫Du dx ' where D is a very general linear operator, the ideas are widely applicable 20. An Equal-Order DG Method for the Incompressible Navier-Stokes Equations KAUST Repository Cockburn, Bernardo 2008-12-20 We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equal-order approach is ensured by a pressure stabilization term. A simple element-by-element post-processing procedure is used to provide globally divergence-free velocity approximations. For small data, we prove the existence and uniqueness of discrete solutions and carry out an error analysis of the method. A series of numerical results are presented that validate our theoretical findings. © 2008 Springer Science+Business Media, LLC. 1. Factors Influencing Contraceptive Choice and Discontinuation ... African Journals Online (AJOL) Erah 2010-03-30 women indicated that their HIV status dictated contraceptive decisions, particularly with ... Women reported method discontinuation because of side effects, having met desired parity, ...... Washington, D.C., 2009. ... Accessed March 30, 2010. 2. Discontinuity effects in dynamically loaded tilting pad journal bearings DEFF Research Database (Denmark) Thomsen, Kim; Klit, Peder; Vølund, Anders 2011-01-01 This paper describes two discontinuity effects that can occur when modelling radial tilting pad bearings subjected to high dynamic loads. The first effect to be treated is a pressure build-up discontinuity effect. The second effect is a contact-related discontinuity that disappears when a contact...... force is included in the theoretical model. Methods for avoiding the pressure build-up discontinuity effect are proposed.... 3. Adaptive stochastic Galerkin FEM with hierarchical tensor representations KAUST Repository Eigel, Martin 2016-01-08 PDE with stochastic data usually lead to very high-dimensional algebraic problems which easily become unfeasible for numerical computations because of the dense coupling structure of the discretised stochastic operator. Recently, an adaptive stochastic Galerkin FEM based on a residual a posteriori error estimator was presented and the convergence of the adaptive algorithm was shown. While this approach leads to a drastic reduction of the complexity of the problem due to the iterative discovery of the sparsity of the solution, the problem size and structure is still rather limited. To allow for larger and more general problems, we exploit the tensor structure of the parametric problem by representing operator and solution iterates in the tensor train (TT) format. The (successive) compression carried out with these representations can be seen as a generalisation of some other model reduction techniques, e.g. the reduced basis method. We show that this approach facilitates the efficient computation of different error indicators related to the computational mesh, the active polynomial chaos index set, and the TT rank. In particular, the curse of dimension is avoided. 4. Peaks, plateaus, numerical instabilities, and achievable accuracy in Galerkin and norm minimizing procedures for solving Ax=b Energy Technology Data Exchange (ETDEWEB) Cullum, J. [IBM T.J. Watson Research Center, Yorktown Heights, NY (United States) 1994-12-31 Plots of the residual norms generated by Galerkin procedures for solving Ax = b often exhibit strings of irregular peaks. At seemingly erratic stages in the iterations, peaks appear in the residual norm plot, intervals of iterations over which the norms initially increase and then decrease. Plots of the residual norms generated by related norm minimizing procedures often exhibit long plateaus, sequences of iterations over which reductions in the size of the residual norm are unacceptably small. In an earlier paper the author discussed and derived relationships between such peaks and plateaus within corresponding Galerkin/Norm Minimizing pairs of such methods. In this paper, through a set of numerical experiments, the author examines connections between peaks, plateaus, numerical instabilities, and the achievable accuracy for such pairs of iterative methods. Three pairs of methods, GMRES/Arnoldi, QMR/BCG, and two bidiagonalization methods are studied. 5. A strongly conservative finite element method for the coupling of Stokes and Darcy flow KAUST Repository Kanschat, G. 2010-08-01 We consider a model of coupled free and porous media flow governed by Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition. This model is discretized using divergence-conforming finite elements for the velocities in the whole domain. Discontinuous Galerkin techniques and mixed methods are used in the Stokes and Darcy subdomains, respectively. This discretization is strongly conservative in Hdiv(Ω) and we show convergence. Numerical results validate our findings and indicate optimal convergence orders. © 2010 Elsevier Inc. 6. Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients KAUST Repository Beck, Joakim 2011-12-22 In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new effective class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. 7. A cubic B-spline Galerkin approach for the numerical simulation of the GEW equation Directory of Open Access Journals (Sweden) S. Battal Gazi Karakoç 2016-02-01 Full Text Available The generalized equal width (GEW wave equation is solved numerically by using lumped Galerkin approach with cubic B-spline functions. The proposed numerical scheme is tested by applying two test problems including single solitary wave and interaction of two solitary waves. In order to determine the performance of the algorithm, the error norms L2 and L∞ and the invariants I1, I2 and I3 are calculated. For the linear stability analysis of the numerical algorithm, von Neumann approach is used. As a result, the obtained findings show that the presented numerical scheme is preferable to some recent numerical methods. 8. Numerical solution of the Neutron Transport Equation using discontinuous nodal methods at X-Y geometry; Solucion numerica de la ecuacion de transporte de neutrones usando metodos nodales discontinuos en geometria X-Y Energy Technology Data Exchange (ETDEWEB) Delfin L, A 1997-12-31 The purpose of this work is to solve the neutron transport equation in discrete-ordinates and X-Y geometry by developing and using the strong discontinuous and strong modified discontinuous nodal finite element schemes. The strong discontinuous and modified strong discontinuous nodal finite element schemes go from two to ten interpolation parameters per cell. They are describing giving a set D{sub c} and polynomial space S{sub c} corresponding for each scheme BDMO, RTO, BL, BDM1, HdV, BDFM1, RT1, BQ and BDM2. The solution is obtained solving the neutron transport equation moments for each nodal scheme by developing the basis functions defined by Pascal triangle and the Legendre moments giving in the polynomial space S{sub c} and, finally, looking for the non singularity of the resulting linear system. The linear system is numerically solved using a computer program for each scheme mentioned . It uses the LU method and forward and backward substitution and makes a partition of the domain in cells. The source terms and angular flux are calculated, using the directions and weights associated to the S{sub N} approximation and solving the angular flux moments to find the effective multiplication constant. The programs are written in Fortran language, using the dynamic allocation of memory to increase efficiently the available memory of the computing equipment. (Author). 9. Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1 KAUST Repository Jin, Bangti; Lazarov, Raytcho; Pasciak, Joseph; Zhou, Zhi 2013-01-01 We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that Ω ⊂ ℝd , d = 1,2,3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L2- and H1-norms for initial data in H-s (Ω), 0 ≤ s ≤ 1. We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac δ-function supported on a (d-1)-dimensional manifold. © 2013 Springer-Verlag. 10. Discontinuous finite element and characteristics methods for neutrons transport equation solution in heterogeneous grids; Resolution de l'equation du transport des neutrons par les methodes des elements finis discontinus et des caracteristiques structurees appliquees a des maillages heterogenes Energy Technology Data Exchange (ETDEWEB) Masiello, E 2006-07-01 The principal goal of this manuscript is devoted to the investigation of a new type of heterogeneous mesh adapted to the shape of the fuel pins (fuel-clad-moderator). The new heterogeneous mesh guarantees the spatial modelling of the pin-cell with a minimum of regions. Two methods are investigated for the spatial discretization of the transport equation: the discontinuous finite element method and the method of characteristics for structured cells. These methods together with the new representation of the pin-cell result in an appreciable reduction of calculation points. They allow an exact modelling of the fuel pin-cell without spatial homogenization. A new synthetic acceleration technique based on an angular multigrid is also presented for the speed up of the inner iterations. These methods are good candidates for transport calculations for a nuclear reactor core. A second objective of this work is the application of method of characteristics for non-structured geometries to the study of double heterogeneity problem. The letters is characterized by fuel material with a stochastic dispersion of heterogeneous grains, and until now was solved with a model based on collision probabilities. We propose a new statistical model based on renewal-Markovian theory, which makes possible to take into account the stochastic nature of the problem and to avoid the approximations of the collision probability model. The numerical solution of this model is guaranteed by the method of characteristics. (author) 11. Discontinuous finite element and characteristics methods for neutrons transport equation solution in heterogeneous grids; Resolution de l'equation du transport des neutrons par les methodes des elements finis discontinus et des caracteristiques structurees appliquees a des maillages heterogenes Energy Technology Data Exchange (ETDEWEB) Masiello, E 2006-07-01 The principal goal of this manuscript is devoted to the investigation of a new type of heterogeneous mesh adapted to the shape of the fuel pins (fuel-clad-moderator). The new heterogeneous mesh guarantees the spatial modelling of the pin-cell with a minimum of regions. Two methods are investigated for the spatial discretization of the transport equation: the discontinuous finite element method and the method of characteristics for structured cells. These methods together with the new representation of the pin-cell result in an appreciable reduction of calculation points. They allow an exact modelling of the fuel pin-cell without spatial homogenization. A new synthetic acceleration technique based on an angular multigrid is also presented for the speed up of the inner iterations. These methods are good candidates for transport calculations for a nuclear reactor core. A second objective of this work is the application of method of characteristics for non-structured geometries to the study of double heterogeneity problem. The letters is characterized by fuel material with a stochastic dispersion of heterogeneous grains, and until now was solved with a model based on collision probabilities. We propose a new statistical model based on renewal-Markovian theory, which makes possible to take into account the stochastic nature of the problem and to avoid the approximations of the collision probability model. The numerical solution of this model is guaranteed by the method of characteristics. (author) 12. System dynamics with interaction discontinuity CERN Document Server Luo, Albert C J 2015-01-01 This book describes system dynamics with discontinuity caused by system interactions and presents the theory of flow singularity and switchability at the boundary in discontinuous dynamical systems. Based on such a theory, the authors address dynamics and motion mechanism of engineering discontinuous systems due to interaction. Stability and bifurcations of fixed points in nonlinear discrete dynamical systems are presented, and mapping dynamics are developed for analytical predictions of periodic motions in engineering discontinuous dynamical systems. Ultimately, the book provides an alternative way to discuss the periodic and chaotic behaviors in discontinuous dynamical systems. 13. The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture KAUST Repository Azdoud, Yan 2014-04-19 We introduce a framework that adapts local and non-local continuum models to simulate static fracture problems. Non-local models based on the peridynamic theory are promising for the simulation of fracture, as they allow discontinuities in the displacement field. However, they remain computationally expensive. As an alternative, we develop an adaptive coupling technique based on the morphing method to restrict the non-local model adaptively during the evolution of the fracture. The rest of the structure is described by local continuum mechanics. We conduct all simulations in three dimensions, using the relevant discretization scheme in each domain, i.e., the discontinuous Galerkin finite element method in the peridynamic domain and the continuous finite element method in the local continuum mechanics domain. © 2014 Springer-Verlag Berlin Heidelberg. 14. Rotational discontinuities in anisotropic plasmas International Nuclear Information System (INIS) Omidi, N. 1992-01-01 The kinetic structure of rotational discontinuities (RDs) in anisotropic plasmas with T perpendicular /T parallel > 1 is investigated by using a one-dimensional electromagnetic hybrid code. To form the RD, a new approach is used where the plasma is injected from one boundary and reflected from the other, resulting in the generation of a traveling fast shock and an RD. Unlike the previously used methods, no a priori assumptions are made regarding the initial structure (i.e. width or sense of rotation) of the rotational discontinuity. The results show that across the RD both the magnetic field strength and direction, as well as the plasma density change. Given that such a change can also be associated with an intermediate shock, the Rankine-Hugoniot relations are used to confirm that the observed structures are indeed RDs. It is found that the thickness of RDs is a few ion inertial lengths and is independent of the rotation angle. Also, the preferred sense of rotation is in the electron sense; however, RDs with a rotation angle larger than 180 degree are found to be unstable, changing their rotation to a stable ion sense 15. Galerkin approximations of nonlinear optimal control problems in Hilbert spaces Directory of Open Access Journals (Sweden) Mickael D. Chekroun 2017-07-01 Full Text Available Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated here in terms of optimal control of energy balance climate models posed on the sphere\\mathbb{S}^2\$.

16. A Galerkin discretisation-based identification for parameters in nonlinear mechanical systems

Science.gov (United States)

Liu, Zuolin; Xu, Jian

2018-04-01

In the paper, a new parameter identification method is proposed for mechanical systems. Based on the idea of Galerkin finite-element method, the displacement over time history is approximated by piecewise linear functions, and the second-order terms in model equation are eliminated by integrating by parts. In this way, the lost function of integration form is derived. Being different with the existing methods, the lost function actually is a quadratic sum of integration over the whole time history. Then for linear or nonlinear systems, the optimisation of the lost function can be applied with traditional least-squares algorithm or the iterative one, respectively. Such method could be used to effectively identify parameters in linear and arbitrary nonlinear mechanical systems. Simulation results show that even under the condition of sparse data or low sampling frequency, this method could still guarantee high accuracy in identifying linear and nonlinear parameters.

17. Petrov-Galerkin mixed formulations for bidimensional elasticity

International Nuclear Information System (INIS)

Toledo, E.M.; Loula, A.F.D.; Guerreiro, J.N.C.

1989-10-01

A new formulation for two-dimensional elasticity in stress and displacements is presented. Consistently adding to the Galerkin classical formulation residuals forms of constitutive and equilibrium equations, the original saddle point is transformed into a minimization problem without any restrictions. We also propose a stress post processing technique using both equilibrium and constitutive equations. Numerical analysis error estimates and numerical results are presented confirming the predicted rates of convergence. (A.C.A.S.) [pt

18. A Galerkin approximation for linear elastic shallow shells

Science.gov (United States)

Figueiredo, I. N.; Trabucho, L.

1992-03-01

This work is a generalization to shallow shell models of previous results for plates by B. Miara (1989). Using the same basis functions as in the plate case, we construct a Galerkin approximation of the three-dimensional linearized elasticity problem, and establish some error estimates as a function of the thickness, the curvature, the geometry of the shell, the forces and the Lamé costants.

19. Model Adaptation in Parametric Space for POD-Galerkin Models

Science.gov (United States)

Gao, Haotian; Wei, Mingjun

2017-11-01

The development of low-order POD-Galerkin models is largely motivated by the expectation to use the model developed with a set of parameters at their native values to predict the dynamic behaviors of the same system under different parametric values, in other words, a successful model adaptation in parametric space. However, most of time, even small deviation of parameters from their original value may lead to large deviation or unstable results. It has been shown that adding more information (e.g. a steady state, mean value of a different unsteady state, or an entire different set of POD modes) may improve the prediction of flow with other parametric states. For a simple case of the flow passing a fixed cylinder, an orthogonal mean mode at a different Reynolds number may stabilize the POD-Galerkin model when Reynolds number is changed. For a more complicated case of the flow passing an oscillatory cylinder, a global POD-Galerkin model is first applied to handle the moving boundaries, then more information (e.g. more POD modes) is required to predicate the flow under different oscillatory frequencies. Supported by ARL.

20. Discontinuation Decision in Assisted Reproductive Techniques

Directory of Open Access Journals (Sweden)

Ashraf Moini

2009-01-01

Full Text Available Background: In vitro fertilization (IVF and intra cytoplasmic sperm injection (ICSI are recognizedas established and increasingly successful forms of treatment for infertility, yet significant numbersof couples discontinue treatment without achieving a live birth. This study aims to identify majorfactors that influence the decision to discontinue IVF/ICSI treatments.Materials and Methods: We studied the data of 338 couples who discontinued their infertilitytreatments after three cycles; based on medical records and phone contact. The main measure wasthe reason for stopping their treatments.Results: Economical problems were cited by 212 couples (62.7%, as their mean income wassignificantly less than other couples (p<0.0001. Lack of success was reported as a reason by229 (67.8%, from whom 165 (72% also had economical problems. Achieving independent-ART pregnancy was the reason for discontinuation in 20 (5.9% couples. Psychological stress,depression and anxiety were reported as other cessation factors by 169 (50%, 148 (43.8% and 182(53.8% couples, respectively.Conclusion: This survey suggests that the most common reasons for assisted reproductivetechnique (ART discontinuation after three cycles are: prior unsuccessful cycles, economicaland psychological problems. Therefore, the substantial proportion of couples could benefit frompsychological intervention, increasing awareness of ART outcomes and health funding to copemore adequately with failed treatments.

1. Stabilities of MHD rotational discontinuities

International Nuclear Information System (INIS)

Wang, S.

1984-11-01

In this paper, the stabilities of MHD rotational discontinuities are analyzed. The results show that the rotational discontinuities in an incompressible magnetofluid are not always stable with respect to infinitesimal perturbation. The instability condition in a special case is obtained. (author)

2. Discontinuity formulas for multiparticle amplitudes

International Nuclear Information System (INIS)

Stapp, H.P.

1976-03-01

It is shown how discontinuity formulas for multiparticle scattering amplitudes are derived from unitarity and analyticity. The assumed analyticity property is the normal analytic structure, which was shown to be equivalent to the space-time macrocausality condition. The discontinuity formulas to be derived are the basis of multi-particle fixed-t dispersion relations

3. Physics-based process model approach for detecting discontinuity during friction stir welding

Energy Technology Data Exchange (ETDEWEB)

Shrivastava, Amber; Pfefferkorn, Frank E.; Duffie, Neil A.; Ferrier, Nicola J.; Smith, Christopher B.; Malukhin, Kostya; Zinn, Michael

2015-02-12

The goal of this work is to develop a method for detecting the creation of discontinuities during friction stir welding. This in situ weld monitoring method could significantly reduce the need for post-process inspection. A process force model and a discontinuity force model were created based on the state-of-the-art understanding of flow around an friction stir welding (FSW) tool. These models are used to predict the FSW forces and size of discontinuities formed in the weld. Friction stir welds with discontinuities and welds without discontinuities were created, and the differences in force dynamics were observed. In this paper, discontinuities were generated by reducing the tool rotation frequency and increasing the tool traverse speed in order to create "cold" welds. Experimental force data for welds with discontinuities and welds without discontinuities compared favorably with the predicted forces. The model currently overpredicts the discontinuity size.

4. Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets

Directory of Open Access Journals (Sweden)

2017-10-01

Full Text Available ‎Stochastic fractional differential equations (SFDEs have been used for modeling many physical problems in the fields of turbulance‎, ‎heterogeneous‎, ‎flows and matrials‎, ‎viscoelasticity and electromagnetic theory‎. ‎In this paper‎, ‎an‎ efficient wavelet Galerkin method based on the second kind Chebyshev wavelets are proposed for approximate solution of SFDEs‎. ‎In ‎this ‎app‎roach‎‎, ‎o‎perational matrices of the second kind Chebyshev wavelets ‎are used ‎for reducing SFDEs to a linear system of algebraic equations that can be solved easily‎. ‎C‎onvergence and error analysis of the proposed method is ‎considered‎.‎ ‎Some numerical examples are performed to confirm the applicability and efficiency of the proposed method‎.

5. A higher order space-time Galerkin scheme for time domain integral equations

KAUST Repository

Pray, Andrew J.; Beghein, Yves; Nair, Naveen V.; Cools, Kristof; Bagci, Hakan; Shanker, Balasubramaniam

2014-01-01

Stability of time domain integral equation (TDIE) solvers has remained an elusive goal formany years. Advancement of this research has largely progressed on four fronts: 1) Exact integration, 2) Lubich quadrature, 3) smooth temporal basis functions, and 4) space-time separation of convolutions with the retarded potential. The latter method's efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was previously reported for first-order surface descriptions (flat elements) and zeroth-order functions as the temporal basis. In this work, we develop the methodology necessary to extend the scheme to higher order surface descriptions as well as to enable its use with higher order basis functions in both space and time. These basis functions are then used in a space-time Galerkin framework. A number of results are presented that demonstrate convergence in time. The viability of the space-time separation method in producing stable results is demonstrated experimentally for these examples.

6. 22nd International Conference on Domain Decomposition Methods

CERN Document Server

Gander, Martin; Halpern, Laurence; Krause, Rolf; Pavarino, Luca

2016-01-01

These are the proceedings of the 22nd International Conference on Domain Decomposition Methods, which was held in Lugano, Switzerland. With 172 participants from over 24 countries, this conference continued a long-standing tradition of internationally oriented meetings on Domain Decomposition Methods. The book features a well-balanced mix of established and new topics, such as the manifold theory of Schwarz Methods, Isogeometric Analysis, Discontinuous Galerkin Methods, exploitation of modern HPC architectures, and industrial applications. As the conference program reflects, the growing capabilities in terms of theory and available hardware allow increasingly complex non-linear and multi-physics simulations, confirming the tremendous potential and flexibility of the domain decomposition concept.

7. Galerkin algorithm for multidimensional plasma simulation codes. Informal report

International Nuclear Information System (INIS)

Godfrey, B.B.

1979-03-01

A Galerkin finite element differencing scheme has been developed for a computer simulation of plasmas. The new difference equations identically satisfy an equation of continuity. Thus, the usual current correction procedure, involving inversion of Poisson's equation, is unnecessary. The algorithm is free of many numerical Cherenkov instabilities. This differencing scheme has been implemented in CCUBE, an already existing relativistic, electromagnetic, two-dimensional PIC code in arbitrary separable, orthogonal coordinates. The separability constraint is eliminated by the new algorithm. The new version of CCUBE exhibits good stability and accuracy with reduced computer memory and time requirements. Details of the algorithm and its implementation are presented

8. Determining the Locations and Discontinuities in the Derivatives of Functions

International Nuclear Information System (INIS)

Archibald, Richard K.; Gelb, Anne; Yoon, Jungho

2007-01-01

We introduce a method for detecting discontinuities in piecewise smooth functions and in their derivatives. The method is constructed from a local stencil of grid point values and is based on a polynomial annihilation technique. By varying the order of the method and the arrangement of the corresponding stencils, the jump discontinuities of a function and its derivatives can be identified with high order accuracy. The method is efficient and robust and can be applied to non-uniform distributions in one dimension

9. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media

KAUST Repository

Wang, Yi

2017-01-25

Fast prediction modeling via proper orthogonal decomposition method combined with Galerkin projection is applied to incompressible single-phase fluid flow in porous media. Cases for different configurations of porous media, boundary conditions and problem scales are designed to examine the fidelity and robustness of the model. High precision (relative deviation 1.0 x 10(-4)% similar to 2.3 x 10(-1)%) and large acceleration (speed-up 880 similar to 98454 times) of POD model are found in these cases. Moreover, the computational time of POD model is quite insensitive to the complexity of problems. These results indicate POD model is especially suitable for large-scale complex problems in engineering.

10. Application of stochastic Galerkin FEM to the complete electrode model of electrical impedance tomography

International Nuclear Information System (INIS)

Leinonen, Matti; Hakula, Harri; Hyvönen, Nuutti

2014-01-01

The aim of electrical impedance tomography is to determine the internal conductivity distribution of some physical body from boundary measurements of current and voltage. The most accurate forward model for impedance tomography is the complete electrode model, which consists of the conductivity equation coupled with boundary conditions that take into account the electrode shapes and the contact resistances at the corresponding interfaces. If the reconstruction task of impedance tomography is recast as a Bayesian inference problem, it is essential to be able to solve the complete electrode model forward problem with the conductivity and the contact resistances treated as a random field and random variables, respectively. In this work, we apply a stochastic Galerkin finite element method to the ensuing elliptic stochastic boundary value problem and compare the results with Monte Carlo simulations

11. A Simple Stochastic Differential Equation with Discontinuous Drift

DEFF Research Database (Denmark)

Simonsen, Maria; Leth, John-Josef; Schiøler, Henrik

2013-01-01

In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. We apply two approaches: The Euler-Maruyama method and the Fokker-Planck equation and show that a candidate density function based on the Euler-Maruyama method approximates a candidate density...... function based on the stationary Fokker-Planck equation. Furthermore, we introduce a smooth function which approximates the discontinuous drift and apply the Euler-Maruyama method and the Fokker-Planck equation with this input. The point of departure for this work is a particular SDE with discontinuous...

12. Application of meshless EFG method in fluid flow problems

Meshless method; element-free Galerkin method; steady state analysis; transient ... ﬂuid ﬂow problems using the meshless element-free Galerkin method. The unknown function of velocity u ( x ) is approximated by moving least square ...

13. Automatic extraction of discontinuity orientation from rock mass surface 3D point cloud

Science.gov (United States)

Chen, Jianqin; Zhu, Hehua; Li, Xiaojun

2016-10-01

This paper presents a new method for extracting discontinuity orientation automatically from rock mass surface 3D point cloud. The proposed method consists of four steps: (1) automatic grouping of discontinuity sets using an improved K-means clustering method, (2) discontinuity segmentation and optimization, (3) discontinuity plane fitting using Random Sample Consensus (RANSAC) method, and (4) coordinate transformation of discontinuity plane. The method is first validated by the point cloud of a small piece of a rock slope acquired by photogrammetry. The extracted discontinuity orientations are compared with measured ones in the field. Then it is applied to a publicly available LiDAR data of a road cut rock slope at Rockbench repository. The extracted discontinuity orientations are compared with the method proposed by Riquelme et al. (2014). The results show that the presented method is reliable and of high accuracy, and can meet the engineering needs.

14. Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

International Nuclear Information System (INIS)

Gao, Kai; Fu, Shubin; Gibson, Richard L.; Chung, Eric T.; Efendiev, Yalchin

2015-01-01

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale medium property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system

15. Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

Energy Technology Data Exchange (ETDEWEB)

Gao, Kai, E-mail: kaigao87@gmail.com [Department of Geology and Geophysics, Texas A& M University, College Station, TX 77843 (United States); Fu, Shubin, E-mail: shubinfu89@gmail.com [Department of Mathematics, Texas A& M University, College Station, TX 77843 (United States); Gibson, Richard L., E-mail: gibson@tamu.edu [Department of Geology and Geophysics, Texas A& M University, College Station, TX 77843 (United States); Chung, Eric T., E-mail: tschung@math.cuhk.edu.hk [Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT (Hong Kong); Efendiev, Yalchin, E-mail: efendiev@math.tamu.edu [Department of Mathematics, Texas A& M University, College Station, TX 77843 (United States); Numerical Porous Media SRI Center (NumPor), King Abdullah University of Science and Technology, Thuwal (Saudi Arabia)

2015-08-15

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale medium property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.

16. Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

KAUST Repository

Gao, Kai

2015-04-14

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both boundaries and the interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale medium property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.

17. Management applications of discontinuity theory

Science.gov (United States)

Angeler, David G.; Allen, Craig R.; Barichievy, Chris; Eason, Tarsha; Garmestani, Ahjond S.; Graham, Nicholas A.J.; Granholm, Dean; Gunderson, Lance H.; Knutson, Melinda; Nash, Kirsty L.; Nelson, R. John; Nystrom, Magnus; Spanbauer, Trisha; Stow, Craig A.; Sundstrom, Shana M.

2015-01-01

Human impacts on the environment are multifaceted and can occur across distinct spatiotemporal scales. Ecological responses to environmental change are therefore difficult to predict, and entail large degrees of uncertainty. Such uncertainty requires robust tools for management to sustain ecosystem goods and services and maintain resilient ecosystems.We propose an approach based on discontinuity theory that accounts for patterns and processes at distinct spatial and temporal scales, an inherent property of ecological systems. Discontinuity theory has not been applied in natural resource management and could therefore improve ecosystem management because it explicitly accounts for ecological complexity.Synthesis and applications. We highlight the application of discontinuity approaches for meeting management goals. Specifically, discontinuity approaches have significant potential to measure and thus understand the resilience of ecosystems, to objectively identify critical scales of space and time in ecological systems at which human impact might be most severe, to provide warning indicators of regime change, to help predict and understand biological invasions and extinctions and to focus monitoring efforts. Discontinuity theory can complement current approaches, providing a broader paradigm for ecological management and conservation.

18. Stacking by electroinjection with discontinuous buffers in capillary zone electrophoresis.

Science.gov (United States)

Shihabi, Zak K

2002-08-01

The work presented here demonstrates that electroinjection can be performed using discontinuous buffers, which can result in better stacking than that obtained by hydrodynamic injection. The sample can be concentrated at the tip of the capillary leaving practically the whole capillary for sample separation. This results in several advantages, such as better sample concentration, higher plate number and shorter time of stacking. However, sample introduction by electromigration is suited for samples free or low in salt content. Samples, which are high in salt content, are better introduced by the hydrodynamic injection for stacking by the discontinuous buffers. Different simple methods to introduce the discontinuity in the buffer for electroinjection are discussed.

19. Discontinuities and the magnetospheric phenomena

International Nuclear Information System (INIS)

Rajaram, R.; Kalra, G.L.; Tandon, J.N.

1978-01-01

Wave coupling at contact discontinuities has an important bearing on the transmission of waves from the solar wind into the magnetosphere across the cusp region of the solar wind-magnetosphere boundary and on the propagation of geomagnetic pulsations in the polar exosphere. Keeping this in view, the problems of wave coupling across a contact discontinuity in a collisionless plasma, described by a set of double adiabatic fluid equations, is examined. The magnetic field is taken normal to the interface and it is shown that total reflection is not possible for any angle of incidence. The Alfven and the magneto-acoustic waves are not coupled. The transmission is most efficient for small density discontinuities. Inhibition of the transmission of the Alfven wave by the sharp density gradients above the F2-peak in the polar exosphere appears to account for the decrease in the pulsation amplitude, on the ground, as the poles are approached from the auroral zone. (author)

20. The structure of rotational discontinuities

International Nuclear Information System (INIS)

Neugebauer, M.

1989-01-01

This study examines the structures of a set of rotational discontinuities detected in the solar wind by the ISEE-3 spacecraft. It is found that the complexity of the structure increases as the angle θ between the propagation vector k and the magnetic field decreases. For rotational discontinuities that propagate at a large angle to the field with an ion (left-hand) sense of rotation, the magnetic hodograms tend to be flattened, in agreement with prior numerical simulations. When θ is large, angular overshoots are often observed at one or both ends of the discontinuity. When the propagation is nearly parallel to the field (i.e., when θ is small), many different types of structure are seen, ranging from straight lines, the S-shaped curves, to complex, disorganized shapes

1. Socio-Economic Differentials in Contraceptive Discontinuation in India

Directory of Open Access Journals (Sweden)

Kiran Agrahari

2016-05-01

Full Text Available Fertility divergence amid declining in use of modern contraception in many states of India needs urgent research and programmatic attention. Although utilization of antenatal, natal, and post-natal care has shown spectacular increase in post National Rural Health Mission (NRHM period, the contraceptive use had shown a declining trend. Using the calendar data from the National Family Health Survey–3, this article examines the reasons of contraceptive discontinuation among spacing method users by socio-economic groups in India. Bivariate and multivariate analyses and life table discontinuation rates are used in the analyses. Results suggest that about half of the pill users, two fifths of the condom users, one third of traditional method users, and one fifth of IUD users discontinue a method in first 12 months of use. However, the discontinuation of all three modern spacing methods declines in subsequent period (within 12-36 months. The probability of method failure was highest among traditional method users and higher among poor and less educated that may lead to unwanted/mistimed birth. Although discontinuation of condom declines with economic status, it does not show any large variation for pill users. The contraceptive discontinuation was significantly associated with duration of use, age, parity, contraceptive method, religion, and contraceptive intention. Based on these findings, it is suggested that follow-up services to modern spacing method users, increasing counseling for spacing method users, motivating the traditional method user to use modern spacing method, and improving the overall quality of family planning services can reduce the discontinuation of spacing method.

2. A collocation--Galerkin finite element model of cardiac action potential propagation.

Science.gov (United States)

Rogers, J M; McCulloch, A D

1994-08-01

A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, we observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques.

3. A mixed Fourier–Galerkin–finite-volume method to solve the fluid dynamics equations in cylindrical geometries

International Nuclear Information System (INIS)

Núñez, Jóse; Ramos, Eduardo; Lopez, Juan M

2012-01-01

We describe a hybrid method based on the combined use of the Fourier Galerkin and finite-volume techniques to solve the fluid dynamics equations in cylindrical geometries. A Fourier expansion is used in the angular direction, partially translating the problem to the Fourier space and then solving the resulting equations using a finite-volume technique. We also describe an algorithm required to solve the coupled mass and momentum conservation equations similar to a pressure-correction SIMPLE method that is adapted for the present formulation. Using the Fourier–Galerkin method for the azimuthal direction has two advantages. Firstly, it has a high-order approximation of the partial derivatives in the angular direction, and secondly, it naturally satisfies the azimuthal periodic boundary conditions. Also, using the finite-volume method in the r and z directions allows one to handle boundary conditions with discontinuities in those directions. It is important to remark that with this method, the resulting linear system of equations are band-diagonal, leading to fast and efficient solvers. The benefits of the mixed method are illustrated with example problems. (paper)

4. Discrete modeling of multiple discontinuities in rock mass using XFEM

OpenAIRE

Das, Kamal C.; Ausas, Roberto Federico; Carol, Ignacio; Rodrigues, Eduardo; Sandeep, Sandra; Vargas, P. E.; Gonzalez, Nubia Aurora; Segura, Josep María; Lakshmikantha, Ramasesha Mookanahallipatna; Mello,, U.

2017-01-01

Modeling of discontinuities (fractures and fault surfaces) is of major importance to assess the geomechanical behavior of oil and gas reservoirs, especially for tight and unconventional reservoirs. Numerical analysis of discrete discontinuities traditionally has been studied using interface element concepts, however more recently there are attempts to use extended finite element method (XFEM). The development of an XFEM tool for geo-mechanical fractures/faults modeling has significant industr...

5. Regge calculus from discontinuous metrics

International Nuclear Information System (INIS)

Khatsymovsky, V.M.

2003-01-01

Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface defined by continuity conditions. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the δ-function-like phase factor. The requirement that continuity conditions be imposed in a 'face-independent' way fixes this factor uniquely. The term 'face-independent' means that this factor depends only on the (hyper)plane spanned by the face, not on it's form and size. This requirement seems to be natural from the viewpoint of existence of the well-defined continuum limit maximally free of lattice artefacts

6. Efficient numerical methods for fluid- and electrodynamics on massively parallel systems

Energy Technology Data Exchange (ETDEWEB)

Zudrop, Jens

2016-07-01

In the last decade, computer technology has evolved rapidly. Modern high performance computing systems offer a tremendous amount of computing power in the range of a few peta floating point operations per second. In contrast, numerical software development is much slower and most existing simulation codes cannot exploit the full computing power of these systems. Partially, this is due to the numerical methods themselves and partially it is related to bottlenecks within the parallelization concept and its data structures. The goal of the thesis is the development of numerical algorithms and corresponding data structures to remedy both kinds of parallelization bottlenecks. The approach is based on a co-design of the numerical schemes (including numerical analysis) and their realizations in algorithms and software. Various kinds of applications, from multicomponent flows (Lattice Boltzmann Method) to electrodynamics (Discontinuous Galerkin Method) to embedded geometries (Octree), are considered and efficiency of the developed approaches is demonstrated for large scale simulations.

7. A higher order space-time Galerkin scheme for time domain integral equations

KAUST Repository

Pray, Andrew J.

2014-12-01

Stability of time domain integral equation (TDIE) solvers has remained an elusive goal formany years. Advancement of this research has largely progressed on four fronts: 1) Exact integration, 2) Lubich quadrature, 3) smooth temporal basis functions, and 4) space-time separation of convolutions with the retarded potential. The latter method\\'s efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was previously reported for first-order surface descriptions (flat elements) and zeroth-order functions as the temporal basis. In this work, we develop the methodology necessary to extend the scheme to higher order surface descriptions as well as to enable its use with higher order basis functions in both space and time. These basis functions are then used in a space-time Galerkin framework. A number of results are presented that demonstrate convergence in time. The viability of the space-time separation method in producing stable results is demonstrated experimentally for these examples.

8. Variational Multiscale Finite Element Method for Flows in Highly Porous Media

KAUST Repository

Iliev, O.; Lazarov, R.; Willems, J.

2011-01-01

We present a two-scale finite element method (FEM) for solving Brinkman's and Darcy's equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes' equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy's equations. In order to reduce the "resonance error" and to ensure convergence to the global fine solution, the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems. © 2011 Society for Industrial and Applied Mathematics.

9. Variational Multiscale Finite Element Method for Flows in Highly Porous Media

KAUST Repository

Iliev, O.

2011-10-01

We present a two-scale finite element method (FEM) for solving Brinkman\\'s and Darcy\\'s equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes\\' equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy\\'s equations. In order to reduce the "resonance error" and to ensure convergence to the global fine solution, the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems. © 2011 Society for Industrial and Applied Mathematics.

10. Level set methods for detonation shock dynamics using high-order finite elements

Energy Technology Data Exchange (ETDEWEB)

Dobrev, V. A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Grogan, F. C. [Univ. of California, San Diego, CA (United States); Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Kolev, T. V. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Rieben, R [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Tomov, V. Z. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2017-05-26

Level set methods are a popular approach to modeling evolving interfaces. We present a level set ad- vection solver in two and three dimensions using the discontinuous Galerkin method with high-order nite elements. During evolution, the level set function is reinitialized to a signed distance function to maintain ac- curacy. Our approach leads to stable front propagation and convergence on high-order, curved, unstructured meshes. The ability of the solver to implicitly track moving fronts lends itself to a number of applications; in particular, we highlight applications to high-explosive (HE) burn and detonation shock dynamics (DSD). We provide results for two- and three-dimensional benchmark problems as well as applications to DSD.

11. Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

Science.gov (United States)

Mabuza, Sibusiso; Shadid, John N.; Kuzmin, Dmitri

2018-05-01

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank-Nicolson scheme and backward Euler scheme are utilized.

12. What happens when people discontinue taking medications? Lessons from COMBINE.

Science.gov (United States)

Stout, Robert L; Braciszewski, Jordan M; Subbaraman, Meenakshi Sabina; Kranzler, Henry R; O'Malley, Stephanie S; Falk, Daniel

2014-12-01

We use intensive longitudinal data methods to illuminate processes affecting patients' drinking in relation to the discontinuation of medications within an alcohol treatment study. Although previous work has focused on broad measures of medication adherence, we focus on dynamic changes in drinking both before and after patients discontinue. We conducted secondary data analyses using the COMBINE (Combined Pharmacotherapies and Behavioral Interventions for Alcohol Dependence) study, focused on participants who discontinued medications prior to the planned end of treatment. Using an interrupted time-series analysis, we analysed drinking in the weeks before and after discontinuation and also studied outcomes at the end of the COMBINE follow-up. Unites States of America. We describe the subsample of COMBINE participants who discontinued medications (n = 450), and compare them with those who were medication-adherent (n = 559) and with those who discontinued but had substantial missing data (n = 217). The primary outcomes were percentage of days abstinent (PDA) and percentage of heavy drinking days (PHDD). Medication adherence data were used to approximate the date of discontinuation. For many patients, an increase in drinking began weeks before discontinuation (PDA: F(1,4803) = 19.07, P < 0.001; PHDD: F(1,4804) = 8.58, P = 0.003) then escalated at discontinuation (PDA: F(1,446) = 5.05, P = 0.025; PHDD: F(1,446) = 4.52, P = 0.034). Among other effects, the amount of change was moderated by the reason for discontinuation (e.g. adverse event; PDA: F(2,4803) = 3.85, P = 0.021; PHDD: F(2,4804) = 5.36, P = 0.005) and also whether it occurred in the first or second half of treatment (PDA: F(1,4803) = 5.23, P = 0.022; PHDD: F(1,4804) = 8.79, P = 0.003). A patient's decision to stop taking medications during alcohol treatment appears to take place during a weeks-long process of disengagement from treatment. Patients who discontinue medications early in treatment or without

13. Factors affecting IUCD discontinuation in Nepal

DEFF Research Database (Denmark)

Thapa, Subash; Paudel, Ishwari Sharma; Bhattarai, Sailesh

2015-01-01

Information related to contraception discontinuation, especially in the context of Nepal is very limited. A nested case-control study was carried out to determine the factors affecting discontinuation of intrauterine contraceptive devices (IUCDs). A total of 115 cases (IUCD discontinuers) and 115...

14. The Effect of Levothyroxine Discontinuation Timing on Postoperative Hypothyroidism after Hemithyroidectomy for Papillary Thyroid Microcarcinoma

Directory of Open Access Journals (Sweden)

Tae Kwun Ha

2016-01-01

Full Text Available Objective. No previous studies regarding the appropriate timing of thyroid hormone discontinuation after hemithyroidectomy have been published. This study aimed to identify the appropriate timing for levothyroxine discontinuation after hemithyroidectomy among patients with papillary thyroid microcarcinoma (PTMC. Methods. This study retrospectively evaluated 304 patients who underwent ≥1 attempt to discontinue levothyroxine after hemithyroidectomy for treating PTMC between January 2008 and December 2013. Fifty-three patients were excluded because of preoperative hypothyroidism or hyperthyroidism, a history of thyroid hormone or antithyroid therapy, no available serological data, or a postoperative follow-up of <24 months. We evaluated the associations of successful levothyroxine discontinuation with patient age, sex, preoperative serological data, underlying thyroid gland histopathology, anteroposterior diameter of the residual thyroid gland, number of discontinuation attempts, and initial discontinuation timing. Results. Among the 251 included patients, 125 patients (49.8% achieved successful levothyroxine discontinuation during the follow-up period after hemithyroidectomy. There was a significant difference in the outcomes for patients who underwent an initial discontinuation attempt at ≤3 months and ≥4 months after hemithyroidectomy (p < 0.001. There were significant differences in the discontinuation outcomes according to underlying thyroid histopathology (p = 0.001, preoperative thyroid-stimulating hormone levels (p < 0.001, and number of discontinuation attempts (p < 0.001. Conclusions. Among patients with PTMC, the initial levothyroxine discontinuation attempt is recommended at ≥4 months after hemithyroidectomy.

15. Quantitative Estimation of Transmitted and Reflected Lamb Waves at Discontinuity

International Nuclear Information System (INIS)

Lim, Hyung Jin; Sohn, Hoon

2010-01-01

For the application of Lamb wave to structural health monitoring(SHM), understanding its physical characteristic and interaction between Lamb wave and defect of the host structure is an important issue. In this study, reflected, transmitted and mode converted Lamb waves at discontinuity of a plate structure were simulated and the amplitude ratios are calculated theoretically using Modal decomposition method. The predicted results were verified comparing with finite element method(FEM) and experimental results simulating attached PZTs. The result shows that the theoretical prediction is close to the FEM and the experimental verification. Moreover, quantitative estimation method was suggested using amplitude ratio of Lamb wave at discontinuity

16. A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations

Science.gov (United States)

Zhang, Guoyu; Huang, Chengming; Li, Meng

2018-04-01

We consider the numerical simulation of the coupled nonlinear space fractional Schrödinger equations. Based on the Galerkin finite element method in space and the Crank-Nicolson (CN) difference method in time, a fully discrete scheme is constructed. Firstly, we focus on a rigorous analysis of conservation laws for the discrete system. The definitions of discrete mass and energy here correspond with the original ones in physics. Then, we prove that the fully discrete system is uniquely solvable. Moreover, we consider the unconditionally convergent properties (that is to say, we complete the error estimates without any mesh ratio restriction). We derive L2-norm error estimates for the nonlinear equations and L^{∞}-norm error estimates for the linear equations. Finally, some numerical experiments are included showing results in agreement with the theoretical predictions.

17. Discontinuous approximate molecular electronic wave-functions

International Nuclear Information System (INIS)

Stuebing, E.W.; Weare, J.H.; Parr, R.G.

1977-01-01

Following Kohn, Schlosser and Marcus and Weare and Parr an energy functional is defined for a molecular problem which is stationary in the neighborhood of the exact solution and permits the use of trial functions that are discontinuous. The functional differs from the functional of the standard Rayleigh--Ritz method in the replacement of the usual kinetic energy operators circumflex T(μ) with operators circumflex T'(μ) = circumflex T(μ) + circumflex I(μ) generates contributions from surfaces of nonsmooth behavior. If one uses the nabla PSI . nabla PSI way of writing the usual kinetic energy contributions, one must add surface integrals of the product of the average of nabla PSI and the change of PSI across surfaces of discontinuity. Various calculations are carried out for the hydrogen molecule-ion and the hydrogen molecule. It is shown that ab initio calculations on molecules can be carried out quite generally with a basis of atomic orbitals exactly obeying the zero-differential overlap (ZDO) condition, and a firm basis is thereby provided for theories of molecular electronic structure invoking the ZDO aoproximation. It is demonstrated that a valence bond theory employing orbitals exactly obeying ZDO can provide an adequate account of chemical bonding, and several suggestions are made regarding molecular orbital methods

18. On the stability of rotational discontinuities

International Nuclear Information System (INIS)

Richter, P.; Scholer, M.

1989-01-01

The stability of symmetric rotational discontinuities in which the magnetic field rotates by 180 degree is investigated by means of a one-dimensional self-consistent hybrid code. Rotational discontinuities with an angle Θ > 45 degree between the discontinuity normal direction and the upstream magnetic field are found to be relatively stable. The discontinuity normal is in the x direction and the initial magnetic field has finite y component only in the transition region. In the case of the ion (left-handed) sense of rotation of the tangential magnetic field, the transition region does not broaden with time. In the case of the electron (right-handed) sense of rotation, a damped wavetrain builds up in the B y component downstream of the rotational discontinuity and the discontinuity broadens with time. Rotational discontinuities with smaller angles, Θ, are unstable. Examples for a rotational discontinuity with Θ = 30 degree and the electron sense of rotation as well as a rotational discontinuity with Θ = 15 degree and the ion sense of rotation show that these discontinuities into waves. These waves travel approximately with Alfven velocity in the upstream direction and are therefore phase standing in the simulation system. The magnetic hodograms of these disintegrated discontinuities are S-shaped. The upstream portion of the hodogram is always right-handed; the downstream portion is always left-handed

19. General Practitioners’ Decisions about Discontinuation of Medication

DEFF Research Database (Denmark)

Nixon, Michael Simon; Vendelø, Morten Thanning

2016-01-01

insights about decision making when discontinuing medication. It also offers one of the first examinations of how the institutional context embedding GPs influences their decisions about discontinuation. For policymakers interested in the discontinuation of medication, the findings suggest that de......Purpose – The purpose of this paper is to investigate how general practitioners’ (GPs) decisions about discontinuation of medication are influenced by their institutional context. Design/methodology/approach – In total, 24 GPs were interviewed, three practices were observed and documents were...... a weak frame for discontinuation. Three reasons for this are identified: the guidelines provide dominating triggers for prescribing, they provide weak priming for discontinuation as an option, and they underscore a cognitive constraint against discontinuation. Originality/value – The analysis offers new...

20. Recent Developments on Discontinuous Precipitation

Directory of Open Access Journals (Sweden)

Zięba P.

2017-06-01

Full Text Available The discontinuous precipitation (DP belongs to a group of diffusive solid state phase transformations during which the formation of a new phase is heterogeneous and limited to a migrating reaction front (RF. The use of analytical electron microscopy provided reliable information that there is no differences in the diffusion rate at the stationary grain boundary and moving RF of DP reaction. On the other hand, the use of “in situ” transmission electron microscopy observations indicated the importance of stop-go motion or oscillatory movement of the RF.

1. Adaptive spacetime method using Riemann jump conditions for coupled atomistic-continuum dynamics

International Nuclear Information System (INIS)

Kraczek, B.; Miller, S.T.; Haber, R.B.; Johnson, D.D.

2010-01-01

We combine the Spacetime Discontinuous Galerkin (SDG) method for elastodynamics with the mathematically consistent Atomistic Discontinuous Galerkin (ADG) method in a new scheme that concurrently couples continuum and atomistic models of dynamic response in solids. The formulation couples non-overlapping continuum and atomistic models across sharp interfaces by weakly enforcing jump conditions, for both momentum balance and kinematic compatibility, using Riemann values to preserve the characteristic structure of the underlying hyperbolic system. Momentum balances to within machine-precision accuracy over every element, on each atom, and over the coupled system, with small, controllable energy dissipation in the continuum region that ensures numerical stability. When implemented on suitable unstructured spacetime grids, the continuum SDG model offers linear computational complexity in the number of elements and powerful adaptive analysis capabilities that readily bridge between atomic and continuum scales in both space and time. A special trace operator for the atomic velocities and an associated atomistic traction field enter the jump conditions at the coupling interface. The trace operator depends on parameters that specify, at the scale of the atomic spacing, the position of the coupling interface relative to the atoms. In a key finding, we demonstrate that optimizing these parameters suppresses spurious reflections at the coupling interface without the use of non-physical damping or special boundary conditions. We formulate the implicit SDG-ADG coupling scheme in up to three spatial dimensions, and describe an efficient iterative solution scheme that outperforms common explicit schemes, such as the Velocity Verlet integrator. Numerical examples, in 1dxtime and employing both linear and nonlinear potentials, demonstrate the performance of the SDG-ADG method and show how adaptive spacetime meshing reconciles disparate time steps and resolves atomic-scale signals in

2. Adaptative mixed methods to axisymmetric shells

International Nuclear Information System (INIS)

Malta, S.M.C.; Loula, A.F.D.; Garcia, E.L.M.

1989-09-01

The mixed Petrov-Galerkin method is applied to axisymmetric shells with uniform and non uniform meshes. Numerical experiments with a cylindrical shell showed a significant improvement in convergence and accuracy with adaptive meshes. (A.C.A.S.) [pt

3. NUMERICAL SIMULATION OF SHOCK WAVE REFRACTION ON INCLINED CONTACT DISCONTINUITY

Directory of Open Access Journals (Sweden)

P. V. Bulat

2016-05-01

Full Text Available We consider numerical simulation of shock wave refraction on plane contact discontinuity, separating two gases with different density. Discretization of Euler equations is based on finite volume method and WENO finite difference schemes, implemented on unstructured meshes. Integration over time is performed with the use of the third-order Runge–Kutta stepping procedure. The procedure of identification and classification of gas dynamic discontinuities based on conditions of dynamic consistency and image processing methods is applied to visualize and interpret the results of numerical calculations. The flow structure and its quantitative characteristics are defined. The results of numerical and experimental visualization (shadowgraphs, schlieren images, and interferograms are compared.

4. Discontinuities in an axisymmetric generalized thermoelastic problem

Directory of Open Access Journals (Sweden)

2005-06-01

Full Text Available This paper deals with discontinuities analysis in the temperature, displacement, and stress fields of a thick plate whose lower and upper surfaces are traction-free and subjected to a given axisymmetric temperature distribution. The analysis is carried out under three thermoelastic theories. Potential functions together with Laplace and Hankel transform techniques are used to derive the solution in the transformed domain. Exact expressions for the magnitude of discontinuities are computed by using an exact method developed by Boley (1962. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, both propagating with finite speeds with exponential attenuation, and a third which is called shear wave, propagating with constant speed but with no exponential attenuation. The Hankel transforms are inverted analytically. The inversion of the Laplace transforms is carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical results are presented graphically along with a comparison of the three theories of thermoelasticity.

5. Diffusion piecewise homogenization via flux discontinuity ratios

International Nuclear Information System (INIS)

Sanchez, Richard; Dante, Giorgio; Zmijarevic, Igor

2013-01-01

We analyze piecewise homogenization with flux-weighted cross sections and preservation of averaged currents at the boundary of the homogenized domain. Introduction of a set of flux discontinuity ratios (FDR) that preserve reference interface currents leads to preservation of averaged region reaction rates and fluxes. We consider the class of numerical discretizations with one degree of freedom per volume and per surface and prove that when the homogenization and computing meshes are equal there is a unique solution for the FDRs which exactly preserve interface currents. For diffusion sub-meshing we introduce a Jacobian-Free Newton-Krylov method and for all cases considered obtain an 'exact' numerical solution (eight digits for the interface currents). The homogenization is completed by extending the familiar full assembly homogenization via flux discontinuity factors to the sides of regions laying on the boundary of the piecewise homogenized domain. Finally, for the familiar nodal discretization we numerically find that the FDRs obtained with no sub-mesh (nearly at no cost) can be effectively used for whole-core diffusion calculations with sub-mesh. This is not the case, however, for cell-centered finite differences. (authors)

6. An A Posteriori Error Analysis of Mixed Finite Element Galerkin Approximations to Second Order Linear Parabolic Problems

KAUST Repository

Memon, Sajid; Nataraj, Neela; Pani, Amiya Kumar

2012-01-01

In this article, a posteriori error estimates are derived for mixed finite element Galerkin approximations to second order linear parabolic initial and boundary value problems. Using mixed elliptic reconstructions, a posteriori error estimates in L∞(L2)- and L2(L2)-norms for the solution as well as its flux are proved for the semidiscrete scheme. Finally, based on a backward Euler method, a completely discrete scheme is analyzed and a posteriori error bounds are derived, which improves upon earlier results on a posteriori estimates of mixed finite element approximations to parabolic problems. Results of numerical experiments verifying the efficiency of the estimators have also been provided. © 2012 Society for Industrial and Applied Mathematics.

7. Time-Domain Full-Wave Modeling of Nonlinear Air Breakdown in High-Power Microwave Devices and Systems

Science.gov (United States)

2017-09-30

distribution is unlimited. 13. SUPPLEMENTARY NOTES 14. ABSTRACT This final report describes the effort to develop a discontinuous Galerkin time ...Approved for public release; distribution is unlimited. 1.0 SUMMARY In this report, a discontinuous Galerkin time -domain (DGTD) method is developed...public release; distribution is unlimited. [23] S. Yan, C.-P. Lin, R. R. Arslanbekov, V. I. Kolobov, and J.-M. Jin, “A discontinuous Galerkin time

8. Correction of distortions in a discontinuous image

International Nuclear Information System (INIS)

Geagan, M.J.; Chase, B.B.; Muehllehner, G.

1994-01-01

Large area position-sensitive NaI detectors have been successfully applied to positron emission tomography (PET). Typical PET studies involve detector singles rates in excess of 500 kcps, which can lead to pile-up and image degradation as a function of countrate. Better high countrate performance can be achieved with a local centroid algorithm, in which the position of each event is calculated from a small group of photomultipliers (PMTs) immediately surrounding the PMT with the highest signal (the peak PMT). The local centroid contains most of the light from the scintillation. If a local centroid of only seven PMTs is used, the position resolution becomes quite stable at high countrates, however, discontinuities appear in the detector flood image as events cluster around each PMT. We therefore developed a method for distortion correction of a discontinuous flood image. For each PMT on the detector, a peak PMT domain is defined. The peak PMT domain is the area on the detector where that PMT could have the highest signal. The peak PMT domains overlap slightly, so that all combinations of peak PMT and position are represented. A collimated source is moved through a regular grid of points - a template - over each peak PMT region. A short collection is performed at each point, and the real and measured position data are recorded. For each point in the spatial range of a given PMT, distortion correction offsets are computed by interpolating between the data points which correspond to that PMT. This new method has been implemented and evaluated. System resolution has been measured at low and high countrates. The high countrate resolution is better with the new method, with no degradation in low countrate resolution. The axial sensitivity profile is also more stable at high countrates, compared to the previously developed method. 3-D brain phantom images show a clear improvement in image quality at high countrates. ((orig.))

9. Solutions of the Wheeler-Feynman equations with discontinuous velocities.

Science.gov (United States)

de Souza, Daniel Câmara; De Luca, Jayme

2015-01-01

We generalize Wheeler-Feynman electrodynamics with a variational boundary value problem for continuous boundary segments that might include velocity discontinuity points. Critical-point orbits must satisfy the Euler-Lagrange equations of the action functional at most points, which are neutral differential delay equations (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann continuity conditions for the partial momenta and the partial energies. We study a special setup having the shortest time-separation between the (infinite-dimensional) boundary segments, for which case the critical-point orbit can be found using a two-point boundary problem for an ordinary differential equation. For this simplest setup, we prove that orbits can have discontinuous velocities. We construct a numerical method to solve the Wheeler-Feynman equations together with the Weierstrass-Erdmann conditions and calculate some numerical orbits with discontinuous velocities. We also prove that the variational boundary value problem has a unique solution depending continuously on boundary data, if the continuous boundary segments have velocity discontinuities along a reduced local space.

10. The optimal time of discontinuing methimazole before radioiodine therapy

International Nuclear Information System (INIS)

Moosavi, Z.; Zakavi, R.

2001-01-01

Hyperthyroidism is a common disease and one of the best methods for its treatment is radioiodine therapy with Treatment with antithyroid drugs brings patients to euthyroidism before radioiodine therapy. Antithyroid drugs should be discontinued before radioiodine therapy to increase thyroid uptake. The purpose of this study was to determine the optimal time of methimazole discontinuation. One hundred eighty four patients, who were referred for radioiodine therapy were classified in 3 groups according to the duration of methimazole discontinuation before thyroid uptake (RAIU) measurement. Group 1,2 and 3 were patients who discontinued methimazole (48-72 h rs), (72-120 h rs) and more than 120 h rs before RAIU measurement, respectively. Mean thyroid uptake in group 1, 2 and 3 was (64±151.1%), (60.1±14.1%) and (59.3±12.8), respectively. No significant difference was noted in thyroid uptake between these groups (F= 1.83, P<0.16). This study shows that 48-72 h rs of methimazole discontinuation before radioiodine therapy is enough and longer term abstention is not associated with higher uptake

11. Rock discontinuity surface roughness variation with scale

Science.gov (United States)

Bitenc, Maja; Kieffer, D. Scott; Khoshelham, Kourosh

2017-04-01

hypothesize that roughness can increase or decrease with the joint size, depending on the large scale roughness (or waviness), which is entering the roughness calculation once the discontinuity size increases. Therefore, our objective is to characterize roughness at various spatial scales, rather than at changing surface size. Firstly, the rock surface is interpolated into a grid on which a Discrete Wavelet Transform (DWT) is applied. The resulting surface components have different frequencies, or in other words, they have a certain physical scale depending on the decomposition level and input grid resolution. Secondly, the Grasselli Parameter is computed for the original and each decomposed surface. Finally, the relative roughness change is analyzed with respect to increasing roughness wavelength for four different rock samples. The scale variation depends on the sample itself and thus indicates its potential mechanical behavior. References: - Barton, N. and V. Choubey (1977). "The shear strength of rock joints in theory and practice." Rock Mechanics and Rock Engineering 10(1): 1-54. - Grasselli, G. (2001). Shear strength of rock joints based on quantified surface description. École Polytechnique Fédérale de Lausanne. Lausanne, EPFL. - Tatone, B. S. A. and G. Grasselli (2009). "A method to evaluate the three-dimensional roughness of fracture surfaces in brittle geomaterials." Review of Scientific Instruments 80(12) - Tatone, B. and G. Grasselli (2012). "An Investigation of Discontinuity Roughness Scale Dependency Using High-Resolution Surface Measurements." Rock Mechanics and Rock Engineering: 1-25.

12. Organising medication discontinuation: a qualitative study exploring the views of general practitioners toward discontinuing statins.

Science.gov (United States)

Nixon, Michael; Kousgaard, Marius Brostrøm

2016-07-07

Discontinuing medications is a complex decision making process and an important medical practice. It is a tool in reducing polypharmacy, reducing health system expenditure and improving patient quality of life. Few studies have looked at how general practitioners (GPs) discontinue a medication, in agreement with the patients, from a professional perspective. Three research questions were examined in this study: when does medication discontinuation occur in general practice, how is discontinuing medication handled in the GP's practice and how do GPs make decisions about discontinuing medication? Twenty four GPs were interviewed using a maximum variation sample strategy. Participant observations were done in three general practices, for one day each, totalling approximately 30 consultations. The results show that different discontinuation cues (related to the type of consultation, medical records and the patient) create situations of dissonance that can lead to the GP considering the option of discontinuation. We also show that there is a lot of ambiguity in situations of discontinuing and that some GPs trialled discontinuing as means of generating more information that could be used to deal with the ambiguity. We conclude that the practice of discontinuation should be conceptualised as a continually evaluative process and one that requires sustained reflection through a culture of systematically scheduled check-ups, routinely eliciting the patient's experience of taking drugs and trialling discontinuation. Some policy recommendations are offered including supporting GPs with lists or handbooks that directly address discontinuation and by developing more person centred clinical guidelines that discuss discontinuation more explicitly.

13. Method of moving frames to solve time-dependent Maxwell's equations on anisotropic curved surfaces: Applications to invisible cloak and ELF propagation

Science.gov (United States)

Chun, Sehun

2017-07-01

Applying the method of moving frames to Maxwell's equations yields two important advancements for scientific computing. The first is the use of upwind flux for anisotropic materials in Maxwell's equations, especially in the context of discontinuous Galerkin (DG) methods. Upwind flux has been available only to isotropic material, because of the difficulty of satisfying the Rankine-Hugoniot conditions in anisotropic media. The second is to solve numerically Maxwell's equations on curved surfaces without the metric tensor and composite meshes. For numerical validation, spectral convergences are displayed for both two-dimensional anisotropic media and isotropic spheres. In the first application, invisible two-dimensional metamaterial cloaks are simulated with a relatively coarse mesh by both the lossless Drude model and the piecewisely-parametered layered model. In the second application, extremely low frequency propagation on various surfaces such as spheres, irregular surfaces, and non-convex surfaces is demonstrated.

14. Factors associated with use and discontinuation of Implanon ...

African Journals Online (AJOL)

Factors associated with use and discontinuation of Implanon contraceptive in ... The low CPR is the direct cause of the high total fertility of 5.7 in Nigeria. ... This is especially more true with the use of long acting reversible contraceptive methods like implanon which ... Their mean age was 31 years with a range of 16 to 53.

15. Apnoea of prematurity – discontinuation of methylxanthines in a ...

African Journals Online (AJOL)

Background. Methylxanthines such as caffeine have been proven to reduce apnoea of prematurity and are often discontinued at 35 weeks' corrected gestational age (GA). Objective. To ascertain whether a caffeine protocol based on international guidelines is applicable in our setting, where GA is often uncertain. Methods.

16. Exploration of lateral discontinuities with shallow seismic reflection

International Nuclear Information System (INIS)

Kose, K.; Ecevitoglu, B.

2007-01-01

Shallow seismic reflection method is used to for research of mine, fault and stratigraphy and these researches have been result successfully. Survey parameters should be determined with detailed pre-research before data collection. In this study, limestone-fly sch discontinuity experienced at surface and its extension was researched with collected reflection data

17. Discontinuities during UV writing of waveguides

DEFF Research Database (Denmark)

Svalgaard, Mikael; Harpøth, Anders; Andersen, Marc

2005-01-01

UV writing of waveguides can be hampered by discontinuities where the index change process suddenly shuts down. We show that thermal effects may account for this behaviour.......UV writing of waveguides can be hampered by discontinuities where the index change process suddenly shuts down. We show that thermal effects may account for this behaviour....

18. Vertebral Fractures After Discontinuation of Denosumab

DEFF Research Database (Denmark)

Cummings, Steven R; Ferrari, Serge; Eastell, Richard

2018-01-01

. We analyzed the risk of new or worsening vertebral fractures, especially multiple vertebral fractures, in participants who discontinued denosumab during the FREEDOM study or its Extension. Participants received ≥2 doses of denosumab or placebo Q6M, discontinued treatment, and stayed in the study ≥7...... months after the last dose. Of 1001 participants who discontinued denosumab during FREEDOM or Extension, the vertebral fracture rate increased from 1.2 per 100 participant-years during the on-treatment period to 7.1, similar to participants who received and then discontinued placebo (n = 470; 8.5 per 100....... Therefore, patients who discontinue denosumab should rapidly transition to an alternative antiresorptive treatment. Clinicaltrails.gov: NCT00089791 (FREEDOM) and NCT00523341 (Extension). © 2017 American Society for Bone and Mineral Research....

19. Special discontinuities in nonlinearly elastic media

Science.gov (United States)

Chugainova, A. P.

2017-06-01

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.

20. Some error estimates for the lumped mass finite element method for a parabolic problem

KAUST Repository

Chatzipantelidis, P.; Lazarov, R. D.; Thomé e, V.

2012-01-01

for the standard Galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation. We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods