Algorithmic aspects of enriched time domain boundary element methods

Gimperlein, Heiko; Stark, David B.

doi:10.1016/j.enganabound.2018.02.010

Cited by 7 publications

(8 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While this approach has compelling advantages, the design of such high-dimensional quadrature methods for hyperbolic problems is complicated due to the nonlinear behavior of x → Q(x). This is the reason why typical quadrature schemes employed in classical semidiscretizations of RPBIEs treat these integrals separately [31,32]. The present paper stays in line with these approaches in the sense that the inner and outer integral are treated individually.…”

Section: Introductionmentioning

confidence: 68%

On the space-time discretization of variational retarded potential boundary integral equations

Pölz,

Schanz

2021

Preprint

View full text Add to dashboard Cite

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Finally, we glimpse at algorithmic aspects regarding the efficient implementation of retarded potentials in the space-time setting. The proposed methods are verified by means of numerical experiments, which illustrate their capacity.

show abstract

Section: Introductionmentioning

confidence: 68%

On the space-time discretization of variational retarded potential boundary integral equations

Pölz,

Schanz

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In the literature decompositions like (25) and (26) in polar coordinates are also expressed in Cartesian coordinates or a mixture of Cartesian and polar coordinates. The Appendix of [14] shows the equivalence of these descriptions, also remarked in Corollary 4 of [44].…”

Section: Singularities For Polygonal Screens and Approximationmentioning

confidence: 99%

“…The situation for the Dirichlet problem of the Laplace operator on an infinite wedge Ω = R × K is analysed in Theorem 7 in [45] and in a polyhedral cone Ω in Theorem 8 in [45] with the limit case of a screen in Example 4 in [45]. Redoing the derivation of the singularity terms after Theorem 24 for the limit case of a screen with corresponding λ −k and Φ k one obtains for a circular screen the expansion (24) and for a polygonal screen the expansion (26). Here one first must modify the decomposition (39) of Theorem 26 for the solution u of (20) in a polyhedral cone.…”

Section: Consider the Dirichlet Problem In The Infinite Cylindermentioning

confidence: 99%

See 1 more Smart Citation

hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Gimperlein

Özdemir

Stark

et al. 2019

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R 3 exhibit singular behavior from the edges and corners. We present quasi-optimal hp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an hp-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.This article initiates the study of high-order boundary elements in the time domain. For elliptic problems, p-and hp-versions of the finite element method give rise to fast approximations of both smooth solutions and geometric singularities. These methods converge to the solution by increasing the polynomial degree p of the elements, possibly in combination with reducing the mesh size h of the quasi-uniform mesh. They were first investigated in the group of Babuska [3,4,17,18]. See [49] for a comprehensive analysis for 2d problems.The analogous p-and hp-versions of the boundary element method go back to [2,53,54]. More recent optimal convergence results for boundary elements on screens and polyhedral surfaces covering 3d problems have been obtained, for example, in [9,10,11,12,13].Boundary element methods for time dependent problems have recently become of interest [48]. In this article we introduce a space-time hp-version of the time domain boundary element method for the wave equation with non-homogeneous Dirichlet or Neumann boundary conditions.

show abstract

“…It approximates the algebraic system using extrapolation and may be used as either a preconditioner for the full space-time equation or as a fast, quasi-exact stand-alone solver of the integral equation for large numbers of degrees of freedom. Our method inherits the approximation properties and long-time stability of the Galerkin method and has been used, but not thoroughly documented in recent works [9,11,14]. The rigorous numerical analysis of the surprisingly good stability properties of the proposed method for the standard h-TDBEM under mesh refinements remains open.…”

Section: Introductionmentioning

confidence: 99%

On a preconditioner for time domain boundary element methods

Gimperlein¹,

Stark²

2018

Engineering Analysis with Boundary Elements

Self Cite

View full text Add to dashboard Cite

We propose a time stepping scheme for the space-time systems obtained from Galerkin time-domain boundary element methods for the wave equation. Based on extrapolation, the method proves stable, becomes exact for increasing degrees of freedom and can be used either as a preconditioner, or as an efficient standalone solver for scattering problems with smooth solutions. It also significantly reduces the number of GMRES iterations for screen problems, with less regularity, and we explore its limitations for enriched methods based on non-polynomial approximation spaces.

show abstract

Algorithmic aspects of enriched time domain boundary element methods

Cited by 7 publications

References 30 publications

On the space-time discretization of variational retarded potential boundary integral equations

On the space-time discretization of variational retarded potential boundary integral equations

hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

On a preconditioner for time domain boundary element methods

Contact Info

Product

Resources

About