Time Domain Boundary Element Methods for the Neumann Problem: Error Estimates and Acoustic Problems

Gimperlein, Heiko; Oezdemir, Ceyhun; Stephan, Ernst P.

doi:10.4208/jcm.1610-m2016-0494

Cited by 18 publications

(24 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Together with the a priori estimates for the time domain boundary element methods on screens [23,24], our results imply convergence rates for the p-version Galerkin approximations which are twice those observed for the quasi-uniform h-method in [22].…”

supporting

confidence: 58%

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

supporting

confidence: 58%

“…, 5. The error remains stable over the time interval, reflecting the space-time variational discretization used [24]. A second right hand side investigates a plane-wave f 2 (t, (x, y, z) T ) = exp(−2/t 2 ) cos(ωt − k(x, y, z) T ) , with k = (3, 0.5, 0.1) and ω = |k|.…”

Section: Wave Equation Outside An Icosahedronmentioning

confidence: 99%

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

show abstract

“…Compute right hand side f n−1 − f n − n−1 m=1 V n−m ψ m Solve system of linear equations (34) Store solution ψ n end for see [15] for details. We use ansatz functions in V 1,1 h,∆t .…”

Section: Algorithmic Considerationsmentioning

confidence: 99%

“…Let ε > 0. a) Let φ be the solution to the hypersingular integral equation W φ = g and φ β h,∆t the best approximation in the norm of H r σ (R + , H Note that the energy norm associated to the weak form of the single layer integral equation (7) is weaker than the norm of H 1 σ (R + , H − 1 2 (Γ)) and stronger than the norm of H 0 σ (R + , H − 1 2 (Γ)), according to the coercivity and continuity properties of V on screens [13]. Similarly, for the weak form of the hypersingular integral equation (10), the energy norm is weaker than the norm of H 1 σ (R + , H Remark C. Together with the a priori estimates for the time domain boundary element methods on screens [13,15], Corollary B implies convergence rates for the Galerkin approximations, which recover those for smooth solutions (up to an arbitrarily small ε > 0) provided the grading parameter β is chosen sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Boundary elements with mesh refinements for the wave equation

et al. 2018

Self Cite

View full text Add to dashboard Cite

The solution of the wave equation in a polyhedral domain in R 3 admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as equivalent boundary integral equations in time domain, study the regularity properties of their solutions and the numerical approximation. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates known for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann operator and applications to the sound emission of tires.

show abstract

“…. , N t (see [24] for further details). From the convolution structure in time, we obtain: For arbitrary n ∈ {1, .…”

Section: 1 Marching-on-in-time Scheme For the Variational Equalitymentioning

confidence: 99%

Time domain boundary elements for dynamic contact problems

Gimperlein

Özdemir

Stephan

2018

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

This article considers a unilateral contact problem for the wave equation. The problem is reduced to a variational inequality for the Dirichlet-to-Neumann operator for the wave equation on the boundary, which is solved in a saddle point formulation using boundary elements in the time domain. As a model problem, also a variational inequality for the single layer operator is considered. A priori estimates are obtained for Galerkin approximations both to the variational inequality and the mixed formulation in the case of a flat contact area, where the existence of solutions to the continuous problem is known. Numerical experiments demonstrate the performance of the proposed mixed method. They indicate the stability and convergence beyond flat geometries.

show abstract

Time Domain Boundary Element Methods for the Neumann Problem: Error Estimates and Acoustic Problems

Cited by 18 publications

References 14 publications

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

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

Boundary elements with mesh refinements for the wave equation

Time domain boundary elements for dynamic contact problems

Contact Info

Product

Resources

About