A residual a posteriori error estimate for the time–domain boundary element method

Gimperlein, Heiko; Özdemir, Ceyhun; Stark, David B.; Stephan, Ernst P.

doi:10.1007/s00211-020-01142-y

Cited by 13 publications

(14 citation statements)

References 39 publications

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“…The resulting MOT schemes are described in [5]. They can be combined into a stable scheme for the Dirichlet-to-Neumann operator S from (14), with σ = 0, using the representation…”

Section: Algorithmic Considerationsmentioning

confidence: 99%

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Boundary elements with mesh refinements for the wave equation

et al. 2018

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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.

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“…The resulting MOT schemes are described in [5]. They can be combined into a stable scheme for the Dirichlet-to-Neumann operator S from (14), with σ = 0, using the representation…”

Section: Algorithmic Considerationsmentioning

confidence: 99%

“…They provide a key example for efficient approximations of the solution of transient wave equations by time-independent, adapted meshes. Such meshes also arise in adaptive algorithms based on time-integrated a posteriori error estimates [14].…”

Section: Introductionmentioning

confidence: 99%

Boundary elements with mesh refinements for the wave equation

et al. 2018

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“…The current work provides a first, rigorous step towards efficient boundary elements for dynamic contact. Future work will focus on the a posteriori error analysis, which is essential for adaptive mesh refinements to resolve the singularities of the solution in space and time [25], as well as on stabilized mixed space-time formulations [8,12]. For applications to traffic noise [9], also the nonsmooth variational inequalities for frictional contact will be of interest.…”

Section: Discussionmentioning

confidence: 99%

“…For both stationary and dynamic contact, the relevance of adaptive methods to approximate the non-smooth solutions is well-known [29,38,46]. Recent advances in the a posteriori error analysis and resulting adaptive mesh refinement procedures for time domain boundary elements [25,26] will therefore be of interest for the dynamic contact considered here, with a particular view towards tire dynamics [9].…”

Section: Introductionmentioning

confidence: 99%

Time domain boundary elements for dynamic contact problems

Gimperlein

Özdemir

Stephan

2018

Computer Methods in Applied Mechanics and Engineering

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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.

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“…In practice, the different choices of ansatz and test functions for MOT schemes may lead to instabilities. Proper Galerkin methods are not only provably stable, but they have also attracted interest from at least three different perspectives: Rigorous a posteriori error estimates give rise to efficient adaptive mesh refinement procedures [11,15,16,17,20,25]; non-polynomial basis functions and efficient assembly of the algebraic system [21,25,24]; formulations based on the physical energy [2,3,4]. In this context, efficient preconditioners have been of current interest.…”

Section: Introductionmentioning

confidence: 99%

On a preconditioner for time domain boundary element methods

Gimperlein¹,

Stark²

2018

Engineering Analysis with Boundary Elements

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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.

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A residual a posteriori error estimate for the time–domain boundary element method

Cited by 13 publications

References 39 publications

Boundary elements with mesh refinements for the wave equation

Boundary elements with mesh refinements for the wave equation

Time domain boundary elements for dynamic contact problems

On a preconditioner for time domain boundary element methods

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