Time domain boundary elements for dynamic contact problems

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

doi:10.1016/j.cma.2018.01.025

Cited by 18 publications

(15 citation statements)

References 43 publications

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“…With a view towards contact problems [12], we also consider an equation for the Dirichlet-to-Neumann operator S σ . For σ > 0 and given boundary data u σ , we consider…”

Section: Boundary Integral Operators and Sobolev Spacesmentioning

confidence: 99%

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Boundary elements with mesh refinements for the wave equation

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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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“…With a view towards contact problems [12], we also consider an equation for the Dirichlet-to-Neumann operator S σ . For σ > 0 and given boundary data u σ , we consider…”

Section: Boundary Integral Operators and Sobolev Spacesmentioning

confidence: 99%

“…They can be combined into a stable scheme for the Dirichlet-to-Neumann operator S from (14), with σ = 0, using the representation S = W − (K ′ − 1 2 I)V −1 (K − 1 2 I) in terms of layer potentials. As in [12], the Dirichlet-to-Neumann equation (15), Su = h, is equivalently reformulated as follows:…”

Section: Algorithmic Considerationsmentioning

confidence: 99%

Boundary elements with mesh refinements for the wave equation

et al. 2018

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“…for 1 ≤ n ≤ N t and 1 ≤ j ≤ N s ′ . The resulting discretization of the Poincaré-Steklov operator has been tested in [15,16], and corresponding results are obtained for more natural discretizations with piecewise constant λ h,△t . Piecewise linear and higher order test functions are considered in [19].…”

Section: Numerical Resultsmentioning

confidence: 99%

A time–dependent FEM-BEM coupling method for fluid–structure interaction in 3d

Gimperlein

Özdemir

Stephan

2020

Applied Numerical Mathematics

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We consider the well-posedness and a priori error estimates of a 3d FEM-BEM coupling method for fluid-structure interaction in the time domain. For an elastic body immersed in a fluid, the exterior linear wave equation for the fluid is reduced to an integral equation on the boundary involving the Poincaré-Steklov operator. The resulting problem is solved using a Galerkin boundary element method in the time domain, coupled to a finite element method for the Lamé equation inside the elastic body. Based on ideas from the timeindependent coupling formulation, we obtain an a priori error estimate and discuss the implementation of the proposed method. Numerical experiments illustrate the performance of our scheme for model problems.

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“…The singular expansion for the inhomogeneous wave equation in (21) in R + t × K x implies an expansion for inhomogeneous boundary conditions in (20). The argument is as for elliptic problems [45,Section 5]: For Dirichlet conditions u| Γ = g on R + t ×∂K, we choose an extension…”

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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

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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.

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Time domain boundary elements for dynamic contact problems

Cited by 18 publications

References 43 publications

Boundary elements with mesh refinements for the wave equation

Boundary elements with mesh refinements for the wave equation

A time–dependent FEM-BEM coupling method for fluid–structure interaction in 3d

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

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