Boundary element methods for the wave equation based on hierarchical matrices and adaptive cross approximation

Abstract

Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equation. We use $${\mathscr {H}}^2$$ H 2 -matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.

Realizations of the generalized adaptive cross approximation in an acoustic time domain boundary element method

“…The stopping criterion requires a norm evaluation of  (𝓁) . As stated in Seibel [14], under the assumption of monotonicity, the norm can be computed iteratively by…”

“…The stopping criterion requires a norm evaluation of $$\mathcal {C}^{(\ell )}$$. As stated in Seibel [14], under the assumption of monotonicity, the norm can be computed iteratively by $$$$\begin{equation} {\left\Vert \mathcal {C}^{(\ell )}\right\Vert} _F^2 = \sum \limits _{d,d^{\prime }}^\ell {\left(\sum \limits _{i,j} H_d[i,j]\,\,\, \overline {\!\!\!H_{d^{\prime }}[i,j]\!\!\! }\,\right)} {\left(\sum \limits _k f_d[k] \overline {\!f_{d^{\prime }}[k]\!$$…”

“…For details, see Bebendorf et al. [11] or the application using $$\mathcal {H}^2$$‐matrices in the slices [14]. But in the current case, there is an open point regarding the determination of a maximal element in an $$\mathcal {H}$$‐matrix with low rank blocks.…”

“…It is based on a Tucker decomposition [12] and can be traced back to the group of Tyrtyshnikov [13]. The application of the 3D‐ACA can as well be found in Seibel [14]. Different to the approach here, it has been applied on a standard convolution quadrature‐based formulation and $$\mathcal {H}^2$$‐matrices are used.…”

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

“…Both space‐time Galerkin and convolution quadrature methods have been developed, see References 20–22 for an overview. Recent developments in the directions of the current work include stable formulations, 23–25 efficient discretizations, 26–31 compression of the dense matrices 32,33 as well as complex coupled and interface problems 34–41 …”