A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

Abstract

Three-dimensional (3D) rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretization (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretization) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the -transform and the convolution quadrature method, we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency-domain BEMs. Then, taking advantage of a well-designed high-frequency approximation, we drastically reduce the number of frequency-domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regard to the time discretization and O(N log N) for the spacial discretization, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid-structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity and the scattering of an abrupt wave by simple and realistic geometries.

Improvement of hierarchical matrices for 3D elastodynamic problems with a complex wavenumber

“…We can then extend this result to the whole BIE (28). It involves the numerical resolution of distinct BIEs in the complex frequency domain given by the discrete values of s: s p = p(ξ p )/∆t with ξ p = ρe 2iπp/L , L complex numbers taken on the circle of radius ρ in the complex space.…”

“…5.2 Efficiency of the approach in the context of the convolution Quadrature Method for 3D time-domain elastodynamics Another interesting configuration in which purely elastodynamic problems are consider with a complex wave number is when a CQM-based approach is used to reformulate the time-domain BIE in terms of BIEs in the (complex) frequency domain. The approach can conveniently be presented by focusing on the evaluation of the single-layer integral operator G{f } for a given causal density f (see [28] for more details in the context of Helmholtz problems). It is based on a numerical approximation of convolution integrals such as:…”

“…The inverse Z-transform is used to express the discrete time-domain solution obtained once the BIE is solved. This procedure is detailed in an acoustic configuration in [28].…”

“…To present the Z-BEM approach [28] to solve (28) in the time domain we focus on the evaluation of the single-layer integral operator:…”

Investigation of linear difference equations with random effects

“…transform method commonly used in solving differential equations [12]. Some of the studies in this area are given in [14][15][16][17][18][19][20][21][22][23].…”