Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

Abstract

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

“…Our main perspective for this work actually concerns other kind of methods. The Padé-type HABCs are used in acceleration techniques for solving boundary integral solvers [6,21,26] and in domain de decomposition methods for parallel finite element solvers [18,64]. The improvement of these methods thanks to the strategies proposed here is currently investigated.…”

“…Unfortunately, because of the highly oscillatory nature of the wave fields, these solvers lead to discretizations with a large number of unknowns and require the solution of large poorly-conditioned linear systems. Research on accurate and computationally efficient methods is very active: we can mention for instance recent works on high-frequency boundary element methods [7,21,22], high-order finite element methods [14,16,57,68] and domain decomposition methods [9,10,34,77].…”

“…They can offer geometric flexibility when choosing the computational domain in application contexts, eventually leading to more appropriate shapes and smaller computational costs. Let us mention that the Padé-type HABCs are also used in acceleration techniques for boundary integral solvers [6,21,26] and domain decomposition methods [18]. The analysis proposed here could also improve these acceleration techniques, which is currently investigated.…”

A fast multipole accelerated indirect boundary element method for broadband scattering of elastic waves in a fluid‐saturated poroelastic domain

“…Liu et al developed an FM‐IBEM for multidomain scattering problems in an infinite domain based on the Taylor series expansion of wave potential functions. Recently, Chaillat et al proposed a fast iterative FM‐BEM for high‐frequency scattering problems. Wang et al proposed a line integration method to calculate the domain integrals combined with the fast multipole method for large‐scale computation.…”

On-surface radiation condition for multiple scattering of waves

“…The OSRCs have been studied by many researchers including Antoine and collaborators who formulated them in a differential geometric setting in order to handle non-canonical surfaces [6,7,8,9,10]. Other improvements have been accomplished by Jones [11,12,13], Ammari [14,15], Calvo et al [16,17], Barucq et al [18,19,20], Chaillat et al [21,22] and Darbas et al [2,3,4,5,23]. See also [24,25,26,27,28,29,30,31,25,32].…”