A high order spectral algorithm for elastic obstacle scattering in three dimensions

Abstract

To cite this version:Frédérique Le Louër. A high order spectral algorithm for elastic obstacle scattering in three dimensions. Journal of Computational Physics, Elsevier, 2014, 279, pp.1-17. <hal-00700779v4> A high order spectral algorithm for elastic obstacle scattering in three dimensionsFrédérique Le Louër AbstractIn this paper we describe a high order spectral algorithm for solving the time-harmonic Navier equations in the exterior of a bounded obstacle in three space dimensions, with Dirichlet or Neumann boundary conditions. Our approach is based on combined-field boundary integral equation (CFIE) reformulations of the Navier equations. We extend the spectral method developed by Ganesh and Hawkins [20] -for solving second kind boundary integral equations in electromagnetism -to linear elasticity for solving CFIEs that commonly involve integral operators with a strongly singular or hypersingular kernel. The numerical scheme applies to boundaries which are globally parameterised by spherical coordinates. The algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods. The computational performances of the proposed spectral algorithm are demonstrated using numerical examples for a variety of three-dimensional convex and non-convex smooth obstacles.

A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles

“…The numerical implementation is briefly described in Section 4 and numerical results on the convergence rate of the method are presented for smooth and non smooth obstacles. The mathematical analysis renders even possible the implementation of hypersingular kernels [37,38]. However, in the extreme low-frequency regime, as ω → 0, numerical methods that use only tangential basis functions are not sufficient to avoid the low-frequency breakdown [19].…”

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

“…The principal part of is . To recover the principal parts of the other terms in the operator N , we consider the following form for the operator T x . We have to apply T x to these two terms …”

“…We consider integral representations of the operators D and D 0 obtained in [32] .D /. By composition, the principal part of the aforementioned expression is i 2 n C Ä 2 s I 1 2 div I t .…”

“…To recover the principal parts of the other terms in the operator N, we consider the following form for the operator T x [32].…”

An accurate boundary element method for the exterior elastic scattering problem in two dimensions

“…To effectively reduce the BIE into a linear system, many different solvers including the boundary element method (BEM) ( [13]), the Nyströme method ( [16]), the fast multipole method [5,17] and the spectral method ( [12]) have been considered.…”