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Journal Articles SN Partial Differential Equations and Applications Year : 2021

Analytical preconditioners for Neumann elastodynamic Boundary Element Methods

Abstract

Recent works in the Boundary Element Method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. Analytical precon-ditioners 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 propose new analytical preconditioners to treat Neumann exterior scattering problems in 2D and 3D elasticity. These preconditioners are local approximations of the adjoint Neumann-to-Dirichlet map. We propose three approximations with different orders. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). An analytical spectral study confirms the expected behavior of the preconditioners, i.e., a better eigenvalue clustering especially in the elliptic part contrary to the standard CFIE of the first-kind. We provide various 2D 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 independent of the density of discretization points per wavelength which is not the case of the standard CFIE. In addition, it is less sensitive to the frequency.
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Dates and versions

hal-02512652 , version 1 (19-03-2020)
hal-02512652 , version 2 (06-10-2021)

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Stéphanie Chaillat, Marion Darbas, Frédérique Le Louër. Analytical preconditioners for Neumann elastodynamic Boundary Element Methods. SN Partial Differential Equations and Applications, 2021, 2 (22), ⟨10.1007/s42985-021-00075-x⟩. ⟨hal-02512652v2⟩
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