Analytical preconditioners for Neumann elastodynamic boundary element methods

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...

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Bibliographic Details
Published in:SN partial differential equations and applications 2021-04, Vol.2 (2), Article 22
Main Authors: Chaillat, Stéphanie, Darbas, Marion, Louër, Frédérique Le
Format: Article
Language:English
Subjects:
Analysis
Mathematical and Computational Biology
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Partial Differential Equations
Review Paper
Theoretical
Waves 2019 – invited papers
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Summary: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 preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead of 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 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 for the standard CFIE. In addition, it is less sensitive to the frequency.
ISSN:2662-2963
2662-2971
DOI:10.1007/s42985-021-00075-x