Comptes Rendus
Research article - Partial differential equations, Mechanics
Boundary integral equation methods for Lipschitz domains in linear elasticity
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 453-467.

A review of stable boundary integral equation methods for solving the Navier equation with either Dirichlet or Neumann boundary conditions in the exterior of a Lipschitz domain is presented. The conventional combined-field integral equation (CFIE) formulations, that are used to avoid spurious resonances, do not give rise to a coercive variational formulation for nonsmooth geometries anymore. To circumvent this issue, either the single layer or the double layer potential operator is composed with a compact or a Steklov–Poincaré type operator. The later can be constructed from the well-know elliptic boundary integral operators associated to the Laplace equation and Gårding’s inequalities are satisfied. Some Neumann interior eigenvalue computations for the unit square and cube are presented for forthcoming numerical investigations.

Published online:
DOI: 10.5802/crmath.317
Classification: 35J25, 35P25, 35C15, 74B05, 65N12
Keywords: Boundary integral equation, Linear elasticity, Lipschitz domains, Gårding’s inequality, Eigenvalues

Frédérique Le Louër 1

1 Alliance Sorbonne Université, Université de Technologie de Compiègne, LMAC EA2222 Laboratoire de Mathématiques Appliquées de Compiègne - CS 60319 - 60203 Compiègne cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Fr\'ed\'erique Le Lou\"er},
     title = {Boundary integral equation methods for {Lipschitz} domains in linear elasticity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {453--467},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.317},
     language = {en},
AU  - Frédérique Le Louër
TI  - Boundary integral equation methods for Lipschitz domains in linear elasticity
JO  - Comptes Rendus. Mathématique
PY  - 2024
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PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.317
LA  - en
ID  - CRMATH_2024__362_G4_453_0
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%A Frédérique Le Louër
%T Boundary integral equation methods for Lipschitz domains in linear elasticity
%J Comptes Rendus. Mathématique
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%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.317
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Frédérique Le Louër. Boundary integral equation methods for Lipschitz domains in linear elasticity. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 453-467. doi : 10.5802/crmath.317.

[1] Mikhail S. Agranovich; B. A. Amosov; Maria Levitin Spectral problems for the Lamé system with spectral parameter in boundary conditions on smooth or nonsmooth boundary, Russ. J. Math. Phys., Volume 6 (1999) no. 3, pp. 247-281 | Zbl

[2] Annalisa Buffa; Martin Costabel; Dongwoo Sheen On traces for H(curl,Ω) in Lipschitz domains, J. Math. Anal. Appl., Volume 276 (2002) no. 2, pp. 845-867 | DOI | MR | Zbl

[3] Annalisa Buffa; Ralf Hiptmair Regularized combined field integral equations, Numer. Math., Volume 100 (2005) no. 1, pp. 1-19 | DOI | MR | Zbl

[4] David Colton; Rainer Kress Integral equation methods in scattering theory, Classics in Applied Mathematics, 72, Society for Industrial and Applied Mathematics, 2013, xvi+271 pages | DOI

[5] Martin Costabel; Frédérique Le Louër On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body, SIAM J. Appl. Math., Volume 71 (2011) no. 2, pp. 635-656 | DOI | MR | Zbl

[6] Martin Costabel Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal., Volume 19 (1988) no. 3, pp. 613-626 | DOI | MR | Zbl

[7] Martin Costabel; Ernst P. Stephan Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J. Numer. Anal., Volume 27 (1990) no. 5, pp. 1212-1226 | DOI | MR

[8] Martin Costabel; Ernst P. Stephan Integral equations for transmission problems in linear elasticity, J. Integral Equations Appl., Volume 2 (1990) no. 2, pp. 211-223 | DOI | MR

[9] Marion Darbas; Frédérique Le Louër Well-conditioned boundary integral formulations for high-frequency elastic scattering problems in three dimensions, Math. Methods Appl. Sci., Volume 38 (2015) no. 9, pp. 1705-1733 | DOI | MR

[10] S. Engleder; Olaf Steinbach Modified boundary integral formulations for the Helmholtz equation, J. Math. Anal. Appl., Volume 331 (2007) no. 1, pp. 396-407 | DOI | MR | Zbl

[11] S. Engleder; Olaf Steinbach Stabilized boundary element methods for exterior Helmholtz problems, Numer. Math., Volume 110 (2008) no. 2, pp. 145-160 | DOI | MR | Zbl

[12] Günter K. Gächter; Marcus J. Grote Dirichlet-to-Neumann map for three-dimensional elastic waves, Wave Motion, Volume 37 (2003) no. 3, pp. 293-311 | DOI | MR

[13] Denis S. Grebenkov; Binh-Thanh Nguyen Geometrical structure of Laplacian eigenfunctions, SIAM Rev., Volume 55 (2013) no. 4, pp. 601-667 | DOI | MR

[14] Antoine Henrot; Michel Pierre Variation et optimisation de formes. Une analyse géométrique, Mathématiques & Applications, 48, Springer, 2005, xii+334 pages | DOI

[15] George C. Hsiao; Wolfgang L. Wendland Boundary integral equations, Applied Mathematical Sciences, 164, Springer, 2008, xx+618 pages | DOI

[16] Olha Ivanyshyn; Frédérique Le Louër An inverse parameter problem with generalized impedance boundary condition for two-dimensional linear viscoelasticity, SIAM J. Appl. Math., Volume 81 (2021) no. 4, pp. 1668-1690 | MR

[17] Viktor D. Kupradze; Tengiz G. Gegelia; Mikheil O. Basheleĭshvili; T. V. Burčuladze Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland, 1979, xix+929 pages (Edited by V. D. Kupradze)

[18] Frédérique Le Louër A high order spectral algorithm for elastic obstacle scattering in three dimensions, J. Comput. Phys., Volume 279 (2014), pp. 1-17 | DOI | MR

[19] Frédérique Le Louër A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems, Inverse Probl., Volume 31 (2015) no. 11, 115006, 27 pages | DOI | MR

[20] Frédérique Le Louër; R. Rais On the coupling between finite elements and integral representation for linear elastic waves scattering problems: analysis and simulation (2023) (submitted)

[21] William McLean Strongly elliptic systems and boundary integral equations, Cambridge University Press, 2000, xiv+357 pages

[22] Jean-Claude Nédélec; Jacques Planchard Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans R 3 , Rev. Franc. Automat. Inform. Rech. Operat., Volume 7 (1973) no. R-3, pp. 105-129 | Zbl

[23] Günther Of; Olaf Steinbach A fast multipole boundary element method for a modified hypersingular boundary integral equation, Analysis and simulation of multifield problems. Selected papers of the international conference on multifield problems (Stuttgart, 2002) (Lecture Notes in Applied and Computational Mechanics), Volume 12, Springer, 2003, pp. 163-169 | Zbl

[24] Olaf Steinbach A note on the ellipticity of the single layer potential in two-dimensional linear elastostatics, J. Math. Anal. Appl., Volume 294 (2004) no. 1, pp. 1-6 | DOI | MR

[25] Olaf Steinbach; Markus Windisch Modified combined field integral equations for electromagnetic scattering, SIAM J. Numer. Anal., Volume 47 (2009) no. 2, pp. 1149-1167 | DOI | MR | Zbl

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