Boundary integral equation methods for Lipschitz domains in linear elasticity

Frédérique Le Louër

doi:10.5802/crmath.317

Research article - Partial differential equations, Mechanics

Boundary integral equation methods for Lipschitz domains in linear elasticity

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

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 453-468.

Abstract

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.

Received: 2021-05-25
Accepted: 2021-12-20
Published online: 2024-05-16

DOI: 10.5802/crmath.317

Classification: 35J25, 35P25, 35C15, 74B05, 65N12
Keywords: Boundary integral equation, Linear elasticity, Lipschitz domains, Gårding’s inequality, Eigenvalues

Author's affiliations:

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

@article{CRMATH_2024__362_G4_453_0,
     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--468},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.317},
     language = {en},
}

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Frédérique Le Louër. Boundary integral equation methods for Lipschitz domains in linear elasticity. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 453-468. doi : 10.5802/crmath.317. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.317/

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