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

Marion Darbas (1) , Frédérique Le Louër (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne

We construct and analyze a family of well-conditioned boundary integral equations for the Krylov iterative solution of three-dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well-known Brakhage-Werner and Combined Field Integral Equation formulations. We use a suitable approximation of the Dirichlet-to-Neumann (DtN) map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate DtN map is inspired by the On-Surface Radiation Conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided.

Mots clés: boundary integral equations; analytic preconditioning; Elastic scattering; Navier equation; Dirichlet condition

Domaines: Analyse numérique [math.NA]

Marion Darbas, Frédérique Le Louër. Well-conditioned boundary integral formulations for high-frequency elastic scattering problems in three dimensions. Mathematical Methods in the Applied Sciences, 2015, 38 (9), pp.1705-1733. ⟨10.1002/mma.3179⟩. ⟨hal-00839653v4⟩