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

Abstract

Abstract. 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.

High order surface radiation conditions for time-harmonic waves in exterior domains

“…For the Neumann problem, ∂ ν u is prescribed and u approximated using the NtD-OSRC N operator. Then the computation of the far-field pattern reduces to numerical quadrature to approximate the integral in (31). The behavior of the error in approximating the far-field pattern with the proposed method is displayed in Figure 5 for increasing order N. This relative error is measured 11 in the L 2 -norm on the unit sphere (domain of the far-field pattern) for both the DtN-OSRC N and NtD-OSRC N methods for the same surfaces and parameters discussed in Section 5 and Figure 3.…”

“…The original conception of the OSRC is due to Kriegsmann et al [13]. Over the years, improvements and extensions have been devised, including the works of Jones [14,15,16], Ammari [17,18], Calvo et al [19,20], Antoine, Barucq et al [21,22,23,24,25,26,27,28] and Darbas et al [29,10,24,11,30,12,31,32]. See also [33,34,35,36,37,38,39,40,28].…”

“…Absolute value of the far-field pattern(31) for the Dirichlet problem on the squash-like surface. The exact boundary data is u = F| Γ and ∂ ν u = ∂ ν F, and F is given as in Section 5.…”

“…The exact boundary data is u = F| Γ and ∂ ν u = ∂ ν F, and F is given as in Section 5. For the DtN-OSRC approximation, in(31) we replace ∂ ν u with [λ + ] 8 f Dir which is computed according to Algorithm 1. The wavenumber k = 20π.…”

A Boundary Algebraic Formulation for Plane Strain Elastodynamic Scattering

“…In solid mechanics, less attention has been paid probably due to relatively high damping required to model some of the usual materials (especially in geological applications involving the soil). However, there exist several methods to overcome fictitious eigenfrequencies for BEM in elastodynamics [20,19,32,23,39,37,8].…”