Regularized integral equation methods for elastic scattering problems in three dimensions

Abstract

This paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three-dimensional space. The proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high-order singularintegration methods [12] introduced recently for the acoustic case. In order to obtain formulations leading to iterative solvers (GMRES) which converge in small numbers of iterations we investigate, theoretically and computationally, the character of the spectra of various operators associated with the elastic-wave Calderón relation-including some of their possible compositions and combinations. In particular, by relying on the fact that the eigenvalues of the composite operator N S are bounded away from zero and infinity, new uniquely-solvable, low-GMRES-iteration integral formulation for the closed-surface case are presented. The introduction of corresponding low-GMRES-iteration equations for the open-surface equations additionally requires, for both spectral quality as well as accuracy and efficiency, use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges. Several numerical examples demonstrate the accuracy and efficiency of the proposed methodology.proposed methods extend directly to the somewhat less challenging elastic problems with Dirichlet (displacement) boundary conditions. For the Neumann problem of scattering by closed-surfaces the proposed method represents the elastic-field on the basis of a combination of single-layer and double-layer potentials [21], which ensures the validity of the critical property of unique solvability; the resulting integral equation includes contributions from the tractions of both the single-layer and double-layer potentialswhich result in strongly singular and hyper-singular kernels, respectively, and which, unlike the single layer operator (whose kernel is weakly singular), are only defined in the sense of Cauchy principle value and Hadamard finite part [26], respectively. For the problem of scattering by open-surfaces, in turn, a representation leading to a hyper-singular integral operator is used [2,20]. In both cases we propose an efficient high-order singular-integration method that extends the "rectangular-polar" methodology [12] introduced recently for the acoustic case, and which, as demonstrated below in this paper, can efficiently produce solutions of very high accuracy.The presence of the elastic hyper-singular operator in the integral-equation formulations presents difficulties concerning spectral character and accurate operator evaluation, both of which arise from the highly singular character of the associated integral kernel. Indeed, as it is well known, the eigenvalues of the hyper-singular operators accumulate at infinity and, hence, the solution of these integral equations by means of the GMRES solver often requires large numbers of iterations for convergence-and thus, large computing costs, specia...

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