Regularized integral equation methods for elastic scattering problems in three dimensions

TL;DR

This paper introduces new integral equation methods for simulating elastic wave scattering in three dimensions, achieving efficient iterative solutions with high accuracy for both open and closed surfaces.

Contribution

It extends high-order singular-integration techniques to elastic waves and develops low-GMRES-iteration formulations based on spectral analysis of elastic operators.

Findings

Spectral analysis shows eigenvalues of key operators are bounded away from zero and infinity.

New formulations enable rapid convergence of iterative solvers.

Numerical examples confirm high accuracy and efficiency.

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 singular-integration methods~\cite{BG18} 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 Calder\'on relation---including some of their possible compositions and combinations. In particular, by relying on the fact that the eigenvalues of the composite operator $NS$ 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.