A boundary-field formulation for elastodynamic scattering

TL;DR

This paper develops a boundary-field formulation for elastodynamic scattering, establishing a nonlocal boundary problem in the Laplace domain, analyzing its well-posedness, and deriving time domain estimates for numerical applications.

Contribution

It introduces a novel boundary-field equation method for elastodynamic scattering, formulating a nonlocal boundary problem and proving its well-posedness in Sobolev spaces.

Findings

Existence and uniqueness of solutions are established in the Laplace domain.

Time domain stability bounds are derived from Laplace domain analysis.

The results facilitate numerical discretization using Convolution Quadrature.

Abstract

An incoming elastodynamic wave impinges on an elastic obstacle is embedded in an infinite elastic medium. The objective of the paper is to examine the subsequent elastic fields scattered by and transmitted into the elastic obstacle. By applying a boundary-field equation method, we are able to formulate a nonlocal boundary problem (NBP) in the Laplace transformed domain, using the field equations inside the obstacle and boundary integral equations in the exterior elastic medium. Existence, uniqueness and stability of the solutions to the NBP are established in Sobolev spaces for two different integral representations. The corresponding results in the time domain are obtained. The stability bounds are translated into time domain estimates that can serve as the starting point for a numerical discretization based on Convolution Quadrature.