Inverse Obstacle Scattering for Elastic Waves in Three Dimensions

TL;DR

This paper addresses the direct and inverse scattering problems for elastic waves in three dimensions, introducing a boundary condition approach and a frequency continuation method, with numerical validation of the inverse obstacle reconstruction.

Contribution

It develops a new variational formulation for the direct problem and proposes a frequency continuation method for the inverse problem, validated through numerical experiments.

Findings

Unique weak solution for the direct problem established.

Effective frequency continuation method for inverse obstacle reconstruction.

Numerical experiments demonstrate the method's effectiveness.

Abstract

Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle. The scattering problem is reformulated into a boundary value problem by introducing a transparent boundary condition. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.