Direct and Inverse Elastic Scattering From Anisotropic Media

TL;DR

This paper develops a variational approach for elastic wave scattering in anisotropic media, establishing theoretical foundations and proposing a numerical method for interface reconstruction from near-field data.

Contribution

It introduces a variational formulation for elastic scattering in anisotropic media, proves solution uniqueness, and designs a descent algorithm for interface recovery.

Findings

The variational formulation ensures well-posedness of the scattering problem.

The descent algorithm effectively reconstructs interfaces from near-field data.

Numerical examples demonstrate the method's accuracy in 2D cases.

Abstract

Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show the uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fr\'echet derivative of the near-field solution operator with respect to the boundary of the scatterer is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are presented to show the validity and accuracy of our methods.