Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles

TL;DR

This paper develops a mathematical framework for analyzing and solving direct and inverse elastic scattering problems involving extended bodies and point-like obstacles, including explicit representations and a factorization-based inverse method.

Contribution

It introduces a point-interaction model for the Lamé operator, derives explicit scattered field representations, and adapts the factorization method for simultaneous shape and location recovery.

Findings

Explicit scattered field formulas for point and extended obstacles.

Validation of the model with known methods like Foldy's and renormalization.

Numerical examples demonstrating the inverse reconstruction schemes.

Abstract

This paper concerns the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lam\'{e} operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions, we derive an explicit representation of the scattered field. In the case of isotropic and local point-interactions, our result is consistent with the ones previously obtained by Foldy's formal method as well as by the renormalization technique. In the case of multiple scattering with pointwise and extended obstacles, we show that the scattered field consists of two parts: one is due to the diffusion by the extended scatterer and the other one is a linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. As to the inverse problem, the factorization method by Kirsch is adapted to recover simultaneously the shape of an extended elastic body and the location of point-like scatterers in the case of isotropic and local interactions. The inverse problems using only one type of elastic waves (i.e. pressure or shear waves) are also investigated and numerical examples are present to confirm the inversion schemes.