Factorization method for the biharmonic scattering problem for an absorbing penetrable scatterer

TL;DR

This paper extends the factorization method to inverse biharmonic scattering problems in elastic plates, providing a rigorous shape reconstruction criterion and analyzing the Born approximation's validity for weak scatterers.

Contribution

It introduces a rigorous factorization method for biharmonic scattering in elastic plates and analyzes the Born approximation's effectiveness for weak scatterers.

Findings

The factorization method provides a binary criterion for shape reconstruction.

Numerical analysis shows the Born approximation's relative error for weak scatterers.

The method distinguishes inside/outside points using spectral data of the far-field operator.

Abstract

This work extends the factorization method to the inverse scattering problem of reconstructing the shape and location of an absorbing penetrable scatterer embedded in a thin infinite elastic (Kirchhoff--Love) plate. With the assumption that the plate thickness is small compared to the wavelength of the incident wave, the propagation of flexural perturbations is modeled by the two--dimensional biharmonic wave equation in the frequency domain. Within this setting, we provide a rigorous justification of the factorization method and demonstrate that it yields a binary criterion for distinguishing whether a sampling point lies inside or outside the scatterer, using only the spectral data of the far--field operator. In addition, we numerically analyze the Born approximation for weak scatterers in this biharmonic scattering context and compute the relative error against exact far--field data for sample weak scatterers, thereby quantifying its validity as a limited but useful approximation.