Lamb modes and Born approximation for small shape defects inversion in elastic plates

TL;DR

This paper develops theoretical tools and a reconstruction method for identifying small shape defects in elastic plates using multi-frequency wavefield measurements and a Born approximation, extending acoustic waveguide techniques to elastic waveguides.

Contribution

It introduces a rigorous framework for elastic wave propagation, characterizes critical frequencies, and generalizes shape reconstruction methods to 2D elastic waveguides.

Findings

Proved well-posedness of elastic wave PDEs in waveguides.

Characterized frequencies where Lamb mode decomposition fails.

Developed a stable mode-by-mode Fourier inversion reconstruction method.

Abstract

The aim of this work is to present theoretical tools to study wave propagation in elastic waveguides and perform multi-frequency scattering inversion to reconstruct small shape defects in a 2D and 3D elastic plate. Given surface multi-frequency wavefield measurements, we use a Born approximation to reconstruct localized defect in the geometry of the plate. To justify this approximation, we introduce a rigorous framework to study the propagation of elastic wavefield generated by arbitrary sources. By studying the decreasing rate of the series of inhomogeneous Lamb mode, we prove the well-posedness of the PDE that model elastic wave propagation in 2D and 3D planar waveguides. We also characterize the critical frequencies for which the Lamb decomposition is not valid. Using these results, we generalize the shape reconstruction method already developed for acoustic waveguide to 2D elastic waveguides and provide a stable reconstruction method based on a mode-by-mode spacial Fourier inversion given by the scattered field.