Detection of a piecewise linear crack with one incident wave

TL;DR

This paper introduces two non-iterative methods and a Newton iteration approach for uniquely detecting and accurately reconstructing piecewise linear cracks in two-dimensional acoustic scattering problems using minimal incident wave data.

Contribution

It proves the unique determination of a piecewise linear crack from a single incident wave and develops hybrid imaging methods combining contrast sampling, factorization, and Newton's iteration.

Findings

Successful reconstruction of crack location and shape demonstrated by numerical examples.

Effective hybrid imaging methods with minimal incident wave data.

High accuracy achieved with Newton's iteration refinement.

Abstract

This paper is concerned with inverse crack scattering problems for time-harmonic acoustic waves. We prove that a piecewise linear crack with the sound-soft boundary condition in two dimensions can be uniquely determined by the far-field data corresponding to a single incident plane wave or point source. We propose two non-iterative methods for imaging the location and shape of a crack. The first one is a contrast sampling method, while the second one is a variant of the classical factorization method but only with one incoming wave. Newton's iteration method is then employed for getting a more precise reconstruction result. Numerical examples are presented to show the effectiveness of the proposed hybrid method.