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Detection of a piecewise linear crack with one incident wave

Xiaoxu Xu, Guanqiu Ma, Guanghui Hu

TL;DR

The paper tackles inverse scattering by a 2D piecewise linear sound-soft crack using a single incident wave, proving uniqueness and proposing a hybrid imaging framework. It combines a contrast sampling method for rough localization, a one-wave factorization technique for shape-rough localization, and Newton iterations for precise reconstruction, with careful handling of limited aperture data and noise. The approach is validated through numerical experiments that demonstrate accurate localization and refinement of crack geometry from both plane-wave and point-source data. The work advances data-efficient crack imaging, showing that meaningful location and shape recovery is possible with a single observation, complemented by robust regularization and tangent-update strategies.

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.

Detection of a piecewise linear crack with one incident wave

TL;DR

The paper tackles inverse scattering by a 2D piecewise linear sound-soft crack using a single incident wave, proving uniqueness and proposing a hybrid imaging framework. It combines a contrast sampling method for rough localization, a one-wave factorization technique for shape-rough localization, and Newton iterations for precise reconstruction, with careful handling of limited aperture data and noise. The approach is validated through numerical experiments that demonstrate accurate localization and refinement of crack geometry from both plane-wave and point-source data. The work advances data-efficient crack imaging, showing that meaningful location and shape recovery is possible with a single observation, complemented by robust regularization and tangent-update strategies.

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.
Paper Structure (12 sections, 5 theorems, 64 equations, 14 figures)

This paper contains 12 sections, 5 theorems, 64 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Uniqueness results
  3. Contrast sampling method to determine a rough location
  4. Comparison with point sources
  5. Comparison with disks
  6. The one-wave factorization method
  7. Preliminary results from the factorization method
  8. The factorization method with a single wave
  9. A more precise result by Newton's iteration method
  10. Computation of Fréchet derivatives
  11. Tangential updates of two crack tips
  12. Numerical examples

Key Result

Theorem 2.1

Assume $\Gamma_j$ is a sound-soft crack such that $\Gamma_j\subset\partial\Omega_j$, where $\partial\Omega_j$ denotes the boundary of some convex polygon $\Omega_j\subset{\mathbb R}^2$, $j=1,2$, as shown in Figure Case1. (i) Let $d_0\in{\mathbb S}$ be an arbitrary fixed incident direction. If the fa where ${\mathbb S}_0$ is an open subset of ${\mathbb S}$, then $\Gamma_1=\Gamma_2$. (ii) Let $y_0\i

Figures (14)

  • Figure 1: An example for geometry of two cracks. Black '-': $\Gamma_1$; red '- -': $\Gamma_2$.
  • Figure 2: Geometry of Theorem \ref{['t1']}.
  • Figure 3: Numerical results for Example \ref{['example0']} with $\Gamma$ denoted by the black thick line.
  • Figure 4: Numerical results for Example \ref{['example1']} (a) with $\Gamma$ denoted by the black solid line.
  • Figure 5: Numerical results for Example \ref{['example1']} (b) with $\Gamma$ denoted by the black solid line.
  • ...and 9 more figures

Theorems & Definitions (19)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Theorem 4.4
  • Remark 4.5
  • Theorem 4.6
  • ...and 9 more