A direct sampling method for inverse elastic scattering in cavity structure

Jiahui Gao; Shuxin Li; Junliang Lv

doi:10.3934/ipi.2025034

Article Contents

2026, Volume 21: 48-72. Doi: 10.3934/ipi.2025034

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A direct sampling method for inverse elastic scattering in cavity structure

School of Mathematics, Jilin University, Changchun, 130012, China

^*Corresponding author: Shuxin Li
^*Corresponding author: Shuxin Li

Received: December 2024

Revised: July 2025

Early access: July 17, 2025

Published online: July 17, 2025

The third author is supported by the National Natural Science Foundation of China grant 12271209.

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Abstract

This paper is concerned with the inverse elastic scattering problem to determine the shape and location of a cavity with Dirichlet boundary condition from the near-field data. A fast and stable direct sampling method is proposed for such inverse problem. We construct an indicator function and prove that the indicator function vanishes only at the boundary of the cavity. In the reconstruction process, we approximate the scattered field using a truncated Fourier series based on the Helmholtz decomposition, and establish an error estimate. Numerical experiments are presented to illustrate the feasibility and effectiveness of our algorithm, and to analyze the influence of various factors, such as noise level, frequency, polarization direction, radius of the measurement curve, and number of point sources, on numerical solutions.

Keywords:

Mathematics Subject Classification: 78A46, 65N21.

Citation:

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Figure 1. The geometry of elastic scattering in cavity.

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Figure 2. Image of $ \Lambda_0 $.

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Figure 3. Image of $ \Lambda_1 $.

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Figure 4. Image of $ \Lambda_2 $.

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Figure 5. (a) is the true cavity boundary, and (b)-(f) correspond to noise levels $ \delta = 0 $, $ 1\% $, $ 5\% $, $ 10\% $, $ 15\% $, respectively.

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Figure 6. (a) is the true cavity boundary; (b)-(d) respectively correspond to polarization angle $ \alpha = 0 $, $ \pi/4 $, $ \pi/2 $; (e) is reconstruction for the superposition of polarization angles $ 0, \pi/2 $, $ \pi $, $ 2\pi/3 $, $ 2\pi $; (f) is reconstruction for the superposition of polarization angles $ 0 $, $ \pi/4 $, $ \pi/2 $, $ 3\pi/4 $, $ \pi $, $ 5\pi/4 $, $ 2\pi/3 $, $ 7\pi/4 $, $ 2\pi $.

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Figure 7. (a) is the true cavity boundary, and (b)-(f) are reconstructions for measurement circles of radius $ R = 0.02 $, $ 0.1 $, $ 0.2 $, $ 0.5 $, $ 0.7 $, respectively.

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Figure 8. (a) is the true cavity boundary; (b)-(e) respetively correspond to frequency $ \omega = 1 $, $ 3 $, $ 5 $, $ 7 $; (f) is the the multi-frequency reconstructions.

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Figure 9. (a) is the true cavity boundary, and (b)-(f) are reconstructions for different numbers of point sources $ M_2 = 2 $, $ 4 $, $ 8 $, $ 12 $, $ 24 $, respectively.

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Figure 10. Reconstructions for different truncated numbers $ N $ with $ \delta = 0 $.

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Figure 11. Reconstructions for different truncated numbers $ N $ with $ \delta = 1\% $.

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Figure 12. Reconstructions for the flower-shaped cavity without noise using different sampling method. (a) is the true cavity boundary; (b) The direct sampling method; (c) The decomposition method; (d) The factorization method.

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Figure 13. Reconstructions for small kited-shaped cavity with $ \delta = 5\% $ using different sampling method. (a) is the true cavity boundary; (b) The direct sampling method; (c) The decomposition method; (d) The factorization method.

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