Fourier method for reconstructing elastic body force from the coupled-wave field

Xianchao Wang; Jiaqi Zhu; Minghui Song; Wei Wu

doi:10.3934/ipi.2021052

Article Contents

2022, Volume 16, Issue 2: 325-340. Doi: 10.3934/ipi.2021052

This issue Previous Article Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations Next Article PCA reduced Gaussian mixture models with applications in superresolution

Fourier method for reconstructing elastic body force from the coupled-wave field

1.
National Key Laboratory of Science and Technology on Advanced Composites in Special Environments, Harbin Institute of Technology, Harbin 150080, China
2.
Key Laboratory of Micro-systems and Micro-structures Manufacturing Ministry of Education, Harbin Institute of Technology, Harbin 150080, China
3.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
4.
Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

^* Corresponding author: Jiaqi Zhu and Minghui Song
^* Corresponding author: Jiaqi Zhu and Minghui Song

Received: March 2021

Revised: March 2021

Early access: July 16, 2021

Published online: July 16, 2021

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Abstract

This paper is concerned with the inverse source problem of the time-harmonic elastic waves. A novel non-iterative reconstruction scheme is proposed for determining the elastic body force by using the multi-frequency Fourier expansion. The key ingredient of the approach is to choose appropriate admissible frequencies and establish an relationship between the Fourier coefficients and the coupled-wave field of compressional wave and shear wave. Both theoretical justifications and numerical examples are presented to verify the validity and robustness of the proposed method.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 65M32; Secondary: 65T50.

Citation:

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Figure 1. Contour plots of the reconstructed source function with different measured directions, where the dotted red line denotes the angle of observation directions. (a) Exact $ F_1 $, (b) exact $ F_2 $, (c) reconstructed $ F_1 $, (d) reconstructed $ F_2 $

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Figure 2. Contour plots of reconstructed source function with limit-view data, where the dotted red line denotes the angle of observation directions. Left column: reconstructed $ F_1 $, right column: reconstructed $ F_2 $

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Figure 3. Slice plots of the exact source functions, where the left column is sliced at $ x = 0 $; center column is sliced at $ y = 0 $; right column is sliced at $ z = 0 $. Top row: exact $ F^{(1)} $; center row: exact $ F^{(2)} $; bottom row: exact $ F^{(3)} $

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Figure 4. Slice plots of the reconstructed source functions, where the left column is sliced at $ x = 0 $; center column is sliced at $ y = 0 $; right column is sliced at $ z = 0 $. Top row: reconstructed $ F^{(1)} $; center row: reconstructed $ F^{(2)} $; bottom row: reconstructed $ F^{(3)} $

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