# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021052
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## 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

Received  March 2021 Revised  March 2021 Early access July 2021

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.

Citation: Xianchao Wang, Jiaqi Zhu, Minghui Song, Wei Wu. Fourier method for reconstructing elastic body force from the coupled-wave field. Inverse Problems & Imaging, doi: 10.3934/ipi.2021052
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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 $F_2$
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 $F_2$
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 $F^{(3)}$
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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