April  2022, 16(2): 325-340. doi: 10.3934/ipi.2021052

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 Published  April 2022 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 and Imaging, 2022, 16 (2) : 325-340. doi: 10.3934/ipi.2021052
References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, J. Math. Anal. Appl., 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.

[2]

R. Albanese and P. B. Monk, The inverse source problem for Maxwell's equations, Inverse Probl., 22 (2006), 1023-1035.  doi: 10.1088/0266-5611/22/3/018.

[3]

C. J. S. Alves, N. F. M. Martins and N. C. Roberty, Identification and reconstruction of elastic body forces, Inverse Probl., 30 (2014), 055015. doi: 10.1088/0266-5611/30/5/055015.

[4]

H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Vol. 52, Princeton University Press, 2015. doi: 10.1515/9781400866625.

[5]

T. S. AngelA. Kirsch and R. E. Kleinmann, Antenna control and optimization, Proc. IEEE, 79 (1991), 1559-1568.  doi: 10.1109/5.104230.

[6]

T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Probl., 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.

[7]

G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Probl., 31 (2015), 093001. doi: 10.1088/0266-5611/31/9/093001.

[8]

E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.  doi: 10.1137/18M1182048.

[9]

E. Blåsten and Y. -H. Lin, Radiating and non-radiating sources in elasticity, Inverse Probl., 35 (2019), 015005. doi: 10.1088/1361-6420/aae99e.

[10]

E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, (2020), in press.

[11]

E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, SIAM J. Math. Anal., (2021), in press.

[12]

N. Bleistein and J. K. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18 (1977), 194-201.  doi: 10.1063/1.523256.

[13]

P. Bolan, 3D Shepp-Logan Phantom, MATLAB Central File Exchange, (2021). Available from: https://www.mathworks.com/matlabcentral/fileexchange/50974-3d-shepp-logan-phantom.

[14]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[16]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Diff. Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.

[17]

M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Probl., 25 (2009), 115005. doi: 10.1088/0266-5611/25/11/115005.

[18]

R. Griesmaier and C. Schmiedecke, A factorization method for multifrequency inverse source problems with sparse far field measurements, SIAM J. Imaging Sci., 10 (2017), 2119-2139.  doi: 10.1137/17M111290X.

[19]

S. Kusiak and J. Sylvester, The scattering support, Commun. Pur. Appl. Math., 56 (2003), 1525-1548.  doi: 10.1002/cpa.3038.

[20]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.

[21]

J. LiH. Liu and S. Ma, Determining a random Schrödinger operator: both potential and source are random, Commun. Math. Phys., 381 (2021), 527-556.  doi: 10.1007/s00220-020-03889-9.

[22]

J. LiH. Liu and H. Sun, On a gesture-computing technique using electromagnetic waves, Inverse Probl. Imag., 12 (2018), 677-696.  doi: 10.3934/ipi.2018029.

[23]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Probl., 31 (2015), 105005. doi: 10.1088/0266-5611/31/10/105005.

[24]

H. LiuY. Wang and C. Yang, Mathematical design of a novel gesture-based instruction/input device using wave detection, SIAM J. Imaging Sci., 9 (2016), 822-841.  doi: 10.1137/16M1063551.

[25]

G. WangF. MaY. Guo and J. Li, Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differ. Equations, 265 (2018), 417-443.  doi: 10.1016/j.jde.2018.02.036.

[26]

X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Probl., 33 (2017), 035001. doi: 10.1088/1361-6420/aa573c.

[27]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving emitter, Inverse Probl., 33 (2017), 105009. doi: 10.1088/1361-6420/aa873f.

[28]

D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Probl., 31 (2015), 035007. doi: 10.1088/0266-5611/31/3/035007.

[29]

D. ZhangY. GuoJ. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Comm. Comput. Phys., 25 (2019), 1328-1356.  doi: 10.4208/cicp.oa-2018-0020.

show all references

References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, J. Math. Anal. Appl., 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.

[2]

R. Albanese and P. B. Monk, The inverse source problem for Maxwell's equations, Inverse Probl., 22 (2006), 1023-1035.  doi: 10.1088/0266-5611/22/3/018.

[3]

C. J. S. Alves, N. F. M. Martins and N. C. Roberty, Identification and reconstruction of elastic body forces, Inverse Probl., 30 (2014), 055015. doi: 10.1088/0266-5611/30/5/055015.

[4]

H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Vol. 52, Princeton University Press, 2015. doi: 10.1515/9781400866625.

[5]

T. S. AngelA. Kirsch and R. E. Kleinmann, Antenna control and optimization, Proc. IEEE, 79 (1991), 1559-1568.  doi: 10.1109/5.104230.

[6]

T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Probl., 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.

[7]

G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Probl., 31 (2015), 093001. doi: 10.1088/0266-5611/31/9/093001.

[8]

E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.  doi: 10.1137/18M1182048.

[9]

E. Blåsten and Y. -H. Lin, Radiating and non-radiating sources in elasticity, Inverse Probl., 35 (2019), 015005. doi: 10.1088/1361-6420/aae99e.

[10]

E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, (2020), in press.

[11]

E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, SIAM J. Math. Anal., (2021), in press.

[12]

N. Bleistein and J. K. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18 (1977), 194-201.  doi: 10.1063/1.523256.

[13]

P. Bolan, 3D Shepp-Logan Phantom, MATLAB Central File Exchange, (2021). Available from: https://www.mathworks.com/matlabcentral/fileexchange/50974-3d-shepp-logan-phantom.

[14]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[16]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Diff. Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.

[17]

M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Probl., 25 (2009), 115005. doi: 10.1088/0266-5611/25/11/115005.

[18]

R. Griesmaier and C. Schmiedecke, A factorization method for multifrequency inverse source problems with sparse far field measurements, SIAM J. Imaging Sci., 10 (2017), 2119-2139.  doi: 10.1137/17M111290X.

[19]

S. Kusiak and J. Sylvester, The scattering support, Commun. Pur. Appl. Math., 56 (2003), 1525-1548.  doi: 10.1002/cpa.3038.

[20]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.

[21]

J. LiH. Liu and S. Ma, Determining a random Schrödinger operator: both potential and source are random, Commun. Math. Phys., 381 (2021), 527-556.  doi: 10.1007/s00220-020-03889-9.

[22]

J. LiH. Liu and H. Sun, On a gesture-computing technique using electromagnetic waves, Inverse Probl. Imag., 12 (2018), 677-696.  doi: 10.3934/ipi.2018029.

[23]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Probl., 31 (2015), 105005. doi: 10.1088/0266-5611/31/10/105005.

[24]

H. LiuY. Wang and C. Yang, Mathematical design of a novel gesture-based instruction/input device using wave detection, SIAM J. Imaging Sci., 9 (2016), 822-841.  doi: 10.1137/16M1063551.

[25]

G. WangF. MaY. Guo and J. Li, Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differ. Equations, 265 (2018), 417-443.  doi: 10.1016/j.jde.2018.02.036.

[26]

X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Probl., 33 (2017), 035001. doi: 10.1088/1361-6420/aa573c.

[27]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving emitter, Inverse Probl., 33 (2017), 105009. doi: 10.1088/1361-6420/aa873f.

[28]

D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Probl., 31 (2015), 035007. doi: 10.1088/0266-5611/31/3/035007.

[29]

D. ZhangY. GuoJ. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Comm. Comput. Phys., 25 (2019), 1328-1356.  doi: 10.4208/cicp.oa-2018-0020.

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 $
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 $
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)} $
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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