# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2022012
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## Recovering a bounded elastic body by electromagnetic far-field measurements

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China 2 LSEC and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 3 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

*Corresponding author: Jiaqing Yang

Received  October 2021 Revised  February 2022 Early access March 2022

Fund Project: J. Yang is supported in part by the NNSF of China grant No. 12122114 and The Young Talent Support Plan of Xi'an Jiaotong University

This paper is concerned with the scattering of a time-harmonic electromagnetic wave by a three-dimensional elastic body. The general transmission conditions are considered to model the interaction between the electromagnetic field and the elastic body on the interface by Voigt's model. The existence of a unique solution is first proved in an appropriate Sobolev space by employing the variational method with the classical Fredholm alternative. The inverse problem is then investigated to recover the elastic body by the scattered wave-field data. It is shown that the shape and location of the body is uniquely determined by the fixed energy magnetic (or electric) far-field measurements corresponding to incident plane waves with all polarizations.

Citation: Tielei Zhu, Jiaqing Yang, Bo Zhang. Recovering a bounded elastic body by electromagnetic far-field measurements. Inverse Problems and Imaging, doi: 10.3934/ipi.2022012
##### References:
 [1] A. Bernardo, A. Marquez and S. Meddahi, Analysis of an interaction problem between an electromagnetic field and an elastic body, Int. J. Numer. Anal. Model., 7 (2010), 749-765. [2] F. Cakoni and G. C. Hsiao, Mathematical model of the interaction problem between electromagnetic field and elastic body, Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, (eds. A. Wirgin), World Scientific, (2002), 48–54. doi: 10.1142/9789812704405_0009. [3] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{rd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [4] G. N. Gatica, G. C. Hsiao and S. Meddahi, A coupled mixed finite element method for the interaction problem between an electromagnetic field and an elastic body, SIAM J. Numer. Anal., 48 (2010), 1338-1368.  doi: 10.1137/090754212. [5] P. Hähner, On the uniqueness of the shape of a penetrable, anisotropic obstacle, J. Comput. Appl. Math., 116 (2010), 167-180.  doi: 10.1016/S0377-0427(99)00323-4. [6] G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., 218 (2000), 139-163.  doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S. [7] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.  doi: 10.1093/acprof:oso/9780199213535.001.0001. [8] A. Kirsch and A. Ruiz, The factorization method for an inverse fluid-solid interaction scattering problem, Inverse Probl. Imag., 6 (2012), 681-695.  doi: 10.3934/ipi.2012.6.681. [9] C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math., 55 (1995), 904-922.  doi: 10.1137/S0036139993259027. [10] P. Monk, Finite Element Method for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001. [11] P. Monk and V. Selgas, An inverse fluid-solid interaction problem, Inverse Probl. Imag., 3 (2009), 173-198.  doi: 10.3934/ipi.2009.3.173. [12] D. Natroshvili, S. Kharibegashvili and Z. Tediashvili, Direct and inverse fluid structure interaction problems, Rend. Mat. Appl., 20 (2000), 57-92. [13] F. Qu, J. Yang and B. Zhang, Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements, Inverse Problems, 34 (2018), 015002, 8 pp. doi: 10.1088/1361-6420/aa9c26. [14] J. Yang, B. Zhang and H. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacle with embedded objects, J. Differ. Equations, 265 (2018), 6352-6383.  doi: 10.1016/j.jde.2018.07.033.

show all references

##### References:
 [1] A. Bernardo, A. Marquez and S. Meddahi, Analysis of an interaction problem between an electromagnetic field and an elastic body, Int. J. Numer. Anal. Model., 7 (2010), 749-765. [2] F. Cakoni and G. C. Hsiao, Mathematical model of the interaction problem between electromagnetic field and elastic body, Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, (eds. A. Wirgin), World Scientific, (2002), 48–54. doi: 10.1142/9789812704405_0009. [3] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{rd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [4] G. N. Gatica, G. C. Hsiao and S. Meddahi, A coupled mixed finite element method for the interaction problem between an electromagnetic field and an elastic body, SIAM J. Numer. Anal., 48 (2010), 1338-1368.  doi: 10.1137/090754212. [5] P. Hähner, On the uniqueness of the shape of a penetrable, anisotropic obstacle, J. Comput. Appl. Math., 116 (2010), 167-180.  doi: 10.1016/S0377-0427(99)00323-4. [6] G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., 218 (2000), 139-163.  doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S. [7] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.  doi: 10.1093/acprof:oso/9780199213535.001.0001. [8] A. Kirsch and A. Ruiz, The factorization method for an inverse fluid-solid interaction scattering problem, Inverse Probl. Imag., 6 (2012), 681-695.  doi: 10.3934/ipi.2012.6.681. [9] C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math., 55 (1995), 904-922.  doi: 10.1137/S0036139993259027. [10] P. Monk, Finite Element Method for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001. [11] P. Monk and V. Selgas, An inverse fluid-solid interaction problem, Inverse Probl. Imag., 3 (2009), 173-198.  doi: 10.3934/ipi.2009.3.173. [12] D. Natroshvili, S. Kharibegashvili and Z. Tediashvili, Direct and inverse fluid structure interaction problems, Rend. Mat. Appl., 20 (2000), 57-92. [13] F. Qu, J. Yang and B. Zhang, Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements, Inverse Problems, 34 (2018), 015002, 8 pp. doi: 10.1088/1361-6420/aa9c26. [14] J. Yang, B. Zhang and H. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacle with embedded objects, J. Differ. Equations, 265 (2018), 6352-6383.  doi: 10.1016/j.jde.2018.07.033.
Interaction between electromagnetic wave and a bounded elastic body
Two different elastic bodies
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