American Institute of Mathematical Sciences

June  2019, 13(3): 545-573. doi: 10.3934/ipi.2019026

Inverse obstacle scattering for elastic waves in three dimensions

 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA

* Corresponding author: Peijun Li

Received  May 2018 Revised  October 2018 Published  March 2019

Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle. The scattering problem is reformulated into a boundary value problem by introducing a transparent boundary condition. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Citation: Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026
References:
 [1] H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Princeton University Press, New Jersey, 2015. doi: 10.1515/9781400866625. Google Scholar [2] G. Bao, G. Hu, J. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301. doi: 10.1016/j.matpur.2018.01.007. Google Scholar [3] G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21PP. doi: 10.1088/0266-5611/31/9/093001. Google Scholar [4] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159. Google Scholar [5] M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), 1-50. doi: 10.1088/0266-5611/21/2/R01. Google Scholar [6] J. H. Bramble and J. E. Pasciak, A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: A priori estimates in $\boldsymbol H^1$, J. Math. Anal. Appl., 345 (2008), 396-404. doi: 10.1016/j.jmaa.2008.04.028. Google Scholar [7] J. H. Bramble, J. E. Pasciak and D. Trenev, Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem, Math. Comp., 79 (2010), 2079-2101. doi: 10.1090/S0025-5718-10-02355-0. Google Scholar [8] Z. Chen, X. Xiang and X. Zhang, Convergence of the PML method for elastic wave scattering problems, Math. Comp., 85 (2016), 2687-2714. doi: 10.1090/mcom/3100. Google Scholar [9] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. Google Scholar [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5. Google Scholar [11] J. Elschner and M. Yamamoto, Uniqueness in inverse elastic scattering with finitely many incident waves, Inverse Problems, 26 (2010), 045005, 8PP. doi: 10.1088/0266-5611/26/4/045005. Google Scholar [12] G. K. Gächter and M. J. Grote, Dirichlet-to-Neumann map three-dimensional elastic waves, Wave Motion, 37 (2003), 293-311. doi: 10.1016/S0165-2125(02)00091-4. Google Scholar [13] D. Givoli and J. B. Keller, Non-reflecting boundary conditions for elastic waves, Wave Motion, 12 (1990), 261-279. doi: 10.1016/0165-2125(90)90043-4. Google Scholar [14] H. Haddar and R. Kress, On the Fréchet derivative for obstacle scattering with an impedance boundary condition, SIAM J. Appl. Math., 65 (2004), 94-208. doi: 10.1137/S0036139903435413. Google Scholar [15] P. Hähner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534. doi: 10.1088/0266-5611/9/5/002. Google Scholar [16] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013. Google Scholar [17] G. Hu, A. Kirsch, and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21PP. doi: 10.1088/0266-5611/29/1/015009. Google Scholar [18] G. Hu, J. Li, H. Liu and H. Sun, Inverse elastic scattering for multiscale rigid bodies with a single far-field pattern, SIAM J. Imaging Sci., 7 (2014), 1799-1825. doi: 10.1137/130944187. Google Scholar [19] X. Jiang, P. Li, J. Lv and W. Zheng, An adaptive finite element PML method for the elastic wave scattering problem in periodic structures, ESAIM: Math. Model. Numer. Anal., 51 (2017), 2017-2047. doi: 10.1051/m2an/2017018. Google Scholar [20] M. Kar and M. Sini, On the inverse elastic scattering by interfaces using one type of scattered waves, J. Elast., 118 (2015), 15-38. doi: 10.1007/s10659-014-9474-5. Google Scholar [21] A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96. doi: 10.1088/0266-5611/9/1/005. Google Scholar [22] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1959. Google Scholar [23] F. Le Louër, A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems, Inverse Problems, 31 (2015), 115006, 27PP. doi: 10.1088/0266-5611/31/11/115006. Google Scholar [24] F. Le Louër, On the Fréchet derivative in elastic obstacle scattering, SIAM J. Appl. Math., 72 (2012), 1493-1507. doi: 10.1137/110834160. Google Scholar [25] P. Li, Y. Wang, Z. Wang and Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Problems, 32 (2016), 115018, 24PP. doi: 10.1088/0266-5611/32/11/115018. Google Scholar [26] P. Monk, Finite Element Methods for Maxwell Equations, Oxford University Press, New York, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar [27] G. Nakamura and G. Uhlmann, Inverse problems at the boundary of elastic medium, SIAM J. Math. Anal., 26 (1995), 263-279. doi: 10.1137/S0036141093247494. Google Scholar [28] J.-C. Nédélec, Acoustic and Electromagnetic Equations Integral Representations for Harmonic Problems, Springer, 2000.Google Scholar [29] R. Potthast, Domain derivatives in electromagnetic scattering, Math. Meth. Appl. Sci., 19 (1996), 1157-1175. doi: 10.1002/(SICI)1099-1476(199610)19:15<1157::AID-MMA814>3.0.CO;2-Y. Google Scholar [30] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. Google Scholar

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References:
 [1] H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Princeton University Press, New Jersey, 2015. doi: 10.1515/9781400866625. Google Scholar [2] G. Bao, G. Hu, J. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301. doi: 10.1016/j.matpur.2018.01.007. Google Scholar [3] G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21PP. doi: 10.1088/0266-5611/31/9/093001. Google Scholar [4] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159. Google Scholar [5] M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), 1-50. doi: 10.1088/0266-5611/21/2/R01. Google Scholar [6] J. H. Bramble and J. E. Pasciak, A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: A priori estimates in $\boldsymbol H^1$, J. Math. Anal. Appl., 345 (2008), 396-404. doi: 10.1016/j.jmaa.2008.04.028. Google Scholar [7] J. H. Bramble, J. E. Pasciak and D. Trenev, Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem, Math. Comp., 79 (2010), 2079-2101. doi: 10.1090/S0025-5718-10-02355-0. Google Scholar [8] Z. Chen, X. Xiang and X. Zhang, Convergence of the PML method for elastic wave scattering problems, Math. Comp., 85 (2016), 2687-2714. doi: 10.1090/mcom/3100. Google Scholar [9] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. Google Scholar [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5. Google Scholar [11] J. Elschner and M. Yamamoto, Uniqueness in inverse elastic scattering with finitely many incident waves, Inverse Problems, 26 (2010), 045005, 8PP. doi: 10.1088/0266-5611/26/4/045005. Google Scholar [12] G. K. Gächter and M. J. Grote, Dirichlet-to-Neumann map three-dimensional elastic waves, Wave Motion, 37 (2003), 293-311. doi: 10.1016/S0165-2125(02)00091-4. Google Scholar [13] D. Givoli and J. B. Keller, Non-reflecting boundary conditions for elastic waves, Wave Motion, 12 (1990), 261-279. doi: 10.1016/0165-2125(90)90043-4. Google Scholar [14] H. Haddar and R. Kress, On the Fréchet derivative for obstacle scattering with an impedance boundary condition, SIAM J. Appl. Math., 65 (2004), 94-208. doi: 10.1137/S0036139903435413. Google Scholar [15] P. Hähner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534. doi: 10.1088/0266-5611/9/5/002. Google Scholar [16] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013. Google Scholar [17] G. Hu, A. Kirsch, and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21PP. doi: 10.1088/0266-5611/29/1/015009. Google Scholar [18] G. Hu, J. Li, H. Liu and H. Sun, Inverse elastic scattering for multiscale rigid bodies with a single far-field pattern, SIAM J. Imaging Sci., 7 (2014), 1799-1825. doi: 10.1137/130944187. Google Scholar [19] X. Jiang, P. Li, J. Lv and W. Zheng, An adaptive finite element PML method for the elastic wave scattering problem in periodic structures, ESAIM: Math. Model. Numer. Anal., 51 (2017), 2017-2047. doi: 10.1051/m2an/2017018. Google Scholar [20] M. Kar and M. Sini, On the inverse elastic scattering by interfaces using one type of scattered waves, J. Elast., 118 (2015), 15-38. doi: 10.1007/s10659-014-9474-5. Google Scholar [21] A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96. doi: 10.1088/0266-5611/9/1/005. Google Scholar [22] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1959. Google Scholar [23] F. Le Louër, A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems, Inverse Problems, 31 (2015), 115006, 27PP. doi: 10.1088/0266-5611/31/11/115006. Google Scholar [24] F. Le Louër, On the Fréchet derivative in elastic obstacle scattering, SIAM J. Appl. Math., 72 (2012), 1493-1507. doi: 10.1137/110834160. Google Scholar [25] P. Li, Y. Wang, Z. Wang and Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Problems, 32 (2016), 115018, 24PP. doi: 10.1088/0266-5611/32/11/115018. Google Scholar [26] P. Monk, Finite Element Methods for Maxwell Equations, Oxford University Press, New York, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar [27] G. Nakamura and G. Uhlmann, Inverse problems at the boundary of elastic medium, SIAM J. Math. Anal., 26 (1995), 263-279. doi: 10.1137/S0036141093247494. Google Scholar [28] J.-C. Nédélec, Acoustic and Electromagnetic Equations Integral Representations for Harmonic Problems, Springer, 2000.Google Scholar [29] R. Potthast, Domain derivatives in electromagnetic scattering, Math. Meth. Appl. Sci., 19 (1996), 1157-1175. doi: 10.1002/(SICI)1099-1476(199610)19:15<1157::AID-MMA814>3.0.CO;2-Y. Google Scholar [30] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. Google Scholar
Example 1: A bean-shaped obstacle. (a) the exact surface; (b) the initial guess; (c) the reconstructed surface; (d)–(f) the corresponding cross section of the exact surface along plane $x_1 = 0, x_2 = 0, x_3 = 0$, respectively; (g)–(i) the corresponding cross section of the reconstructed surface along plane $x_1 = 0, x_2 = 0, x_3 = 0$, respectively
Example 2: A cushion-shaped obstacle. (a) the exact surface; (b) the initial guess; (c) the reconstructed surface; (d)–(f) the corresponding cross section of the exact surface along the plane $x_1 = 0, x_2 = 0, x_3 = 0$, respectively; (d)–(f) the corresponding cross section of the reconstructed surface along the plane $x_1 = 0, x_2 = 0, x_3 = 0$, respectively
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