Numerical solution of an inverse obstacle scattering problem for elastic waves via the Helmholtz decomposition. (English) Zbl 1473.78010
Summary: Consider an inverse obstacle scattering problem in an open space which is filled with a homogeneous and isotropic elastic medium. 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, a new approach is proposed for numerical solution of the inverse problem. By introducing two scalar potential functions, the method uses the Helmholtz decomposition to split the displacement of the elastic wave equation into the compressional and shear waves, which satisfy a coupled boundary value problem of the Helmholtz equations. The domain derivative is studied for the coupled Helmholtz system. In particular, we show that the domain derivative of the potentials is the Helmholtz decomposition of the domain derivative of the displacement for the elastic wave equation. Numerical results are presented to demonstrate the effectiveness of the proposed method.
MSC:
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
Keywords:
elastic wave equation; inverse obstacle scattering; domain derivative; Helmholtz decompositionSoftware:
FreeFem++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. · Zbl 1332.35002 |
| [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. · Zbl 1397.35094 |
| [3] | G. Bao, P. Li, J. Lin, and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Probl., 31(2015), 093001. · Zbl 1332.78019 |
| [4] | G. Bao, P. Li, and H. Wu, A computational inverse diffraction grating problem, J. Opt. Soc. Am. A, 29(2012), 394-399. |
| [5] | G. Bao, J. Lv, and P. Li, Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30(2013), 293-299. |
| [6] | M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Probl., 21(2005), 1-50 · Zbl 1070.35118 |
| [7] | A. Charalambopoulos, On the Fréchet differentiability of the boundary integral operators in the inverse elastic scattering problems, Inverse Probl., 11(1995), 1137-1161. · Zbl 0847.35144 |
| [8] | J. Chen and G. Huang, A Direct Imaging Method for Inverse Electromagnetic Scattering Problem in Rectangular Waveguide, Commun. Comput. Phys., 23(2017), 1415-1433. · Zbl 1488.65582 |
| [9] | R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, An improved operator expansion algorithm for direct and inverse scattering computations, Waves Random Media, 9(1999), 441-457. · Zbl 0973.76080 |
| [10] | Z. Chen and G. Huang, Phaseless Imaging by Reverse Time Migration: Acoustic Waves, Numer. Math.-Theory Methods Appl., 10(2017),1-21. · Zbl 1389.78006 |
| [11] | J. Elschner and M. Yamamoto, Uniqueness in inverse elastic scattering with nitely many incident waves, Inverse Probl., 26(2010), 045005. · Zbl 1189.35374 |
| [12] | H. Haddar and R. Kress, On the Fréchet derivative for obstacle scattering with an impedance boundary condition, SIAM J. Appl. Math., 65(2004), 194-208. · Zbl 1084.35103 |
| [13] | P. Hahner and G. C. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Probl., 9(1993), 525-534. · Zbl 0789.35172 |
| [14] | F. Hecht, New development in FreeFem++, J. Numer. Math. 20(2012), 251-265. · Zbl 1266.68090 |
| [15] | F. Hettlich, Fréchet derivatives in inverse obstacle scattering, Inverse Probl., 11(1995), 371-382. Erratum: Fréchet derivatives in inverse obstacle scattering, Inverse Probl., 14(1998), 209-210. · Zbl 0821.35147 |
| [16] | G. Hu. A. Kirsch, and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Probl., 29(2013), 015009. · Zbl 1334.74047 |
| [17] | 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. · Zbl 1305.74046 |
| [18] | M. Kar and M. Sini, On the inverse elastic scattering by interfaces using one type of scattered waves, J. Elast., 118(2015), 15-38 · Zbl 1316.35299 |
| [19] | A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Probl., 9(1993), 81-96. · Zbl 0773.35085 |
| [20] | R. Kress, Inverse elastic scattering from a crack, Inverse Probl., 12(1996), 667-684. · Zbl 0864.35118 |
| [21] | L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Oxford: Pergamon 1986. · Zbl 0014.37504 |
| [22] | F. Le Louër, A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems, Inverse Probl., 31(2015), 115006. · Zbl 1329.35353 |
| [23] | F. Le Louër, On the Fréchet derivative in elastic obstacle scattering SIAM J. Appl. Math., 72(2012), 1493-1507 · Zbl 1259.35154 |
| [24] | P. Li, Y. Wang, Z. Wang, and Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Probl., 32(2016), 115018. · Zbl 1433.74064 |
| [25] | P. Li P, Y. Wang, and Y. Zhao, Inverse elastic surface scattering with near-field data, Inverse Probl., 31(2015), 035009 · Zbl 1325.35284 |
| [26] | P. Li and X. Yuan, Inverse obstacle scattering for elastic waves in three dimensions, Inverse Probl., 32(2017), 115018. · Zbl 1435.74045 |
| [27] | G. Nakamura and G. Uhlmann, Inverse problems at the boundary of elastic medium, SIAM J. Math. Anal., 26(1995), 263-279. · Zbl 0840.35122 |
| [28] | R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scat-tering, Inverse Probl., 10(1994), 431-447. · Zbl 0805.35157 |
| [29] | R. Potthast, Domain derivatives in electromagnetic scattering, Math. Meth. Appl. Sci., 19(1996), 1157-1175. · Zbl 0866.35019 |
| [30] | Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, New York, 1949. · Zbl 0034.35702 |
| [31] | M. Sini and N. T. Thanh, Inverse acoustic obstacle scattering problems using multifrequency measurements, Inverse Probl. Imaging, 6(2012), 749-773. · Zbl 1261.35154 |
| [32] | Andre Roger, Newton-Kantorovitch algorithm applied to an electromagnetic inverse prob-lem, IEEE Trans. Antennas Propag., 29(1981), 232-238. · Zbl 0947.65503 |
| [33] | M. Zhang and J. Lv, Numerical method of profile reconstruction for a periodic transmission problem from single-sided data, Commun. Computat. Phys., 24(2018), 435-453. · Zbl 1488.35630 |
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