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A Numerical Study of Complex Reconstruction in Inverse Elastic Scattering

Part of: Equations of mathematical physics and other areas of application Waves Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  17 May 2016

Guanghui Hu ,
Jingzhi Li ,
Hongyu Liu  and
Qi Wang
Guanghui Hu*
Affiliation:
Beijing Computational Science Research Center, Beijing 100094, P.R. China
Jingzhi Li*
Affiliation:
Beijing Computational Science Research Center, Beijing 100094, P.R. China
Hongyu Liu*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Qi Wang*
Affiliation:
Department of Computing Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, P.R. China
*
*Corresponding author., Email addresses:, hu@csrc.ac.en(G. Hu), li.jz@sustc.edu.en(J. Li), hongyu. liuip@gmail.com(H. Liu), qi.wang.xjtumath@gmail.com(Q. Wang)
*Corresponding author., Email addresses:, hu@csrc.ac.en(G. Hu), li.jz@sustc.edu.en(J. Li), hongyu. liuip@gmail.com(H. Liu), qi.wang.xjtumath@gmail.com(Q. Wang)
*Corresponding author., Email addresses:, hu@csrc.ac.en(G. Hu), li.jz@sustc.edu.en(J. Li), hongyu. liuip@gmail.com(H. Liu), qi.wang.xjtumath@gmail.com(Q. Wang)
*Corresponding author., Email addresses:, hu@csrc.ac.en(G. Hu), li.jz@sustc.edu.en(J. Li), hongyu. liuip@gmail.com(H. Liu), qi.wang.xjtumath@gmail.com(Q. Wang)
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Abstract

The purpose of this paper is to numerically realize the inverse scattering scheme proposed in [19] of reconstructing complex elastic objects by a single far-field measurement. The unknown elastic scatterers might consist of both rigid bodies and traction-free cavities with components of multiscale sizes presented simultaneously. We conduct extensive numerical experiments to show the effectiveness and efficiency of the imaging scheme proposed in [19]. Moreover, we develop a two-stage technique, which can significantly speed up the reconstruction to yield a fast imaging scheme.

Keywords

Inverse elastic scatteringcomplex scattererscavities and rigid elastic bodiessingle-shot method

MSC classification

Secondary: 74J20: Wave scattering 74J25: Inverse problems 35Q74: PDEs in connection with mechanics of deformable solids 35R30: Inverse problems
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1265 - 1286
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Abubakar, I., Scattering of plane elastic waves at rough surface I, Proc. Cambridge Philos. Soc., 58 (1962), 136–157.Google Scholar
[2]Alves, C. and Ammari, H., Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium, SIAM J. Appl. Math., 62 (2001), 503–531.Google Scholar
[3]Ammari, H., Bretin, E., Garnier, J., Jing, W., Kang, H., and Wahab, A., Localization, stability, and resolution of topological derivative based imaging functionals in elasticity, SIAM Journal on Imaging Sciences, 6 (2013), 2174–2212.Google Scholar
[4]Ammari, H., Kang, H., Nakamura, G., and Tanuma, K., Complete asymptotic expansions of solu-tions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elasticity, 67 (2002), 97–129.Google Scholar
[5]Ammari, H. and Kang, H., Polarization and moment tensors. With applications to inverse problems and effective medium theory, Appl. Math. Sci., 162, Springer, New York, 2007.Google Scholar
[6]Ammari, H., Kang, H., and Lee, H.,Asymptotic expansions for eigenvalues of the Lamé system in the presence of small inclusions, Comm. Part. Diff. Equ., 32 (2007), 1715–1736.Google Scholar
[7]Alves, C. J. and Kress, R., On the far-field operator in elastic obstacle scattering, IMA J. Appl. Math., 67(2002), 1–21.Google Scholar
[8]Bao, G., Liu, H. Y. and Zou, J., Nearly cloaking the full Maxwell equations: cloaking active contents with general conducting layers,J. Math. Pures Appl., in press, 2013.Google Scholar
[9]Benites, R., Aki, K. and Yomogida, K., Multiple Scattering of SH waves in 2D media with many cavities, Pageoph, 138 (1992), 353–390.Google Scholar
[10]Challa, D. P. and Sini, M., The Foldy-Lax approximation of the scattered waves by many small bodies for the Lame system, arXiv: 1308.3072Google Scholar
[11]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer-Verlag, Berlin, 1998.Google Scholar
[12]Elschner, J. and Yamamoto, M., Uniqueness in inverse elastic scattering with finitely many incident waves, Inverse Problems, 26 (2010), 045005.Google Scholar
[13]Fokkema, J. T., Reflection and transmission of elastic waves by the spatially periodic interface between two solids (theory of the integral-equation method), Wave Motion, 2 (1980), 375–393.Google Scholar
[14]Fokkema, J. T. and Van den Berg, P. M., Elastodynamic diffraction by a periodic rough surface (stress-free boundary), J. Acoust. Soc. Am., 62 (1977), 1095–1101.Google Scholar
[15]Gintides, D. and Sini, M., Identification of obstacles using only the scattered P-waves or the Scattered S-waves, Inverse Problems and Imaging, 6 (2012), 39–55.Google Scholar
[16]Griesmaier, R., Multi-frequence orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2001), 085005.Google Scholar
[17]Häner, P., Hsiao, G. C., Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525–534.Google Scholar
[18]Hu, G., Kirsch, A. and Sini, M., Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems 29 (2013), 015009.Google Scholar
[19]Hu, G., Li, J. and Liu, H., Recovering complex elastic scatterers by a single far-field pattern, J. Differential Equations, 257 (2014), 469–489.Google Scholar
[20]Hu, G., Li, J., Liu, H. and Sun, H., Inverse elastic scattering for multiscale rigid bodies with a single far-field pattern, SIAM J. Imaging Sciences, 7 (2014), 1799–1825.Google Scholar
[21]Hsiao, G. C. and Wendland, W. L., Boundary Integral Equations, Appl. Math. Sci., 164, Springer, Berlin Heidelberg, 2008.Google Scholar
[22]Kress, R., Linear Integral Equations, Berlin, Springer, 1989.Google Scholar
[23]Kupradze, V. D.et al, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam, North-Holland, 1979.Google Scholar
[24]Landau, L.D., Lifshitz, E.M., Theory of Elasticity, Pergamon Press, 1986.Google Scholar
[25]Li, J. and J. Zou, , A direct sampling method for inverse scattering using far-field data, Inverse Problems and Imaging, 7 (2013), 757–775.Google Scholar
[26]Liu, H., On near-cloak in acoustic scattering, J. Differential Equations, 254 (2013), 1230–1246.Google Scholar
[27]Liu, H. and Sun, H., Enhanced near-cloak by FSH lining, J. Math. Pures Appl., 99 (2013), 17–42.Google Scholar
[28]Nakamura, G. and Uhlmann, G., Identification of Lamé parameters by boundary measurements, Amer. J. Math., 115 (1993), 1161–1187.Google Scholar
[29]Nakamura, G. and Uhlmann, G., Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 118 (1994), 457–474. Erratum, 152 (2003), 205-207.Google Scholar
[30]Nakamura, G. and Uhlmann, G., Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal., 26 (1995), 263–279.Google Scholar
[31]Nakamura, G., Kazumi, T. and Uhlmann, G., Layer stripping for a transversely isotropic elastic medium, SIAM J. Appl. Math., 59 (1999), 1879–1891.Google Scholar
[32]Rachele, L. V., Uniqueness of the density in an inverse problem for isotropic elastodynamics, Trans. Amer. Math. Soc., 355 (2003), 4781–4806.Google Scholar
[33]Sherwood, J. W. C., Elastic wave propagation in a semi-infinite solid medium, Proc. Phys. Soc., 71 1958, 207–219.CrossRefGoogle Scholar