American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The broken ray transform in $n$ dimensions with flat reflecting boundary
  • IPI Home
  • This Issue
  • Next Article
    Finite-dimensional attractors for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation in image inpainting
February  2015, 9(1): 127-141. doi: 10.3934/ipi.2015.9.127

Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type

Johannes Elschner 1, , Guanghui Hu 2, and  Masahiro Yamamoto 3,

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin
2. 

Weierstrass Institute, Mohrenstr. 39, 10117 Berlin
3. 

Graduate School of Mathematical Sciences University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914

Received  June 2014 Revised  September 2014 Published  January 2015

Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.
Keywords: Inverse scattering, rough surface, Navier equation, uniqueness, diffraction grating., linear elasticity, Dirichlet boundary condition.
Mathematics Subject Classification: Primary: 74J25; Secondary: 74J20, 35R30, 35Q7.
Citation: Johannes Elschner, Guanghui Hu, Masahiro Yamamoto. Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type. Inverse Problems & Imaging, 2015, 9 (1) : 127-141. doi: 10.3934/ipi.2015.9.127
References:
[1]

G. Bao, H. Zhang and J. Zou, Unique determination of periodic polyhedral structures by scattered electromagnetic fields II: The resonance case,, Trans. Amer. Math. Soc., 366 (2014), 1333.  doi: 10.1090/S0002-9947-2013-05761-3.  Google Scholar

[2]

A. Charalambopoulos, D. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures,, Inverse Problems, 17 (2001), 1923.  doi: 10.1088/0266-5611/17/6/323.  Google Scholar

[3]

M. Costabel, M. Dauge and Y. Lafranche, Fast semi-analytic computation of elastic edge singularities,, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 2111.  doi: 10.1016/S0045-7825(00)00226-7.  Google Scholar

[4]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer, (1998).  doi: 10.1007/978-3-662-03537-5.  Google Scholar

[5]

J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings,, Math. Meth. Appl. Sci., 33 (2010), 1924.  doi: 10.1002/mma.1305.  Google Scholar

[6]

J. Elschner and G. Hu, Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/11/115002.  Google Scholar

[7]

J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions,, Meth. Appl. Anal., 18 (2011), 215.  doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar

[8]

J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles,, Commun. Comput. Phys., 12 (2012), 1434.   Google Scholar

[9]

J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces,, Appl. Anal., 94 (2015), 251.  doi: 10.1080/00036811.2014.887695.  Google Scholar

[10]

J. Elschner, G. Schmidt and M. Yamamoto, Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number,, J. Inverse Ill-Posed Probl., 11 (2003), 235.  doi: 10.1515/156939403769237024.  Google Scholar

[11]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures,, Inverse Problems, 13 (1997), 351.  doi: 10.1088/0266-5611/13/2/010.  Google Scholar

[12]

G. Hu, Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation,, Appl. Anal., 91 (2012), 703.  doi: 10.1080/00036811.2011.587811.  Google Scholar

[13]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[14]

P. Grisvard, Singularités en élasticité,, Arch. Rational Mech. Anal., 107 (1989), 157.  doi: 10.1007/BF00286498.  Google Scholar

[15]

A. Kirsch, Diffraction by periodic structures,, In 'Proc. Lapland Conf. Inverse Problems' (ed. L. Päivärinta et al), 422 (1993), 87.  doi: 10.1007/3-540-57195-7_11.  Google Scholar

[16]

C. B. Morrey, Jr. and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations,, Communications on Pure and Applied Mathematics, 10 (1957), 271.  doi: 10.1002/cpa.3160100204.  Google Scholar

[17]

S. Nakagawa, K. T. Nihei, L. R. Myer and E. L. Majer, Three-dimensional elastic wave scattering by a layer containing vertical periodic fractures,, J. Acoust. Soc. Am., 113 (2003), 3012.  doi: 10.1121/1.1572139.  Google Scholar

[18]

A. Rössle, Corner singularities and regularity of weak solution for the two-dimensional Lamé equations on domains with angular corners,, Journal of Elasticity, 60 (2000), 57.  doi: 10.1023/A:1007639413619.  Google Scholar

show all references

References:
[1]

G. Bao, H. Zhang and J. Zou, Unique determination of periodic polyhedral structures by scattered electromagnetic fields II: The resonance case,, Trans. Amer. Math. Soc., 366 (2014), 1333.  doi: 10.1090/S0002-9947-2013-05761-3.  Google Scholar

[2]

A. Charalambopoulos, D. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures,, Inverse Problems, 17 (2001), 1923.  doi: 10.1088/0266-5611/17/6/323.  Google Scholar

[3]

M. Costabel, M. Dauge and Y. Lafranche, Fast semi-analytic computation of elastic edge singularities,, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 2111.  doi: 10.1016/S0045-7825(00)00226-7.  Google Scholar

[4]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer, (1998).  doi: 10.1007/978-3-662-03537-5.  Google Scholar

[5]

J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings,, Math. Meth. Appl. Sci., 33 (2010), 1924.  doi: 10.1002/mma.1305.  Google Scholar

[6]

J. Elschner and G. Hu, Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/11/115002.  Google Scholar

[7]

J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions,, Meth. Appl. Anal., 18 (2011), 215.  doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar

[8]

J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles,, Commun. Comput. Phys., 12 (2012), 1434.   Google Scholar

[9]

J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces,, Appl. Anal., 94 (2015), 251.  doi: 10.1080/00036811.2014.887695.  Google Scholar

[10]

J. Elschner, G. Schmidt and M. Yamamoto, Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number,, J. Inverse Ill-Posed Probl., 11 (2003), 235.  doi: 10.1515/156939403769237024.  Google Scholar

[11]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures,, Inverse Problems, 13 (1997), 351.  doi: 10.1088/0266-5611/13/2/010.  Google Scholar

[12]

G. Hu, Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation,, Appl. Anal., 91 (2012), 703.  doi: 10.1080/00036811.2011.587811.  Google Scholar

[13]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[14]

P. Grisvard, Singularités en élasticité,, Arch. Rational Mech. Anal., 107 (1989), 157.  doi: 10.1007/BF00286498.  Google Scholar

[15]

A. Kirsch, Diffraction by periodic structures,, In 'Proc. Lapland Conf. Inverse Problems' (ed. L. Päivärinta et al), 422 (1993), 87.  doi: 10.1007/3-540-57195-7_11.  Google Scholar

[16]

C. B. Morrey, Jr. and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations,, Communications on Pure and Applied Mathematics, 10 (1957), 271.  doi: 10.1002/cpa.3160100204.  Google Scholar

[17]

S. Nakagawa, K. T. Nihei, L. R. Myer and E. L. Majer, Three-dimensional elastic wave scattering by a layer containing vertical periodic fractures,, J. Acoust. Soc. Am., 113 (2003), 3012.  doi: 10.1121/1.1572139.  Google Scholar

[18]

A. Rössle, Corner singularities and regularity of weak solution for the two-dimensional Lamé equations on domains with angular corners,, Journal of Elasticity, 60 (2000), 57.  doi: 10.1023/A:1007639413619.  Google Scholar

[1]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[2]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
[3]

Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013
[4]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[5]

Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033
[6]

H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure & Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907
[7]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295
[8]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095
[9]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153
[10]

Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
[11]

Victor Isakov. On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data. Inverse Problems & Imaging, 2008, 2 (1) : 151-165. doi: 10.3934/ipi.2008.2.151
[12]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[13]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139
[14]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803
[15]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[16]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[17]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431
[18]

Yang Yang, Jian Zhai. Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity. Inverse Problems & Imaging, 2019, 13 (6) : 1309-1325. doi: 10.3934/ipi.2019057
[19]

Shi Jin, Dongsheng Yin. Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction. Kinetic & Related Models, 2011, 4 (1) : 295-316. doi: 10.3934/krm.2011.4.295
[20]

Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences