Advances in Computational Mathematics

, Volume 44, Issue 2, pp 453–476 | Cite as

The inverse scattering problem by an elastic inclusion

  • Roman Chapko
  • Drossos Gintides
  • Leonidas Mindrinos


In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.


Linear elasticity Inverse scattering problem Integral equation method 

Mathematics Subject Classification (2010)

35P25 35R30 45G05 65N35 74J20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alessandrini, G., Morassi, A., Rosset, E.: Detecting an inclusion in an elastic body by boundary measurements. SIAM Rev. 46(3), 477–498 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Altundag, A., Kress, R.: On a two-dimensional inverse scattering problem for a dielectric. Appl. Anal. 91(4), 757–771 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alves, C.J.S., Martins, N.F.M.: The direct method of fundamental solutions and the inverse kirsch-kress method for the reconstruction of elastic inclusions or cavities. J. Integral Equations Appl. 21, 153–178 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chapko, R.: On the numerical solution of a boundary value problem in the plane elasticity for a bouble-connected domain. Math. Comput. Simulat. 66, 425–438 (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chapko, R., Ivanyshyn, O., Protsyuk, O.: On a nonlinear integral equation approach for the surface reconstruction in semi-infinite-layered domains. Inverse Prob. Sci. Eng. 21(3), 547–561 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chapko, R., Kress, R., Mönch, L.: On the numerical solution of a hypersingular integral equation for elastic scattering from a planar crack. IMA J. Numer. Anal. 20, 601–619 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Charalambopoulos, A.: On the fréchet differentiability of boundary integral operators in the inverse elastic scattering problem. Inv. Probl. 11, 1137–1161 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Charalambopoulos, A., Kirsch, A., Anagnostopoulos, K., Gintides, D., Kiriaki, K.: The factorization method in inverse elastic scattering from penetrable bodies. Inv. Probl. 23(1), 27–51 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gintides, D., Midrinos, L.: Inverse scattering problem for a rigid scatterer or a cavity in elastodynamics. ZAMM Z. Angew. Math. Mech. 91(4), 276–287 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gintides, D., Sini, M.: Identification of obstacles using only the scattered p-waves or the scattered s-wave. Inverse Probl. Imag. 6, 39–55 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hähner, P., Hsiao, G.: Uniqueness theorems in inverse obstacle scattering of elastic waves. Inv. Probl. 9, 525–534 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hu, G., Kirsch, A., Sini, M.: Some inverse problems arising from elastic scattering by rigid obstacles. Inv. Probl. 29(1), 015009 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ivanyshyn, O., Johansson, B. T.: Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle. J. Integral Equations Appl. 19(3), 289–308 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ivanyshyn, O., Johansson, B.T.: Boundary integral equations for acoustical inverse sound-soft scattering. J. Inv. Ill-posed Probl. 16(1), 65–78 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ivanyshyn, O., Kress, R.: Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks. J. Integral Equations Appl. 18 (1), 13–38 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Johansson, B.T., Sleeman, B.: Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern. IMA J. Appl. Math. 72, 96–112 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kar, M., Sini, M.: On the inverse elastic scattering by interfaces using one type of scattered waves. J. Elast. 118(1), 15–38 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Knops, R.J., Payne, L.E.: Uniqueness theorems in linear elasticity. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kress, R.: On the numerical solution of a hypersingular integral equation in scattering theory. J. Comput. Appl. Math. 61(3), 345–360 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kress, R.: Inverse elastic scattering from a crack. Inv. Probl. 12, 667–684 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kress, R.: Linear Integral Equations. 3rd edn. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kress, R., Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inv. Probl. 21, 1207–1223 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kupradze, V.: Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland Publishing Co., New York (1979)Google Scholar
  24. 24.
    Le Louër, F.: A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems. Inv. Probl. 31(11), 115006 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lee, K.: Inverse scattering problem from an impedance crack via a composite method. Wave Motion 56, 43–51 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, J., Sun, G.: A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces. Inverse Probl. Sci. Eng. 23(4), 557–577 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Martin, P.: On the scattering of elastic waves by an elastic inclusion in two dimensions. Quart. J. Mech. and Appl. Math. 43(3), 275–291 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pelekanos, G., Kleinman, R., van den Berg, P.: Inverse scattering in elasticity – a modified gradient approach. Wave Motion 32(1), 57–65 (2000)CrossRefzbMATHGoogle Scholar
  29. 29.
    Pelekanos, G., Sevroglou, V.: Inverse scattering by penetrable objects in two-dimensional elastodynamics. J. Comput. Appl. Math. 151(1), 129–140 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Qin, H.H., Cakoni, F.: Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem. Inv. Probl. 27, 035005 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sevroglou, V.: The far-field operator for penetrable and absorbing obstacles in 2d inverse elastic scattering. Inv. Probl. 21(2), 717–738 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Applied Mathematics and InformaticsIvan Franko National University of LvivLvivUkraine
  2. 2.Department of MathematicsNational Technical University of AthensZografouGreece
  3. 3.Computational Science CenterUniversity of ViennaViennaAustria

Personalised recommendations