On the Determination of a Robin Boundary Coefficient in an Elastic Cavity Using the MFS

  • Carlos J. S. Alves
  • Nuno F. M. MartinsEmail author
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 11)

In this work, we address a problem of recovering a boundary condition on an elastic cavity from a single boundary measurement on an external part of the boundary. The boundary condition is given by a Robin condition and we aim to identify its Robin coefficient (matrix). We discuss the uniqueness question for this inverse problem and present several numerical simulations, based on two different reconstruction approaches: An approach by solving the Cauchy problem and an iterative Newton type approach (that requires the computation of several direct problems). To solve the mentioned (direct and inverse) problems, we propose the Method of Fundamental Solutions (MFS) whose properties will be discussed.


Inverse problems Robin boundary conditions Lamé system method of fundamental solutions 


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  1. 1.
    C. J. S. Alves and C. S. Chen, A new method of fundamental solutions applied to nonhomoge-neous elliptic problems, Adv. Comp. Math., 23, 125–142, 2005zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    C. J. S. Alves and N. F. M. Martins, The Direct Method of Fundamental Solutions and the Inverse Kirsch-Kress Method for the Reconstruction of Elastic Inclusions or Cavities, Preprint (22), Department of Mathematics, FCT/UNL, Caparica, Portugal, 2007Google Scholar
  3. 3.
    C. J. S. Alves and N. F. M. Martins, Reconstruction of inclusions or cavities in potential problems using the MFS, Submitted Google Scholar
  4. 4.
    A. Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22, 644–669, 1985zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    K. Bryan and L. Caudill, An inverse problem in thermal imaging, SIAM J. Appl. Math., 56, 715–735, 1996zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. Chaabane, C. Elhechmi and M. Jaoua, A stable recovery method for the Robin inverse problem, Math. Comput. Simulat., 66, 367–383, 2004zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Problems and Imaging, 1(2), 229–245, 2007zbMATHMathSciNetGoogle Scholar
  8. 8.
    G. Chen and J. Zou, Boundary Element Methods, Academic, London, 1992zbMATHGoogle Scholar
  9. 9.
    G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9, 69–95, 1998zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    G. Inglese, An inverse problem in corrosion detection, Inverse Probl., 13, 977–994, 1997zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. Kaup and F. Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, J. Nondestruct. Eval., 14, 127–136, 1995CrossRefGoogle Scholar
  12. 12.
    V. D. Kupradze and M. A. Aleksidze, The method of functional equations for the aproximate solution of certain boundary value problems, Zh. vych. mat., 4, 683–715, 1964MathSciNetGoogle Scholar
  13. 13.
    K. Madsen, H. B. Nielsen and O. Tingleff, Methods for non-linear least squares problems, IMM, 60 pages, Denmark, 2004Google Scholar
  14. 14.
    L. Marin and D. Lesnic, The method of fundamental solutions for the Cauchy problem in two-dimensional linear elastiCity, Int. J. Solids Struct., 41(13), 3425–3438, 2004zbMATHCrossRefGoogle Scholar

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© Springer Science + Business Media B.V 2009

Authors and Affiliations

  1. 1.CEMAT-IST and Departamento de Matemática, Faculdade de Ciências e TecnologiaUniv. Nova de Lisboa, Quinta da TorreCaparicaPortugal
  2. 2.CEMAT-IST and Departamento de Matemática, Instituto Superior Técnico, TULisbonAvenida Rovisco PaisLisboa CodexPortugal

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