Topological Sensitivity Analysis for the Anisotropic Laplace Problem

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DOI: https://doi.org/10.28924/2291-8639-19-2021-949
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Imen Kallel, Topological Sensitivity Analysis for the Anisotropic Laplace Problem, Int. J. Anal. Appl., 19 (6) (2021), 949-969.

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Imen Kallel

Abstract

This paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. The aim of this paper is to propose an alternative approach based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. We derive a topological asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm.

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References

  1. M. Abdelwahed, M. Hassine and M. Masmoudi, Optimal shape design for fluid flow using topological perturbation technique, J. Math. Anal. Appl. 356(2) (2009), 548-563.
  2. M. Abdelwahed and M. Hassine, Topological optimization method for a geometric control problem in stokes flow, Appl. Numer. Math. 59(8) (2009), 1823-1838.
  3. L. Afraites, M. Dambrine, K. Eppler and D. Kateb, Detecting perfectly insulated obstacles by shape optimization techniques of order two, Discrete Contin. Dyn. Syst. Ser. B, 8(2) (2007), 389-416.
  4. S. Amstutz, Aspects th´eoriques et num´eriques en optimisation de forme topologique, PhD thesis, Toulouse, INSA, 2003.
  5. S. Amstutz, The topological asymptotic for the navier-stokes equations, ESAIM: Control, Optim. Calc. Var. 11(3) (2005), 401-425.
  6. S. Andrieux, T. Baranger and A. Ben Abda, Solving cauchy problems by minimizing an energy-like functional. Inverse Probl. 22(1) (2006), 115-133.
  7. M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Meth. Appl. Sci. 21(10) (2011), 2069-2101.
  8. S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for pde systems: the elasticity case, SIAM J. Control Optim. 39(6) (2001), 1756-1778.
  9. P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the dirichlet problem, SIAM J. Control Optim. 41(4) (2002), 1042-1072.
  10. A. Hamdi, A non-iterative method for identifying multiple unknown timedependent sources compactly supported occurring in a 2d parabolic equation,Inverse Probl. Sci. Eng. 26(5) (2018), 744-772.
  11. M. Hassine and I.Kallel, Kohn-vogelius formulation and topological sensitivity analysis based method for solving geometric inverse problems, Arab J. Math. Sci. 24(1) (2018), 43-62.
  12. M. Hassine and I. Kallel, One-iteration reconstruction algorithm for geometric inverse problems, Appl. Math. E-Notes, 18 (2018), 43-50.
  13. L. Jackowska-Strumillo, J. Sokolowski and A. Zochowski, The topological derivative method in shape optimization, In: Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No. 99CH36304), volume 1, pages 674-679. IEEE, 1999.
  14. E. Kim, H. Kang and K. Kim, Anisotropic polarization tensors and detection of an anisotropic inclusion, SIAM J. Appl. Math. 63(4) (2003), 1276-1291.
  15. R. V. Kohn and M. Vogelius, Relaxation of a variational method for impedance computed tomography, Commun. Pure Appl. Math. 40(6) (1987), 745-777.
  16. M. Hassine, Shape optimization for the stokes equations using topological sensitivity analysis, Rev. Afr. Rech. Inform. Math. Appl. 5 (2016), 216-229.
  17. M. Masmoudi, The topological asymptotic, computational methods for control applications, ed. H. Kawarada and J. P´eriaux. International Series, Gakuto, 2002.
  18. M. Masmoudi, J. Pommier and B. Samet, The topological asymptotic expansion for the maxwell equations and some applications, Inverse Probl. 21(2) (2005), 547.
  19. A. Novotny and J. Soko lowski, Topological derivatives in shape optimization, Springer Science & Business Media, 2012.
  20. J. Pommier and B. Samet, The topological asymptotic for the helmholtz equation with dirichlet condition on the boundary of an arbitrarily shaped hole, SIAM J. Control Optim. 43(3) (2004), 899-921.
  21. B. Samet, S. Amstutz and M. Masmoudi, The topological asymptotic for the helmholtz equation, SIAM J. Control Optim. 42(5) (2003), 1523-1544.
  22. W. Yan, M. Liu and F. Jing, Shape inverse problem for stokes- brinkmann equations, Appl. Math. Lett. 88 (2019), 222-229.