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FM-BEM and Topological Derivative Applied to Acoustic Inverse Scattering

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Book cover Boundary Element Analysis

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 29))

Abstract

This study is set in the framework of inverse scattering of scalar (e.g. acoustic) waves. A qualitative probing technique based on the distribution of topological sensitivity of the cost functional associated with the inverse problem with respect to the nucleation of an infinitesimally-small hard obstacle is formulated. The sensitivity distribution is expressed as a bilinear formula involving the free field and an adjoint field associated with the cost function. These fields are computed by means of a boundary element formulation accelerated by the Fast Multipole method. A computationally fast approach for performing a global preliminary search based on the available overspecified boundary data is thus defined. Its usefulness is demonstrated through results of numerical experiments on the qualitative identification of a hard obstacle in a bounded acoustic domain, for configurations featuring O(105) nodal unknowns and O(106) sampling points.

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References

  1. M. Bonnet: Boundary Integral Equations Methods for Solids and Fluids. John Wiley and Sons, 1999.

    Google Scholar 

  2. M. Bonnet, B. B. Guzina: Sounding of finite solid bodies by way of topological derivative. Internat. J. Numer. Methods Engrg. 61 (2004) 2344–2373.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Colton, A. Kirsch: A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12 (1996) 383–393.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Colton, R. Kress: Integral Equation Method in Scattering Theory. John Wiley and Sons, 1983.

    Google Scholar 

  5. E. Darve: The fast multipole method: numerical implementation. J. Comp. Phys. 160 (2000) 195–240.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. A. Eschenauer, V. V. Kobelev, A. Schumacher: Bubble method for topology and shape optimization of structures. Structural Optimization 8 (1994) 42–51.

    Article  Google Scholar 

  7. G. R. Feijóo: A new method in inverse scattering based on the topological derivative. Inverse Problems 20 (2004) 1819–1840.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Gallego, G. Rus: Identification of cracks and cavities using the topological sensitivity boundsary integral equation. Comp. Mech. 33 (2004) 154–163.

    Article  MATH  Google Scholar 

  9. S. Garreau, P. Guillaume, M. Masmoudi: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Opt. 39 (2001) 1756–1778.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Greenbaum: Iterative methods for solving linear systems. SIAM, Philadelphia, 1997.

    MATH  Google Scholar 

  11. L. Greengard, V. Rokhlin: A fast algorithm for particle simulations. J. Comp. Phys. 73 (1987) 325–348.

    Article  MATH  MathSciNet  Google Scholar 

  12. N. A. Gumerov, R. Duraiswami: Fast multipole methods for the Helmholtz equation in three dimensions. Elsevier Series in Electromagnetism, 2005.

    Google Scholar 

  13. B. B. Guzina, M. Bonnet: Topological derivative for the inverse scattering of elastic waves. Quart. J. Mech. Appl. Math. 57 (2004) 161–179.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. B. Guzina, I. Chikichev: From imaging to material identification: a generalized concept of topological sensitivity. In: Proceedings of ECCOMAS 2004 (P. Neittanmäki, et al. eds.).

    Google Scholar 

  15. A. Kirsch: The domain derivative and two applications in inverse scattering theory. Inverse Problems 9 (1993) 81–96.

    Article  MATH  MathSciNet  Google Scholar 

  16. Z. Michalewicz, D. B. Fogel: How to solve it: modern heuristics. Springer, 2004.

    Google Scholar 

  17. S. Nintcheu Fata, B. B. Guzina: A linear sampling method for near-field inverse problems in elastodynamics. Inverse Problems 20 (2004) 713–736.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Nintcheu Fata, B. B. Guzina, M. Bonnet: A computational basis for elastodynamic cavity identification in a semi-infinite solid. Comp. Mech. 32 (2003) 370–380.

    Article  Google Scholar 

  19. N. Nishimura: Fast multipole accelerated boundary integral equation methods. Appl. Mech. Rev. 55 (2002) 299–324.

    Article  Google Scholar 

  20. R. Potthast: On the convergence of a new Newton-type method in inverse scattering. Inverse Problems 17 (2001) 1419–1434.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Potthast: A survey on sampling and probe methods for inverse problems. Inverse Problems 22 (2006) R1–R47.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Schumacher: Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien. Ph.D. thesis, Universität Siegen, 1995.

    Google Scholar 

  23. G. Sylvand: La méthode multipôle rapide en électromagnétisme: performances, parallélisation, applications. Ph.D. thesis, Ecole Nationale des Ponts et Chaussées, Noisy le Grand, 2002.

    Google Scholar 

  24. G. Sylvand: Complex industrial computations in electromagnetism using the fast multipole method. In: Mathematical and numerical methods aspects of wave propagation (G. Cohen, E. Heikkola, P. Joly, P. Neittaanmäki eds.), pp. 657–662, Springer, 2003.

    Google Scholar 

  25. A. Tarantola: Inverse problem theory and methods for model parameter estimation. SIAM, 2005.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Solid Mechanics Laboratory (UMR CNRS 7649), Ecole Polytechnique, 91128, Palaiseau, France

    Marc Bonnet & Nicolas Nemitz

Authors
  1. Marc Bonnet
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  2. Nicolas Nemitz
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Editor information

Editors and Affiliations

  1. Institut für Baumechanik, Technische Universität Graz, Technikerstr. 4, 8010, Graz, Austria

    Martin Schanz

  2. Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010, Graz, Austria

    Olaf Steinbach

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Bonnet, M., Nemitz, N. (2007). FM-BEM and Topological Derivative Applied to Acoustic Inverse Scattering. In: Schanz, M., Steinbach, O. (eds) Boundary Element Analysis. Lecture Notes in Applied and Computational Mechanics, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-47533-0_8

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  • DOI: https://doi.org/10.1007/978-3-540-47533-0_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-47465-4

  • Online ISBN: 978-3-540-47533-0

  • eBook Packages: EngineeringEngineering (R0)

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