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On the Existence and Uniqueness of a Solution to the Interior Transmission Problem for Piecewise-Homogeneous Solids

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Abstract

The interior transmission problem (ITP), which plays a fundamental role in inverse scattering theories involving penetrable defects, is investigated within the framework of mechanical waves scattered by piecewise-homogeneous, elastic or viscoelastic obstacles in a likewise heterogeneous background solid. For generality, the obstacle is allowed to be multiply connected, having both penetrable components (inclusions) and impenetrable parts (cavities). A variational formulation is employed to establish sufficient conditions for the existence and uniqueness of a solution to the ITP, provided that the excitation frequency does not belong to (at most) countable spectrum of transmission eigenvalues. The featured sufficient conditions, expressed in terms of the mass density and elasticity parameters of the problem, represent an advancement over earlier works on the subject in that (i) they pose a precise, previously unavailable provision for the well-posedness of the ITP in situations when both the obstacle and the background solid are heterogeneous, and (ii) they are dimensionally consistent, i.e., invariant under the choice of physical units. For the case of a viscoelastic scatterer in an elastic solid it is further shown, consistent with earlier studies in acoustics, electromagnetism, and elasticity that the uniqueness of a solution to the ITP is maintained irrespective of the vibration frequency. When applied to the situation where both the scatterer and the background medium are viscoelastic, i.e., dissipative, on the other hand, the same type of analysis shows that the analogous claim of uniqueness does not hold. Physically, such anomalous behavior of the “viscoelastic-viscoelastic” case (that has eluded previous studies) has its origins in a lesser known fact that the homogeneous ITP is not mechanically insulated from its surroundings—a feature that is particularly cloaked in situations when either the background medium or the scatterer are dissipative. A set of numerical results, computed for ITP configurations that meet the sufficient conditions for the existence of a solution, is included to illustrate the problem. Consistent with the preceding analysis, the results indicate that the set of transmission values is indeed empty in the “elastic-viscoelastic” case, and countable for “elastic-elastic” and “viscoelastic-viscoelastic” configurations.

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References

  1. Arens, T.: Linear sampling methods for 2D inverse elastic wave scattering. Inverse Probl. 17, 1445–1464 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Baganas, K., Guzina, B.B., Charalambopoulos, A., Manolis, G.D.: A linear sampling method for the inverse transmission problem in near-field elastodynamics. Inverse Probl. 22, 1835–1853 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Bonnet, M., Guzina, B.B.: Sounding of finite solid bodies by way of topological derivative. Int. J. Numer. Meth. Eng. 61, 2344–2373 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Burton, A.J., Miller, G.F.: The application of the integral equation methods to the numerical solution of some exterior boundary-value problems. Proc. R. Soc. Lond. A 323, 201–210 (1971)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Cakoni, F., Colton, D.: Qualitative Methods in Inverse Scattering Theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  6. Cakoni, F., Haddar, H.: The linear sampling method for anisotropic media: Part 2. Preprints 26, MSRI Berkeley, California (2001)

  7. Cakoni, F., Haddar, H.: A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Probl. Imaging 1, 443–456 (2007)

    MATH  MathSciNet  Google Scholar 

  8. Cakoni, F., Colton, D., Haddar, H.: The linear sampling method for anisotropic media. J. Comput. Appl. Math. 146, 285–299 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carcione, J.M., Cavallini, F.: Energy balance and fundamental relations in anisotropic-viscoelastic media. Wave Motion 18, 11–20 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Charalambopoulos, A.: On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity. J. Elast. 67, 149–170 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Charalambopoulos, A., Anagnostopoulos, K.A.: On the spectrum of the interior transmission problem in isotropic elasticity. J. Elast. 90, 295313 (2008)

    Article  MathSciNet  Google Scholar 

  12. Charalambopoulos, A., Gintides, D., Kiriaki, K.: The linear sampling method for the transmission problem in three-dimensional linear elasticity. Inverse Probl. 18, 547558 (2002)

    Article  MathSciNet  Google Scholar 

  13. Charalambopoulos, A., Kirsch, A., Anagnostopoulos, K.A., Gintides, D., Kiriaki, K.: The factorization method in inverse elastic scattering from penetrable bodies. Inverse Probl. 23, 27–51 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Colton, D., Coyle, J., Monk, P.: Recent developments in inverse acoustic scattering theory. SIAM Rev. 42, 369–414 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Colton, D., Kress, R.: Using fundamental solutions in inverse scattering. Inverse Probl. 22, R49–R66 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Colton, D., Kirsch, A., Päivärinta, L.: Far-field patterns for acoustic waves in an inhomogeneous medium. SIAM J. Math. Anal. 20, 1472–1483 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  18. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin (1998)

    MATH  Google Scholar 

  19. Colton, D., Paivarinta, L., Sylvester, J.: The interior transmission problem. Inverse Probl. Imaging 1, 13–28 (2007)

    MATH  MathSciNet  Google Scholar 

  20. Findley, W.N., Lai, J.S., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials. Dover, New York (1989)

    Google Scholar 

  21. Flügge, W.: Viscoelasticity. Springer, Berlin (1975)

    MATH  Google Scholar 

  22. Guzina, B.B., Madyarov, A.I.: A linear sampling approach to inverse elastic scattering in piecewise-homogeneous domains. Inverse Probl. 23, 1467–1493 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Haddar, H.: The interior transmission problem for anisotropic Maxwell’s equations and its applications to the inverse problem. Math. Meth. Appl. Sci. 27, 2111–2129 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hähner, P.: On the uniqueness of the shape of a penetrable, anisotropic obstacle. J. Comput. Appl. Math. 116, 167–180 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kirsch, A.: An integral equation approach and the interior transmission problem for Maxwell’s equations. Inverse Probl. Imaging 1, 107–127 (2007)

    MathSciNet  Google Scholar 

  26. Kirsch, A.: On the existence of transmission eigenvalues. Inverse Probl. Imaging 3, 155–172 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kirsch, A., Grinberg, N.: The Factorization Method for Inverse Problems. Oxford University Press, New York (2008)

    MATH  Google Scholar 

  28. Knowles, J.K.: On the representation of the elasticity tensor for isotropic media. J. Elast. 39, 175–180 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kupradze, V.D.: Potential Methods in the Theory of Elasticity. Israel Program for Scientific Translations (1965)

  30. Liu, Y., Rizzo, F.J.: Hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions. Comput. Meth. Appl. Mech. Eng. 107, 131–144 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Madyarov, A.I., Guzina, B.B.: A radiation condition for layered elastic media. J. Elast. 82, 73–98 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  32. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs (1969)

    Google Scholar 

  33. Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (1994)

    Google Scholar 

  34. Mataraezo, G.: Irreversibility of time and symmetry property of relaxation function in linear viscoelasticity. Mech. Res. Commun. 28, 373–380 (2001)

    Article  Google Scholar 

  35. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  36. Mehrabadi, M.M., Cowin, S.C., Horgan, C.O.: Strain energy density bounds for linear anisotropic elastic materials. J. Elast. 30, 191–196 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam (1981)

    MATH  Google Scholar 

  38. Nintcheu Fata, S., Guzina, B.B.: Elastic scatterer reconstruction via the adjoint sampling method. SIAM J. Appl. Math. 67, 1330–1352 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  40. Päivärinta, L., Sylvester, J.: Transmission eigenvalues. SIAM J. Math. Anal. 40, 738–753 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Potthast, R.: A survey on sampling and probe methods for inverse problems. Inverse Probl. 22, R1–47 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  42. Pritz, T.: The Poisson’s loss factor of solid viscoelastic materials. J. Sound Vib. 306, 790–802 (2007)

    Article  ADS  Google Scholar 

  43. Pyl, L., Clouteau, D., Degrande, G.: A weakly singular boundary integral equation in elastodynamics for heterogeneous domains mitigating fictitious eigenfrequencies. Eng. Anal. Bound. Elem. 28, 1493–1513 (2004)

    Article  MATH  Google Scholar 

  44. Rynne, B.P., Sleeman, B.D.: The interior transmission problem and inverse scattering from inhomogeneous media. SIAM J. Math. Anal. 22, 1755–1762 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. Shter, I.M.: Generalization of Onsager’s principle and its application. J. Eng. Phys. Thermophys. 25, 1319–1323 (1973)

    Google Scholar 

  46. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  47. Yosida, K.: Functional Analysis. Springer, Berlin (1980)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, University of Minnesota, Minneapolis, MN, 55455, USA

    Cédric Bellis & Bojan B. Guzina

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

    Cédric Bellis

Authors
  1. Cédric Bellis
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  2. Bojan B. Guzina
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Correspondence to Bojan B. Guzina.

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Bellis, C., Guzina, B.B. On the Existence and Uniqueness of a Solution to the Interior Transmission Problem for Piecewise-Homogeneous Solids. J Elast 101, 29–57 (2010). https://doi.org/10.1007/s10659-010-9242-0

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  • DOI: https://doi.org/10.1007/s10659-010-9242-0

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