Advertisement

Journal of Elasticity

, Volume 101, Issue 1, pp 29–57 | Cite as

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

  • Cédric Bellis
  • Bojan B. Guzina
Article

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.

Keywords

Interior transmission problem Transmission eigenvalues Piecewise-homogeneous media Anisotropic viscoelasticity Existence Uniqueness 

Mathematics Subject Classification (2000)

35P25 74D05 74H20 74H25 74J25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arens, T.: Linear sampling methods for 2D inverse elastic wave scattering. Inverse Probl. 17, 1445–1464 (2001) zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 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) zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 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) zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Cakoni, F., Colton, D.: Qualitative Methods in Inverse Scattering Theory. Springer, Berlin (2006) zbMATHGoogle Scholar
  6. 6.
    Cakoni, F., Haddar, H.: The linear sampling method for anisotropic media: Part 2. Preprints 26, MSRI Berkeley, California (2001) Google Scholar
  7. 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) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cakoni, F., Colton, D., Haddar, H.: The linear sampling method for anisotropic media. J. Comput. Appl. Math. 146, 285–299 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Carcione, J.M., Cavallini, F.: Energy balance and fundamental relations in anisotropic-viscoelastic media. Wave Motion 18, 11–20 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Charalambopoulos, A.: On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity. J. Elast. 67, 149–170 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Charalambopoulos, A., Anagnostopoulos, K.A.: On the spectrum of the interior transmission problem in isotropic elasticity. J. Elast. 90, 295313 (2008) CrossRefMathSciNetGoogle Scholar
  12. 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) CrossRefMathSciNetGoogle Scholar
  13. 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) zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Colton, D., Coyle, J., Monk, P.: Recent developments in inverse acoustic scattering theory. SIAM Rev. 42, 369–414 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Colton, D., Kirsch, A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Probl. 12, 383–393 (1996) zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Colton, D., Kress, R.: Using fundamental solutions in inverse scattering. Inverse Probl. 22, R49–R66 (2006) zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin (1998) zbMATHGoogle Scholar
  19. 19.
    Colton, D., Paivarinta, L., Sylvester, J.: The interior transmission problem. Inverse Probl. Imaging 1, 13–28 (2007) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Findley, W.N., Lai, J.S., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials. Dover, New York (1989) Google Scholar
  21. 21.
    Flügge, W.: Viscoelasticity. Springer, Berlin (1975) zbMATHGoogle Scholar
  22. 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) zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hähner, P.: On the uniqueness of the shape of a penetrable, anisotropic obstacle. J. Comput. Appl. Math. 116, 167–180 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kirsch, A.: An integral equation approach and the interior transmission problem for Maxwell’s equations. Inverse Probl. Imaging 1, 107–127 (2007) MathSciNetGoogle Scholar
  26. 26.
    Kirsch, A.: On the existence of transmission eigenvalues. Inverse Probl. Imaging 3, 155–172 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kirsch, A., Grinberg, N.: The Factorization Method for Inverse Problems. Oxford University Press, New York (2008) zbMATHGoogle Scholar
  28. 28.
    Knowles, J.K.: On the representation of the elasticity tensor for isotropic media. J. Elast. 39, 175–180 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Kupradze, V.D.: Potential Methods in the Theory of Elasticity. Israel Program for Scientific Translations (1965) Google Scholar
  30. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Madyarov, A.I., Guzina, B.B.: A radiation condition for layered elastic media. J. Elast. 82, 73–98 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs (1969) Google Scholar
  33. 33.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (1994) Google Scholar
  34. 34.
    Mataraezo, G.: Irreversibility of time and symmetry property of relaxation function in linear viscoelasticity. Mech. Res. Commun. 28, 373–380 (2001) CrossRefGoogle Scholar
  35. 35.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000) zbMATHGoogle Scholar
  36. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam (1981) zbMATHGoogle Scholar
  38. 38.
    Nintcheu Fata, S., Guzina, B.B.: Elastic scatterer reconstruction via the adjoint sampling method. SIAM J. Appl. Math. 67, 1330–1352 (2004) CrossRefMathSciNetGoogle Scholar
  39. 39.
    Nintcheu Fata, S., Guzina, B.B.: A linear sampling method for near-field inverse problems in elastodynamics. Inverse Probl. 20, 713–736 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  40. 40.
    Päivärinta, L., Sylvester, J.: Transmission eigenvalues. SIAM J. Math. Anal. 40, 738–753 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Potthast, R.: A survey on sampling and probe methods for inverse problems. Inverse Probl. 22, R1–47 (2006) zbMATHCrossRefMathSciNetADSGoogle Scholar
  42. 42.
    Pritz, T.: The Poisson’s loss factor of solid viscoelastic materials. J. Sound Vib. 306, 790–802 (2007) CrossRefADSGoogle Scholar
  43. 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) zbMATHCrossRefGoogle Scholar
  44. 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) zbMATHCrossRefMathSciNetADSGoogle Scholar
  45. 45.
    Shter, I.M.: Generalization of Onsager’s principle and its application. J. Eng. Phys. Thermophys. 25, 1319–1323 (1973) Google Scholar
  46. 46.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1992) Google Scholar
  47. 47.
    Yosida, K.: Functional Analysis. Springer, Berlin (1980) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Civil EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.Solid Mechanics Laboratory (UMR CNRS 7649)Ecole PolytechniquePalaiseauFrance

Personalised recommendations