Journal of Integral Equations and Applications

Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering

Marc Bonnet

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This article investigates the solvability of volume integral equations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certain limitations for the background medium. The volume integral equation associated with this problem is first derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we also propose a definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution is found to be equivalent to the Sommerfeld-Kupradze radiation conditions. Moreover, solvability for anisotropic elastostatics, directly related to known results on the equivalent inclusion method, is recovered as a by-product.

Article information

J. Integral Equations Applications, Volume 28, Number 2 (2016), 169-203.

First available in Project Euclid: 1 July 2016

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Zentralblatt MATH identifier

Primary: 35J15: Second-order elliptic equations 45F15: Systems of singular linear integral equations 65R20: Integral equations 74J20: Wave scattering

Volume integral equation elastodynamics anisotropy scattering


Bonnet, Marc. Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering. J. Integral Equations Applications 28 (2016), no. 2, 169--203. doi:10.1216/JIE-2016-28-2-169.

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  • H. Ammari and H. Kang, Polarization and moment tensors with applications to inverse problems and effective medium theory, Appl. Math. Sci. 162, Springer-Verlag, New York, 2007.
  • M. Bonnet and G. Delgado, The topological derivative in anisotropic elasticity, Quart. J. Mech. Appl. Math. 66 (2013), 557–586.
  • M. Born and E. Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, Cambridge University Press, Cambridge, 1999.
  • V.T. Buchwald, Elastic waves in anisotropic media, Proc. Roy. Soc. Lond. 253 (1959), 563–580.
  • D.E. Budreck and J.H. Rose, Three-dimensional inverse scattering in anisotropic elastic media, Inverse Prob. 6 (1990), 331–348.
  • R. Burridge, P. Chadwick and A.N. Norris, Fundamental elastodynamic solutions for anisotropic media with ellipsoidal slowness surfaces, Proc. Roy. Soc. Lond. 440 (1993), 655–681.
  • F. Cakoni and D. Colton, Qualitative methods in inverse scattering theory, Springer-Verlag, New York, 2006.
  • W.C. Chew, Waves and fields in inhomogeneous media. IEEE Press, New York, 1995.
  • D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, New York, 1998.
  • M. Costabel, E. Darrigrand and E.H. Koné, Volume and surface integral equations for electromagnetic scattering by a dielectric body, J. Comp. Appl. Math. 234 (2010), 1817–1825.
  • M. Costabel, E. Darrigrand and H. Sakly, The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body, Comp. Rend. Math. 350 (2012), 193–197.
  • M. Costabel and M. Dauge, On representation formulas and radiation conditions, Math. Meth. Appl. Sci. 20 (1997), 133–150.
  • M. Costabel and E.P. Stephan, Integral equations for transmission problems in linear elasticity, J. Int. Equations Appl. 2 (1990), 211–223.
  • A.C. Eringen and E.S. Suhubi, Elastodynamics, Volume II–Linear theory, Academic Press, New York, 1975.
  • D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, New York, 2001.
  • D. Gintides and K. Kiriaki, Solvability of the integrodifferential equation of Eshelby's equivalent inclusion method, Quart. J. Mech. Appl. Math. 68 (2015), 85–96.
  • D. Gridin, Far-field asymptotics of the Green's tensor for a transversely isotropic solid, Proc. Roy. Soc. Lond. 456 (2000), 571–591.
  • P. Hähner, A uniqueness theorem in inverse scattering of elastic waves, IMA J. Appl. Math. 51 (1993), 201–215.
  • L. Hörmander, The analysis of linear partial differential operators, III, Springer-Verlag, New York, 1985.
  • G.C. Hsiao and W.L. Wendland, Boundary integral equations, Springer, New York, 2008.
  • A. Kirsch, An integral equation approach and the interior transmission problem for Maxwell's equations, Inv. Prob. Imag. 1 (2007), 107–127.
  • ––––, An integral equation for Maxwell's equations in a layered medium with an application to the factorization method, J. Integral Equations Appl. 19 (2007), 333–358.
  • ––––, An integral equation for the scattering problem for an anisotropic medium and the factorization method, in Advanced Topics in scattering and biomedical engineering, A. Charalambopoulos, D.I. Fotiadis and D. Polyzos, eds., World Scientific, Singapore, 2008.
  • A. Kirsch and A. Lechleiter, The operator equations of Lippmann-Schwinger type for acoustic and electromagnetic scattering problems in $L^2$, Appl. Anal. 88 (2009), 807–830.
  • V.D. Kupradze, ed., Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North Holland, Amsterdam, 1979.
  • W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients, Trans. Amer. Math. Soc. 123 (1966), 449–459.
  • A. Madyarov and B.B. Guzina, A radiation condition for layered elastic media, J. Elast. 82 (2006), 73–98.
  • P.A. Martin, Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math. 64 (2003), 297–308.
  • W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
  • T. Mura, Micromechanics of defects in solids, Martinus Nijhoff, Dordrecht, 1987.
  • D. Natroshvili, Boundary integral equation method in the steady state oscillation problems for anisotropic bodies, Math. Meth. Appl. Sci. 20 (1997), 95–119.
  • G. Pelekanos, A. Abubakar and P.M. Van den Berg, Contrast source inversion methods in elastodynamics, J. Acoust. Soc. Amer. 114 (2003), 2825–2834.
  • R. Potthast, Electromagnetic scattering from an orthotropic medium, J. Integral Equations Appl. 11 (1999), 197–215.
  • J.M. Richardson, Scattering of elastic waves from symmetric inhomogeneities at low frequencies, Wave Motion 6 (1984), 325–336.
  • T. Touhei, A fast volume integral equation method for elastic wave propagation in a half space, Int. J. Solids Struct. 48 (2011), 3194–3208.
  • B.R. Vainberg, Principles of radiation, limit absorption and limit amplitude in the general theory of partial differential equations, Russian Math. Surv. 21 (1966), 115–193.
  • C.Y. Wang and J.D. Achenbach, Three-dimensional time-harmonic elastodynamic Green's functions for anisotropic solids, Proc. Roy. Soc. Lond. 449 (1995), 441–458.
  • J.R. Willis, A polarization approach to the scattering of elastic waves–II. Multiple scattering from inclusions, J. Mech. Phys. Solids 28 (1980), 307–327.
  • ––––, Polarization approach to the scattering of elastic waves–I. Scattering by a single inclusion, J. Mech. Phys. Solids 28 (1980), 287–305.