Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering

Abstract

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 certainme 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. 1. Introduction. Volume integral equations, also known as Lippmann-Schwinger integral equations, arise naturally when considering the scattering of waves by penetrable inhomogeneities embedded in a homogeneous background medium, for which a fundamental solution is known. They have been developed and used in various areas of physics such as electromagnetism and optics [3, Chap. 2] [8, Chap. 9], acoustics [28] [9, Chap. 8], or elastodynamics [34, 35, 39], since a long time. If the penetrable object has homogeneous properties, scattering may alternatively be modelled using coupled surface integral equations, see e.g. [13] for elastodynamics. Variational formulations, combined with appropriate handling of the solution behavior at infinity, can also be applied to such scattering problems.Volume integral equations have a geometrical support restricted to the spatial region where material properties differ from the background. This feature makes them useful e.g. for deriving asymptotic or homogenized models involving inhomogeneities of low contrast or vanishing size [2,38]. Moreover, as they provide a direct mathematical link between unknown inhomogeneities and remote measurements, they are also convenient for medium imaging inverse problems [5], for instance providing a foundation for contrast source inversion methods [32] or allowing rather explicit expressions of far-field patterns.In contrast with the vast existing literature on the mathematical aspects of boundary integral equations and their application to scattering by impenetrable obstacles characterized by Dirichlet, Neumann or impedant boundary conditions, comparatively few studies are available regarding the mathematical properties of volume integral equations. The well-posedness of volume integral formulations for various electromagnetic scattering problems is addressed in [33] f...

The Half-Space Matching method for elastodynamic scattering problems in unbounded domains

“…The first material is isotropic (λ = 16, µ = 2), the second and the third materials are taken from [14] and are orthotropic materials, whereas the last one is a general anisotropic medium. The slowness curves for an isotropic material are two circles which illustrates (22). In general, the slowness curves are not circles anymore but they have some symmetries, which illustrates (20) for general materials and ( 21) for orthotropic materials.…”

“…For anisotropic materials, a radiation condition similar to (10) is not known up to our knowledge. Let us mention [22] which proposed a well-posed volume integral equation formulation based on the Green's function of the anisotropic background. In this work, we propose another definition of the outgoing solution of (1) which is coherent with (10) for isotropic materials.…”

“…since by definition of κ I (ξ), F(±κ I (ξ), ξ, ω) = 0 (where the dispersion relation is given by ( 17), or by (22) for isotropic materials). One speaks of P waves (or pressure waves) when I = P , and of S waves (or shear waves), when I = S. As is well-known, P waves are longitudinal waves and S waves are transverse waves.…”

A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

“…Conversely, any solution of the FSTP (2a-d) satisfies the VIE (10a) and the representation formula (10c), so that problems (2a-d) and (10a,c) are equivalent (a proof of this statement, similar to that given in [1] for the elastodynamic scattering case, is provided for completeness in Sec. 6.1).…”

“…In a previous work [1], to which we refer the reader for bibliography related to the above opening remarks, we established the solvability of a VIE formulation for anisotropic elastodynamic scattering. Therein the VIE problem for an imaginary frequency was shown to be coercive, the real-frequency scattering problems of interest then being found to be compact perturbations of the former.…”