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

Abstract

ABSTRACT. This work addresses the solvability and solution of volume integrodifferential equations (VIEs) associated with 3D free-space transmission problems (FSTPs) involving elastic or conductive inhomogeneities. A modified version of the singular volume integral equation (SVIE) associated with the VIE is introduced and shown to be of second kind involving a contraction operator, i.e. solvable by Neumann series, implying the wellposedness of the initial VIE. Then, the solvability of VIEs for frequency-domain FSTPs (modelling the scattering of waves by compactly-supported inhomogeneities) follows by a compact perturbation argument. This approach extends work by Potthast (1999) on 2D electromagnetic problems (transverse-electric polarization conditions) involving orthotropic inhomogeneities in a isotropic background, and contains recent results on the solvability of Eshelby's equivalent inclusion problem as special cases. The proposed modified SVIE is also useful for iterative solution methods, as Neumannn series converge (i) unconditionally for static problems and (ii) on some inhomogeneity configurations for which divergence occurs with the usual SVIE for wave scattering problems.

Analysis of topological derivative as a tool for qualitative identification

“…We derive an explicit expression of the topological derivative, T (z), for the scattering by anisotropic media embedded in anisotropic background, with anisotropic trial inhomogeneity of arbitrary shape and near field measurements. Taking advantage of a recently-proposed reformulation of such volume integral equation [7], we provide a symmetric factorization for T (z) where the middle operator contains the material contrast. For the case of isotropic media and background, and isotropic trial inhomogeneity we rigorously prove the sign heuristic for T (z).…”

“…with R 0 := I + 2a∇W 0 , and where the volume potential W 0 is defined as in (13) except that Φ κ is replaced with the zero-frequency fundamental solution Φ 0 , given by Φ 0 (r) = 1/(4πa|r|). Since q z R 0 < 1 for any q z > −1 [7] and R 0 defines a real symmetric L 2 (B; R 3 ) → L 2 (B; R 3 ) operator, the operator I − q z R 0 is symmetric and positive definite, implying that the polarization tensor can be recast in the form…”

“…Our main aim is to establish conditions under which the usual heuristic for TD imaging is valid. Towards this aim, we formulate the forward scattering problem as a volume integral equation, and take advantage of a recently-proposed reformulation of such volume integral equation [7] which allows to express T (z) separately in terms of the material contrast and a contrastindependent normalized integral operator; this in particular facilitates the handling of material anisotropy. Some of our main findings are similar in nature to those of [6]; in particular the sign component of the TD heuristic is again found to be valid within a "moderate enough scatterer" condition, here expressed in terms of the norm of the normalized integral operator.…”

“…Only the fields inside the bounded region circumscribed by the excitation or measurement surface (whatever bounds the larger region) matter in our analysis, hence the discussion presented here includes the case when the scattering problem is formulated in the whole space or in a bounded region, with obvious changes in the fundamental solution. Section 3 is dedicated to the derivation of explicit expressions for the topological derivative, where a new volume integral equation for anisotropic media recently obtained in [7] plays an essential role in obtaining a symmetric factorization of TD. We consider two cases: isotropic scatterers in Section 4 and anisotropic scatterer in Section 5.…”

Inverse acoustic scattering using high-order small-inclusion expansion of misfit function

“…Arbitrary-order expansions are given in [3] for zero-frequency problems in conducting or elastic media. To derive the required solution expansion for the present case (bounded acoustic medium with a penetrable obstacle, Neumann boundary conditions), this work exploits a volume integral equation (VIE) setting [11,25], which is natural for modelling many inhomogeneity problems. The geometrical support of the VIE is B a ; this facilitates the use of coordinate rescaling commonly used in the derivation of asymptotic models and, in combination with the adjoint solution approach, makes the inner expansion in B a play a key role for both the derivation and the justification of the expansion of u a .…”