On the small-defect perturbation and sampling of heterogeneous solids

Abstract

This paper aims to extend the application of topological sensitivity, developed for elastic-wave imaging of defects in a homogeneous background medium, to the class of heterogeneous (background) bodies with piecewise-analytic viscoelastic and mass density coefficients. Founded on the premise of time-harmonic wave propagation, the approach employs dimensional analysis to formally establish the expected, yet previously unavailable result wherein the leading behavior of the scattered field caused by a vanishing inclusion is governed by (1) the illuminating visco-elastodynamic field and its adjoint companion, both computed for the background body, at the sampling point; (2) the geometry and material properties of the vanishing defect, and (3) local properties of the (heterogeneous) background solid at the sampling point. By virtue of this result, an explicit formula for the topological sensitivity is obtained by an asymptotic expansion of a misfit-type cost functional with respect to the nucleation of a trial inclusion in the given background, i.e. reference solid. Through numerical examples, it is shown that such defined sensitivity provides a computationally effective platform for preliminary sounding of heterogeneous solids by viscoelastic waves. This is accomplished by computing the topological sensitivity over an arbitrary grid of sampling points inside the reference solid, and identifying those regions where the topological sensitivity attains pronounced negative values with the support of a hidden defect (or a set thereof). The results further highlight the potential of topological sensitivity to expose not only the geometry, but also the nature of internal defects through a local, point-wise identification of “optimal” inclusion properties that minimize the topological sensitivity at a given sampling location.