On the generalized concept of topological sensitivity

To establish an alternative analytical framework for the elastic-wave identification of material defects, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and the idea of cavity nucleation, to 3D elastodynamics involving germination of solid obstacles. The main result of the proposed generalization is an expression for topological sensitivity, explicit in terms of the elastodynamic fundamental solution, obtained by an asymptotic expansion of the misfit-type cost functional with respect to the creation of an infinitesimal elastic inclusion in an otherwise homogeneous semi-infinite solid. A set of numerical results is included to illustrate the potential of generalized topological derivative as a computationally efficient tool for exposing the material characteristics of subsurface obstacles through a point-wise identification of "optimal" inclusion properties that minimize the topological sensitivity. Consistent with the infinitesimal-obstacle assumption, the results indicate that the approach is most effective when used at wave lengths exceeding the inclusion diameter. Although being presented within the context of elastodynamics, the methodology proposed is directly extensible to acoustic and electromagnetic problems.