Generalized topological derivative for the Navier equation and inverse scattering in the time domain

In this work, the concept of generalized topological sensitivity, developed recently to deal with inverse elastodynamic scattering by penetrable obstacles in the frequency domain, is extended to permit non-invasive defect reconstruction and material characterization by means of transient elastic waves. This quantity, which signifies the first-order perturbation of a given cost functional due to nucleation of an infinitesimal elastic defect in the reference (defect-free) solid, is intended to be used as an obstacle indicator though an assembly of interior sampling points where it attains pronounced negative values. From an asymptotic analysis of the transient scattered field emanating from a defect with vanishing size in the reference elastic solid, the expression for topological sensitivity, explicit in terms of (i) the Green’s function and (ii) the free (i.e. incident) field for the reference domain, is obtained by employing the so-called direct approach and a convolution-based boundary integral representation of the featured elastodynamic wavefields. For generality, the proposed formula is recast in terms of its adjoint-field counterpart that caters for general reference domains where the Green’s function may not be available. Through numerical simulations, it is shown that the generalized topological sensitivity in the time domain provides an effective preliminary tool for the 3D reconstruction and material identification of discrete internal heterogeneities. The results also demonstrate that the defect “illumination” by transient elastic waveforms results in a higher quality of reconstruction relative to its time-harmonic (monotonal) counterpart.