Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for wave-based identification of finite-sized objects. This approach remains however largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort towards bridging this gap, this study focuses on a topological derivative approach applied to the L 2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full scattering model. Topological derivativebased imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other wellknown qualitative methods.

International audienceA time-domain topological sensitivity (TS) approach is developed for elastic-wave imaging of media of arbitrary geometry. The TS, which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal hole, is expressed in terms of the time convolution of the free field and a supplementary adjoint field as a function of that specified location. Following previous studies performed under (mostly) static or time-harmonic conditions, the TS field is here considered as a natural and computationally efficient approach for defining a defect indicator function. This study emphasizes the implementation and exploitation of TS fields using standard displacement-based FEM approaches, a straightforward task once the correct sensitivity formulation is available. A comprehensive set of numerical experiments on 3-D and 2-D elastodynamic and acoustic configurations is reported, allowing to assess and highlight many features of the proposed TS-based fast qualitative identification such as its ability to identify multiple defects and its robustness against data noise

The interior transmission problem (ITP), which plays a fundamental role in inverse scattering theories involving penetrable defects, is investigated within the framework of mechanical waves scattered by piecewise-homogeneous, elastic or viscoelastic obstacles in a likewise heterogeneous background solid. For generality, the obstacle is allowed to be multiply connected, having both penetrable components (inclusions) and impenetrable parts (cavities). A variational formulation is employed to establish sufficient conditions for the existence and uniqueness of a solution to the ITP, provided that the excitation frequency does not belong to (at most) countable spectrum of transmission eigenvalues. The featured sufficient conditions, expressed in terms of the mass density and elasticity parameters of the problem, represent an advancement over earlier works on the subject in that (i) they pose a precise, previously unavailable provision for the well-posedness of the ITP in situations when both the obstacle and the background solid are heterogeneous, and (ii) they are dimensionally consistent, i.e., invariant under the choice of physical units. For the case of a viscoelastic scatterer in an elastic solid it is further shown, consistent with earlier studies in acoustics, electromagnetism, and elasticity that the uniqueness of a solution to the ITP is maintained irrespective of the vibration frequency. When applied to the situation where both the scatterer and the background medium are viscoelastic, i.e., dissipative, on the other hand, the same type of analysis shows that the analogous claim of uniqueness does not hold. Physically, such anomalous behavior of the "viscoelastic-viscoelastic" case (that has eluded previous studies) has its origins in a lesser known fact that the homogeneous ITP is not mechanically insulated from its surroundings-a feature that is particularly cloaked in situations when either the background medium or the scatterer are dissipative. A set of numerical results, computed for ITP configurations that meet the sufficient conditions for the existence of a solution, is included to illustrate the problem. Consistent with the preceding analysis, the results indicate that the set of transmission values is indeed empty in the "elastic-