The present work is concerned with the extension of the factorization method to the inverse elastic scattering problem by penetrable isotropic bodies embedded in an isotropic host environment for time-harmonic plane wave incidence. Although the former method has been successfully employed for the shape reconstruction problem in the field of elastodynamic scattering by rigid bodies or cavities, no corresponding results have been recorded, so far, for the very interesting (both from a theoretical and a practical point of view) case of isotropic elastic inclusions. This paper aims at closing this gap by developing the theoretical framework which is necessary for the application of the factorization method to the inverse transmission problem in elastodynamics. As in the previous works referring to the particular reconstruction method, the main outcome is the construction of a binary criterion which determines whether a given point is inside or outside the scattering obstacle by using only the spectral data of the far-field operator.
In this paper the sampling method for the shape reconstruction of a penetrable scatterer in three-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. We establish the interior transmission problem in the weak sense and consider the case where the nonhomogeneous boundary data are generated by a dyadic source point located in the interior domain. Assuming that the inclusion has absorbing behaviour, we prove the existence and uniqueness of the weak solution of the interior transmission problem. In this framework the main theorem for the shape reconstruction for the transmission case is established. As for the cases of the rigid body and the cavity an approximate far-field equation is derived with the known dyadic Green function term with the source point an interior point of the inclusion. The inversion scheme which is proposed is based on the unboundedness for the solution of an equation of the first kind. More precisely, the support of the body can be found by noting that the solution of the integral equation is not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interior points.
Abstract. An incident longitudinal, or transverse, plane wave is scattered by a bounded region immersed in an infinite isotropic and homogeneous elastic medium. The region could be either a rigid scatterer or a cavity. Integral representations for the total displacement field, as well as for the introduced spherical scattering amplitudes are given explicitely in a compact form. Representations for the scattering cross-section whenever the incident wave is a longitudinal or a transverse wave are also provided. Using Papkovich potentials and low-frequency techniques the scattering problems are reduced to an iterative sequence of potential problems which can be solved successively in terms of expansions in appropriate harmonic functions. In each one of the four cases (longitudinal and transverse incidence on rigid scatterer and cavity) the corresponding exterior boundary value problems that specify the approximations as well as the analytic expressions for the scattering amplitudes and the scattering cross-section are given explicitly. The leading low-frequency term of the scattering cross-section for a rigid scatterer is independent of the wave number while for the case of a cavity it is proportional to the fourth power of the wave number. The low-frequency limit of the displacement field which corresponds to the static problem when the scatterer is a cavity, does not depend on the geometrical characteristics of the scatterer and it is always a constant.
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