The linear sampling method for the transmission problem in three-dimensional linear elasticity

Abstract

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.

Inverse problems in elasticity

“…Now, let ϕ[z] be any vector function satisfying (38), g[z] the corresponding Galerkin potential given by (37) and…”

Using mesh adaption for the identification of a spatial field of material properties

“…The shape of the introduced cavity in (11) can be seen in both figures in the bottom left of the studied domain. Figure 13 also demonstrates that the successive refinements occur near the location of the sought cavity, as well as close to the surface of the ground, where the solution of the adjoint problem is mainly located and influences the optimality Equation (9). After these six iterations, the relative discrepancy indicator is e exp = 0.235%, which shows the improvement of the identification by means of the iterative strategy (compared with 1.37% when using the mesh M H and no refinement).…”

“…The minimization problem eventually consists of solving three PDEs with unknowns (u(E), z(E), E): the forward problem (1), the adjoint problem (7) and the optimality equations (9). The identification process results in the solution of a system, which is highly non-linear in the spatially variable unknown field E. The straightforward solution of this system first consists of choosing finite-dimensional subspaces V h ⊂ V, V 0,h ⊂ V 0 and P h ⊂ P using a typical FE discretization with a given mesh M h .…”

“…The best trade-off is to introduce two distinct meshes: a sufficiently refined mesh M h for the solution of the forward and adjoint problems (1) and (7) and a coarse mesh M H for the discretization 215 of the sought field E and the solution of the optimality equation (9). Then the discrete problem consists of finding (…”

“…For example, the shape of the cavity can be assumed, or the spatial Young's modulus can be sought as a piecewise constant field in a fixed spatial discretization. Techniques such as the linear sampling method [9,10] or the topological derivative [11] also allow a preliminary probing of the medium and can provide an interesting initial guess. In a general way, however, regularization can be difficult to set or adapt.…”