Identification of an object in an elastic layer of constant thickness: the antiplane problem

The inverse geometrical problem of determining the position and size of an unknown cavity in a layer of constant thickness is considered. For the antiplane problem with a harmonic behaviour with time, it is suggested that a certain known concentrated force is applied to the upper boundary of the layer. It is then possible to construct an algorithm for reconstructing the position and size of the cavity, based on a measurement of the deformation of the boundary in a certain finite interval of the same upper boundary of the layer. Examples of the reconstruction considered confirm the stability of the proposed algorithm. To realise it in practice a special Green's function is constructed, which automatically satisfies the boundary conditions on the sides of the layer, by means of which the direct problem is converted to an integral equation on the boundary of the recessed object. The inverse problem consists of identifying the geometry of the recessed object from the measured input data. The proposed algorithm is based on a minimization of the discrepancy functional. A number of examples of the reconstruction of elliptic and circular objects is given.