Inverse problems in the presence of inclusions and unilateral constraints: a boundary element approach

The authors present a boundary element method (BEM) numerical procedure for the solution of 2D non-destructive identification problems in the presence of unilateral boundary conditions. Firstly, the position of a deformable inclusion in frictionless unilateral contact with the matrix is identified on the basis of measurements surveyed at some sensor points on the external boundary where given static loads are applied (identification problem). Then the procedure used is extended to the identification of the position which a rigid/deformable inclusion must occupy within the matrix in order to maximise the structural stiffness of the matrix-inclusion system under prescribed external loads (optimisation problem). Matrix and deformable inclusion are both considered linear elastic. A minimisation problem is stated with design variables representing the size and the shape of the inclusion. The cost function is an error function that evaluates the difference between computed and observed displacements at the sensor points in the identification problem and the strain energy accumulated by the matrix-inclusion system in the optimisation problem. The minimisation is performed by using a first-order nonlinear optimisation technique in which the cost function gradient is computed by implicit differentiation. Some simple but meaningful examples are presented and discussed in order to show the applicability of the proposed technique.