On optimal shape design of systems governed by mixed Dirichlet-Signorini boundary value problems

A problem for finding optimal shape for systems governed by the mixed unilateral boundary value problem of Dirichlet-Signorini-type is considered. Conditions for the solvability of the problem are stated when a variational inequality formulation and when a penalty method is used for solving the state problem in question. The asymptotic relation of design problems based on these two formulations is presented. The optimal shape design problem is discretized by means of finite element method. The convergence results for the approximation are proved. The discretized versions are then formulated as a non-linear programming problem. Results of practical computations of the problem in question are reported.