On the stability of sizing optimization problems for a class of nonlinearly elastic materials

In this work we will deal with a stability aspect of sizing optimization problems for a class of nonlinearly elastic materials, where the underlying state problem is nonlinear in both the displacements and the stresses. In numerical examples the nonlinear state problem has to be solved iteratively, and therefore it can be solved only up to some small error. The question of interest is how this affects the optimal solution, respectively the set of solutions, of the design problem. We will show with the theory of point-to-set mappings that if the material is not too nonlinear, then the optimal design depends continuously on the error.