On Stochastic Structural Topology Optimization

We consider structural topology optimization problems including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or noncompression conditions for elastic ropes. To construct more realistic models and to hedge off possible failures or inefficient behaviour of optimal structures, we allow parameters (for example, loads) defining the problem to be stochastic. The resulting nonsmooth stochastic optimization problem is an instance of stochastic mathematical programs with equilibrium constraints (MPEC), or stochastic bilevel programs. The existence as well as the continuity of optimal solutions with respect to the lower bounds on the design variables are established. The question of continuity of optimal solutions with respect to small changes in probability measure is analyzed. For a subclass of the problems considered the answer is affirmative, thus showing the robustness of optimal solutions.