Risk Averse Shape Optimization

Risk-averse optimization has attracted much attention in nite-dimensional stochastic programming. In this paper, we propose a risk-averse approach in the in nite dimensional context of shape optimization. We consider elastic materials under stochastic loading. As measures of risk awareness we investigate the expected excess and the excess probability. The developed numerical algorithm is based on a regularized gradient ow acting on an implicit description of the shapes based on level sets. We incorporate topological derivatives to allow for topological changes in the shape optimization procedure. Numerical results in 2D demonstrate the impact of the risk-averse modeling on the optimal shapes and on the cost distribution over the set of scenarios.