Risk Averse Shape Optimization

Abstract

Risk averse optimization has attracted much attention in finite dimensional stochastic programming. In this paper, we propose a risk averse approach in the infinite 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 flow 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 two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios.