Multi-Scale Shape Optimization under Uncertainty

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We will deal with micro-structured materials with macroscopic boundary conditions and stochastic loading in structural mechanics. We will bring together the analytical treatment of multi-scale problems via homogenization, the reliable numerical solution of PDE problems on complex domains described via level sets, and risk neutral and risk averse two-stage stochastic programming approaches. Homogenization and shape optimization in mechanics have already been treated extensively, but usually with deterministic objective functions and without a focus on spatially varying micro-structures. Level set methods are renowned as a flexible tool in shape and topology optimization but usually without accurate, non averaged solution of the involved problems on complex domains. There is a rich theory and algorithmics in two-stage stochastic programming in finite dimensions. But up to now, to the best of our knowledge, not in a PDE optimization context. Uncertain spatial distribution, uncertain modulation of loading and cost functions involving risk of material failure are a characteristic feature in many situations. In particular, we will consider:- two shells with perforated connecting material, loaded in bending, where the locally optimal geometry will vary along the shell,- a simplified model for micro-structured materials in nature with a hard shell filled with porous material, such as bones and micro-structured plant tissue.