Two-stage stochastic optimization meets two-scale simulation

Risk averse stochastic optimization is investigated in the context of elastic shape optimization, allowing for microstructures in the admissible shapes. In particular, a two-stage model for shape optimization under stochastic loading with risk averse cost functionals is combined with a two-scale approach for the simulation of microstructured materials. The microstructure is composed of an elastic material with geometrically simple perforations located on a regular periodic lattice. Different types of microscopic geometries are investigated and compared to each other. In addition they are compared to optimal nested laminates, known to realize the optimal lower bound of compliance cost functionals. We combine this two-scale approach to elastic shapes with a two-stage stochastic programming approach to risk averse shape optimization, dealing with risk neutral and risk averse cost functionals in the presence of stochastic loadings. Mathematics Subject Classification (2010). 90C15, 74B05, 74P05, 74Q05, 74S05, 74S15, 49M29.