P. Atwal, S. Conti, B. Geihe, M. Pach, M. Rumpf, and R. Schultz.
On shape optimization with stochastic loadings.
In G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher,
V. Schulz, M. Ulbrich, and S. Ulbrich, editors, Constrained Optimization
and Optimal Control for Partial Differential Equations, volume 160 of
International Series of Numerical Mathematics, chapter 2, pages 215-243.
Springer, Basel, 2012.
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This article is concerned with different approaches to elastic shape optimization under stochastic loading. The underlying stochastic optimization strategy builds upon the methodology of two-stage stochastic programming. In fact, in the case of linear elasticity and quadratic objective functionals our strategy leads to a computational cost which scales linearly in the number of linearly independent applied forces, even for a large set of realizations of the random loading. We consider, besides minimization of the expectation value of suitable objective functionals, also two different risk averse approaches, namely the expected excess and the excess probability . Numerical computations are performed using either a level set approach representing implicit shapes of general topology in combination with composite finite elements to resolve elasticity in two and three dimensions, or a collocation boundary element approach, where polygonal shapes represent geometric details attached to a lattice and describing a perforated elastic domain. Topology optimization is performed using the concept of topological derivatives. We generalize this concept, and derive an analytical expression which takes into account the interaction between neighboring holes. This is expected to allow efficient and reliable optimization strategies of elastic objects with a large number of geometric details on a fine scale.