Deterministic approaches for shape optimization under random uncertainties (INVITED)

Abstract eng:
We consider shape and topology optimization problems with uncertainties in the loadings, the material properties or the geometry. In view of minimizing the CPU cost of solving such problems, we propose two deterministic approximation methods, based on the assumption of small uncertainties. The first one solves the so-called worst-case design scenario for a linearized approximation. The computational cost is at most twice that of the unperturbed case since it involves three adjoint equations on top of the state equation. The second one minimizes averaged objective functions (mean value, variance) of second-order Taylor expansions of standard cost functions, under the additional assumption that the (small) uncertainties are generated by a finite number N of random variables. The computational cost is now similar to that of a multiple load problems where the number of loads is N . We demonstrate the effectiveness of our approach on various geometric optimization problems in 2-d linearized elasticity. We rely on a gradient algorithm with shape derivatives in a level set framework.

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