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A deterministic approximation method in shape optimization under random uncertainties. (English) Zbl 1416.74080

Summary: This paper is concerned with the treatment of uncertainties in shape optimization. We consider uncertainties in the loadings, the material properties, the geometry and the vibration frequency, both in the parametric and geometric optimization setting. We minimize objective functions which are mean values, variances or failure probabilities of standard cost functions under random uncertainties. By assuming that the uncertainties are small and generated by a finite number \(N\) of random variables, and using first- or second-order Taylor expansions, we propose a deterministic approach to optimize approximate objective functions. The computational cost is similar to that of a multiple load problems where the number of loads is N. We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions.

MSC:

74P20 Geometrical methods for optimization problems in solid mechanics
49Q10 Optimization of shapes other than minimal surfaces
65K10 Numerical optimization and variational techniques
90C15 Stochastic programming
65C20 Probabilistic models, generic numerical methods in probability and statistics
74S30 Other numerical methods in solid mechanics (MSC2010)

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