Use of Metamodels in the Multi-Objective Optimization of Mechanical Structures with Uncertainties

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In this paper, we propose a method which takes into account the propagation of uncertainties in the finite element models in a multi-objective optimization procedure. This method is based on the coupling of the Stochastic Response Surface Method (SRSM) and a genetic algorithm of NSGA type (Non-dominated Sorting Genetic Algorithm). The SRSM is based on the use of Stochastic Finite Element Method (SFEM) via the use of the perturbation method. Thus, we can avoid the use of Monte Carlo simulation, whose cost is prohibitive in the optimization problems, especially when the finite element models are large and with a considerable number of design parameters. The objective of this study is, on the one hand, to quantify efficaciously the effects of these uncertainties on the variability of responses which we wish to optimize, and on the other hand, to calculate solutions which are both optimal and robust resulting from the numerical simulation. At the end of a multi-objective optimization procedure, the space of optimal solutions is generally of a large dimension. The solutions obtained are practically non-exploitable by the designer. To facilitate this interpretation, a study of sensitivity a posteriori can be exploited in order to eliminate the non-significant design parameters. The use of the clusters resulting from the Self-Organizing Maps of Kohonen (SOM) is also suggested for a rational management of the design space. The importance of the methodology that we have used along with suggestions for its performances are highlighted by two numerical examples. The criterion of quality selected consists in obtaining the best compromise: the minimal computing time versus the maximum precision of results.