Keywords: uncertainties, robustness, optimization, stochastic finite element method, variability, response surface method.

The study in this paper treats the problem linked to the consideration and the propagation of the uncertainties in linear and non-linear predicted calculations in dynamic lower frequencies of mechanical structures. The variability of the behaviour is associated with the dispersion of the modal characteristics, the damping sources and the internal and external excitations sources. Very often, the physical origin of this variability are the geometric default inherent to the fabrication, the uncertainties that affects the mechanical proprieties and the boundary conditions. The consideration of the dispersion phenomenon in the structural dynamics, remains again an open problem. In this domain, several methods have been proposed in the literature, in particular those based on the probabilistic description of the uncertain parameters. The principal use of these methods is the study of the propagation and the quantification of the impact of the uncertainties of the input parameters on the model outputs. Among these methods, the Monte Carlo (MC) simulation method is often considered to be the most popular due to its simplicity and reliability, but the calculation time associated to its practice often remains prohibitive. Thus, its exploitation in several real cases remains limited. It is often used by way of reference to test other methods.

Multi objective optimization is a help tool to decisions in the design of the structures in pre-project or the project phase. The procedures which are employed frequently in optimization use several reanalyses that are necessary for the calculations of the performance functions or the generalised cost. But the complexity of models and the associated modelling (finite elements models with a great size, non-linear models, etc.), make these reanalyses long and sometimes they can not be undertaken becuase of their long calculation times. Furthermore, the consideration of uncertainties in the optimization procedures necessitates the introduction of robust indicators of the solutions against the uncertainties in view of obtaining the most stable solutions. Classically, the current approach consists of considering margin on the imposed constraints, then to verify a posteriori that the solution obtained by deterministic optimization remains stable when the different variables describe the estimated tolerance intervals. These tolerances intervals depend simultaneously on the fabrication process and the modelling errors. The robust design is evaluated only at the end of the optimization process. This idea supposes that the deterministic design space contains the robust solutions that must be selected by introducing the stochastic criteria. The major inconvenient of this approach is that it enables only robust zones to be obtained but not optimal and robust solutions. Moreover, it is costly, because the evaluation of the robustness [1], which is based generally on the mean and the standard deviation, obtained using the MC method.

In the direct calculation of the variability of the responses or in the optimization procedure, one can propose in this paper a method based on the exploitation of stochastic metamodel [2] where the random polynomial coefficients are characterized by the stochastic finite elements method (SFEM) using the projection on the polynomial chaos (PC) [3] or the perturbation method [4]. The advantage of the proposed method is to reduce the cost of the uncertainty analysis, classically obtained by the MC method. The integration of this method in the robust multi objective optimization procedure leads to a drastic reduction of the calculation time while conserving a sufficient accuracy. The robustness of the optimal solution is obtained during the optimization thanks to a fast evaluation of the stochastic metamodel implemented. The interest of this approach in the direct problem or in the optimization procedure is illustrated through simulation examples of non-linear structures. Some difficulties linked to the use of the SFEM as a function of the number of uncertain parameters, the uncertainty levels and the degree of non-linearity. Finally, the interest of the proposed method and its performances, are illustrated through the comparison with the classic calculation methods of the response variability or the multi objective optimization.

1

S.S. Isukapalli, "Uncertainty analysis of transport-transformation models, PhD dissertation", The State University of New Jersey, New Brunswick, 1999.

2

K.H. Lee, G.J. Park, "Robust optimization considering tolerances of design variables", Computers and Structures. 79 (2001) 77-86. doi:10.1016/S0045-7949(00)00117-6

3

R.G. Ghanem, P.D. Spanos, "Stochastic finite elements - A spectral approach", Spring Verlag, 1991.

4

B. Van den Nieuwenhof, J.-P. Coyette, "Modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties", Computer Methods in Applied Mechanics and Engineering, 192 (2003) 3705-3729. doi:10.1016/S0045-7825(03)00371-2

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