Keywords: genetic algorithms, nonparametric approach, parametric approach, uncertainties, multi-objective optimization, robustness analysis.

In this paper we propose a stochastic optimization method based on the introduction of the robustness of the objective functions with respect to uncertainties on the design parameters. This robustness is introduced as additional objective function in the multi-objective design space. An evolutionary genetic algorithm is used in order to find an optimal design space whose the solutions are more robust than the solutions resulting from a deterministic optimization.

Optimization methods can be classified in two main categories: deterministic approaches which only considering nominal values of the design variables and stochastic approaches which consider uncertainties and variations on design variables. In engineering, uncertainties are due to defects in materials properties (Young modulus, density, ...), and manufacturing processes (thickness, other geometrical variables, ...). In the preliminary design phase, these uncertainties are introduced to take into account the much of knowledge inherent in certain design variables.

In the field of mechanical engineering, much efforts has been devoted to multi- objective evolutionary algorithms (MOEA) since they provide a unique opportunity to address global tradeoffs between multiple objective functions by sampling a number of Pareto solutions [1].

The goal of the proposed here method is to find the robust solutions by introducing additional cost functions (known as robustness functions) for each original cost functions. This robustness function is defined to have the dispersion of the original cost function and during the optimization process, the robustness functions and original functions are evaluated simultaneously. A Monte Carlo sampling is used to take into account uncertainties on the design parameters. This is performed by using an evolutionary multi-objective optimisation with sharing [2] in order to find all Pareto optimal solutions.

To introduce uncertainties, we can use the parametric stochastic approach which directly uses the variation on design variables, or a nonparametric stochastic approach [3] which uses a random matrix of the finite element model. Compared with the parametric approach, the principal advantages of the nonparametric approach are reduced time calculation costs and easy with model dynamic condensation and component modal synthesis methods.

We illustrate the interest of the suggested method on two numerical applications. In the first academic example, we use the parametric approach to verify the interest of the robust optimization which shows the effectiveness of the proposed method and the difference between the deterministic optimal solution and the stochastic optimal solution. In the second example a rotor dynamics problem is solved and we show the interest of the robust approach to deal with of stochastic optimization problem. The objective functions to be minimized are: the response static, the unbalanced response and the critical speed of the rotor.

1

E. Zitzler and L. Thiele. "An evolutionary algorithm for Multiobjective Optimization: the strength Pareto approach", Technical Report 43, Computer Engineering and Communication Networks Lab (TIK), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, May 1998.

2

D.E. Goldberg and J. Richardson, "Genetic Algorithms with Sharing for multimodal function optimization", pp.41-49.

3

C. Soize, "A nonparametric model of random uncertainties for reduced matrix models in structural dynamics", Probabilistic Engineering Mechanics, 15(3), pp.277-294, 2000. doi:10.1016/S0266-8920(99)00028-4

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