M. Guedri

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

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.

Collective Dynamics of Disordered Two Coupled Nonlinear Pendulums

The effect of disorder on the collective dynamics of two coupled nonlinear pendulums is investigated in this paper. The disorder is introduced by slightly perturbing the length of some pendulums in the nearly periodic structure. A generic discrete analytical model combining the multiple scales method and a standing-wave decomposition is proposed and adapted to the presence of disorder. The proposed model leads to a set of coupled complex algebraic equations which are written according to the number and positions of disorder in the structure. The impact of the disorder on the collective dynamics of two coupled pendulums structure is analyzed through the frequency responses and the basins of attraction. Results show that, in presence of disorder, the multimode solutions are enhanced and the multistability domain is wider. The disorder introduced by reducing the length of one pendulum favors modal localization on its response. In practice, the energy localization generated by disorder is suitable for several engineering applications such as vibration energy harvesting.

Uncertainties Propagation through Robust Reduced Model

Designing large-scale systems in which parametric uncertainties and localized nonlinearities are incorporated requires the implementation of both uncertainty propagation and robust model condensation methods. In this context, we propose to propagate uncertainties through a model, which combines the statistical Latin Hypercube Sampling (LHS) technique and a robust condensation method. The latter is based on the enrichment of a truncated eigenvectors bases using static residuals taking into account parametric uncertainty and localized nonlinearity effects. The efficiency, in terms of accuracy and time consuming, of the proposed method is evaluated on the nonlinear time response of a 2D frame structure.

Uncertainty Propagation Combining Robust Condensation and Generalized Polynomial Chaos Expansion

Among probabilistic uncertainty propagation methods, the generalized Polynomial Chaos Expansion (gPCE) has recently shown a growing emphasis. The numerical cost of the non-intrusive regression method used to compute the gPCE coefficient depends on the successive Latin Hypercube Sampling (LHS) evaluations, especially for large size FE models, large number of uncertain parameters, presence of nonlinearities and when using iterative techniques to compute the dynamic responses. To overcome this issue, the regression technique is coupled with a robust condensation method adapted to the Craig-Bampton component mode synthesis approach leading to computational cost reduction without significant loss of accuracy. The performance of the proposed method and its comparison to the LHS simulation are illustrated by computing the time response of a structure composed of several coupled-beams containing localized nonlinearities and stochastic design parameters.