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This paper compares different probabilistic optimization methods dealing with uncertainties. Reliability-Based Design Optimization is presented as well as various approaches to calculate the probability of failure. They are compared in terms of precision and number of evaluations on mathematical and electromagnetic design problems to highlight the most effective methods.

Reliability-Based Design Optimization (RBDO) in electromagnetic field problems requires the calculation of probability of failure leading to a huge computational cost in the case of expensive models. Three different RBDO approaches using kriging surrogate model are proposed to overcome this difficulty by introducing an approximation of the objective function and constraints. These methods use different infill sampling criteria (ISC) to add samples in the process of optimization or/and in the reliability analysis. Several enrichment criteria and strategies are compared in terms of number of evaluations and accuracy of the solution.

Reliability-Based Design Optimization (RBDO) in electromagnetic field problems requires the calculation of probability of failure leading to huge computational cost in the case of expensive models. Three different types of RBDO approaches using kriging surrogate model are proposed to overcome this difficulty by introducing an approximation of the objective and of the constraints. These methods use different infill searching criteria to add new samples in the process of optimization or/and in the reliability analysis. The enrichment criteria and the best suited enrichment strategies are discussed in this communication. These approaches are compared in terms of number of evaluations and accuracy of the solution.

This paper compare different probabilistic optimization methods dealing with uncertainties. Methods of Robust Design Optimization (RDO), Reliability-Based Design Optimization (RBDO) and Reliability-Based Robust Design Optimization (RBRDO) are presented and the approaches to calculate the moments and probability of failure are introduced. Then mathematical and electromagnetic design problems are tested to show the effectiveness of these methods.

Méthodes de conception par optimisation robuste et fiable de dispositifs électrotechniques Résumé: Cette thèse porte sur les méthodes d'optimisation robuste et fiable. Les différentes catégories de méthodes d'optimisation stochastique pour traiter les incertitudes sur les dimensions et les matériaux sont présentées. Ces méthodes visent à trouver une solution plus robuste et/ou fiable en minimisant la variance de l'objectif et/ou en réduisant la probabilité de violer les contraintes de faisabilité. Cependant, ces méthodes augmentent le nombre d'évaluations par rapport à une optimisation déterministe et nécessitent le gradient qui peut être bruité pour des modèles éléments finis. Ainsi, des stratégies basées sur des méta-modèles de kriging sont proposées pour approcher des fonctions complexes et donner des dérivées sans bruit. Des critères d’enrichissement sont utilisés pour affiner la précision tout en cherchant l’optimum. Différentes stratégies comprenant le choix du critère et le positionnement de l'enrichissement sont comparées pour mettre en évidence les plus efficaces. Enfin, les stratégies d’optimisation développées dans cette thèse sont appliquées à l’optimisation d'un transformateur modélisé par des équations analytiques puis par la méthode des éléments finis. Mots-clefs: optimisation, conception, incertitude, robustesse, fiabilité, méta-modèle, transformateur, modèle d’éléments finis Methods for Robust and Reliability-based Design Optimization of Electromagnetic Devices Abstract: This PhD thesis deals with the robust and reliability-based optimization problems under uncertainty on dimensions and material properties. First, the different categories of stochastic optimization methods to treat the uncertainty are presented. These methods aim to find a more robust and/or reliable solution by minimizing the variance of objective and/or reducing the probability to violate the constraints of feasibility. However, as these methods increase the number of evaluations compared to deterministic optimization and need the gradient information that may be noisy when provided by finite element models, they are not suitable for the time-consuming models. So kriging-based meta-model strategies are proposed as they can approximate complex functions and give noise-free derivatives. Infill sampling criteria are used to increase their precision while searching for the optimal solution. Different strategies including the choice of the criterion and the positioning of sample enrichment are compared to highlight the most effective ones. Then, the optimization approaches developed within this research work are applied to the optimization problem of a transformer modelled with analytic equations and finite element models. Keywords: design optimization, uncertainty,electromagnetic device, finite element model