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Optimal design under uncertainty has gained much attention in the past ten years due to the ever increasing need for manufacturers to build robust systems at the lowest cost. Reliability-based design optimization (RBDO) allows the analyst to minimize some cost function while ensuring some minimal performances cast as admissible failure probabilities for a set of performance functions. In order to address real-world engineering problems in which the performance is assessed through computational models (e.g. finite element models in structural mechanics) metamodeling techniques have been developed in the past decade. This paper introduces adaptive Kriging surrogate models to solve the RBDO problem. The latter is cast in an augmented space that ``sums up'' the range of the design space and the aleatory uncertainty in the design parameters and the environmental conditions. The surrogate model is used (i) for evaluating robust estimates of the failure probabilities (and for enhancing the computational experimental design by adaptive sampling) in order to achieve the requested accuracy and (ii) for applying a gradient-based optimization algorithm to get optimal values of the design parameters. The approach is applied to the optimal design of ring-stiffened cylindrical shells used in submarine engineering under uncertain geometric imperfections. For this application the performance of the structure is related to buckling which is addressed here by means of a finite element solution based on the asymptotic numerical method.

This thesis is a contribution to the resolution of the reliability-based design optimization problem. This probabilistic design approach is aimed at considering the uncertainty attached to the system of interest in order to provide optimal and safe solutions. The safety level is quantified in the form of a probability of failure. Then, the optimization problem consists in ensuring that this failure probability remains less than a threshold specified by the stakeholders. The resolution of this problem requires a high number of calls to the limit-state design function underlying the reliability analysis. Hence it becomes cumbersome when the limit-state function involves an expensive-to-evaluate numerical model (e.g. a finite element model). In this context, this manuscript proposes a surrogate-based strategy where the limit-state function is progressively replaced by a Kriging meta-model. A special interest has been given to quantifying, reducing and eventually eliminating the error introduced by the use of this meta-model instead of the original model. The proposed methodology is applied to the design of geometrically imperfect shells prone to buckling., Cette thèse est une contribution à la résolution du problème d’optimisation sous contrainte de fiabilité. Cette méthode de dimensionnement probabiliste vise à prendre en compte les incertitudes inhérentes au système à concevoir, en vue de proposer des solutions optimales et sûres. Le niveau de sûreté est quantifié par une probabilité de défaillance. Le problème d’optimisation consiste alors à s’assurer que cette probabilité reste inférieure à un seuil fixé par les donneurs d’ordres. La résolution de ce problème nécessite un grand nombre d’appels à la fonction d’état-limite caractérisant le problème de fiabilité sous-jacent. Ainsi,cette méthodologie devient complexe à appliquer dès lors que le dimensionnement s’appuie sur un modèle numérique coûteux à évaluer (e.g. un modèle aux éléments finis). Dans ce contexte, ce manuscrit propose une stratégie basée sur la substitution adaptative de la fonction d’état-limite par un méta-modèle par Krigeage. On s’est particulièrement employé à quantifier, réduire et finalement éliminer l’erreur commise par l’utilisation de ce méta-modèle en lieu et place du modèle original. La méthodologie proposée est appliquée au dimensionnement des coques géométriquement imparfaites soumises au flambement.

Scikit-learn is a Python module integrating a wide range of state-of-the-art machine learning algorithms for medium-scale supervised and unsupervised problems. This package focuses on bringing machine learning to non-specialists using a general-purpose high-level language. Emphasis is put on ease of use, performance, documentation, and API consistency. It has minimal dependencies and is distributed under the simplified BSD license, encouraging its use in both academic and commercial settings. Source code, binaries, and documentation can be downloaded from http://scikit-learn.org., Update authors list and URLs

National audience; See http://hal.archives-ouvertes.fr/docs/00/59/27/95/ANNEX/r_S5P52J4K.pdf

See http://hal.archives-ouvertes.fr/docs/00/59/29/33/ANNEX/r_W171C8UF.pdf

Colloque avec actes et comité de lecture. Internationale.; International audience; Le calcul de la charge critique de flambage est réalisé par la Méthode Asymptotique Numérique (MAN) couplée à un modèle éléments finis de coques au sein du code EVE. Le modèle stochastique prend en compte la variabilité spatiale et aléatoire des propriétés matériaux et de l’épaisseur des coques grâce à la théorie des champs aléatoires dont la représentation est basée sur leur développement de Karhunen-Loève. Le défaut de forme initial de la structure est modélisé comme une combinaison aléatoire de ses modes de flambement les plus critiques. L’estimation de la probabilité de défaillance, représentative du niveau de sûreté, utilise une méthode de simulation à variance réduite connue sous le nom de Subset Simulation et implémentée dans le code fiabiliste FERUM.

Structural reliability methods aim at computing the probability of failure of systems with respect to some prescribed performance functions. In modern engineering such functions usually resort to running an expensive-to-evaluate computational model (e.g. a finite element model). In this respect simulation methods, which may require $10^{3-6}$ runs cannot be used directly. Surrogate models such as quadratic response surfaces, polynomial chaos expansions or kriging (which are built from a limited number of runs of the original model) are then introduced as a substitute of the original model to cope with the computational cost. In practice it is almost impossible to quantify the error made by this substitution though. In this paper we propose to use a kriging surrogate of the performance function as a means to build a quasi-optimal importance sampling density. The probability of failure is eventually obtained as the product of an augmented probability computed by substituting the meta-model for the original performance function and a correction term which ensures that there is no bias in the estimation even if the meta-model is not fully accurate. The approach is applied to analytical and finite element reliability problems and proves efficient up to 100 random variables., Comment: 20 pages, 7 figures, 2 tables. Preprint submitted to Probabilistic Engineering Mechanics

In the field of structural reliability, the Monte-Carlo estimator is considered as the reference probability estimator. However, it is still untractable for real engineering cases since it requires a high number of runs of the model. In order to reduce the number of computer experiments, many other approaches known as reliability methods have been proposed. A certain approach consists in replacing the original experiment by a surrogate which is much faster to evaluate. Nevertheless, it is often difficult (or even impossible) to quantify the error made by this substitution. In this paper an alternative approach is developed. It takes advantage of the kriging meta-modeling and importance sampling techniques. The proposed alternative estimator is finally applied to a finite element based structural reliability analysis., Comment: 8 pages, 3 figures, 1 table. Preprint submitted to ICASP11 Mini-symposia entitled "Meta-models/surrogate models for uncertainty propagation, sensitivity and reliability analysis"

8 pages, 5 figures, 2 tables. Preprint submitted to ICASP11 Mini-symposia entitled "Meta-models/surrogate models for uncertainty propagation, sensitivity and reliability analysis"; Reliability-based design optimization (RBDO) has gained much attention in the past fifteen years as a way of introducing robustness in the process of designing structures and systems in an optimal manner. Indeed classical optimization (e.g. minimize some cost under mechanical constraints) usually leads to solutions that lie at the boundary of the admissible domain, and that are consequently rather sensitive to uncertainty in the design parameters. In contrast, RBDO aims at designing the system in a robust way by minimizing some cost function under reliability constraints. Thus RBDO methods have to mix optimization algorithms together with reliability calculations. The classical approach known as "double-loop" consists in nesting the computation of the failure probability with respect to the current design within the optimization loop. It is not applicable to industrial models (e.g. finite element models) due to the associated computational burden. In contrast, methods based on the approximation of the reliability (e.g. FORM) may not be applicable to real-world problems. In this context, an original method has been developed that tries to circumvent the abovementioned drawbacks of the existing approaches. It is based on the adaptive construction of a meta-model for the expensive-to-evaluate mechanical model, and on the subset simulation technique for the efficient and accurate computation of the failure probability and its sensitivities with respect to the design variables. The proposed methodology is briefly described in this paper before it is applied to the reliability-based design of an imperfect submarine pressure hull.