en fr Coupled finite element and probalistic transformation method Couplage éléments finis et méthode de transformation probaliste

1 LAMI - Laboratoire de Mécanique et Ingénieries

Abstract : The modeling of mechanical systems can be defined as the mathematical idealization of the physical phenomena controlling it. This implies to define the input variables geometrical parameters, loading conditions

. and the output variables displacements, stresses

., allowing to understand the evolution of the system. The behavior models are more and more complex and the difficulty lies is the identification of the input parameters. As a matter of fact, we cannot admit to use the deterministic models where only the average parameters are considered, because it generally leads to wrong representation of the reality. Hence, it is interesting to introduce the uncertainties in the parameter evaluation and to consider their variability. The fundamental issue of probabilistic studies is therefore to take into account the uncertain character and the spatial variability of the parameters. The reliability methods have for main objective the determination of the structural safety level, based on some assumptions related to the uncertainties, and by defining the state of failure. Therefore, the failure probability can be evaluated along the structure life span and consequently, the design can be verified with respect to safety considerations. The application of probabilistic methods in design requires the use of efficient tools to evaluate the reliability of the considered structure. When the mechanical behavior is given by explicit models, the reliability analysis becomes easy, due to the large number of available methods which can be efficiently used. However, when the mechanical model is numerical finite element method for example, a method allowing the combination of mechanical and probability models must be applied: it is the goal of mechanical-reliability coupling. The mechanical-reliability coupling is defined by the combination of finite element models and reliability algorithms, in such a way that the solution can be efficiently obtained. In this kind of approach, the reliability code drives the finite element analysis procedures and ensures the convergence. The objective of this thesis is therefore to analyze and to study the probabilistic response of mechanical systems with uncertain parameters. Contrary to other methods, the proposed technique couples the deterministic finite element method and the probabilistic transformation method, in order to evaluate the probability density function of the response, in a closed-form or in a semi-analytical form. To show the advantage of the proposed method, we have carried out different applications to cover several structural engineering fields: static, dynamic, reliability and optimization.

Résumé : La modélisation des systèmes mécaniques consiste en l-idéalisation mathématique des phénomènes physiques qui les commandent, en reliant les variables d-entrée paramètres géométriques, conditions de chargement

. aux variables de sortie déplacements, contraintes

Les méthodes probalistes, permettent de prendre en compte le caractère aléatoire et la variabilité spatiale et temporelle, dans l-évaluation de la réponse mécanique. L-application de ces méthodes en vue du dimensionnement nécessite de disposer d-un outil efficace permettant d-évaluer la fiabilité des structures concernées. Lorsque le comportement mécanique d-une structure est décrit par un modèle explicite, son étude fiabiliste est aisée grâce à un nombre important de méthodes qui ont montré leur efficacité. Par contre, lorsque la modélisation mécanique est numérique méthode des éléments finis par exemple, une méthode permettant le couplage des modélisations mécanique et probabiliste doit être utilisée. Dans ce contexte, l-objet de cette thèse consiste à proposer une méthode probabiliste de la réponse d-un système mécanique avec des paramètres aléatoires. La technique proposée dans ce travail est basée sur le couplage des modèles éléments finis et de la méthode de transformation probabiliste, en vue de l-évaluation, sous forme analytique ou semi-analytique, de la fonction de densité de la réponse. Cette méthode est appliquée à différents types de problèmes en vue de démontrer ses avantages et ses limites

Mots-clés : mécanique matrice

Source: https://hal.archives-ouvertes.fr/