Model Order Reduction based on Proper Generalized Decomposition for the Propagation of Uncertainties in Structural Dynamics

International audienceA priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The Proper Generalized Decomposition method is used to construct a quasi-optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. The dynamic response being highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension

A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties

International audienceWe here propose a multiscale numerical method for the solution of stochastic partial di erential equations with localized uncertainties. It is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales. An efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale, involving a deterministic operator, and local problems at a micro scale for which we have the possibility to use ne approximation spaces. Global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions. Convergence properties of these tensor based methods, which are closely related to spectral decompositions, benefit from the separation of scales. Different types of uncertainties are considered at the micro level. They may be associated with some variability in the operator or source terms, or even with some geometrical variability. In the latter case, specific reformulations of local problems using fictitious domain methods are introduced

Méthode multi-échelle avec patchs pour la propagation d'incertitudes localisées dans les modèles stochastiques

National audienceNous présentons une stratégie basée sur une méthode multiéchelle avec patch afin de traiter des problèmes stochastiques où les sources d'incertitudes sont nombreuses et dont les modèles associés sont des modèles multiéchelles complexes de grande dimension stochastique. La méthode exploite l'aspect localisé des incertitudes en séparant les échelles ce qui permet d'améliorer à la fois le conditionnement du problème et la convergence des méthodes d'approximation de tenseur utilisées pour résoudre les problèmes stochastiques de grande dimension aux niveaux local et global

Statistical properties of effective elastic moduli of random cubic polycrystals

The homogenized elastic properties of polycrystals depend on the grain morphology and crystallographic orientations. For simplification purposes, the orientations of the grains are usually considered three independent Euler angles. However, experimental investigations reveal spatial correlations in these angles. The Karhunen–Loève expansion is used to generate random fields of Euler angles having exponential kernel functions with varying correlation lengths. The effective elastic moduli for numerically generated statistically equiaxed cubic polycrystals are estimated via the classical Eshelby–Kröner Self-Consistent homogenization model. The influence of the correlation lengths of the orientations’ random fields on the statistical properties of the effective elastic moduli has been investigated. Our results show that spatially correlated Euler angles could increase the variability of the homogenized elastic properties compared to the ones having uncorrelated Euler angles. Nevertheless, using independent random variables for Euler angles remains valid when correlation lengths are close to the average grain size

Analyse de la variabilité spatiale géométrique pour la fiabilité des cordons de soudure

Les joints soudés sont présents dans de nombreuses structures tels que les ponts ou les structures marines et sous-marines. La géométrie de ces joints a un impact important sur l'initiation et la propagation de fissure. Toutefois la mesure des paramètres est délicate et longue. Des procédés laser ont permis de disposer de nombreuses trajectoires de différents paramètres géométriques. Cet article se propose d'étoffer les travaux précédents avec des analyses de corrélation spatiale et de corrélation entre paramètres. Des modélisations de distribution sont aussi proposées

Sur une nouvelle approche en calcul dynamique transitoire, incluant les basses et les moyennes fréquences

The computational method proposed is a frequency approach in which the low-frequency part is obtained through a classical technique, while the medium-frequency part is handled using the Variational Theory of Complex Rays (VTCR) initially introduced for forced vibrations.La stratégie de calcul proposée est une approche fréquentielle dans laquelle la partie basses fréquences est traitée grâce aux éléments finis classiques, alors que les moyennes fréquences sont calculées grâce à la Théorie Variationnelle des Rayons Complexes (TVRC) initialement introduite pour des problèmes de vibration forcée