The Proper Generalized Decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics

In this work, we present a new approach to solve linear elastic crack problems in plates using the so-called Proper Generalized Decomposition (PGD). In contrast to the standard FE method, the method enables to solve the crack problem in an efficient way by obtaining a single solution in which the Poisson's ratio v and the plate thickness B are non-fixed parameters. This permits to analyze the influence of v and B in the 3D solutions at roughly the cost of a series expansion of 2D analyses. Computationally, the PGD solution is less expensive than a full 3D standard FE analysis for typical discretizations used in practice to capture singularities in 3D crack problems. In order to verify the effectiveness of the proposed approach, the method is applied to cracked plates in mode I with a straight-through crack and a quarter-elliptical corner crack, validating J-integral results with different reference solutions.The authors thank the Ministerio de Ciencia y Tecnologia for the support received in the framework of the projects DPI2010-20990, DPI2010-20542 and to the Generalitat Valenciana, Programme PROMETEO 2012/023.Giner Maravilla, E.; Bognet, B.; Ródenas, J.; Leygue, A.; Fuenmayor Fernández, FJ.; Chinesta Soria, FJ. (2013). The Proper Generalized Decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics. International Journal of Solids and Structures. 50(10):1710-1720. https://doi.org/10.1016/j.ijsolstr.2013.01.039S17101720501

A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains

The present paper extends in-plane-out-of-plane separated representations successfully used for addressing fully 3D model solutions defined in plate-like domain, to dynamics. Common time integration are performed using explicit or implicit strategies. Even if the implementation of implicit integration schemes into a 3D in-plane-out-of-plane separated representation does not imply major difficulties, the use of explicit integration preferable in many applications becomes a tricky issue. In fact the mesh employed for discretizing the out-of-plane dimension (thickness) determines the maximum time-step ensuring stability. In this paper we introduce a new efficient hybrid explicit/implicit in-plane-out-of-plane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-plane discretization

First steps towards an advanced simulation of composites manufacturing by automated tape placement

International audienceComposite materials and their related manufacturing processes involve many modeling and simulation issues, mainly related to their multi-physics and multi-scale nature, to the strong couplings and the complex geometries. In our former works we developed a new paradigm for addressing the solution of such complex models, the so-called Proper Generalized Decomposition based model order reduction. In this work we are summarizing the most outstanding capabilities of such methodology and then all these capabilities will be put together for addressing efficiently the simulation of a challenging composites manufacturing process, the automated tape placement

Stratégies numériques avancées pour la simulation de modèles définis sur des géométries de plaques et coques : solutions 3D avec une complexité 2D

Nowadays, most of the engineering products for transports (naval, aeronautical, automotive, ...), energy (wind power, ...), and civil engineering widely uses parts of small thickness: plates end shells. Metallic materials are still very used although composite materials are more and more used. The design and dimensioning of metal and composite parts therefore requires adapted and efficient simulation tools. The here chosen approach is to perform 3D mechanical simulations combined with a PGD (Proper Generalized Decomposition) based model reduction method to solve the problem in separated space variables. This method consists in looking for the solution under the form of a finite sum of products of functions involving the mean surface's coordinates and functions involving the coordinate of the thickness. The finite element method is used to solve the 2D (based on the coordinates of the mean surface) and 1D (depending on the thickness coordinate) problems from the variables separation. Thanks to this method, the 3D solution of the problem is iteratively built, with a complexity that scales like the complexity of a 2D problem. Additional variables are added as coordinates of the problem in order to include possible uncertainties, variabilities, design parameters or process parameters in the simulations. These multidimensional simulations therefore provide numeric charts, which can then be used for design and optimization.La plupart des produits d'ingénierie actuels, que ce soit dans le domaine des transports (naval, aéronautique, automobile, ...), de l'énergie (éolien, ...) ou du génie civil, font massivement appel à des pièces de faible épaisseur de formes variées : les plaques et les coques. Les matériaux métalliques sont toujours très utilisés, bien que l'utilisation des matériaux composites augmente fortement. La conception et le dimensionnement des pièces métalliques et composites nécessite par conséquent des outils de calculs adaptés et performants. L'approche retenue est d'effectuer des simulations mécaniques 3D et d'utiliser la méthode de réduction de modèle PGD (Proper Generalized Decomposition) pour résoudre le problème en variables d'espace séparées. Cette méthode consiste à chercher la solution sous la forme d'une somme finie de produits de fonctions des coordonnées de la surface moyenne et de fonctions de la coordonnée de l'épaisseur. La résolution par la méthode des éléments finis des problèmes 2D (fonction des coordonnées de la surface moyenne) et 1D (fonction de la coordonnée de l'épaisseur) issus de la séparation des variables permet de construire de façon itérative la solution 3D du problème avec une complexité qui reste celle d'un problème 2D. Des variables supplémentaires sont également ajoutées en tant que coordonnées du problème afin d'inclure dans les simulations d'éventuelles incertitudes, variabilités, des paramètres de conception ou des paramètres du procédé d'élaboration. Ces simulations multidimensionnelles fournissent donc des abaques numériques qui peuvent ensuite être utilisées pour la conception et l'optimisation

PGD and Separated Space Variables Representation for Linear Elasticity in Plate Domains

International audienceIn this paper, we focus on the simulation of linear elastic behaviour of plates using a 3D approach which numerical cost only scales like a 2D one. In the case of plates, the kinematic hypothesis introduced in plate theories to go from 3D to 2D is usually unsatisfactory where one cannot rely on St Venant’s principle (usually close to the plate edges). We propose to apply the PGD (Proper Generalized Decomposition) method [1] to the simulation of the linear elastic behavior of plates. This method allows us to separately search for the in‐plane and the out‐of plane contributions to the 3D solution, yielding significant savings in computational cost. The method is validated on a simple case and its full potential is then presented for the simulation of the behavior of laminated composite plates

Proper Generalized Decomposition (PGD) et séparation de variables spatiales pour la résolution en thermoélasticité linéaire appliquée à des plaques composites

National audienceSee http://hal.archives-ouvertes.fr/docs/00/59/26/84/ANNEX/r_4U9F7ZH2.pd