A stochastic LATIN method for stochastic and parameterized elastoplastic analysis

The LATIN method has been developed and successfully applied to a variety of deterministic problems, but few work has been developed for nonlinear stochastic problems. This paper presents a stochastic LATIN method to solve stochastic and/or parameterized elastoplastic problems. To this end, the stochastic solution is decoupled into spatial, temporal and stochastic spaces, and approximated by the sum of a set of products of triplets of spatial functions, temporal functions and random variables. Each triplet is then calculated in a greedy way using a stochastic LATIN iteration. The high efficiency of the proposed method relies on two aspects: The nonlinearity is efficiently handled by inheriting advantages of the classical LATIN method, and the randomness and/or parameters are effectively treated by a sample-based approximation of stochastic spaces. Further, the proposed method is not sensitive to the stochastic and/or parametric dimensions of inputs due to the sample description of stochastic spaces. It can thus be applied to high-dimensional stochastic and parameterized problems. Four numerical examples demonstrate the promising performance of the proposed stochastic LATIN method

Coupling multi-fidelity kriging and model-order reduction for the construction of virtual charts

International audienceThis article presents the coupling between multi-fidelity kriging and a database generated on-the-fly by model reduction to accelerate the generation of a surrogate model. The two-level multi-fidelity kriging method Evofusion is used for data fusion. The remarkable point is the generation of low-fidelity and high-fidelity observations from the same solver using the Proper Generalized Decomposition, a model-order reduction method. A 17× speedup is obtained here on an elasto-viscoplastic test case

A LATIN-based model reduction approach for the simulation of cycling damage

International audienceThe objective of this article is to introduce a new method including model order reduction for the life prediction of structures subjected to cycling damage. Contrary to classical incremental schemes for damage computation, a non-incremental technique, the LATIN method, is used herein as a solution framework. This approach allows to introduce a PGD model reduction technique which leads to a drastic reduction of the computational cost. The proposed framework is exemplified for structures subjected to cyclic loading, where damage is considered to be isotropic and micro-defect closure effects are taken into account. A difficulty herein for the use of the LATIN method comes from the state laws which can not be transformed into linear relations through an internal variable transformation. A specific treatment of this issue is introduced in this work

Abaques virtuels pour l'optimisation géométrique de structures

Malgré le progrès constant des moyens informatiques ces dernières années, les essais expérimentaux conservent une place prépondérante lors de la conception de structures dans l’industrie, car la résolution numérique de modèles complexes de grande taille demeure encore souvent hors de portée. Et même lorsque la simulation est possible, chaque nouvelle structure conçue en bureau d’études est abordée comme un nouveau problème, traité de manière indépendante des cas déjà étudiés, ce qui conduit à un très grand nombre de simulations. Le recours à des essais expérimentaux ainsi qu’à ces multiples simulations entraîne des coûts temporels et financiers importants dont la réduction est un enjeu crucial. L’idée développée ici, consiste à regrouper les structures semblables (qui ne diffèrent que par les valeurs données à un certain jeu de paramètres) en « familles » et à précalculer la solution générale pour chacune des familles sous forme paramétrée. Les quantités d’intérêt utiles au dimensionnement sont alors stockées dans des « abaques virtuels » qui seront utilisés en quasi temps réel par l’ingénieur lors de la phase de conception en particularisant les solutions pour les valeurs de paramètres considérées. La construction de ces abaques est basée sur la méthode de réduction de modèle PGD (Proper Generalized Decomposition [1][2]) qui permet de générer la solution d’un problème paramétré pour l’ensemble des jeux de paramètres. Dans le cadre de cette étude avec ASTRIUM-ST, les abaques virtuels sont créés pour prendre en compte les variations de géométrie, qui sont un des points clés dans le processus de conception. Il s’agit alors de considérer ces variations géométriques comme des paramètres dans la méthode PGD. Cette approche a été validée dans un premier temps sur des exemples académiques bi-dimensionnels. Elle sera appliquée par la suite à des structures plus complexes afin de se rapprocher des exigences industrielles d’ASTRIUM-ST. [1] P. Ladevèze. Nonlinear Computational Structural Mechanics—New Approaches and Non-Incremental Methods of Calculation, Springer Verlag, 1999. [2] P. Ladevèze, J.C. Passieux, D. Néron. The LATIN multiscale computational method and the Proper Generalized Decomposition. Computer Methods in Applied Mechanics and Engineering, 199:1287-1296, 2010

A Model Reduction Technique in Space and Time for Fatigue Simulation

International audienceThe simulation of mechanical responses of structures subjected to cyclic loadings for a large number of cycles remains a challenge. The goal herein is to develop an innovative computational scheme for fatigue computations involving non-linear mechanical behaviour of materials, described by internal variables. The focus is on the Large Time Increment (LATIN) method coupled with a model reduction technique, the Proper Generalized Decomposition (PGD). Moreover, a multi-time scale approach is proposed for the simulation of fatigue involving large number of cycles. The quantities of interest are calculated only at particular pre-defined cycles called the “nodal cycles” and a suitable interpolation is used to estimate their evolution at the intermediate cycles. The proposed framework is exemplified for a structure subjected to cyclic loading, where damage is considered to be isotropic and micro-defect closure effects are taken into account. The combination of these techniques reduce the numerical cost drastically and allows to create virtual S-N curves for large number of cycles

A multi-temporal scale model reduction approach for the computation of fatigue damage

International audienceOne of the challenges of fatigue simulation using continuum damage mechanics framework over the years has been reduction of numerical cost while maintaining acceptable accuracy. The extremely high numerical expense is due to the temporal part of the quantities of interest which must reflect the state of a structure that is subjected to exorbitant number of load cycles. A novel attempt here is to present a non-incremental LATIN-PGD framework incorporating temporal model order reduction. LATIN-PGD method is based on separation of spatial and temporal parts of the mechanical variables, thereby allowing for separate treatment of the temporal problem. The internal variables, especially damage, although extraneous to the variable separation, must also be treated in a tactical way to reduce numerical expense. A temporal multi-scale approach is proposed that is based on the idea that the quantities of interest show a slow evolution along the cycles and a rapid evolution within the cycles. This assumption boils down to a finite element like discretisation of the temporal domain using a set of "nodal cycles" defined on the slow time scale. Within them, the quantities of interest must satisfy the global admissibility conditions and constitutive relations with respect to the fast time scale. Thereafter, information of the "nodal cycles" can be interpolated to simulate the behaviour on the whole temporal domain. This numerical strategy is tested on different academic examples and leads to an extreme reduction in numerical expense

Sur une vision non intrusive de la méthode LaTIn-PGD en non linéaire

International audienceA reformulation of the LaTIn-PGD method in the form of generalized quantities is proposed in order to couple model order reduction methods and industrial codes using finite elements. This work proposes to operate directly at the level of the non-linear solver in order to be able to manipulate reduced models in an intrinsic way. The key point of the method is that the introduction of the PGD natively in commercial calculation codes, such as Samcef software developed by SIEMENS, requires an approximation of the solution on the space-time-parameter set at each step of the resolution, which is naturally provided by the LaTIn method. A test case on an elasto-visco-plastic behavioural law is performed with a reduction of calculation time.Une reformulation de la méthode LaTIn-PGD sous forme de grandeurs généralisées est proposée dans l'optique de coupler méthodes de réduction de modèles et codes éléments finis industriels. Ces travaux proposent d'intervenir directement au niveau du solveur non linéaire afin de pouvoir manipuler des modèles réduits de manière intrinsèque. Le point clé de la méthode réside dans le fait que l'introduction de la PGD en natif dans les codes de calculs commerciaux tels que le logiciel Samcef, développé par SIEMENS, nécessite de connaitre une approximation de la solution sur l'ensemble espace-temps-paramètres à chaque étape de la résolution, ce que fournit naturellement la méthode LaTIn. Un cas test sur une loi de comportement élasto-visco-plastique permet d'appréhender un premier gain de temps CPU