Intégrateurs temporels hétérogènes asynchrones pour la dynamique computationnelle des structures

La dynamique computationnelle des structures joue un rôle essentiel dans la simulation de systèmes mécaniques linéaires et non linéaires. Dans ce contexte, les caractéristiques numériques de l’intégrateur temporel ont une incidence critique sur la faisabilité du calcul. Pour aller au-delà de l'approche classique (un intégrateur de temps unique et un pas de temps unique), les travaux initiateurs de Belytschko et ses collègues ont consisté à développer des intégrateurs en temps mixtes implicite explicite pour la dynamique des structures. Dans un premier temps, l’analyse de la stabilité des intégrateurs hétérogènes avec un unique pas de temps a été effectuée pour une large classe d'intégrateurs temporels. Dans second temps, l’analyse de la stabilité des intégrateurs hétérogènes avec différentes échelles de temps a été réalisée en détail mais pour des cas particuliers. Cependant, une analyse générale de la stabilité impliquant différents pas de temps et différents intégrateurs temporels dans différentes parties de la structure considérée est encore une question ouverte pour la dynamique des structures. Ici, un état de l'art des intégrateurs temporels hétérogènes (intégrateurs temporels différents) asynchrones (différents pas de temps) (HATI) pour le calcul des structures en dynamique est effectué. Enfin, une approche alternative basée sur des considérations énergétiques (avec la continuité de la vitesse à l'interface) est proposée afin de développer une classe générale de HATI pour la dynamique des structures

A reduced order modeling for uncertainty propagation in the coupled biomechanical model of bone–implant healing process

International audienceUncertainty quantification is a useful approach for robust design and prediction in numerical simulation of engineering problems. Indeed, once a reliable model is available, it can be subjected to variability or lack of knowledge on one or several input parameters, considered as random variables. The model output is then also a random variable and the user is therefore interested in obtaining stochastic information on this model prediction. Biomechanical problems are prone to such uncertainty, eventually with a large dispersion on the input parameters. The target problem is herein a transient chemical poromechanical coupled problem of parabolic non-linear convection-diffusion-reaction serving to predict the bone healing around an implant. The goal is to provide a tool for implant design and a help to surgical decision. In this contribution focus is on the case where the deterministic model is considered as a black box: the stochastic analysis should therefore be non-intrusive. Amongst non-intrusive approaches, Monte Carlo simulations and polynomial chaos expansions with collocation are reference approaches. The inputs are given with their cumulative density functions (cdf), and we wish to build the cdf of the output quantity of interest (QoI). A reduced order model (ROM) can serve as reducing the cost of the overall analysis. Its efficiency is strongly related to the capability of the QoI to be represented in a smaller subspace. This ability can be improved for geometrical input parameters using a morphing approach. We therefore propose to couple ROM with geometric morphing and stochastic analysis, and assess the effectivity of the approach on the bone healing problem, both from an accuracy and efficiency points of view

A multi-model X-FEM strategy dedicated to frictional crack growth under cyclic fretting fatigue loadings.

International audienceA 2D X-FEM/LATIN numerical model (eXtended Finite Element Method/Large Time Increment method) is proposed in this paper for the analysis of fretting fatigue problems and the simulation of the crack propagation under such loadings. The half-analytical two-body contact analysis allows to capture accurately the pressure and the cyclic tractions exerted at the interface that induce non-proportional multi-axial loading. These distributions are then used as input data for determining critical location for crack initiation and crack inclination based on Dang Van's criterion. The frictional contact conditions of the fretting fatigue cracks have an important impact on the crack behaviour. In this respect, contact with friction between the crack faces is finely modeled within the X-FEM frame. The obtained results are compared and validated with a half-analytical reference model. The numerical simulations reveal the robustness and the efficiency of the proposed approach for a wide range of fretting loadings and friction coefficients values along crack faces. The crack growth directions are then predicted accurately based on the use of criteria adapted to multi-axial non-proportional fatigue. Four cases dealing with crack propagation are then presented. It is shown how the crack length, the tangential loading modify the crack path during the propagation process

Une méthode variationelle en temps pour le couplage de schémas hétérogènes en dynamique transitoire

International audienceCet article présente très succinctement les derniers résultats de l'équipe sur le cou-plage de codes en dynamique. La méthode proposée est très générale et elle est basée sur une vision faible de l'intégration temporelle des équations d'équilibre dynamique. Cette manière de comprendre les algorithmes de calcul en dynamique permet de mettre en place une stratégie de couplage qui assure que le couplage ne perturbe en aucune façon les bilans d'énergie des sous do-maines assemblés. Il en résulte que la méthode proposée permet d'assembler des sous domaines ou des codes différenets utilisant leur propre intégrateur temporel, leur propre maillage, et leur propre pas de temps. Un exemple est proposé pour illustrer le propos

Level set X-FEM non matching meshes: application to dynamic crack propagation in elastic-plastic media

International audienceThis paper develops two aspects improving crack propagation modeling with the X-FEM method. On the one hand, it explains how one can use at the same time a regular structured mesh for a precise and efficient level set update and an unstructured irregular one for the mechanical model. On the other hand, a new numerical scheme based on the X-FEM method is proposed for dynamic elastic-plastic situations. The simulation results are compared with two experiments on PMMA for which crack speed and crack path are provided

Modèle thermique hyper-réduit avec source mobile : formulation dans le repère mobile

La technique d'hyper-réduction de modèle est proposée pour le calcul de la charge thermique avec source mobile, où la réduction de modèle est effectuée sur le volume de contrôle avec le repère mobile. Dans cet article, le volume de contrôle est déterminé par le gradient de température dans la direction de charge mobile. Le modèle hyper-réduit a été appliqué à un modèle EF 3D, avec un gain CPU de l'ordre de 5 (facteur 10^5)

Experimental and numerical analysis of fretting crack formation based on 3D X-FEM frictional contact fatigue crack model

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Crack repulsion and attraction in Linear Elastic Fracture Mechanics

Predicting the direction of a crack submitted to mixed mode loading is key in a variety of domains ranging from earth science to structure safety. In the context of Linear Elastic Fracture Mechanics (LEFM), many bifurcation criteria have been proposed overtime: the Principle of Local Symmetry (PLS) [1], the Maximum Tangential Stress (MTS)[2], the Strain Energy Density (SED) criterion[3], ... The MTS and PLS are the most widely used and give extremely similar results [4]. Whether the theoretical framework of LEFM+PLS is sufficient to study propagation paths of interacting cracks is still under debate. The simplest case of crack interaction is sometimes referred in the literature as en-passant cracks pairs (EP-cracks): two initially straight, parallel and offset cracks which are submitted to far-field opening stress. It was observed experimentally many time that EP-cracks propagate straight ahead until their inner tips overlap, where they begin to attract one another. This behavior can be reproduced correctly assuming the PLS [5]. However, it has been observed that cracks may repel one another in some instances [6]. In this case, it seems that the PLS systematically underestimates the angle of repulsion [7]. We present a simple iterative method, based on FEM computation of the stress state at a given time and determination of the SIF after an infinitesimally small propagation step, to compute quickly and accurately the crack propagation direction in the context of LEFM and under the assumption of PLS. Applying it to the case of EP-cracks, we were able to determine under which conditions the trajectories were initially repulsive or attractive. We find surprising results, among which the fact that perfectly aligned cracks do not interact and that repulsion is a non-monotonous function of the inner tips' separation distances. This method is robust and fast enough to be iterated in order to compute full crack paths. We numerically reproduced the trajectories observed in the experiments presented in an earlier paper [6]. This study allowed us to provide further insights into quantifying the impact of boundary and initial conditions on the shape of crack paths, and into the limitations of applying the LEFM+PLS framework to the case of interacting cracks.   References: [1] B. Cotterell and J.R. Rice. Slightly curved or kinked cracks. International Journal of Fracture, 16(2):155?169, 1980. [2] F Erdogan and G C Sih. On the crack extension in plane loading and transverse shear. Journal Basic Engr., 85:519?527, 1963. [3] G.C. Sih. Some basic problems in fracture mechanics and new concepts. Engineering Fracture Mechanics, 5(2):365?377, 1973. [4] J.B. Leblond. Mécanique de la rupture fragile et ductile. Hermes Science publication, Editions Lavoisier, 2003. [5] Melissa L. Fender, Frédéric Lechenault, and Karen E. Daniels. Universal shapes formed by two interacting cracks. Physical Review Letters, 105(12):2?5, 2010. [6] Marie-Julie Dalbe, Juha Koivisto, Loïc Vanel, Amandine Miksic, Osvanny Ramos, Mikko Alava, and Stéphane Santucci. Repulsion and Attraction between a Pair of Cracks in a Plastic Sheet. Physical review letters, 114(20):205501, 2015

Domain integral formulation for 3-D curved and non-planar cracks with the extended finite element method

The computation of stress intensity factors (SIFS) in curved and non-planar cracks using domain integrals introduces some difficulties related to the use of curvilinear gradients. Several approaches exist in the literature that consider curvilinear corrections within a finite element framework, but these depend on each particular crack configuration and they are not general. In this work, we introduce the curvilinear gradient correction within the extended finite element method framework (XFEM), based only on the level set information used for the crack description and the local coordinate system definition. Our formulation depends only on the level set coordinates and, therefore, an explicit analytical description of the crack is not needed. It is shown that this curvilinear correction improves the results and enables the study of generic cracks. In addition, we have introduced a simple error indicator for improving the SIF computed via the interaction integral, thanks to the better behavior of the J-integral as it does not need auxiliary extraction fields.This work has been carried out within the framework of the research projects DPI2007-66995-C03-02 and DPI2010-20990 financed by the Ministerio de Economia y Competitividad. The support of the Generalitat Valenciana, Programme PROMETEO 2012/023 is also acknowledged.González Albuixech, VF.; Giner Maravilla, E.; Tarancón Caro, JE.; Fuenmayor Fernández, FJ.; Gravouil, A. (2013). Domain integral formulation for 3-D curved and non-planar cracks with the extended finite element method. Computer Methods in Applied Mechanics and Engineering. 264:129-144. https://doi.org/10.1016/j.cma.2013.05.016S12914426

METHODE MULTI-ECHELLES EN TEMPS ET EN ESPACE AVEC DECOMPOSITION DE DOMAINES POUR LA DYNAMIQUE NON-LINEAIRE DES STRUCTURES

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