Treating volumetric inequality constraint in a continuum media with a coupled X-FEM/Level-Set strategy

International audienceSome mechanical problems involve inequality kinematic constraint. This study deals with an original approach to handle those difficult problems. The main issue is the treatment of the variational inequalities due to the fact that the constrained area is a priori unknown. The method, introduced here, is to find the exact constrained area iteratively starting from an initial trial one. Thanks to numerical tools such as level-set and X-FEM we turn the constrained minimization problem into a shape equilibrium problem

The inequality level-set approach to handle contact: membrane case

International audienceBackground: Contact mechanics involves models governed by inequality constraints. Even for the simplest contact problem, inequalities arise from the lack of information on the contact zone position. In addition to increasing the difficulty to solve such problems, an unknown contact zone makes it difficult to use an appropriate mesh and to represent efficiently phenomena on the contact zone boundary. Nevertheless these phenomena are often crucial to have an accurate representation of the problem such as weak discontinuity of the displacement.Methods: In this paper, we propose a method specifically designed to solve inequality constraint problems linked to an unknown domain without remeshing. In order to do so, level sets coupled with X-FEM is used to define the unknown domain and take into account the specific behavior at the contact zone boundary. The key idea of the method is to split the problem involving inequality constraints into two problems. In the first problem, the unknown domain is set and therefore it only involves equalities. Nevertheless, the constraints might be violated, meaning the set domain has to be changed. Then, the other problem is a shape optimization of this domain and leads to an updated set domain. These two problems are iterated up to convergence of the algorithm. Moreover, the addition of adhesion to the problem will be considered.Results: The studied case in this paper is a membrane in the context of small deformations. First, a 1D example will be given to illustrate the method with and without adhesion. Then 2D cases will be studied. Finally an example with an evolving load will be given. Comparison will be made with a classical active-set method.Conclusions: The ILS is proved to be an efficient method giving a convincing accuracy for the contact boundary without need of re-meshing. It is also able to naturally handle adhesion

A two-scale solver for linear elasticity problems in the context of parallel message passing

This paper pushes further the intrinsic capabilities of the GFEM$^{gl}$ global-local approach introduced initially in [1]. We develop a distributed computing approach using MPI (Message Passing Interface) both for the global and local problems. Regarding local problems, a specific scheduling strategy is introduced. Then, to measure correctly the convergence of the iterative process, we introduce a reference solution that revisits the product of classical and enriched functions. As a consequence, we are able to propose a purely matrix-based implementation of the global-local problem. The distributed approach is then compared to other parallel solvers either direct or iterative with domain decomposition. The comparison addresses the scalability as well as the elapsed time. Numerical examples deal with linear elastic problems: a polynomial exact solution problem, a complex micro-structure, and, finally, a pull-out test (with different crack extent). 1: C. A. Duarte, D.-J. Kim, and I. Babu\v{s}ka. A global-local approach for the construction of enrichment functions for the generalized fem and its application to three-dimensional cracks. In Advances in Meshfree Techniques, Dordrecht, 2007. SpringerComment: To be published in Computer Methods in Applied Mechanics and Engineering. Revision: mainly introduction, conclusion and Fig3 update

Modèles d'endommagement par level-set épaisse : solutions analytiques et simulations

Titre du résumé joint : Modélisation de l'endommagement par l'évolution d'une fonction de niveau le modèle TLS (Thick Level Set)National audienceEn endommagement brutal, la rupture se fait le long d'une frontière mobile accompagnée de discontinuités fortes du gradient du déplacement. On propose un modèle d'endommagement plus régulier fondé sur le mouvement d'une couche de largeur finie lc au sein de laquelle l'endommagement varie de façon continue. Cette approche permet dans un même cadre d'initier et de propager des défauts.See http://hal.archives-ouvertes.fr/docs/00/59/27/32/ANNEX/r_E3IGVTN6.pd

Mécanique des milieux continus

1 — Pourquoi la mécanique des milieux continus ? 1.1 Du point matériel aux milieux continus 1.2 Mécanique des milieux continus et disciplines de l'ingénieur 1.3 Notions de milieu continu et d'échelle d'observation 1.4 Principales hypothèses 1.5 Système d'unités 2 — Éléments de calcul tensoriel 2.1 Convention de sommation d'Einstein 2.2 Symbole de Kronecker 2.3 Symbole de permutation de Lévi-Civita 2.4 Changement de base 2.5 Scalaire 2.6 Vecteur 2.7 Tenseur d'ordre deux 2.8 Intégration par partie 2.9 Systèmes de coordonnées curvilignes orthogonales 3 — Cinématique d'un milieu continu 3.1 Trajectoire et dérivées temporelles 3.2 Gradient de la transformation 3.3 Définition des tenseurs de déformation 3.4 Interprétation des composantes des tenseurs de déformations 3.5 Décomposition polaire 3.6 Changement de volume 3.7 Changement de surface 3.8 Taux de déformation 3.9 Déformations en petites perturbations 4 — Lois de bilan 4.1 Forme globale des lois de bilan 4.2 Forme locale des lois de bilan 4.3 Conséquences des lois de bilan 5 — Tenseur des contraintes 5.1 Introduction du tenseur des contraintes 5.2 Tenseur des contraintes et principe des puissances virtuelles 5.3 Propriétés locales du tenseur des contraintes 6 — Théorie de l'élasticité linéaire isotrope 6.1 Équations 6.2 Théorèmes de l'énergie potentielle 6.3 Techniques de résolution analytique 6.4 Techniques de résolution numérique 6.5 Thermoélasticité 7 — Problèmes classiques d'élasticité 7.1 Cylindre sous pression 7.2 Traction d'un barreau prismatique 7.3 Torsion d'un barreau prismatique 8 — Thermodynamique et lois de comportement 8.1 Premier principe 8.2 Second principeEngineering schoolLa mécanique des milieux continus doit être utilisée à la place de la mécanique des solides indéformables lorsque: — des déformations interviennent; — le comportement du milieu, fluide ou solide, doit être pris en compte. Il faut connaître la relation entre la déformation du corps et les efforts mis en jeu; — des phénomènes thermiques interviennent.Ce document aborde la plupart des notions utiles à l'ingénieur à travers de nombreux exemples

Constitutive relation error estimators for (visco) plastic finite element analysis with softening

International audienceA posteriori error estimators based on constitutive relation residuals have been developed for 20 years, in particular at Cachan. This approach has a strong physical meaning and is quite general. Here, we introduce an extended constitutive relation error estimators family able to measure the quality of finite element computations of structures which exhibit plastic/viscoplastic behavior with softening. These measures take into account, over the studied time interval, all the classical error sources involved in the computation: the space discretization (the mesh), the time discretization and the iterative technique used to solve the nonlinear discrete problem

The macroeconomic implications of healthcare. Bruegel Policy Contribution Issue n˚11 | August 2018

Health-care systems play a crucial role in supporting human health. They also have major macroeconomic implications, an aspect that is not always properly acknowledged. Countries spend very different amounts on healthcare, with spending in North America (Canada and the United States) more than twice as much per capita as in the European Union on average, and there are significant differences between EU countries too. Various explanatory factors such as income levels, population age structures and epidemiological profiles cannot explain the differences between countries. Decisions on the optimal level of spending should also consider various others factor, including the macroeconomic implications of health-care systems. Whatever amount is spent on health care, it should be spent efficiently, in order not to waste resources and to improve the macroeconomic impacts. We demonstrate that there are threshold effects whereby certain quantitative indicators of health tend to improve with increased spending only up to certain amount of spending, but not further. Using a standard method to measure efficiency, data envelopment analysis (DEA), we find significant differences between countries, suggesting that not all countries use existing technologies and best practices to their full potential. This finding calls for policy responses. Health-care systems matter for the macroeconomy because of their large size in output, employment and research. They also have direct fiscal implications in terms of the long-term sustainability of public finances, while health-care spending decisions influence short-term economic development through the fiscal multiplier effect, which is substantial. Most southern European countries cut health-care spending aggressively in recent years, likely amplifying the depth of their recessions and possibly causing hysteresis effects from long-term unemployment and reduced productivity. Fiscal consolidation strategies should aim to preserve spending items that have large fiscal multipliers, including health-care expenditures. Health-care systems also influence labour force participation, productivity and human capital formation through various channels, and thereby have an influence on overall macroeconomic outcomes. They also play an important role in inequality, and we find that inequality of access to health care is particularly high in about one-third of EU countries, which calls for policy responses. It is essential that discussions of health systems consider both the opportunity cost and the economic value of investing in health. Such an approach can help policymakers resist the temptation to default to the potentially inefficient status quo

Pilotage du chargement en formulation X-FEM: application aux lois cohésives

National audienceSee http://hal.archives-ouvertes.fr/docs/00/59/28/21/ANNEX/r_22HF3P0A.pd

An X-FEM and Level Set computational approach for image-based modeling. Application to homogenization.

International audienceThe advances in material characterization by means of imaging techniques require powerful computational methods for numerical analysis. The present contribution focuses on highlighting the advantages of coupling the Extended Finite Elements Method (X-FEM) and the level sets method, applied to solve microstructures with complex geometries. The process of obtaining the level set data starting from a digital image of a material structure and its input into an extended finite element framework is presented. The coupled method is validated using reference examples and applied to obtain homogenized properties for heterogeneous structures. Although the computational applications presented here are mainly two dimensional, the method is equally applicable for three dimensional problems