HAL (Le Centre pour la Communication Scientifique Directe), May 21, 2007
Nous nous intéressons à l'utilisation d'une méthode d'éléments finis étendue XFEM lorsque la sing... more Nous nous intéressons à l'utilisation d'une méthode d'éléments finis étendue XFEM lorsque la singularité du problème n'est pas connue ou bien encore est très compliquée (par exemple, dans le cas d'une fissure interfaciale). Dans ce cas, l'idée d'utiliser la singularité pour enrichir la base éléments finis est inexploitable ou au moins très coûteuse. On considère une variante de XFEM qui consiste à enrichir les éléments finis sur un maillage donné par des solutions pré-calculées sur un maillage raffiné. Un résultat mathématique de convergence est donné et illustré par quelques simulations numériques. ABSTRACT. The extended finite element method XFEM is very efficient to solve problems where the nonsmooth behavior of the solution is known and numerically workable (e.g. a fracture in an isotopic material). In order to consider others situations (interfacial fractures, for instance), we study a variant of XFEM where the finite element basis is enriched with some pre-computed finite element solutions on a refined mesh. An optimal error estimate is given and corroborated by numerical tests.
The present paper is concerned with the unilateral contact model in linear elastostatics, the so-... more The present paper is concerned with the unilateral contact model in linear elastostatics, the so-called Signorini problem. (Our results can also be applied to the scalar Signorini problem.) A standard continuous linear finite element approximation is first chosen to approach the two-dimensional problem. We develop a new error analysis in the H 1 -norm using estimates on Poincaré constants with respect to the size of the areas of the noncontact sets. In particular we do not assume any additional hypothesis on the finiteness of the set of transition points between contact and noncontact. This approach allows us to establish better error bounds under sole H τ assumptions on the solution: if 3/2 < τ < 2 we improve the existing rate by a factor h (τ -3/2) 2 and if τ = 2 the existing rate (h 3/4 ) is improved by a new rate of h | ln(h)|. Using the same finite element spaces as previously we then consider another discrete approximation of the (nonlinear) contact condition in which the same kind of analysis leads to the same convergence rates as for the first approximation.
In this work, we will presente a comparison of two formulation for the discretization of elastody... more In this work, we will presente a comparison of two formulation for the discretization of elastodynamic contact problems. The first approach consists on a midpoint scheme and a contact condition expressed in terms of velocity. This approach gives an energy conserving scheme. The second one we propose is a new distribution of the solid mass. The problem expressed with the new mass matrix is well posed, energy conserving and has a lipschitz solution. Finally, some numerical results are presented.
European Journal of Mechanics A-solids, Sep 1, 2008
This paper is devoted to a new method dealing with the semi-discretized finite element unilateral... more This paper is devoted to a new method dealing with the semi-discretized finite element unilateral contact problem in elastodynamics. This problem is ill-posed mainly because the nodes on the contact surface have their own inertia. We introduce a method based on an equivalent redistribution of the mass matrix such that there is no inertia on the contact boundary. This leads to a mathematically well-posed and energy conserving problem. Finally, some numerical tests are presented.
Journal of Computational and Applied Mathematics, Dec 1, 2022
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
HAL (Le Centre pour la Communication Scientifique Directe), May 15, 2017
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Lecture Notes in Computational Science and Engineering, 2017
We summarize recent achievements in applying Nitsche's method to some contact and friction proble... more We summarize recent achievements in applying Nitsche's method to some contact and friction problems. We recall the setting of Nitsche's method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are detailed: consistency, well-posedness, fully optimal convergence in H 1 (Ω )-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multibody contact, contact in large transformations and contact in elastodynamics are presented as well.
The SMAI journal of computational mathematics, 2016
In this paper, we develop and analyze a finite element fictitious domain approach based on Nitsch... more In this paper, we develop and analyze a finite element fictitious domain approach based on Nitsche's method for the approximation of frictionless contact problems of two deformable elastic bodies. In the proposed method, the geometry of the bodies and the boundary conditions, including the contact condition between the two bodies, are described independently of the mesh of the fictitious domain. We prove that the optimal convergence is preserved. Numerical experiments are provided which confirm the correct behavior of the proposed method. Math. classification. 65N85, 35M85, 74M15.
Dans cet article nous étudions les possibilités d'application de la méthode des éléments finis ét... more Dans cet article nous étudions les possibilités d'application de la méthode des éléments finis étendue (XFEM) au cas des plaques minces fissurées en flexion. Nous supposons le matériau homogène isotrope et la fissure traversante. La déformation de la plaque est régie par le modèle de Kirchhoff-Love, pour lequel on utilise l'élément triangulaire HCT réduit ou son équivalent en quadrangle. Deux stratégies d'enrichissement sont présentées : ajout, dans une zone fixe, des singularités de fond de fissure soit sur tous les noeuds de cette zone, soit de façon globale avec raccord intégral à la frontière de la zone d'enrichissement. Des tests numériques montrent que la méthode conduit à une précision optimale, dans le sens où l'ordre de convergence de l'erreur numérique est comparable à celui d'une méthode d'éléments finis classique sur un problème régulier.
An optimal fictitious domain method inspired by the extended finite element method
We present a new fictitious domain approach inspired by the extended finite element method (Xfem)... more We present a new fictitious domain approach inspired by the extended finite element method (Xfem) of Moës, Dolbow and Belytschko [4]. An optimal method is obtained thanks to an additional stabilization technique which is an adaptation of the one of Barbosa-Hughes [1]. Some a priori estimates are estab-lished and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward. The stabilization proposed allows the method to converge whatever is the intersection of the domain with the mesh. Figure 1: The test domain on which the numerical experiments are performed. A Neumann condition is prescribed on with Γ N and a Dirichlet one on Γ D .
This paper focuses on a one-dimensional elastodynamic contact problem and aims to give some new n... more This paper focuses on a one-dimensional elastodynamic contact problem and aims to give some new numerical results. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. An approximation of this evolutionary problem combining the nite element method as well as the mass redistribution method that consists on a redistribution of the body mass such that there is no inertia at the contact node, is introduced. Then two benchmark problems (one being new) with their analytical solutions are presented and some possible discretizations using di erent time{ integration schemes are described. Finally, numerical experiments are reported and analyzed.
Nous nous intéressons à l'utilisation d'une méthode d'éléments finis étendue XFEM lorsque la sing... more Nous nous intéressons à l'utilisation d'une méthode d'éléments finis étendue XFEM lorsque la singularité du problème n'est pas connue ou bien encore est très compliquée (par exemple, dans le cas d'une fissure interfaciale). Dans ce cas, l'idée d'utiliser la singularité pour enrichir la base éléments finis est inexploitable ou au moins très coûteuse. On considère une variante de XFEM qui consiste à enrichir les éléments finis sur un maillage donné par des solutions pré-calculées sur un maillage raffiné. Un résultat mathématique de convergence est donné et illustré par quelques simulations numériques. ABSTRACT. The extended finite element method XFEM is very efficient to solve problems where the nonsmooth behavior of the solution is known and numerically workable (e.g. a fracture in an isotopic material). In order to consider others situations (interfacial fractures, for instance), we study a variant of XFEM where the finite element basis is enriched with some pre-computed finite element solutions on a refined mesh. An optimal error estimate is given and corroborated by numerical tests.
The present paper is concerned with the unilateral contact model in linear elastostatics (or the ... more The present paper is concerned with the unilateral contact model in linear elastostatics (or the equivalent scalar Signorini problem). A standard continuous conforming linear finite element approximation is first chosen to approach the two-dimensional problem. We develop a new error analysis in the H 1 -norm using estimates on Poincaré constants with respect to the size of the areas of the noncontact sets. In particular we do not assume any additional hypothesis on the finiteness of the set of transition points between contact and noncontact. This approach allows us to establish better error bounds under sole H τ assumptions on the solution: if 3/2 < τ < 2 we improve the existing rate by a factor h (τ -3/2) 2 and if τ = 2 the existing rate (h 3/4 ) is improved by a new rate of h | ln(h)|. We then consider a continuous (nonconforming) linear finite element approximation in which the same kind of analysis leads to the same convergence rates as for the first approximation.
A Posteriori Error Estimations for Frictional Contact Problems Approximated by the Extended Finite Element Method
The benefits of computational methods using classical finite element strategies are limited when ... more The benefits of computational methods using classical finite element strategies are limited when solv-ing problems defined over cracked domains. Indeed the mesh should be sufficiently refined around the crack tip to model the singular strain and the domain should be remeshed step by step according to the geometry of the crack propagation. To overcome these difficulties and to make the finite element methods more flexible, Moës, Dolbow and Belytschko ([12,13]) have introduced in 1999 the XFEM (eXtended Finite Element Method). The idea of XFEM consists in enriching the basis of the classical finite element method by a step function along the crack line to take into consideration the discontinuity of the displacement field accross the crack and by some non-smooth functions representing the asymp-totic displacement around the crack tip. This enrichment strategy allows the use of a mesh independent of the crack geometry. The main novelty in our work consists in taking into account the fr...
Journal of Computational and Applied Mathematics, 2019
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
The purpose of this paper is to present a new fictitious domain approach inspired by the extended... more The purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by Moës, Dolbow and Belytschko in . An optimal method is obtained thanks to an additional stabilization technique. Some a priori estimates are established and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward.
Lecture Notes in Computational Science and Engineering, 2019
The aim of this paper is to provide some mathematical results for the discrete problem associated... more The aim of this paper is to provide some mathematical results for the discrete problem associated to contact with Coulomb friction, in linear elasticity, when finite elements and Nitsche method are considered. We consider both static and dynamic situations. We establish existence and uniqueness results under appropriate assumptions on physical (friction coefficient) and numerical parameters. These results are complemented by a numerical assessment of convergence.
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Papers by Yves Renard