A quasi-optimal convergence result for fracture mechanics with XFEM
TL;DR: Chahine et al. as discussed by the authors gave a convergence result for a variant of the eXtended Finite Element Method (XFEM) on cracked domains using a cut-off function to localize the singular enrichment area.
About: This article is published in Comptes Rendus Mathematique.The article was published on 2006-04-01 and is currently open access. It has received 33 citations till now. The article focuses on the topics: Extended finite element method & Rate of convergence.
Summary (1 min read)
Jump to: [1. Introduction] – [2. Model problem] – [3. XFEM: description and discretized problem] – [4. Error estimate] and [5. Concluding remarks]
1. Introduction
- Classical finite element methods used for modeling crack propagation are subjected to several constraints: the mesh should match the crack geometry, should always evolve with the crack growth and should be refined near the crack tip.
- This motivated Moës, Dolbow and Belytschko to introduce an approach called XFEM (eXtended Finite Element Method) in 1999 (see [11]).
- The idea is to add singular functions to the finite element basis taking into account the singular behavior around the crack tip, and a step function modeling the discontinuity of the displacement field across the crack.
- This is not an improvement of the convergence order of the classical finite element method solution (see [8,13]).
- This later method can even realize better convergence results for the computation of the stress intensity factors (see [3]).
2. Model problem
- The authors consider the linear elasticity problem on this domain for an isotropic material.
- The normal (respectively tangential) component of the function uI (respectively uII) is discontinuous along the crack.
- They both correspond to the well known I and II opening modes for a bi-dimensional crack (see [9,10]).
3. XFEM: description and discretized problem
- The idea of XFEM is to use a classical finite element space enriched by some additional functions.
- These functions result from the product of global enrichment functions and some classical finite element functions (see [11]).
4. Error estimate
- Let the displacement field u, solution to problem (1), satisfy the condition (2).
- Note that a similar work has been done in [7], but for a domain totally cut by the crack, which means that the domain does not contain a crack tip.
- Thus the interpolation operator the authors defined allows us to make a classical interpolation over each part of the triangle, and to have the same optimal rate of convergence obtained in the classical global interpolation theorem (see [2,4,12]).
5. Concluding remarks
- (ii) Let us note that the work presented in [12] is applied to a mesh respecting the crack geometry.
- Thus it does not involve the problem presented here of the triangles partially enriched by the Heaviside function.
- On the other hand, this note offers an improvement for the ‘classical’ XFEM method where the convergence rate remains of order √ h for some reasons detailed in [8].
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Citations
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TL;DR: In this article, a residual a posteriori error estimator for Laplace's equa- tion in two space dimensions approximated by the eXtended Finite Element Method (XFEM) is presented.
Abstract: This paper presents and studies a residual a posteriori error estimator for Laplace's equa- tion in two space dimensions approximated by the eXtended Finite Element Method (XFEM). The XFEM allows to perform finite element computations on multi-cracked domains by using meshes of the non-cracked domain. The main idea consists of adding supplementary basis functions of Heaviside type and singular functions in order to take into account the crack geometry and the singularity at the crack tip respectively. Resume. Dans ce travail on propose et onetudie un estimateur d'erreur par residu pour l'´equation de Laplace en deux dimensions d'espace discretisee par la methode d'´elements finis ´ (XFEM). La XFEM permet de realiser des simulations parements finis sur des domaines multi-fissures en utilisant des maillages du domaine non fissure. L'idee principale de la methode consisteajouter des fonctions de base supplementaires de type Heaviside et des fonctions singulieres afin de prendre en compte la geometrie de la fissure et la singularite en pointe de fissure.
5 citations
TL;DR: The X-FEM as discussed by the authors is an extension to the classical finite element method (FEM) using the concepts of partition of unity and meshless approaches, which is specifically designed to improve the performance of the conventional FEM, while keeping the computational costs at an acceptable level, and avoiding the cumbersome remeshing of FEM in crack propagation problems.
Abstract: The extended finite element method (X-FEM) is reviewed and some new developments for fracture analysis of structures is presented. The X-FEM is an extension to the classical finite element method (FEM), using the concepts of partition of unity and meshless approaches. It is specifically designed to improve the performance of the conventional finite element method, while keeping the computational costs at an acceptable level, and avoiding the cumbersome remeshing of FEM in crack propagation problems. The simplicity, flexibility in handling several cracks and crack propagation patterns on a fixed mesh, and the level of accuracy with minimum additional degrees of freedom have transformed X-FEM into the most efficient numerical procedure in the arena of computational fracture mechanics.
5 citations
Dissertation•
20 Nov 2009
TL;DR: In this article, a modelisation, l'analyse and the simulation of problemes de contact intervenant en mecanique des solides and des fluides are discussed.
Abstract: La modelisation des problemes de contact pose de serieuses difficultes qu'elles soient conceptuelles, mathematiques ou informatiques. Motives par le role fondamental que jouent les phenomenes de contact, nous nous interessons a la modelisation, l'analyse et la simulation de problemes de contact intervenant en mecanique des solides et des fluides. Dans une premiere partie theorique, on etudie le comportement asymptotique de solutions de problemes variationnels dependant du temps issus de la mecanique du contact frottant. La deuxieme partie est consacree au controle de la qualite des calculs en mecanique des solides. Guides par la recherche de la formulation et l'etude du contact dans la methode des elements finis etendus (XFEM), nous etudions notamment les estimateurs d'erreur par residu pour la methode XFEM dans le cas lineaire, ceux pour le probleme de contact unilateral avec frottement de Coulomb approches par une methode d'elements finis standard et l'extension au cas de methodes mixtes stabilisees (i.e., ne necessitant pas de condition inf-sup). Cette partie s'acheve par la definition du probleme de contact avec XFEM suivie d'une estimation a priori de l'erreur. La troisieme partie concerne la simulation numerique en mecanique des fluides, plus precisement du probleme de contact de la dynamique des globules rouges evoluant dans un fluide regi par les equations de Navier-Stokes en dimension deux.
4 citations
01 Jan 2007
TL;DR: In this article, the XFEM method for fracture mechanics is revisited using an enlarged fixed enriched subdomain around the crack tip and a bonding condition for the corresponding degrees of freedom.
Abstract: The XFEM method for fracture mechanics is revisited. A first improvement is considered using an enlarged fixed enriched subdomain around the crack tip and a bonding condition for the corresponding degrees of freedom. An efficient numerical integration rule is introduced for the nonsmooth enrichment functions. The lack of accuracy due to the transition layer between the enrichment area and the rest of the domain leads to consider a pointwise matching condition at the boundary of the subdomain. An optimal rate of convergence is then obtained, numerically and theoretically, even for high degree polynomial approximation.
3 citations
Dissertation•
15 May 2009
TL;DR: In this article, the authors propose a technique a meme de fournir un encadrement conservatif des FIC evalues par une methode elements finis classique and par la XFEM.
Abstract: La prevision de la tenue des structures fissurees necessite le calcul du taux de restitution d'energie ou des facteurs d'intensite de contrainte (FIC) en pointe de fissure. Ces quantites sont generalement evaluees apres une analyse elements finis. Plus recemment l'apparition de la XFEM a permis d'ameliorer la description des champs en pointe fissure et de s'affranchir des remaillages successifs apres chaque pas de propagation. Neanmoins, la solution ainsi calculee demeure une solution approchee de la solution du probleme de reference. Il est donc important de pouvoir evaluer la pertinence de ces calculs. Ces travaux de these proposent une technique a meme de fournir un encadrement conservatif des FIC evalues par une methode elements finis classique et par la XFEM. L'utilisation des techniques d'evaluation d'erreur sur les quantites d'interet et de l'erreur en relation de comportement permet dans un premier temps de fournir des bornes de bonne qualite pour les FIC. On propose ensuite une methode permettant d'evaluer l'erreur globale commise lors d'une analyse XFEM. Elle fait intervenir l'erreur en relation de comportement et des techniques de construction de champs de contrainte adequates. On est alors en mesure de proposer un encadrement assez fin des FIC pour un cout numerique tres raisonnable. L'estimation d'erreur peut finalement etre envisagee comme un moyen de determiner les quantites d'interet avec precision.
2 citations
References
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Book•
01 Jan 1989
TL;DR: In this article, the methodes are numeriques and the fonction de forme reference record created on 2005-11-18, modified on 2016-08-08.
Abstract: Keywords: methodes : numeriques ; fonction de forme Reference Record created on 2005-11-18, modified on 2016-08-08
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01 Jan 1978TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.
8,407 citations
Book•
01 Apr 2002
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Abstract: From the Publisher:
This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained.
About the Author
Philippe G. Ciarlet is a Professor at the Laboratoire d'Analyse Numerique at the Universite Pierre et Marie Curie in Paris. He is also a member of the French Academy of Sciences. He is the author of more than a dozen books on a variety of topics and is a frequent invited lecturer at meetings and universities throughout the world. Professor Ciarlet has served approximately 75 visiting professorships since 1973, and he is a member of the editorial boards of more than 20 journals.
8,052 citations
TL;DR: In this article, a displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method.
Abstract: SUMMARY An improvement of a new technique for modelling cracks in the nite element framework is presented. A standard displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Copyright ? 1999 John Wiley & Sons, Ltd.
5,815 citations