Optimal convergence analysis for the extended finite element method

International Journal for Numerical Methods in Engineering

Volumn 86, Issue 4-5, 2011, Pages 528-548

  Nicaise, Serge ^a   Renard, Yves ^b   Chahine, Elie ^c  

^a Université de Valenciennes et du Hainaut Cambrésis   (France)

^b UNIVERSITÉ LYON 1   (France)

^c PAUL SCHERRER INSTITUTE   (Switzerland)

Author keywords

Error estimates; Extended finite element method; Stress intensity factors

Indexed keywords

STRESS INTENSITY FACTORS;

ERROR ESTIMATES; EXTENDED FINITE ELEMENT METHOD; ISOTROPIC ELASTICITY; LAPLACE PROBLEM; NUMERICAL EXPERIMENTS; OPTIMAL A PRIORI ERROR ESTIMATES; OPTIMAL CONVERGENCE;

FINITE ELEMENT METHOD;

EID: 79953092804     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.3092     Document Type: Article

References (39)

1. Melenk JM, Babuška I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996; 139:289-314.
2. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46:131-150.
3. Moës N, Belytschko T. X-FEM: nouvelles frontiéres pour les eléments finis. Revue Européenne des Éléments Finis (Calcul des Structures GIENS'01) 2001/2002; 11:305-318.
4. Chahine E, Laborde P, Renard Y. Crack-tip enrichment in the Xfem method using a cut-off function. International Journal for Numerical Methods in Engineering 2008; 75(6):629-646.
5. Strang G, Fix G. An Analysis of the Finite Element Method. Prentice-Hall: Englewood Cliffs, 1973.
6. Blum H, Dobrowolski M. On finite element methods for elliptic equations on domains with corners. Computing 1982; 28(1):53-63.
7. Dobrowolski M. Numerical Approximation of Elliptic Interface and Corner Problems. Habilitationsschrift: Bonn, 1981.
8. Bourlard M, Dauge M, Lubuma M-S, Nicaise S. Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III. Finite element methods on polygonal domains. SIAM Journal on Numerical Analysis 1992; 29(1):136-155.
9. Ciarlet Jr P, He J. The singular complement method for 2d scalar problems. Comptes Rendus de l'Académie des Sciences Paris 2003; 336(4):353-358.
10. Moussaoui M-A. Sur l'approximation des solutions du probléme de Dirichlet dans un ouvert avec coins. In Singularities and Constructive Methods for Their Treatment (Oberwolfach, 1983). Springer: Berlin, 1983; 199-206.
11. Arfaoui M, Mansouri K, Rezgui A. An asymptotic finite plane deformation analysis of the elastostatic fields at a notch vertex of an incompressible hyperelastic material. Comptes Rendus Mécanique 2008; 336(9):737-743.
12. Béchet E, Minnebo H, Moës N, Burgardt B. Improved implementation and robustness study of the X-FEM for stress analysis around cracks. International Journal for Numerical Methods in Engineering 2005; 64:1033-1056.
13. Bordas S, Conley JGC, Moran B, Gray J, Nichols E. A simulation-based design paradigm for complex cast components. Engineering with Computers 2007; 23(1):25-37.
14. Bordas S, Moran B. Enriched finite elements and level sets for damage tolerance assessment of complex structures. Engineering Fracture Mechanics 2006; 73:1176-1201.
15. Bordas S, Nguyen VP, Dunant C, Nguyen-Dang H, Guidoum A. An extended finite element library. International Journal for Numerical Methods in Engineering 2007; 71(6):703-732.
16. Hansbo A, Hansbo P. A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Computer Methods in Applied Mechanics and Engineering 2004; 193:3523-3540.
17. Laborde P, Renard Y, Pommier J, Salaun M. High order extended finite element method for cracked domains. International Journal for Numerical Methods in Engineering 2005; 64:354-381.
18. Wyart E, Coulon D, Martiny P, Pardoen T, Remacle J-F, Lani F. A substructured FE/XFE method for stress-intensity factors computation in an industrial structure. Revue Européenne de Mécanique Numérique 2007; 16(2):199-212.
19. Wyart E, Coulon D, Duflot M, Pardoen T, Remacle JF, Lani F. A substructured FEshell/XFE-3D method for crack analysis in thin walled structures. International Journal for Numerical Methods in Engineering 2007; 72:757-779.
20. Ventura G. On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method. International Journal for Numerical Methods in Engineering 2006; 66:761-795.
21. Xiao QZ, Karihaloo BL. Improving the accuracy of Xfem crack tip fields using higher order quadrature and statically admissible stress recovery. International Journal for Numerical Methods in Engineering 2006; 66:1378-1410.
22. Haslinger J, Renard Y. A new fictitious domain approach inspired by the extended finite element method. SIAM Journal on Numerical Analysis 2009; 47(2):1474-1499.
23. Béchet E, Moës N, Wolhmuth B. A stable Lagrange Multiplier Space for interface conditions within the extended finite element method. International Journal for Numerical Methods in Engineering 788: 2009; 931-954.
24. Chahine E. Etude mathématique et numérique de méthodes d'ĺents finis étendues pour le calcul en domaines fissurés. Ph.D. Thesis, INSA Toulouse, France, 2008.
25. Chahine E, Laborde P, Renard Y. Spider-xfem, an extended finite element variant for partially unknown crack-tip displacement. European Journal of Computational Mechanics 2008; 15(5-7):625-636.
26. Chahine E, Laborde P, Renard Y. A reduced basis enrichment for the extended finite element method. Mathematical Modelling Natural Phenomena 2009; 4(1):88-105.
27. Grisvard P. Singularities in Boundary Value Problems. Masson: Paris, 1992.
28. Grisvard P. Problémes aux limites dans les polygones-mode d'emploi. EDF Bulletin de la Direction des Etudes et Recherches Série C. Mathhatiques et Informatique, vol. 1, MR 87g:35073, 1986; 21-59.
29. Ciarlet PG. The Finite Element Method for Elliptic Problems. North-Holland: Amsterdam, 1979.
30. Adams RA. Sobolev Spaces. Academic Press: New York, 1975.
31. Stazi FL, Budyn E, Chessa J, Belytschko T. An extended finite element method with higher-order elements for curved cracks. Computational Mechanics 2003; 31:38-48.
32. Ern A, Guermond JL. Eléments Finis: Théorie, Applications, Mise Enœuvre. Springer: Berlin, 2001.
33. Bourlard M, Dauge M, Nicaise S. Error estimates on the coefficients obtained by the singular function method. Numerical Functional Analysis and Optimization 1989; 10(11&12):1077-1113.
34. Zlámal M. On the finite element method. Numerische Mathematik 1968; 12:394-409.
35. Renard Y, Pommier J. An open source generic C ++ library for finite element methods. Available from.
36. Chahine E, Laborde P, Renard Y. A non-conformal eXtended Finite Element approach: Integral Matching Xfem, doi: 10.1016/j.apnum.2010.10.009.
37. Destuynder Ph, Djaoua M. Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile. Mathematical Methods in the Applied Sciences 1981; 3:70-87.
38. Szabó BA, Yosibash Z. Numerical analysis of singularities in two dimensions. II. Computation of generalized flux/stress intensity factors. International Journal for Numerical Methods in Engineering 1996; 39(3):409-434.
39. Szabó BA, Yosibash Z. Superconvergent extraction of flux intensity factors and first derivatives from finite element solutions. Computer Methods in Applied Mechanics and Engineering 1996; 129(4):349-370.