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A Reduced Basis Enrichment for the eXtendedFinite Element Method

Published online by Cambridge University Press:  27 January 2009

E. Chahine ,
P. Laborde  and
Y. Renard
E. Chahine
Affiliation:
Institut de Mathématiques, UMR CNRS 5215, GMM INSA Toulouse, Complexe scientifique de Rangueil, 31077 Toulouse Cedex 4, France
P. Laborde
Affiliation:
Institut de Mathématiques, UMR CNRS 5215, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France
Y. Renard*
Affiliation:
Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, LaMCoS UMR5259, F-69621, Villeurbanne, France
*
[email protected]
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Abstract

This paper is devoted to the introduction of a new variant of the extendedfinite element method (Xfem) for the approximation of elastostatic fractureproblems. This variant consists in a reduced basis strategy for the definitionof the crack tip enrichment. It is particularly adapted when the asymptoticcrack-tip displacement is complex or even unknown. We give a mathematical resultof quasi-optimal a priori error estimate and some computational tests includinga comparison with some other strategies.

Keywords

fracturefinite element methodXfemreduced basiserrorestimates
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 1: Modelling and numerical methods in contact mechanics , 2009 , pp. 88 - 105
Copyright
© EDP Sciences, 2009

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References

R.A. Adams. Sobolev Spaces. Academic Press, 1975.
Béchet, E., Minnebo, H., Moës, N., Burgardt, B.. Improved implementation and robustness study of the x-fem for stress analysis around cracks. Int. J. Numer. Meth. Engng., 64 (2005), 1033–1056. CrossRef
Belytschko, T., Moës, N., Usui, S., Parimi, C.. Arbitrary discontinuities in finite elements. Int. J. Numer. Meth. Engng., 50 (2001), 993–1013. 3.0.CO;2-M>CrossRef
E. Chahine. Etude mathématique et numérique de méthodes d'éléments finis étendues pour le calcul en domaines fissurés. Thèse de Doctorat de l'INSA de Toulouse, 2008.
E. Chahine, P. Laborde, Y. Renard. Crack-tip enrichment in the Xfem method using a cut-off function. To appear in Int. J. Numer. Meth. Engng.
E. Chahine, P. Laborde, Y. Renard. Spider Xfem: an extended finite element variant for partially unknown crack-tip displacement. To appear in Europ. J. of Comp. Mech.
E. Chahine, P. Laborde, Y. Renard. The extended finite element method with an integral matching condition. Submitted.
P.G. Ciarlet. The finite element method for elliptic problems. Studies in Mathematics and its Applications No 4, North Holland, 1978.
H. Ben Dhia. Multiscale mechanical problems : the Arlequin method. C. R. Acad. Sci., série I, Paris, 326 (1998), 899–904.
Dupeux, M.. Mesure des énergies de rupture interfaciale: problématique et exemples de résultats d'essais de gonflement-d'écollement. Mécanique et industrie, 5 (2004), 441–450. CrossRef
A. Ern, J.-L. Guermond. Éléments finis: théorie, applications, mise en œuvre. Mathématiques et Applications 36, SMAI, Springer-Verlag, 2002.
R. Glowinski, J. He, J. Rappaz, J. Wagner. Approximation of multi-scale elliptic problems using patches of elements. C. R. Math. Acad. Sci., Paris, 337 (2003), 679–684.
P. Grisvard. Problèmes aux limites dans les polygones - mode d'emploi. EDF Bull. Dirctions Etudes Rech. Sér. C. Math. Inform. 1, MR 87g:35073 (1986), 21–59.
P. Grisvard. Singularities in boundary value problems. Masson, 1992.
D.B.P. Huynh, A.T. Patera. Reduced basis approximation and a posteriori error estimation for stress intensity factors. Int. J. Numer. Meth. Engng. 72 (2007), 1219–1259.
Laborde, P., Renard, Y., Pommier, J., Salaün, M.. High order extended finite element method for cracked domains. Int. J. Numer. Meth. Engng., 64 (2005), 354–381. CrossRef
J. Lemaitre, J.-L. Chaboche. Mechanics of Solid Materials. Cambridge University Press, 1994.
J.L. Lions, E. Magenes. Problèmes aux limites non homogènes et applications, volume 1. Dunod, 1968.
Y. Maday, E.M. Rønquist. A reduced-basis element method. J. Sci. Comput., 17 (2002), (1-4), 447–459.
Melenk, J.M., Babuška, I.. The partition of unity finite element method: Basic theory and applications. Comput. Meths. Appl. Mech. Engrg., 139 (1996), 289–314. CrossRef
Moës, N., Belytschko, T.. X-fem: Nouvelles frontières pour les éléments finis. Revue européenne des éléments finis, 11 (1999), 131–150.
Moës, N., Dolbow, J., Belytschko, T.. A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Engng., 46 (1999), 131–150. 3.0.CO;2-J>CrossRef
Noor, A.K., Peters, J.M.. Reduced basis technique for nonlinear analysis of structures. AIAA Journal, 18 (2002), No. 4, 455–462.
Y. Renard, J. Pommier. Getfem++. An open source generic C++ library for finite element methods, http://home.gna.org/getfem.
G. Strang, G. Fix. An Analysis of the finite element method. Prentice-Hall, Englewood Cliffs, 1973.
Strouboulis, T., Babuska, I., Copps, K.. The design and analysis of the generalized finite element method. Comput. Meths. Appl. Mech. Engrg., 181 (2000), 43–69. CrossRef
Strouboulis, T., Babuska, I., Copps, K.. The generalized finite element method: an example of its implementation and illustration of its performance. Int. J. Numer. Meth. Engng., 47 (2000), 1401–1417. 3.0.CO;2-8>CrossRef
Sukumar, N., Huang, Z. Y., Prévost, J.-H., Suo, Z.. Partition of unity enrichment for bimaterial interface cracks. Int. J. Numer. Meth. Engng., 59 (2004), 1075–1102. CrossRef