Spider XFEM, an extended finite element variant for partially unknown crack-tip displacement
DOI:
https://doi.org/10.13052/REMN.17.625-636Keywords:
fracture, Xfem, error estimates, numerical convergence rateAbstract
In this paper, we introduce a new variant of the extended finite element method (Xfem) allowing an optimal convergence rate when the asymptotic displacement is partially unknown at the crack tip. This variant consists in the addition of an adapted discretization of the asymptotic displacement. We give a mathematical result of quasi-optimal a priori error estimate which allows to analyze the potentialities of the method. Some computational tests are provided and a comparison is made with the classical Xfem.
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References
Adams R., 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., vol. 64,
p. 1033-1056, 2005.
Bendhia H., “ Multiscale mechanical problems : the Arlequin method”, C. R. Acad. Sci., série
I, Paris, vol. 326, p. 899-904, 1998.
Chahine E., Laborde P., Renard Y., “ Quasi-optimal convergence result in fracture mechanics
with XFEM”, C.R. Math. Acad. Sci. Paris, vol. 342, p. 527-532, 2006.
Chahine E., Laborde P., Renard Y., “ Crack tip enrichment in the XFEM method using a cut-off
function”, Int. J. Numer. Meth. Engng., to appear.
Chang J., Xu J.-Q., “ The singular stress field and stress intensity factors of a crack terminating
at a bimaterial interface”, Int. J. Mech. Sci., vol. 49, p. 888-897, 2007.
Ciarlet P., The finite element method for elliptic problems, Studies in Mathematics and its Applications
No 4, North Holland, 1978.
Ern A., Guermond J.-L., Éléments finis: théorie, applications, mise en oeuvre, Mathématiques
et Applications 36, SMAI, Springer-Verlag, 2002.
Glowinski R., He J., Rappaz J., Wagner J., “ Approximation of multi-scale elliptic problems
using patches of elements”, C. R. Math. Acad. Sci., Paris, vol. 337, p. 679-684, 2003.
Grisvard P., “ Problèmes aux limites dans les polygones - Mode d’emploi”, EDF Bull. Dirctions
Etudes Rech. Sér. C. Math. Inform. 1, vol. MR 87g:35073, p. 21-59, 1986.
Grisvard P., Singularities in boundary value problems, Masson, 1992.
Laborde P., Renard Y., Pommier J., Salaün M., “ High Order Extended Finite Element Method
For Cracked Domains”, Int. J. Numer. Meth. Engng., vol. 64, p. 354-381, 2005.
Lemaitre J., Chaboche J.-L., Mechanics of Solid Materials, Cambridge University Press, 1994.
Melenk J., Babu·ska I., “ The partition of unity finite element method: Basic theory and applications”,
Comput. Meths. Appl. Mech. Engrg., vol. 139, p. 289-314, 1996.
Moës N., Belytschko T., “ X-FEM: Nouvelles Frontières Pour les Eléments Finis”, Revue européenne
des éléments finis, vol. 11, p. 131-150, 1999a.
Moës N., Dolbow J., Belytschko T., “ A finite element method for crack growth without remeshing”,
Int. J. Numer. Meth. Engng., vol. 46, p. 131-150, 1999b.
Renard Y., Pommier J., Getfem++, An open source generic C++ library for finite element methods,
Rice J., “ Elastic fracture mechanics concepts for interfacial cracks”, Journal of Applied Mechanics,
vol. 55, p. 98-103, 1988.
Strang G., Fix G., An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs,
Strouboulis T., Babuska I., Copps K., “ The design and analysis of the Generalized Finite Element
Method”, Comput. Meths. Appl. Mech. Engrg., vol. 181, p. 43-69, 2000.
