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A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies

Published online by Cambridge University Press:  03 February 2012

Saber Amdouni ,
Patrick Hild ,
Vanessa Lleras ,
Maher Moakher  and
Yves Renard
Saber Amdouni
Affiliation:
Laboratoire LAMSIN, École Nationale d’Ingénieurs de Tunis, Université Tunis El Manar, B.P. 37, 1002 Tunis-Belvédère, Tunisia. Saber.Amdouni@insa-lyon.fr ; maher.moakher@enit.rnu.tn Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, 69621 Villeurbanne, France
Patrick Hild
Affiliation:
Laboratoire de Mathématiques de Besançon, CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France; patrick.hild@univ-fcomte.fr
Vanessa Lleras
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, CNRS UMR 5149, Université de Montpellier 2, Case courrier 51, Place Eugène Bataillon, 34095 Montpellier Cedex, France; vlleras@math.univ-montp2
Maher Moakher
Affiliation:
Laboratoire LAMSIN, École Nationale d’Ingénieurs de Tunis, Université Tunis El Manar, B.P. 37, 1002 Tunis-Belvédère, Tunisia. Saber.Amdouni@insa-lyon.fr ; maher.moakher@enit.rnu.tn
Yves Renard
Affiliation:
Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, LaMCoS UMR5259, 69621 Villeurbanne, France; Yves.Renard@insa-lyon.fr
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Abstract

The purpose of this paper is to provide a priori error estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM) for two dimensional elastic bodies. This method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a finite-element method on the non-cracked domain, avoiding the definition of a specific mesh of the crack. Additionally, we present numerical experiments which confirm the efficiency of the proposed method.

Keywords

Extended finite element method (Xfem)crackunilateral contactSignorini’s problem

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012 , pp. 813 - 839
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

R. Adams, Sobolev spaces. Academic Press, New York (1975).
Alart, P. and Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92 (1991) 353–375. Google Scholar
Barbosa, H.J. and Hughes, T., The finite element method with Lagrange multipliers on the boundary : circumventing the Babuška-Brezzi condition. Comput. Methods Appl. Mech. Eng. 85 (1991) 109–128. Google Scholar
Barbosa, H.J. and Hughes, T., Boundary Lagrange multipliers in finite element methods : error analysis in natural norms. Numer. Math. 62 (1992) 1–15. Google Scholar
Barbosa, H.J. and Hughes, T., Circumventing the Babuška-Brezzi condition in mixed finite element approximations of elliptic variational inequalities. Comput. Methods Appl. Mech. Eng. 97 (1992) 193–210. Google Scholar
Becker, R., Hansbo, P. and Stenberg, R., A finite element method for domain decomposition with non-matching grids. ESAIM : M2AN 37 (2003) 209–225. Google Scholar
Ben Belgacem, F., Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. SIAM J. Numer. Anal. 37 (2000) 1198–1216. Google Scholar
Ben Belgacem, F. and Renard, Y., Hybrid finite element methods for the Signorini problem. Math. Comp. 72 (2003) 1117–1145. Google Scholar
Ben Dhia, H. and Zarroug, M., Hybrid frictional contact particles in elements. Revue Européenne des Éléments Finis 9 (2002) 417–430. Google Scholar
Bordas, S. and Duflot, M., Derivative recovery and a posteriori error estimate for extended finite elements. Comput. Methods Appl. Mech. Eng. 196 (2007) 3381–3399. Google Scholar
Bordas, S. and Duflot, M., A posteriori error estimation for extended finite elements by an extended global recovery. Int. J. Numer. Methods Eng. 76 (2008) 1123–1138. Google Scholar
Bordas, S., Duflot, M. and Le, P., A simple error estimator for extended finite elements. Commun. Numer. Methods Eng. 24 (2008) 961–971. Google Scholar
Chahine, E., Laborde, P. and Renard, Y., Crack-tip enrichment in the XFEM method using a cut-off function. Int. J. Numer. Methods Eng. 75 (2008) 629–646. Google Scholar
P. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis. Part 1, edited by P. Ciarlet and J. Lions, North Holland II (1991) 17–352.
Dolbow, J., Moës, N. and Belytschko, T., An extended finite element method for modelling crack growth with frictional contact. Int. J. Numer. Methods Eng. 46 (1999) 131–150. Google Scholar
S. Géniaut, Approche XFEM pour la fissuration sous contact des structures industrielles. Thèse, École Centrale Nantes (2006).
Géniaut, S., Massin, P. and Moës, N., A stable 3D contact formulation for cracks using XFEM. Revue Européenne de Mécanique Numérique, Calculs avec Méthodes sans Maillage 16 (2007) 259–275. Google Scholar
P. Grisvard, Elliptic problems in nonsmooth domains. Pitman (1985).
Hansbo, P., Lovadina, C., Perugia, I. and Sangalli, G., A Lagrange multiplier method for the finite element solution of elliptic interface problems using nonmatching meshes. Numer. Math. 100 (2005) 91–115. Google Scholar
Haslinger, J. and Renard, Y., A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 47 (2009) 1474–1499. Google Scholar
J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis. Part 2, edited by P. Ciarlet and J.-L. Lions, North Holland IV (1996) 313–485.
Heintz, P. and Hansbo, P., Stabilized Lagrange multiplier methods for bilateral elastic contact with friction. Comput. Methods Appl. Mech. Eng. 195 (2006) 4323–4333. Google Scholar
Hild, P., Numerical implementation of two nonconforming finite element methods for unilateral contact. Comput. Methods Appl. Mech. Eng. 184 (2000) 99–123. Google Scholar
Hild, P. and Renard, Y., An error estimate for the Signorini problem with Coulomb friction approximated by finite elements. SIAM J. Numer. Anal. 45 (2007) 2012–2031. Google Scholar
Hild, P. and Renard, Y., A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math. 15 (2010) 101–129. Google Scholar
P. Hild, V. Lleras and Y. Renard, A residual error estimator for the XFEM approximation of the elasticity problem. Submitted.
Hüeber, S., Wohlmuth, B.I., An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal. 43 (2005) 156–173. Google Scholar
Khenous, H., Pommier, J. and Renard, Y., Hybrid discretization of the Signorini problem with Coulomb friction, theoretical aspects and comparison of some numerical solvers. Appl. Numer. Math. 56 (2006) 163–192. Google Scholar
Khoei, A. and Nikbakht, M., Contact friction modeling with the extended finite element method (XFEM). J. Mater. Proc. Technol. 177 (2006) 58–62. Google Scholar
Khoei, A. and Nikbakht, M., An enriched finite element algorithm for numerical computation of contact friction problems. Int. J. Mech. Sci. 49 (2007) 183–199. Google Scholar
N. Kikuchi and J. Oden, Contact problems in elasticity. SIAM, Philadelphia (1988).
Laborde, P. and Renard, Y., Fixed point strategies for elastostatic frictional contact problems. Math. Methods Appl. Sci. 31 (2008) 415–441. Google Scholar
Moës, N., Dolbow, J. and Belytschko, T., A finite element method for cracked growth without remeshing. Int. J. Numer. Methods Eng. 46 (1999) 131–150. Google Scholar
Moussaoui, M. and Khodja, K., Regularité des solutions d’un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Partial Differential Equations 17 (1992) 805–826. Google Scholar
Nicaise, S., Renard, Y. and Chahine, E., Optimal convergence analysis for the extended finite element method. Int. J. Numer. Methods Eng. 86 (2011) 528–548. Google Scholar
Pierres, E., Baietto, M.-C. and Gravouil, A., A two-scale extended finite element method for modeling 3D crack growth with interfacial contact. Comput. Methods Appl. Mech. Eng. 199 (2010) 1165–1177. Google Scholar
J. Pommier and Y. Renard, Getfem++, an open source generic C++ library for finite element methods. Available on : http://download.gna.org/getfem/html/homepage/userdoc/index.html, December, 23rd (2011).
Rodenas, J.J., Gonzales-Estrada, O.A. and Tarancon, J.E., A recovery-type error estimator for the extended finite element method based on singular plus smooth stress field splitting. Int. J. Numer. Methods Eng. 76 (2008) 545–571. Google Scholar
Stenberg, R., On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63 (1995) 139–148. Google Scholar
G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Englewood Cliffs (1973).