A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, p. 813-839

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.

DOI : https://doi.org/10.1051/m2an/2011072
Classification:  74M15,  74S05,  35M85
Keywords: extended finite element method (Xfem), crack, unilateral contact, Signorini's problem
@article{M2AN_2012__46_4_813_0,
     author = {Amdouni, Saber and Hild, Patrick and Lleras, Vanessa and Moakher, Maher and Renard, Yves},
     title = {A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {4},
     year = {2012},
     pages = {813-839},
     doi = {10.1051/m2an/2011072},
     zbl = {1271.74354},
     mrnumber = {2891471},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_4_813_0}
}
Amdouni, Saber; Hild, Patrick; Lleras, Vanessa; Moakher, Maher; Renard, Yves. A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, pp. 813-839. doi : 10.1051/m2an/2011072. http://www.numdam.org/item/M2AN_2012__46_4_813_0/

[1] R. Adams, Sobolev spaces. Academic Press, New York (1975). | MR 450957 | Zbl 1098.46001

[2] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92 (1991) 353-375. | MR 1141048 | Zbl 0825.76353

[3] H.J. Barbosa and T. Hughes, The finite element method with Lagrange multipliers on the boundary : circumventing the Babuška-Brezzi condition. Comput. Methods Appl. Mech. Eng. 85 (1991) 109-128. | MR 1088318 | Zbl 0764.73077

[4] H.J. Barbosa and T. Hughes, Boundary Lagrange multipliers in finite element methods : error analysis in natural norms. Numer. Math. 62 (1992) 1-15. | MR 1159042 | Zbl 0765.65102

[5] H.J. Barbosa and T. Hughes, Circumventing the Babuška-Brezzi condition in mixed finite element approximations of elliptic variational inequalities. Comput. Methods Appl. Mech. Eng. 97 (1992) 193-210. | MR 1167712 | Zbl 0768.65033

[6] R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM : M2AN 37 (2003) 209-225. | Numdam | MR 1991197 | Zbl 1047.65099

[7] F. Ben Belgacem, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. SIAM J. Numer. Anal. 37 (2000) 1198-1216. | MR 1756422 | Zbl 0974.74055

[8] F. Ben Belgacem and Y. Renard, Hybrid finite element methods for the Signorini problem. Math. Comp. 72 (2003) 1117-1145. | MR 1972730 | Zbl 1023.74043

[9] H. Ben Dhia and M. Zarroug, Hybrid frictional contact particles in elements. Revue Européenne des Éléments Finis 9 (2002) 417-430. | Zbl 1120.74814

[10] S. Bordas and M. Duflot, Derivative recovery and a posteriori error estimate for extended finite elements. Comput. Methods Appl. Mech. Eng. 196 (2007) 3381-3399. | MR 2335273 | Zbl 1173.74401

[11] S. Bordas and M. Duflot, A posteriori error estimation for extended finite elements by an extended global recovery. Int. J. Numer. Methods Eng. 76 (2008) 1123-1138. | MR 2466773 | Zbl 1195.74171

[12] S. Bordas, M. Duflot and P. Le, A simple error estimator for extended finite elements. Commun. Numer. Methods Eng. 24 (2008) 961-971. | MR 2474664 | Zbl 1156.65093

[13] E. Chahine, P. Laborde and Y. Renard, Crack-tip enrichment in the XFEM method using a cut-off function. Int. J. Numer. Methods Eng. 75 (2008) 629-646. | MR 2442683 | Zbl 1195.74167

[14] 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. | Zbl 0875.65086

[15] J. Dolbow, N. Moës and T. Belytschko, An extended finite element method for modelling crack growth with frictional contact. Int. J. Numer. Methods Eng. 46 (1999) 131-150. | Zbl 0955.74066

[16] S. Géniaut, Approche XFEM pour la fissuration sous contact des structures industrielles. Thèse, École Centrale Nantes (2006).

[17] S. Géniaut, P. Massin and N. Moës, A stable 3D contact formulation for cracks using XFEM. Revue Européenne de Mécanique Numérique, Calculs avec Méthodes sans Maillage 16 (2007) 259-275. | Zbl 1208.74096

[18] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman (1985). | MR 775683 | Zbl 0695.35060

[19] P. Hansbo, C. Lovadina, I. Perugia and G. Sangalli, A Lagrange multiplier method for the finite element solution of elliptic interface problems using nonmatching meshes. Numer. Math. 100 (2005) 91-115. | MR 2129703 | Zbl 1066.65125

[20] J. Haslinger and Y. Renard, A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 47 (2009) 1474-1499. | MR 2497337 | Zbl 1205.65322

[21] 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. | MR 1422506 | Zbl 0873.73079

[22] P. Heintz and P. Hansbo, Stabilized Lagrange multiplier methods for bilateral elastic contact with friction. Comput. Methods Appl. Mech. Eng. 195 (2006) 4323-4333. | MR 2229843 | Zbl 1123.74045

[23] P. Hild, Numerical implementation of two nonconforming finite element methods for unilateral contact. Comput. Methods Appl. Mech. Eng. 184 (2000) 99-123. | MR 1752624 | Zbl 1009.74062

[24] P. Hild and Y. Renard, An error estimate for the Signorini problem with Coulomb friction approximated by finite elements. SIAM J. Numer. Anal. 45 (2007) 2012-2031. | MR 2346368 | Zbl 1146.74050

[25] P. Hild and Y. Renard, A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math. 15 (2010) 101-129. | MR 2594343 | Zbl 1194.74408

[26] P. Hild, V. Lleras and Y. Renard, A residual error estimator for the XFEM approximation of the elasticity problem. Submitted.

[27] S. Hüeber, B.I. Wohlmuth, An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal. 43 (2005) 156-173. | MR 2177139 | Zbl 1083.74047

[28] H. Khenous, J. Pommier and Y. Renard, Hybrid discretization of the Signorini problem with Coulomb friction, theoretical aspects and comparison of some numerical solvers. Appl. Numer. Math. 56 (2006) 163-192. | MR 2200937 | Zbl 1089.74046

[29] A. Khoei and M. Nikbakht, Contact friction modeling with the extended finite element method (XFEM). J. Mater. Proc. Technol. 177 (2006) 58-62.

[30] A. Khoei and M. Nikbakht, An enriched finite element algorithm for numerical computation of contact friction problems. Int. J. Mech. Sci. 49 (2007) 183-199.

[31] N. Kikuchi and J. Oden, Contact problems in elasticity. SIAM, Philadelphia (1988). | MR 961258 | Zbl 0685.73002

[32] P. Laborde and Y. Renard, Fixed point strategies for elastostatic frictional contact problems. Math. Methods Appl. Sci. 31 (2008) 415-441. | MR 2387416 | Zbl 1132.74032

[33] N. Moës, J. Dolbow and T. Belytschko, A finite element method for cracked growth without remeshing. Int. J. Numer. Methods Eng. 46 (1999) 131-150. | Zbl 0955.74066

[34] M. Moussaoui and K. Khodja, Regularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Partial Differential Equations 17 (1992) 805-826. | MR 1177293 | Zbl 0806.35049

[35] S. Nicaise, Y. Renard and E. Chahine, Optimal convergence analysis for the extended finite element method. Int. J. Numer. Methods Eng. 86 (2011) 528-548. | MR 2815989 | Zbl 1216.74029

[36] E. Pierres, M.-C. Baietto and A. Gravouil, A two-scale extended finite element method for modeling 3D crack growth with interfacial contact. Comput. Methods Appl. Mech. Eng. 199 (2010) 1165-1177. | MR 2594832 | Zbl 1227.74088

[37] 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).

[38] J.J. Rodenas, O.A. Gonzales-Estrada and J.E. Tarancon, 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. | Zbl 1195.74194

[39] R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63 (1995) 139-148. | MR 1365557 | Zbl 0856.65130

[40] G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Englewood Cliffs (1973). | MR 443377 | Zbl 0356.65096