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Locking-Free Finite Elements for Unilateral CrackProblems in Elasticity

Published online by Cambridge University Press:  27 January 2009

Z. Belhachmi ,
J.-M. Sac-Epée  and
S. Tahir
Z. Belhachmi*
Affiliation:
LMAM UMR7122, Université Paul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France
J.-M. Sac-Epée
Affiliation:
LMAM UMR7122, Université Paul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France
S. Tahir
Affiliation:
LMAM UMR7122, Université Paul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France
*
belhach@univ-metz.fr
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Abstract

We consider mixed and hybrid variational formulations to the linearizedelasticity system in domains with cracks. Inequality type conditions areprescribed at the crack faces which results in unilateral contact problems. Thevariational formulations are extended to the whole domain including the crackswhich yields, for each problem, a smooth domain formulation. Mixedfinite element methods such as PEERS or BDM methods are designed to avoidlocking for nearly incompressible materials in plane elasticity. We study andimplement discretizations based on such mixed finite element methods for thesmooth domain formulations to the unilateral crack problems. We obtainconvergence rates and optimal error estimates and we present some numericalexperiments in agreement with the theoretical results.

Keywords

crack problemsvariational inequalitiessmooth domain methodmixedfinite elementsa priori estimates
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 1: Modelling and numerical methods in contact mechanics , 2009 , pp. 1 - 20
Copyright
© EDP Sciences, 2009

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References

J. Alberty, C. Carstensen, S. A. Funken, R. Klose. Matlab Implementation of the Finite Element Method in Elasticity. Berichtsreihe des Mathematischen Seminars Kiel, 00-21 (2000).
D. N. Arnold, F. Brezzi, J. Douglas. PEERS: A new finite element for plane elasticity Japan J. Appl. Math., No. 1 (1984), 347–367.
Belhachmi, Z., Ben Belgacem, F.. Quadratic finite element for Signorini problem. Math. Comp., 72 (2003), No. 241, 83–104. CrossRef
Belhachmi, Z., Sac-Epée, J.M., Sokolowski, J.. Mixed finite element methods for a smooth domain formulation of a crack problem. SIAM J. Numer. Anal., 43 (2005), No. 3, 1295–1320. CrossRef
F. Ben Belgacem. Numerical simulation of some variational inequalities arisen from unilateral contact problems by finite element method. Siam J. Numer. Anal, 37 (2000),No. 4, 1198–1216.
Ben Belgacem, F., Hild, P., Laborde, P.. Extension of the mortar finite element method to a variational inequality modelling unilateral contact. Math. Models Methods Appl. Sci., 9 (1999), No. 2, 287–303. CrossRef
Ben Belgacem, F., Renard, Y.. Hybrid finite element methods for the Signorini problem. Math. Comput., 72 (2003), No. 243, 1117–1145. CrossRef
Bernardi, C., Girault, V.. A local regularization operator for triangular and quadrilateral finite elements. SIAM. J. Numer. Anal., 35 (1998), No. 5, 1893–1916. CrossRef
Braess, D., Klaas, O., Niekamp, R., Stein, E., Wobschal, F.. Error Indicators For Mixed Finite Elements in 2-dimensional Linear Elasticity. Comput. Methods. Appl. Mech. Engrg., 127 (1995), No. 1-4, 345–356. CrossRef
Brezzi, F., Douglas Jr, J., Marini, L.D.. Two families of mixed finite elements for second order elliptic problems. Numer. Math., 47 (1985), No. 2, 217–235. CrossRef
F. Brezzi, M. Fortin. Mixed and hybrid finite element methods. Springer Verlag, New York, Springer Series in Computational Mathematics, 15, 1991.
Carstensen, C., Dolzmann, G., Funken, S.A., Helm, D.S.. Locking-free adaptive mixed finite element in linear elasticity. Comput. Methods. Appl.Mech. Engrg., 190 (2000), No. 13-14, 1701–1718. CrossRef
P.G. Ciarlet. Basic Error Estimates for Elliptic Problems. In the Handbook of Numerical Analysis, Vol II, P.G. Ciarlet & J.-L. Lions eds, North-Holland, (1991), 17–351.
P. Coorevits, P. Hild, K. Lhalouani, T. Sassi. Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comp., 71, (2001), No. 237, 1–25.
G. Duvaut, J.-L. Lions. Les inéquations en mécanique et en physique. Dunod, 1972.
V. Girault, P.-A. Raviart. Finite element methods for the Navier-Stokes equations, Theory and algorithms. Springer-Verlag 1986.
R. Glowinski. Lectures on numerical methods for nonlinear variational problems. Springer, Berlin, 1980.
J. Haslinger, I. Hlaváček. Contact between Elastic Bodies -2.Finite Element Analysis, Aplikace Matematiky, 26 (1981), 263–290.
J. Haslinger, I. Hlaváček, J. Nečas. Numerical Methods for Unilateral Problems in Solid Mechanics, in the Handbook of Numerical Analysis, Vol IV, Part 2, P.G. Ciarlet & J.-L. Lions eds, North-Holland, 1996.
F. Hecht, O. Pironneau. FreeFem++, www.freefem.org
Hild, P., Renard, Y.. An error estimates for the Signorini problem with Coulomb friction approximated by finite elements. Siam J. Numer. Anal., 45 (2007), No. 5, 2012–2031. CrossRef
S. Hüeber, B.I. Wohlmuth. An optimal a priori error estimates for nonlinear multibody contact problems. SIAM J. Numer. Anal., 43 (2005), No. 1, 156–173
Khludnev, A.M., Sokolowski, J.. Smooth domain method for crack problems. Quarterly of Applied Mathematics., 62 (2004), No. 3, 401–422. CrossRef
N. Kikuchi, J. Oden. Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM, 1988.
D. Kinderlehrer, G. Stamppachia. An introduction to variational inequalities and their applications, Academic Press, 1980.
Lhalouani, K., Sassi, T.. Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Numer. Math., 7 (1999), No. 1, 23–30.
Slimane, L., Bendali, A., Laborde, P.. Mixed formulations for a class of variational inequalities. M2AN, 38 (2004), 1, 177–201. CrossRef
Stenberg, R.. A family of mixed finite elements for the elasticity problem. Numer. Math., 53 (1988), 5, 513–538. CrossRef
S. Tahir. Méthodes d'approximation par éléments finis et analyse a posteriori d'inéquations variationnelles modélisant des problèmes de fissures unilatérales en élasticité linéaire. Ph.D. Thesis, University of Metz, France (2006).
S. Tahir, Z. Belhachmi. Mixed finite elements discretizations of some variational inequalities arising in elasticity problems in domains with cracks. Electron. J. Diff. Eqns., Conference 11 (2004), 33–40.
Z.-H. Zhong. Finite Element Procedures for Contact-Impact Problems. Oxford. University. Press, Oxford 1993.