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Mixed finite element approximation of 3D contactproblems with given friction: Error analysisand numerical realization

Published online by Cambridge University Press:  15 June 2004

Jaroslav Haslinger  and
Taoufik Sassi
Jaroslav Haslinger
Affiliation:
Department of Numerical Mathematics, Charles University 12116 Praha 2, Czech Republic. mfkfk@met.mff.cuni.cz.
Taoufik Sassi
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR UFR Sciences Campus II, Bd Maréchal Juin, 14032 Caen Cedex, France. sassi@math.unicaen.fr.
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Abstract

This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution is smooth enough. The numerical realization of such problems will be discussed and results of a model example will be shown.

Keywords

Mixed finite element methodsunilateral contact problems with frictiona priori error estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 3 , May 2004 , pp. 563 - 578
Copyright
© EDP Sciences, SMAI, 2004

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References

R.A. Adams, Sobolev Spaces. Academic Press (1975).
G. Amontons, Sur l'origine de la résistance dans les machines. Mémoires de l'Académie Royale (1699) 206–222.
L. Baillet and T. Sassi, Méthodes d'éléments finis avec hybridisation frontière pour les problèmes de contact avec frottement. C.R. Acad. Sci. Paris, Ser. I 334 (2002) 917–922.
G. Bayada, M. Chambat, K. Lhalouani and T. Sassi, Éléments finis avec joints pour des problèmes de contact avec frottement de Coulomb non local. C.R. Acad. Sci. Paris, Ser. I 325 (1997) 1323–1328.
P.-G. Ciarlet, The finite element method for elliptic problems, Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. 2, Part 1, North-Holland (1991) 17–352.
Coulomb, C.A., Théorie des machines simples. Mémoire de Mathématique et de Physique de l'Académie Royale 10 (1785) 145–173.
Dostál, Z., Box constrained quadratic programming with proportioning and projections. SIAM J. Opt. 7 (1997) 871–887. CrossRef
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).
R. Glowinski, Numerical methods for nonlinear variational problems. Springer, New York (1984).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monogr. Studies Math., Pitman 24 (1985).
J. Haslinger and I. Hlaváček, Approximation of the Signorini problem with friction by mixed finite element method, J. Math. Anal. Appl. 86 (1982) 99–122. CrossRef
Haslinger, J. and Panagiolopoulas, P.D., Approximation of contact problems with friction by reciprocal variational formulations. Proc. Roy. Soc. Edinburgh 98A (1984) 365–383. CrossRef
J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. 4, Part 2, North-Holland (1996) 313–485.
Haslinger, J., Kučera, R. and Dostál, Z., An algorithm for numerical realization of 3D contact problems with Coulomb friction. J. Comput. Appl. Math. 164-165 (2004) 387–408. CrossRef
P. Hild, À propos d'approximation par éléments finis optimale pour les problèmes de contact unilatéral. C.R. Acad. Sci. Paris, Ser. I 326 (1998) 1233–1236.
N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988).
D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications. Academic Press (1980).
Lhalouani, K. and Sassi, T., Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Numer. Math. 7 (1999) 23–30.