Mixed finite element approximation of 3D contact problems with given friction : error analysis and numerical realization
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 563-578.

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.

DOI : 10.1051/m2an:2004026
Classification : 65N30, 74M15
Mots-clés : mixed finite element methods, unilateral contact problems with friction, a priori error estimates 
@article{M2AN_2004__38_3_563_0,
     author = {Haslinger, Jaroslav and Sassi, Taoufik},
     title = {Mixed finite element approximation of {3D} contact problems with given friction : error analysis and numerical realization},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {563--578},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/m2an:2004026},
     mrnumber = {2075760},
     zbl = {1080.74046},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2004026/}
}
TY  - JOUR
AU  - Haslinger, Jaroslav
AU  - Sassi, Taoufik
TI  - Mixed finite element approximation of 3D contact problems with given friction : error analysis and numerical realization
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2004
SP  - 563
EP  - 578
VL  - 38
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2004026/
DO  - 10.1051/m2an:2004026
LA  - en
ID  - M2AN_2004__38_3_563_0
ER  - 
%0 Journal Article
%A Haslinger, Jaroslav
%A Sassi, Taoufik
%T Mixed finite element approximation of 3D contact problems with given friction : error analysis and numerical realization
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2004
%P 563-578
%V 38
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2004026/
%R 10.1051/m2an:2004026
%G en
%F M2AN_2004__38_3_563_0
Haslinger, Jaroslav; Sassi, Taoufik. Mixed finite element approximation of 3D contact problems with given friction : error analysis and numerical realization. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 563-578. doi : 10.1051/m2an:2004026. http://archive.numdam.org/articles/10.1051/m2an:2004026/

[1] R.A. Adams, Sobolev Spaces. Academic Press (1975). | MR | Zbl

[2] G. Amontons, Sur l'origine de la résistance dans les machines. Mémoires de l'Académie Royale (1699) 206-222.

[3] 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. | Zbl

[4] 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. | Zbl

[5] 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. | Zbl

[6] C.A. Coulomb, Théorie des machines simples. Mémoire de Mathématique et de Physique de l'Académie Royale 10 (1785) 145-173.

[7] Z. Dostál, Box constrained quadratic programming with proportioning and projections. SIAM J. Opt. 7 (1997) 871-887. | Zbl

[8] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | MR | Zbl

[9] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). | MR | Zbl

[10] R. Glowinski, Numerical methods for nonlinear variational problems. Springer, New York (1984). | MR | Zbl

[11] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monogr. Studies Math., Pitman 24 (1985). | MR | Zbl

[12] 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. | Zbl

[13] J. Haslinger and P.D. Panagiolopoulas, Approximation of contact problems with friction by reciprocal variational formulations. Proc. Roy. Soc. Edinburgh 98A (1984) 365-383. | Zbl

[14] 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. | Zbl

[15] J. Haslinger, R. Kučera and Z. Dostál, An algorithm for numerical realization of 3D contact problems with Coulomb friction. J. Comput. Appl. Math. 164-165 (2004) 387-408. | Zbl

[16] 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. | Zbl

[17] N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988). | MR | Zbl

[18] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications. Academic Press (1980). | MR | Zbl

[19] K. Lhalouani and T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Numer. Math. 7 (1999) 23-30. | Zbl

Cité par Sources :