Bayada, Guy; Sabil, Jalila; Sassi, Taoufik
Convergence of a Neumann-Dirichlet algorithm for two-body contact problems with non local Coulomb's friction law
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 2 , p. 243-262
Zbl 1133.74042 | MR 2405147
doi : 10.1051/m2an:2008003
URL stable : http://www.numdam.org/item?id=M2AN_2008__42_2_243_0

Classification:  65N30,  65N55,  65K05
In this paper, the convergence of a Neumann-Dirichlet algorithm to approximate Coulomb's contact problem between two elastic bodies is proved in a continuous setting. In this algorithm, the natural interface between the two bodies is retained as a decomposition zone.

Bibliographie

[1] P. Alart, M. Barboteu, P. Le Tallec and M. Vidrascu, Méthode de Schwarz additive avec solveur grossier pour problèmes non symétriques. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 399-404. MR 1784922 | Zbl 0961.65108

[2] L. Baillet and T. Sassi, Simulations numériques de différentes méthodes d'éléments finis pour les problèmes contact avec frottement. C. R. Acad. Sci. Paris Sér. II B 331 (2003) 789-796. Zbl 1134.74401

[3] L. Baillet and T. Sassi, Mixed finite element method for the Signorini problem with friction. Numer. Methods Partial Differential Equations 22 (2006) 1489-1508. MR 2257645 | Zbl 1105.74041

[4] G. Bayada, J. Sabil and T. Sassi, Algorithme de Neumann-Dirichlet pour des problèmes de contact unilatéral: résultat de convergence. C. R. Math. Acad. Sci. Paris 335 (2002) 381-386. MR 1931521 | Zbl 1044.74031

[5] A.B. Chandhary and K.J. Bathe, A solution method for static and dynamic analysis of three-dimensional contact problems with friction. Comput. Struc. 24 (1986) 855-873. Zbl 0604.73116

[6] P.W. Christensen, A. Klarbring, J.S. Pang and N. Strömberg, Formulation and comparison of algorithms for frictional contact problems. Internat. J. Numer. Methods Engrg. 42 (1998) 145-173. MR 1618014 | Zbl 0917.73063

[7] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques 21. Dunod, Paris (1972). MR 464857 | Zbl 0298.73001

[8] C. Eck and B. Wohlmuth, Convergence of a contact-Neumann iteration for the solution of two-body contact problems. Math. Models Methods Appl. Sci. 13 (2003) 1103-1118. MR 1998817 | Zbl 1053.74044

[9] C. Farhat and F.X. Roux, Implicit parallel processing in structural mechanics. Computational Mechanics Advances 1 (1994) 1-124. MR 1280753 | Zbl 0805.73062

[10] R. Glowinski, J.-L. Lions and R. Trémolières, Numerical analysis of variational inequalities, Studies in Mathematics and its Applications 8. North-Holland Publishing Co., Amsterdam (1981). Translated from the French. MR 635927 | Zbl 0463.65046

[11] J. Haslinger, Z. Dostál and R. Kučera, On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2261-2281. MR 1903144 | Zbl 1131.74344

[12] N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988). MR 961258 | Zbl 0685.73002

[13] R. Kornhuber and R. Krause, Adaptive multigrid methods for Signorini's problem in linear elasticity. Comput. Vis. Sci. 4 (2001) 9-20. MR 1936249 | Zbl 1051.74045

[14] R.H. Krause, Monotone multigrid methods for Signorini's problem with friction. Ph.D. thesis, University of Berlin, Germany (2001).

[15] R.H. Krause and B.I. Wohlmuth, Nonconforming domain decomposition techniques for linear elasticity. East-West J. Numer. Math. 8 (2000) 177-206. MR 1807260 | Zbl 1050.74046

[16] R.H. Krause and B.I. Wohlmuth, A Dirichlet-Neumann type algorithm for contact problems with friction. Comput. Vis. Sci. 5 (2002) 139-148. MR 1950338 | Zbl 1099.74536

[17] P. Le Tallec, Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1 (1994) 121-220. MR 1263805 | Zbl 0802.73079

[18] L. Lusternik and V. Sobolev, Précis d'analyse fonctionnelle. MIR, Moscow (1989).

[19] B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain decomposition, Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (1996). MR 1410757 | Zbl 0857.65126

[20] G. Zavarise and P. Wriggers, A superlinear convergent augmented Lagrangian procedure for contact problems. Engrg. Comput. 16 (1999) 88-119. Zbl 0947.74079