Numerical implementation of two nonconforming finite element methods for unilateral contact. (English) Zbl 1009.74062

The paper addresses the nonforming finite element approximation of unilateral contact between elastic bodies. As the meshes usually do not match, two methods of coupling the finite element meshes are investigated. One is a mortar-element type approach, the other uses explicit contact conditions between nodes and line-segments – this only works in two dimensions. Error estimates for both approaches are given, the numerical matrix formulation is presented, and finally a numerical example comparing the two approaches is discussed.


74S05 Finite element methods applied to problems in solid mechanics
74M15 Contact in solid mechanics
Full Text: DOI


[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Alart, P., Méthode de Newton généralisée en mécanique du contact, J. math. pures appl., 76, 83-108, (1997) · Zbl 0868.49021
[3] Bayada, G.; Chambat, M.; Lhalouani, K.; Sassi, T., Eléments finis avec joints pour des problèmes de contact avec frottement de Coulomb non local, C. R. acad. sci. Paris, 325, Série I, 1323-1328, (1997) · Zbl 0898.73059
[4] F. Ben Belgacem, Discrétisations 3-D non conformes par la méthode de décomposition de domaine des éléments avec joints: Analyse mathématique et mise en œuvre pour le problème de Poisson, Thèse de l’Université Pierre et Marie Curie, Paris 6, 1993
[5] Ben Belgacem, F.; Hild, P.; Laborde, P., Approximation of the unilateral contact problem by the mortar finite element method, C. R. acad. sci. Paris, 324, Série I, 123-127, (1997) · Zbl 0872.65057
[6] F. Ben Belgacem, P. Hild, P. Laborde, Extension of the mortar finite element method to a variational inequality modeling unilateral contact, Internal report of MIP, IR96.18, Math. Mod. and Meth. in the Appl. Sci. 9 (2), (1999) 287-303 · Zbl 0940.74056
[7] Ben Belgacem, F.; Hild, P.; Laborde, P., The mortar finite element method for contact problems, Math. comput. model., 28, 263-271, (1998) · Zbl 1098.74682
[8] C. Bernardi, Y. Maday, A.T. Patera, A new nonconforming approach to domain decomposition: The mortar element method, in: H. Brezis, J.-L. Lions, Pitman (Eds.), Collège de France Seminar, 1994, pp. 13-51 · Zbl 0797.65094
[9] Brezzi, F.; Hager, W.W.; Raviart, P.A., Error estimates for the finite element solution of variational inequalities, Numer. math., 28, 431-443, (1997) · Zbl 0369.65030
[10] Canon, M.D.; Cullum, C.D., A tight upper bound on the rate of convergence of the Frank-Wolfe algorithm, SIAM J. on control, 6, 509-516, (1968) · Zbl 0186.24002
[11] Ciarlet, P.-G., The finite element method for elliptic problems, (1978), North Holland Amsterdam · Zbl 0383.65058
[12] P. Coorevits, P. Hild, J.-P. Pelle, Contrôle des calculs par éléments finis pour un problème de contact unilatéral, Internal report of LMT-Cachan, IR203, submitted to Revue Européenne des Eléments Finis, 1998
[13] G. Duvaut, J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 · Zbl 0298.73001
[14] Frank, M.; Wolfe, P., An algorithm for quadratic programming, Naval research logist quarterly, 3, 95-110, (1956)
[15] Haslinger, J.; Hlaváček, I., Contact between elastic bodies - 2. finite element analysis, Aplikace matematiky, 26, 263-290, (1981) · Zbl 0465.73144
[16] J. Haslinger, I. Hlaváček, J. Nečas, Numerical methods for unilateral problems in solid mechanics, in: P.G. Ciarlet, J.-L. Lions (Eds.), Handbook of Numerical Analysis, vol. IV, Part 2, North-Holland, Amsterdam, 1996, pp. 313-485
[17] P. Hild, Problèmes de contact unilatéral et maillages éléments finis incompatibles, Thèse de l’Université Paul Sabatier, Toulouse 3, 1998
[18] Hild, P., Eléments finis non conformes pour un problème de contact unilatéral avec frottement, C. R. acad. sci. Paris, 324, Série I, 707-710, (1997) · Zbl 0873.73071
[19] Hild, P., A propos d’approximation par éléments finis optimale pour LES problèmes de contact unilatéral, C. R. acad. sci. Paris, 326, Série I, 1233-1236, (1998) · Zbl 0914.73060
[20] J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimisation Algorithms, vol. I and II, in: Grundlehren der Mathematischen Wissenschaften (305,306), Springer, Berlin, 1993
[21] Kikuchi, N.; Oden, J.T., Contact problems in elasticity: A study of variational inequalities and finite element methods, (1988), SIAM Philadelphia · Zbl 0685.73002
[22] Le Tallec, P.; Sassi, T., Domain decomposition with nonmatching grids: augmented Lagrangian approach, Math. of comp., 64, 1367-1396, (1995) · Zbl 0849.65087
[23] C. Licht, E. Pratt, M. Raous, Remarks on a numerical method for unilateral contact including friction, in: International Series of Numerical Mathematics, vol. 101, Birkhäuser, 1991, pp. 129-144 · Zbl 0762.73076
[24] M. Minoux, Programmation mathématique, théorie et algorithmes, Tome 1, Dunod, 1983
[25] K. Lhalouani, T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems, Internal report of Equipe d’Analyse Numérique Lyon/Saint-Etienne, IR286, 1998
[26] P. Wolfe, Convergence theory in nonlinear programming, in: J. Abadie (Ed.), Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970, pp. 1-36 · Zbl 0336.90045
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