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Mixed finite element methods for unilateral problems: convergence analysis and numerical studies
Author(s):
Patrice
Coorevits;
Patrick
Hild;
Khalid
Lhalouani;
Taoufik
Sassi.
Journal:
Math. Comp.
71
(2002),
1-25.
MSC (2000):
Primary 65N30, 74M15
Posted:
May 21, 2001
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Abstract:
In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.
References:
-
- 1.
- R. A. Adams, Sobolev Spaces, Academic Press, 1975. MR 56:9247
- 2.
- F. Ben Belgacem, Méthodes d'éléments finis pour des inéquations variationnelles de contact unilatéral. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 811-816. MR 2000j:74083
- 3.
- F. Ben Belgacem, P. Hild and P. Laborde, Extension of the mortar finite element method to a variational inequality modeling unilateral contact. Math. Models Methods Appl. Sci. 9 (1999), 287-303. MR 99m:73066
- 4.
- C. Bernardi, Y. Maday and A. T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, Collège de France Seminar, H. Brezis, J. L. Lions, Pitman, 1994, pp. 13-51. MR 95a:65201
- 5.
- F. Brezzi, W. W. Hager and P. A. Raviart, Error Estimates for the Finite Element Solution of Variational Inequalities. Numer. Math. 28 (1977), 431-443. MR 56:7254
- 6.
- F. Brezzi, W. W. Hager and P. A. Raviart, Error estimates for the finite element solution of variational inequalities. Part 2: Mixed Methods, Numer. Math. 31 (1978), 1-16. MR 80b:65135
- 7.
- M. D. Canon and C. D. Cullum, A tight upper bound on the rate of convergence of the Frank-Wolfe algorithm. SIAM J. Control Optim. 6 (1968), 509-516. MR 39:3827; MR 39:2462
- 8.
- P. G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis, Volume II, Part 1, Eds. P. G. Ciarlet and J. L. Lions, North Holland, 17-352, 1991. MR 58:25001
- 9.
- G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972. MR 57:4778
- 10.
- M. Frank and P. Wolfe, An algorithm for quadratic programming. Naval Res. Logist. 3 (1956), 95-110.
- 11.
- J. Haslinger, I. Hlavácek and J. Necas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Volume IV, Part 2, Eds. P. G. Ciarlet and J. L. Lions, North Holland, 313-485, 1996. CMP 97:05
- 12.
- P. Hild, Numerical Implementation of two nonconforming finite element methods for unilateral contact. Comput. Methods Appl. Mech. Engrg. 184 (2000) 99-123. MR 2000k:74058
- 13.
- P. Hild, A propos d'approximation par éléments finis optimale pour les problèmes de contact unilatéral. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 1233-1236. MR 99h:65191
- 14.
- N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM, Philadelphia, 1988. MR 89j:73097
- 15.
- D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, 1980. MR 81g:49013
- 16.
- P. Le Tallec and T. Sassi, Domain decomposition with nonmatching grids: augmented lagrangian approach. Math. Comp. 64 (1995), 1367-1396. MR 95m:65212
- 17.
- K. Lhalouani and T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Numer. Math. 7 (1999), 23-30. MR 2000b:65217
- 18.
- F. Natterer, Optimale
 -Konvergenz finiter Elemente bei Variationsungleichungen. Bonner Math. Schriften 89 (1976), 1-12. MR 56:10070
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Additional Information:
Patrice
Coorevits
Affiliation:
Laboratoire de Mécanique et CAO, Université de Picardie - Jules Verne, IUT, 48 rue d'Ostende, 02100 Saint-Quentin, France
Email:
patrice.coorevits@insset.u-picardie.fr
Patrick
Hild
Affiliation:
Laboratoire de Mathématiques, Université de Savoie / CNRS EP 2067, 73376 Le Bourget Du Lac, France
Email:
hild@univ-savoie.fr
Khalid
Lhalouani
Affiliation:
Laboratoire de Modélisation Mathématique et de Calcul Scientifique, INSA de Lyon / CNRS UMR 5585, 20 avenue Albert Einstein, 69621 Villeurbanne, France
Email:
khalid@laninsa.insa-lyon.fr
Taoufik
Sassi
Affiliation:
Laboratoire de Modélisation Mathématique et de Calcul Scientifique, INSA de Lyon / CNRS UMR 5585, 20 avenue Albert Einstein, 69621 Villeurbanne, France
Email:
sassi@laninsa.insa-lyon.fr
DOI:
10.1090/S0025-5718-01-01318-7
PII:
S 0025-5718(01)01318-7
Keywords:
Mixed finite element methods,
unilateral contact problems,
a priori error estimates
Received by editor(s):
November 5, 1999
Received by editor(s) in revised form:
March 14, 2000
Posted:
May 21, 2001
Copyright of article:
Copyright
2001,
American Mathematical Society
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