Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T22:57:00.622Z Has data issue: false hasContentIssue false
  • English
  • Français

A finite element discretization of the contactbetween two membranes

Published online by Cambridge University Press:  16 October 2008

Faker Ben Belgacem ,
Christine Bernardi ,
Adel Blouza  and
Martin Vohralík
Faker Ben Belgacem
Affiliation:
L.M.A.C. (E.A. 2222), Département de Génie Informatique, Université de Technologie de Compiègne, Centre de Recherches de Royallieu, B.P. 20529, 60205 Compiègne Cedex, France. faker.ben-belgacem@utc.fr
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr; vohralik@ann.jussieu.fr
Adel Blouza
Affiliation:
Laboratoire de Mathématiques Raphaël Salem (U.M.R. 6085 C.N.R.S.), Université de Rouen, avenue de l'Université, B.P. 12, 76801 Saint-Étienne-du-Rouvray, France. Adel.Blouza@univ-rouen.fr
Martin Vohralík
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr; vohralik@ann.jussieu.fr
Get access

Abstract

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

Keywords

Unilateral contactvariational inequalitiesfinite elementsa priori and a posteriori analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 1 , January 2009 , pp. 33 - 52
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ainsworth, M., Oden, J.T. and Lee, C.Y., Local a posteriori error estimators for variational inequalities. Numer. Methods Partial Differential Equations 9 (1993) 23–33. CrossRef
Ali Mehmeti, F. and Nicaise, S., Nonlinear interaction problems. Nonlinear Anal. Theory Methods Appl. 20 (1993) 27–61. CrossRef
C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Collection Mathématiques & Applications 45. Springer-Verlag (2004).
Brezis, H. and Stampacchia, G., Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Math. France 96 (1968) 153–180. CrossRef
F. Brezzi, W.W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, II. Mixed methods. Numer. Math. 31 (1978-1979) 1–16.
Chen, Z. and Nochetto, R.H., Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527–548. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, New York, Oxford (1978).
P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.-L. Lions Eds., North-Holland, Amsterdam (1991) 17–351.
P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 R2 (1975) 77–84.
I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod & Gauthier-Villars (1974).
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985).
J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Vol. IV, P.G. Ciarlet and J.-L. Lions Eds., North-Holland, Amsterdam (1996) 313–485.
Hild, P. and Nicaise, S., Residual a posteriori error estimators for contact problems in elasticity. ESAIM: M2AN 41 (2007) 897–923. CrossRef
Lions, J.-L. and Stampacchia, G., Variational inequalities. Comm. Pure Appl. Math. 20 (1967) 493–519. CrossRef
Nochetto, R.H., Siebert, K.G. and Veeser, A., Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95 (2003) 163–195. CrossRef
Raugel, G., Résolution numérique par une méthode d'éléments finis du problème de Dirichlet pour le laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. A-B 286 (1978) A791–A794.
Slimane, L., Bendali, A. and Laborde, P., Mixed formulations for a class of variational inequalities. ESAIM: M2AN 38 (2004) 177–201. CrossRef
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996).
Wohlmuth, B.I., An a posteriori error estimator for two body contact problems on non-matching meshes. J. Sci. Computing 33 (2007) 25–45. CrossRef