On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization and Mixed Reformulation

Issue		Math. Model. Nat. Phenom. Volume 4, Number 1, 2009 Modelling and numerical methods in contact mechanics


Page(s)		21 - 43
DOI		10.1051/mmnp/20094102
Published online		27 January 2009

Math. Model. Nat. Phenom. Vol. 4, No. 1, 2009, pp. 21-43
DOI: 10.1051/mmnp/20094102

On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization and Mixed Reformulation

F. Ben Belgacem¹, C. Bernardi², A. Blouza³ and M. Vohralík²

¹ 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
² Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
³ 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

bernardi@ann.jussieu.fr

Published online: 27 January 2009

Abstract
The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first perform the analysis of this system. We then propose a discretization, where the displacements are approximated by standard finite elements and the action by a local postprocessing. Such a discretization admits an equivalent mixed reformulation. We prove the well-posedness of the discrete problem and establish optimal a priori error estimates.

Mathematics Subject Classification. 65N30, 73K10, 73T05

Key words: unilateral contact -- elastic membranes -- variational inequalities

© EDP Sciences 2009

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