Keywords: contact, friction, equilibrium finite elements, dual formulation, condensation, red-black Gauss Seidel.

This paper deals with dual methods for solving unilateral contact problems with friction. Variational formulations and the discretization using the equilibrium finite elements method are presented. The contact is governed by Signorini and Coulomb contact conditions. The problem is condensed on the contact boundary and solved using a Gauss Seidel red-black relaxation algorithm. Numerical results obtained using this dual method are in good agreement with results obtained using the classical one and the stress field is more accurate.

The mechanical problem is related to an elastic body, subjected to prescribed forces and displacements, and in contact with friction with a rigid obstacle. Two formulations in terms of stress are given, one in the volume, the other on the contact boundary. These formulations are respectively named the dual formulation and the condensed dual formulation. The condensed dual formulation can be both obtained by dualizing the classical primal formulation (written in terms of the displacement) or by condensating the dual formulation. This second manner seems to be the more convenient and is thus adopted.

The discretization of the dual problem is reduced to that of complementary energy. The problem is then written as one without any contact or friction. Since the stress field has to satisfy local equilibrium, the problem is discretized using equilibrium finite elements, which have been introduced by de Veubeke in [1] for the case without contact. The problem is discretized using the Bogner-Fox-Schmidt element, which was originally developed for plate bending problems [2]. On this element, a regular Airy stress function is interpolated. The stress field is then obtained by differentiating the Airy stress function and is thus in equilibrium. The elementary complementary energy functional is written using this element, we show how the prescribed and contact forces are linked to the degrees of freedom. Some convergence results for the finite element method are given in a simplified framework [3].

The regularity of the Airy function and the system assembly allows the continuity of the stress vector between two neighbouring elements to be satisfied. The global system is thus obtained. The prescribed and contact forces are introduced into the system using Lagrangian multipliers. This system is then condensed on the contact boundary and solved using a Gauss Seidel red-black algorithm. This algorithm enables the solution of a succession of local problems and to apply Signorini and Coulomb conditions.

The dual method has been implemented in the computer code LMGC90 [4] and applied to a classical benchmark: a compressed elastic block in contact with friction on a rigid obstacle [5]. This benchmark has been treated, for various values of forces and friction coefficient, using the dual and primal methods. Results from both methods have been compared, on the contact boundary and in the volume, and show very good agreement. Results show that at a similar cost for both approaches, the stress field values obtained using the primal approach are more accurate than using the primal approach. Finally, it has been shown that the friction coefficient has low influence on the algorithm convergence.

Future works will be related to the development of other elements and to the use of the primal and dual methods for mesh refinement and limit analysis.

1

B. Fraeijs de Veubeke, "Displacement and equilibrium Models in the Finite Element Method", international Journal for Numerical Methods in Engineering, 52, 287-342, 2001. doi:10.1002/nme.339

2

P.G. Ciarlet, "The finite element method for elliptic problems", North Holland publishing company, 1979.

3

A. Capatina, F. Lebon, editor M. Mihailescu-Suliciu, "Remarks on the equilibrium finite element method for frictional contact problems", New Trends in Continuum Mechanics, Theta, 25-33, 2005.

4

http://www.lmgc.univ-montp2.fr/~dubois/LMGC90

5

P. Bisegna, F. Lebon, F. Maceri, "Relaxation procedures for solving Signorini-Coulomb contact problems", Advances in Engineering Software, 35, 595-600, 2004. doi:10.1016/j.advengsoft.2004.03.018

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