Keywords: contact problem, domain decomposition, Schur complement, interface preconditioner, seminorm.

The purpose of the research, described in this paper, is to study the quasistatic two-body contact problem for small strains with friction. The mechanical interaction between the bodies is modelled, subject to the assumption of small displacement, the bilateral or unilateral contact condition, and Coulomb's friction law relating the contact force and the displacement [1].

The main difficulties of contact problems are: the non-penetration of the bodies, the friction effect, and that the contact surfaces are unknown in the problem. An algorithm is introduced to solve the resulting finite element system using a non-overlapping domain decomposition method, which consists of a suitable iteration based on the solution of the elasticity equations for each body separately and the solution of a smaller problem for the contact surface. The central aspect of this paper is the adaptation of a preconditioner construction developed in [2,3] for a non-overlapping Dirichlet type domain decomposition method to the contact problem. The circulant matrix representation of the H^1/2 seminorm has been proved to be spectrally equivalent to the Schur Complement in [3]. Using this equivalence, the interface problem is transformed to an equivalent problem which is solved by a two-stage iterative technique consisting of solving consecutively a problem with prescribed tangential force and a problem with prescribed normal force [4]. Each problem is solved with adequate mathematical programming methods.

The paper is organized in the following way: in Section 2, the contact problem with friction is discussed. In Section 3, the variational formulation of the problem is presented. The finite element method is used to construct approximation spaces and an algorithm based on domain decomposition is presented in Section 4. In Section 5, a preconditioning technique for the resulting interface problem based on the circulant matrix representations of the H^1/2 seminorm is suggested. Some numerical examples are presented in the last section.

1

N. Kikuchi, J.T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods", SIAM, Philadelphia, 1988.

2

B. Kiss, G. Molnárka, "A preconditioned Domain Decomposition Algorithm for the Solution of the Elliptic Neumann Problem", In W. Hackbush, (Editor), "Parallel Algorithm for PDE", Proceeding of 6th GAMM Seminar, Kiel, 119-129, 1990.

3

B. Kiss, A. Krebsz, "On the Schur Complement Preconditioners", Computers and Structures, 73, 537-544, 1999. doi:10.4203/ccp.38.8.1

4

P.D. Panagiotopoulos, "A nonlinear programming approach to the unilateral contact - and friction- boundary value problem in the theory of elasticity", Ing. Archiv, 44, 1975. doi:10.1007/BF00534623

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