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

In this paper we discuss the numerical resolution of frictionless contact problems in linear elasticity. The mechanical interation between the bodies is modelled, with the assumption of small displacements, using the bilateral or unilateral contact condition [1]. We consider a mixed formulation in which the unknowns are the displacement field and the normal stress in the contact zone. An algorithm is introduced to solve the resulting finite element system by a non-overlapping domain decomposition method. The global problem is transformed to a independant local problems posed in each body and a problem posed on the contact surface (the interface problem). The central aspect of this work is to construct preconditioners for the interface problem and the adaptation of a preconditioner construction developed in [2] for the non-overlapping decomposition domain method applied to the contact problem. The circulant matrix representations of the H^1/2 seminorm has been proved to be spectrally equivalent to the Schur Complement in [3] and this matrix can be used as the preconditioner for the Schur Complement. The advantage of this preconditioner construction is, that its preconditioning property is optimal and this technique allows us to reduce the storage and the matrix-vector multiplication costs. Using this equivalence, the interface problem is transformed to an equivalent problem which is solved using mathematical programming methods [4,5]. The developed algorithm is validated for two- and three-dimensional problems.

The paper is organized in the following way: In Section 2, the frictionless contact problem 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 ed., 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:10.1016/S0045-7949(98)00258-2

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.

5

R. Kucera, J. Haslinger, Z. Dostál, "The FETI based domain decomposition method for solving 3D-multibody contact problems with Coulomb friction", In: R. Kornhuber, et al. ed., Domain Decomposition Methods in Science and Engineering, Springer, Berlin Heidelberg, 2005.

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