Keywords: contact, domain decomposition, non-linear, semi-coercive problems, structural analysis, finite element method.

The solution to contact problems between solid bodies poses difficulties to finite element systems because neither the distributions of the contact tractions throughout the surface areas currently in contact nor mutual positions of these areas are known a priori until we have run the problem. These salient features of general contact problems imply that the contact inherently is strongly nonlinear.

One of new methods which can successfully be applied to solution to contact problems is the FETI (Finite Element Tearing and Interconnecting) method, which is based on decomposition of a spatial domain into a set of totally disconnected non-overlapping sub-domains. Its novelty consists in the fact that the Lagrangian multipliers, or forces in this context, were introduced to enforce the compatibility at the interface nodes. They are also called the dual variables in contrast to the primal variables, which are nodal displacements with the displacement based finite element analysis. By eliminating the primal variables the original problem is reduced to a relatively small and well conditioned quadratic programming problem with bound and equality constraints. These ideas were published for the first time in [1].

Algorithms based on the FETI method have proved to be ones of the most successful algorithms for parallel solution of problems governed by elliptic partial differential equations. The idea that every individual sub-domain, into which the body is partitioned, interacts with its neighbours in terms of the Lagrangian multipliers, with physical meaning of forces, can naturally be applied to solution to contact problems [2]. In addition in static cases, this approach renders possible the solution to the semi-coercive problems, i.e. the structures with some floating sub-domains.

The most recently the algorithm was proved to enjoy linear complexity [3].

While the FETI method is directly applicable to the solution to linear elastic and frictionless contact problems with small displacements and rotations, any other non-linearity, in addition to the contact, necessitates introduction of additional outer iteration loop as a consequence of geometrically or materially nonlinear behaviour of structural systems [4]. In our case the non-linearity we take into account, in addition to the contact, is the one caused by large displacements and finite rotations. To this end we use the total Lagrangian formulation.

The algorithms stemming from the FETI method were implemented in our in-house general purpose finite element computational system PMD (Package for Machine Design) [5], with which we carried out all numerical experiments.

The first part of the paper describes principles of the FETI method and outlines basic theory, including exposition, which the semi-coercive problem consists in. The results of numerical experiments are presented in the second part. They show (a) Comparison of the numerical solution with the analytical one for a classic Hertzian problem of contact of two cylindric bodies with parallel axes; (b) Comparison of the numerical solution with the analytical one for the contact of a cylinder in a cylindric hole with parallel axes while their radii are nearly of the same magnitudes (Persson's problem); (c) Computation of a geometrically nonlinear problem with the same geometry as in the case (a) but with loads large enough that displacements cannot be regarded as the small ones.

1

Ch. Farhat, F.-X. Roux, "A method of finite element tearing and interconnecting and its parallel solution algorithm", Int. J. Numer. Methods Engng., 32, 1205-1227, 1991. doi:10.1002/nme.1620320604

2

Z. Dostál, F.A.M. Gomes, and S. A. Santos, "Solution of contact problems by FETI domain decomposition with natural coarse space projection", Comp. Meth. in Appl. Mech. and Eng., 190, 1611-1627, 2000. doi:10.1016/S0045-7825(00)00180-8

3

Z. Dostál, and D. Horák, "On scalable algorithms for numerical solution of a variational inequalities based on FETI and semi-monotonic augmented Lagrangians", submitted to Proc. of DDM15, Berlin, 2003.

4

Dostál, D. Horák, R. Kucera, V. Vondrák, J. Haslinger, J. Dobiáš, S. Pták, "FETI based algorithms for contact problems: scalability, large displacements and 3D Coulomb friction", submitted to Computer Meth. in Appl. Mech. and Engineering. doi:10.1016/j.cma.2004.05.015

5

PMD, "Manuals on" http://www.it.cas.cz/manual/pmd

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