Keywords: contact problem, domain decomposition, numerical scalability, geometric non-linearity, material non-linearity, finite element method.

Contact modelling is still a challenging problem of non-linear computational mechanics. The complexity of such problems is related to the a priori unknown contact interface and contact tractions. Their evaluation have to be part of the solution. In addition, the solution across the contact interface is non-smooth.

In 1991 Farhat and Roux came up with a novel domain decomposition method called the finite element tearing and interconnecting (FETI) method [1]. This method belongs to the class of non-overlapping totally disconnected spatial decompositions. Its key concept stems from the idea that the spatial sub-domains, into which the domain is partitioned, are 'glued' by Lagrange multipliers, or forces in this context. After eliminating the primal variables, which are displacements in the displacement based analysis, the original problem is reduced to a small, relatively well conditioned, typically equality constrained quadratic programming problem that is solved iteratively. The CPU time that is necessary for both the elimination and iterations can be reduced nearly proportionally to the number of the processors, so that the algorithm exhibits parallel scalability. This method has proved to be one of the most successful algorithms for parallel solution of problems governed by elliptic partial differential equations. Observing that the equality constraints may be used to define so called 'natural coarse grid', Farhat, Mandel and Roux [2] modified the basic FETI algorithm so that they were able to prove its numerical scalability, i.e. asymptotically linear complexity.

If the FETI method is applied to the contact problems, the same methodology can be used to prescribe conditions of non-penetration between bodies. We shall obtain a new minimization problem with additional non-negativity constraints which replace more complex general non-penetration conditions [3]. It turned out that the scalability of the FETI methods may be preserved even for the solution of contact problems [3,5].

In this paper we are concerned with application of one new variant of the FETI domain decomposition method, called TFETI (Total FETI) method, to the solution of contact problems. Both compatibility between adjacent sub-domains and Dirichlet boundary conditions are enforced by Lagrange multipliers acting along the boundaries or mutual interfaces. Such an approach is very advantageous especially from the computational point of view, because the stiffness matrices of all sub-domains exhibit the same defect which is, in addition, known beforehand. See [4] for discussion of this topic.

We describe theoretical foundation of the TFETI algorithm and its implementation into the inner loop of the code which treats the material and geometric non-linear effects in the outer loop. To solve a contact problem by the FETI or TFETI method, we use modified proportioning with the reduced gradient projection (MPRGP) algorithm. More details can be found in [6].

Numerical experiments were carried out with our in-house general purpose finite element package PMD (package for machine design) [7] and they include solutions of contact problems both with and without geometric and material non-linear effects. To demonstrate the ability of the method to solve contact problems, we computed the problem of contact of two cylindric bodies with parallel axes.

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

Ch. Farhat, J. Mandel, F.-X. Roux, "Optimal convergence properties of the FETI domain decomposition method", Comp. Meth. Appl. Mech. Eng., 115, 365-385, 1994. doi:10.1016/0045-7825(94)90068-X

3

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", Comp. Meth. Appl. Mech. Eng., 194, 395-409, 2005. doi:10.1016/j.cma.2004.05.015

4

Ch. Farhat, M. Géradin, "On the general solution by a direct method of a large-scale singular system of linear equations: application to the analysis of floating structures", Int. J. Numer. Methods Engng., 41, 675-696, 1998. doi:10.1002/(SICI)1097-0207(19980228)41:4<675::AID-NME305>3.3.CO;2-#

5

Z. Dostál and D. Horák, "Scalable FETI with optimal dual penalty for a variational inequality", Numerical Linear Algebra with Applications, 11, 455 - 472, 2004. doi:10.1002/nla.355

6

Z. Dostál, J. Schöberl, "Minimizing quadratic functions over non-negative cone with the rate of convergence and finite termination", Comput. Optim. Appl., 30, 23-43, 2005. doi:10.1007/s10589-005-4557-7

7

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

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