Keywords: dynamics, domain decomposition, geometric non-linearity, material non-linearity, contact non-linearity, finite element method, parallel processing.

The paper is concerned with application of a novel variant of the FETI (finite element tearing and interconnecting) domain decomposition method, called the Total FETI (TFETI) [1], for the solution of dynamic non-linear problems of deformable bodies. As the non-linear effects we consider geometric and material ones and also contact phenomena. The basic theory and relationships that the TFETI method stems from are briefly reviewed and the time stepping scheme, solution algorithm and results of numerical experiments are shown.

The TFETI method belongs to the class of non-overlapping totally disconnected spatial decompositions. Its key concept is based on the idea that the compatibility between the spatial sub-domains, into which the original domain is partitioned, is ensured by Lagrange multipliers with the physical meaning of forces in this context. The primal variables, which are displacements in the displacement based finite element analysis, are eliminated so that we solve the problem for the dual variables or Lagrange multipliers. The difference between the FETI and TFETI is that the latter method, unlike the former one, also enforces the Dirichlet boundary conditions in terms of the Lagrange multipliers, which has a significant bearing on some segments of computations because of lesser sensitivity to round-off errors.

The TFETI technique converts the original problem to a quadratic programming one and if applied to the contact problems, this approach transforms the general non-penetration constraints to the simple bound constraints [2]. After eliminating the primal variables, the original problem is reduced to a small and relatively well conditioned one which is solved iteratively.

For application of the method to the high performance computers, it is essential that TFETI exhibits both parallel and numerical scalabilities [2].

In this paper we combine our optimal algorithms for the solution of bound and equality constrained problems with geometric and material non-linearities.

Our computational results show that the good convergence properties of the time independent TFETI are preserved in the time dependent case and that the method exhibits parallel scalability.

Numerical experiments were carried out with our in-house general purpose finite element package PMD [3]. They included solutions to various dynamic problems with non-linear effects computed in parallel. Figures documenting the quality of solution and parallel scalability are shown.

1

Z. Dostál Z, D. Horák, R. Kucera, "Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE", Communications in Numerical Methods in Engineering, 22, 1155-1162, 2006. doi:10.1002/cnm.881

2

Z. Dostál, "Inexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for convex bound and equality constrained quadratic programming", SIAM J. Numerical Analysis, 43, 96-115, 2005. doi:10.1137/S0036142903436393

3

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

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