Keywords: domain decomposition, FETI-DP, quadratic programming, scalable algorithms, contact problems.

Domain decomposition methods are powerful methods for solving systems of algebraic equations arising from finite element discretization. The speed up due to parallel implementation is nearly proportional to the number of processors. This feature, called parallel scalability, can be eliminated by decreasing of discretization parameter size resulting in an increase of number of iterations. Thus we face a new challenge to design and to develop algorithms that are independent of this discretization parameter. This feature is called numerical scalability. The FETI-1 method is based on the decomposition into non-overlapping subdomains that are "glued" by Lagrange multipliers. The efficiency was further improved by introducing special projectors and preconditioners. By projecting the Lagrange multipliers in each iteration onto an auxiliary space to enforce continuity of the primal solutions at the crosspoints, a faster converging method for plate and shell problems was obtained - FETI-2. A similar effect was achieved by a variant called FETI-DP. The continuity of the primal solution at crosspoints is implemented directly into the formulation of the primal problem.

FETI methods are even more successful for the solution of variational inequalities. The reason is that duality transforms the general inequality into the nonnegativity constraints so that efficient algorithms that exploit cheap projections and other tools may be exploited. For nonlinear problems the scalability of the FETI-DPC based on active set strategies with additional planning steps was established by Farhat et al. [1] only experimentally. Dostál et al. [2,3,4] proved this scalability theoretically. Our research concerns the development of the scalable FETI-based methods for contact problems. Numerical scalability for FETI-DP algorithm for coercive problems was proven theoretically and experimentally in [2]. Later, the result was extended to include mortar disctretization [3] and for semi-coercive problems [4].

Farhat et al. [1] observed that the corner nodes on the contact interface cause difficulties and recommended avoiding them. The generalization of method for corners on a contact interface is not trivial. This paper describes a modification of the FETI-DP for the corner nodes used on the contact interface for both coercive and semi-coercive variational inequalities through the additional condition that preserves the non-penetration in the Lagrange multipliers. This richer corner mesh results in better convergence of the method because of better error propagation across the non-linear interface.

1

P. Avery, G. Rebel, M. Lesoinne, C. Farhat, "A numerically scalable dual-primal substructuring method for the solution of contact problems - part I: the frictionless case", Comput. Methods Appl. Mech. Engrg., 193, 2403-2426, 2004. doi:10.1016/j.cma.2004.01.016

2

Z. Dostál, D. Horák, D. Stefanica, "A scalable FETI-DP algorithm for a coercive variational inequalities", IMACS J. Appl. Numer. Math., vol. 54, 3-4, 378-390, 2005. doi:10.1016/j.apnum.2004.09.009

3

Z. Dostál, D. Horák, D. Stefanica, "A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface", In SIAM Jour. of Scient. Comp., 2005.

4

Z. Dostál, D. Horák, D. Stefanica, "A Scalable FETI-DP Algorithm for Semi-coercive Variational Inequality", Computer Methods in Applied Mechanics and Engineering, Vol. 196, 8, 1369-1379, 2007. doi:10.1016/j.cma.2006.03.025

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