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

Our research concerns the preconditioning of FETI-based methods for contact problems. The standard FETI-DP is based on the decomposition into non-overlapping subdomains, where the continuity of the primal solution at crosspoints is implemented directly into the formulation of the primal problem so that one degree of freedom is considered at each crosspoint and the continuity of the solution on auxiliary interfaces is enforced using Lagrange multipliers. The duality transforms the general inequality into the nonnegativity constraints. After eliminating the corners, the problem reduces to a small, relatively well conditioned strictly convex quadratic programming (QP) problem with a simple bound for Lagrange multipliers that is solved iteratively by efficient algorithms that exploit cheap projections and other tools. For semi-coercive problems the efficiency of the FETI-DP can be further improved by introducing special projectors onto an auxiliary space related to rigid body modes of floating bodies and preconditioners - lumped and Dirichlet's. The preconditioners can be applied only to the linear part and their efficiency is very poor. Once the Lagrange multipliers are known, we solve the linear problem for corners.

Corner nodes on contact interface cause difficulties and it is recommended to avoid them. These difficulties can be overcome through the additional condition that preserves the nonpenetration using Lagrange multipliers, and moreover in this way it is possible to improve rate of convergence. This richer corner mesh results in better convergence of the method because of better error propagation across the nonlinear interface and in better preconditioning of nonlinear steps using standard FETI-DP preconditioners. We showed experimentally that for unpreconditioned and preconditioned FETI-DP using no corners on the contact zone the number of conjugate gradient (CG) iterations increases much more rapidly with an increasing number of subdomains along the contact interface in comparison to the case where we use corners on the contact zone, when the numbers of CG iterations vary very moderately. The results demonstrate that for a given decomposition the use of corners always significantly reduces number of CG iterations for both unpreconditioned and preconditioned systems and this effect is magnified with an increasing number of subdomains along the contact interface, i.e. with increasing number of corners on the the contact zone.

A significant modification, creating a new type of FETI-DP method, is based on the definition of all nodes on the contact zone as the corners, i.e. constraint matrix with inequality conditions considers only corner nodes. This approach enables us to split the problem into a very small nonlinear one with Lagrange multipliers for inequalities as unknowns and a linear one with the Lagrange multipliers for equalities. We eliminate the Lagrange multipliers for equalities first and the problem reduces to very small QP problem with inequality constraint for corner unknowns and after applying the duality it reduces to well conditioned strictly convex QP problem, with a simple bound for Lagrange multipliers for inequalities, that is solved iteratively by efficient algorithms similar to the standard version. Once the Lagrange multipliers for inequalities are known, we reconstruct corner unknowns, then we solve the linear problem to find the solution for the Lagrange multipliers for equalities. Moreover we can significantly reduce the number of CG interations by applying the standard FETI-DP preconditioners. In this paper we give the results of numerical experiments that confirm scalability of the algorithm. It is interesting, that total number of iterations for the solution of bound contrained problem with Lagrange multipliers for standard FETI-DP is significantly larger than the sum of iterations for the solution of bound constrained nonlinear problem with Lagrange multipliers for inequalities and the linear problem with Lagrange multipliers for equalities, both subproblems have dimensions in sum equal to the dimension of original dual problem.

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