Keywords: domain decomposition, natural coarse grid, elasto-plasticity, scalability, Matsol library, von Mises.

are often solved using an incremental finite element method [1]. For the time-discretisation we can use the explicit or implicit Euler methods or the return mapping concept. Each time-step problem may be formulated in different ways by variational equalities or inequalities described in terms of stress, plastic strain, hardening parameter and displacements. In this paper, we consider the time-step problems formulated by nonlinear variational equations in terms of displacements. To treat nonlinearity and non-smoothness we use the semi-smooth Newton method introduced in [2] and used in [3] for elasto-plastic problems.

In each Newton iteration we have to solve an auxiliary (possibly of large size) linear system of algebraic equations. The key idea of our approach is to use for the numerical solution of the linear systems arising in each Newton step the FETI method introduced by Farhat and Roux [4] for parallel solution of linear problems. Using this approach, a body is partitioned into non-overlapping subdomains, an elliptic problem with Neumann boundary conditions is defined for each subdomain, and intersubdomain field continuity is enforced using Lagrange multipliers. The Lagrange multipliers are evaluated by solving a relatively well conditioned dual problem of small size that may be efficiently solved by a suitable variant of the conjugate gradient algorithm. The first practical implementations exploited only the favorable distribution of the spectrum of the matrix of the smaller problem, known also as the dual Schur complement matrix, but such algorithm was efficient only with a small number of subdomains. Later, Farhat, Mandel, and Roux introduced a "natural coarse problem" whose solution was implemented by auxiliary projectors so that the resulting algorithm became in a sense optimal [5]. In our approach, we use the Total-FETI variant of the FETI domain decomposition method, where even the Dirichlet boundary conditions are enforced using the Lagrange multipliers. Hence all subdomain stiffness matrices are singular with a-priori known kernels which is a great advantage in the numerical solution.

1

R. Blaheta, "Numerical methods in elasto-plasticity", Documenta Geonica 1998, Peres Publishers, Prague, 1999.

2

P.G. Gruber, J. Valdman, "Solution of One-Time Step Problems in Elastoplasticity by a Slant Newton Method", SIAM J. Sci. Comp., 31, 1558-1580, 2009. doi:10.1137/070690079

3

L. Qi, J. Sun, "A nonsmooth version of Newtons method", Mathematical Programming, 58, 353-367, 1993. doi:10.1007/BF01581275

4

C. Farhat, F-X. Roux, "An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems", SIAM J. Sci. Stat. Comput., 13, 379-396, 1992. doi:10.1137/0913020

5

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

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