Keywords: Total-FETI, elasto-plasticity, kinematic hardening, Trilinos.

Elasto-plastic processes describe the behaviour of solid continuum beyond reversible elastic deformations. They are typically described by hysteresis models with time memory [1]. The rigorous mathematical analysis of elasto-plastic problems and the numerical methods for their solution started to appear in the late 1970s and in the early 1980s. Since then, many mathematical contributions to computational plasticity have been written [2,3].

In this paper, we focus on the efficient parallel solution of elasto-plastic problems based on our implementation of the TFETI domain decomposition method in the Trilinos framework [4]. More specifically, we consider an associated elasto-plasticity with the von Mises plastic criterion and the kinematic hardening law [2,3,5]. The corresponding elasto-plastic constitutive model is discretized using the implicit Euler method in time and consequently a nonlinear stress-strain operator is implemented using the return mapping concept. This approach leads to the solution of a nonlinear variational equation with respect to the primal unknown displacement in each time step.

By a finite element space discretization of the one time step problem, we obtain a system of nonlinear equations. The corresponding nonlinear operator is nondifferentiable but strongly semismooth. Therefore, the semismooth Newton method for solving the system is suitable because the strong semismoothness together with other properties ensures local quadratic convergence.

In each Newton iteration, it is necessary to solve the respective linearized problem. The linear solver considered in this paper is based on a FETI type domain decomposition method enabling its efficient parallel implementation. The standard FETI method (FETI-1) was originally introduced by Farhat and Roux [6]. Here we use the Total-FETI (TFETI) variant, where also the Dirichlet boundary conditions are enforced by Lagrange multipliers [7]. Hence all subdomain stiffness matrices are singular with a-priori known kernels which is a great advantage in the numerical solution. With a known kernel basis we can regularize effectively the stiffness matrix without an extra fill in and use any standard sparse Cholesky type decomposition method for nonsingular matrices [8].

1

M. Brokate, J. Sprekels, "Hysteresis and Phase Transitions", Springer-Verlag New York, 1996.

2

W. Han, B.D. Reddy, "Plasticity: mathematical theory and numerical analysis", Springer, 1999.

3

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

4

Trilinos web page. trilinos.sandia.gov

5

E.A. de Souza Neto, D. Peric, D.R.J. Owen, "Computational methods for plasticity: theory and application", Wiley, 2008.

6

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.

7

Z. Dostál, 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 (12), 1155-1162, 2006.

8

T. Kozubek, V. Vondrák, M. Menšík, D. Horák, Z. Dostál, V. Hapla, P. Kabelíková, M. Cermák, "Total FETI domain decomposition method and its massively parallel implementation", Advances in Engineering Software, 2012, accepted.

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