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This paper illustrates how to implement effectively solvers for elasto-plastic problems. We consider the time step problems formulated by nonlinear variational equations in terms of displacements. To treat nonlinearity and nonsmoothnes we use semismooth Newton method. In each Newton iteration we have to solve linear system of algebraic equations and for the numerical solution of the linear systems we use TFETI algorithm. In our benchmark we compute von Misses plasticity with isotropic hardening and use return mapping concept.

This paper deals with a fast method for solving large-scale algebraic saddle-point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside the original domain. This approach has a significantly higher convergence rate; however, the algebraic systems resulting from finite element discretizations are typically non-symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle-point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved using a projected Krylov subspace method for non-symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non-projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy.