Jan Valdman

Solution of One-Time-Step Problems in Elastoplasticity by a Slant Newton Method

We discuss a solution algorithm for quasi-static elastoplastic problems with hardening. Such problems can be described by a time dependent variational inequality, where the displacement and the plastic strain fields serve as primal variables. After discretization in time, one variational inequality of the second kind is obtained per time step and can be reformulated as a minimization problem with a convex energy functional which depends smoothly on the displacement and nonsmoothly on the plastic strain. There exists an explicit formula for minimizing the energy functional with respect to the plastic strain for a given displacement. By substitution, the energy functional can be written as a functional depending only on the displacement. The theorem of Moreau from convex analysis states that the energy functional is differentiable with an explicitly computable first derivative. The second derivative of the energy functional does not exist, due to the lack of smoothness of the plastic strain across the elastoplastic interface, which separates the continuum in elastically and plastically deformed parts. A Newton-like method exploiting slanting functions of the energy functionals first derivative instead of the nonexistent second derivative is applied. Such a method is called a slant Newton method for short. The local superlinear convergence of the algorithm in the discrete case is shown, and sufficient regularity assumptions are formulated, which would guarantee the local superlinear convergence also in the continuous case.

Implementation of an elastoplastic solver based on the Moreau-Yosida Theorem

We discuss a technique for solving elastoplastic problems with hardening. The one time-step elastoplastic problem can be formulated as a convex minimization problem with a continuous but non-smooth objective. We actually show that its objective structure satisfies conditions of the Moreau-Yosida Theorem known from convex analysis. Therefore, the substitution of the non-smooth plastic-strain p as a function of the total strain @?(u) yields an already smooth functional in the displacement u only. The second derivative of such functional exists in all continuum points apart from interfaces where elastic and plastic zones intersect. The numerical experiment states super-linear convergence of a Newton method or even quadratic convergence as long as the interface is detected sufficiently.

Numerical solution of the two-yield elastoplastic minimization problem

SummaryThis paper concentrates on fast calculation techniques for the two-yield elastoplastic problem, a locally defined, convex but non-smooth minimization problem for unknown plastic-strain increment matrices P1 and P2. So far, the only applied technique was an alternating minimization, whose convergence is known to be geometrical and global. We show that symmetries can be utilized to obtain a more efficient implementation of the alternating minimization. For the first plastic time-step problem, which describes the initial elastoplastic transition, the exact solution for P1 and P2 can even be obtained analytically. In the later time-steps used for the computation of the further development of elastoplastic zones in a continuum, an extrapolation technique as well as a Newton-algorithm are proposed. Finally, we present a realistic example for the first plastic and the second time-steps, where the new techniques decrease the computation time significantly.

Newton-Like Solver for Elastoplastic Problems with Hardening and its Local Super-Linear Convergence

We discuss a solution algorithm for quasi-static elastoplastic problems with hardening. Such problems can be described by a time dependent variational inequality, where the displacement and the plastic strain fields serve as primal variables. After discretization in time, one variational inequality of the second kind is obtained per time step and can be reformulated as each one minimization problem with a convex energy functional which depends smoothly on the displacement and non-smoothly on the plastic strain. There exists an explicit formula how to minimize the energy functional with respect to the plastic strain for a given displacement. By substitution, the energy functional can be written as a functional depending only on the displacement. The theorem of Moreau from convex analysis states that this energy functional is differentiable with an explicitly computable first derivative. The second derivative of the energy functional does not exist, hence the plastic strain minimizer is not differentiable on the elastoplastic interface, which separates the continuum in elastically and plastically deformed parts. A Newton-like method exploiting slanting functions of the energy functional’s first derivative instead of the nonexistent second derivative is applied.

An Efficient Solution Algorithm for Elastoplasticity and its First Implementation Towards Uniform h- and p- Mesh Refinements

Special Research Program SFB F013, ’Numerical and Symbolic ScientificComputing’, Johannes Kepler University Linz,{johanna.kienesberger,jan.valdman}@sfb013.uni-linz.ac.atSummary. The main subject of this paper is the detailed description of an algo-rithm solving elastoplastic deformations. Our concern is a one time-step problem,for which the minimization of a convex but non-smooth functional is required. Wepropose a minimization algorithm based on the reduction of the functional to aquadratic functional in the displacement and the plastic strain increment omittinga certain nonlinear dependency. The algorithm also allows for an easy extension tohigher order finite elements. A numerical example in 2D reports on first results foruniform h- and p- mesh refinements.

Multi-yield Elastoplastic Continuum-Modeling and Computations

The quasi-static evolution of an elastoplastic body with a multi-surface constitutive law of linear kinematic hardening type allows the modeling of curved stress-strain relations. It generalises classical small-strain elastoplasticity from one to various plastic phases. Firstly, we briefly recall a mathematical model represented by an initial-boundary value problem in the form a variational inequality. Then, the main concern of this paper is focused on an efficient numerical implementation of a one time-step problem. Based on the minimisation problem we describe an iterative non-linear algorithm whose linear subsystems are solved by a geometrical multigrid method. Finally, the numerical computations in 2D and 3D are presented.

Fast Solvers and A Posteriori Error Estimates in Elastoplasticity

The paper reports some results on computational plasticity obtained within the Special Research Program “Numerical and Symbolic Scientific Computing” and within the Doctoral Program “Computational Mathematics” both supported by the Austrian Science Fund FWF under the grants SFB F013 and DK W1214, respectively. Adaptivity and fast solvers are the ingredients of efficient numerical methods. The paper presents fast and robust solvers for both 2D and 3D plastic flow theory problems as well as different approaches to the derivations of a posteriori error estimates. In the last part of the paper higher-order finite elements are used within a new plastic-zone concentrated setup according to the regularity of the solution. The theoretical results obtained are well supported by the results of our numerical experiments.