Overview

In this last chapter we present some results on the numerical analysis for the dual formulation of the elastoplasticity problem. For various numerical approximation schemes, we will derive error estimates under sufficient regularity assumptions on the solution and prove the convergence under the basic solution regularity condition. In Section 13.1 we study a family of generalized midpoint schemes for the stress problem. For the dual problem, we analyze several time-discrete schemes in Section 13.2 and fully discrete schemes in Section 13.3.

We then turn our attention to the implementation of numerical methods for solving the dual problem. For simplicity in notation, the discussion will be given in the context of the solution of temporal semidiscrete schemes. The extension of the discussion to fully discrete schemes is straightforward; one needs only to change infinite-dimensional spaces or their subsets to corresponding finite element spaces or their subsets in the argument. At each time level, one needs to solve a variational inequality system for the current state of the generalized stress and the displacement (or velocity). A common practice in engineering is to use an iteration procedure to update the generalized stress and the displacement separately, thus breaking a large-scale problem into two subproblems. Such an iteration procedure is termed a predictor-corrector method. Analysis of some predictor-corrector methods are given in Section 13.4. The main work required to carry out one step of a corrector-predictor method is the solution of a constrained variational inequality for updating the generalized stress. The problem can...