Offene Fragen und Ausblick

We have presented a systematic approach to reliable error control and mesh design for finite element models in linear-elastic perfect plasticity. The mesh refinement or coarsening is based on weighted a posteriori error estimates which are derived by solving linearized dual problems. In this way most economical meshes and error bounds can be generated at acceptable cost. In fact, the evaluation of the a posteriori error bounds only amounts to less than 30% of the total cost as it requires only to solve a few linear dual problems within a Newton iteration for the given nonlinear problem. The efficiency and reliability of this technique has been demonstrated at a (plane strain) model problem using a primal-mixed formulation of Hencky and Prandtl-Reuss plasticity. More test calculations including a benchmark problem ,,disk with a hole'' are reported in [34]. Our approach is not dimension dependent and can be applied also to three-dimensional problems. Here, the gain in efficiency due to more economical meshes may be even greater than in the two-dimensional case. We plan to perform such a test in the near future for the three-dimensional analogue of the ,,disk with a hole'' benchmark.

In this study we have neglected the effect of the error due to the incremental loading (time-discretization error) as it seems to be of minor importants compared to that of the spatial discretization. However, being forced, for safety, to work with very small load increments may cause a significant loss of efficiency for the whole solution process. Hence, we will investigate the extension of our adaptive method to also include control of the error in the loading process. This may become very costly as then the dual problems have to be solved in the space-time domain requiring to store the computed solution over the whole time interval.

A further extension of the present study will include the effect of hardening by considering an additional interior variable which describes the change in the flow rule during the loading process. We expect our method for a posteriori error estimation to work even better in this case, as due to the hardening effect the dual problems become more regular and easier to solve.