Mesh refinement strategy

We describe the strategy used for local mesh refinement on the basis of the error estimators introduced above. By local averaging the values of the norm  |$\sigma_{h}^{D}$|  of the stress deviator are known in each vertex of the triangulation. We call a cell elastic, if  |$\sigma_{h}^{}$| < $\sigma_{0}^{}$  at each vertex, and plastic, if  |$\sigma_{h}^{}$| = $\sigma_{0}^{}$  at each vertex. The remaining cells represent the transition zone between elastic and plastic behaviour.

The mesh refinement process in each adaptive step is based on error bounds of the form

$$\eta \sum_{K\in\mathbb T_h}^{} \eta_{K}^{} \eta \Big( \sum_{K\in\mathbb T_h}^{} \eta^{2}_{K} \Big)^{1/2}_{}$$

Our goal is to minimize the degree of equidistribution $\cal {E}$ of the error indicators  $\eta_{K}^{}$ , i.e., the ratio of  $\max_{K}^{}${$\eta_{K}^{}$}  and  $\min_{K}^{}${$\eta_{K}^{}$} , and, at the same time, to keep the total number of cells below a fixed (prescribed) number  N_max . To this end, we use the following strategy:

Mesh adaptiation Strategy: The elements are ordered according to the size  $\eta_{K}^{}$. A fixed portion (say 20%) of cells K with smallest contributions $\eta_{K}^{}$ to the indicator value $\eta$ is marked to be deleted. Then, we refine those cells with largest $\eta_{K}^{}$, so that the desired number of cells  N_max  is almost reached.