# Improvement of the Zienkiewicz-Zhu Error Recovery Technique Using a Patch Configuration
## 8 Plastic Sheet under Stretching Process Problem  
To demonstrate the quality of the improved ZZ stress recovery under large deformations, the problem of the axisymmetric sheet stretching process was analyzed using adaptive finite element analysis. The finite element formulation, error recovery, and adaptive procedures are followed, as explained in [26]. The sheet material is considered to be rigid plastic. The punch and die used in the process are assumed to be rigid. Friction is present at the punch-sheet and die-sheet interfaces. The downward displacement of the punch is modeled in incremental steps. The schematic diagram of the axisymmetric sheet forming process is shown in Figure 5. Due to symmetry, only one half of the sheet is modeled. The input parameters for the process are as follows:  
Radius of sheet, \(\text{R}_{\text{b}}=66.0\) mm and radius of the die corner, \(\text{R}_{\text{d}}=6.35\) mm;  
Velocity of punch V = 1 mm/sec and sheet thickness (h) = 2 mm;  
Blank to punch radius ratio (\(\text{R}_{\text{b}}/\text{R}_{\text{p}}\)) = 1.33 and the stress-strain relation is given as:  
\[\overline{\sigma}=589\ [0.0001+\overline{\epsilon}\ ]^{0.216}, \tag{27}\]  
where \(\overline{\sigma}\) and \(\overline{\epsilon}\) are effective stress and effective strain, respectively. The sheet is discretized using six nodded triangular elements. The number of elements of uniform mesh that have four layers of elements are 737 and the degrees of freedom  
Figure 5: Schematic diagram of sheet stretching process. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Mesh**} & \multicolumn{2}{c}{**ZZ (Standard Patch)**} & \multicolumn{2}{c}{**ZZ (Modified Patch)**} & \multicolumn{2}{c}{**LS (Modified Patch)**} \\ \hline \multirow{2}{*}{**Elem.**} & \multirow{2}{*}{**Dof**} & **FEM Error** (\(\times\)**10\({}^{-3}\))** & **Error (Stress)** & **Effectivity** & **Error (\(\times\)10\({}^{-3}\))** & **Effectivity** & **Error (Displacement)** (\(\times\)10\({}^{-3}\))** & **Effectivity** \\ \hline
68 & 318 & 2.391 & 1.845 & 0.83612 & 3.389 & 1.41650 & 8.194 & 3.29350 \\
131 & 588 & 1.698 & 1.363 & 0.67366 & 1.630 & 0.90573 & 4.559 & 2.56915 \\
355 & 1516 & 1.495 & 1.247 & 0.56854 & 1.332 & 0.73770 & 3.133 & 1.96228 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Solution error and global effectivity for the plate with an opening problem using stress/displacement recovery and patch configuration (quadratic triangular element). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Mesh**} & \multicolumn{2}{c}{**ZZ (Standard Patch)**} & \multicolumn{2}{c}{**ZZ (Modified Patch)**} & \multicolumn{2}{c}{**LS (Modified Patch)**} \\ \hline \multirow{2}{*}{**Elem.**} & \multirow{2}{*}{**Dof**} & **FEM Error** (\(\times\)**10\({}^{-3}\))** & **Error (Stress)** & **Effectivity** & **Error (Stress)** & **Effectivity** & **Error (Displacement)** (\(\times\)10\({}^{-3}\))** & **Effectivity** \\  & & & (\(\times\)**10\({}^{-3}\))** & **Effectivity** & **Error (\(\times\)10\({}^{-3}\))** & **Effectivity** & **Error (Displacement)** (\(\times\)10\({}^{-3}\))** & **Effectivity** \\ \hline
188 & 430 & 5.206 & 2.853 & 0.93465 & 3.521 & 0.99604 & 4.123 & 1.05608 \\
358 & 786 & 4.070 & 1.878 & 0.94003 & 2.049 & 0.96181 & 2.615 & 1.06141 \\
1217 & 2558 & 2.367 & 1.536 & 0.87953 & 1.349 & 0.83577 & 1.628 & 0.88195 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Solution error and global effectivity for the plate with an opening problem using stress/displacement recovery and patch configuration (linear quadrilateral element). are 3228 (Figure 6a). The target accuracy limit was taken as 14% of the global error. An adaptively modified mesh is generated during the analysis, based on the computed error by uniform distribution of the square of error in the elements of the domain, to attain the predefined solution accuracy.