# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 5 Effectivity of the Method in Model Problems
### Numerical Examples
#### 5.2.4 An Octant of a Membrane-Dominated Shell  
Data  
A symmetric octant of the cylindrical shell and its initial discretization, based on \(p=2\), \(q=2\), and \(h=l/2\), are presented in Figure 27. A very coarse initial mesh can be seen in this figure. The thinness ratio equals \(l/t=3.1415\times 10^{-2}/0.03\times 10^{-2}\). The error data controlling the _hp_-adaptation are the admissible relative error value on the final mesh \(\gamma_{T}=0.01\) and the ratio of the admissible error values on the intermediate and final meshes \(\gamma_{I}/\gamma_{T}=2\). Results  
As expected, the automatic tools for detection and/or assessment of the locking have not indicated the appearance of the phenomenon. Because of this, the automatic mode of adaptation corresponds to standard _hp_-approach. The final, _hp_-adapted mesh is presented in Figure 28. Note that the intermediate (_h_-adapted) mesh is not presented but it can easily be obtained from the results in Figures 27 and 28 by taking approximation orders \(p\) and _h_-division pattern from these two figures, respectively. It can be seen in the presented figure that, in the vicinity of the curved edge with the rotations constrained, the higher error level and the resultant higher approximation orders are present due to the local bending along \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Adaptive Method**} & \multirow{2}{*}{**Result Quantity**} & \multicolumn{4}{c}{**Mesh Type**} \\ \cline{3-6}  & & **Initial** & **Modified** & **Intermediate** & **Final** \\ \hline \multirow{4}{*}{standard \(hp\)-adaptivity} & mesh figure no. & Figure 21 & – & – & Figure 22 \\  & dofs number \(N\)[1] & 81 & – & 918 & 8145 \\  & \(\log(U_{r}-U)\) [N/m] & 8.015996 & – & 8.014184 & 6.184208 \\  & \((U_{r}-U)/U_{r}\) [\%] * & 99.9 & – & 99.5 & 1.47 \\ \hline \multirow{4}{*}{standard \(hp\) after detection} & mesh figure no. & Figure 21 & – & Figure 23 & – \\  & dofs number \(N\)[1] & 81 & 729 & 3294 & – \\  & \(\log(U_{r}-U)\) [N/m] & 8.015996 & 6.330842 & 5.930839 & – \\  & \((U_{r}-U)/U_{r}\) [\%] * & 99.9 & 2.06 & 0.82 & – \\ \hline \multirow{4}{*}{standard \(hp\) after optimization} & mesh figure no. & Figure 21 & Figure 24 & – & Figure 25 \\  & dofs number N[1] & 81 & 324 & 5265 & 10,512 \\  & \(\log(U_{r}-U)\) [N/m] & 8.015996 & 7.978775 & 6.039104 & 5.449802 \\  & \((U_{r}-U)/U_{r}\) [\%] * & 99.9 & 91.73 & 1.05 & 0.27 \\ \hline \hline \end{tabular}  
* admissible relative error value \(\gamma_{T}=0.7\) %.  
\end{table} 
Table 3: Result summary—a quarter of the bending-dominated shell. the constrained boundary. The membrane and bending strain energies are of the same order along this boundary. In the rest of the shell octant, the membrane strains dominate over bending ones and the error level is lower and evenly distributed. Because of this, the resultant approximation orders are also lower and evenly distributed. Figure 28: An octant of a membrane-dominated shell—\(hp\)-adapted (final) mesh. Figure 27: An octant of a membrane-dominated shell—initial mesh. The convergence curve corresponding to the standard three-step \(hp\)-adaptivity is shown in Figure 29. The first point on the curve corresponds to the initial mesh. The second point of the curve is for the intermediate mesh, while the third point for the final mesh. The admissible error value was confidently achieved in the corresponding three steps. Finally, in Table 4, the summarizing results of the absolute error \(U_{r}-U\) and the relative error \((U_{r}-U)/U_{r}\) are presented versus the number \(N\) of degrees of freedom (dofs) for three points of the convergence curve. The mesh figure numbers corresponding to these points are also included in the presented table. Discussion  
It can seen that the locking detection and/or assessment tools are capable of not only detecting the phenomena but also recognizing the problems where the phenomenon is not present. Such a result of the detection is consistent with theory presented in the literature (compare [32]).