# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 5 Effectivity of the Method in Model Problems
### Numerical Examples
#### 5.2.3 A Quarter of a Bending-Dominated Shell  
Data  
This example concerns a symmetric quarter of the half-cylindrical shell. The thinness ratio for this structure is equal to \(1/t=3.1415\times 10^{-2}/0.003\times 10^{-2}\). We assume the following discretization parameters: the longitudinal order of approximation \(p=2\), the transverse one \(q=2\), and the element size \(h=1/2\). The shell quarter and its initial discretization is illustrated in Figure 21. The error analysis is based on the following assumptions: \(\gamma_{T}=0.007\) and \(\gamma_{I}/\gamma_{T}=3\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Adaptive Method**} & \multirow{2}{*}{**Result Quantity**} & \multicolumn{5}{c}{**Mesh Type**} \\ \cline{3-6}  & & **Initial** & **Modified** & **Intermediate** & **Final** \\ \hline \multirow{3}{*}{standard \(hp\)-adaptivity} & mesh figure no. & Figure 15 & – & – & Figure 16 \\  & dofs number \(N\)[1] & 36 & – & 225 & 6381 \\  & \(\log(U_{r}-U)\) [N/m] & \(1.060969\) & – & \(1.059175\) & \(-1.010223\) \\  & \((U_{r}-U)/U_{r}\) [\%] \({}^{*}\) & 100. & – & 99.6 & 0.85 \\ \hline \multirow{3}{*}{standard \(hp\) after detection} & mesh figure no. & Figure 15 & Figure 17 & – & – \\  & dofs number \(N\)[1] & 36 & 729 & – & – \\  & \(\log(U_{r}-U)\) [N/m] & \(1.060969\) & \(-0.527871\) & – & – \\  & \((U_{r}-U)/U_{r}\) [\%] \({}^{*}\) & 100. & 2.58 & – & – \\ \hline \multirow{3}{*}{standard \(hp\) after optimization} & mesh figure no. & Figure 15 & Figure 18 & – & Figure 19 \\  & dofs number \(N\)[1] & 36 & 324 & 5265 & 9279 \\ \cline{1-1}  & \(\log(U_{r}-U)\) [N/m] & \(1.060969\) & \(0.927962\) & \(-0.817269\) & \(-1.338752\) \\ \cline{1-1}  & \((U_{r}-U)/U_{r}\) [\%] \({}^{*}\) & 100. & 73.6 & 1.32 & 0.40 \\ \hline \hline \end{tabular}  
* admissible relative error value \(\gamma_{T}=2.0\) %.  
\end{table} 
Table 2: Result summary—the bending-dominated plate. Figure 21: A quarter of a bending-dominated shell—initial mesh. Results  
As in the previous two bending-dominated examples, also here three courses of the adaptation are performed. The first adaptation is the standard procedure composed of \(h\)- and \(p\)-steps. The mesh after \(hp\)-adaptation can be seen in Figure 22. The second and third types of the adaptation are composed of the \(hp\)-adaptation which follows the modification of the initial mesh. In the second type, the modification is based on the maximum possible value of the longitudinal order of approximation \(p=8\) (the corresponding figure is not displayed). In the case of the second type of adaptation, the modified mesh has been \(h\)-adapted further (see Figure 23). In the third type, the modification takes advantage of the optimized value of \(p=5\) (Figure 24). In the second type, the \(p\)-adaptation has not been possible as the maximum value of \(p=8\) has already been applied. In the third type of adaptation, the modified mesh has been \(hp\)-adapted. The final mesh is presented in Figure 25. The not revealed intermediate mesh possesses the same division pattern as the final mesh and the uniform order of approximation as in the modified mesh, namely \(p=5\). Three convergence curves resulting from the described adaptations are presented in Figure 26. The standard \(hp\)-adaptivity convergence curve consists of two sections and three points indicating the estimated error level for the initial, intermediate, and final meshes. In the case of the \(hp\)-adaptation following the modification based on the maximum value of \(p=8\), the curve consists of two sections and three points corresponding to the initial, modified, and \(h\)-adapted (intermediate) meshes. The shear-membrane locking has been removed--the first section of the curve is not horizontal. In the last case of the \(hp\)-adaptivity following the modification based on the optimized value of \(p=5\), the curve is composed of three sections and four points--the initial, modified, intermediate, and final meshes have been generated.