# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 5 Effectivity of the Method in Model Problems
### Numerical Examples
#### 5.2.1 A Quarter of a Bending Dominated Plate  
Data  
The dimensions of a quarter of the plate are such that the thinness ratio of the plate is \(l/t=3.1415\times 10^{-2}/0.03\times 10^{-2}\). The initial mesh is coarse. Its data are as follows: the longitudinal approximation order \(p=1\), the transverse approximation order \(q=2\), and the element size \(h=l/2\). This mesh is shown in Figure 9. In the error analysis, the target error \(\gamma_{T}=0.01\) and the ratio \(\gamma_{I}/\gamma_{T}=3\). Results  
Three different courses of the adaptation are presented for this example. The first one corresponds to the standard \(hp\)-adaptivity composed of the \(h\)- and \(p\)-adaptation steps only. The final mesh for this first course is presented in Figure 10. The second course is the \(hp\)-adaptation preceded by the modification step based on the detection algorithm from Section 3.1.3, where, after the detection, the modification of the initial mesh is performed with the maximum setting \(p=8\) (Figure 11). No further automatic \(h\)-adaptation is performed by the adaptivity control algorithm. In addition, no further \(p\)-adaptationcan be performed due to the maximum value of \(p\) applied in the modification step after the detection. The last course is the \(hp\)-adaptation performed after the initial mesh modification based on setting the longitudinal order of approximation to its optimized value \(p=4\) (Figure 12) obtained from the algorithm of Section 3.2.1. The final mesh completing this course of adaptation is presented in Figure 13. The not presented intermediate mesh possesses the same division pattern as the final mesh but the order of approximation is uniform and taken from the modified mesh (\(p=4\)). The adaptive convergence curves for three described cases are presented in Figure 14. In the case of the standard \(hp\)-adaptivity, the curve consists of two sections and three points corresponding to the initial, intermediate (\(h\)-adapted) and final (\(hp\)-adapted) meshes. The influence of the shear locking is visible in the first section of the curve--this section is almost horizontal. In the case of the \(hp\)-adaptivity performed after detection of the locking and based on the modified value of \(p=8\), two points of the convergence curve correspond to the initial and modified meshes. The locking has been removed--the only section of the curve is not horizontal. In the case of the \(hp\)-adaptation performed after the detection and optimization of the value of \(p\), the convergence curve consists of four points (the initial, modified, intermediate, and final meshes) and three sections. The first not horizontal section of the curve reflects locking removal, while the next two sections correspond to \(h\)-refinement and \(p\)-enrichment. Finally, it is worth mentioning that the automatic choice of the program may be the two courses with the initial mesh modification. These two automatic courses correspond to either the assumed maximum or determined optimized value of \(p\) applied in the modification step. The enforced course corresponds to the standard \(hp\)-adaptation. The relations between the number \(N\) of degrees of freedom (dofs) and the absolute \(U_{r}-U\) and relative \((U_{r}-U)/U_{r}\) errors are summarized in Table 1 for the consecutive points of the three mentioned convergence curves. The mesh figure numbers corresponding to these points are also indicated. Discussion  
The shear locking has been detected in both cases which include modification of the initial mesh based on the maximum and optimized values of the longitudinal order \(p\). One can see that the standard \(hp\)-adaptation leads to the final error value below the desired admissible value of \(\gamma_{T}\).