# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 5 Effectivity of the Method in Model Problems
### Numerical Examples
#### 5.2.2 A Bending-Dominated Plate  
Data  
In this example, the entire plate is considered. Thus, the thinness ratio is equal to \(l/2t=1.57075\times 10^{-2}/0.03\times 10^{-2}\). The discretization parameters of the coarse initial mesh, i.e., the element longitudinal and transverse approximation orders and the element size, are \(p=1\), \(q=2\), and \(h=l/2\), respectively. These parameters can be seen in Figure 15. The data for the error analysis are assumed as: the admissible target error \(\gamma_{T}=0.02\) and the ratio of the admissible intermediate to admissible target errors \(\gamma_{I}/\gamma_{T}=3\).  
Results  
As in the previous test, three courses of adaptation are of our interest. The first course corresponds to the standard \(hp\)-adaptivity where the initial mesh is \(h\)- and then \(p\)-adapted. The \(hp\)-adapted final mesh is shown in Figure 16. The next course of the adaptation is based on the \(hp\)-adaptation which is preceded by the modification of the initial mesh. This modification (Figure 17) lies in setting the longitudinal order of approximation as equal to the maximum possible value removing the locking, i.e., \(p=8\), after the phenomenon has been detected by means of the algorithm of Section 3.1.3. As in the previous example, no further adaptive actions have been performed. The \(h\)-division has not been made due to the estimated approximation error level and discretized character of performance of the adaptivity control algorithm. The \(p\)-enrichment has not been possible as the maximum value of \(p=8\) has already been applied in the modified mesh. The third course of adaptation consists of the \(hp\)-adaptation following the modification of the initial mesh by adopting the optimized value (\(p=5\)) of the longitudinal order of approximation (Figure 18). This value was established by the algorithm from Section 3.2.1. The final mesh for this adaptation is presented in Figure 19. The not displayed intermediate mesh has the same division pattern as the final mesh and the longitudinal approximation order as in the modified mesh, i.e., \(p=5\).  
Figure 15: A bending-dominated plate—initial mesh.  
Figure 16: A bending-dominated plate—_hp_-adapted (final) mesh.  
Figure 17: A bending-dominated plate—mesh after simple detection, no _hp_-adaptation.  
The convergence curves corresponding to the performed adaptations are displayed in Figure 20. In the case of the standard \(hp\)-adaptivity, the curve consists of three points and two section. These three  
Figure 19: A bending-dominated plate—mesh after optimization and \(hp\)-adaptation.  
Figure 18: A bending-dominated plate—mesh after optimized modification.  
points correspond to three meshes generated within this course: the initial, intermediate (or \(h\)-adapted), and final (or \(hp\)-adapted) ones. The presence of the shear locking is reflected by the first horizontal section of the curve. In the case of the \(hp\)-adaptivity performed after detection of the locking and applying the modified maximum value of the longitudinal order of approximation, i.e., \(p=8\), the convergence curve consists of two points and one section. These two points correspond to the initial and modified meshes. In the only section of the curve that is not horizontal, he locking has been removed. In the third course of adaptation, based on the modification of the initial mesh by the optimized value of the approximation order \(p=5\), the convergence curve consists of three sections and four points--the initial,