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 16.** A bending-dominated plate—*hp*-adapted (final) mesh.

**Figure 17.** A bending-dominated plate—mesh after simple detection, no *hp*-adaptation.

**Figure 19.** A bending-dominated plate—mesh after optimization and *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

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, modified, intermediate, and final meshes have been generated. It can be seen that the locking has been removed as the first section is not horizontal.

**Figure 20.** A bending-dominated plate—convergence for three adaptation cases.

It should be stressed that the modes including modification step are the automatic choice of the adaptive algorithm, while the standard *hp*-adaptivity is the enforced mode.

In Table 2, the relations between the number *N* of degrees of freedom and the absolute and relative errors, respectively, *Ur* − *U* and (*Ur* − *<sup>U</sup>*)/*Ur*, are presented for the consecutive points of the convergence curves of three adaptation modes. The mesh figure numbers corresponding to these points are also indicated in the table.
