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.

**Figure 22.** A quarter of a bending-dominated shell—*hp*-adapted (final) mesh.

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. The locking has not been removed from the modified mesh, however the value of *p* = 5 has appeared sufficient for removal of the locking from the *h*-adapted (intermediate) mesh—the second section of the convergence curve is not horizontal.

**Figure 23.** A quarter of a bending-dominated shell—after simple detection and *h*-adaptation

**Figure 24.** A quarter of a bending-dominated shell—mesh after optimized modification.

Note that the adaptations including the described mesh modifications, based on either the maximum (*p* = 8) or optimized (*p* = 5) values of the longitudinal order of approximation, may be performed automatically by the program, while the standard *hp* course has to be enforced by a user.

**Figure 25.** A quarter of a bending-dominated shell—after optimization and *hp*-adaptation.

**Figure 26.** A quarter of a bending-dominated shell—convergence for three adaptation cases.

As in the previous numerical examples, the absolute and relative error values, respectively, log(*Ur* − *U*) and (*Ur* − *<sup>U</sup>*)/*Ur*, are presented in Table 3 versus the number *N* of degrees of freedom (dofs). For the points of three adaptive convergence curves, the corresponding mesh figure numbers are included in the table.


**Table 3.** Result summary—a quarter of the bending-dominated shell.

> admissible relative error value *γT* = 0.7 %.

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