Results

As expected, the automatic tools for detection and/or assessment of the locking have not indicated the appearance of the phenomenon. Because of this, the automatic mode of adaptation corresponds to standard *hp*-approach. The final, *hp*-adapted mesh is presented in Figure 28. Note that the intermediate (*h*-adapted) mesh is not presented but it can easily be obtained from the results in Figures 27 and 28 by taking approximation orders *p* and *h*-division pattern from these two figures, respectively. It can be seen in the presented figure that, in the vicinity of the curved edge with the rotations constrained, the higher error level and the resultant higher approximation orders are present due to the local bending along p,q

p,q

the constrained boundary. The membrane and bending strain energies are of the same order along this boundary. In the rest of the shell octant, the membrane strains dominate over bending ones and the error level is lower and evenly distributed. Because of this, the resultant approximation orders are also lower and evenly distributed.

**Figure 27.** An octant of a membrane-dominated shell—initial mesh.

**Figure 28.** An octant of a membrane-dominated shell—*hp*-adapted (final) mesh.

The convergence curve corresponding to the standard three-step *hp*-adaptivity is shown in Figure 29. The first point on the curve corresponds to the initial mesh. The second point of the curve is for the intermediate mesh, while the third point for the final mesh. The admissible error value was confidently achieved in the corresponding three steps.

Finally, in Table 4, the summarizing results of the absolute error *Ur* − *U* and the relative error (*Ur* − *U*)/*Ur* are presented versus the number *N* of degrees of freedom (dofs) for three points of the convergence curve. The mesh figure numbers corresponding to these points are also included in the presented table.

**Figure 29.** An octant of a membrane-dominated shell—convergence for *hp*-adaptation.


**Table 4.** Result summary—an octant of the membrane-dominated shell.

 admissible relative error value *γT* = 1.0 %.

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