## 5 Numerical Tests
### Ability to Remove High Stress Gradients  
In order to present the ability of three (classical, modified, and enhanced) types of the transition elements to remove the high stress gradients between the transition elements and the neighboring basic (hierarchical and first-order) elements, exemplary (\(p\equiv\pi=6\)) distributions of the global normal stress \(\sigma_{33}\) (marked as \(szz\)) for three model problems and three types of the transition elements are presented. These distributions correspond to the mixed models from Figures 8-10. The stress distributions for the plate problem, with the classical, modified and transition elements employed, are presented in Figures 11-13, respectively. Figures 14-16 display the analogous three stress distributions for the bending-dominated shell. The last figures, Figures 17-19, correspond to the membrane-dominated problem with three types of the transition elements applied. Note that, in the figures, the stress values are written in numerical form not in analytical form due to capabilities of the numerical code used. In the case of the plate and membrane problems, high solution gradients are visible when the classical transition elements are employed. These gradients are reduced for the mixed models with the modified and enhanced elements applied. In the case of the bent shell, high gradients are not visible for the classical elements as the displayed global normal stress component is a combination of the transverse and longitudinal stresses. In order to quantify the presented graphical results concerning stress gradients, the Table 1 is presented. Therein, gradients (jumps/differences) of the global stress \(\sigma_{33}\) between the face centers of the neighboring elements are displayed for three model structures. Five models for each of three structures are included: the entirely hierarchical (MI) shell model (\(r=0.0\)), three mixed (RM/MI) models (\(r=0.5\)) with the classical, modified, and enhanced (respectively, TR1, TR2, TR3) transition models, and entirely first-order (RM) model (\(r=1.0\)). In the case of the bending-dominated (bent) plate, the chosen elements 31 and 24 correspond to 3D model and TR1 or TR2 or TR3 model. Both elements 24 and 23 correspond to either TR1 or TR2 or TR3 model, while the elements 23 and 16 to either TR1 or TR2 or TR3 model and RM model. In the case of the bending-dominated (bent) shell, the corresponding couples of elements are: 25 and 28, 28 and 27, and 27 and 30. In the case of the membrane (membrane-dominated shell), the corresponding couples are: 6 and 13, 13 and 14, 14 and 21. In the case of the plate, the third normal stress \(\sigma_{33}\) corresponds to the transverse normal stress. In the entirely hierarchical (MI) shell model, the observed values of gradients are small and due to finite element discretization. In the entirely first-order (RM) model, the jumps are extremely small (correspond to the numerical zero), as the plane stress assumption holds in all elements. In the case of the mixed TR1 model, the jump between the transition and first-order elements 23 and 16 results from the three-dimensional stress state and plane stress state in the neighboring elements. Such jumps are diminished to numerical zero for the cases of the TR2 and TR3 models, as the plane stress assumption holds on both sides of the neighboring faces of the transition and first-order elements. In both shell cases, the presented jumps of \(\sigma_{33}\) result from both discretization and plane stress assumption. The exact analysis for the transverse direction requires taking jumps for the stress components \(\sigma_{11}\) and \(\sigma_{13}=\sigma_{31}\) into account. Such an analysis (not revealed here) leads to exactly the same observations as in the case of the plate.