**6. Conclusions**

Three types of the transition piezoelectric elements joining the basic piezoelectric elements are possible. The joined basic elements are the three-dimensional piezoelectric elements (or hierarchical symmetric-thickness piezoelectric elements) and the first-order symmetric-thickness piezoelectric elements.

The first (classical) transition element guarantees continuity of the displacement and electric potential fields between the basic elements. It also guarantees that the assumption of no elongation of the normals to the mid-surface holds on the boundary between the transition elements and the first-order elements.

The second (modified) transition element, additionally, guarantees the gradual change from the plane stress to three-dimensional stress state between the basic elements of the first-order and three-dimensional (or hierarchical) character, respectively.

The third (enhanced) transition element, additionally, assures the gradual change of the assumption of no elongation of the normals to the mid-surface between the basic elements. The latter assumption holds on the boundary with the first-order element and is gradually switched off towards the boundary with the three-dimensional (or hierarchical) element. On this boundary, this assumption is completely off.

The gradual change of the stress state from the plane to three-dimensional is based on introduction of the blending functions that determine contributions of the basic models to the transition model. Implementation of this idea on the element level needs modifications of the strain vector, and the piezoelectric and dielectric constant matrices as well.

The gradual change of the assumption of no elongation of the normals needs utilization of the blending functions, which play the role of gradually switching functions. On the element level, the nodal values of the complement (to unity) of the gradually switching functions appear in the penalty stiffness matrix responsible for the gradual change of the constraints.

Comparing the stress distributions within the transition zones for three model problems, one can notice that high stress gradients appear on the boundary between the classical transition elements and the first-order elements. In the case of the enhanced transition elements, the gradual change of the stress state can be observed. In the case of the modified transition element, the stress distribution is intermediate with respect to the previous two distributions.

It can be observed that the discrete models of the three analyzed model structures, with three types of the transition elements employed, produce the convergence curves of slightly higher convergence for the case of the modified elements in comparison to the classical elements. The highest convergence, however, is obtained in the case of the enhanced transition elements.

It can be concluded that the enhanced transition element should be recommended in the analysis of mixed models of piezoelectrics, as this element gives the same or better convergence than the other transition models, and it removes the high stress gradients between the classical transition elements and the first-order piezoelectric elements.

Even though all three transition models can be applied to error-controlled adaptivity due to monotonicity of the applied error measure, the enhanced transition elements deliver convergence curves with the error level and convergence rates just between the curves for the basic models.

The presented research needs the following next steps. Firstly, the error estimation methods and adaptive procedures for the new transition elements have to be prepared and verified. Next, application of the proposed transition models to parametric and/or adaptive analysis of electro-mechanical systems should be performed and critical factors influencing quality of such an analysis should be determined.

**Author Contributions:** Conceptualization, G.Z. and M.Z.; methodology, G.Z. and M.Z.; writing and verifying software, G.Z. and M.Z.; validation, G.Z. and M.Z.; formal analysis, G.Z. and M.Z.; investigation, G.Z. and M.Z.; resources, G.Z. and M.Z.; writing—original draft preparation, G.Z. and M.Z.; writing—review and editing, G.Z. and M.Z.; projects administration, G.Z. and M.Z.; funding acquisition, G.Z. and M.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partly funded by Scientific Research Committee Poland under research grants 7 T07A 004 14 and 5 T07A 040 24 and by National Science Center Poland under research gran<sup>t</sup> N N504 515340.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
