## 2 Materials and Methods
### Geometry and Finite Element Mesh  
The electromechanical coupling coefficient is calculated using the relation (1), i.e., based on the knowledge of the resonance and antiresonance frequency of the respective oscillation. The resonance and antiresonance frequencies are determined by the procedure in the introduction of Section 2. The thickness oscillation that is symmetric to the resonator symmetry axis will be examined. The resonator has the cylindrical shape, so the axis of symmetry is the axis of the cylinder representing the resonator. It is therefore appropriate that the chosen discretization of the resonator area reflects the symmetry of the oscillation and is therefore homogeneous. The use of inhomogeneous grids may lead to:  
* An axially asymmetrical shape of the oscillation that does not reflect reality;
* Inaccuracies in the calculation of the resonance frequency.  
We can say that this is a form of discretization error. By using a finite element mesh, the edges of which will have a comparable size in the whole discretized area, we eliminate the mentioned errors in the obtained solution. At the same time, identical finite element meshes will be used to calculate the resonance and antiresonance frequencies of resonators with wrap-around electrodes and to calculate the resonance and antiresonance frequencies of a fully electroded resonator. The use of identical meshes has the following benefits:  
* The same discretization error can be expected for the calculation of the resonance frequency of a resonator with fully electroded circular surfaces, as well as for resonators with wrap-around electrodes;
* A very similar numerical error in the calculation of resonance frequencies can be expected for all analyzed cases.  
Under these circumstances, the results of the calculation of the resonance frequencies obtained for each case can be compared. The differences between the results will be negligible because:  
* Resonators fully electroded on circular surfaces and resonators with wrap-  
The only difference between the analysed variants is due to the anisotropy of the elastic coefficients tensor. In the case of a resonator fully electroded on circular surfaces, the homogeneous polarizing electric field results in an almost perfect orientation of the individual ceramic grains in the direction of the applied electric field, i.e., in the direction perpendicular to the thickness of the resonator.  
In the case of resonators with wrap-around electrodes, we will consider the orientation of the ceramic grains in the directions calculated in the work [3]. Thus, the difference of the individual analyzed variants will be apparent only in those elements of the stiffness matrix \(\mathbf{K}\), which will correspond to the finite elements in which the material properties differ.  
The thickness oscillation we are analyzing is symmetric along the resonator axis. Therefore, the geometry corresponding to the symmetric half of the resonator can be used for the calculation, taking into account the boundary conditions  
\[\mathrm{D}_{\mathrm{k}}n_{k} = 0, \tag{24}\] \[\mathrm{T}_{\mathrm{i}}\eta_{i} = 0, \tag{25}\]  
in the plane of symmetry. The finite element meshes used are shown in Figure 3. They have a sufficient number of nodes for the calculation of the fundamental thickness oscillation over the base diameter, both variants (a) and (b) contain a total of about 14,000 nodes.