# FEM Analysis of Piezoelectric Resonator Polarization Process
## 2 Materials and Methods
### Simulation of the Polarization Process  
A crucial step in the development of the model is to take into account the material properties of the real piezoceramic resonator. As already stated, the subject of the simulation is the very event of polarization of the transducer. Let us begin with the fact that before the polarization process starts, the transducer is macroscopically non-piezoelectric. That is due to the different orientation of the individual ceramic grains. At the level of the finite element method model, it is not possible to work with geometrically simpler and smaller entities than the individual finite elements, the set of which approximates the volume of the piezoelectric transducer. According to [19], the grain size of PZT type piezoelectric ceramics is in the range (1-10) \(\upmu\)m. Piezoelectric transducers typically have a diameter in the range of (1-5) cm and a thickness in the interval of (0.5-5) mm. If we consider a piezoelectric transducer with a diameter of 20 mm, thickness of 2 mm, ceramic grain size of 5 \(\upmu\)m, the discretization will contain the number of nodes in the order of \(10^{9}\). Outside of the nodes where the Dirichlet boundary condition is specified, we determine four degrees of freedom, three components of the displacement vector and the electric potential. The resulting set of algebraic equations will therefore have a dimension four times the number of discretization nodes. Such a system of equations is already at the verge of solvability on a common computing station and in acceptable time (within a time frame of max. tens of minutes). As stated above, the model considers only linear phenomena, omitting nonlinearities that arise from the existence of domain interfaces. Therefore, we do not have to consider domain interfaces in the model. We perform the analysis of the electric field distribution to determine the areas in which polarization occurs in the resonator, resp. polarization in the direction identical to the thickness of the transducer, which is the direction that has a fundamental effect on the electromechanical coupling coefficient of the thickness oscillation. The model must therefore capture the distribution of the electric field well enough, especially in areas where the field will be inhomogeneous, i.e., in the area where the electrodes wrap around. This can be achieved by adjusting the discretization parameter, by refining the finite element mesh, in the mentioned areas. Based on the above stated facts, it can be assumed that the model will not contain the same number of elements as there are in a real ceramic grain transducer. We replace the random orientation of the individual grains with a random orientation of the material property tensors in the individual finite elements of the model. We will treat each element as a grain of ceramics and assign it material tensors transformed into a randomly chosen Cartesian coordinate system. This idea is implemented by the program code in the following way:  
1. Define the coordinate systems into which the material tensors will be transformed. Thirty-six coordinate systems were created, derived from the basic Cartesian coordinate system of rotations about the \(x\)-coordinate axis (\(x_{1}\)) in \(10^{\circ}\) increments. That is, rotations of \(0^{\circ}\), \(10^{\circ}\), \(20^{\circ}\)\(\ldots 350^{\circ}\). In each rotation around the coordinate axis \(x\), create 35 more coordinate systems, rotating about the axis \(y\) (\(x_{2}\)) by \(10^{\circ}\), i.e., rotations by \(10^{\circ}\), \(20^{\circ}\)\(\ldots 350^{\circ}\). We have 36 \(\times\) 36 = 1296 coordinate systems in total. 2. Using transformation equations, e.g., [20], transform material tensors into a specified coordinate system. The material tensors are transformed into the coordinate systems defined inpoint 1. In the case of piezoelectrics, these are tensors of elastic modules, piezoelectric coefficients, permittivity. 3. Assign a unique identifier to the transformed material constants that refers to the material properties.