## 2 Materials and Methods
### Multiphysics Modeling Using ANSYS Mechanical  
After selecting two optimal materials at the material selection stage, piezoelectric material properties required to fully define the selected materials were collected for modeling using ANSYS Multiphysics. The properties of piezoelectric material were available in the Institute of Electrical and Electronics Engineers (IEEE)'s standard matrix formats, which were not ANSYS compatible. Conversion of property matrices was required, so various matrix operations were performed to obtain the properties in the required ANSYS format. The order of rows and columns in ANSYS matrices tensor notation is x, y, z, xy, yz, xz instead of x, y, z, yz, xz, xy, which is the order of rows and columns in standard IEEE forms based on [34; 35]. Then, geometric models of two plates of different dimensions and of three different paired plates were generated using SOLIDWORKS. Finally, static response analyses were performed on two separate plates made of two selected piezoelectric materials and on three different pairs of the two plates to determine their longitudinal (3-3) free displacements by applying a 1000V DC voltage difference on each plate in all analyses. Of the three pairs of plates, two were the potential piezoelectric plate actuators listed in Figure 1, and the third pair, made of plates of the same materials and dimensions, served as a reference. After comparing the longitudinal (3-3 mode) free displacement results of the candidate actuators, the optimal actuator having higher longitudinal free displacement was selected and modal and harmonic response analyses were performed on it to determine its frequency characteristics. In terms of piezoelectric finite element formulations and applications, early studies focused on 3D elements with nodal electric potential degrees of freedom. They take into account the surface characteristics and full electromechanical coupling inherent in piezoelectric models. According to the source, these were too thick to model very thin structures. Thus, in recent years, attention has been focused on 2D and 1D elementssuch as shells, plates, and beams, despite the difficulty in using conventional midplane formulations that take into account potentials on upper and lower surfaces. An overview of the development of FEM based on element types and degrees of freedom is presented in [36]. According to the above investigations, the FEM implementation in this work was based on the standard nodal approach extended for piezoelectric materials by using 3D elements. In order to account for electrical-mechanical coupling in piezoelectric materials, an additional degree of freedom for electrical potential was added to each node in those elements, together with their standard displacement unknowns. Specifically, hexahedral elements are considered in this work due to their superior accuracy and robustness compared to tetrahedral elements [37]. Tetrahedral elements are more effective in handling complex geometries [37] often encountered in practical applications, but due to the structural simplicity of our problem, they are not taken into account in this work. The thicknesses of the piezoelectric plates in this article are 8 mm and 10 mm, which are sufficient for modeling using 3D finite elements. In order to perform the abovementioned analyses, two different methods were followed for each analysis and the results from both methods were compared. Method 1: Using Ansys Parametric Design Language (APDL) command snippets in ANSYS Mechanical. Command snippets are small pieces of code in the ANSYS Mechanical APDL language that help to take advantage of the undisclosed capabilities of the ANSYS Mechanical interface. They are added to the project tree as commands for ANSYS Mechanical models. When the solver file is written, ANSYS Mechanical inserts the snippets into the command stream.