The discrepancies stem from experimental measurement errors, the fineness of the mesh used in our method, and the accuracy of the properties of barium titanate. These proximities clearly suggest that our method is correct and accurate. The simulated acoustic displacement fields at the upper surface of the disk with two different aspect ratios are plotted and compared in Fig. 15 The simulated and measured frequencies are plotted in Fig. The properties (elasticity matrix, coupling matrix, and relative permittivity) of barium titanate ceramic were obtained from Dent. A barium titanate cylinder was used in the simulation, whose height was set as 5 mm and radius changed from 2.5 to 17.5 mm. The same modes were simulated using our method and the corresponding resonant frequencies and acoustic displacement fields were compared with Shaw’s results. Shaw 14 measured the frequency of axial symmetric normal modes on barium titanate disks with different aspect ratios. Verification of the accuracy of our model was performed on fundamental modes. In reality, the torsional mode is unlikely to appear on a disk with a very small aspect ratio so the magnetic coupling coefficient should not be very high in this system. The highest coupling coefficient in this model is 2.6 × 10 − 7 for a cylinder with a small aspect ratio of ( h / r = 0.1). The relation among those three quantities is shown in Fig. It is found that when the aspect ratio decreases, both the magnetic coupling coefficient and frequency will increase. The acoustic displacement field, electric field, and magnetic field on the central YZ cross section are shown in Fig. A continuous boundary condition is applied to the cylinder’s surfaces, while, an absorbing boundary condition is applied on the sphere’s outer surface. In each simulation, the cylinder resides at the center of a sphere, representing the surrounding environment. Simulations were conducted on a series of PZT-5H cylinders, varying in height from 1 to 5 mm, all with a fixed radius of 10 mm. COMSOL’s built-in material properties are used. In this simulation, z 0 is chosen to be air. For some metallized surfaces, an electric-wall boundary condition needs to be applied on surface ⑥. The electromagnetic continuity boundary condition is applied on surface ⑥, whereas this boundary is set as a free boundary for the acoustic field. The Rayleigh mode is evanescent in this direction so that 300 μm, which is three times as long as the wavelength, is set as the distance from the surface to the fixed constraint. The fixed constrain is applied in the z direction. Since the Rayleigh mode is uniform in this direction, the distance between the two surfaces does not influence the form of the solution. Another periodic boundary condition is applied in the y direction. The distance between these two surfaces is set as 100 μm, and this distance will become the wavelength of the Rayleigh mode. Boundary conditions suitable for this mode are applied: A periodic boundary condition is applied on the propagating direction. A schematic diagram of the Rayleigh mode is shown in Fig. The verification of our method thus focuses on its simulation. METHODOLOGYĬompared to others, a Rayleigh mode generates a relatively strong and moreover analytically calculable magnetic field. Definitions of variables, example codes and various details and tips relevant to the building of working simulations are provided in the supplementary material. The results corresponding to the torsional mode on a cylindrical piezoelectric antenna showcase the capability of our method to simulate complicated devices. Results for a Rayleigh mode propagating in the X-direction on the Z-cut surface of a LiNbO 3 crystal are compared against both experimental results and analytical solutions, so as to validate our method’s correctness and accuracy. COMSOL is used here to provide explicit examples. Our open method is applicable to any FEM simulation software where weak forms define the whole system of coupled partial differential equations. The prime purpose of this paper is to remedy this frustrating inadequacy. Alas, many it not all of the popular finite-element-method (FEM) based simulation platforms, like COMSOL Multiphysics (both its structural mechanics and MEMs modules), implicitly invoke the quasi-static approximation, meaning that they cannot be used to simulate the complete electromagnetic behavior of piezo-mechanical devices. 6,7 These methods are limited to very specific spatial geometries and crystal classes (point group symmetry)-they cannot be extended to simulate arbitrary 3D structures or piezoelectric materials of a different class. Examples include the superposition-of-partial-waves method, 2 transverse-resonance method, 5 and Kirchhoff-plate-theory method. Certain known analytical methods can precisely simulate the EMA coupling without invoking the quasi-static approximation.
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