FEM modelling of resonant frequencies of in-plane parallel piezoelectric resonator

In the contribution, we introduce an application of finite element model of the piezoelectric resonator. The model is based on the physical description of piezoelectric materials, using linear piezoelectric state equations. Weak formulation and discretization of the problem lead to a large and sparse linear algebraic system, which results in a generalized eigenvalue problem. Resonant frequencies, the most important parameters of the resonator, are subsequently found by solving this algebraic problem. Depending on the discretization parameters, this problem may become large. Typically, we are not interested in all eigenvalues (resonant frequencies). For determination of several of them it seems therefore appropriate to consider the Krylov subspace methods (namely the implicitly restarted Arnoldi method implemented in the ARPACK library). For coarser meshes, we compute the complete spectra and we find the frequencies of dominant oscillation modes (the selection is made according to their electromechanical coupling coefficients). Then we focus on the part of the spectra near to the chosen dominant frequency and repeat the computation for refined meshes. From the results, we can also find out intervals between the dominant resonant frequencies (which is other important parameter describing the behavior of the resonator). The model was tested on the problem of thickness-shear vibration of the in-plane parallel quartz resonator. The results, compared with the measurement, will be given in the contribution.