Thickness-shear vibration analysis of circular quartz crystal plates by a differential quadrature hierarchical finite element method

Piezoelectric crystals are widely used to make acoustic wave resonators. A large portion of quartz crystal resonators are operated with the so-called thickness-shear (TS) modes of a plate, which is characterized by high frequency vibration that needs both huge computational cost and high accuracy. The situation is even more difficult for circular resonators because the scalar differential equation is not separable in polar coordinates due to material anisotropy. In this paper, TS vibration analysis of circular quartz plates was carried out by a differential quadrature hierarchical finite element method (DQHFEM). The DQHFEM adopts the formulations of the DQFEM that are compact and highly accurate. The numerical stability problem of the high-order or very high-order hierarchical basis is overcome by using the recursion formula of Legendre polynomials and more than 2000th order can be reached. The high frequency thickness-shear vibration of a circular quartz plate can be modeled by using only one or several DQHFEM elements, which greatly simplifies pre- and post-processing. Numerical comparison of the DQHFEM results with the results in literatures validated the high accuracy of the DQHFEM.