Three-dimensional finite elements and their relationships to Mindlin´s higher order plate theory in quartz crystal plate resonators

The finite element analysis of quartz crystal plate resonators is a computationally intensive task which also uses large computer memory. We are constantly in need of reducing the computational complexity of this task. The use of Mindlin´s higher order plate equations to reduce the size of the fem model is popular. In this paper, we explore the use of three-dimensional finite elements and compare their results to the finite element model of Mindlin´s higher order plate equations. We found that if we model the quartz plate with one layer of the 3-D finite elements, we can compare directly the number of nodes in the 3-D elements with the order of Mindlin´s plate equations. For example, a 4×4×4 nodes 3-D element using Lagrangian polynomial shape functions is the same as a 4×4 nodes 2-D element of the third order Mindlin´s plate equations without correction factors. The size of the stiffness and mass matrices are identical for the two finite element models. A 2×2×2 nodes (8-node) 3-D element is equivalent to a 2×2 modes 2-D element of the first order Mindlin´s plate equations without correction factors. The issue of whether we should use more layers of lower order elements versus using a single layer of higher order element is resolved in favor of the higher order element. This is because the finite element method only guarantees displacement continuity over the multi-layers, and stress continuity was neglected. Derived equations and results from frequency spectra are presented to support the findings. We also demonstrate that for large plate length to thickness ratios, the higher order 3-D elements or higher order Mindlin´s plate equations are needed to provide accurate frequency spectra. Efficient means of modeling the electrodes are presented