An analysis of quartz crystal resonators with mindlin plate theory based parallel finite element method on a Linux cluster

Finite element method has been applied to quartz crystal resonator engineers due to its long history in research based on Mindlin plate and three-dimensional approaches for piezoelectric plates with considerations of complication factors such as electrodes, thermal effect, mounting, and packaging, among others. The earlier efforts have been dampened, however, by the fact that the high vibration frequency in the thickness-shear mode has caused significant increase of the problem size in terms of number of equations, or total degree of freedom, in the linear system resulted. It is typical that an accurate analysis of quartz crystal resonator vibrating at the fundamental thickness-shear mode may require solving a linear system around one million for both free and forced vibrations. What we can get from the analysis are the frequency spectra, which are the relationship between frequencies and geometry, and mode shapes of the resonator structure with mountings. This, not surprisingly, is beyond the computing capabilities for many industrial engineers who do not have access to resources like supercomputers widely available to academic researchers. On the other hand, the finite element method is an excellent tool for crystal resonator analysis we should utilize for quick and precise prototyping process. In order to overcome the challenge of computing power, the finite element program development has been taking steps through employing Mindlin plate theory, efficient eigenvalue solvers, and sparse matrix handling algorithms, as ways to aggressively improve the efficiency and reduce stringent requirements on hardware. In this continuing research, we have replaced our earlier version of the finite element program with ARPACK as the eigenvalue solver, added sparse matrix handling functions, and implemented parallel computing capability on a cost-effective Linux cluster. The computing capability has been improved with the utilization of multiple processors significantly while t- - he cost of infrastructure and operations is within the reach of the frequency control industry. We shall introduce the core improvements on algorithms, parallel features and implementation, hardware requirements, and computing power gain based on our current program configuration. The program has been developed in partnership with major industry players to demonstrate the effectiveness of supercomputing and finite element method as an important design tool for resonator design and improvement.