Skip to main content
SpringerLink
Log in

Parallel finite element analysis of high frequency vibrations of quartz crystal resonators on Linux cluster

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

Quartz crystal resonators are typical piezoelectric acoustic wave devices for frequency control applications with mechanical vibration frequency at the radio-frequency (RF) range. Precise analyses of the vibration and deformation are generally required in the resonator design and improvement process. The considerations include the presence of electrodes, mountings, bias fields such as temperature, initial stresses, and acceleration. Naturally, the finite element method is the only effective tool for such a coupled problem with multi-physics nature. The main challenge is the extremely large size of resulted linear equations. For this reason, we have been employing the Mindlin plate equations to reduce the computational difficulty. In addition, we have to utilize the parallel computing techniques on Linux clusters, which are widely available for academic and industrial applications nowadays, to improve the computing efficiency. The general principle of our research is to use open source software components and public domain technology to reduce cost for developers and users on a Linux cluster. We start with a mesh generator specifically for quartz crystal resonators of rectangular and circular types, and the Mindlin plate equations are implemented for the finite element analysis. Computing techniques like parallel processing, sparse matrix handling, and the latest eigenvalue extraction package are integrated into the program. It is clear from our computation that the combination of these algorithms and methods on a cluster can meet the memory requirement and reduce computing time significantly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lee,P.C.Y., Zee,C. and Brebbia,C.A., Thickness-shear, thickness-twist, and flexural vibration of rectangular AT-cut quartz plates with patch electrodes. In: Proceedings of the 32nd Annual Symposium on Frequency Control, 1978: 108–119.

  2. Yong,Y.-K., Three-dimensional finite element solution of the Lagrangean equations for the frequency-temperature behavior of Y-cut and NT-cut bars. In: Proceedings of the 40th IEEE Annual Symposium on Frequency Control, 1986: 179–186.

  3. Yong, Y.-K. and Zheng Zhang, A perturbation method for the finite element modeling of piezoelectric vibrations in quartz crystal resonators. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1993, 40(5): 551–562.

    Article  Google Scholar 

  4. Yong, Y.-K., Wang, J. and Imai, T., On the accuracy of Mindlin plate predictions for the frequency-temperature behavior of resonator modes in AT- and SC-cut quartz plates. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1999, 46(1): 1–13.

    Article  Google Scholar 

  5. Yong, Y.-K., Patel, M.S. and Tanaka, M., Estimation of quartz resonator Q and other figures of merit by an energy sink method. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2007, 54(7): 1386–1398.

    Article  Google Scholar 

  6. Yong, Y.-K., Patel, M. S., and Tanaka, M., Effects of Thermal Stresses on the Frequency-Temperature Behavior of Piezoelectric Resonators. Journal of Thermal Stresses, 2007, 30(6): 639–661. [DOI: 10.1080/01495730701274252]

    Article  Google Scholar 

  7. Yong,Y.-K., Patel,M.S., Srivastava,S., Tanaka,M. and Imai,T., The impact of finite element analysis on the design of quartz resonators. In: Proceedings of the 2006 International Frequency Control Symposium and Exposition, 22 September. [DOI: 10.1109/FREQ.2006.275345]

  8. Lerch, R., Simulation of Piezoelectric Devices by Two- and Three-Dimensional Finite Elements. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1990, 37(3): 233–247.

    Article  Google Scholar 

  9. Ha, Y. and Cho, S., Design sensitivity analysis and topology optimization of eigenvalue problems for piezoelectric resonators. Smart Material and Structure, 2006, 15: 1513–1524. [DOI:10.1088/0964-1726/15/6/002]

    Article  Google Scholar 

  10. Wang, J., Yong, Y.-K. and Imai, T., Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory. International Journal of Solids and Structure, 1999, 36(15), 2303–2319.

    Article  Google Scholar 

  11. Wang, J., Yu, J.-D., Yong, Y.-K. and Imai, T., A new theory for electroded piezoelectric plates and its finite element application for the forced vibration of quartz crystal resonators. International Journal of Solids and Structure, 2000, 37: 5653–5673.

    Article  Google Scholar 

  12. Wang, J. and Yang, J.S., Higher-order theories of piezoelectric plates and applications. Applied Mechanics Reviews, 2000, 53(4): 87–99.

    Article  Google Scholar 

  13. Wang,J., Hu,W., Zhao,W., Du,J.K. and Huang,D., Finite element analysis of quartz crystal resonators with Mindlin plate theory and parallel computing techniques on computer clusters. In: Proceedings of the 2007 IEEE International Ultrasonics Symposium, 1878–1881.

  14. Wang, J., Yu, J.-D., Yong, Y.-K. and Imai, T., A finite element analysis of frequency-temperature relations of AT-cut quartz crystal resonators with higher-order Mindlin plate theory. Acta Mechanica, 2008, 199(1–4): 117–130. [DOI: 10.1007/s00707-007-0538-5]

    Article  Google Scholar 

  15. Mindlin, R.D., An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. J.S. Yang (ed.). New Jersey: World Scientific, 2007.

    Book  Google Scholar 

  16. Söderkvist, J., Using FEA to treat piezoelectric low-frequency resonators. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1998, 45(3): 815–823.

    Article  Google Scholar 

  17. Southin, J.E.A. and Whatmore, R.W., Finite element modelling of nanostructured piezoelectric resonators (NAPIERs). IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2004, 51(6): 654–662.

    Article  Google Scholar 

  18. Wang,J., Zhao,W. and Bian,T., A method for the fast analysis of vibrations of Mindlin first-order plates for resonator design applications. In: Proceedings of the 2004 IEEE International Frequency Control Symposium and Exposition, 596–599.

  19. Yong, Y.-K. and Patel, M.S., Application of a DC-bias to reduce acceleration sensitivity in quartz resonators. International Journal of Applied Electromagnetics and Mechanics, 2005, 22(1–2): 69–82.

    Google Scholar 

  20. EerNisse,E.P. and Puccio,D., Suppression of the B-mode in SC-Cuts. In: Proceedings of the 2007 IEEE International Frequency Control Symposium, 1132–1136.

  21. Wang, Z., Zhu, H., Dong, Y., Wang, J. and Feng, G., Force-frequency coefficient of symmetrical incomplete circular quartz crystal resonator. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2001, 48(5): 1471–1479.

    Article  Google Scholar 

  22. Pao,S.Y., Chao,M.K., Lam,C.S. and Chang,P.Z., An efficient numerical method in calculating the electrical impedance different modes of AT-cut quartz crystal resonator. In: Proceedings of the 2004 IEEE International Frequency Control Symposium, 396–400.

  23. Pao,S.Y., Chao,M.K., Chiu,C.H., Lam,C.S. and Chang,P.Z., Beveling AT-cut quartz resonator design by an efficient numerical method. In: Proceedings of 2005 IEEE International Ultrasonics Symposium, 1848–1852.

  24. Patel,M.S., Yong,Y.-K., Tanaka,M. and Imai,T., Drive level dependency in quartz resonators. In: Proceedings of 2005 IEEE International Frequency Control Symposium, 793–801.

  25. Lehoucq,R.B., Sorensen,D.C., Maschhoff,K. and Yang,C., ARPACK Software. [2008-7-16]. http://www.caam.rice.edu/software/ARPACK/

  26. Lehoucq,R.B., Sorensen,D.C., Vu,P.A. and Yang,C., ARPACK: Fortran subroutines for solving large scale eigenvalue problems (Release 2.1), 1997 [2008-7-16]. http://ftp.caam.rice.edu/pub/people/sorensen/ARPACK

  27. Lehoucq,R.B., Maschhoff,K., Sorensen,D.C. and Yang,C., Parallel ARPACK Home Page. [2008-7-16]. http://www.caam.rice.edu/kristyn/parpack_home.html

  28. Maschhoff,K.J. and Sorensen,D.C., PARPACK: An efficient portable large scale eigenvalue package for distributed memory parallel architectures. In: Applied Parallel Computing Industrial Computation and Optimization (LNCS 1184), Springer, 1996: 478–486.

    Chapter  Google Scholar 

  29. BLAS (Basic Linear Algebra Subprograms). [2008-7-16]. http://www.netlib.org/blas/

  30. LAPACK (Linear Algebra PACKage) (Version 3.2). [2008-7-16]. http://www.netlib.org/lapack/

  31. Saad,Y., A basic tool-kit for sparse matrix computations (Version 2). 2005 [2008-7-16]. http://www-users.cs.umn.edu/saad/software/SPARSKIT/sparskit.html

  32. Chen, P., Zheng, D., Sun, S. and Yuan, M., High performance sparse static solver in finite element analyses with loop-unrolling. Advances in Engineering Software, 2003, 34(4): 203–215.

    Article  Google Scholar 

  33. Chen, P., Runesha, H., Nguyen, D.T., Tong, P. and Chang, T.Y.P., Sparse algorithms for indefinite system of linear equations. Computational Mechanics, 2000, 25(1): 33–42.

    Article  MathSciNet  Google Scholar 

  34. Portable, Extensible Toolkit for Scientific Computation (Version 2.3.3). 2007 [2008-7-16]. http://www-unix.mcs.anl.gov/petsc/petsc-as/

  35. The Trilinos Project (Release 9.0). 2008 [2008-7-16]. http://trilinos.sandia.gov/index.html

Download references

Author information

Authors and Affiliations

  1. Piezoelectric Device Laboratory, School of Engineering, Ningbo University, Ningbo, 315211, China

    Ji Wang, Yu Wang, Wenke Hu, Jianke Du & Dejin Huang

  2. Vectron International (China), Inc., 308 Fengju Road, Plant 8, Waigaoqiao Free Trade Zone, Shanghai, 200131, China

    Wenhua Zhao

Authors
  1. Ji Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenke Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wenhua Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jianke Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Dejin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji Wang.

Additional information

Project supported by a grant from the Science and Technology Division, Zhejiang Provincial Government, under the Key Project of the International Collaborative Program (Grant No.2006C14021). Additional support is from the K.C. Wong Magna Fund of Ningbo University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J., Wang, Y., Hu, W. et al. Parallel finite element analysis of high frequency vibrations of quartz crystal resonators on Linux cluster. Acta Mech. Solida Sin. 21, 549–554 (2008). https://doi.org/10.1007/s10338-008-0866-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10338-008-0866-6

Key words

Search

Navigation