Makkonen T, Holappa A, Ellä J, Salomaa M M
Materials Physics Laboratory, Helsinki University of Technology, Finland.
IEEE Trans Ultrason Ferroelectr Freq Control. 2001 Sep;48(5):1241-58. doi: 10.1109/58.949733.
A finite element method (FEM) formulation is presented for the numerical solution of the electroelastic equations that govern the linear forced vibrations of piezoelectric media. A harmonic time dependence is assumed. Both of the approaches, that of solving the field problem (harmonic analysis) and that of solving the corresponding eigenvalue problem (modal analysis), are described. A FEM software package has been created from scratch. Important aspects central to the efficient implementation of FEM are explained, such as memory management and solving the generalized piezoelectric eigenvalue problem. Algorithms for reducing the required computer memory through optimization of the matrix profile, as well as Lanczos algorithm for the solution of the eigenvalue problem are linked into the software from external numerical libraries. Our FEM software is applied to detailed numerical modeling of thin-film bulk acoustic wave (BAW) composite resonators. Comparison of results from 2D and full 39 simulations of a resonator are presented. In particular, 3D simulations are used to investigate the effect of the top electrode shape on the resonator electrical response. The validity of the modeling technique is demonstrated by comparing the simulated and measured displacement profiles at several frequencies. The results show that useful information on the performance of the thin-film resonators can be obtained even with relatively coarse meshes and, consequently, moderate computational resources.
本文提出了一种有限元方法（FEM）公式，用于求解控制压电介质线性强迫振动的电弹性方程的数值解。假设时间依赖为谐波形式。文中描述了两种方法，即求解场问题（谐波分析）和求解相应的特征值问题（模态分析）。我们从头创建了一个有限元软件包。文中解释了有限元高效实现的核心重要方面，如内存管理和求解广义压电特征值问题。通过优化矩阵轮廓来减少所需计算机内存的算法，以及用于求解特征值问题的兰索斯算法，都从外部数值库链接到了该软件中。我们的有限元软件应用于薄膜体声波（BAW）复合谐振器的详细数值建模。给出了谐振器二维和全三维模拟结果的比较。特别是，三维模拟用于研究顶部电极形状对谐振器电响应的影响。通过比较几个频率下的模拟和测量位移轮廓，证明了建模技术的有效性。结果表明，即使使用相对粗糙的网格，进而使用适度的计算资源，也能获得有关薄膜谐振器性能的有用信息。
