有限弹性体中声表面波的并行有限元计算方法及其应用

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论文题目：有限弹性体中声表面波的并行有限元计算方法及其应用

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论文提交日期：2008年 11月 20日

公开 11646

TB12

G06B08010402 王 羽 工程力学 工学院王骥教授

A Thesis Submitted to Ningbo University for the Master’s Degree

The Parallel Finite Element Analysis and Applications of Surface Acoustic Waves in Finite Elastic Solids

Candidate：Yu Wang

Supervisors：Professor Ji Wang

Faculty of Engineering

Ningbo University

Ningbo 315211, Zhejiang P.R.CHINA

Date: November 20, 2008

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有限弹性体中声表面波的并行有限元计算方法及其应用

摘 要

声表面波在有限压电弹性体中的传播特性一直是声表面波器件设计的主要问题，是高效和精确设计产品的重要理论依据。对声表面波器件进行有限元分析，由此计算弹性波在器件中的传播特性是非常必要的，特别有助于反映结构参数改变对声表面波传播特性的影响。本文运用并行有限元方法对声表面波的传播特性进行分析，从而对声表面波在有限弹性体中的传播问题进行计算。

本文首先介绍了声表面波三维有限元方程的实现过程，它最终归结为大规模线性方程组的特征值计算和求解，通过频率高这一特征解释了问题难度超过常见计算能力的原因，说明了传统有限元技术为何没有得到广泛应用。本研究主要利用集群及高效的数值软件ARPACK/PARPACK和PETSc来扩大计算规模，提高计算效率。本文着重介绍了特征值求解包ARPACK/PARPACK，它是基于隐式重启动Arnoldi方法，特别适合大规模稀疏矩阵指定区间特征值计算；ARPACK/PARPACK逆向通信接口的特点使得矩阵存储格式、矩阵向量积和线性方程组求解方式的选择更加灵活，因此本文使用压缩行存储代替了一维变带宽存储，实现了矩阵的非零存储。同时介绍了线性方程求解包PETSc，它几乎提供了所有求解线性方程组的高效求解器，既有串行求解也有并行求解，既有直接法求解也有迭代法求解。PETSc函数多态的特性使得调用接口简单且模式化，方便程序的移植。随后将PETSc中矩阵向量积和线性方程组的求解功能移植到特征值计算程序中，并在集群系统上进行测试，大大提高了计算效率，取得了很好的加速比。将特征值计算程序与有限元程序相结合，最终形成了完整的并行有限元程序，并将有限元程序用于梁板振动问题的分析，通过与ANSYS计算结果和理论分析结果对比，验证了有限元程序的正确性。最后，将有限元程序应用于覆盖有周期性电极的各向异性有限弹性体中声表面波传播特性的分析，研究了声表面波的频率随电极厚度的增加而改变的规律。

基于集群技术的三维有限元并行计算方法将更加精确的分析有限弹性体中声表面波的传播特性，为声表面波器件的设计和优化提供重要的理论依据，有着重要的实用价值。

关键词：声表面波，有限元，特征值，并行计算，PETSc

The Parallel Finite Element Analysis and Applications of Surface Acoustic Waves in Finite Elastic Solids

Abstract

The characteristics of acoustic wave propagation in finite elastic solids is a major issue in surface acoustic wave (SAW) devices, which are designed based on accurate analysis. As a result, it is essential in engineering applications to analyze the surface acoustic wave propagation in finite piezoelectric solids. Since the exact results from analytical solutions are not available, the finite element analysis based on the three-dimensional (3D) implementation of elasticity and piezoelectricity theory will provide accurate and precise results for the design and optimization of SAW resonators with wide engineering applications.

First, the process of establishing the 3D finite element equation is introduced, and the problem of SAW analysis is reduced to the calculation of eigenvalues of the extremely large linear equations. In fact, the difficulties are beyond the capacity of available computing resources, and it is also the reason why the general-purpose finite element technology has not been widely used. In this study, the computer cluster and numerical computing packages ARPACK/PARPACK and PETSc are used to improve the scale and efficiency of the finite element calculation. Second, the ARPACK/PARPACK, which is based on Implicitly Restarted Arnoldi method, is introduced. Then the compressed sparse row (CSR) storage format is used to replace the Symmetric Skyline (SSK) storage method. In the meantime, the PETSc package, which provides almost all efficient solvers of linear equations, including serial, parallel, direct and iterative method is introduced. The multi-state properties of PETSc functions make interfacing and transplanting simple and easy. The PETSc is incorporated into the eigenvalue extraction program for the calculation of the matrix vector product and the solution of the linear equations. The complete program is tested on a Linux cluster. The improvement of computing efficiency is clear and significant. Third, the eigenvalue extraction program and the finite element analysis were integrated to complete a parallel finite element program, and it is used for the vibration analysis of beams and plates as a required verification and validation process. The numerical results are consistent with the ones from the ANSYS and analytical results. Finally, the finite element program is used to analyze the wave propagation in finite elastic solids with periodic electrodes to study the influence of electrodes on the SAW propagation.