A Numerical Method for Estimating the Nonlinear Eigenvalue Numbers of Boundary Element

一种估计边界元非线性特征值数目的数值方法

School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China

Received: 2 March 2018

Abstract

Recently, some new proposed methods for solving nonlinear eigenvalue problems (NEPs) have promoted the development of large-scale modal analysis using BEM. However, the efficiency and robustness of such methods are generally still dependent on input parameters, especially on the parameters related to the number of eigenvalues to be solved. This limitation obviously restricts the popularization of the practical engineering application of modal analysis using BEM. Therefore, this paper develops a numerical method for estimating the number of nonlinear eigenvalues of the boundary element method. Firstly, the interpolation method based on the discretized Cauchy integral formula of analytic function is used for obtaining the BEM matrix's derivative with regard to frequency, and this method is easily combined with the mainstream fast algorithm libraries of BEM. Secondly, the method for evaluating the eigenvalue number of BEM under various boundary conditions is obtained by combining the interpolation method with the analytic formula to obtain the eigenvalue number, while the unbiased estimation is used to determine the trace of matrix. Finally, a series of typical examples are used to explore the principle for selecting optimal input parameters in this method, and then a set of optimal input parameters are determined. The overall excellent performance of this method is verified by a complex large-scale example.

摘要

近年来，一些新提出的非线性特征值解法很好地推动了大规模边界元模态分析问题的发展，但这类方法的效率和鲁棒性还普遍依赖于输入参数的选取，特别是与待求解特征值数目相关的参数，这一局限性也明显制约了边界元模态分析方法向实际工程应用的推广。围绕这一问题，发展了一种通用的边界元非线性特征值数目估计方法。首先，针对特征值数目解析计算公式中边界元矩阵关于频率求导困难的问题，采用解析函数的Cauchy积分公式，发展了一种非常易于同主流边界元快速算法库结合的边界元矩阵关于频率导数的插值计算格式；其次，将该插值计算格式与特征值数目解析计算公式结合，并通过无偏估计确定矩阵的迹，获得了适用于各类边界条件的边界元非线性特征值数目估计方法；最后，采用典型算例探索新方法最优输入参数的选取原则，并确定出了一组最优输入参数，而且通过大规模复杂算例对该方法具有的整体优异性能进行了验证。