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

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.