# A rational approximation method for solving acoustic nonlinear eigenvalue problems

## TL;DR

This paper introduces two efficient rational approximation methods, using Cauchy integral and Chebyshev interpolation, for solving large-scale nonlinear acoustic eigenvalue problems derived from boundary element methods.

## Contribution

It develops two novel approximation techniques with a parallelizable Rayleigh-Ritz procedure for large-scale acoustic eigenvalue problems.

## Key findings

- Effective for problems with up to two million degrees of freedom.
- Demonstrated high accuracy on benchmark and industrial applications.
- Methods outperform traditional approaches in computational efficiency.

## Abstract

We present two approximation methods for computing eigenfrequencies and eigenmodes of large-scale nonlinear eigenvalue problems resulting from boundary element method (BEM) solutions of some types of acoustic eigenvalue problems in three-dimensional space. The main idea of the first method is to approximate the resulting boundary element matrix within a contour in the complex plane by a high accuracy rational approximation using the Cauchy integral formula. The second method is based on the Chebyshev interpolation within real intervals. A Rayleigh-Ritz procedure, which is suitable for parallelization is developed for both the Cauchy and the Chebyshev approximation methods when dealing with large-scale practical applications. The performance of the proposed methods is illustrated with a variety of benchmark examples and large-scale industrial applications with degrees of freedom varying from several hundred up to around two million.