A rational approximation method for solving acoustic nonlinear eigenvalue problems

TL;DR

Two approximation methods for computing eigenfrequencies and eigenmodes of large-scale nonlinear eigen value problems resulting from boundary element method (BEM) solutions of some types of acoustic eigenvalue problems in three-dimensional space are presented.

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.

Figures (11)

• Figure 4.1: Left: Approximation error versus the order of the approximation $m$ inside a unit circle. Right: Approximation error versus the order of the approximation $m$ inside an ellipse centered at $c = 0$ with semi-major axis $r_x = 1$ and semi-minor axis $r_y = 0.2$.
• Figure 4.2: Left: The eigenvalues of (\ref{['eq:Helmholtz3D']}) with homogeneous Dirichlet boundary conditions inside a circle centered at $c = 8.5$ with radius $r = 3.5$ (circles) computed via (\ref{['eq:sc']}) (plus) and Chebyshev interpolation method inside the real interval $[5,12]$ (squares). Right: The relative residuals $\|T(\lambda)u\|_2/\|u\|_2$ of the computed eigenpairs.
• Figure 4.3: Left: The eigenvalues of (\ref{['eq:Helmholtz3D']}) with homogeneous Dirichlet boundary conditions inside an ellipse centered at $c = 8.5$ with semi-major axis $r_x = 4.5$ and semi-minor axis $r_y = 0.2$ (circles) computed computed via (\ref{['eq:sc']}) (plus) and Chebyshev interpolation method inside the real interval $[5,12]$ (squares). Right: The relative residuals $\|T(\lambda)u\|_2/\|u\|_2$ of the computed eigenpairs.
• Figure 4.4: Relative residuals $\|T(\lambda)u\|_2/\|u\|_2$ of the $17$ eigenvalues of the Laplace eigenvalue problem on the unit cube. Left: after $10$ outer iterations of the reduced approach using Cauchy approximation. Right: after $7$ outer iterations using Chebyshev approximation.
• Figure 4.5: Approximation errors versus the order of the approximation $m$ inside an ellipse centered at $c =0$ with semi-major axis $r_x = 1$ and semi-minor axis $r_y = 0.2$ for the spherical BEM problem.