A nonsingular integral formulation for the Helmholtz eigenproblems of a circular domain

Abstract

A nonsingular integral formulation for the Helmholtz eigenproblem is developed in this paper. This novel method contains only imaginary‐part kernels instead of complex‐part kernels in the complex‐valued BEM. Based on the imaginary‐part formulation without singular source, no singular or hypersingular integrals are present. Although this formulation avoids the computation of singular and hypersingular integrals, this approach results in spurious eigensolutions. After comparing the results from the dual formulation, the true and spurious solutions can be separated. An analytical example for the eigensolutions of a two‐dimensional circular domain is studied. The continuous system can be transformed to a discrete system with circulants. Based on the spectral properties of circulants, the true and spurious solutions for the eigenvalues, boundary modes, interior modes and multiplicities are all examined. The possible failure of Hutchinson's sorting technique of looking at modal shapes is also discussed.

Key words :

• imaginary‐part dual BEM
• spurious eigenvalues and eigenmodes
• Bessel function and Helmholtz equation

Notes

Correspondence addressee