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.本文針對Helmholtz特徵值問題發展一非奇異積分推導解法。此法僅採用負數核函數中的虛部。基於此不含奇異元的輔助系統，奇異與超奇異積分將不會出現。然而，卻會導致假的特徵解。配合對偶架構後，真假特徵解可被分辨出來。利用循環對稱矩陣可解析的特性，一個圓形範例，將可解析推導並說明真假特徵解的發生機制。同時，包括真假特徵值、真假特徵邊界狀態、真假內域模態與真假重根數均有所探討。另Hutchinson由模態鑑別真假解的技巧可行性，在本文亦加以討論。