Analytical study and numerical experiments for spurious eigensolutions of interior problem and the fictitious wave number of exterior acoustic problem using BEM

Abstract

In this report the reason why the spurious solution occurs in the interior eigenproblem using real-part BEM and why the fictitious solution occurs in numerical computations of the exterior Helmholtz integral equation at certain characteristic frequencies is investigated. It was recently found that the real-part BEM for the interior problem results in spurious eigensolutions. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation results in fictitious solutions for the exterior problem. Both the two problems stem from the rank deficiency of the influence matrix. By using the circulant properties and degenerate kernels, an analytical scheme in a discrete system of a circular case is achieved. Numerical experiments are found to agree with the analytical results. 本研究報告在於討論以積分方程求解內域或外域Helmholtz 場在數值上產生的真假特徵值及虛擬頻率的問題。在解外域聲場時於某些特徵頻率會得到不唯一的解。而在求解內域特徵值時如果僅採用實部邊界元素法的奇異積分方程式，則會得到假特徵值。而內域實數邊界元素法得到假根與外域問題得到虛擬頻率的原因是類似的。兩個問題的產生皆由於影響矩陣的秩數不足所致。藉由圓形循環矩陣的特性及退化核函數，可得知為何採用實部邊界元素法會產生假根是因為少了虛數部的限制而產生了假根。而虛擬頻率(或波數)的產生則是數學上 0/0 的問題。在解析上我們以二維圓形問題來證明,並以設計的數值實驗結果來驗證我們理論的正確性。