Analytical Study and Numerical Experiments for Spurious Eigensolutions of Interior Problem and Fictitious Wave Numbers of Exterior Problem Using BEM

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Project title

Code/計畫編號

NSC89-2211-E019-003

Translated Name/計畫中文名

邊界元素法求解內域假根與外域虛擬波數的理論探討與數值研究

Project Coordinator/計畫主持人

Jeng-Tzong Chen

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=527750

Year

2000

Start date/計畫起

01-08-1999

Expected Completion/計畫迄

31-07-2000

Bugetid/研究經費

252千元

ResearchField/研究領域

土木水利工程
數學
機械工程

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。In this report the reason why the spurious solution occurs in the interior eigenproblem using realpart 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. Based on 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.

Keyword(s)

假特徵值
實部邊界元素法
虛擬頻率
圓形循環矩陣
退化核函數
spurious eigenvalue
real part BEM
fictitious frequency
circulant
degenerate kernels

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Analytical Study and Numerical Experiments for Spurious Eigensolutions of Interior Problem and Fictitious Wave Numbers of Exterior Problem Using BEM

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