Applications of dual boundary integral equations to exterior acoustic problems

Abstract

This paper presents the mechanism why the irregular frequencies are imbedded in the exterior acoustics using the dual BEM The relation between the matrices of in uence coeffi cients for the interior and exterior acoustic problems is examined Also the irregular (fictitious) frequencies embedded in the singular or hypersingular integral equations are discussed respec tively It is found that the irregular values depend on the kernels in the integral representation for the solution A two- dimensional dual BEM program for the exterior acoustic problems was developed Numerical experiments using the program are conducted to check the validity in comparison with the theoretical proof of the independence of boundary conditions which have been shown by Chen using the degenerate kernels Both the radiation and scattering problems are considered Two cases including the Dirichlet and Neumann boundary conditions show that the singular integral equation results in the fi ctitious frequencies which are associated with the eigenfrequencies of the interior Dirichlet problem while the hypersingular integral equation results in the fictitious frequencies which are associated with the interior Neumann problem Burton and Miller approach is employed to deal with the problem of fictitious frequencies. 本文探討對偶邊界元素法之虛擬頻率產生的機制，並建立聲場上內域問題及外域問題中影響係數矩陣之間的關係。同時個別討論了隱藏於奇異積分方程及超強奇異積分方程中的不規則(或稱之虛擬)頻率，並發現不規則頻率的產生與積分方程表示式中的核函數有關。而陳以可分解核函數從理論及數值實驗上比較並證明不規則值的產生與邊界條件無關。本研究以外域聲場輻射與散射問題為例，並考慮Dirichlet及Neumann邊界條件得到當以奇異積分方程求解時產生的虛擬特徵值會對應到內域的Dirichlet問題之特徵值，當以超強奇異積分方程求解時產生的虛擬特徵值會對應到內域的Neumann問題的特徵值，而與邊界條件無關。為了克服此數值問題，本文採用Burton與Miller法予以克服虛擬頻率。