Null-field integral equation approach for Helmholtz (interior and exterior acoustic) problems with circular boundaries

Abstract

The Helmholtz (interior and exterior acoustics) problems with circular boundaries are studied by using the null-field integral equations in conjunction with degenerate kernels and Fourier series to avoid calculating the Cauchy and Hadamard principal values. Adaptive observer system of polar coordinate is considered to fully employ the property of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradient of potential is carefully considered. In interior acoustic problems, direct-searching scheme is employed to detect the eigenvalues by using the singular value decomposition (SVD) technique. Two approaches to overcome spurious eigenvalus, SVD updating technique and Burton & Miller methods are employed to suppress the appearance of spurious eigenvalue. Several examples are demonstrated to see the validity of the present formulation and numerical results indicate the better accuracy than BEM in predicting the spurious eigenvalues. In exterior acoustic problems, the radiation and scattering problems with multiple circular cylinders are also examined successfully. 本文使用零場積分方程，搭配退化核(分離核)及傅立葉級數來解決含圓形邊界的赫姆茲(內外域聲)問題以避免柯西及哈達馬主值的計算。自適性觀察的極座標系統可充分展現退化核(分離核)的特性。而在使用超奇異式時，對勢能場的法線及切線方向的向量分解均必須小心處理。在內域聲場問題，直接使用奇異值分解技巧，可求得特徵值與特徵模態。為了克服假根問題，我們使用 SVD 補充行與補充列法 及 Burton & Miller 法可克服假根產生的問題。幾個例子驗證了本方法的正確性。數值結果顯示，本方法對於假根所預測座落的位置比邊界元素法更為精確。在外域聲場問題，含多圓柱的輻射及散射問題都已測試成功。