A new method for eigenproblems with circular boundaries

Abstract

In this paper, the eigenproblems with circular boundaries are studied by using null-field integral equations in conjunction with degenerate kernels and Fourier series. Direct-searching scheme is employed to detect the eigenvalues by using singular value decomposition (SVD) technique. It is analytically verified that an inner circle results in the spurious eigenvalue and it appears in the numerical experiment. Also, the spurious eigenequation due to the inner circle is examined. Several examples are demonstrated to see the validity of the present formulation. 本文以勢能理論為基礎，提出以退化核與傅立葉級數展開搭配零場積分方程求解含多孔洞二維特徵值問題，此方法可視為半解析法。邊界未知勢能與流通量使用有限項傅立葉級數來近似求得。利用退化核與傅立葉展開可導得一線性代數方法 而無須對邊界離散。再靠奇異值分解法來求得特徵值及傅立葉係數。文中以幾個不同邊界條件的特徵值問題進行測試。所得結果無論與邊界元素法的數值結果或有限元素法的數值結果比較，均可驗證本方法的正確性。