Eigensolution of annular membrane using the method of fundamental solutions

Abstract

In this paper, the method of fundamental solutions (MFS) for solving the eigenfrequencies of annular membrane is proposed. By employing the fundamental solution, the coefficients of influence matrices are easily determined. The spurious eigensolution in conjunction with the true eigensolution appears. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the sources are distributed. To verify this finding, the true and spurious eigenvalues in an annular domain are analytically studied using the degenerate kernel and circulant. In order to obtain the true eigensolution, the singular value decomposition (SVD) updating technique and the Burton & Miller method are utilized to filter out the spurious eigensolutions. One example is demonstrated analytically and numerically to see the validity of the present method.本文主要是用基本解法求解同心圓薄膜之特徵值問題。利用基本解法，我們可以容易地求得影響係數矩陣。在求解的過程中，假根常常伴隨著真根出現，我們也發現到假根產生的機制是受到內部源點分佈位置的影響。為了證明這個發現，我們將以一個同心圓薄膜為例進行解析探討及數值結果的比較，其中我們將會使用退化核及循環矩陣這兩個數學的工具來進行分析。為了獲得真根，採用奇異值補充行及 Burton & Miller 法這兩個技巧來濾除假根。文中以一個例子來論證我們這個方法的正確性。