True and spurious eigensolutions of elliptical membranes by using null-field boundary integral equations

Abstract

In this paper, the true and spurious eigensolutions of elliptical membranes appearing in boundary element method are examined by using the null-field boundary integral equation. To analytically study the eigenproblems with elliptical boundaries, the elliptic coordinates and Mathieu functions are adopted. The fundamental solutions are expanded into the degenerate kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration and they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that if we only apply the real or the imaginary-part kernel to deal with a simply-connected elliptical membrane, spurious eigensolutions may appear. Even though we employ the complex-valued kernel, the spurious eigensolutions also occur in the case of a confocal elliptical annulus. Spurious eigenvalues depend on the geometry of inner boundary and the approach used. These two findings agree with those corresponding to the circular and annular cases, respectively. To verify the findings, the boundary element method is also implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide eigensolutions for comparisons. It is found that good agreement is obtained.本文使用零場邊界積分方程法來探討使用邊界元素法求解橢圓形薄膜特徵值問題時所產生的真假根問題。為了能夠解析橢圓形邊界的特徵值問題，則需採用橢圓座標及Mathieu 函數來分析。將基本解在橢圓座標下展開成退化核，邊界密度則使用特徵函數展開。Jacobian 項會存在於退化核、邊界密度和邊界積分裡，但會互相對消。因此正交關係在邊界積分裡是被保留的。有趣的是我們發現假若只使用實部或虛部核函數處理單連通的橢圓形薄膜，亦會有假根的產生。即使我們使用複數核函數，在共焦點的多連通橢圓薄膜也是有假根的產生。假根的產生是取決於內邊界的幾何形狀和所使用的方法。上述的兩個發現分別與圓形和同心圓環薄膜的結論是相同的。使用邊界元素法與有限元素法的套裝軟體 ABAQUS 所得數值結果亦驗證本文的正確性，且均可得到一致的結果。