邊界元素法中退化問題的統一推導

Abstract

本文提出邊界元素法中有關退化問題的統一觀點，包含了退化邊界、退化尺度、假根、虛擬頻率及角點問題。對於這些退化問題我們可以歸納均源自於影響係數矩陣的秩降現象，為探討這些解不唯一問題採用 Fredholm 二擇一定理以及奇異值分解法的技巧來處理，而 SVD 補充行與補充列技巧亦將同時被考慮。邊界元素法所建構的影響係數矩陣在經過奇異值分解後，產生之左、右酉矩陣與真假根、虛擬頻率與虛擬模態之相關性，以 Fredholm 二擇一定理來檢驗。我們成功地舉數個例題來驗證統一推導的可行性。吳大猷獎助計畫的第一年計畫針對退化邊界以及退化尺度的問題，第二年則著重於單連通及多連通問題的真假特徵值問題。最後，虛擬頻率的問題將於第三年來探討。本文將研究成果所發表論文作一彙整。 We provide a perspective on the degenerate problems, including degenerate boundary, degenerate scale, spurious eigensolution, fictitious frequency and corner problems, in the boundary integral formulation. All the degenerate problems are found to originate from the rank deficiency in the influence matrix. Both the Fredholm alternative theorem and singular value decomposition (SVD) technique are employed to study the degenerate problems. Updating terms and updating documents of the SVD technique are utilized. The roles of right and left unitary matrices of SVD in BEM and their relations to true, spurious and fictitious modes are examined by using the Fredholm alternative theorem. A unified method for dealing with the degenerate problems in BEM is proposed. Several examples are given to show the validity of the unified formulation. In the first year, of Wu Ta-You Award Project, degenerate boundary and degenerate scale problems were studied. For the second year, we focused on the spurious eigenvalues for both simply-connected and multiply-connected problems. Finally, the fictitious frequencies were revisited in the third year. The published papers are summarized in the references.