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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/24563

Title: 間接式Trefftz法求解Poisson方程式問題
Indirect Trefftz method for solving problem of Poisson equation
Authors: 張建仁
Contributors: NTOU:Department of Systems Engineering and Naval Architecture
國立臺灣海洋大學:系統工程暨造船學系
Keywords: Trefftz 法;基底函數;Poisson 方程式;數值劣化;多連通領域
Date: 2006
Issue Date: 2011-10-20T08:12:45Z
Publisher: 國科會計畫報告
Abstract: 摘要:在本研究計畫中,吾人擬推導出Trefftz 間接法來求解Poisson 方程式問題。 欲求解Poisson 方程式有兩個步驟:首先必須找出特解,然後再求解剩餘的 Laplace 方程式。吾人將採用徑向基底函數法來尋求Poisson 方程式的特解,然 後兩種Trefftz 法,即T-Trefftz 法與F-Trefftz 法再引入來求解剩餘的Laplace 方程 式。一如先前之研究結論,使用無奇異源正規型邊界元素法如傳統之Trefftz 法, 用於求解Laplace 方程式問題時,雖然沒有奇異積分之問題但因需要更高階之基 底函數,卻碰到數值劣化行為的影響。雖有其它學者提初以降階做為改善此種病 態問題之方式,卻未能有效改善。在本研究中,擬採用Truncated 奇異值分解法 結合L 曲線技巧來解決數值污染之現象,同時提出一套自動搜尋機制來克服前 法在求解過程中需仰賴人工觀察與判斷進之缺點,進而求得真實解。此外,在本 研究中,針對Trefftz 法所定義之原點落在領域內之幾何中心或偏離中心或領域 之外,對求解Poisson 方程式問題是否會出現相同之劣化行為,也將一併深入解 析。而針對多孔洞之多連通領域所面臨需採用高階基底函數之問題,本研究也將 提出一套有效的方法來解決。同時,對於Trefftz 原點落在領域內與領域外該如 何採用適當之之基底函數,也會設計數值算例,有深入之探討與解析。最後,我 們將針對上述諸多問題所提之解決方法,進一部設計出許多有趣的數值算例來加 以說明與驗證,期盼對一系列有關Trefftz 法求解Poisson 方程式之研究,提出 有系統的研究結論,為往後之應用性研究建構出一個雙向溝通橋樑。
Abstract:In this research, the indirect Trefftz method has been developed to deal with problems of the Poisson equation. The Poisson equation can be solved by the beginning of finding a particular solution and then to solve the resulting Laplace Equation. The radial basis function approach is then used to find an approximate particular solution for the Poisson equation and then, two kinds of Trefftz methods, the T-Trefftz method and F-Trefftz one, are adopted to solve the resulting Laplace equation. As previous research conclusions, when boundary element methods (BEMs) of the regular type, such as various Trefftz methods, are applied to solve the Laplace equation problems, the ill-posed behavior owing to necessary uses of higher order basis functions in formulations is encountered although such a regular type BEM has no problems of singular integrals. Even some researchers had proposed the concept of reducing orders of basis functions as a remedy to overcome the ill-posed problem; however, an effective numerical scheme to fix the problem is still not available. In this study, the truncated singular value decomposition method in conjunction with the L-curve technique is adopted to cope with the numerical instability and an automatically numerical searching technique is developed to improve the shortcomings of the L-curve technique, in which an artificial judgment is still necessary for determining the true values. Besides, aimed at the Trefftz origin locating inside or outside the physical domain for solving the Poisson equation, the question whether the numerical instability is encountered will be answered and further studied in detail. Moreover, for a multiply connected domain problem of uses of higher order basis functions, a systematic discussion with a remedy is also proposed. Meanwhile, how to choose the appropriate basis function for the location of the Trefftz origin is discussed. Several designed numerical examples are provided for validation of the proposed approach.
Relation: NSC95-2221-E019-076
URI: http://ntour.ntou.edu.tw/handle/987654321/24563
Appears in Collections:[系統工程暨造船學系] 研究計畫

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