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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/24479

Title: Trefftz直接法與間接法在修正後荷姆霍茲方程式問題之研究
A Study of the Direct and Indirect Trefftz Methods for Solving Problems of the Modified Helmholtz 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;修正後荷姆霍茲方程式;數值劣化
Date: 2005-08
Issue Date: 2011-10-20T08:12:27Z
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 直接法與間接法來求解修正後荷姆 霍茲方程式問題。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,擬先採用Tikhonov 正規化法結合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,對求解修正後荷姆霍茲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允許原點可以移動而得以降低基底階數之移動式Trefftz 法來解決。最後,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。
Abstract:In this report, a direct and an indirect Trefftz boundary-type methods for solving problems of the modified Helmholtz equation is proposed. It is found that the ill-posed nature, i.e., the numerical instability, exists in the direct Trefftz method when the boundary element number increases. In the contrast, when the indirect Trefftz method is adopted, the numerical instability still exists; however, the indirect Trefftz method can effectively deal with a multiply-connected domain of genus 1. To overcome the numerical instability existing in both the direct and indirect Trefftz methods, the singular-value decomposition and the Tikhonov’s regularization method are suggested. In particular, putting source point in or out of the domain and how to select suitable bases functions to treat simply and multiply connect domain are also critically discussed and analyzed. Finally, several numerical examples are included to show the validity of the proposed approach.
Relation: NSC94-2611-E019-020
URI: http://ntour.ntou.edu.tw/handle/987654321/24479
Appears in Collections:[系統工程暨造船學系] 研究計畫

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