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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/30886

Title: 二元樹調適網格結合空間時間最小平方有限元素法之應用
Applications of Adaptive Binary Mesh Based on Space-Time Least-Square Finite Element Method
Authors: Chiung-Yang Lan
藍烱揚
Contributors: NTOU:Department of Marine Environmental Informatics
國立臺灣海洋大學:海洋環境資訊學系
Keywords: 二元樹調適網格;空間時間元素;最小平方有限元素法;對流擴散方程式;等寬方程式
bi-tree adaptive mesh;space-time element;least square finite element method;convection diffusion equation;equal width equation
Date: 2012
Issue Date: 2012-04-13T07:55:57Z
Abstract: 本論文以二元樹調適網格為基礎,發展一套空間時間最小有限元素法求解偏微分方程式之演算法。演算法以最小平方有限元素法所提供的殘餘值作為調適的依據。首先,以簡單的移動孤立波驗證二元樹調適網格的可行性,結果顯示調適網格可確實針對高變化量的區域加密網格,同時減少低變化量區域的網格密度。其次,對穩態對流擴散方程式,使用直接離散求解(U-form)以及引入輔助變數離散求解(UW-form)兩種方法,驗證輔助變數的使用與否,對於矩陣條件數與計算結果的影響。結果顯示使用輔助變數以及線性內插函數所得的矩陣條件數隨元素量的成長率較小,約與O((Δx)-1) 成比例,其中Δx代表最小網格尺度。最後,將模式應用於非穩態對流擴散方程式以及等寬方程式之求解,以驗證二元樹調適網格對於動態物理現象的效能,並與最小網格尺度相等之均勻網格及其他計算方法比較,結果顯示本模式可減少更多的網格數量,並維持與均勻網格相近的精準度。
A space-time least-squares finite element method (STLSFEM) with adaptive bi-tree mesh is developed. Distribution of the bi-tree mesh is based on the square of residual of approximations. A translating solitary wave is used to validate the developed algorithm first. It is found that distribution of meshes can adapt to large variation of residual and solution. Therefore, computer resources is saved and high accuracy of approximation is achieved. Various forms of least-squares FEM can be formulated for the steady convection-diffusion equation (CDE), depending on the choice of auxiliary variable (without an auxiliary variable named U-form or with an auxiliary variable named UW-form), interpolation function, and possible treatment of boundary conditions. It reveals that with introducing an auxiliary variable (U-form), condition number increases with number of meshes, i.e. approximate to O((Δx)-1). The algorithm is then applied to unsteady CDE and Equal Width equation (EWE) to illustrate the solution-adaptive feature of the method.
URI: http://ethesys.lib.ntou.edu.tw/cdrfb3/record/#G0M98810008
http://ntour.ntou.edu.tw/handle/987654321/30886
Appears in Collections:[海洋環境資訊系] 博碩士論文

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