兩種迭代最小平方有限元素法求解不可壓縮那維爾-史托克方程組之研究

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王韻詞(Yun-Tsz Wang) 查詢紙本館藏

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論文名稱

兩種迭代最小平方有限元素法求解不可壓縮那維爾-史托克方程組之研究
(On Two Iterative Least-Squares Finite Element Schemes for Solving the Incompressible Navier-Stokes Equations)

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摘要(中)

本篇論文主要研究在均勻網格中使用兩種迭代最小平方有限元素法求解具速度邊界值的穩態不可壓縮那維爾-史托克方程組。引入旋度當作新的未知變數，那維爾-史托克問題可改寫成一個擬線性速度-旋度-壓力一階方程組。我們提出兩種皮卡型迭代最小平方有限元素法求取此非線性一階問題的數值近似解，在每次迭代過程中使用一般的L2最小平方法或加權L2最小平方法去求解其相對應的歐辛問題。我們主要專注於此兩種迭代最小平方法在均勻網格上使用連續片狀多項式有限元素求解二維模型問題。數值實驗證明，對於具有平滑正確解的相同問題，在雷諾數較小的時候，L2最小平方解比加權L2最小平方解更準確；然而當雷諾數相對較大時，加權L2最小平方近似解似乎比L2最小平方近似解更好。最後，我們報告凹槽驅動流場的數值結果以驗證此類迭代最小平方有限元素法之效力。

摘要(英)

This thesis is devoted to a numerical study of two iterative least-squares finite element schemes on uniform meshes for solving the stationary incompressible Navier-Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, the Navier-Stokes problem can be recast as a first-order quasilinear velocity-vorticity-pressure system. Two Picard-type iterative least-squares finite element schemes are proposed for approximating the solution to the nonlinear first-order problem. In each iteration, we apply the usual L2 least-squares scheme or a weighted L2 least-squares scheme to solve the corresponding Oseen problem. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that, for the same test problem with smooth exact solution, the L2 least-squares solutions are more accurate than the weighted L2 least-squares solutions for low Reynolds number flows, while for flows with relatively higher Reynolds numbers the weighted L2 least-squares approximations seem to be better than the L2 least-squares approximations. Finally, numerical results for driven cavity flows are also given to demonstrate the effectiveness of the iterative least-squares finite element approach.

關鍵字(中)

★ 有限元素法
★ 那維爾-史托克方程組
★ 迭代法
★ 最小平方
★ 凹槽驅動流場
★ 歐辛方程組

關鍵字(英)

★ driven cavity flows
★ Oseen equations
★ least squares
★ finite element methods
★ iterative methods
★ Navier-Stokes equations

論文目次

中文摘要 ……………………………………………………………… i
英文摘要 …………………………………………………………… ii
目錄 ………………………………………………………………… iii
Abstract ……………………………………………………………… 1
1. Problem formulation …………………………………………… 2
2. Least-squares finite element schemes …………………… 6
3. Analysis of the least-squares finite element schemes for the Oseen problem …………………………………………… 10
4. Numerical experiments ……………………………………… 13
5. Numerical results of driven cavity flows ……………… 17
6. Conclusions …………………………………………………… 23
References ………………………………………………………… 24

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指導教授

楊肅煜(Suh-Yuh Yang)

審核日期

2008-5-18

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