調適性最小平方/間斷 Galerkin 方法在黏彈性流體問題上的應用

Adaptive Least-squares/Discontinuous Galerkin Approximations To Viscoelastic Flow Problems

劉家禎(Liu, Chiachen)

指導教授 ： 陳慈芬

國立中正大學/理學院/數學系應用數學研究所/博士(2012年)

摘要

本論文探討結合加權最小平方法與間斷 Galerkin 方法解黏彈性流體的問題。我們使用數值疊代法，將方程組中的本構方程式與動量及連續方程式分別求解，也就是以加權最小平方法解 Stokes 問題，再以間斷 Galerkin 方法求本構方程式之解。在論文中將推導出此方法之事前估計，並且提供數值結果以印證我們所推導的理論。
探討的模型是平面流體問題與四對一收縮流問題。主要的工作包括利用網格重新分布建構最佳網格來解決高梯度區域、轉角奇異點與窩流的形成等問題。我們將呈現牛頓流體與 Oldroyd-B 流體的數值結果，也會介紹平面流體建構最佳網格的演算法。為了解決四對一收縮區域的轉角奇異點問題，此演算法所建構的網格不只沿著高梯度區域加密也沿著流線增加網格。此外，對於四對一收縮流問題，藉著在壁面增加網格，將會形成平滑的窩流。最後將探討結合加權最小平方法與 Galerkin 最小平方有限元法的應用。

關鍵字

調適性網格 ； 間斷 Galerkin 法 ； 最小平方法 ； 黏彈性流體

並列摘要

This work concerns solutions of viscoelastic flow problems by a combination of the weighted least squares (WLS) method and the discontinuous Galerkin (DG) finite element method. The constitutive equation is decoupled from the momentum and continuity equations and the approximate solution is computed using iterations; alternatively, the Stokes problem is solved by WLS and a linearized constitutive
equation by DG. An a priori estimate for the WLS/DG method is derived and numerical results supporting the estimate are presented.
Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. One major purpose of this work is to construct optimal adaptive mesh redistribution algorithm to resolve the high gradient region, the corner singularities and formation of vortices in the model problem. Numerical results of both Newtonian (Stokes) and Oldroyd-B model equations are presented.
Optimal grid construction of the algorithm for the planar flow problem is illustrated. To resolve the corner singularity in the 4-to-1 contraction domain,
the mesh generated by the algorithm not only refines the mesh near the high gradient region but also along the flow lines. In addition, for a 4-to-1 contraction problem, a smooth vortex is formed by suitable mesh refinement near the wall boundary and the domain. A combination of the weighted least-squares (WLS) method and the Galerkin least-squares(GLS) finite element method for the model problems are also investigated.

並列關鍵字

Adaptive mesh ； Discontinuous Galerkin method ； Least-squares method ； Viscoelastic flow

參考文獻

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