利用區域最大熵值有限元素法求解流場之研究

Abstract

本論文的主旨在於對區域最大熵值有限元素法(local maximum-entropy finite element method)的利用與研究，先將方法用於二維不同Peclet數的穩態的對流擴散問題計算，確認其可行性，接著為了進一步確認其改善，我們把此方法用於奈維爾-史托克斯方程式同時利用運算子拆解法求解，測試各種可能的加密佈點方式，再將之與現有文獻的數據加以比較，確認其因最大熵值有限元素法而提升了精度，最後為了進一步證明此方法能應用於多種幾何場域，再測試了不規則形狀的流場，並以高密度網格之有限元素法求解作為指標，再與使用最大熵值有限元素法之結果加以比較，亦能發現結果與效率有顯著的提升，以證明此方法能夠在複雜幾何流場與高梯度的問題中提升其精度與效率。

This thesis is concerned with the study of local maximum-entropy finite element method (LME-FEM) on flow field problems. On this study the method is first used to solve steady advection-diffusion problems at various Peclet numbers for two-dimensional conditions. After verifying the capability of this method to simulate the advection-diffusion equation, we next apply the scheme to solve Navier-Stokes equations with the operator splitting procedure by a two-step projection method. Testing a variety of refinement method, we tried to demonstrate that the procedure of adding extra points in the elements would increase the accuracy of numerical computation. All the numerical results of this study are compared favorably with the existing reference data. In addition, we further tried to do the same problem of cavity flow but with a hole in the domain. Comparing the results with the mesh independent solution, reasonably good agreements and better efficiency can be observed through present LME-FEM algorithm. It is proved that LME-FEM will increase the efficiency even under the unfavorable conditions of high gradient and complex geometry.