# 臺灣博碩士論文加值系統

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 This thesis is an application of finite element analysis, the goal is to discuss the influence of the mesh, and build a computer model for Saybolt viscosity measurement. At first, we will introduce the method to measure the Saybolt viscosity and the transform between the Saybolt viscosity, kinematic viscosity, and absolute viscosity. And we will point out the model we build, and the difference between our model and real condition. In this paper, we use 2-D incompressible steady-state Navier-Stokes equations as the base function to solve our problem. Secondly, will discuss the method we used — the penalty method. Besides, introduce some basic theory of finite element method, and use some tips to simplify the problem we met. And then, an introduction is given on conjugate gradient method we use to iterate the variation of system. Finally, we will discuss some fluid problems and the advantage or disadvantage present in our computer model, and compare with the real condition.
 中文摘要 ……………………………………………………………i 英文摘要 ……………………………………………………………ii 誌謝 …………………………………………………………………iii 目錄 …………………………………………………………………iv 表目錄 ………………………………………………………………vi 圖目錄 ………………………………………………………………vii 符號說明 ……………………………………………………………ix 一、緒論 ……………………………………………………………1 二、文獻回顧 ………………………………………………………3 三、賽氏通用黏度計 ………………………………………………9 3.1 測量裝置與方法 …………………………………………9 3.2 賽氏黏度與絕對黏度之轉換 ……………………………10 3.3 電腦模型與實際情況之關係 ……………………………11 四、有限單元數值計算與納維爾─史托克方程 …………………13 4.1 Babuska-Brezzi Condition ………………………………13 4.2 推導 ……………………………………………………13 4.3 線性三角單元 …………………………………………18 4.4 對流加速項的處理 ……………………………………19 4.5 有限單元的組合 ………………………………………20 4.6 Conjugate Gradient Method ……………………………23 五、程式之建構與計算 ……………………………………………27 5.1 產生網格及點資料 ……………………………………27 5.2 代入邊界條件 …………………………………………29 5.3 計算系統驅動力及勁度矩陣 …………………………29 5.4 代入Conjugate Gradient法計算速度變量 ……………30 5.5 將速度代回對流加速項作計算 ………………………30 六、實際範例與計算結果 …………………………………………31 6.1 Lid-Driven 2D Cavity Flow ……………………………31 6.2 Step Flow ………………………………………………35 6.3 Saybolt Viscosity Measurement …………………………40 七、結論 ……………………………………………………………44 參考文獻 ……………………………………………………………46 附錄 …………………………………………………………………48
 [1] Dudley D. Fuller, “Theory and Practice of Lubrication for Engineers”, 2nd Edition, John Wiley and Sons, 1984.[2] Robert D. Cook, David S. Malkus, Michael E. Plesha, “Concepts and Applications of Finite Element Analysis”, 3rd Edition, John Wiley and Sons, 1989.[3] Richard H. Gallagher, “Finite Element Analysis, Fundamentals”, Prentice-Hall, Inc., 1975.[4] O. C. Zienkiewicz, FRS, R. L. Taylor, “The Finite Element Method”, 4th Edition, Vol.1, McGraw-Hill, 1988.[5] I. Babuska, “The Finite Element Method with Penalty”, Tech. Note BN-710, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, August 1971.[6] O. C. Zienkiewicz, “Constrained Variational Principles and Penalty Function Methods in Finite Element Analysis”, Lecture Notes in Mathematics : Conference on the Numerical Solution of Differential Equation, Edited by G. A. Watson, Springer-Verlag, Berlin, 207-214, 1974.[7] J. N. Reddy, “Penalty Finite Element Analysis of 3-D Navier-Stokes Equations”, Computer Methods in Applied Mechanics and Engineering, 35, 87-106, 1982.[8] Ralston Anthony, “A First Course in Numerical Analysis”, 1969.[9] Alexander N. Brooks, J. R. Hughes, ”Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations”, Computer Methods in Applied Mechanics and Engineering, 32, 199-259, 1982.[10] Alastair Cameron, “Basic Lubrication Theory”, 3rd Edition, Ellis Horwood, 1976.[11] P. A. B. De Sampaio, “A Petrov-Galerkin Formulation for the Incompressible Navier-Stokes Equations Using Equal Order Interpolation for Velocity and Pressure”, International Journal for Numerical Methods in Engineering, vol
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