本文使用晶格波茲曼法（Lattice Boltzmann Method，LBM），利用其介觀尺度的模擬方法，模擬高分子成型的過程中，長型微流道內的速度場與壓力分佈的情形，本文主要以原先的正方格為基礎，將計算方式透過一階內插以及二階內插之方式發展矩形網格的模擬方式，應用到具有大的長高比之長型流道模擬中；透過調整網格的內插計算方式，在不提高模擬的結果誤差的前提下，利用格點的合理分佈，大幅減少模擬所耗時間，使模擬結果達到時間短且不失準度的優勢。
文中，主要以插值法，改變分佈函數的位置，從而達到建立矩形網格的效果，透過內插的方式，能夠續用方格之分佈函數，巧妙避開修改分佈函數的限制，避免不符合等向性的缺點，也使程式撰寫以及模擬過程有較高的簡便性。並在此基礎下，應用矩形格點到高分子成型的研究中，改善大的長高比之長型流道的模擬問題中，並提供讀者如何內插正方形網格的分佈函數，以建立對應的矩行網格模型。本研究也將同時分析不同內插方式下，對模擬出的速度以及壓力之準確度的影響。
在矩形網格的內插算法中，每個碰撞步驟都以與在正方形網格中相同的方式進行，在碰撞結束時，可以使用線性或二次插值法來計算矩形網格上的分佈函數。由於正方形和矩形網格的格點對位問題，本文中將說明建立矩形網格的修改條件有哪些，包括調整入口速度邊界條件以及模擬完畢後的壓力分佈處理方式。為了簡化計算過程，我們只用插值法來擴展水平網格的長度。加入一個新的參數「a」(aspect ratio)來控制不同網格的長高比；不直接改變縱向的網格長度，除了能夠使程式撰寫較容易，也使模擬精度更加穩定。
本研究的方法旨在不提高模擬誤差的前提下，提升模擬的速度，讀者可以利用本文所提到的方式選擇出最適合的網格長高比大小「a」(aspect ratio)值，從而改善過往模擬耗時及長管流中誤差較大的缺點，因此，網格長高比大小「a」(aspect ratio)的最佳值並沒有一定的答案，端看讀者模擬的物理模型以及準確度的要求來決定。

Lattice Boltzmann method is widely used on fluid flow simulations. Most of simulations were based on uniform lattices, e.g., square grid or cubic lattices. Additionally, rectangular and orthorhombic lattices are also suitable for several simulation cases. In this research, we extend the lattice from square grids to rectangular grids using both the linear and quadratic interpolation methods. In this algorithm, every collision step takes place in the same way as in the square grids. In the end of the collision, linear or quadratic interpolation method can be used to estimate the distribution functions on the rectangular grid. Because of the matching problem between square and rectangular scheme, inlet boundary condition should be adjusted for the rectangular grids. In order to simplify the computational process, we only extend the calculations with interpolation method in the horizontal grid. It keeps the numerical stability on vertical grid direction. The methodology about how to adopt the square grid distribution function to establish the corresponding rectangular grid model was demonstrated. With this algorithm, we can efficiently perform the flow simulation in a flow field that has a large length to height aspect ratio.

[1] J. Hardy, O. De Pazzis, and Y. Pomeau, "Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions," Physical review A, vol. 13, p. 1949, 1976.
[2] U. Frisch, B. Hasslacher, and Y. Pomeau, "Lattice-gas automata for the Navier-Stokes equation," Physical review letters, vol. 56, p. 1505, 1986.
[3] G. R. McNamara and G. Zanetti, "Use of the Boltzmann equation to simulate lattice-gas automata," Physical review letters, vol. 61, p. 2332, 1988.
[4] H. Chen, S. Chen, and W. H. Matthaeus, "Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method," Physical Review A, vol. 45, p. R5339, 1992.
[5] Y. Qian, D. d'Humières, and P. Lallemand, "Lattice BGK models for Navier-Stokes equation," EPL (Europhysics Letters), vol. 17, p. 479, 1992.
[6] P. L. Bhatnagar, E. P. Gross, and M. Krook, "A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems," Physical review, vol. 94, p. 511, 1954.
[7] D. P. Ziegler, "Boundary conditions for lattice Boltzmann simulations," Journal of Statistical Physics, vol. 71, pp. 1171-1177, 1993.
[8] T. Inamuro, M. Yoshino, and F. Ogino, "A non‐slip boundary condition for lattice Boltzmann simulations," Physics of Fluids, vol. 7, pp. 2928-2930, 1995.
[9] R. S. Maier, R. S. Bernard, and D. W. Grunau, "Boundary conditions for the lattice Boltzmann method," Physics of Fluids, vol. 8, pp. 1788-1801, 1996.
[10] Q. Zou and X. He, "On pressure and velocity boundary conditions for the lattice Boltzmann BGK model," Physics of fluids, vol. 9, pp. 1591-1598, 1997.
[11] R. Mei, L.-S. Luo, and W. Shyy, "An accurate curved boundary treatment in the lattice Boltzmann method," Journal of computational physics, vol. 155, pp. 307-330, 1999.
[12] P. Lallemand and L.-S. Luo, "Lattice Boltzmann method for moving boundaries," Journal of Computational Physics, vol. 184, pp. 406-421, 2003.
[13] S.-C. Chang, Y.-S. Hsu, and C.-L. Chen, "Lattice Boltzmann simulation of fluid flows with fractal geometry: An unknown-index algorithm," J. Chin. Soc. Mech. Eng, vol. 32, pp. 523-531, 2011.
[14] X. He, L.-S. Luo, and M. Dembo, "Some progress in lattice Boltzmann method. Part I. Nonuniform mesh grids," Journal of computational Physics, vol. 129, pp. 357-363, 1996.
[15] M. h. Bouzidi, D. d'Humières, P. Lallemand, and L.-S. Luo, "Lattice Boltzmann equation on a two-dimensional rectangular grid," Journal of Computational Physics, vol. 172, pp. 704-717, 2001.
[16] X. Nie, X. Shan, and H. Chen, "Galilean invariance of lattice Boltzmann models," EPL (Europhysics Letters), vol. 81, p. 34005, 2008.
[17] J. G. Zhou, "Rectangular lattice Boltzmann method," Physical Review E, vol. 81, p. 026705, 2010.
[18] S. Chikatamarla and I. Karlin, "Comment on “Rectangular lattice Boltzmann method”," Physical Review E, vol. 83, p. 048701, 2011.
[19] L. A. Hegele Jr, K. Mattila, and P. C. Philippi, "Rectangular lattice-Boltzmann schemes with BGK-collision operator," Journal of Scientific Computing, vol. 56, pp. 230-242, 2013.
[20] Y. Zong, C. Peng, Z. Guo, and L.-P. Wang, "Designing correct fluid hydrodynamics on a rectangular grid using MRT lattice Boltzmann approach," Computers & Mathematics with Applications, vol. 72, pp. 288-310, 2016.
[21] C. Peng, H. Min, Z. Guo, and L.-P. Wang, "A hydrodynamically-consistent MRT lattice Boltzmann model on a 2D rectangular grid," Journal of Computational Physics, vol. 326, pp. 893-912, 2016.
[22] C. Peng, Z. Guo, and L.-P. Wang, "A lattice-BGK model for the Navier–Stokes equations based on a rectangular grid," Computers & Mathematics with Applications, 2016.
[23] M. Sukop, "DT Thorne, Jr. Lattice Boltzmann Modeling Lattice Boltzmann Modeling," 2006.
[24] H. Huang, M. Sukop, and X. Lu, Multiphase lattice Boltzmann methods: Theory and application: John Wiley & Sons, 2015.
[25] A. A. Mohamad, Lattice Boltzmann method: fundamentals and engineering applications with computer codes: Springer Science & Business Media, 2011.
[26] D. d’Humieres and P. Lallemand, "Numerical simulations of hydrodynamics with lattice gas automata in two dimensions," Complex Systems, vol. 1, p. 599, 1987.
[27] T. Krüger, "Unit conversion in LBM," in LBM Workshop 2011.
[28] B. He, W.-B. Feng, W. Zhang, and Y.-M. Cheng, "Parallel Simulation of Compressible Fluid Dynamics Using Lattice Boltzmann Method," in The first International Symposium on Optimization and System Biology (ISM'07), 2007, pp. 451-58.
[29] E. Waring, "Problems concerning Interpolations. By Edward Waring, MDFRS and of the Institute of Bononia, Lucasian Professor of Mathematics in the University of Cambridge," Philosophical transactions of the royal society of London, vol. 69, pp. 59-67, 1779.
[30] 林晃業, "微奈米結構射出成型之研究," 2010.
[31] 林百會, "微射出成型微結構之尺寸變形研究," 成功大學航空太空工程學系學位論文, pp. 1-74, 2013.
[32] 楊爲凱, "應用晶格波茲曼法模擬樹脂滲透纖維之流動行為," 成功大學航空太空工程學系學位論文, pp. 1-98, 2017.
[33] 陳螢萱, "應用晶格波茲曼法於微射出成型之研究," 成功大學航空太空工程學系學位論文, pp. 1-105, 2017.
[34] 郭照立 and 鄭楚光, "格子 Boltzmann 方法的原理及應用," ed: 北京: 科學出版社, 2008.
[35] 陳維霖, "應用晶格波茲曼法結合大渦法模擬紊流強制對流熱傳問題," 成功大學機械工程學系學位論文, pp. 1-131, 2012.
[36] 李建鋐, "微流道射出成型充模研究," 成功大學航空太空工程學系學位論文, pp. 1-94, 2007.
[37] 李宏偉, "晶格波茲曼法在微流體之應用," 成功大學工程科學系學位論文, pp. 1-62, 2009.