本篇論文的重點為在晶格波茲曼法的架構下引入沉浸邊界法去模擬三維複雜幾何形狀的流場，分別對兩個主題做討論，首先驗證晶格波茲曼法的數值模擬準確性，接著討論以沉浸邊界法的觀念模擬複雜幾何形狀流場的能耐。此複雜幾何形狀是由拉格朗日標記點來表示，且外力被施加在拉格朗日標記點上以準確地滿足邊界上指定的速度。接著透過一個適當的離散 delta 函數，將拉格朗日標記點上的外力分配到尤拉網格點上。藉著尤拉網格點上已知的外力，用晶格波茲曼法以計算沉浸在卡氏計算範圍內，流經複雜幾何形狀的流場。目前提出四種方法計算三維的流場來檢驗：(1) 在方管中的完全發展流場，(2) 三維頂蓋拉動的方腔流場，(3) 圓柱置放於方管中非對稱位置的流場，(4) 對三維頂蓋拉動的方腔流場進行平行計算處理。雖然目前方法僅有一階精度，但所有計算出來的結果與解析解或參考解都相當吻合。

The focus of the present thesis is to develop an immersed boundary based scheme to model 3-D complex geometry flows within the lattice Boltzmann method framework. Two key issues are addressed. Firstly is the numerical accuracy of the lattice Boltzmann models, and secondly is the capability of the immersed boundary concept to model complex flows. Here, the complex geometry is represented by Lagrangian markers and forces are exerted at the Lagrangian markers in order to satisfy exactly the prescribed velocity of the boundary. This force at the Lagrangian markers is then distributed to the Eulerian grid by a well-chosen discretized delta function. With the known force field in the Eulerian grid to mimic the boundary, the lattice Boltzmann method is used to compute the flow field where the complex geometry is immersed inside the Cartesian computational domain. The computational accuracy and efficiency of the immersed boundary based lattice Boltzmann method are examined by computing 3-D flows: (1) Fully developed flows in a square duct, (2) 3-D lid-driven cavity flow, (3) Flow over an asymmetrically placed cylinder in a square duct. (4) Parallel Computation of 3-D lid-driven cavity flow. Reasonable agreements with the analytical solution or the benchmark solution are obtained, though the present scheme is shown to be first order accurate.

1. U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56,1505 (1986); S. Wolfram, J. Stat. Phys. 45, 471 (1986).
2. F. J. Higuera, S. Succi, and R. Benzi, Europhys. Lett. 9, 345 (1989); F. J. Higuera and J. Jeme´nez, ibid. 9, 663 (1989).
3. S. Harris, “An Introduction to the Theory of the Boltzmann Equation”, Holt, Rinehart and Winston, New York, (1971).
4. U. Frisch, D. d’Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, “Lattice gas hydrodynamics in two and three dimensions”, Complex Syst. 1, 649 (1987).
5. D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery, “Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics”, Phys. Fluids 6, 1285 (1994).
6. X. He, Q. Zou, L.-S. Luo, M. Dembo, “Analytic solutions and analysis on non-slip boundary condition for the lattice Boltzmann BGK model”, J. Statist. Phys. 87, 115 – 136 (1997)
7. Q. Zou, X. He, “On pressure and velocity boundary conditions for the lattice Boltzmann BGK model”, Phys. Fluids 9, 1591 – 1598 (1997)
8. O. Filippova, D. Hänel, “Grid refinement for lattice-BGK models”, J. Comput. Phys. 147, 219 (1998)
9. R. Mei, L.-S. Luo, W. Shyy, “An Accurate Curved Boundary Treatment in the Lattice Boltzmann Method”, J. Comput. Phys. 155, 307 – 330 (1999)
10. P. Lallemand, L.-S. Luo, “Lattice Boltzmann method for moving boundaries”, J. Comput. Phys. 184, 406 – 421 (2003)
11. M. Bouzidi, M. Firdaouss, P. Lallemand, “Momentum transfer of a lattice-Boltzmann fluid with boundaries”, Phys. Fluids 13, 3452 – 3459 (2002)
12. R. Mei, W. Shyy, D.Yu, L. S. Luo, “Lattice Boltzmann method for 3-D flows with curved boundary”, J. Comput. Phys. 161, 680 – 699 (2000)
13. J. M. Buick and C. A. Greated, “Gravity in a lattice Boltzmann model”, Physical Review E, 61, NUMBER 5, (MAY 2000)
14. Li-Shi Luo, “Theory of the lattice Boltzmann method: Lattice Boltzmann models for nonideal gases”, Physical Review E, 62, NUMBER 4, (OCTOBER 2000)
15. Dieter A. Wolf-Gladrow, “Lattice-Gas Cellular Automata and Lattice Boltzmann Models - An Introduction”, Springer, Berlin, Lecture Notes in Mathematics (2000)
16. C. S. Peskin, “Flow patterns around heart valves: a numerical method”, J. Comp. Phys. 10, 252 (1972)
17. M.-C. Lai and C. S. Peskin, “An immersed boundary method with formal second order accuracy and reduced numerical viscosity”, J. Comp. Phys. 160, 705 (2000)
18. N. Satofuka, T. Nishioka, “Parallelization of lattice Boltzmann method for incompressible flow computations. ”, Comp. mech. 23, 164 – 171 (1999)
19. Tamas I. Gombosi, “Gaskinetic Theory”, Cambridge University Press, (1994)
20. X. He, L.-S. Luo, “Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation”, Phys. Rev. E 56, 6811 – 6817 (1997).
21. R. S. Maier, R. S. Bermard, and D. W. Grunau, “Boundary condition for the lattice Boltzmann method”, Phys. Fluid. 8, 1788 (1996)
22. Shen-Wei Su, Ming-Chih Lai, Chao-An Lin, “A simple immersed boundary technique for simulating complex flows with rigid boundary”, submitted to Computer and Fluids.
23. M. Schafer and S. Turek, in “Flow Simulation with High-Performance Computer II”, edited by E. H. Hirschel, Notes in Numerical Fluid Mechanics (Vieweg, Braunschweig, 1996), 52, 547-566
24. K. Y. Hu and C. A. Lin, Lattice Boltzmann method for complex and moving boundaries, master thesis, National Tsing Hua University, Taiwan, (2003).
25. D. J. Chen and C. A. Lin, Immersed boundary method based LBM to simulate complex geometry, master thesis, National Tsing Hua University, Taiwan, (2003).