對於晶格波茲曼法在模擬高雷諾數時，均勻網格無法精確的解析出渦流以及局部流場會出現非自然現象的震盪。因此，在物理變化劇烈的區域採用比較密的網格能有效解決上述的問題。在本論文，是在圖形顯示卡上應用晶格波茲曼法計算整個非均勻網格。雖然前後有許多學者已經提出以單鬆弛及多鬆弛的方式加密局部網格。然而，本論文利用二維以及三維拉板驅動空穴來驗證單鬆弛及多鬆弛的方式局部加密方法以及空間差分方法，並將三維方面的問題推廣到一維切割平行運算。在結果部分，本文探討了在不同雷諾數下粗網格以及細網格的交界面都能符合質、動量守恆。並且驗證了局部加密網格確實能改善局部渦旋的位置和震盪的現象。

In its original form, lattice Boltzmann method is based on regular lattices. However, in order to enhance resolution at high Reynolds number or capture vortices accurately, regular grid may not be sufficient. One of the approaches to overcome this problem is to adopt local grid refinement. In this thesis, local grid refinement using lattice Boltzmann method on GPU is developed. Single relaxation time and multi-relaxation time LBM model with locally refined grid are adopted to compute two dimensional lid driven cavity flow. For three dimensional cavity flow, simulation are computed on multi-GPU with single relaxation model. Besides, a local cubic spatial interpolation which satisfies the mass and momentum continuity is used. Several cases are tested to investigate the method of grid refined scheme, interface structure and spatial interpolation between coarse and fined grids. The present strategy can greatly improve the accuracy of vortices and the impact of oscillation which is from singularity for viscous flow.

Contents
Introduction 1
1.1 Introduction to lattice Boltzmann equation 1
1.2 Literature survey 3
1.2.1 Lattice Boltzmann models 3
1.2.2 Grid Refinement for Lattice-BGK Models 4
1.2.3 Multi-block on lattice Boltzmann method 4
1.2.4 Boundary conditions 5
1.2.5 2D and 3D lid driven cavity flows 6
1.2.6 GPU implementation 6
1.3 Motivation and objective 7
Methodology 8
2.1 The Boltzmann equation 8
2.2 The BGK approximation 9
2.3 The low-Mach-number approximation 11
2.4 Discretization of the Boltzmann equation 12
2.4.1 Discretization of time 12
2.4.2 Discretization of phase space 14
2.5 The Chapman-Enskog expansion 16
2.6 The single relaxation time and multi relaxation time lattice Boltzmann model 17
Numerical algorithm 20
3.1 Simulation procedure 20
3.1.1 SRT and MRT Lattice Boltzmann method 20
3.2 Boundary condition implementations 21
3.3 Local Grid Refinement 23
3.3.1 The grid refinement of singe-relaxation-time scheme 23
3.2.2 The grid refinement of multi-relaxation-time scheme 23
3.2.3 The Interface Structure 23
3.2.4 Cubic interpolation Scheme 23
3.2.5 The Algorithm of Multi-Blocks Scheme 23
3.4 GPU implementation 23
3.5 One dimensional domain decomposition 24
Numerical results 34
4.1 Two dimension lid-driven cavity flow 34
4.1.1 Numerical validation in a square cavity 34
4.1.2 Flow structure in 2D cavity flow 35
4.1.3 Cubic spatial interpolation 35
4.2 Three dimensional cavity flow 36
4.2.1 Multi-GPU implementation 36
Conclusions 54

U. Firsch, B. Hasslacher, and Y. Pomeau, "Lattice-gas automata for the Navier-Stokes equation," Phys. Rev. Lett. 56, 1505, (1986).

S. Wolfram, "Cellular automata fluids 1: Basic theory,"J. Stat. Phys. 45, 471(1986).

S. Succi, E. Foti and F. J. HIGUERA, "Three-Dimensional Flows in Complex
Geometries with the Lattice Boltzmann Method", Europhysics Letters 9, 345,
(1989).

F. J. Higuera, and J. Jemenez, "Boltzmann approach to lattice gas simulations", Europhysics Letters 9, 663, (1989).
P. L. Bhatnagar, E. P. Gross, and M. Grook, "A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems", Physics Reviews E 94, 511, (1954).

S. Harris, \An introduction to the throry of the Boltzmann equation", Holt,Rinehart and Winston, New York, (1971). 17, 479, (1992).

S. Chen, H. Chen, D. O. Martinez, and W. H. Matthaeus, "Lattic Boltzmann model for simulation of magnet hydrodynamics," Phys. Rev. Lett. 67, 3776,
(1991).

Y. H. Qian, D. d'Humieres, and P. Lallemand, "Lattice BGK model for Navier-Stokes equation", Europhysics Letters 17, 479, (1992).

S. Chen and G. D. Doolen, "Lattice Boltzmann method for fluid flow", Annual Reviews of Fluid Mechanics 30 329, (1998).

H. Chen, S. Chen, and W. H. Matthaeus, "Recovery of the Navier-Stokesquations using a lattice gas Boltzmann method", Physics Reviews A 45, R5339,(1992).

S. Ansumali,and I. V. Karlin, "Entropy function approach to Lattice Boltzmann Method\, Journal of Statical Physicas 107, Nos.1/2, (2002).

I. V. Karlin, S. Succi, and S. S. Chikatamarla,\Comment on "Numerical of the lattice Boltzmann method: Effects of collision models on lattice Boltzmann simulations", American Physical Society Physical review 84, 098701, (2011).

L. S. Lin, H. W. Chang, C. A. Lin, "Multi relaxation time lattice Boltzmann simulations of transition in deep 2D lid driven cavity using GPU", Computers and Fluids 80, 381, (2013).

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "Multi-GPU implementation of a hybrid thermal lattice Boltzmann solver using the TheLMA framework", Comput. Fluids 80, 269, (2013).

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "Multi-GPU implementation of the lattice Boltzmann method", Comput. Math. Appl 65, 252, (2013).

X.Wang, T. Aoki, "Multi-GPU performance of incompressible flow computation by lattice Boltzmann method on GPU cluster", Parallel. Computing 37, 521, (2011).

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "Scalable lattice Boltzmann solvers for CUDA GPU clusters", Comput. Fluids 39, 259, (2013).

C. Obrecht, F. Kuznik, B. Tourancheau, J. J. Roux, "A new approach to the lattice Boltzmann method for graphics processing units ", Computers and Mathematics with Applications 61, 3628, (2011).

F. Kuznik, C. Obrecht, G. Rusaouen, J. J. Roux, "LBM based flow simulation using GPU computing processor", Computers and Mathematics with Applications 59, 2380, (2010).

J. Tolke, "Implementation of a lattice Boltzmann kernel using the compute unified device architecture developed by NVIDIA", Comput. Vis. Sci 13, 29, (2010).

G. R. McNamara, and G. Zanetti, "Use of the Boltzmann equation to simulatelattice-gas automata", Phys. Rev. Lett. 61, 2332, (1988).

F. J. Higuera, and J. Jimenez, "Boltzmann approach to lattice gas simulations", Europhys. Lett. 9, 663, (1989).

H. Chen, S. Chen, and W. H. Matthaeus, "Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method," Phys. Rev. A. 45, 5339,(1992).

Y. H. Qian, D. d'Humieres, and P. Lallemand, "Lattice BGK models for Navier-Stokes equation," Europhys. Lett. 17, 479, (1992).

X. He, and L. S. Luo, "Theory of the lattice Boltzmann method: From theBoltzmann equation to the lattice Boltzmann equation," Phys. Rev. E 56,6811, (1997).

X. He, and L. S. Luo, "A priori derivation of the lattice Boltzmann equation,"Phys. Rev. E 55, 6333, (1997).

K. Kono, T. Ishizuka, H. Tsuda, and A. Kurosawa, "Application of latticeBoltzmann model to multiphase flows with phase transition," Comput. Phys.Commun. 129, 110, (2000).

S. Hou, X. Shan, Q. Zou, G. D. Doolen, and W. E. Soll, "Evaluation of two lattice Boltzmann models for multiphase flows," J. Comput. Phys. 138, 695,(1997).

X. He, S. Chen, and R. Zhang, "A lattice Boltzmann scheme forincompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability," J. Comput. Phys. 152, 642, (1999).

C. H. Shih, C. L. Wu, L. C. Chang, and C. A. Lin, "Lattice Boltzmann simulations of incompressible liguid-gas system on partial wetting surface," Phil. Trans. R. Soc. A 369, 2510, (2011).

M. Krafczyk, M. Schulz, and E. Rank, "Lattice-gas simulations of two-phase flow in porous media," Commun. Numer. Meth. Engng 14, 709, (1998).

J. Bernsdorf, G. Brenner, and F. Durst, "Numerical analysis of the pressure drop in porous media flow with lattice Boltzmann (BGK) automata," Comput. Phys. Commun. 129, 247, (2000).

D. M. Freed, "Lattice-Boltzmann method for macroscopic porous media modeling," Int. J. Mod. Phys. C 9, 1491, (1998).

Y. Hashimoto, and H. Ohashi, "Droplet dynamics using the lattice-gas method," Int. J. Mod. Phys. 8, 977, (1997).

H. Xi, and C. Duncan, "Lattice Boltzmann simulations of three-dimensionalsingle droplet deformation and breakup under simple shear flow," Phys. Rev.E 59, 3022, (1999).

Olga Filippova and Dieter H¨anel "Grid Refinement for Lattice-BGK Models " J. Comput . Phys. 147,219-228(1998)

M. J. Berger and P. Collela, "Local adaptive mesh refinement for shock hydrodynamics", J. Comput. Phys. 82, 67 (1989).

J. J. Quirk, "An adaptive grid algorithm for computational shock hydrodynamics", Ph.D. thesis, Cranfield Inst. of Technology, UK, 1991.

Alexandre Dupuis and Bastien Chopard "Theory and applications of an alternative lattice Boltzmann grid refinement algorithm" Phys. Rev E67,066707(2003)

Dazhi Yu, Renwei Mei and Wei Shyy, "A multi-block lattice Boltzmann method for viscous fluid flows", Int. J. Numer. Meth. Fluids 2002; 39:99–120 (DOI: 10.1002/fld.280)

X. He, Q. Zou, L. S. Luo, and M. Dembo, "Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model," J. Stat. Phys. 87, 115, (1997).

C. F. Ho, C. Chang, K. H. Lin, and C. A. Lin, "Consistent boundary conditions for 2D and 3D lattice Boltzmann simulations," CMES 44, 137, (2009).

P. A. Skordos, "Initial and boundary conditions for the lattice Boltzmann method," Phys. Rev. E 48, 4823, (1993).

D. R. Noble, S. Chen, J. G. Georgiadis, and R. O. Buckius, "A consistent hydrodynamics boundary condition for the lattice Boltzmann method," Phys. Fluids 7, 203, (1995).

T. Inamuro, M. Yoshino, and F. Ogino, "A non-slip boundary condition for lattice Boltzmann simulations," Phys. Fluids 7, 2928, (1995).
 

J. Tolke, "Implementation of a lattice Boltzmann kernel using the compute unified device architecture developed by NVIDIA," Comput. Visual Sci. 13,29, (2008).

J. Tolke, and M. Krafczyk, "TeraFLOP computing on a desktop PC with GPUs for 3D CFD," Int. J. Comput. Fluid D. 22, 443, (2008)

E. Riegel, T. Indinger, and N. A. Adams, "Implementation of a Lattice-Boltzmann method for numerical fluid mechanics using the NVIDIA CUDA technology," CSRD 23, 241, (2009).

F. Kuznik, C. Obrecht, G. Rusaouen, and J. J. Roux, "LBM based flow simulation using GPU computing processor," Comput. Math. Appl. 59, 2380, (2010).

J. Habich, T. Zeiser, G. Hager, and G. Wellein, "Performance analysis and optimization strategies for a D3Q19 lattice Boltzmann kernel on NVIDIA GPUs using CUDA," Adv. Eng. Softw. 42, 266, (2011).

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "A new approach to the lattice Boltzmann method for graphics processing units," Comput. Math. Appl. 61, 3628, (2011).

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "Multi-GPU implementation of a hybrid thermal lattice Boltzmann solver using the TheLMA framework," Comput. Fluids. 80, 269, (2013).

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "Multi-GPU implementation of the lattice Boltzmann method," Comput. Math. Appl. 65, 252, (2013).

X. Wang, T. Aoki, "Multi-GPU performance of incompressible flow computation by lattice Boltzmann method on GPU cluster," Parallel. Computing. 37, 521, (2011).

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "Scalable lattice Boltzmann solvers for CUDA GPU clusters," Comput. Fluids. 39, 259, (2013).

H. W. Chang, P.Y. Hong, L.S. Lin and C.A. Lin, "Simulations of three- dimensional cavity flows with multi relaxation time lattice Boltzmann method and graphic processing units," Procedia Engineering. 61, 94, (2013)

P. Y. Hong "Lattice Boltzmann model with multi-GPU implementation," Master's thesis, National Tsing Hua University, (2013)

T. I. Gombosi, "Gas kinetic theory," Cambridge University Press, (1994).

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," Phys. Rev. 94, 511, (1954).

S. Harris, "An introduction to the theory of the Boltzmann equation," Holt, Rinehart and Winston, New York, (1971).
T. I. Gombosi, "Gas kinetic theory," Cambridge University Press, (1994).

X. He, and L. S. Luo, "Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation," Phys. Rev. E 56, 6811, (1997).

H. Chen, S. Chen, and W. H. Matthaeus, "Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method," Phys. Rev. A. 45, 5339, (1992).

J. S. Wu, and Y. L. Shao, "Simulation of lid-driven cavity flows by parallel lattice Boltzmann method using multi-relaxation-time scheme," Int. J. Numer. Meth. Fluids 46, 921, (2004).

P. Lallemand, and L. S. Luo, "Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability," Phys. Rev. E 61, 6546, (2000).

C. F. Ho, C. Chang, K. H. Lin, and C. A. Lin, "Consistent boundary conditions for 2D and 3D lattice Boltzmann models in three dimensions, " CMES 44, 137,(2009).

S. Hou, Q. Zou, S. Chen, G doolen, and A.C. Cogly, "Simulation of cavity flow by the lattice Boltzmann method, " J. COmput. Phys. 118, 329, (1995)

Olga Filippova and Dieter H ̈anel "Grid Refinement for Lattice-BGK Models" J. Comput.Phy 147, 219–228 (1998)

Dazhi Yu, Renwei Mei and Wei Shyy "A multi-block lattice Boltzmann method for viscous fluid flows" Int. J. Numer. Meth. Fluids 2002; 39:99–120 (DOI: 10.1002/fld.280)

Y.peng, C.Shu, Y.T. Chew, X.D. Niu, X.Y. Lu "Application of multi-block approach in the immersed boundary-lattice Boltzmann method for viscous fluid flows" Int J. Compt. Physics 218 (2006) 460-478

D. Lagravaa , O. Malaspinasb,a, J. Latta, B. Chopard "Advances in multi-domain lattice Boltzmann grid refinement"

Qisu Zou and Xiaoyi He "On pressure and velocity boundary conditions for the lattice Boltzmann BGK model"1997 American Institute of Physics. [S1070-6631(97)03406-5 ]

C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, "A new approach to the lattice Boltzmann method for graphics processing units," Comput. Math. Appl. 61, 3628, (2011).
U. Ghia, K. N. Ghia, and C. T. Shin "High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multi-grid Method" Journal of Computational Physics 48, 387-411 (1982).

S. Albensoeder, H. C. Kuhlmann, "Accurate three-dimensional lid-driven flow, " J. Comput. Phys., 206,5036,(2005)