晶格波茲曼法一般是使用均勻網格，但是在高雷諾數下物理變化較大的區域需要更精密的網格，局部加密的網格能改善這問題。本論文中，利用晶格波茲曼法及局部加密網格，在多張圖顯示卡上計算，加速模擬效率。前後許多學者已提出關於晶格波茲曼法的加密方式，然而在本論文中模擬二維及三維平板流，來驗證單鬆弛時間的局部網格加密之間的傳值、差分方法以及邊界條件在完全發展的平板流中與解析解相符合。此外本論文也探討局部加密網格相較於完全均勻網格能在計算中能節省的記憶體，以及一維切割的平行計算下，在不同圖形顯示卡中的平行效率。

In general, the lattice Boltzmann method is based on the uniform lattice. However, in order to simulate at high Reynolds number or in fine grid size, the local refinement strategy is advantageous. In this thesis, the local grid refinement and the multi-block method are employed here. The local grid refinement scheme does not only improve the accuracy but also economize the computing resource. The single relaxation time model is adopted to compute two and three-dimensional channel flow here, and the simulation in three dimensions is calculated on multi-GPU. Both the results of 2-D and 3-D match the analytical solutions. The relation between different blocks, interface structure, and the spatial interpolation would be investigated in this thesis. Additionally, the economized memory and the parallel performance tested by strong scaling tests are also discussed.

Introduction 1
1.1 Introduction 1
1.2 Literature survey 3
1.2.1 Theory of lattice Boltzmann models 3
1.2.2 Local Grid Refinement for LBGK Models 3
1.2.3 Multi-block on lattice Boltzmann method 4
1.2.4 Boundary conditions 5
1.2.5 2D and 3D Poiseuille flow 6
1.2.6 GPU implementation 6
1.3 Motivation 7
Methodology 8
2.1 The Boltzmann equation 8
2.2 The BGK approximation 9
2.3 The low-Mach-number approximation 12
2.4 Discretization of the Boltzmann equation 13
2.4.1 Discretization of time 13
2.4.2 Discretization of phase space 14
2.5 The Chapman-Enskog expansion 17
Numerical Algorithm 19
3.1 Simulation procedure 19
3.1.1 Single relaxation time lattice Boltzmann method 19
3.2 Local Grid Refinement 20
3.2.1 The grid refinement of singe-relaxation-time scheme 20
3.2.2 The Interface Structure and Computational Procedure 23
3.2.3 Interpolation Scheme 23
3.2.4 The Algorithm of Multi-Blocks Scheme 25
3.3 Boundary condition implementations 26
3.4 The external forcing term 27
3.5 GPU implementation 27
3.6 One dimensional domain decomposition 28
Numerical results 36
4.1 Two dimensional Poiseuille flow 36
4.2 Three dimensional Poiseuille flow 37
4.2.1 Memory savings 38
4.3 Parallel performance 38
Conclusions 48
Bibliography 49

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