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研究生：	邵冠崴 Shao, Kuan-Wei
論文名稱：	使用GPU平行化運算自適應網格的有限體積法於流線渦度方程式 Adaptive Mesh Finite Volume Method Applied to Steam-Vorticity based formulation of the Navier-Stokes Equations using GPU parallelization
指導教授：	李汶樺 Matthew Smith
學位類別：	碩士 Master
系所名稱：	工學院 - 機械工程學系 Department of Mechanical Engineering
論文出版年：	2018
畢業學年度：	106
語文別：	英文
論文頁數：	189
中文關鍵詞：	計算流體力學、自適應性網格演算法、流線渦度方程式、中央差分法、對流向二次迎風差值法、圖形處理器、CUDA
外文關鍵詞：	Computational Fluid Dynamics (CFD), Adaptive Mesh Refinement (AMR), streamfunction-voriticy formulation, central difference scheme (CDS), Quadratic Upstream Interpolation for Convective Kinematics (QUICK), Parallel Computing, Graphic Processing Unit (GPU), CUDA
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大多數的人使用結構性的卡式座標的網格來處理計算流體力學的問題，因為這種方不僅簡單而且也能增加平行的效率。然而，隨著真實的模擬的模型越來越複雜，我們需要更高的解析度去得到更好的幾何模型，再加上很多流體力學的問題同時包含需要大尺度跟小尺度的物理現象，因此使用結構性的網格需要更多的網格來解決物理模型，相對的耗費的時間就更多。為了克服這些問題，本研究適運用自適應網格方法(Adaptive Mesh Refinement)來模擬暫態的不可壓縮流的問題，自適應網格不但能有效的減少網格數來解析物理模型，也能有效的解析在暫態中重要的物理現象。
此研究主要使用暫態的流線渦度方程式來解決納維－斯托克斯方程(Navier-Stokes equation)。此方法能夠簡化二維的納維－斯托克斯方程，在不需要計算壓力的情況下來求得數值解。我們使用網格大小加權渦度 wdx^(3/2)的變異數來判別流場中哪部分的網格需要細化或粗化來達成減少網格的目的。雖然網格數減少許多，但是自適應網格方法所需的時間比結構性網的時間還要多，因為在計算過程中花費大量的時間在細化跟粗化。
在此研究中主要使用圖型處理器(GPU)來加速運算的時間，雖然兩種網格的加速結果都很高，但是結構性的網格還是遠小於自適應網格，主要的原因來有兩個，其一是我們自適應網格使用的資料結構相當的不佳導致GPU花更多時間在傳遞資料。第二個是在自適應化網格的過程中，我們難以平行化造成需要更多的時間在CPU跟GPU之間的資料傳。另一個研究目的在比較自適應網格的影響，使用中央差分法(CDS)和對流項二次迎風插值法(QUICK)來模擬一些基準模擬和太陽能集熱器。最後我們的自適應網格不能降低計算時間。

Many Computational Fluid Dynamics implementations employ structured Cartesian grids as part of their implementation due to increased parallel efficiency and ease of implementation. However, with increasingly complex geometries for real-life modern applications, we need to employ high resolutions to correctly capture the flow physics within the model. Moreover, a lot of Computational Fluid Dynamic problems exist with mixtures of not only large scales but also small scale physical phenomenon. Thus, many cells may be required in a structured Cartesian grid, with a large potential computation expense. Presented in this work is an investigation of the application of Adaptive Mesh Refinement (AMR) for solving transient incompressible flows, with the goal of not only resolving complex geometry but also capturing important phenomena in transient flow.
The goal of the thesis is to apply Adaptive Mesh Refinement (AMR) algorithm for the transient stream function vorticity formulation of the Navier-Stokes equations. The stream function vorticity formulation is the simplification of the two-dimensional Navier-Stoke equation which does not require the direct solution for pressure. During computation, cell refinement (and coarsening) is computed using the variance of size-weighted vorticity wdx^(3/2) with the goal that the total cell number employed in the AMR solver is relatively less than the structured grid result. In this work, AMR implementation requires more time than structure grid implementation since excessive time is allocated to the procedure of refining and merging.
In this thesis, we apply Graphics Processing Units (GPUs) parallel computing architectures to accelerate the simulation time. Although the computed speed-up of both grid methods are high, the calculation time for the structured grid method is still dramatically less than the AMR solver. The main causes are the bad data structure of the current AMR implementation, which takes more time to access data in our grid, and the procedure of AMR which are hard to parallelize, resulting in host mesh refinement which takes extra time due to host-device data transfer bottlenecks.
Another goal of this research is to observe the influence of AMR on the result obtained by the central difference scheme (CDS), Quadratic Upstream Interpolation for Convective Kinematics (QUICK) and other relative flux calculations. Several commonly used benchmark tests are investigated, and then applied to the simulation of a solar air collector. The performance of the AMR code in the practical problem does not justify the increased complexity and decreased computational performance associated with the method.

中文摘要	I
ABSTRACT	III
ACKNOWLEDGEMENTS	V
CONTENTS	VI
CHAPTER 1 – INTRODUCTION	1
1.1	MOTIVATION AND BACKGROUND	1
1.2 GOVERNING EQUATION	2
1.2.1 Convection-Diffusion Equation	2
1.2.2 Navier-Stokes and Stream function Vorticity Formulation	3
1.2.3 Incompressible Flow of Energy Equation	4
1.2.4 Analytical Solution	5
1.3 COMPUTATIONAL FLUID DYNAMICS	7
1.3.1	Finite Volume Method	8
1.3.1.1 	First Order Upwind (FOU)	10
1.3.1.2 	Central Difference Scheme (CDS)	10
1.3.1.3  Quadratic Upstream Interpolation for Convective Kinematics (QUICK)	11
1.3.1.4  Diffusion Equation	11
1.3.1.5	Dissipation Term-Source Term	12
1.3.1.6 	Poisson Equation- Jacob’s Method	12
1.3.2	Stability	13
1.3.3	Adaptive Mesh Refinement Method	14
1.4 PARALLEL COMPUTING	15
1.5	GENERAL-PURPOSE COMPUTING USING GRAPHIC PROCESS UNITS	16
1.5.1	GPU Memory	17
1.5.2	CUDA threads, Blocks, Grids	18
1.5.3	CUDA API	21
1.5.4	Kepler Architecture	22
1.5.5	Maxwell Architecture	23
1.6	SOLAR AIR COLLECTOR	24
CHAPTER 2 – METHODOLOGY	25
2.1 IMPLEMENTATION	25
2.1.1	Stream function Vorticity Formulation	25
2.1.1.1 	Diffusion Equation	25
2.1.1.2	Convection Equation	25
2.1.1.3 	Poisson Equation	26
2.1.1.4	Velocity	26
2.1.1.5 Boundary Condition	26
2.1.2	Energy Equation	30
2.1.2.1	Dissipation Term	30
2.1.2.2 	Boundary Condition	31
2.1.3 	AMR Implementation	32
2.1.3.1	Refining and Merging Criterion	32
2.1.3.2	Cell Splitting Procedure	33
2.1.3.3	Cell Merging Procedure	34
2.1.3.4	Organization Cell in GPU	34
2.1.3.5	Reordering	35
2.1.3.7	Flux Calculation	35
2.1.3.8	Relative Difference	39
2.2 DATA STRUCTURE	40
2.2.1	Cartesian Grid	40
2.2.2	Adaptive Mesh Refinement Techniques	40
2.3	GPU PARALLELIZATION	41
2.3.1	Memory Management on the GPU using CUDA API	41
2.3.2	CPU-Launched GPU Kernels (Global Function)	43
2.3.3	GPU-Launched GPU Kernels (Device Function)	44
2.3.4	Parallel Reduction	44
2.3.5	Compiling and Building Solver	45
CHAPTER 3 – RESULT	46
3.1	ADVECTION TEST	46
3.2	DIFFUSION TEST	47
3.3 POISSON EQUATION	47
3.4 LID-DRIVEN CAVITY	47
3.5 INTERNAL FLOW OF PARALLEL PLATES TEST	50
3.6 BACKWARD FACING STEP	51
3.7 SOLAR AIR COLLECTOR	53
3.8 ACCURACY AND PARALLEL PERFORMANCE OF AMR	55
CHAPTER 4 – CONCLUSION	57
REFERENCE	59

Table (1. 1): The classification of GPU memory.	63
Table (3. 1): The average difference of structure grid and AMR in diffusion test which minimum size is the same.	63
Table (3. 2): The average difference of structure grid and AMR in Poisson test which minimum size is the same.	63
Table (3. 3): The Lid-driven cavity data of Ghia [36].(a) u-velocity (b) v-velocity	64
Table (3. 4): The CDS mean neighbors, standard deviation neighbor and max difference in Re 100 Lid-driven cavity in each block.	65
Table (3. 5) The QUICK mean neighbors, standard deviation neighbor and max difference in Re 100 Lid-driven cavity in each block.	68
Table (3. 6): Average difference of lid-driven cavity in Re100.	71
Table (3. 7): Average difference of lid-driven cavity in Re400.	71
Table (3. 8): Average difference of lid-driven cavity in Re1000.	71
Table (3. 9): Average difference of backward facing step in Re100.	72
Table (3. 10): Average difference of backward facing step in Re389.	72
Table (3. 11): Average difference of backward facing step in Re1000.	72
Table (3. 12) : Comparison of lid-driven cavity speed up between different level and same minimum size of structured	73
Table (3. 13) : Comparison of backward-facing step speed up between different level and same minimum size of structured.	73

Figure (1. 1): The boundary condition of steady state of two-dimensional heat conduction.	74
Figure (1. 2): The motion of convection equation.	75
Figure (1. 3): The difference between the cell-centered FVM (right) and the cell-vertex FVM (left).	75
Figure (1. 4): Structured Cartesian cell with the index i, j.	76
Figure (1. 5): the quadtree data structure for a regular mesh condition.	76
Figure (1. 6) Flynn’s Taxonomy (top) the classification (bottom) the execution flow chart.	77
Figure (1. 7): The memory storage location and the related access mechanism	78
Figure (1. 8): The hierarchy of the grids, blocks and threads. Both blocks and threads are declared as three-dimensional.	79
Figure (1. 9): The memory access model.	80
Figure (1. 10) A single SMX composition in the Kepler Architecture	81
Figure (1. 11) Comparison of operating procedure of Fermi (left) and Kepler (right) Architecture.	82
Figure (1. 12): A single SMX composition in Maxwell Architecture.	83
Figure (2. 1): The flowchart of procedure of streamfunction vorticity formulation.	84
Figure (2. 2) Code of diffusion term for finite volume method.	84
Figure (2. 3) Code of advection term for finite volume method.	85
Figure (2. 4) Code of Poisson equation by Jacob’s method and the criterion	86
Figure (2. 5) code of velocity calculation.	86
Figure (2. 6) the concave corner and convex corner	86
Figure (2. 7) the flow chart of Energy equation.	87
Figure (2. 8) the code of the dissipation term	87
Figure (2. 9): the flowchart of reconstructing cell situation.	88
Figure (2. 10): the splitting figures with indices assigning	89
Figure (2. 11): the code of merge procedure.	90
Figure (2. 12): the code of organization	90
Figure (2. 13): the part of code of reordering.	91
Figure (2. 14): the TVD monotonicity region on r-ψ diagram in Sweby[29].	92
Figure (2. 15): the demonstration of advection notation in [15]	92
Figure (2. 16) : the data structure of structure grid	93
Figure (2. 17): (a) the function perform on GPU kernel. (b) the method calls corresponding GPU kernel	93
Figure (2. 18) : the demonstration of parallel reduction.	94
Figure (2. 19) : the demonstration of parallel reduction. Add all value of blockId together.	94
Figure (2. 20) : the code of parallel reduction	95
Figure (2. 21) : the instruction of Makefile.	95
Figure (3. 1): the initial condition of advection test…………………………………….96
Figure (3. 2): the condition of mesh for advection test	96
Figure (3. 3) the comparison between the structured gird and AMR in FOU	97
Figure (3. 4) the comparison between the structured gird and AMR in QUICK.	98
Figure (3. 5) the analytical result of diffusion test	99
Figure (3. 6) the structure result of diffusion test	99
Figure (3. 7) the AMR with level 1 of diffusion test	100
Figure (3. 8) the AMR with level 2 of diffusion test	101
Figure (3. 9) the analytical result of Poisson equation	102
Figure (3. 10) the structured grid result of Poisson equation	102
Figure (3. 11) the AMR with level 1 of Poisson equation	103
Figure (3. 12) the AMR with level 2 of Poisson equation	104
Figure (3. 13): Boundary condition of Lid-Driven Cavity.	105
Figure (3. 14): The Velocity result of Re100 Lid-Driven Cavity by CDS.	106
Figure (3. 15): The Temperature result of Re100 Lid-Driven Cavity by CDS.	107
Figure (3. 16): The streamline of Re100 Lid-Driven Cavity by CDS	108
Figure (3. 17): The Velocity result of Re100 Lid-Driven Cavity by QUICK.	109
Figure (3. 18): The Temperature result of Re100 Lid-Driven Cavity by QUICK.	110
Figure (3. 19): The streamline of Re100 Lid-Driven Cavity by QUICK	111
Figure (3. 20): The comparison of Re100 lid driven cavity velocity profile between structure and AMR. (Top) v-profile (bottom) u-profile	112
Figure (3. 21): The comparison of Re100 lid driven cavity Temperature profile between structure and AMR. (Top) horizontal line (bottom) vertical line	113
Figure (3. 22): The comparison of Re100 lid driven cavity velocity profile between different level. (Top) v profile (bottom) u profile.	114
Figure (3. 23): The comparison of Re100 lid driven cavity Temperature profile between different level.(Top) horizontal line (bottom) vertical line	115
Figure (3. 24): The final grid of Re100 lid driven cavity with level 3	116
Figure (3. 25): The Velocity result of Re400 Lid-Driven Cavity by CDS.	117
Figure (3. 26): The Temperature result of Re400 Lid-Driven Cavity by CDS	118
Figure (3. 27): The streamline of Re100 Lid-Driven Cavity by CDS	119
Figure (3. 28): The Velocity result of Re400 Lid-Driven Cavity by QUICK.	120
Figure (3. 29): The Temperature result of Re400 Lid-Driven Cavity by QUICK	121
Figure (3. 30): The streamline of Re400 Lid-Driven Cavity by QUICK	122
Figure (3. 31): The comparison of Re400 lid driven cavity velocity profile between structure and AMR. (Top) v-profile (bottom) u-profile	123
Figure (3. 32): The comparison of Re400 lid driven cavity Temperature profile between structure and AMR. (Top) horizontal line (bottom) vertical line	124
Figure (3. 33): The comparison of Re400 lid driven cavity velocity profile between different level. (Top) v profile (bottom) u profile.	125
Figure (3. 34): The comparison of Re400 lid driven cavity Temperature profile between different level.(Top) horizontal line (bottom) vertical line	126
Figure (3. 35): The final grid of Re400 lid driven cavity with level 3	127
Figure (3. 36): The Velocity result of Re1000 Lid-Driven Cavity by CDS.	128
Figure (3. 37): The temperature result of Re1000 Lid-Driven Cavity by CDS	129
Figure (3. 38): The streamline of Re1000 Lid-Driven Cavity by CDS	130
Figure (3. 39): The Velocity result of Re1000 Lid-Driven Cavity by QUICK.	131
Figure (3. 40): The Temperature result of Re1000 Lid-Driven Cavity by QUICK	132
Figure (3. 41): The streamline of Re1000 Lid-Driven Cavity by QUICK	133
Figure (3. 42): The comparison of Re1000 lid driven cavity velocity profile between structure and AMR. (Top) v-profile (bottom) u-profile	134
Figure (3. 43): The comparison of Re1000 lid driven cavity Temperature profile between structure and AMR. (Top) horizontal line (bottom) vertical line	135
Figure (3. 44): The comparison of Re400 lid driven cavity velocity profile between different level. (Top) v profile (bottom) u profile.	136
Figure (3. 45): The comparison of Re1000 lid driven cavity Temperature profile between different level.(Top) horizontal line (bottom) vertical line	137
Figure (3. 46): The final grid of Re400 lid driven cavity with level 3	138
Figure (3. 47): The comparison of Velocity solver driven (Top) Green Gauss (bottom) Least square.	139
Figure (3. 48): The final mesh of lid-driven cavity which is moving wall is on the right	140
Figure (3. 49): The comparison of top wall and right wall lid-driven cavity. The right wall result is y-velocity along horizontal line through the cell center.	141
Figure (3. 50): The comparison of top wall and right wall lid-driven cavity. The right wall result is x-velocity along vertical line line through the cell center.	141
Figure (3. 51): The model of Internal Flow of Parallel Plates	142
Figure (3. 52): The result of Re 100 inflow test (Top) u-velocity (Bottom) Temprature.	142
Figure (3. 53): The comparison of Re 100 velocity between sturctured and AMR.	143
Figure (3. 54): The comparison of Re 100 Temperature between sturctured and AMR.	143
Figure (3. 55): The final mesh of Re 100 inflow test	144
Figure (3. 56): The result of Re 500 inflow test (Top) u-velocity (Bottom) Temprature.	144
Figure (3. 57): The result of Re 1000 inflow test (Top) u-velocity (Bottom) Temprature.	145
Figure (3. 58): The comparison of Re 500 velocity between sturctured and AMR.	145
Figure (3. 59): The comparison of Re 500 Temperature between sturctured and AMR.	146
Figure (3. 60): The comparison of Re 1000 velocity between sturctured and AMR.	146
Figure (3. 61): The comparison of Re 1000 Temperature between sturctured and AMR.	147
Figure (3. 62): The final mesh of Re 500 inflow test	147
Figure (3. 63): The final mesh of Re 1000 inflow test	147
Figure (3. 64): The flow demonstration of backward-facing step.	148
Figure (3. 65): The experimental data for the location of recirculation regions in Armaly et al[2].	148
Figure (3. 66) The u contour of backward Re = 100	149
Figure (3. 67): Re 100 velocity profile at x/S = 0.0 (left) and 2.55 (Right)	150
Figure (3. 68): Re 100 velocity profile at x/S = 3.06 (left) and 4.18(Right)	150
Figure (3. 69): Final mesh of backward-facing step with maximum level 2.	150
Figure (3. 70) The u contour of backward Re = 389	151
Figure (3. 71): Re 389 velocity profile at x/S = 0.0 (left) and 2.55 (Right)	152
Figure (3. 72): Re 389 velocity profile at x/S = 3.06 (left) and 4.18 (Right)	152
Figure (3. 73): Final mesh of backward-facing step with maximum level 2.	152
Figure (3. 74) The u contour of backward Re = 1000	153
Figure (3. 75) : Re 1000 velocity profile at x/S = 0.0 (left) and 2.55 (Right)	154
Figure (3. 76): Final mesh of backward-facing step with maximum level 2.	154
Figure (3. 77): The geometry of solar air collector , Mahfoud et al.[28][29].	155
Figure (3. 78) The boundary condition of part of solar air collector model.	155
Figure (3. 79): The result of 1 chicane in Re 100 (left) Structured (Right) AMR.	156
Figure (3. 80): The result of 1 chicane in Re 557 (left) Structured (Right) AMR.	157
Figure (3. 81): The result of 2 chicane in Re 100 (left) Structured (Right) AMR.	158
Figure (3. 82): The result of 2 chicane in Re 557 (left) Structured (Right) AMR	159
Figure (3. 83): The comparison of velocity profile in Re 557 with 2 chiacane	160
Figure (3. 84): The velocity profile in Mahfoud et al. [28][29]	161
Figure (3. 85): The comparison of temperature profile in Re 557 with 2 chiacane	162
Figure (3. 86): The Temperature profile in Mahfoud et al. [28][29]	163
Figure (3. 87): The level 1 relative difference of Lid-driven cavity in Re 100 with CDS	164
Figure (3. 88): The level 2 relative difference of Lid-driven cavity in Re 100 with CDS	165
Figure (3. 89): The level 3 relative difference of Lid-driven cavity in Re 100 with CDS	166
Figure (3. 90): The level 1 relative difference of Lid-driven cavity in Re 100 with QUICK	167
Figure (3. 91): The level 2 relative difference of Lid-driven cavity in Re 100 with QUICK	168
Figure (3. 92): The level 3 relative difference of Lid-driven cavity in Re 100 with QUICK	169
Figure (3. 93): The level 1 relative difference of Lid-driven cavity in Re 400 with CDS	170
Figure (3. 94): The level 2 relative difference of Lid-driven cavity in Re 400 with CDS	171
Figure (3. 95): The level 3 relative difference of Lid-driven cavity in Re 400 with CDS	172
Figure (3. 96): The level 1 relative difference of Lid-driven cavity in Re 400 with QUICK	173
Figure (3. 97): The level 2 relative difference of Lid-driven cavity in Re 400 with QUICK	174
Figure (3. 98): The level 3 relative difference of Lid-driven cavity in Re 400 with QUICK	175
Figure (3. 99): The level 1 relative difference of Lid-driven cavity in Re 1000 with CDS	176
Figure (3. 100): The level 2 relative difference of Lid-driven cavity in Re 1000 with CDS	177
Figure (3. 101): The level 3 relative difference of Lid-driven cavity in Re 1000 with CDS	178
Figure (3. 102): The level 1 relative difference of Lid-driven cavity in Re 1000 with QUICK	179
Figure (3. 103): The level 2 relative difference of Lid-driven cavity in Re 1000 with QUICK	180
Figure (3. 104): The level 3 relative difference of Lid-driven cavity in Re 1000 with QUICK	181
Figure (3. 105): The level 1 relative difference of backward-facing step in Re 100 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	182
Figure (3. 106): The level 2 relative difference of backward-facing step in Re 100 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	182
Figure (3. 107): The level 3 relative difference of backward-facing step in Re 100 with CDS (Top) Mesh (Middle) u velocity (Bottom) stream function	182
Figure (3. 108): The level 1 relative difference of backward-facing step in Re 100 with QUICK	183
Figure (3. 109): The level 2 relative difference of backward-facing step in Re 100 with QUICK	183
Figure (3. 110): The level 3 relative difference of backward-facing step in Re 100 with QUICK	183
Figure (3. 111): The level 1 relative difference of backward-facing step in Re 389 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	184
Figure (3. 112): The level 2 relative difference of backward-facing step in Re 389 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	184
Figure (3. 113): The level 3 relative difference of backward-facing step in Re 389 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	184
Figure (3. 114): The level 1 relative difference of backward-facing step in Re 389 with QUICK (Top) Mesh (Middle) u velocity (Bottom) streamfunction	185
Figure (3. 115): The level 1 relative difference of backward-facing step in Re 389 with QUICK (Top) Mesh (Middle) u velocity (Bottom) streamfunction	185
Figure (3. 116): The level 1 relative difference of backward-facing step in Re 389 with QUICK (Top) Mesh (Middle) u velocity (Bottom) streamfunction	185
Figure (3. 117): The level 1 relative difference of backward-facing step in Re 1000 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	186
Figure (3. 118): The level 2 relative difference of backward-facing step in Re 1000 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	186
Figure (3. 119): The level 3 relative difference of backward-facing step in Re 1000 with CDS (Top) Mesh (Middle) u velocity (Bottom) streamfunction	186
Figure (3. 120): The level 1 relative difference of backward-facing step in Re 1000 with QUICK (Top) Mesh (Middle) u velocity (Bottom) streamfunction	187
Figure (3. 121): The level 2 relative difference of backward-facing step in Re 1000 with QUICK (Top) Mesh (Middle) u velocity (Bottom) streamfunction	187
Figure (3. 122): The level 3 relative difference of backward-facing step in Re 1000 with QUICK (Top) Mesh (Middle) u velocity (Bottom) streamfunction	187
Figure (3. 123): The ratio of Lid-driven cavity time cost in Re 1000 for each different AMR level using QUICK	188
Figure (3. 124): The ratio of backward facing step time cost in Re 1000 for each different AMR level using QUICK.	189



 

                                    

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