本論文旨在使用叢集圖形顯示卡(GPU)計算壓力驅動方管流。 GPU的設計理念與中央處理器(CPU)不同。 而且在編程的概念上也需要做很多的改變。由於圖形顯示卡的普及，近年來對 GPU 中並行計算的研究也越來越多。 然而，在數值模擬中，以往的研究大多是直接解矩陣進行的，不適合GPU執行。 因此，我們在數值模擬中求解矩陣時使用了Fraction step和搭配顯式迭代求解柏松方程式。 在矩陣求解器中，我們在 GPU 中應用低延遲共享記憶體並進行子迭代以提高收斂速度。 我們發現子迭代的收斂效率可以增加五倍。 且湍流統計結果與 Benchmark 相比也有很好的一致性，雖然流場中央有些許誤差。最後本文也進行的平行測試得到約90%的平行效率。

This study aimed to simulate Poiseuille duct flow by using multiple graphics processing units (GPUs). The design concept of the GPU is different from that of the central processing unit. Owing to the popularity of GPUs, research on GPU-based parallel computing has been increasing in recent years. In particular, in fluid dynamics, most of the previous studies is conducted direct solver, which is unsuitable for GPU programming. In this study, we used a fractional step and Jacobi iteration for solving matrices in the numerical simulation. In the matrix solver, we used low-latency shared memory in the GPU and subiteration to increase the convergence rate. We found that the subiteration can increase five times. While the turbulent statistic result showed good agreement with that of Moser et al[1], there was a small error at the center of the flow field. A strong scaling test showed the parallel performance to be 90%.

1 Introduction 1
1.1 Introduction 1
1.2 Literature survey 2
1.2.1 Turbulent Poiseuille duct flow 2
1.2.2 Fractional step method 3
1.2.3 Parallel computing 3
1.2.4 Jacobi iteration 4
1.2.5 GPU implementation 5
1.3 GPU structure 5
1.4 Objective and motivation 7
2 Numerical method 8
2.1 Governing equation 9
2.2 Grid Generation 9
2.3 Fractional step method 11
2.4 Discretization of the Momentum Equation 12
2.4.1 Spatial discretization 13
2.4.2 Temporal discretization 14
2.5 Pressure Poisson equation 14
2.6 Parallelization 16
2.7 GPU implementation 16
2.8 Jacobi iteration 17
3 Result 23
3.1 GPU implementation on Poisson solver 23
3.1.1 Effect of subiteration 24
3.1.2 Effect of block size 24
3.2 Laminar Poiseuille duct flow 26
3.3 Turbulent Poiseuille duct flow 29
3.3.1 Convergence test in turbulent flow 29
3.3.2 Turbulent flow field 29
3.3.3 Turbulent statistic 31
3.4 Performance of program 37
4 Conclusion and Future work 38
4.1 Conclusion 38
4.2 Future work 39

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