臺灣博碩士論文加值系統

分子層級的流體性質經常以全原子模擬(all-atom simulations)的方法進行研究,該方法以古典力學描述每一個原子的動態行為,因此為目標研究系統提供最清楚的動態資訊。另一方面,流體在巨觀的行為則視流體為一連續的變量,並藉由流體動力學方程式(hydrodynamic equations)預測隨時間下流體的表現。奈米層級下的探討則需要結合介於上述兩種尺度下的模型來進行分析。此一新的耦合系統對每個粒子配置了與其相對應的網格。粒子對場變數、或反之場變數對粒子的配對是透過粒子與場網格之內插來計算。然而此一耦合演算法尚未建立於高效能運算架構之下。

近年來,圖形處理器(graphics processing units, GPUs)儼然成為科學計算上富競爭的平台。原本GPU是為電腦繪圖而設計,但是GPU架構現在被優化於處理計算密集型任務與高通量資料。比起傳統的高效能運算叢集系統,上述特性使得GPU更為吸引人且計算更有效率。因此新的運算架構即選擇了通用圖形處理器(GPU–CPU)之架構來進行混合模型之高效能運算。

本篇碩士論文討論以兩種力學的混合模型在GPU加速模擬下的設計與實踐。目的是要將原先之CPU演算架構改建置於可大量同步控制、以達到最高加速運算可能性之GPU演算架構。此一新的GPU演算架構使用共享記憶體為暫存區域以對耦合系統進行快速地局部內插運算。兩階段執行緒對應將額外內存空間之使用降到最低以達到最大限度地提高運算處理量。藉由徹底地增加模擬的計算效率,利用混合模型所探索的時空間尺度將可以被大幅的增加。

Fluid properties at the molecular scale are often investigated using all-atom simulations, which provide the highest level of detail attainable using classical mechanics. On the other hand, the behavior of fluids at the macroscopic scale is modeled by approximating the fluid as a continuous quantity and tracking its evolution by hydrodynamic equations. At the nanoscale both of these modeling paradigms are necessary. A hybrid model implementing molecular dynamics and hydrodynamics has previously been designed for simulations of nanoscale fluids. It implements a novel coupling scheme that associates a collocating grid with each particle. The mapping of particle to field variables and vice versa is then achieved through interpolation of particle and field grids. However, the coupling algorithm has not yet been adapted for high-performance computing (HPC).

In recent years, graphics processing units (GPUs) have emerged as a competitive platform for scientific computations. Originally designed for computer graphics, the GPU architecture is optimized for computationally intensive tasks and high data throughput. These features make them attractive and cost-effective alternatives compared to traditional HPC clusters. Therefore, a GPU–CPU framework was chosen as the HPC platform for the hybrid model.

This thesis thus presents the design and implementation of a GPU-accelerated simulation of the hybrid model. The objective was to reformulate the original CPU algorithms to expose massive concurrency, implement them on the GPU and achieve the highest computational speedup possible. A novel GPU algorithm was designed for the coupling scheme that uses shared memory as a staging area to perform fast local interpolations. To maximize computational throughput, a two-stage thread mapping was employed with a minimal amount of additional memory overhead. By drastically increasing the computational efficiency of simulations, the spatial and temporal scales that can be explored using the hybrid model were greatly expanded.

Chinese Abstract i
English Abstract ii
Acknowledgements iii
Table of Contents v
List of Figures viii
List of Tables ix
List of Listings x
I Introduction 1
II Mathematical Modeling of Complex Molecular Systems at the Nanoscale 4
2.1 Fluctuating Hydrodynamics 4
2.1.1 Governing Equations of Conservation Laws 5
2.1.2 Constitutive Equations 5
2.1.3 Conservation of Mass 6
2.1.4 Conservation of Momentum 6
2.1.5 The Stress Tensor 7
2.1.6 The Fluctuating Stress Tensor 8
2.1.7 Summary 9
2.2 Molecular Dynamics 9
2.2.1 Equations of Motion 10
2.2.2 Force Fields 10
2.2.3 Periodic Boundary Conditions 11
2.3 The Particle–Solvent Coupling Scheme 12
2.3.1 Free-Energy Densities 12
2.3.2 Friction Forces 15
2.4 Electrostatic Interactions 16
2.4.1 Charged Particles 16
2.4.2 The Polarization Density 17
2.4.3 Poisson’s Equation 18
2.5 Summary 19
III Numerical Solution of the Hybrid Model 23
3.1 Auxiliary Fluids 23
3.2 Staggered Grids 24
3.2.1 Definitions 25
3.2.2 Discretization over Staggered Grids 26
3.2.3 Calculation of Gradients and Divergences 28
3.2.4 Calculation of Laplacians 30
3.2.5 Multiplication and Division of Scalar and Vector Fields 31
3.3 Lagrangian Grids 31
3.3.1 Occupied Volumes 32
3.3.2 Particle –Solvent Forces 34
3.4 Numerical Solution of Poisson’s Equation 35
3.4.1 The Fourier Transform 35
3.4.2 The Discrete Fourier Transform 36
3.4.3 The Fast Fourier Transform Algorithm 37
3.5 Neighbor Lists 37
3.5.1 Force Truncation 38
3.5.2 Cell Lists 38
3.6 Stochastic Fluxes 39
3.7 Temporal Propagation 40
IV GPU Architecture and the CUDA Execution Model 43
4.1 Overview of the GPU Architecture 43
4.2 The CUDA Execution Model 44
4.2.1 CUDA Threads 44
4.2.2 Thread Blocks and Grids 45
4.2.3 Streaming Multiprocessors 47
4.2.4 Branch Divergence 48
4.3 The CUDA Memory Model 49
4.3.1 The Memory Hierarchy 50
4.3.2 Global Memory and Local Memory 51
4.3.3 Shared Memory 52
4.3.4 Constant Memory 53
4.3.5 Atomic Operations 53
4.3.6 Address Coalescing 54
V Design and Implementation of the GPU-Accelerated Simulation 55
5.1 Design Objectives and Principles 55
5.1.1 Performance 55
5.1.2 Memory Efficiency 57
5.1.3 Scalability 57
5.1.4 Modularity 57
5.2 The Grid-Stride Loop 58
5.3 Grid Tiling 61
5.3.1 Tiling Strategy 63
5.3.2 The Block-Stride Loop 64
5.3.3 The Tile-Stride Loop 64
5.3.4 Summary 65
5.4 Neighbor List Generation 66
5.4.1 Construction of Cell Lists 66
5.4.2 Construction of Neighbor Lists 67
5.4.3 List Overflow 67
5.5 The Interpolation Scheme from Lagrangian to Eulerian Grids 68
5.5.1 Original Implementation 68
5.5.2 GPU Implementation 71
5.5.3 Auxiliary Data for Interpolation 74
5.5.4 Summary 80
5.6 GPU-Accelerated Libraries 81
VI Results and Discussion 83
6.1 Experimental Setup 83
6.2 Fluctuating Hydrodynamics Performance 83
6.3 Neighbor List Performance 89
6.4 Occupied Volume Interpolation Performance 91
6.5 Simulation Performance 92
VII Conclusion 94
Bibliography 95

[1] M. P. Allen and D. J. Tildesley. “Computer simulation of liquids”. Oxford: Clarendon Press, 1987.
[2] J. A. Anderson, C. D. Lorenz, and A. Travesset. “General purpose molecular dynamics simulations fully implemented on graphics processing units”. In: “Journal of Computational Physics” 227.10 (2008), pp. 5342–5359.
[3] Boost Community. “Boost C++ Libraries”. url: http://www.boost.org/ (visited on 07/25/2016).
[4] B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus. “CHARMM: A program for macromolecular energy, minimization, and dynamics calculations”. In: “Journal of Computational Chemistry” 4.2 (Jan. 1983), pp. 187–217.
[5] J. Cheng, M. Grossman, and T. McKercher. “Professional CUDA C Programming”. Indianapolis, Indiana: John Wiley & Sons, Inc., 2014.
[6] W. M. Deen. “Analysis of transport phenomena”. Oxford: Oxford University Press, 2013.
[7] Free Software Foundation Inc. “GNU Compiler Collection”. 2016. url: https://gcc.gnu.org/.
[8] D. Frenkel and B. Smit. “Understanding molecular simulation: from algorithms to applications”. New York: Academic Press, 2002.
[9] J. Glaser, T. D. Nguyen, J. A. Anderson, P. Lui, F. Spiga, J. A. Millan, D. C. Morse, and S. C. Glotzer. “Strong scaling of general-purpose molecular dynamics simulations on GPUs”. In: “Computer Physics Communications” 192 (2015), pp. 97–107.
[10] A. Grama, A. Gupta, G. Karypis, and V. Kumar. “Introduction to Parallel Computing; 2nd Edition”. Upper Saddle River, New Jersey: Addison-Wesley, 2003.
[11] M. Harris. “CUDA Pro Tip: Write Flexible Kernels with Grid-Stride Loops”. 2013. url: https://devblogs.nvidia.com/parallelforall/cuda-pro-tip-write-flexible-kernels-grid-stride-loops/ (visited on 07/25/2016).
[12] M. J. Harris. “Fast Fluid Dynamics Simulation on the GPU”. In: “GPU gems: programming techniques, tips, and tricks for real-time graphics”. Upper Saddle River, New Jersey: Addison-Wesley, 2004.
[13] J. L. Hennessy and D. A. Patterson. “Computer architecture: a quantitative approach”. Boston: Elsevier, 2012.
[14] D. Kirk and W.-M. W. Hwu. “Programming Massively Parallel Processors: A Hands-on Approach”. Boston: Elsevier, 2010.
[15] NVIDIA Corporation. “CUDA C Programming Guide”. September. 2015. url: http://docs.nvidia.com/cuda/cuda-c-programming-guide/ (visited on 07/25/2016).
[16] NVIDIA Corporation. “CUDA Toolkit”. 2016. url: https://developer.nvidia.com/cuda-toolkit (visited on 07/26/2016).
[17] NVIDIA Corporation. “NVIDIA’s Next Generation CUDA Compute Architecture: Fermi”. In: (2009). url: http://www.nvidia.com.tw/content/PDF/fermi_white_papers/NVIDIA_Fermi_Compute_Architecture_Whitepaper.pdf.
[18] R. A. X. Persson, N. K. Voulgarakis, and J.-W. Chu. “Dynamic mesoscale model of dipolar fluids via fluctuating hydrodynamics”. In: “Journal of Chemical Physics” 141.17 (2014).
[19] S. Popinet and S. Zaleski. “A front-tracking algorithm for accurate representation of surface tension”. In: “International Journal for Numerical Methods in Fluids” 30.6 (1999), pp. 775–793.
[20] J. Sanders and E. Kandor. “CUDA By Example”. Upper Saddle River, New Jersey: Addison-Wesley, 2010.
[21] N. K. Voulgarakis and J.-W. Chu. “Bridging fluctuating hydrodynamics and molecular dynamics simulations of fluids”. In: “Journal of Chemical Physics” 130.13 (2009).
[22] N. K. Voulgarakis, B. Z. Shang, and J.-W. Chu. “Linking hydrophobicity and hydrodynamics by the hybrid fluctuating hydrodynamics and molecular dynamics methodologies”. In: “Physical Review E - Statistical, Nonlinear, and Soft Matter Physics” 88.2 (2013).