臺灣博碩士論文加值系統

有限體積法(FVM)的應用在計算流體力學中有普遍增加的趨勢。其中有許多方法專注於通量計算，像是早期的平衡通量法(Equilibrium Flux Method, EFM)，其通量計算是經由積分速度機率分佈函數而取得。然而，在數值計算中會產生指數函數與誤差函數，在計算上這不僅費時並且還有可能導致截斷誤差的發生。本研究的目的在於探討三角化平衡通量法(Triangular Equilibrium Flux Method, TEFM)，此方法利用較為簡單的分佈函數總和近似馬克斯威爾-波茲曼速度機率分佈，進而取代原本計算昂貴的方程式，並使用MPI與A系列的AMD APU於平行計算中。為了要進一步提高計算效能，使用多重圖型處理器(MPI-CUDA)平行模式以達更高的模擬效率。透過已存在的數值模擬結果，來驗證一階與二階空間精準度的差別與正確性。

Kinetic-theory based Finite Volume Methods (FVM) have become increasingly common in Computational Fluid Dynamics (CFD). Many of these methods focus on computation of fluxes through integrating a velocity probability distribution function, such as the early Equilibrium Flux Method (EFM). However, analytical computation of these moments results in the exponential function and error function which is not only time-consuming but may also introduce error through truncation. The purpose of this study is to investigate an alternative – the Triangular Equilibrium Flux Method (TEFM) which uses the sum of simpler distribution functions to approximate the Maxwell-Boltzmann equilibrium velocity probability distribution function – applied to parallel computing using MPI with a specific focus on the A-series of AMD APU’s. To further enhance the performance, the hybrid MPI-CUDA parallelization paradigm is employed with multiple GPUs to achieve higher simulation efficiency. Furthermore, several benchmarks are used to verify both first and second order numerical results

中文摘要 i
Abstract iii
Acknowledgements v
Table of Contents vi
List of Tables ix
List of Figures x
Nomenclature xvi
Chapter 1 Introduction 1
1.1 Computational Fluid Dynamics 1
1.2 Governing Equations 1
1.2.1 Navier-Stokes Equation and Euler Equation 1
1.3 Finite Volume Methods 3
1.3.1 CFL number 5
1.4 High Order Scheme 6
1.5 Equilibrium Flux Method 11
1.6 True Direction Equilibrium Flux Method 13
1.7 Uniform Distribution Equilibrium Flux Method 18
1.8 Parallel Computing 23
1.8.1 Parallel Computing Theory 23
1. 9 Message Passing Interface (MPI) 28
1.10 Graphical Processing Unit 31
1.10.1 CUDA Memory 31
1.10.2 CUDA Threads, Blocks and Grids 32
1.10.3 Using Separate Compilation in CUDA 35
Chapter 2 Methodology 36
2.1 Uniform Equilibrium Flux Method 36
2.2 Triangular Equilibrium Flux Method 39
2.3 Determination of Weighting Fractions and Characteristic Thermal Velocities 43
2.3.1 UEFM 43
2.3.2 TEFM 45
2.4 Analysis of Dissipative Qualities of First Order TEFM 47
2.5 Second Order Extension of Spatial Accuracy 49
2.6 MPI Parallelization 50
2.6.1 MPI Implementation 50
2.6.2 Compiling and Executing MPI Program in Linux 52
2.7 Hybrid MPI-CUDA Parallelization 53
2.7.1 Hybrid MPI-CUDA Implementation 54
2.7.2 Compiling and Building Hybrid MPI-CUDA Applications in Linux 54
Chapter 3 Numerical Results and Parallel Performance 56
3.1 One-Dimensional Shock Tube Problem 56
3.2 Shock Bubble Interaction 58
3.3 Euler Four Shocks Problem 60
3.4 Euler Four Contact Problem 62
3.5 Analysis of Parallel Performance 63
3.5.1 MPI Parallel Performance 63
3.5.2 Parallel Performance using Multi-GPUs 65
Chapter 4 Conclusion 66
References 68
Tables 71
Figures 78

[1] Toro E.F., “Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin, 2009.
[2] Sod G.A., “A Survey of Serval Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws, J. Comp. Phys., Vol. 27, pp. 1-31, 1978.
[3] C ̌ada M. and Torrilhon M., “Compact Third-Order Limiter Functions for Finite Volume Methods, J. Comp. Phys., Vol. 228, pp. 4118-4145, 2009.
[4] Schulz-Rinne C.W., Collins J.P., and H.M. Glaz, “Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics, SIAM J. Sci Compt., Vol. 14, pp. 1394-1414, 1993.
[5] LeVeque R.J., “Wave Propagation Algorithms for Multidimensional Hyperbolic Systems, J. Comp. Phys., Vol. 131, pp. 327-353, 1997.
[6] Ferguson A., Smith M.R. and Wu J.-S., “Accurate True Direction Solutions to the Euler Equations Using a Uniform Distribution Equilibrium Method, CMES: Computer Modeling in Engineering ＆ Sciences, Vol. 63, No. 1, 2010.
[7] Macrossan M.N., “The Equilibrium Flux Method for the Calculation of Flows with Non-equilibrium Chemical Reactions, J, Comp. Phys., Vol. 80, pp. 204-231., 1989.
[8] Roe P.L., “Characteristic-Based Schemes for the Euler Equations, Ann. Rev. Fluid Mech. Vol. 18, pp. 337-365, 1986.
[9] Van Leer B., “Towards the Ultimate Conservative Difference Scheme III. Upstream-Centered Finite-Difference Schemes for Ideal Compressible Flow, J. Comp. Phys., Vol. 23, Issue 3, pp. 263-275, 1977.
[10] Su C.-C., Smith M.R., Kuo F.-A., Wu J.-S., Hsieh C.-W., Tseng K.-C., “Large-scale simulations on multiple Graphics Processing Units (GPUs) for the direct simulation Monte Carlo method, J. Comp. Phys., Vol. 231, pp. 7932-7958, 2012.
[11] Chen Y.-C., Smith M.R., Ferguson A., “Analysis of the Second Order Accurate Uniform Equilibrium Flux Method and its graphics processing unit acceleration, Journal of Computers and Fluids, 110: pp. 9-18, 2015.
[12] Versteeg H.K., Malalaserkera W., “An introduction to computational fluid dynamics The finite volume method, Longman Scientific ＆ Technical, 1995.
[13] Cebeci T., Shao J.P., Kafyeke F. and Laurendeau E., “Computational Fluid Dynamics for Engineers From Panel to Navier-Stokes Methods with Computer Programs. Horizons Publishing Inc., California, 2005.
[14] C ̌ada M. and Torrilhon M., “Compact Third-Order Limiter Functions for Finite Volume Methods, J. Comp. Phys., Vol. 288, pp. 4118-4145, 2009.
[15] Hirsch, Charles. “Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics: The Fundamentals of Computational Fluid Dynamics. Butterworth-Heinemann, 2007.
[16] Smith M.R., “The True Direction Equilibrium Flux Method and its Application, The University of Queensland, 2008.
[17] Pullin D.I., “Direct Simulation Methods for Compressible Inviscid Ideal-Gas Flow, J. Comp. Phys., Vol. 34, pp. 231-244, 1980.
[18] Rusanov V.V., “The Calculation of the Interaction of Non-stationary Shock Waves and Obstacles, J. Comput. Math. Phys. USSR, Vol. 1, pp. 267-279, 1961.
[19] Smith M.R., Macrossan M.N. and Abdel-jawad M.M., “Effects of Direction Decoupling in Flux Calculation in Euler Solvers, J. Comp. Phys., Vol. 227, pp. 4142-4161, 2008.
[20] Harten A., “High Resolution Schemes for Hyperbolic Conservation Laws , J. Comp. Phys., Vol. 49, pp. 357-393, 1983.
[21] Smith M.R., “Introduction to GPU and Multi-Core Computation, 2014.
[22] Amdahl G.M., “Validity of the Single Processor Approach to Achieving Large-Scale Computing Capabilities, AFIPS Conference Proceedings, Vol. 30, pp. 483-485, 1967.
[23] Gustafson J.L., “Reevaluating Amdahl’s Law, Communications of the ACM, Vol. 31, Number 5, pp. 532-533, 1988.
[24] Sweby P.K., “High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws, SIAM J. Numer. Anal., Vol. 21, pp. 995-1011, 1984.
[25] Lax P.D. and Wendroff B., “Systems of Conservation Laws, Comm. Pure Appl. Math., Vol. 13, pp. 217-237, 1960.
[26] Van Leer B., “Towards the Ultimate Conservative Difference Scheme II. Monotonicity and Conservation Combined in a Second Order Scheme, J. Comp. Phys., Vol. 14, pp. 361-370, 1974.
[27] Pacheco, Peter S. “Parallel programming with MPI. Morgan Kaufmann, 1997.
[28] Hwang, Kai, Jack Dongarra, and Geoffrey C. Fox. Distributed and cloud computing: from parallel processing to the internet of things. Morgan Kaufmann, 2013.
[29] Benzi, John; Damodaran, M., “Parallel Three Dimensional Direct Simulation Monte Carlo for Simulating Micro Flows. Parallel Computational Fluid Dynamics 2007: Implementations and Experiences on Large Scale and Grid Computing. Parallel Computational Fluid Dynamics. Springer. p. 95.