MPI平台下二维欧拉方程数值解法
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摘要
我们知道，由于流场的复杂性，要有足够多的网格点才能分辨出流场的细微结构。另外，在非定常流的模拟中，需要获得流动随时间的变化过程，这使得计算工作量变得十分巨大。为了使冗长的求解过程达到工程上可以接受的程度，以利于实际应用，必须缩短计算时间提高计算效率。目前解决问题的主要途径有两条：一是改善数值方法，利用各种收敛技术来减少计算时间；二是利用并行环境，实现软件的多机并行计算，达到大幅度降低计算时间的目的，使得某些巨型计算成为可能。
本论文采用欧拉方程作为控制方程，利用中心有限体积法进行空间离散，得到对时间变量的常微分方程组，采用龙格库塔多步法进行时间积分，加入人工粘性以消除激波和驻点附近的压力振荡等方法来对NACA0012翼型的实际流动进行并行数值模拟。
本论文的工作主要包括以下三个方面：
(1) 给出单机完成流场计算的例子，作为与并行计算比较的基础。
(2) 利用以太网来连接一组微机，以Linux操作系统和消息传递接口(Message Passing Interface, MPI)为并行程序开发环境，建立了集群式的并行计算系统。
(3) 采用MPI并行编程方法实现流场数值计算的并行化，并对影响并行效率的负载分配、通信等因素进行了研究。
As we know, much more grid points are required in order to differentiate the structure of flow field, which is extremely complex. In addition, unsteady flow simulation needs to gain flow movement along time. As a result, the computing task of the numerical simulation is very great. In order to make the long computing can be expediently applied to engineering, the computing time have to be decreased and the efficiency must be increased. There are two main ways that will be effective to solve the question currently: one is to improving numerical algorithm by all kinds of skills to accelerate convergence; the other is to introducing parallel computing.
In order to work out the numerical simulation of flow in NACA0012, Euler equations are selected as the control equations of the flow field in this paper. The main numerical method of this code is coming from scheme (Jameson, Schimit and Turkel): using cell-centered finite volume method as spatial discretization tools, and a system of ordinary differential equations for time variable is obtained, which is solved by utilizing five-step Runge-Kutta scheme as time marching method , introducing artificial dissipation to damp high frequency oscillations near the shock and stagnation point.
The thesis is focused on the following aspects:
1 .As the base of comparison with parallel computing, an example which is completed on single processor is given.
2.Cluster of workstation (COW) has been established as a parallel computing platform.
3.A parallel computing program of two-dimension flow numerical calculation has been completed. Some important techniques such as communication among nodes and load balancing are analyzed.
本论文采用欧拉方程作为控制方程，利用中心有限体积法进行空间离散，得到对时间变量的常微分方程组，采用龙格库塔多步法进行时间积分，加入人工粘性以消除激波和驻点附近的压力振荡等方法来对NACA0012翼型的实际流动进行并行数值模拟。
本论文的工作主要包括以下三个方面：
(1) 给出单机完成流场计算的例子，作为与并行计算比较的基础。
(2) 利用以太网来连接一组微机，以Linux操作系统和消息传递接口(Message Passing Interface, MPI)为并行程序开发环境，建立了集群式的并行计算系统。
(3) 采用MPI并行编程方法实现流场数值计算的并行化，并对影响并行效率的负载分配、通信等因素进行了研究。
As we know, much more grid points are required in order to differentiate the structure of flow field, which is extremely complex. In addition, unsteady flow simulation needs to gain flow movement along time. As a result, the computing task of the numerical simulation is very great. In order to make the long computing can be expediently applied to engineering, the computing time have to be decreased and the efficiency must be increased. There are two main ways that will be effective to solve the question currently: one is to improving numerical algorithm by all kinds of skills to accelerate convergence; the other is to introducing parallel computing.
In order to work out the numerical simulation of flow in NACA0012, Euler equations are selected as the control equations of the flow field in this paper. The main numerical method of this code is coming from scheme (Jameson, Schimit and Turkel): using cell-centered finite volume method as spatial discretization tools, and a system of ordinary differential equations for time variable is obtained, which is solved by utilizing five-step Runge-Kutta scheme as time marching method , introducing artificial dissipation to damp high frequency oscillations near the shock and stagnation point.
The thesis is focused on the following aspects:
1 .As the base of comparison with parallel computing, an example which is completed on single processor is given.
2.Cluster of workstation (COW) has been established as a parallel computing platform.
3.A parallel computing program of two-dimension flow numerical calculation has been completed. Some important techniques such as communication among nodes and load balancing are analyzed.
引文
[1] 傅德薰，“流体力学数字模拟”，国防工业出版社，1993．
[2] 苏铭德，黄素逸，“计算流体力学基础”，清华大学出版社，1997．
[3] 朱自强等，“应用计算流体力学”，北京航空航天大学出版社，1998．
[4] 郑纬民，“并行计算机之路”，计算机世界，2000．1
[5] J.Y. Yang, "Numerical Simulation of Transient Inviscid Shock Tube Flows",AIAA J.,Vol,25,No.2,1987,pp245-251
[6] J.Y. Yang, "Third-order Nonoscillatory Schemes for the EulerEquation",AIAA J. ,Vol.29, No, 10,1992, pp161-1618
[7] J.Y. Yang, "High-Resolution,Nonosillatory Schemes for Unsteady Compressible Flows",AIAA J.,Vol.30,No.6,1992,pp1570-1575
[8] J.Y. Yang, "Second-and Third-order Upwind Difference Schemes for Hyperbolic Conservation Laws', NASA TM 85959,1984
[9] A.Jamson, "implicit Schemes and LU Decompositions",Math,of Comp,Vol. 37, No 156,1981.pp385-397
[10] A.Jamson, "LU Implicit Schenmes with Multiple Grids for the Euler Equations",AIAA P. 86-0105
[11] A.Jamson, "Lower-Upper Implicit Schemes with Multiple Grids for Euler Equations",AIAA J.,Vol.25,No.7,1987
[12] S.Yoon, "Lower-Upper Symmetric-Gauss-Seidel Method for the Euler and Navier-Stoker Equations",AIAA J.,Voll.26,No.9,1988
[13] J.S.Shuen, "Numerical Study of Chemically Reacting Flows Using a Lower-Upper Symmetric Successive Overrelaxation Scheme",AIAA J.,Vol.27,No.12,1989
[14] S.Yoon, "An Implicit Three-Dimensional Navier-Stokes Solver for Compressible FIows",AIAA 10 Computational Fluid Dynamic Conference, 1991.
[15] L.T. Tam, "LU-SGS Implicit Schemes for Entry Vehicle Flow Computation and Comparison with Aerodynamic Data",AIAA 92-2671-CP
[16] M.M.Rai, "New Implicit Boundary Procedures: Theory and Applications",AIAA,P. 88-307-CP
[17] M.Visbal, "Evaluation of the Baldwin-Lomax Turbulence Model for Two-Dimensional Shock Wave Boundary Lager Interactions",AIAA
P.83-1697
[18] J.S.Brock, "The Baldwin-Lomax Model for Separated and Wake Flows Using an Entropy Envelope Concept",AIAA P. 92-0148
[19] 周新海，“叶栅激波流场的数值模拟与高分辨率差分格式”，工程热物理学报，Vol．10，No．3，1989．
[20] 张涵信，“无波动，无自由参数的耗散差分格式”，空气动力学报，Vol．6,No．2，(1988)pp．143-165
[21] 李超，“管道中复杂Mach反射流场的高精度数值模拟”，空气动力学报，Vol．10，No3，1992．
[22] 孙家昶，林小波，迟学斌等，“网络并行计算与分布式编程环境”，科学出版社，1996
[23] 陈国良，“并行计算-结构．算法．编程”，高等教育出版社，1999．
[24] 黄铠，徐志伟著，陆鑫达等译，“可扩展并行计算：技术、结构与编程”，机械工业出版社，2000．
[25] 都志辉，“高性能计算并行编程技术-MPI并行程序设计”，清华大学出版社，2001．
[26] 王鼎兴，郑伟民，沈美明，“并行机群的若干关键技术”，清华大学学报(自然科学版)，1998年第38卷第51期．
[27] 李立三，都志辉等，“网络并行超级计算系统THNPSC-1”，计算机学报，2001年第6期．
[28] 赵军锁，周恩强，“消息传递、PVM及MPI”，电脑与信息技术，1998年第2期．
[29] 孟国庆，周盛，“叶轮机械非定常流动研究进展”，力学进展，第27卷第2期．
[30] Message Passing Interface Forum.MPI:A message-passing interface standard. Computer Science Dept. Technical Report CS-94-230, University of Tenessee, Knoxville,TN, 1994.
[31] William Gropp Ewing Lusk, "User'Guide for mpich,a Portable Implementation of MPI",Technical Report ANL-96/6,1996.
[32] William Gropp,Ewing Lusk,Nathan Doss,and Anthony Skjellum, "A highperformance,portable implementation of the MPI Message-Passing Interface standard".Parallel Computing,22(6):789-828,1996.
[33] "Vampir 2.0 User Manual",Pallas Comp.,1999.6
[34] RS/6000 SP:Practical MPI Programming IBM Corp,1999.
[35] 赵书钦，应吉康，“Linux环境下的并行计算”，微型电脑应用，2001年第1期。
[36] 王承尧，王正华，扬晓辉，“计算流体力学及其并行算法”，国防科技大学出版社，2000．
[37] 彭德纯，邱毓兰，林子禹，“分布式并行处理技术导论”，武汉大学出版社，1996．
[38] 王保国，黄虹宾，“叶轮机械跨声速及亚声速流场的计算方法”，国防工业出版社，2000．
[39] 姚吉先，“跨音速压气机流场数值模拟与激波组织研究”，西北工业大学博士论文，1995．
[40] 扬树池，“复杂流动空气动力学数值模拟及并行计算研究”，西北工业大学博士论文，1999。
[41] 李育彬等，“多块网格网络并行计算的负载分配研究”，空气动力学报，第19卷第3期．
[42] 李育彬，“计算空气动力学软件CADR3D的集成并行计算和后处理研究”，西北工业大学博士论文，1999．
[43] 聂铁军等，“数值计算方法”，西北工业大学出版社，1990．
[44] 李栋，“翼身组合体跨音速流动的欧拉方程计算”，西北工业大学硕士论文，1994．
[45] 陈春峰，“叶轮机械三维流动数值模拟的并行算法和集群系统研究”，西北工业大学硕士论文，2002．
[46] 导弹与航天丛书(空气动力学专著)，宇航出版社，1994．
[2] 苏铭德，黄素逸，“计算流体力学基础”，清华大学出版社，1997．
[3] 朱自强等，“应用计算流体力学”，北京航空航天大学出版社，1998．
[4] 郑纬民，“并行计算机之路”，计算机世界，2000．1
[5] J.Y. Yang, "Numerical Simulation of Transient Inviscid Shock Tube Flows",AIAA J.,Vol,25,No.2,1987,pp245-251
[6] J.Y. Yang, "Third-order Nonoscillatory Schemes for the EulerEquation",AIAA J. ,Vol.29, No, 10,1992, pp161-1618
[7] J.Y. Yang, "High-Resolution,Nonosillatory Schemes for Unsteady Compressible Flows",AIAA J.,Vol.30,No.6,1992,pp1570-1575
[8] J.Y. Yang, "Second-and Third-order Upwind Difference Schemes for Hyperbolic Conservation Laws', NASA TM 85959,1984
[9] A.Jamson, "implicit Schemes and LU Decompositions",Math,of Comp,Vol. 37, No 156,1981.pp385-397
[10] A.Jamson, "LU Implicit Schenmes with Multiple Grids for the Euler Equations",AIAA P. 86-0105
[11] A.Jamson, "Lower-Upper Implicit Schemes with Multiple Grids for Euler Equations",AIAA J.,Vol.25,No.7,1987
[12] S.Yoon, "Lower-Upper Symmetric-Gauss-Seidel Method for the Euler and Navier-Stoker Equations",AIAA J.,Voll.26,No.9,1988
[13] J.S.Shuen, "Numerical Study of Chemically Reacting Flows Using a Lower-Upper Symmetric Successive Overrelaxation Scheme",AIAA J.,Vol.27,No.12,1989
[14] S.Yoon, "An Implicit Three-Dimensional Navier-Stokes Solver for Compressible FIows",AIAA 10 Computational Fluid Dynamic Conference, 1991.
[15] L.T. Tam, "LU-SGS Implicit Schemes for Entry Vehicle Flow Computation and Comparison with Aerodynamic Data",AIAA 92-2671-CP
[16] M.M.Rai, "New Implicit Boundary Procedures: Theory and Applications",AIAA,P. 88-307-CP
[17] M.Visbal, "Evaluation of the Baldwin-Lomax Turbulence Model for Two-Dimensional Shock Wave Boundary Lager Interactions",AIAA
P.83-1697
[18] J.S.Brock, "The Baldwin-Lomax Model for Separated and Wake Flows Using an Entropy Envelope Concept",AIAA P. 92-0148
[19] 周新海，“叶栅激波流场的数值模拟与高分辨率差分格式”，工程热物理学报，Vol．10，No．3，1989．
[20] 张涵信，“无波动，无自由参数的耗散差分格式”，空气动力学报，Vol．6,No．2，(1988)pp．143-165
[21] 李超，“管道中复杂Mach反射流场的高精度数值模拟”，空气动力学报，Vol．10，No3，1992．
[22] 孙家昶，林小波，迟学斌等，“网络并行计算与分布式编程环境”，科学出版社，1996
[23] 陈国良，“并行计算-结构．算法．编程”，高等教育出版社，1999．
[24] 黄铠，徐志伟著，陆鑫达等译，“可扩展并行计算：技术、结构与编程”，机械工业出版社，2000．
[25] 都志辉，“高性能计算并行编程技术-MPI并行程序设计”，清华大学出版社，2001．
[26] 王鼎兴，郑伟民，沈美明，“并行机群的若干关键技术”，清华大学学报(自然科学版)，1998年第38卷第51期．
[27] 李立三，都志辉等，“网络并行超级计算系统THNPSC-1”，计算机学报，2001年第6期．
[28] 赵军锁，周恩强，“消息传递、PVM及MPI”，电脑与信息技术，1998年第2期．
[29] 孟国庆，周盛，“叶轮机械非定常流动研究进展”，力学进展，第27卷第2期．
[30] Message Passing Interface Forum.MPI:A message-passing interface standard. Computer Science Dept. Technical Report CS-94-230, University of Tenessee, Knoxville,TN, 1994.
[31] William Gropp Ewing Lusk, "User'Guide for mpich,a Portable Implementation of MPI",Technical Report ANL-96/6,1996.
[32] William Gropp,Ewing Lusk,Nathan Doss,and Anthony Skjellum, "A highperformance,portable implementation of the MPI Message-Passing Interface standard".Parallel Computing,22(6):789-828,1996.
[33] "Vampir 2.0 User Manual",Pallas Comp.,1999.6
[34] RS/6000 SP:Practical MPI Programming IBM Corp,1999.
[35] 赵书钦，应吉康，“Linux环境下的并行计算”，微型电脑应用，2001年第1期。
[36] 王承尧，王正华，扬晓辉，“计算流体力学及其并行算法”，国防科技大学出版社，2000．
[37] 彭德纯，邱毓兰，林子禹，“分布式并行处理技术导论”，武汉大学出版社，1996．
[38] 王保国，黄虹宾，“叶轮机械跨声速及亚声速流场的计算方法”，国防工业出版社，2000．
[39] 姚吉先，“跨音速压气机流场数值模拟与激波组织研究”，西北工业大学博士论文，1995．
[40] 扬树池，“复杂流动空气动力学数值模拟及并行计算研究”，西北工业大学博士论文，1999。
[41] 李育彬等，“多块网格网络并行计算的负载分配研究”，空气动力学报，第19卷第3期．
[42] 李育彬，“计算空气动力学软件CADR3D的集成并行计算和后处理研究”，西北工业大学博士论文，1999．
[43] 聂铁军等，“数值计算方法”，西北工业大学出版社，1990．
[44] 李栋，“翼身组合体跨音速流动的欧拉方程计算”，西北工业大学硕士论文，1994．
[45] 陈春峰，“叶轮机械三维流动数值模拟的并行算法和集群系统研究”，西北工业大学硕士论文，2002．
[46] 导弹与航天丛书(空气动力学专著)，宇航出版社，1994．