三维特征线方法的并行与加速方法研究
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摘要
三维特征线方法能够精确求解任意几何中子输运问题，是下一代反应堆物理分析的重要研究方向之一。但是，特征线方法具有收敛缓慢、计算耗时及内存占用大的不足，用于三维几何求解时计算量变得更加庞大，这些从根本上限制了三维特征线方法的发展与应用。本文针对三维特征线方法的不足，以三维特征线方法的并行和加速方法作为研究对象，主要开展了以下研究：
首先，研究了基于中央处理器（CPU）并行的特征线方法。采用角度并行方式，不引入额外计算量同时保持与串行程序的一致性；提出角度分组的方法，减少了通信，提高了并行效率。
其次，将新兴的高性能计算图形处理器（GPU）用于特征线方法加速计算的研究。相比于CPU，GPU拥有更多的计算核心，计算成本更低，加速效果更好。针对其硬件构造的特殊性，采用射线并行的方式，并结合CPU并行，实现了多GPU系统计算。
再次，在借助计算机硬件加速特征线方法以外，提出了双重广义粗网有限差分加速方法。该方法能够同时对任意形状的网格和能群进行加速。
最后，提出插值方法处理的三维模块式建模，继承了模块式建模占用存储少建模速度快的优点，解决了传统模块式建模无法处理非基本模块的问题。
基于上述研究内容，研制了具有模块式建模功能、采用双重广义粗网有限差分加速、CPU和GPU并行计算的三维特征线方法程序SPIDER。数值计算结果表明：程序具有较高计算精度，通过并行和加速后程序计算时间大大缩短；插值方法处理的模块式建模方法在减少网格划分和射线求交的时间、加快建模速度的基础上实现了对存在水隙等非模块的组件的栅元模块式建模；双重广义粗网有限差分加速方法能够同时从能群和网格上进行加速；角度分组的CPU并行方式不会增加迭代计算次数，同时能够达到较高的并行效率；GPU并行计算能够以较低的计算成本实现更高的加速比。
针对三维特征线方法的特点，本文研究了适用于三维特征线方法的并行算法和加速方法，加快了收敛速度，减少了计算时间，增强了三维特征线方法及程序的工程实用性。
Since the method of characteristics (MOC) can solve neutron transportequation for arbitrary geometry accurately, it becomes one of the most promisingmethods for the next generation in reactor analysis. However, the MOC has somedrawbacks: The convergence speed is slow and very time consuming. There arealso restrictions on the development of the three-dimensional MOC such as thehuge amount of computation in practical three-dimensional geometric computing.In order to resolve these disadvantages, this thesis focuses on the parallelizationand acceleration of three-dimensional MOC, and the research contents are asfollowed.
First, the MOC parallelized by central processing unit (CPU) has beenstudied. In order to keep consistent with serial code and not to increase extracomputational burden, an angular parallel scheme is adopted. Moreover, theangular grouping method is proposed to reduce the communication and thusimprove the efficiency.
Then, the graphics processing unit (GPU) computing, which is a newlyrising high performance computing, is applied to the MOC. As the number of theGPU computing core is much more, a ray parallel scheme is chosen in GPUcomputing. Combined with the CPU parallel method, the MOC is performedunder multiple GPUs.
Besides the parallel computing, another acceleration method calledtwo-level generalized coarse mesh finite difference (GCMFD) is proposed. Thetwo-level GCMFD, which can accelerate the energy group as well as the arbitrarycoarse mesh, is used to accelerate the MOC in this paper.
Finally, because the reactor cores are usually composed of reduplicateconstructions and the water gaps exist between assemblies, an interpolatedthree-dimensional modular ray tracing is proposed. The modular ray tracing canreduce the memory requirement and accelerate geometric modeling while theinterpolated technique can simulate the water gaps between the assemblies.
According to the methods mentioned above, the SPIDER code has beendeveloped. The numerical results show that a good accelerated performance on both modeling and computing of the MOC has been achieved under the highaccuracy.
This thesis focuses on the study of the parallel computing and acceleratedmethod. As a result, it accelerates the three-dimensional MOC, decreases thenumber of iterations, reduces the computing time and enhances its engineeringpracticability.
首先，研究了基于中央处理器（CPU）并行的特征线方法。采用角度并行方式，不引入额外计算量同时保持与串行程序的一致性；提出角度分组的方法，减少了通信，提高了并行效率。
其次，将新兴的高性能计算图形处理器（GPU）用于特征线方法加速计算的研究。相比于CPU，GPU拥有更多的计算核心，计算成本更低，加速效果更好。针对其硬件构造的特殊性，采用射线并行的方式，并结合CPU并行，实现了多GPU系统计算。
再次，在借助计算机硬件加速特征线方法以外，提出了双重广义粗网有限差分加速方法。该方法能够同时对任意形状的网格和能群进行加速。
最后，提出插值方法处理的三维模块式建模，继承了模块式建模占用存储少建模速度快的优点，解决了传统模块式建模无法处理非基本模块的问题。
基于上述研究内容，研制了具有模块式建模功能、采用双重广义粗网有限差分加速、CPU和GPU并行计算的三维特征线方法程序SPIDER。数值计算结果表明：程序具有较高计算精度，通过并行和加速后程序计算时间大大缩短；插值方法处理的模块式建模方法在减少网格划分和射线求交的时间、加快建模速度的基础上实现了对存在水隙等非模块的组件的栅元模块式建模；双重广义粗网有限差分加速方法能够同时从能群和网格上进行加速；角度分组的CPU并行方式不会增加迭代计算次数，同时能够达到较高的并行效率；GPU并行计算能够以较低的计算成本实现更高的加速比。
针对三维特征线方法的特点，本文研究了适用于三维特征线方法的并行算法和加速方法，加快了收敛速度，减少了计算时间，增强了三维特征线方法及程序的工程实用性。
Since the method of characteristics (MOC) can solve neutron transportequation for arbitrary geometry accurately, it becomes one of the most promisingmethods for the next generation in reactor analysis. However, the MOC has somedrawbacks: The convergence speed is slow and very time consuming. There arealso restrictions on the development of the three-dimensional MOC such as thehuge amount of computation in practical three-dimensional geometric computing.In order to resolve these disadvantages, this thesis focuses on the parallelizationand acceleration of three-dimensional MOC, and the research contents are asfollowed.
First, the MOC parallelized by central processing unit (CPU) has beenstudied. In order to keep consistent with serial code and not to increase extracomputational burden, an angular parallel scheme is adopted. Moreover, theangular grouping method is proposed to reduce the communication and thusimprove the efficiency.
Then, the graphics processing unit (GPU) computing, which is a newlyrising high performance computing, is applied to the MOC. As the number of theGPU computing core is much more, a ray parallel scheme is chosen in GPUcomputing. Combined with the CPU parallel method, the MOC is performedunder multiple GPUs.
Besides the parallel computing, another acceleration method calledtwo-level generalized coarse mesh finite difference (GCMFD) is proposed. Thetwo-level GCMFD, which can accelerate the energy group as well as the arbitrarycoarse mesh, is used to accelerate the MOC in this paper.
Finally, because the reactor cores are usually composed of reduplicateconstructions and the water gaps exist between assemblies, an interpolatedthree-dimensional modular ray tracing is proposed. The modular ray tracing canreduce the memory requirement and accelerate geometric modeling while theinterpolated technique can simulate the water gaps between the assemblies.
According to the methods mentioned above, the SPIDER code has beendeveloped. The numerical results show that a good accelerated performance on both modeling and computing of the MOC has been achieved under the highaccuracy.
This thesis focuses on the study of the parallel computing and acceleratedmethod. As a result, it accelerates the three-dimensional MOC, decreases thenumber of iterations, reduces the computing time and enhances its engineeringpracticability.
引文
[1]裴鹿成,张孝泽.蒙特卡罗方法及其在粒子输运问题中的应用[M].北京:科学出版社,1980.
[2]许淑艳.蒙特卡罗方法在实验核物理中的应用[M].北京:原子能出版社,2006.
[3]赵荣安.当前轻水堆堆芯分析程序理论与方法上存在的主要问题及其发展方向
[R].成都:2011.
[4] Cho J-Y, Kim K-S, Lee C-C, et al. Axial SPNand Radial MOC Coupled Whole CoreTransport Calculation[J]. Journal of NUCLEAR SCIENCE and TECHNOLOGY,2007,44(9):1156-1171.
[5] Sugimura N, Yamamoto A, Ushio T, et al. Neutron Transport Models of AEGIS: AnAdvanced Next-Generation Neutronics Design System[J]. Nuclear Science andEngineering,2007,155:276-289.
[6] Yamamoto A, Tabuchi M. Advanced Lattice Physics Code, AEGIS[C]. PHYSOR2010Workshop, May9, Pittsburgh, USA.2010.
[7] Knott D, Edenius M. Validation of the CASMO-4transport solution[C]. Proc. Int.Mtg. Math-ematical Methods and Supercomputing in Nuclear Applications,19-23April, Karlsruhe, Germany.1993,2:547.
[8] Halsall M J. WIMS7, an overview[C]. Proc. Int. Conf. on Physics of Reactors, Physor96,16-20September, Mito, Ibaraki, Japan.1996.
[9] Cho J-Y, Kim K-S, Shim H-J, et al. Whole Core Transport Calculation EmployingHexagonal Modular Ray Tracing and CMFD Formulation[J]. Journal of NUCLEARSCIENCE and TECHNOLOGY,2008,45:740-751.
[10] Tang C, Zhang S. Development and verification of an MOC code employing assemblymodular ray tracing and efficient acceleration techniques[J]. Annals of NuclearEnergy,2009,36:1013-1020.
[11]柴晓明.求解三维中子输运问题的特征线方法研究[博士学位论文].北京:清华大学工程物理系,2010.
[12] Liu Z, Wu H, Cao L, et al. A new three-dimensional method of characteristics for theneutron transport calculation[J]. Annals of Nuclear Energy,2011,38:447-454.
[13] Santandrea S, Sanchez R. Acceleration techniques for the characteristic method inunstructured meshes[J]. Annals of Nuclear Energy,2002,29:323-352.
[14] Wu G J, Roy R. Acceleration techniques for trajectory-based deterministic3Dtransport solvers[J]. Annals of Nuclear Energy,2003,30:567-583.
[15] Yamamoto A. Generalized Coarse-Mesh Rebalance Method for Acceleration ofNeutron Transport Calculations[J]. Nuclear Science and Engineering,2005,151:274-282.
[16] Kim K-S, DeHart M D. Unstructured partial-and net-current based coarse mesh finitedifference acceleration applied to the extend step characteristics method in NEWT[J].Annals of Nuclear Energy,2010,38:527-534.
[17] Dahmani M, Roy R. Parallel Solver Based on the Three Dimensional CharacteristicsMethod-Design and Performance Analysis[J]. Nuclear Science and Engineering,2005,150:155-169.
[18] Yamaji K, Matsumoto H, Sato D, et al. Simple and Efficient Parallelization Methodfor MOC Calculation[J]. Journal of NUCLEAR SCIENCE and TECHNOLOGY,2010,47(1):90-102.
[19] Kirschenmann W, Plagne L, Ploix S, et al. Massively Parallel Solving of3DSimplified PNEquations on Graphics Processing Units[C]. International Conferenceon Mathematics, Computational Methods&Reactor Physics (M&C2009), May3-7,Saratoga Springs, New York.2009.
[20] Nelson A G. Monte Carlo Methods for Neutron Transport on Graphics ProcessingUnits Using CUDA[D]. The Pennsylvania State University Department of Mechanicaland Nuclear Engineering,2009.
[21] Ding A, Liu T, Liang C, et al. Evaluation of Speedup of Monte Carlo Calculations ofTwo Simple Reactor Physics Problems Coded for the GPU-CUDA Environment[C].International Conference on Mathematics and Computational Methods Applied toNuclear Science and Engineering (M&C2011) May8-12, Rio de Janeiro, RJ, Brazi.2011.
[22] Jamelot E, Dubois J, Lautard J J, et al. High Performance3D Neutron Transport onPetascale and Hybrid Architectures within APOLLO3Code[C]. InternationalConference on Mathematics and Computational Methods Applied to Nuclear Scienceand Engineering (M&C2011) May8-12, Rio de Janeiro, RJ, Brazi.2011.
[23] Kodama Y, Tatsumi M, Ohoka Y. Study on GPU Computing for SCOPE2withCUDA[C]. International Conference on Mathematics and Computational MethodsApplied to Nuclear Science and Engineering (M&C2011) May8-12, Rio de Janeiro,RJ, Brazi.2011.
[24] Kosaka S, Saji E. Transport Theory Calculation for a Heterogeneous Multi-AssemblyProblem by Characteristics Method with Direct Neutron Path Linking Technique [J].Journal of NUCLEAR SCIENCE and TECHNOLOGY,2000,37(12):1105-1023.
[25] Askew J R. A Characteristics Formulation of the Neutron Transport Equation inComplicated Geometries[R]. AEEW-M1108,1972.
[26]汤春桃,张少泓. CMFD加速在特征线法输运计算中的应用[J].核动力工程,2009,30(5):8-12.
[27]柴晓明,姚栋,王侃,等.使用广义粗网有限差分方法加速中子输运特征线方法[J].计算物理,2010,27(4):541-547.
[28] Jevremovic T, Vujic J, Tsuda K. ANEMONA—a neutron transport code for generalgeometry reactor assemblies based on the method of characteristics and R-functionsolid modeler[J]. Annals of Nuclear Energy,2001,28:125-152.
[29] Sanchez R, Mao L, Santandrea S. Treatment of Boundary Conditions inTrajectory-Based Deterministic Transport Methods[J]. Nuclear Science andEngineering,2002,140:23-50.
[30] Longoni G. Advanced Quadrature Sets, Acceleration and Preconditioning Techniquesfor the Discrete Ordinates Method in Parallel Computing Environments[D].University of Florida,2004.
[31] Yamamoto A, Tabuchi M, Sugimura N, et al. Derivation of Optimum Polar AngleQuadrature Set for the Method of Characteristics Based on Approximation Error forthe Bickley Function[J]. Journal of NUCLEAR SCIENCE and TECHNOLOGY,2007,44(2):129-136.
[32]谢仲生,邓力.中子输运理论数值计算方法[M].西安:西北工业大学出版社,2005.
[33] Lee G S, Cho N Z, Hong S G. Acceleration and Parallelization of the Method ofCharacteristics for Lattice and Whole-Core Heterogeneous Calculations[C]. PHYSOR2000, May7-11, Pittsburg, USA.2000.
[34] Gelbard E M, Hageman L A. The Synthetic Method as Applied to the SNEquations[J].Nuclear Science and Engineering,1969,37(2):288-298.
[35] Alcouffe R E. Diffusion Synthetic Accelemtion Methods for Diamond-differencedDiscrete-ordinates Equations[C]. ANS Mathematics and Computations TopicalMeeting, March28-30, Tucson Arizona.1977.
[36] Ramone G L, Adams M L. A Transport Synthetic Acceleration Method for TransportIterations[J]. Nuclear Science and Engineering,1997,125:257-283.
[37] Zika M R, Adams M L. Transport Synthetic Acceleration for Long-CharacteristicsAssembly-Level Transport Problems[J]. Nuclear Science and Engineering,2000,134:135-158.
[38] Sanchez R, Chetaine A. A synthetic acceleration for a two-dimensional characteristicmethod in unstructured meshes [J]. Nuclear Science and Engineering,2000,136:122-139.
[39] Santandrea S, Sanchez R. Analysis and improvements of the DPN accelerationtechnique for the method of characteristics in unstructured meshes[J]. Annals ofNuclear Energy,2005,32:163-193.
[40] Wu G J, Roy R. Self-collision rebalancing technique for the MCI characteristicssolver[C]. Twentieth Annual Conf. Canadian Nuclear Society, Montreal.1999.
[41] Suslov I R. An Algebraic Collapsing Acceleration in Long Characteristics TransportTheory[C]. Proc.11th Symp. Atomic Energy Research, Csopak, Hungary.2001.
[42] Tellier R L, Hebert A. An Improved Algebraic Collapsing Acceleration with GeneralBoundary Conditions for the Characteristics Method[J]. Nuclear Science andEngineering,2007,156:121-138.
[43] Park C J, Cho N Z. Convergence Analysis of Additive Angular Dependent RebalanceAcceleration for the Discrete Ordinates Transport Calculations[J]. Nuclear Scienceand Engineering,2002,142:64-74.
[44] Park Y R, Cho N Z. Coarse-Mesh Angular Dependent Rebalence Acceleration ofDiscrete Ordinates Transport Calculations[J]. Nuclear Science and Engineering,2004,148:355-373.
[45] Dahmani M, Tellier R L, Roy R, et al. An efficient preconditioning technique usingKrylov subspace methods for3D characteristics solvers[J]. Annals of Nuclear Energy,2005,32:876-896.
[46] Palmiotti G, Smith M, Rabiti C, et al. UNìC: Ultimate Neutronic InvestigationCode[C]. Joint International Topical Meeting on Mathematics&Computation andSupercomputing in Nuclear Applications (M&C+SNA2007) April15-19,2007,Monterey, California.2007.
[47] Siegel S F, Siegel A R, Rabiti C. UNIC: Code Algorithmic Specification of theMethod of Long Characteristics[R]. Technical Memorandum ANL/MCS-TM-301,Argonne National Laboratory,2008.
[48]汤春桃.中子输运方程特征线解法及嵌入式组件均匀化方法的研究[博士学位论文].上海:上海交通大学机械与动力工程学院,2009.
[49] Zhang H, Wu H, Cao L. An acceleration technique for2D MOC based on Krylovsubspace and dimain decomposition methods[J]. Annals of Nuclear Energy,2011,38(12):2742-2751.
[50] Ohoka Y, Tatsumi M. Study on the acceleration of the neutronics calculation based onGPGPU[C]. Joint International Topical Meeting on Mathematics&Computation andSupercomputing in Nuclear Applications (M&C+SNA2007), April, Monterey,California.2007.
[51] NVIDIA Corporation. NVIDIA CUDA C Programming Guide. In2011.
[52]徐琪,余纲林,吴小飞,等.基于GPU加速的蒙卡中子输运方法研究[C].核反应堆系统设计重点实验室年会,成都.2012.
[53] Suslov I. Improvements in the Long Characteristics Method and Their Efficiency forDeep Penetration Calculations [J]. Progress in Nuclear Energy,2001,39(2):233-243.
[54] Goldberg L, Vujic J, Leonard A. The Method of Characteristics in GeneralGeometry[J]. Transactions of the American Nuclear Society,1995,73:173.
[55] Févotte F, Santandrea S, Sanchez R. Advanced Transverse Integration for the Methodof Characteristics[C]. Joint International Topical Meeting on Mathematics&Computation and Supercomputing in Nuclear Applications (M&C+SNA2007), April15-19, Monterey, California.2007.
[56]柴晓明,姚栋,王侃.基于特征线方法的三维中子输运程序(I)——边界条件的插值处理[J].核动力工程,2010,31(2):11-15.
[57] Joo H G, Cho J Y, Kim K S. Methods and Performance of a Three DimensionalWhole-Core Transport Code DeCART[C]. PHYSOR2004, April25-29, Chicago, ILUSA.2004.
[58] Dahmani M, Roy R, Koclas J. New Computational Methodology for Large3DNeutron Transport Problems [C]. PHYSOR2004, April25-29, Chicago, Illinois.American Nuclear Society,2004.
[59] Smith K S, Rhodes J D. Full-Core,2-D, LWR Core Calculations with CASMO-4E[C].PHYSOR2002, October7-10, Seoul, Korea.2002.
[60]莫则尧,陈军,曹小林译.并行计算综论[M].北京:电子工业出版社,2005.
[61]张林波,迟学斌,莫则尧,等.并行计算导论[M].北京:清华大学出版社,2006.
[62]张舒,褚艳利. GPU高性能运算之CUDA[M].北京:中国水利水电出版社,2009.
[63] Cho N Z, Lee G S, Park C J. Fusion of Method of Characteristics and Nodal Methodfor3-D Whole-Core Transport Calculations[J]. Transactions of the American NuclearSociety,2002,86:322.
[64] Smith K S, Rhodes J D. CASMO-4Characteristics Method for Two-DimensionalPWR and BWR Core Calculations[J]. Transactions of the American Nuclear Society,2000,83:292.
[65]李庆扬,关治,白峰衫.数值计算原理[M].北京:清华大学出版社,2000.
[66] Ushio T, Takeda T, Mori M. Neutron Anisotropic Scattering Effect in HeterogeneousCell Calculations of Light Water Reactors[J]. Journal of NUCLEAR SCIENCE andTECHNOLOGY,2003,40(7):464-480.
[67] Postma T, Vujic J. The Method of Characteristics in General Geometry withAnisotropic Scattering[C]. Proc. Int. Conf. Mathematics and Computation, Reactorphysics and Environmental Analysis in Nuclear Application, Sept.27-30, Madrid,Spain.1999.
[68] Petrovic I, Benoist P, Marleau G. A Quasi-Isotropic Reflecting Boundary Conditionfor the TIBERE Heterogeneous Leakage Model[J]. Nuclear Science and Engineering,1996,122:151-166.
[69] Stepanek J, Auerbach T, Haelg W. Calculation of Four Thermal Reactor Benchmark,Problems in X-Y Geometry[R]. EPRI NP-2855,1983.
[70] OECD/NEA. Benchmark on Deterministic Transport Calculations Without SpatialHomogenisation-A2-D/3-D MOX Fuel Assembly Benchmark[R]. Nuclear ScienceNEA/NSC/DOC,2003.
[71] Okumura K, Kugo T, Kaneko K, et al. SRAC2006: A Comperhensive NeutronicsCalculation Code System[R]. Japan Atomic Energy Agency,2007.
[72] Lau J. Experience with CANFLEX Fuel[R]. AECL,2001.
[73] Takeda T, Ikeda H.3-D Neutron Transport Benchmarks[R]. Osaka: Technical ReportOECD/NEA Committee on Reactor Physics (NEACRP),1991.
[2]许淑艳.蒙特卡罗方法在实验核物理中的应用[M].北京:原子能出版社,2006.
[3]赵荣安.当前轻水堆堆芯分析程序理论与方法上存在的主要问题及其发展方向
[R].成都:2011.
[4] Cho J-Y, Kim K-S, Lee C-C, et al. Axial SPNand Radial MOC Coupled Whole CoreTransport Calculation[J]. Journal of NUCLEAR SCIENCE and TECHNOLOGY,2007,44(9):1156-1171.
[5] Sugimura N, Yamamoto A, Ushio T, et al. Neutron Transport Models of AEGIS: AnAdvanced Next-Generation Neutronics Design System[J]. Nuclear Science andEngineering,2007,155:276-289.
[6] Yamamoto A, Tabuchi M. Advanced Lattice Physics Code, AEGIS[C]. PHYSOR2010Workshop, May9, Pittsburgh, USA.2010.
[7] Knott D, Edenius M. Validation of the CASMO-4transport solution[C]. Proc. Int.Mtg. Math-ematical Methods and Supercomputing in Nuclear Applications,19-23April, Karlsruhe, Germany.1993,2:547.
[8] Halsall M J. WIMS7, an overview[C]. Proc. Int. Conf. on Physics of Reactors, Physor96,16-20September, Mito, Ibaraki, Japan.1996.
[9] Cho J-Y, Kim K-S, Shim H-J, et al. Whole Core Transport Calculation EmployingHexagonal Modular Ray Tracing and CMFD Formulation[J]. Journal of NUCLEARSCIENCE and TECHNOLOGY,2008,45:740-751.
[10] Tang C, Zhang S. Development and verification of an MOC code employing assemblymodular ray tracing and efficient acceleration techniques[J]. Annals of NuclearEnergy,2009,36:1013-1020.
[11]柴晓明.求解三维中子输运问题的特征线方法研究[博士学位论文].北京:清华大学工程物理系,2010.
[12] Liu Z, Wu H, Cao L, et al. A new three-dimensional method of characteristics for theneutron transport calculation[J]. Annals of Nuclear Energy,2011,38:447-454.
[13] Santandrea S, Sanchez R. Acceleration techniques for the characteristic method inunstructured meshes[J]. Annals of Nuclear Energy,2002,29:323-352.
[14] Wu G J, Roy R. Acceleration techniques for trajectory-based deterministic3Dtransport solvers[J]. Annals of Nuclear Energy,2003,30:567-583.
[15] Yamamoto A. Generalized Coarse-Mesh Rebalance Method for Acceleration ofNeutron Transport Calculations[J]. Nuclear Science and Engineering,2005,151:274-282.
[16] Kim K-S, DeHart M D. Unstructured partial-and net-current based coarse mesh finitedifference acceleration applied to the extend step characteristics method in NEWT[J].Annals of Nuclear Energy,2010,38:527-534.
[17] Dahmani M, Roy R. Parallel Solver Based on the Three Dimensional CharacteristicsMethod-Design and Performance Analysis[J]. Nuclear Science and Engineering,2005,150:155-169.
[18] Yamaji K, Matsumoto H, Sato D, et al. Simple and Efficient Parallelization Methodfor MOC Calculation[J]. Journal of NUCLEAR SCIENCE and TECHNOLOGY,2010,47(1):90-102.
[19] Kirschenmann W, Plagne L, Ploix S, et al. Massively Parallel Solving of3DSimplified PNEquations on Graphics Processing Units[C]. International Conferenceon Mathematics, Computational Methods&Reactor Physics (M&C2009), May3-7,Saratoga Springs, New York.2009.
[20] Nelson A G. Monte Carlo Methods for Neutron Transport on Graphics ProcessingUnits Using CUDA[D]. The Pennsylvania State University Department of Mechanicaland Nuclear Engineering,2009.
[21] Ding A, Liu T, Liang C, et al. Evaluation of Speedup of Monte Carlo Calculations ofTwo Simple Reactor Physics Problems Coded for the GPU-CUDA Environment[C].International Conference on Mathematics and Computational Methods Applied toNuclear Science and Engineering (M&C2011) May8-12, Rio de Janeiro, RJ, Brazi.2011.
[22] Jamelot E, Dubois J, Lautard J J, et al. High Performance3D Neutron Transport onPetascale and Hybrid Architectures within APOLLO3Code[C]. InternationalConference on Mathematics and Computational Methods Applied to Nuclear Scienceand Engineering (M&C2011) May8-12, Rio de Janeiro, RJ, Brazi.2011.
[23] Kodama Y, Tatsumi M, Ohoka Y. Study on GPU Computing for SCOPE2withCUDA[C]. International Conference on Mathematics and Computational MethodsApplied to Nuclear Science and Engineering (M&C2011) May8-12, Rio de Janeiro,RJ, Brazi.2011.
[24] Kosaka S, Saji E. Transport Theory Calculation for a Heterogeneous Multi-AssemblyProblem by Characteristics Method with Direct Neutron Path Linking Technique [J].Journal of NUCLEAR SCIENCE and TECHNOLOGY,2000,37(12):1105-1023.
[25] Askew J R. A Characteristics Formulation of the Neutron Transport Equation inComplicated Geometries[R]. AEEW-M1108,1972.
[26]汤春桃,张少泓. CMFD加速在特征线法输运计算中的应用[J].核动力工程,2009,30(5):8-12.
[27]柴晓明,姚栋,王侃,等.使用广义粗网有限差分方法加速中子输运特征线方法[J].计算物理,2010,27(4):541-547.
[28] Jevremovic T, Vujic J, Tsuda K. ANEMONA—a neutron transport code for generalgeometry reactor assemblies based on the method of characteristics and R-functionsolid modeler[J]. Annals of Nuclear Energy,2001,28:125-152.
[29] Sanchez R, Mao L, Santandrea S. Treatment of Boundary Conditions inTrajectory-Based Deterministic Transport Methods[J]. Nuclear Science andEngineering,2002,140:23-50.
[30] Longoni G. Advanced Quadrature Sets, Acceleration and Preconditioning Techniquesfor the Discrete Ordinates Method in Parallel Computing Environments[D].University of Florida,2004.
[31] Yamamoto A, Tabuchi M, Sugimura N, et al. Derivation of Optimum Polar AngleQuadrature Set for the Method of Characteristics Based on Approximation Error forthe Bickley Function[J]. Journal of NUCLEAR SCIENCE and TECHNOLOGY,2007,44(2):129-136.
[32]谢仲生,邓力.中子输运理论数值计算方法[M].西安:西北工业大学出版社,2005.
[33] Lee G S, Cho N Z, Hong S G. Acceleration and Parallelization of the Method ofCharacteristics for Lattice and Whole-Core Heterogeneous Calculations[C]. PHYSOR2000, May7-11, Pittsburg, USA.2000.
[34] Gelbard E M, Hageman L A. The Synthetic Method as Applied to the SNEquations[J].Nuclear Science and Engineering,1969,37(2):288-298.
[35] Alcouffe R E. Diffusion Synthetic Accelemtion Methods for Diamond-differencedDiscrete-ordinates Equations[C]. ANS Mathematics and Computations TopicalMeeting, March28-30, Tucson Arizona.1977.
[36] Ramone G L, Adams M L. A Transport Synthetic Acceleration Method for TransportIterations[J]. Nuclear Science and Engineering,1997,125:257-283.
[37] Zika M R, Adams M L. Transport Synthetic Acceleration for Long-CharacteristicsAssembly-Level Transport Problems[J]. Nuclear Science and Engineering,2000,134:135-158.
[38] Sanchez R, Chetaine A. A synthetic acceleration for a two-dimensional characteristicmethod in unstructured meshes [J]. Nuclear Science and Engineering,2000,136:122-139.
[39] Santandrea S, Sanchez R. Analysis and improvements of the DPN accelerationtechnique for the method of characteristics in unstructured meshes[J]. Annals ofNuclear Energy,2005,32:163-193.
[40] Wu G J, Roy R. Self-collision rebalancing technique for the MCI characteristicssolver[C]. Twentieth Annual Conf. Canadian Nuclear Society, Montreal.1999.
[41] Suslov I R. An Algebraic Collapsing Acceleration in Long Characteristics TransportTheory[C]. Proc.11th Symp. Atomic Energy Research, Csopak, Hungary.2001.
[42] Tellier R L, Hebert A. An Improved Algebraic Collapsing Acceleration with GeneralBoundary Conditions for the Characteristics Method[J]. Nuclear Science andEngineering,2007,156:121-138.
[43] Park C J, Cho N Z. Convergence Analysis of Additive Angular Dependent RebalanceAcceleration for the Discrete Ordinates Transport Calculations[J]. Nuclear Scienceand Engineering,2002,142:64-74.
[44] Park Y R, Cho N Z. Coarse-Mesh Angular Dependent Rebalence Acceleration ofDiscrete Ordinates Transport Calculations[J]. Nuclear Science and Engineering,2004,148:355-373.
[45] Dahmani M, Tellier R L, Roy R, et al. An efficient preconditioning technique usingKrylov subspace methods for3D characteristics solvers[J]. Annals of Nuclear Energy,2005,32:876-896.
[46] Palmiotti G, Smith M, Rabiti C, et al. UNìC: Ultimate Neutronic InvestigationCode[C]. Joint International Topical Meeting on Mathematics&Computation andSupercomputing in Nuclear Applications (M&C+SNA2007) April15-19,2007,Monterey, California.2007.
[47] Siegel S F, Siegel A R, Rabiti C. UNIC: Code Algorithmic Specification of theMethod of Long Characteristics[R]. Technical Memorandum ANL/MCS-TM-301,Argonne National Laboratory,2008.
[48]汤春桃.中子输运方程特征线解法及嵌入式组件均匀化方法的研究[博士学位论文].上海:上海交通大学机械与动力工程学院,2009.
[49] Zhang H, Wu H, Cao L. An acceleration technique for2D MOC based on Krylovsubspace and dimain decomposition methods[J]. Annals of Nuclear Energy,2011,38(12):2742-2751.
[50] Ohoka Y, Tatsumi M. Study on the acceleration of the neutronics calculation based onGPGPU[C]. Joint International Topical Meeting on Mathematics&Computation andSupercomputing in Nuclear Applications (M&C+SNA2007), April, Monterey,California.2007.
[51] NVIDIA Corporation. NVIDIA CUDA C Programming Guide. In2011.
[52]徐琪,余纲林,吴小飞,等.基于GPU加速的蒙卡中子输运方法研究[C].核反应堆系统设计重点实验室年会,成都.2012.
[53] Suslov I. Improvements in the Long Characteristics Method and Their Efficiency forDeep Penetration Calculations [J]. Progress in Nuclear Energy,2001,39(2):233-243.
[54] Goldberg L, Vujic J, Leonard A. The Method of Characteristics in GeneralGeometry[J]. Transactions of the American Nuclear Society,1995,73:173.
[55] Févotte F, Santandrea S, Sanchez R. Advanced Transverse Integration for the Methodof Characteristics[C]. Joint International Topical Meeting on Mathematics&Computation and Supercomputing in Nuclear Applications (M&C+SNA2007), April15-19, Monterey, California.2007.
[56]柴晓明,姚栋,王侃.基于特征线方法的三维中子输运程序(I)——边界条件的插值处理[J].核动力工程,2010,31(2):11-15.
[57] Joo H G, Cho J Y, Kim K S. Methods and Performance of a Three DimensionalWhole-Core Transport Code DeCART[C]. PHYSOR2004, April25-29, Chicago, ILUSA.2004.
[58] Dahmani M, Roy R, Koclas J. New Computational Methodology for Large3DNeutron Transport Problems [C]. PHYSOR2004, April25-29, Chicago, Illinois.American Nuclear Society,2004.
[59] Smith K S, Rhodes J D. Full-Core,2-D, LWR Core Calculations with CASMO-4E[C].PHYSOR2002, October7-10, Seoul, Korea.2002.
[60]莫则尧,陈军,曹小林译.并行计算综论[M].北京:电子工业出版社,2005.
[61]张林波,迟学斌,莫则尧,等.并行计算导论[M].北京:清华大学出版社,2006.
[62]张舒,褚艳利. GPU高性能运算之CUDA[M].北京:中国水利水电出版社,2009.
[63] Cho N Z, Lee G S, Park C J. Fusion of Method of Characteristics and Nodal Methodfor3-D Whole-Core Transport Calculations[J]. Transactions of the American NuclearSociety,2002,86:322.
[64] Smith K S, Rhodes J D. CASMO-4Characteristics Method for Two-DimensionalPWR and BWR Core Calculations[J]. Transactions of the American Nuclear Society,2000,83:292.
[65]李庆扬,关治,白峰衫.数值计算原理[M].北京:清华大学出版社,2000.
[66] Ushio T, Takeda T, Mori M. Neutron Anisotropic Scattering Effect in HeterogeneousCell Calculations of Light Water Reactors[J]. Journal of NUCLEAR SCIENCE andTECHNOLOGY,2003,40(7):464-480.
[67] Postma T, Vujic J. The Method of Characteristics in General Geometry withAnisotropic Scattering[C]. Proc. Int. Conf. Mathematics and Computation, Reactorphysics and Environmental Analysis in Nuclear Application, Sept.27-30, Madrid,Spain.1999.
[68] Petrovic I, Benoist P, Marleau G. A Quasi-Isotropic Reflecting Boundary Conditionfor the TIBERE Heterogeneous Leakage Model[J]. Nuclear Science and Engineering,1996,122:151-166.
[69] Stepanek J, Auerbach T, Haelg W. Calculation of Four Thermal Reactor Benchmark,Problems in X-Y Geometry[R]. EPRI NP-2855,1983.
[70] OECD/NEA. Benchmark on Deterministic Transport Calculations Without SpatialHomogenisation-A2-D/3-D MOX Fuel Assembly Benchmark[R]. Nuclear ScienceNEA/NSC/DOC,2003.
[71] Okumura K, Kugo T, Kaneko K, et al. SRAC2006: A Comperhensive NeutronicsCalculation Code System[R]. Japan Atomic Energy Agency,2007.
[72] Lau J. Experience with CANFLEX Fuel[R]. AECL,2001.
[73] Takeda T, Ikeda H.3-D Neutron Transport Benchmarks[R]. Osaka: Technical ReportOECD/NEA Committee on Reactor Physics (NEACRP),1991.