Krylov子空间方法及先进重水堆在线通量拟构方法初步研究
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摘要
在反应堆物理领域,随着高阶谐波的广泛应用,对高阶谐波求解方法的研究也越来越受到关注。近年来Krylov子空间方法在矩阵特征值问题和线性方程组的求解中得到了很大的研究和发展。在基于Krylov子空间投影的方法中,Rice大学的D.C.Sorensen等人提出的一种隐式重启动的Arnoldi方法(IRAM)可以方便快速地同时求取矩阵的前几阶特征值和特征向量,为此,本文部分的研究工作是对IRAM方法在中子扩散方程高阶谐波求解中的应用加以研究。研究结果表明:该方法在求解高阶谐波时具有和常规的源迭代方法相同的计算精度;其求解速度与所选取的子空间维数有关,选择适当的子空间维数可获得最快的计算速度;对于Krylov子空间方法,求解速度基本不受高阶谐波阶次的影响,而常规源迭代法的求解速度随着高阶谐波阶次的升高而大大降低。因此对于需要求解高阶次谐波的问题,Krylov子空间方法的计算速度远比源迭代法快;同时由于IRAM计算过程中不断的正交化过程使得IRAM方法能够求解具有重根的本征值问题。本文还初步探讨了IRAM方法在非线性本征值问题求解中的应用。
本文第二部分工作是对先进重水堆(ACR)的在线拟构方法进行研究。对于重水堆来说,利用堆芯内探测器的测量信号来在线拟构全堆三维中子通量密度空间分布在反应堆的运行管理中有着非常重要的作用。由于本文研究的在线拟构方法,需在线地进行全堆三维扩散计算,因此是否能够在线拟构取决于全堆芯扩散计算的时间。然而与压水堆相比,ACR的几何尺寸要大得多,且必须进行全堆计算,因此通常的堆芯扩散计算程序无法满足计算速度上的要求,因此开发一种快速堆芯扩散计算方法是非常必要且迫切的。为此,本文探讨了多种方法来提高ACR反应堆全堆扩散计算的速度,包括非线性迭代法、全堆粗细网格交替扫描方法以及粗细网格交替迭代过程中用反照率替代反射层加速的方法。通过研究发现,反照率受堆芯状态的影响较小,因此本文最终采用反照率给定条件下、全堆粗细网格交替迭代的方法,它能够满足ACR反应堆在线通量拟构对计算速度的要求。
最后,本文建立了一种将堆内探测器信号作为扩散计算时的内边界条件、通过三维扩散计算在线拟构中子通量分布的方法,并通过数值验证表明,该方法能够利用探测器的测量值,较有效地矫正中子通量分布的计算值。
Calculation method of harmonics is being widely studied due to its importance in the reactor physics area. Recently, Krylov subspace method is rapidly developed for solving the large-scale linear equation and eigen-value equation. In this paper, Implicitly Restarted Arnoldi Method (IRAM), one of the Krylov subspace method, is applied to solving the higher order harmonics. The numerical results demonstrate that while its accuracy is comparable with the conventional modified source iteration method (MSIM), IRAM method runs much faster than MSIM. Moreover, IRAM is very efficient to solve the duplicate eigenvalue problem, and has the capability of solving nonlinear eigenvalue problem.
Online flux mapping is very important for the operation management of the heavy water reactor. Because the current flux mapping code POWERMAP relies on the previous information, so strictly speaking, it is not a online method. In order to map the neutron flux distribution online, it is necessary to fastly solve neutron diffusion equation. In this paper, three different methods, including the nonlinear iteration method, the coarse mesh and fine mesh alternate sweeping method, and the coarse mesh and fine mesh alternate sweeping method with albedo boundary condition, are investigated. Through the evaluation on the calculation precision and the CPU time, the last method can meet the requirement of the online flux mapping.
Finally, a method of the online flux mapping is developped. Numerical results show that the method has the capability of remedying the calculated neutron flux distribution.
本文第二部分工作是对先进重水堆(ACR)的在线拟构方法进行研究。对于重水堆来说,利用堆芯内探测器的测量信号来在线拟构全堆三维中子通量密度空间分布在反应堆的运行管理中有着非常重要的作用。由于本文研究的在线拟构方法,需在线地进行全堆三维扩散计算,因此是否能够在线拟构取决于全堆芯扩散计算的时间。然而与压水堆相比,ACR的几何尺寸要大得多,且必须进行全堆计算,因此通常的堆芯扩散计算程序无法满足计算速度上的要求,因此开发一种快速堆芯扩散计算方法是非常必要且迫切的。为此,本文探讨了多种方法来提高ACR反应堆全堆扩散计算的速度,包括非线性迭代法、全堆粗细网格交替扫描方法以及粗细网格交替迭代过程中用反照率替代反射层加速的方法。通过研究发现,反照率受堆芯状态的影响较小,因此本文最终采用反照率给定条件下、全堆粗细网格交替迭代的方法,它能够满足ACR反应堆在线通量拟构对计算速度的要求。
最后,本文建立了一种将堆内探测器信号作为扩散计算时的内边界条件、通过三维扩散计算在线拟构中子通量分布的方法,并通过数值验证表明,该方法能够利用探测器的测量值,较有效地矫正中子通量分布的计算值。
Calculation method of harmonics is being widely studied due to its importance in the reactor physics area. Recently, Krylov subspace method is rapidly developed for solving the large-scale linear equation and eigen-value equation. In this paper, Implicitly Restarted Arnoldi Method (IRAM), one of the Krylov subspace method, is applied to solving the higher order harmonics. The numerical results demonstrate that while its accuracy is comparable with the conventional modified source iteration method (MSIM), IRAM method runs much faster than MSIM. Moreover, IRAM is very efficient to solve the duplicate eigenvalue problem, and has the capability of solving nonlinear eigenvalue problem.
Online flux mapping is very important for the operation management of the heavy water reactor. Because the current flux mapping code POWERMAP relies on the previous information, so strictly speaking, it is not a online method. In order to map the neutron flux distribution online, it is necessary to fastly solve neutron diffusion equation. In this paper, three different methods, including the nonlinear iteration method, the coarse mesh and fine mesh alternate sweeping method, and the coarse mesh and fine mesh alternate sweeping method with albedo boundary condition, are investigated. Through the evaluation on the calculation precision and the CPU time, the last method can meet the requirement of the online flux mapping.
Finally, a method of the online flux mapping is developped. Numerical results show that the method has the capability of remedying the calculated neutron flux distribution.
引文
[1]李富等,反应堆中子方程中的高阶谐波,核科学与工程,第16卷第2期,1996
[2]吕栋等,反应堆高阶k本征方程数值计算方法的改进,原子核物理评论,第23卷第2期,2006
[3]戈卢布等,矩阵计算,科学出版社,2005
[4] Saad Y. Krylov subspace methods for solving large unsymmetric linear systems [J]. Mathematics of Comp utations ,1981 ,37 (155) :105～126.
[5] Saad Y, Schultz M A. Conjugate gradient- like algorithm for solving nonsymmetric linear systems [J ] .Mathematics of Comp utations ,1985 ,44 (170) :417～424.
[6] Saad Y, Schultz M A. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems [J ] . SIAM J Sci Comp ut ,1986 ,7 (3) :859～869.
[7]戴华,求解大规模矩阵问题的Krylov子空间方法,南京航空航天大学学报,第33卷第2期,2001
[8] H. Saberi Najafi, A new restarting method in the Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix, Applied Mathematics and Computation 156 (2004) 59–71
[9]漆志鹏等,Krylov子空间算法研究,南昌航空工业学院学报(自然科学版,第20卷第2期,2006
[10]刘晓明, Krylov子空间投影法及其在油藏数值模拟中的应用,应用数学和力学,第21卷第6期,2000
[11]章宗耀,GEDAK-生成INCORE-3D理论数据库的程序系统,核动力工程,第16卷第5期,1996.10
[12] G. Verdu etc., The implicit restarted Arnoldi method, an efficient alternative to solve the neutron diffusion equation, Annals of Nuclear Energy, 26 (1999) 579-593, 1999
[13] James S. Warsa etc., Krylov Subspace Iterations for Deterministic k-Eigenvalue Calculations, Nuclear Science and Engineering, 147, 26–42, 2004
[14] Filippo Zinzani,On the Calculation of the Higher Modes of the Neutron Flux,Dissertation of bachelor, 2006
[15] D. C. SORENSEN,“Implicit Application of Polynomial Filters in a k-Step Arnoldi Method,”SIAM J. Matrix Anal. Appl., 13, 1, 357, 1992
[16] R. B. LEHOUCQ etc., ARPACK User’s Guide, Society for Industrial and AppliedMathematics, 1998
[17]吕栋,张少泓;谐波计算方法及谐波在堆芯换料优化中的应用[A];第二届全国反应堆物理与核材料学术研讨会论文集[C]; 2005年
[18]付学峰,谐波与线性微扰杂交法研究和换料基准题设计,硕士论文,上海交通大学,2008.1
[19]廖成奎,三维节块中子动力学方程组的数值解法及物理与热工水力耦合瞬态过程的数值计算的研究,博士论文,西安交通大学,2002年
[20]谢仲生,核反应堆物理分析(上册),原子能出版社,1996年12月。
[21]谢仲生,核反应堆物理数值计算,原子能出版社,1997年10月
[22] T. Takeda, M. Tanaka, Three-Dimensional Kinetic Calculations With Improved Coarse-Mesh Method for Fast Reactors, Tran. Am. Nucl. Soc.,61:350-351, 1990.
[23]竹生东,非线性迭代先进节块方法的研究及PWR燃料管理计算软件包研制,博士学位论文,西安交通大学,2000年。
[24]谢仲生,压水堆核电厂堆芯燃料管理计算及优化,原子能出版社,2001年6月。
[25]傅学东,非线性迭代节块方法,博士学位论文,中国原子能科学研究院,1998年9月。
[26]谢仲生,反应堆物理分析与计算方法,北京:原子能出版社,1999,29-31。
[27]谢仲生,压水堆核电厂堆芯燃料管理计算及优化,原子能出版社,2001
[28]竹生东,谢仲生,非线性迭代方法在中子扩散计算中的应用核动力工程,21(3),259,2000年
[29]谢仲生,反应堆节块计算中一种数值求解反照率的方法,核科学与工程,1997.9,17-3
[30]李广钧,在线活性区模拟器反照率边界条件的”Powell”最优化技术,核动力工程,1985.12,6-6
[31] H. EZURE,“Estimation of Most Probable Power Distributions in BWRs by Least Squares Method Using In-Core Measurements,”J. Nucl. Sci. Technol., 25, 731,1988
[32] B. G. KIM, I. S. KIM, and S. Y. KIM,“Core Simulation Using Actual Detector Readings for CANDU Reactors,”Nucl. Technol., 93, 138,1991
[33] K.B. LEE and C. H. KIM,“The Least Squares Method for Three-Dimensional Core Power Distribution Monitoring in Pressurized Water Reactors,”,Nucl. Sci. Eng., 143, 268,2003
[34] S. HONG and C. H. KIM,“Advanced Online Flux Mapping of CANDU PHWR by Least-Squares Method”NSE, 150, 299, (2005)
[35] Li Fu etc., High order harmonics and its application for flux mapping, Tsinghua Science and Technology, 1-1, 1996.3
[36] Yaqi Wang etc., On-line monitoring the incore power distribution by using excore ion-chambers, Nuclear Engineering and Design 225 (2003) 315–326
[37] M.E. Pomerantza etc., Nuclear reactor power and flux distribution fitting from a diffusion theory model and experimental data, Annals of Nuclear Energy 29 (2002) 1073–1083
[38] C.J.Jeong etc., Power Mapping in a Canada Deuterium Uranium Reactor Using Kalman Filtering Technique, Nuclear Science and Technology, 758-768,2000.9
[39]刘轩黄,最小二乘递推算法和Kalman滤波算法,华东交通大学学报,1998.6,15-2
[40]唐月红,三元二次样条函数及其计算—计算机表示,南京航空航天大学学报,1997.2,29-1
[41]唐月红,三元二次样条函数及其计算—高维数据拟合,南京航空航天大学学报1996.10,28-5
[2]吕栋等,反应堆高阶k本征方程数值计算方法的改进,原子核物理评论,第23卷第2期,2006
[3]戈卢布等,矩阵计算,科学出版社,2005
[4] Saad Y. Krylov subspace methods for solving large unsymmetric linear systems [J]. Mathematics of Comp utations ,1981 ,37 (155) :105～126.
[5] Saad Y, Schultz M A. Conjugate gradient- like algorithm for solving nonsymmetric linear systems [J ] .Mathematics of Comp utations ,1985 ,44 (170) :417～424.
[6] Saad Y, Schultz M A. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems [J ] . SIAM J Sci Comp ut ,1986 ,7 (3) :859～869.
[7]戴华,求解大规模矩阵问题的Krylov子空间方法,南京航空航天大学学报,第33卷第2期,2001
[8] H. Saberi Najafi, A new restarting method in the Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix, Applied Mathematics and Computation 156 (2004) 59–71
[9]漆志鹏等,Krylov子空间算法研究,南昌航空工业学院学报(自然科学版,第20卷第2期,2006
[10]刘晓明, Krylov子空间投影法及其在油藏数值模拟中的应用,应用数学和力学,第21卷第6期,2000
[11]章宗耀,GEDAK-生成INCORE-3D理论数据库的程序系统,核动力工程,第16卷第5期,1996.10
[12] G. Verdu etc., The implicit restarted Arnoldi method, an efficient alternative to solve the neutron diffusion equation, Annals of Nuclear Energy, 26 (1999) 579-593, 1999
[13] James S. Warsa etc., Krylov Subspace Iterations for Deterministic k-Eigenvalue Calculations, Nuclear Science and Engineering, 147, 26–42, 2004
[14] Filippo Zinzani,On the Calculation of the Higher Modes of the Neutron Flux,Dissertation of bachelor, 2006
[15] D. C. SORENSEN,“Implicit Application of Polynomial Filters in a k-Step Arnoldi Method,”SIAM J. Matrix Anal. Appl., 13, 1, 357, 1992
[16] R. B. LEHOUCQ etc., ARPACK User’s Guide, Society for Industrial and AppliedMathematics, 1998
[17]吕栋,张少泓;谐波计算方法及谐波在堆芯换料优化中的应用[A];第二届全国反应堆物理与核材料学术研讨会论文集[C]; 2005年
[18]付学峰,谐波与线性微扰杂交法研究和换料基准题设计,硕士论文,上海交通大学,2008.1
[19]廖成奎,三维节块中子动力学方程组的数值解法及物理与热工水力耦合瞬态过程的数值计算的研究,博士论文,西安交通大学,2002年
[20]谢仲生,核反应堆物理分析(上册),原子能出版社,1996年12月。
[21]谢仲生,核反应堆物理数值计算,原子能出版社,1997年10月
[22] T. Takeda, M. Tanaka, Three-Dimensional Kinetic Calculations With Improved Coarse-Mesh Method for Fast Reactors, Tran. Am. Nucl. Soc.,61:350-351, 1990.
[23]竹生东,非线性迭代先进节块方法的研究及PWR燃料管理计算软件包研制,博士学位论文,西安交通大学,2000年。
[24]谢仲生,压水堆核电厂堆芯燃料管理计算及优化,原子能出版社,2001年6月。
[25]傅学东,非线性迭代节块方法,博士学位论文,中国原子能科学研究院,1998年9月。
[26]谢仲生,反应堆物理分析与计算方法,北京:原子能出版社,1999,29-31。
[27]谢仲生,压水堆核电厂堆芯燃料管理计算及优化,原子能出版社,2001
[28]竹生东,谢仲生,非线性迭代方法在中子扩散计算中的应用核动力工程,21(3),259,2000年
[29]谢仲生,反应堆节块计算中一种数值求解反照率的方法,核科学与工程,1997.9,17-3
[30]李广钧,在线活性区模拟器反照率边界条件的”Powell”最优化技术,核动力工程,1985.12,6-6
[31] H. EZURE,“Estimation of Most Probable Power Distributions in BWRs by Least Squares Method Using In-Core Measurements,”J. Nucl. Sci. Technol., 25, 731,1988
[32] B. G. KIM, I. S. KIM, and S. Y. KIM,“Core Simulation Using Actual Detector Readings for CANDU Reactors,”Nucl. Technol., 93, 138,1991
[33] K.B. LEE and C. H. KIM,“The Least Squares Method for Three-Dimensional Core Power Distribution Monitoring in Pressurized Water Reactors,”,Nucl. Sci. Eng., 143, 268,2003
[34] S. HONG and C. H. KIM,“Advanced Online Flux Mapping of CANDU PHWR by Least-Squares Method”NSE, 150, 299, (2005)
[35] Li Fu etc., High order harmonics and its application for flux mapping, Tsinghua Science and Technology, 1-1, 1996.3
[36] Yaqi Wang etc., On-line monitoring the incore power distribution by using excore ion-chambers, Nuclear Engineering and Design 225 (2003) 315–326
[37] M.E. Pomerantza etc., Nuclear reactor power and flux distribution fitting from a diffusion theory model and experimental data, Annals of Nuclear Energy 29 (2002) 1073–1083
[38] C.J.Jeong etc., Power Mapping in a Canada Deuterium Uranium Reactor Using Kalman Filtering Technique, Nuclear Science and Technology, 758-768,2000.9
[39]刘轩黄,最小二乘递推算法和Kalman滤波算法,华东交通大学学报,1998.6,15-2
[40]唐月红,三元二次样条函数及其计算—计算机表示,南京航空航天大学学报,1997.2,29-1
[41]唐月红,三元二次样条函数及其计算—高维数据拟合,南京航空航天大学学报1996.10,28-5