CN106202865A

CN106202865A - A kind of calculate the method for arbitrary order coefficient in neutron transport discrete locking nub method

Info

Publication number: CN106202865A
Application number: CN201610472926.XA
Authority: CN
Inventors: 吴宏春; 徐志涛; 李云召; 郑友琦
Original assignee: Xian Jiaotong University
Current assignee: Xian Jiaotong University
Priority date: 2016-06-24
Filing date: 2016-06-24
Publication date: 2016-12-07
Anticipated expiration: 2036-06-24
Also published as: CN106202865B

Abstract

一种计算中子输运离散节块法中任意阶系数的方法，主要内容包括：1.形式简化：通过引入中间系数将复杂的源系数、通量系数和耦合系数进行形式简化；2.积分变换：根据勒让德多项式的性质，将源系数、通量系数和耦合系数中出现的指数函数和勒让德多项式乘积的积分转化为指数函数和一般多项式乘积的积分；3.解析求解：通过数学方法解析地推导出中间积分的精确表达式和递推关系式，这些精确表达式和递推关系式中不包含复杂积分运算，易于计算机编程实现，将计算机精确求解的中间积分值代入积分变换后的源系数、通量系数和耦合系数的表达式，结合作为已知条件的中间系数值，获得任意阶源系数、通量系数和耦合系数的精确值。A method for calculating coefficients of any order in the neutron transport discrete block method, the main contents include: 1. Form simplification: the complex source coefficient, flux coefficient and coupling coefficient are simplified in form by introducing intermediate coefficients; 2. Integral Transformation: According to the properties of the Legendre polynomial, the integral of the product of the exponential function and the Legendre polynomial appearing in the source coefficient, the flux coefficient and the coupling coefficient is converted into the integral of the product of the exponential function and the general polynomial; 3. Analytical solution: by The mathematical method analytically deduces the exact expression and recursive relational expression of the intermediate integral. These exact expressions and recursive relational expressions do not contain complex integral operations, and are easy to be realized by computer programming. The intermediate integral value accurately solved by the computer is substituted into the integral transformation The expressions of the final source coefficient, flux coefficient and coupling coefficient are combined with the intermediate coefficient values as known conditions to obtain the exact values of the source coefficient, flux coefficient and coupling coefficient of any order.

Description

A Method for Calculating Arbitrary Order Coefficients in the Discrete Nodal Method of Neutron Transport

技术领域technical field

本发明涉及核工程中子输运过程数值模拟领域，具体涉及一种计算中子输运离散节块法中任意阶勒让德多项式展开系数的方法。The invention relates to the field of numerical simulation of the neutron transport process in nuclear engineering, in particular to a method for calculating expansion coefficients of Legendre polynomials of any order in the neutron transport discrete block method.

背景技术Background technique

核反应堆是一种实现可控自持中子裂变反应过程的装置。中子裂变反应是中子与可裂变物质原子核发生裂变反应生成裂变碎片、新中子及光子等，同时释放出能量的过程。这些能量可被人们利用，这便是核电站的原理。然而由于自持链式裂变反应的复杂性，及核裂变反应直接及裂变碎片间接释放的中子、光子具有强烈的生物杀伤性，对自持链式裂变反应的数值模拟在核工程领域广泛发展。另一方面，人们往往要知道中子、光子在介质中的分布情况，如核电厂房辐射探测，材料辐照研究，这往往也需要对中子、光子行为的数值模拟。A nuclear reactor is a device that realizes a controlled self-sustaining neutron fission reaction process. Neutron fission reaction is a process in which neutrons and fissionable nuclei undergo fission reactions to generate fission fragments, new neutrons and photons, etc., and release energy at the same time. This energy can be used by people, which is the principle of nuclear power plants. However, due to the complexity of the self-sustaining chain fission reaction, and the neutrons and photons released directly by the nuclear fission reaction and indirectly by the fission fragments have strong biocidal properties, the numerical simulation of the self-sustaining chain fission reaction has been widely developed in the field of nuclear engineering. On the other hand, people often need to know the distribution of neutrons and photons in the medium, such as nuclear power plant radiation detection and material irradiation research, which often require numerical simulation of neutron and photon behavior.

中子输运理论是目前为止最精确地描述中子在介质中的反应、输运迁移过程及最终分布的理论。中子输运理论在数学上可表达为一个关于中子通量分布(简称中子通量)的复杂微积分方程。其中中子通量分布是描述中子在时间、空间、飞行方向及能量不同维度上分布的变量。由于该方程的复杂性，目前数学是无法精确求解的。随着计算机技术的广泛发展和应用，人们通过数值方法近似求解该方程。然而由于其复杂性及计算机发展水平的制约，人们还未能完美地将求解精度和效率优化至工业界可接受的范围内。The neutron transport theory is by far the most accurate theory describing the reaction, transport and migration process and final distribution of neutrons in media. The neutron transport theory can be expressed mathematically as a neutron flux distribution (referred to as neutron flux) complex calculus equation. The neutron flux distribution is a variable that describes the distribution of neutrons in different dimensions of time, space, flight direction and energy. Due to the complexity of this equation, it is currently impossible to solve it precisely mathematically. With the extensive development and application of computer technology, people approximate to solve the equation by numerical method. However, due to its complexity and the constraints of computer development level, people have not been able to perfectly optimize the solution accuracy and efficiency to the acceptable range of the industry.

经调研，中子输运离散节块法是一种精度和效率相对较高的求解中子输运方程的数值方法。经过Ahmed Badruzzaman改进的三维中子输运离散节块方法在保证同样精度的前提下具有更高的效率。该方法的具体过程见文献《An Efficient Algorithm for Nodal-Transport Solutions in Multidimensional Geometry》(以下简称文献)。该方法求解无时间变量的稳态的中子输运方程；通过变量离散，将空间离散成多个网格，这里称每个网格为一个节块，在节块内部，再进一步将空间变量使用勒让德多项式展开，得到对应不同勒让德多项式阶数的通量矩；对飞行方向使用离散纵标法离散，即用空间上多个离散的方向代表连续的空间飞行方向；并将能量变量分成多段，每段为一能群，称之为多能群近似。通过这些数值近似，最终将中子通量分布表达成随不同维度变化的离散变量。通过数值方法求解出这些离散变量的值，便近似求解了中子输运方程。值得注意的是，这些离散变量的离散程度和求解的精度密切相关。比如对于空间变量来说，节块划分的越密越精确，节块内部勒让德多项式的展开阶数越高越精确。After investigation, the neutron transport discrete block method is a relatively high-precision and efficient numerical method for solving the neutron transport equation. The three-dimensional neutron transport discrete block method improved by Ahmed Badruzzaman has higher efficiency under the premise of ensuring the same accuracy. For the specific process of this method, see the document "An Efficient Algorithm for Nodal-Transport Solutions in Multidimensional Geometry" (hereinafter referred to as the document). This method solves the steady-state neutron transport equation without time variable; through variable discretization, the space is discretized into multiple grids, and each grid is called a node here, and inside the node, the space variable is further divided into Use Legendre polynomial expansion to obtain flux moments corresponding to different Legendre polynomial orders; use the discrete ordinate method to discretize the flight direction, that is, use multiple discrete directions in space to represent the continuous space flight direction; and combine the energy The variable is divided into multiple segments, and each segment is an energy group, which is called multi-energy group approximation. Through these numerical approximations, the final neutron flux distribution Expressed as discrete variables that vary with different dimensions. The values of these discrete variables are solved numerically, and the neutron transport equation is approximately solved. It is worth noting that the degree of dispersion of these discrete variables is closely related to the accuracy of the solution. For example, for spatial variables, the denser the block division is, the more accurate it will be, and the higher the expansion order of the Legendre polynomial inside the block, the more accurate it will be.

对中子输运方程离散化后，经过特殊的推导变形，可获得如下如下形式的三个关键系数：After the neutron transport equation is discretized, three key coefficients can be obtained in the following form after special derivation and deformation:

${F f}_{x x n no} = = \frac{Δ Δ x x}{22 μ μ} \frac{22 n no + + 11}{22} exp exp ((\frac{- - {Σ Σ}_{t t} Δ Δ x x}{22 μ μ})) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{n no} ((x x)) exp exp ((\frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) d d x x - - - - - - ((11))$

${G G}_{x x k k} = = exp exp ((\frac{- - {Σ Σ}_{t t} Δ Δ x x}{22 μ μ})) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) d d x x - - - - - - ((22))$

${G G}_{x x k k n no} = = \frac{Δ Δ x x}{22 μ μ} \frac{22 n no + + 11}{22} {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) [[{&Integral; &Integral;}_{- - 11}^{x x} {P P}_{n no} (({x x}^{' '})) exp exp ((\frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} {x x}^{' '})) {dx dx}^{' '}]] d d x x - - - - - - ((33))$

其中，F_xn、G_xk和G_xkn的下标x表示该系数和笛卡尔坐标系中的x方向有关，k和n为非负整数，分别表示节块中中子通量和中子源的勒让德多项式展开阶数，简称展开阶数；括号中的x'和x都表示沿笛卡尔坐标系中x方向的位置变量，单位为cm，为了在积分中区分对x'做了上标；Δx表示节块沿x方向的尺寸cm，exp表示自然指数函数，μ为中子飞行方向与x轴的角度余弦值，该式中μ限于正值，Σ_t为该节块的中子总反应截面，代表反应概率，单位cm^-1，P_n(x)为关于x的n阶勒让德多项式，P_k(x)为关于x的k阶勒让德多项式，P_n(x')为关于x'的n阶勒让德多项式，勒让德多项式具有如下形式：Among them, the subscript x of F _xn , G _xk and G _xkn indicates that the coefficient is related to the x direction in the Cartesian coordinate system, and k and n are non-negative integers, respectively representing the neutron flux and neutron source in the node Legendre polynomial expansion order, referred to as the expansion order; x' and x in brackets represent the position variable along the x direction in the Cartesian coordinate system, the unit is cm, and x' is superscripted in order to distinguish in the integral ; Δx represents the size cm of the nodule along the x direction, exp represents the natural exponential function, μ is the cosine value of the angle between the neutron flight direction and the x-axis, in this formula, μ is limited to positive values, Σ _t is the neutron total of the nodule Reaction cross section, representing the reaction probability, unit cm ^-1 , P _n (x) is the nth order Legendre polynomial about x, P _k (x) is the kth order Legendre polynomial about x, P _n (x') is the Legendre polynomial of order n with respect to x', the Legendre polynomial has the following form:

$\begin{matrix} {P P}_{00} ((x x)) = = 11,, \\ {P P}_{11} ((x x)) = = x x,, \\ {P P}_{22} ((x x)) = = \frac{33 {x x}^{22} - - 11}{22},, \\ {P P}_{33} ((x x)) = = \frac{55 {x x}^{33} - - 33 x x}{22},, \\ ... ... \end{matrix} - - - - - - ((44))$

易验证对于阶数不超过2阶的勒让德多项式满足下面(5)式的递推关系式，更高阶的勒让德多项式则不具有该性质。It is easy to verify that Legendre polynomials with an order not exceeding 2 satisfy the recurrence relation of the following formula (5), while higher order Legendre polynomials do not have this property.

$\frac{{dP dP}_{n no} ((x x))}{d d x x} = = ((22 n no - - 11)) {P P}_{n no - - 11} ((x x)),, n no \leq \leq 22 - - - - - - ((55))$

文献中利用该递推关系式，给出了只适用于2阶以下的系数递推关系式：Using this recursive relation in the literature, a coefficient recursive relation that is only applicable to the second order is given:

${F f}_{x x n no} = = \frac{22 n no + + 11}{22 {Σ Σ}_{t t}} {{[[11 + + {((- - 11))}^{n no + + 11} exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{μ μ}))]] {P P}_{n no} ((11)) - - \frac{44 μ μ}{Δ Δ x x} {F f}_{n no - - 11}}} - - - - - - ((66))$

${F f}_{x x 00} = = \frac{11}{22 {Σ Σ}_{t t}} [[11 - - exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{μ μ}))]]$

$((77))$

${G G}_{x x k k} = = \frac{22 μ μ}{Δ Δ x x} \frac{22}{22 k k + + 11} {((- - 11))}^{k k} {F f}_{k k} - - - - - - ((88))$

${G G}_{k k n no} = = \frac{11}{{Σ Σ}_{t t}} [[{δ δ}_{k k n no} + + {((- - 11))}^{k k + + n no + + 11} ((\frac{22 n no + + 11}{22 k k + + 11})) \frac{22 μ μ}{Δ Δ x x} {F f}_{k k}]] - - ((22 n no + + 11)) \frac{22 μ μ}{{Σ Σ}_{t t} Δ Δ x x} (({G G}_{k k,, n no - - 11} + + {G G}_{k k,, n no - - 33} + + ... ...)) - - - - - - ((99))$

${G G}_{x x 0000} = = \frac{11}{{Σ Σ}_{t t}} ((11 - - {G G}_{00})) - - - - - - ((1010))$

${G G}_{x x 0101} = = \frac{- - 33}{22 {Σ Σ}_{t t}} [[\frac{22}{11 - - exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{μ μ}))} - - \frac{22 μ μ}{{Σ Σ}_{t t} Δ Δ x x} - - 11]] - - - - - - ((1111))$

${G G}_{x x 0101} = = - - \frac{11}{33} {G G}_{0101} - - - - - - ((1212))$

节块内空间变量的展开阶数影响数值求解的精度和效率，通常越高的展开阶数意味着节块尺寸可以取得越大，从而在获得相同精度的前提下提高数值求解的效率。文献中这一计算系数的方法限制了节块内空间变量展开阶数，从而限制了该数值方法的精度和效率。The expansion order of the spatial variables in the nodal block affects the accuracy and efficiency of the numerical solution. Usually, the higher the expansion order, the larger the size of the nodular block can be obtained, thereby improving the efficiency of the numerical solution while obtaining the same accuracy. This method of calculating coefficients in the literature limits the expansion order of the spatial variables in the nodal block, thus limiting the accuracy and efficiency of the numerical method.

发明内容Contents of the invention

为了克服上述现有技术存在的问题，本发明的目的在于提供一种计算中子输运离散节块法中任意阶勒让德多项式展开系数(源系数、通量系数和耦合系数)的方法，该方法打破了原中子输运离散节块法在计算反应堆中子通量密度分布时精度和效率受展开阶数的限制。In order to overcome the problems in the above-mentioned prior art, the object of the present invention is to provide a method for calculating the arbitrary order Legendre polynomial expansion coefficients (source coefficient, flux coefficient and coupling coefficient) in the neutron transport discrete block method, This method breaks the limitation of the expansion order of the accuracy and efficiency of the original neutron transport discrete block method in calculating the reactor neutron flux density distribution.

为了达到上述目的，本发明采用如下技术方案：In order to achieve the above object, the present invention adopts following technical scheme:

一种计算中子输运离散节块法中任意阶系数的方法，主要内容包括：1.形式简化：通过引入中间系数将复杂的源系数、通量系数和耦合系数进行形式简化；2.积分变换：根据勒让德多项式的性质，将源系数、通量系数和耦合系数中的n阶勒让德多项式拆分成n个一般多项式，从而将源系数、通量系数和耦合系数中出现的指数函数和勒让德多项式乘积的积分(简称原始积分)转化为指数函数和一般多项式乘积的积分(简称中间积分)；3.解析求解：通过数学方法解析地推导出中间积分的精确表达式和递推关系式，这些精确表达式和递推关系式中不包含复杂积分运算，易于计算机编程实现，将计算机精确求解的中间积分值代入积分变换后的源系数、通量系数和耦合系数的表达式，结合作为已知条件的中间系数值，获得任意阶源系数、通量系数和耦合系数的精确值。A method for calculating coefficients of any order in the discrete block method of neutron transport, the main contents include: 1. Form simplification: the complex source coefficient, flux coefficient and coupling coefficient are simplified in form by introducing intermediate coefficients; 2. Integral Transformation: According to the properties of the Legendre polynomials, the n-order Legendre polynomials in the source coefficients, flux coefficients and coupling coefficients are split into n general polynomials, so that the source coefficients, flux coefficients and coupling coefficients appear in The integral of the product of the exponential function and the Legendre polynomial (referred to as the original integral) is transformed into the integral of the product of the exponential function and the general polynomial (referred to as the intermediate integral); 3. Analytical solution: the exact expression and sum of the intermediate integral are analytically deduced by mathematical methods Recursive relational expressions, these exact expressions and recursive relational expressions do not contain complex integral operations, which are easy to realize by computer programming, and substitute the intermediate integral values accurately solved by the computer into the expressions of source coefficients, flux coefficients and coupling coefficients after integral transformation Combined with the intermediate coefficient values as known conditions, the exact values of source coefficients, flux coefficients and coupling coefficients of any order can be obtained.

下面具体给出获得中子输运离散节块法中任意阶系数的方法：The method for obtaining the coefficients of any order in the neutron transport discrete nodal method is given in detail below:

第一步，通过引入中间系数，对中子输运离散节块法中的三个关键系数即源系数、通量系数和耦合系数的形式进行简化：In the first step, the forms of the three key coefficients in the neutron transport discrete-block method, namely source coefficient, flux coefficient and coupling coefficient, are simplified by introducing intermediate coefficients:

重写原中子输运离散节块法中的三个关键系数，即源系数F_xn、通量系数G_xk和耦合系数G_xkn如下，Rewrite the three key coefficients in the original neutron transport discrete block method, namely the source coefficient F _xn , the flux coefficient G _xk and the coupling coefficient G _xkn as follows,

${F f}_{x x n no} = = \frac{Δ Δ x x}{22 μ μ} \frac{22 n no + + 11}{22} exp exp ((\frac{- - {Σ Σ}_{t t} Δ Δ x x}{22 μ μ})) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{n no} ((x x)) exp exp ((\frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) d d x x - - - - - - ((1313))$

${G G}_{x x k k} = = exp exp ((\frac{- - {Σ Σ}_{t t} Δ Δ x x}{22 μ μ})) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) d d x x - - - - - - ((1414))$

${G G}_{x x k k n no} = = \frac{Δ Δ x x}{22 μ μ} \frac{22 n no + + 11}{22} {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) [[{&Integral; &Integral;}_{- - 11}^{x x} {P P}_{n no} (({x x}^{' '})) exp exp ((\frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} {x x}^{' '})) {dx dx}^{' '}]] d d x x - - - - - - ((1515))$

其中，F_xn、G_xk和G_xkn的下标x表示该系数和笛卡尔坐标系中的x方向有关，k和n为非负整数，分别表示节块中中子通量和中子源的勒让德多项式展开阶数，简称展开阶数；括号中的x'和x都表示沿笛卡尔坐标系中x方向的位置变量，单位为cm，为了在积分中区分对x'做了上标；Δx表示节块沿x方向的尺寸cm，exp表示自然指数函数，μ为中子飞行方向与x轴的角度余弦值，该式中μ限于正值，Σ_t为该节块的中子总反应截面，代表反应概率，单位cm^-1，P_n(x)为关于x的n阶勒让德多项式，P_k(x)为关于x的k阶勒让德多项式，P_n(x')为关于x'的n阶勒让德多项式；Among them, the subscript x of F _xn , G _xk and G _xkn indicates that the coefficient is related to the x direction in the Cartesian coordinate system, and k and n are non-negative integers, respectively representing the neutron flux and neutron source in the node Legendre polynomial expansion order, referred to as the expansion order; x' and x in brackets represent the position variable along the x direction in the Cartesian coordinate system, the unit is cm, and x' is superscripted in order to distinguish in the integral ; Δx represents the size cm of the nodule along the x direction, exp represents the natural exponential function, μ is the cosine value of the angle between the neutron flight direction and the x-axis, where μ is limited to positive values, Σ _t is the neutron total number of the nodule Reaction cross section, representing the reaction probability, unit cm ^-1 , P _n (x) is the nth order Legendre polynomial about x, P _k (x) is the kth order Legendre polynomial about x, P _n (x') is the nth-order Legendre polynomial about x';

为了简化源系数、通量系数和耦合系数，引入如下的中间系数b、a和c：In order to simplify the source coefficient, flux coefficient and coupling coefficient, the following intermediate coefficients b, a and c are introduced:

$b b = = \frac{Δ Δ x x}{22 | | μ μ | |} - - - - - - ((1616))$

a＝Σ_tb (17)a= _Σtb (17)

c＝e^-a (18)c＝ ^e- a (18)

其中e为自然指数，where e is the natural exponent,

将(16)、(17)和(18)式分别代入(13)、(14)和(15)式得到如下形式的源系数、通量系数和耦合系数：Substitute equations (16), (17) and (18) into equations (13), (14) and (15) respectively to obtain the source coefficient, flux coefficient and coupling coefficient in the following forms:

${F f}_{x x n no} = = b b c c ((n no + + 0.5 0.5)) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{n no} ((x x)) {e e}^{a a x x} d d x x - - - - - - ((1919))$

${G G}_{x x n no} = = c c {&Integral; &Integral;}_{- - 11}^{11} {P P}_{n no} ((x x)) {e e}^{- - a a x x} d d x x - - - - - - ((2020))$

${G G}_{x x k k n no} = = b b ((n no + + 0.5 0.5)) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) {e e}^{- - a a x x} [[{&Integral; &Integral;}_{- - 11}^{x x} {P P}_{n no} (({x x}^{' '})) {e e}^{{ax ax}^{' '}} {dx dx}^{' '}]] d d x x - - - - - - ((21 twenty one))$

在应用中子输运离散节块法求解反应堆中子通量时，中间系数b、a和c当做已知值，源系数、通量系数和耦合系数是勒让德多项式展开阶数n和k的函数，然而函数关系(19)、(20)和(21)式中含有勒让德多项式和指数函数的积分，直接数值求解是耗时和不精确的，下面的步骤采用积分变换和解析求解的思想获得精确的任意阶源系数、通量系数和耦合系数值；When applying the neutron transport discrete-block method to solve the reactor neutron flux, the intermediate coefficients b, a and c are taken as known values, and the source coefficient, flux coefficient and coupling coefficient are Legendre polynomial expansion orders n and k However, the functional relations (19), (20) and (21) contain the integral of the Legendre polynomial and the exponential function, and the direct numerical solution is time-consuming and inaccurate. The following steps use integral transformation and analytical solution Accurate arbitrary-order source coefficients, flux coefficients and coupling coefficients can be obtained by using the idea;

第二步，将勒让德多项式拆分为多项式求和的形式，将源系数、通量系数和耦合系数中出现的指数函数和勒让德多项式乘积的积分简称原始积分，转化为指数函数和一般多项式乘积的积分简称中间积分：In the second step, the Legendre polynomial is split into the form of polynomial summation, and the integral of the product of the exponential function and the Legendre polynomial appearing in the source coefficient, flux coefficient and coupling coefficient is referred to as the original integral, and transformed into an exponential function and Integrals of general polynomial products are referred to as intermediate integrals:

将勒让德多项式写成如下多项式求和的形式：Write the Legendre polynomials in the form of a sum of polynomials as follows:

${P P}_{n no} ((x x)) = = {Σ Σ}_{{n no}^{' '} = = 00}^{n no} {p p}_{{n no}^{' '} - - n no} {x x}^{{n no}^{' '}} - - - - - - ((22 twenty two))$

其中p_n'-n为n阶勒让德多项式的第n'次幂项的系数，如p_0-0＝1、p_0-1＝0、p_1-1＝1等。Where p _n'-n is the coefficient of the n'th power term of the nth-order Legendre polynomial, such as p _0-0 =1, p _0-1 =0, p _1-1 =1 and so on.

将(22)式代入(19)、(20)和(21)式中，得到源系数、通量系数和耦合系数中间积分的关系式：Substituting Equation (22) into Equations (19), (20) and (21), the relational expression of the intermediate integral of source coefficient, flux coefficient and coupling coefficient is obtained:

${F f}_{x x n no} = = b b c c ((n no + + 0.5 0.5)) {Σ Σ}_{{n no}^{' '} = = 00}^{n no} {p p}_{{n no}^{' '} - - n no} {I I}_{{n no}^{' '}}^{a a} - - - - - - ((23 twenty three))$

${G G}_{x x k k} = = c c {Σ Σ}_{{k k}^{' '} = = 00}^{k k} {p p}_{{k k}^{' '} - - k k} {I I}_{{k k}^{' '}}^{- - a a} - - - - - - ((24 twenty four))$

${G G}_{x x k k n no} = = b b ((n no + + 0.5 0.5)) {Σ Σ}_{{n no}^{' '} = = 00}^{n no} {Σ Σ}_{{k k}^{' '} = = 00}^{k k} {p p}_{{n no}^{' '} - - n no} {p p}_{{k k}^{' '} - - k k} {J J}_{{k k}^{' '} {n no}^{' '}} - - - - - - ((2525))$

其中中间积分形式如下：The intermediate integral form is as follows:

${I I}_{{n no}^{' '}}^{a a} = = {&Integral; &Integral;}_{- - 11}^{11} {x x}^{{n no}^{' '}} {e e}^{a a x x} d d x x - - - - - - ((2626))$

${I I}_{{k k}^{' '}}^{- - a a} = = {&Integral; &Integral;}_{- - 11}^{11} {x x}^{{k k}^{' '}} {e e}^{- - a a x x} d d x x - - - - - - ((2727))$

${J J}_{{k k}^{' '} {n no}^{' '}} = = {&Integral; &Integral;}_{- - 11}^{11} {x x}^{{k k}^{' '}} {e e}^{- - a a x x} (({&Integral; &Integral;}_{- - 11}^{x x} {x x}^{' ' {n no}^{' '}} {e e}^{{ax ax}^{' '}} {dx dx}^{' '})) d d x x - - - - - - ((2828))$

第三步，通过分部积分和数学归纳法获得中间积分的解析表达式，从而获得任意阶源系数、通量系数和耦合系数的解析表达式：In the third step, the analytical expression of the intermediate integral is obtained by integration by parts and mathematical induction, so as to obtain the analytical expressions of the source coefficient, flux coefficient and coupling coefficient of any order:

(23)-(28)式已将求任意阶系数的关键转化为求形如和J_kn的指数函数和任意阶多项式积分；通过数学上的分部积分法及数学归纳法，得到如下形式的任意阶中间积分解析表达式：Formulas (23)-(28) have transformed the key to finding coefficients of any order into finding the form and the exponential function of J _kn and the polynomial integral of any order; through the integration by parts method and the mathematical induction method in mathematics, the analytical expression of the intermediate integral of any order in the following form is obtained:

${I I}_{n no}^{a a} = = {((- - 11))}^{n no + + 11} {e e}^{a a} {Σ Σ}_{l l = = 00}^{n no} \frac{{((- - 11))}^{l l + + 11}}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a} {Σ Σ}_{l l = = 00}^{n no} \frac{11}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!},, a a > > 00 - - - - - - ((2929))$

${I I}_{n no}^{a a} = = {{\begin{matrix} 22,, & n no = = 00 \\ 00,, & n no > > 00 \end{matrix},, a a = = 00 - - - - - - ((3030))$

${I I}_{n no}^{- - a a} = = {((- - 11))}^{n no} {I I}_{n no}^{a a} - - - - - - ((3131))$

${J J}_{k k n no} = = {((- - 11))}^{n no + + 11} {Σ Σ}_{l l = = 00}^{n no} \frac{{((- - 11))}^{l l + + 11}}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!} \frac{11 - - {((- - 11))}^{l l + + k k + + 11}}{l l + + k k + + 11} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a} {I I}_{k k}^{- - a a} {Σ Σ}_{l l = = 00}^{n no} \frac{11}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!},, a a > > 00 - - - - - - ((3232))$

${J J}_{k k n no} = = {{\begin{matrix} 44,, & k k + + n no = = 00 \\ 00,, & k k + + n no > > 00 \end{matrix},, a a = = 00 - - - - - - ((3333))$

并且得到如下的递推关系式：And get the following recurrence relation:

${I I}_{00}^{a a} = = (({e e}^{a a} - - {e e}^{- - a a})) / / a a,, a a > > 00 - - - - - - ((3434))$

${I I}_{n no}^{a a} = = - - \frac{n no}{a a} {I I}_{n no - - 11}^{a a} + + (({e e}^{a a} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a})) / / a a,, a a > > 00 - - - - - - ((3535))$

${J J}_{k k 00} = = [[\frac{11 - - {((- - 11))}^{k k + + 11}}{k k + + 11} - - {e e}^{- - a a} {I I}_{k k}^{- - a a}]] / / a a,, a a > > 00 - - - - - - ((3636))$

${J J}_{k k n no} = = - - \frac{n no}{a a} {I I}_{k k ((n no - - 11))} + + [[\frac{11 - - {((- - 11))}^{n no + + k k + + 11}}{n no + + k k + + 11} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a} {I I}_{k k}^{- - a a}]] / / a a,, a a > > 00 - - - - - - ((3737))$

(29)-(33)式为中间积分的解析表达式，对于任意的勒让德多项式展开阶数n和k，能够方便地通过计算机编程求得这些中间积分；将求得的中间积分的值代入经积分变换的源系数、通量系数和耦合系数的表达式(23)、(24)和(25)式中，便获得任意阶源系数、通量系数和耦合系数的计算值，(29)-(33) formula is the analytical expression of intermediate integral, for arbitrary Legendre polynomial expansion order n and k, can obtain these intermediate integrals easily by computer programming; The value of the intermediate integral of finding Substituting the integral transformed source coefficient, flux coefficient and coupling coefficient into the expressions (23), (24) and (25), the calculated values of arbitrary order source coefficient, flux coefficient and coupling coefficient can be obtained,

注意(30)、(31)式给出了a＝0的特殊情况，这对应于中子总反应截面Σ_t为零的情况，这使得中子输运离散节块法在理论上能够处理带真空材料的中子输运问题，如带真空空洞反应堆中子分布计算、反应堆外围中子分布计算。Note that equations (30) and (31) give the special case of a=0, which corresponds to the case that the total neutron reaction cross section Σ _t is zero, which makes the neutron transport discrete-block method theoretically able to deal with Neutron transport problems of vacuum materials, such as calculation of neutron distribution in reactor with vacuum cavity, calculation of neutron distribution in reactor periphery.

上述中子输运离散节块法任意阶系数的计算方法已经通过计算机程序实现，证明了其正确性和有效性，The calculation method of the arbitrary order coefficient of the neutron transport discrete nodal method has been realized by a computer program, and its correctness and effectiveness have been proved.

与现有技术相比，本发明有如下突出优点：Compared with the prior art, the present invention has the following outstanding advantages:

1.打破了传统中子输运离散节块法对节块内空间变量勒让德多项式展开阶数的限制，可以通过解析的方法精确获得任意高阶展开系数，使得节块尺寸可以取得更大，从而在反应堆中子通量分布计算时同时获得较高的精度和效率；1. It breaks the limitation of the traditional neutron transport discrete-block method on the expansion order of the space variable Legendre polynomial in the block, and can accurately obtain any high-order expansion coefficient through the analytical method, so that the size of the block can be made larger , so as to obtain higher accuracy and efficiency in the calculation of reactor neutron flux distribution;

2.给出了零中子反应截面情况的特殊中子输运离散节块法展开系数表达式，使得中子输运离散节块法可以用来计算带真空区域反应堆或反应堆外围通量分布。2. The expansion coefficient expression of the special neutron transport discrete segment method is given for the case of zero neutron reaction cross section, so that the neutron transport discrete segment method can be used to calculate the flux distribution of the reactor with vacuum region or the periphery of the reactor.

具体实施方式detailed description

下面结合具体实施方式对本发明作进一步详细说明：Below in conjunction with specific embodiment the present invention is described in further detail:

本发明方法基于中子输运离散节块法，为了得到中子输运问题中任意阶中子输运离散节块法任意阶展开系数，在中子输运离散节块法的框架下，通过计算机编程将前面的步骤代码实现。The method of the present invention is based on the neutron transport discrete block method, in order to obtain the arbitrary order expansion coefficient of the neutron transport discrete block method in the neutron transport problem, under the framework of the neutron transport discrete block method, through Computer programming implements the preceding steps in code.

编程中用到的最基本的关系式是(23)、(24)和(25)式，观察这三个关系式可知，一方面要获得中间系数a、b和c；另一方面需要获得形如和J_kn的指数函数和任意阶多项式乘积的积分。The most basic relational expressions used in programming are (23), (24) and (25). Observing these three relational expressions, we can know that, on the one hand, the intermediate coefficients a, b, and c must be obtained; on the other hand, the form Such as Integral of the product of an exponential function and polynomials of any order and J _kn .

中间系数a、b、c和节块的尺寸Δx，中子飞行方向与坐标轴的角度余弦值μ，以及节块的中子总反应截面Σ_t有关，这些量根据具体待计算问题的性质并结合网格划分和离散角度划分获得，求解时可当做已知值；The intermediate coefficients a, b, c are related to the size Δx of the nodule, the cosine value μ of the angle between the neutron flight direction and the coordinate axis, and the total neutron reaction section Σ _t of the nodule. These quantities are based on the nature of the specific problem to be calculated and Obtained by combining grid division and discrete angle division, it can be regarded as a known value when solving;

形如和J_kn的任意阶积分通过前面给出的积分表达式或递推关系式获得。由于递推关系式在计算高阶系数时用到已经计算好的低阶系数的值，可以减少计算量，在实践应用中使用递推关系式，具体计算时源展开阶数N和通量展开阶数K的值为给定值(方便讨论期间，这里假设N＝K)，对于中间系数a＝0的情况，直接利用(30)和(33)式计算积分值；对于中间系数a＞0的情况，使用(34)-(37)式递推计算积分值：首先使用(34)式计算然后使用(35)式逐步递推计算然后使用关系式(31)计算然后使用(36)式计算J_k0(k＝1,2,...,K)，最后利用(37)式计算J_kn(k＝1,2,...,K；n＝1,2,...,N)，将这些计算好的各阶中间系数存储备用。Shaped like The arbitrary-order integral of and J _kn is obtained by the integral expression or recurrence relation given above. Since the recursive relational formula uses the calculated low-order coefficient values when calculating the high-order coefficients, the amount of calculation can be reduced, and the recursive relational formula is used in practical applications. The specific calculation is the source expansion order N and the flux expansion The value of order K is a given value (during convenient discussion, assuming N=K here), for the situation of intermediate coefficient a=0, directly utilize (30) and (33) formula to calculate integral value; For intermediate coefficient a＞0 In the case of , use formula (34)-(37) to recursively calculate the integral value: first use formula (34) to calculate Then use formula (35) to recursively calculate step by step Then use the relation (31) to calculate Then use formula (36) to calculate J _k0 (k=1,2,...,K), and finally use formula (37) to calculate J _kn (k=1,2,...,K; n=1,2 ,...,N), store these calculated intermediate coefficients of each order for future use.

最后使用(23)、(24)和(25)式组合中间系数和中间积分，获得最终的任意阶(源展开阶数N和通量展开阶数K可取任意非负整数)源系数，通量系数和耦合系数的值。Finally, use (23), (24) and (25) to combine the intermediate coefficients and intermediate integrals to obtain the final arbitrary order (source expansion order N and flux expansion order K can be any non-negative integer) source coefficient, flux Coefficient and coupling coefficient values.

Claims

1. a method for calculating the arbitrary order coefficient in the neutron transport discrete nodal method, is characterized in that: comprise the steps:

In the first step, the forms of the three key coefficients in the neutron transport discrete-block method, namely source coefficient, flux coefficient and coupling coefficient, are simplified by introducing intermediate coefficients:

Rewrite the three key coefficients in the original neutron transport discrete block method, namely the source coefficient F _xn , the flux coefficient G _xk and the coupling coefficient G _xkn as follows,

{F f}_{x x n no} = = \frac{Δ Δ x x}{22 μ μ} \frac{22 n no + + 11}{22} exp exp ((\frac{- - {Σ Σ}_{t t} Δ Δ x x}{22 μ μ})) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{n no} ((x x)) exp exp ((\frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) d d x x - - - - - - ((1313))

{G G}_{x x k k} = = exp exp ((\frac{- - {Σ Σ}_{t t} Δ Δ x x}{22 μ μ})) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) d d x x - - - - - - ((1414))

{G G}_{x x k k n no} = = \frac{Δ Δ x x}{22 μ μ} \frac{22 n no + + 11}{22} {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) exp exp ((- - \frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} x x)) [[{&Integral; &Integral;}_{- - 11}^{x x} {P P}_{n no} (({x x}^{' '})) exp exp ((\frac{{Σ Σ}_{t t} Δ Δ x x}{22 μ μ} {x x}^{' '})) {dx dx}^{' '}]] d d x x - - - - - - ((1515))

Among them, the subscript x of F _xn , G _xk and G _xkn indicates that the coefficient is related to the x direction in the Cartesian coordinate system, and k and n are non-negative integers, respectively representing the neutron flux and neutron source in the node Legendre polynomial expansion order, referred to as the expansion order; x' and x in brackets represent the position variable along the x direction in the Cartesian coordinate system, the unit is cm, and x' is superscripted in order to distinguish in the integral ; Δx represents the size cm of the nodule along the x direction, exp represents the natural exponential function, μ is the cosine value of the angle between the neutron flight direction and the x-axis, where μ is limited to positive values, Σ _t is the neutron total number of the nodule Reaction cross section, representing the reaction probability, unit cm ^-1 , P _n (x) is the nth order Legendre polynomial about x, P _k (x) is the kth order Legendre polynomial about x, P _n (x') is the nth-order Legendre polynomial about x';

In order to simplify the source coefficient, flux coefficient and coupling coefficient, the following intermediate coefficients b, a and c are introduced:

b b = = \frac{Δ Δ x x}{22 | | μ μ | |} - - - - - - ((1616))

a= _Σtb (17)

c＝ ^e- a (18)

where e is the natural exponent,

Substitute equations (16), (17) and (18) into equations (13), (14) and (15) respectively to obtain the source coefficient, flux coefficient and coupling coefficient in the following forms:

{F f}_{x x n no} = = b b c c ((n no + + 0.5 0.5)) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{n no} ((x x)) {e e}^{a a x x} d d x x - - - - - - ((1919))

{G G}_{x x n no} = = c c {&Integral; &Integral;}_{- - 11}^{11} {P P}_{n no} ((x x)) {e e}^{- - a a x x} d d x x - - - - - - ((2020))

{G G}_{x x k k n no} = = b b ((n no + + 0.5 0.5)) {&Integral; &Integral;}_{- - 11}^{11} {P P}_{k k} ((x x)) {e e}^{- - a a x x} [[{&Integral; &Integral;}_{- - 11}^{x x} {P P}_{n no} (({x x}^{' '})) {e e}^{{ax ax}^{' '}} {dx dx}^{' '}]] d d x x - - - - - - ((21 twenty one))

When applying the neutron transport discrete block method to solve the reactor neutron flux, the intermediate coefficients b, a and c are regarded as known values, and the source coefficient, flux coefficient and coupling coefficient are Legendre polynomial expansion order n and The function of k, however, the functional relations (19), (20) and (21) contain the integral of Legendre polynomial and exponential function, the direct numerical solution is time-consuming and inaccurate, the following steps adopt integral transformation and analysis The idea of solving to obtain accurate arbitrary order source coefficient, flux coefficient and coupling coefficient value;

In the second step, the Legendre polynomial is split into the form of polynomial summation, and the integral of the product of the exponential function and the Legendre polynomial appearing in the source coefficient, flux coefficient and coupling coefficient is referred to as the original integral, and transformed into an exponential function and Integrals of general polynomial products are referred to as intermediate integrals:

Write the Legendre polynomials as a sum of polynomials as follows:

{P P}_{n no} ((x x)) = = {Σ Σ}_{{n no}^{' '} = = 00}^{n no} {p p}_{{n no}^{' '} - - n no} {x x}^{{n no}^{' '}} - - - - - - ((22 twenty two))

Where p _n'-n is the coefficient of the n'th power term of the nth-order Legendre polynomial,

Substituting Equation (22) into Equations (19), (20) and (21), the relational expression of the intermediate integral of source coefficient, flux coefficient and coupling coefficient is obtained:

{F f}_{x x n no} = = b b c c ((n no + + 0.5 0.5)) {Σ Σ}_{{n no}^{' '} = = 00}^{n no} {p p}_{{n no}^{' '} - - n no} {I I}_{{n no}^{' '}}^{a a} - - - - - - ((23 twenty three))

{G G}_{x x k k} = = c c {Σ Σ}_{{k k}^{' '} = = 00}^{k k} {p p}_{{k k}^{' '} - - k k} {I I}_{{k k}^{' '}}^{- - a a} - - - - - - ((24 twenty four))

{G G}_{x x k k n no} = = b b ((n no + + 0.5 0.5)) {Σ Σ}_{n no = = 00}^{n no} {Σ Σ}_{k k = = 00}^{k k} {p p}_{{n no}^{' '} - - n no} {p p}_{{k k}^{' '} - - k k} {J J}_{{k k}^{' '} {n no}^{' '}} - - - - - - ((2525))

The form of the intermediate integral is as follows:

{I I}_{{n no}^{' '}}^{a a} = = {&Integral; &Integral;}_{- - 11}^{11} {x x}^{{n no}^{' '}} {e e}^{a a x x} d d x x - - - - - - ((2626))

{I I}_{{k k}^{' '}}^{- - a a} = = {&Integral; &Integral;}_{- - 11}^{11} {x x}^{{k k}^{' '}} {e e}^{- - a a x x} d d x x - - - - - - ((2727))

{J J}_{{k k}^{' '} {n no}^{' '}} = = {&Integral; &Integral;}_{- - 11}^{11} {x x}^{{k k}^{' '}} {e e}^{- - a a x x} (({&Integral; &Integral;}_{- - 11}^{x x} {x x}^{' ' {n no}^{' '}} {e e}^{{ax ax}^{' '}} {dx dx}^{' '})) d d x x - - - - - - ((2828))

In the third step, the analytical expression of the intermediate integral is obtained by integration by parts and mathematical induction, so as to obtain the analytical expressions of the source coefficient, flux coefficient and coupling coefficient of any order:

Formulas (23)-(28) have transformed the key to finding coefficients of any order into finding the form and the exponential function of J _kn and the polynomial integral of any order; through the integration by parts method and the mathematical induction method in mathematics, the analytical expression of the intermediate integral of any order in the following form is obtained:

{I I}_{n no}^{a a} = = {((- - 11))}^{n no + + 11} {e e}^{a a} {Σ Σ}_{l l = = 00}^{n no} \frac{{((- - 11))}^{l l + + 11}}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a} {Σ Σ}_{l l = = 00}^{n no} \frac{11}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!},, a a > > 00 - - - - - - ((2929))

{I I}_{n no}^{a a} = = \{\begin{matrix} 22,, & n no = = 00 \\ 00,, & n no > > 00 \end{matrix},, a a = = 00 - - - - - - ((3030))

{I I}_{n no}^{- - a a} = = {((- - 11))}^{n no} {I I}_{n no}^{a a} - - - - - - ((3131))

{J J}_{k k n no} = = {((- - 11))}^{n no + + 11} {Σ Σ}_{l l = = 00}^{n no} \frac{{((- - 11))}^{l l + + 11}}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!} \frac{11 - - {((- - 11))}^{l l + + k k + + 11}}{l l + + k k + + 11} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a} {I I}_{k k}^{- - a a} {Σ Σ}_{l l = = 00}^{n no} \frac{11}{{a a}^{n no - - l l + + 11}} \frac{n no!!}{l l!!},, a a > > 00 - - - - - - ((3232))

{J J}_{k k n no} = = \{\begin{matrix} 44,, & k k + + n no = = 00 \\ 00,, & k k + + n no > > 00 \end{matrix},, a a = = 00 - - - - - - ((3333))

And get the following recurrence relation:

{I I}_{00}^{a a} = = (({e e}^{a a} - - {e e}^{- - a a})) / / a a,, a a > > 00 - - - - - - ((3434))

{I I}_{n no}^{a a} = = - - \frac{n no}{a a} {I I}_{n no - - 11}^{a a} + + (({e e}^{a a} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a})) / / a a,, a a > > 00 - - - - - - ((3535))

{J J}_{k k 00} = = [[\frac{11 - - {((- - 11))}^{k k + + 11}}{k k + + 11} - - {e e}^{- - a a} {I I}_{k k}^{- - a a}]] / / a a,, a a > > 00 - - - - - - ((3636))

{J J}_{k k n no} = = - - \frac{n no}{a a} {I I}_{k k ((n no - - 11))} + + [[\frac{11 - - {((- - 11))}^{n no + + k k + + 11}}{n no + + k k + + 11} + + {((- - 11))}^{n no + + 11} {e e}^{- - a a} {I I}_{k k}^{- - a a}]] / / a a,, a a > > 00 - - - - - - ((3737))

(29)-(33) formula is the analytical expression of intermediate integral, for arbitrary Legendre polynomial expansion order n and k, can obtain these intermediate integrals easily by computer programming; The value of the intermediate integral of finding Substituting the integral transformed source coefficient, flux coefficient and coupling coefficient into the expressions (23), (24) and (25), the calculated values of arbitrary order source coefficient, flux coefficient and coupling coefficient can be obtained,

Note that equations (30) and (31) give the special case of a=0, which corresponds to the case that the total neutron reaction cross section Σ _t is zero, which makes the neutron transport discrete-block method theoretically able to deal with Neutron transport problems in vacuum materials.