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

Abstract

A kind of calculating the method for arbitrary order coefficient in neutron transport discrete locking nub method, main contents include: 1. form simplifies: by introducing middle coefficient, complicated source coefficient, sediment flux ratio and the coefficient of coup are carried out form simplification；2. integral transformation: according to the character of Legnedre polynomial, the integration of the exponential function occurred in source coefficient, sediment flux ratio and the coefficient of coup and Legnedre polynomial product is converted into exponential function and the integration of general polynomial product；3. Analytical Solution: analytically derived accurate expression and the recurrence Relation of intermediary intergal by mathematical method, these accurate expressions and recurrence Relation do not comprise complicated integral operation, it is prone to computer programming realize, the intermediary intergal value accurately solved by computer substitutes into the expression formula of source coefficient, sediment flux ratio and the coefficient of coup after integral transformation, it is combined as the middle coefficient value of known conditions, it is thus achieved that arbitrary order source coefficient, sediment flux ratio and the exact value of the coefficient of coup.

Description

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

Technical field

The present invention relates to nuclear engineering neutron transport numerical simulation field, be specifically related to a kind of calculating neutron transport discrete The method of arbitrary order Legendre polynomial expansion coefficient in locking nub method.

Background technology

Nuclear reactor is a kind of device realizing controlled self-holding neutron fission course of reaction.Neutron fission reaction be neutron with Fissile material atomic nucleus generation fission reaction generates fission fragment, new neutron and photon etc., the process simultaneously given off energy. These energy can be utilized by people, and this is the principle of nuclear power station.Yet with the complexity of self-sustaining chain fission reaction, and core Fission reaction is direct and fission fragment discharges indirectly neutron, photon have strong biocidal, fission self-sustaining chain The numerical simulation of reaction is at nuclear engineering field broad development.On the other hand, people often it is to be understood that neutron, photon in media as well Distribution situation, such as nuclear power Factory Building radiation detection, material irradiation research, this is often also required to neutron, the Numerical-Mode of photon behavior Intend.

Neutron transport theory is the most accurately to describe neutron reaction in media as well, transport transition process and The theory of distribution eventually.Neutron transport theory mathematically can be expressed as one about Neutron flux distribution(abbreviation neutron flux) Complicated integro-differential equation.Wherein Neutron flux distribution is to describe neutron at time, space, heading and energy different dimensions The variable of upper distribution.Due to the complexity of the equation, current mathematics cannot accurately solve.Extensive along with computer technology Development and application, people pass through the numerical method approximate solution equation.Yet with its complexity and development of computer level Restriction, people also fail to ideally by solving precision and efficiency optimization to the acceptable scope of industrial quarters.

Through investigation, neutron transport discrete locking nub method be a kind of precision and efficiency of a relatively high solve neutron-transport equation Numerical method.The discrete Nodal method of three-dimensional neutron transport improved through Ahmed Badruzzaman is ensureing same accuracy There is under premise higher efficiency.The detailed process of the method is shown in document " An Efficient Algorithm for Nodal- Transport Solutions in Multidimensional Geometry " (hereinafter referred to as document).The method solves nothing The neutron-transport equation of the stable state of time variable；Discrete by variable, spatial spreading is become multiple grid, referred to here as each grid It is a locking nub, inside locking nub, further space variable is used Legendre polynomial expansion, obtain corresponding different Le and allow The flux square of moral polynomial order；Use DISORT method discrete heading, i.e. with the most multiple discrete direction generations Table continuous print space flight direction；And energy variable is divided into multistage, every section is an energy group, referred to as multipotency group approximation.Pass through These numerical approximations, the most at last Neutron flux distributionIt is expressed as the discrete variable with different dimensions change.Pass through numerical method Solve the value of these discrete variables, just approximate solution neutron-transport equation.It should be noted that these discrete variables from The degree of dissipating is closely related with the precision solved.Such as space variable, it is the closeest more accurate, inside locking nub that locking nub divides The expansion exponent number of Legnedre polynomial is the highest more accurate.

After neutron-transport equation discretization, through special derivation deformation, three passes of following form can be obtained Key coefficient:

$F_{xn}=\frac{\mathrm{\Δ}x}{2\mathrm{\μ}}\frac{2n+1}{2}\exp \left(\frac{-{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}\right)\underset{-1}{\overset{1}{\∫}}{P}_n\left(x\right)\exp \left(\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x\right)dx---\left(1\right)$

$G_{xk}=\exp \left(\frac{-{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}\right)\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x)dx---\left(2\right)$

$G_{xkn}=\frac{\mathrm{\Δ}x}{2\mathrm{\μ}}\frac{2n+1}{2}\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x)\[\underset{-1}{\overset{x}{\∫}}{P}_n\left(x^{\′}\right)\exp \left(\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}{x}^{\′}\right){\mathrm{dx}}^{\′}\]dx---\left(3\right)$

Wherein, F_xn、G_xkAnd G_xknSubscript x represent that this coefficient is relevant with the x direction in cartesian coordinate system, k and n is non- Negative integer, represents neutron flux and the Legendre polynomial expansion exponent number of neutron source in locking nub respectively, is called for short and launches exponent number；Bracket In x' and x all represent the location variable in x direction along cartesian coordinate system, unit is cm, in order to distinguish x' in integration Do subscript；Δ x represents locking nub size cm in the x-direction, and exp represents natural exponential function, and μ is neutron heading and x-axis Angle cosine value, in this formula, μ is limited on the occasion of, Σ_tFor the neutron overall reaction cross section of this locking nub, represent response probability, unit cm^-1, P_nX () is the n rank Legnedre polynomial about x, P_kX () is the k rank Legnedre polynomial about x, P_n(x') it is about x''s N rank Legnedre polynomial, Legnedre polynomial has a following form:

$\begin{array}{c}{P}_0\left(x\right)=1,\\ P_1\left(x\right)=x,\\ P_2\left(x\right)=\frac{3x^2-1}{2},\\ P_3\left(x\right)=\frac{5x^3-3x}{2},\\ \mathrm{...}\end{array}---\left(4\right)$

Easily checking meets the recurrence Relation of following (5) formula, higher order for exponent number less than the Legnedre polynomial on 2 rank Legnedre polynomial the most not there is this character.

$\frac{{\mathrm{dP}}_n\left(x\right)}{dx}=(2n-1)P_{n-1}\left(x\right),n\≤2---\left(5\right)$

Document utilizes this recurrence Relation, gives the coefficient recursion relation formula being only applicable to below 2 rank:

$F_{xn}=\frac{2n+1}{2{\Σ}_t}\{\[1+{(-1)}^{n+1}\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{\mathrm{\μ}})\]P_n\left(1\right)-\frac{4\mathrm{\μ}}{\mathrm{\Δ}x}{F}_{n-1}\}---\left(6\right)$

$F_{x0}=\frac{1}{2{\Σ}_t}\[1-\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{\mathrm{\μ}})\]$

$\left(7\right)$

$G_{xk}=\frac{2\mathrm{\μ}}{\mathrm{\Δ}x}\frac{2}{2k+1}{(-1)}^k{F}_k---\left(8\right)$

$G_{kn}=\frac{1}{{\Σ}_t}\[{\mathrm{\δ}}_{kn}+{(-1)}^{k+n+1}\left(\frac{2n+1}{2k+1}\right)\frac{2\mathrm{\μ}}{\mathrm{\Δ}x}{F}_k\]-(2n+1)\frac{2\mathrm{\μ}}{{\Σ}_t\mathrm{\Δ}x}(G_{k,n-1}+G_{k,n-3}+\mathrm{...})---\left(9\right)$

$G_{x00}=\frac{1}{{\Σ}_t}(1-G_0)---\left(10\right)$

$G_{x01}=\frac{-3}{2{\Σ}_t}\[\frac{2}{1-\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{\mathrm{\μ}})}-\frac{2\mathrm{\μ}}{{\Σ}_t\mathrm{\Δ}x}-1\]---\left(11\right)$

$G_{x01}=-\frac{1}{3}{G}_{01}---\left(12\right)$

In locking nub, the exponent number that launches of space variable affects precision and the efficiency of numerical solution, the highest expansion exponent number meaning Taste locking nub size and can be obtained the biggest, thus improves the efficiency of numerical solution on the premise of obtaining same precision.In document In the method for this design factor limits locking nub, space variable launches exponent number, thus limits precision and the effect of this numerical method Rate.

Summary of the invention

For the problem overcoming above-mentioned prior art to exist, it is an object of the invention to provide a kind of calculate neutron transport from Dissipate the method for arbitrary order Legendre polynomial expansion coefficient (source coefficient, sediment flux ratio and the coefficient of coup), the method in locking nub method Broken virgin neutron transport discrete locking nub method calculate Reactor Neutron Flux Density distribution time precision and efficiency by launch exponent number Restriction.

In order to achieve the above object, the present invention adopts the following technical scheme that

A kind of calculating the method for arbitrary order coefficient in neutron transport discrete locking nub method, main contents include: 1. form simplifies: By introducing middle coefficient, complicated source coefficient, sediment flux ratio and the coefficient of coup are carried out form simplification；2. integral transformation: according to The character of Legnedre polynomial, splits into n one by the n rank Legnedre polynomial in source coefficient, sediment flux ratio and the coefficient of coup As multinomial, thus by the exponential function occurred in source coefficient, sediment flux ratio and the coefficient of coup and Legnedre polynomial product Integration (being called for short original integration) is converted into the integration (abbreviation intermediary intergal) of exponential function and general polynomial product；3. resolve and ask Solve: analytically derived accurate expression and the recurrence Relation of intermediary intergal by mathematical method, these accurate expressions and Recurrence Relation does not comprise complicated integral operation, it is easy to computer programming realizes, the intermediary intergal accurately solved by computer Value substitutes into the expression formula of source coefficient, sediment flux ratio and the coefficient of coup after integral transformation, is combined as the middle system of known conditions Numerical value, it is thus achieved that arbitrary order source coefficient, sediment flux ratio and the exact value of the coefficient of coup.

Provide the method for arbitrary order coefficient in neutron transport discrete locking nub method that obtains in detail below:

The first step, by introducing middle coefficient, three key coefficients i.e. source coefficient in locking nub method discrete to neutron transport, The form of sediment flux ratio and the coefficient of coup simplifies:

Rewrite virgin neutron and transport three key coefficients in discrete locking nub method, i.e. source coefficient F_xn, sediment flux ratio G_xkAnd coupling Coefficient G_xknIt is as follows,

$F_{xn}=\frac{\mathrm{\Δ}x}{2\mathrm{\μ}}\frac{2n+1}{2}\exp \left(\frac{-{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}\right)\underset{-1}{\overset{1}{\∫}}{P}_n\left(x\right)\exp \left(\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x\right)dx---\left(13\right)$

$G_{xk}=\exp \left(\frac{-{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}\right)\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x)dx---\left(14\right)$

$G_{xkn}=\frac{\mathrm{\Δ}x}{2\mathrm{\μ}}\frac{2n+1}{2}\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x)\[\underset{-1}{\overset{x}{\∫}}{P}_n\left(x^{\′}\right)\exp \left(\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}{x}^{\′}\right){\mathrm{dx}}^{\′}\]dx---\left(15\right)$

Wherein, F_xn、G_xkAnd G_xknSubscript x represent that this coefficient is relevant with the x direction in cartesian coordinate system, k and n is non- Negative integer, represents neutron flux and the Legendre polynomial expansion exponent number of neutron source in locking nub respectively, is called for short and launches exponent number；Bracket In x' and x all represent the location variable in x direction along cartesian coordinate system, unit is cm, in order to distinguish x' in integration Do subscript；Δ x represents locking nub size cm in the x-direction, and exp represents natural exponential function, and μ is neutron heading and x-axis Angle cosine value, in this formula, μ is limited on the occasion of, Σ_tFor the neutron overall reaction cross section of this locking nub, represent response probability, unit cm^-1, P_nX () is the n rank Legnedre polynomial about x, P_kX () is the k rank Legnedre polynomial about x, P_n(x') it is about x''s N rank Legnedre polynomial；

In order to simplify source coefficient, sediment flux ratio and the coefficient of coup, introduce following middle coefficient b, a and c:

$b=\frac{\mathrm{\Δ}x}{2\left|\mathrm{\μ}\right|}---\left(16\right)$

A=Σ_tb (17)

C=e^-a (18)

Wherein e is natural Exponents,

(16), (17) and (18) formula is substituted into (13), (14) and (15) formula respectively and obtains the source coefficient of following form, flux Coefficient and the coefficient of coup:

$F_{xn}=bc(n+0.5)\underset{-1}{\overset{1}{\∫}}{P}_n\left(x\right)e^{ax}dx---\left(19\right)$

$G_{xn}=c\underset{-1}{\overset{1}{\∫}}{P}_n\left(x\right)e^{-ax}dx---\left(20\right)$

$G_{xkn}=b(n+0.5)\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)e^{-ax}\[\underset{-1}{\overset{x}{\∫}}{P}_n\left(x^{\′}\right)e^{{\mathrm{ax}}^{\′}}{\mathrm{dx}}^{\′}\]dx---\left(21\right)$

Son transports discrete locking nub method when solving pile neutron flux in the application, and middle coefficient b, a and c are as known Value, source coefficient, sediment flux ratio and the coefficient of coup are the functions of Legendre polynomial expansion exponent number n and k, but functional relationship (19), in (20) and (21) formula containing Legnedre polynomial and the integration of exponential function, direct numerical solution is time-consuming and the most smart True, below step use the thought of integral transformation and Analytical Solution obtain accurate arbitrary order source coefficient, sediment flux ratio and Coupled systemes numerical value；

Second step, is split as the form of multinomial summation, by source coefficient, sediment flux ratio and coupled systemes by Legnedre polynomial The exponential function occurred in number and the integration of Legnedre polynomial product are called for short original integration, are converted into exponential function with the most The integration abbreviation intermediary intergal of item formula product:

Legnedre polynomial is written as multinomial summation form:

$P_n\left(x\right)=\underset{n^{\′}=0}{\overset{n}{\Σ}}{p}_{n^{\′}-n}{x}^{n^{\′}}---\left(22\right)$

Wherein p_n'-nFor n rank Legnedre polynomial n-th ' the coefficient of power item, such as p_0-0=1, p_0-1=0, p_1-1=1 etc..

(22) formula is substituted in (19), (20) and (21) formula, obtains source coefficient, sediment flux ratio and coefficient of coup intermediary intergal Relational expression:

$F_{xn}=bc(n+0.5)\underset{n^{\′}=0}{\overset{n}{\Σ}}{p}_{n^{\′}-n}{I}_{n^{\′}}^a---\left(23\right)$

$G_{xk}=c\underset{k^{\′}=0}{\overset{k}{\Σ}}{p}_{k^{\′}-k}{I}_{k^{\′}}^{-a}---\left(24\right)$

$G_{xkn}=b(n+0.5)\underset{n^{\′}=0}{\overset{n}{\Σ}}\underset{k^{\′}=0}{\overset{k}{\Σ}}{p}_{n^{\′}-n}{p}_{k^{\′}-k}{J}_{k^{\′}{n}^{\′}}---\left(25\right)$

Wherein intermediary intergal form is as follows:

$I_{n^{\′}}^a=\underset{-1}{\overset{1}{\∫}}{x}^{n^{\′}}{e}^{ax}dx---\left(26\right)$

$I_{k^{\′}}^{-a}=\underset{-1}{\overset{1}{\∫}}{x}^{k^{\′}}{e}^{-ax}dx---\left(27\right)$

$J_{k^{\′}{n}^{\′}}=\underset{-1}{\overset{1}{\∫}}{x}^{k^{\′}}{e}^{-ax}\left(\underset{-1}{\overset{x}{\∫}}{x}^{\′n^{\′}}{e}^{{\mathrm{ax}}^{\′}}{\mathrm{dx}}^{\′}\right)dx---\left(28\right)$

3rd step, is obtained the analytical expression of intermediary intergal, thus obtains arbitrarily by integration by parts and mathematical induction Source, rank coefficient, sediment flux ratio and the analytical expression of the coefficient of coup:

(23)-(28) formula will ask the key transformation of arbitrary order coefficient for asking shape such asAnd J_knExponential function and arbitrary order Multinomial integration；By mathematical integration by parts and mathematical induction, obtain the arbitrary order intermediary intergal solution of following form Analysis expression formula:

$I_n^a={(-1)}^{n+1}{e}^a\underset{l=0}{\overset{n}{\Σ}}\frac{{(-1)}^{l+1}}{a^{n-l+1}}\frac{n!}{l!}+{(-1)}^{n+1}{e}^{-a}\underset{l=0}{\overset{n}{\Σ}}\frac{1}{a^{n-l+1}}\frac{n!}{l!},a>0---\left(29\right)$

$I_n^a=\{\begin{array}{cc}2,& n=0\\ 0,& n>0\end{array},a=0---\left(30\right)$

$I_n^{-a}={(-1)}^n{I}_n^a---\left(31\right)$

$J_{kn}={(-1)}^{n+1}\underset{l=0}{\overset{n}{\Σ}}\frac{{(-1)}^{l+1}}{a^{n-l+1}}\frac{n!}{l!}\frac{1-{(-1)}^{l+k+1}}{l+k+1}+{(-1)}^{n+1}{e}^{-a}{I}_k^{-a}\underset{l=0}{\overset{n}{\Σ}}\frac{1}{a^{n-l+1}}\frac{n!}{l!},a>0---\left(32\right)$

$J_{kn}=\{\begin{array}{cc}4,& k+n=0\\ 0,& k+n>0\end{array},a=0---\left(33\right)$

And obtain following recurrence Relation:

$I_0^a=(e^a-e^{-a})/a,a>0---\left(34\right)$

$I_n^a=-\frac{n}{a}{I}_{n-1}^a+(e^a+{\left(-1\right)}^{n+1}{e}^{-a})/a,a>0---\left(35\right)$

$J_{k0}=\[\frac{1-{(-1)}^{k+1}}{k+1}-e^{-a}{I}_k^{-a}\]/a,a>0---\left(36\right)$

$J_{kn}=-\frac{n}{a}{I}_{k(n-1)}+\[\frac{1-{(-1)}^{n+k+1}}{n+k+1}+{(-1)}^{n+1}{e}^{-a}{I}_k^{-a}\]/a,a>0---\left(37\right)$

(29)-(33) formula is the analytical expression of intermediary intergal, for arbitrary Legendre polynomial expansion exponent number n and k, These intermediary intergals can be tried to achieve conveniently by computer programming；The value of the intermediary intergal tried to achieve is substituted into integrated conversion In expression formula (23), (24) and (25) formula of source coefficient, sediment flux ratio and the coefficient of coup, just obtain arbitrary order source coefficient, flux Coefficient and the value of calculation of the coefficient of coup,

Noticing that (30), (31) formula give the special circumstances of a=0, this corresponds to neutron overall reaction cross section Σ_tIt is the feelings of zero Condition, this makes neutron transport discrete locking nub method can process the Neutron Transport Equation of band vacuum material in theory, such as band vacuum Cavity pile neutron distribution calculates, neutron distribution in reactor periphery calculates.

The computational methods of above-mentioned neutron transport discrete locking nub method arbitrary order coefficient are realized by computer program, it was demonstrated that Its correctness and effectiveness,

Compared with prior art, the present invention has a following outstanding advantages:

1. break conventional neutron and transport discrete locking nub method to space variable Legendre polynomial expansion exponent number in locking nub Limit, can accurately obtain any higher-order expansion coefficient by the method resolved so that locking nub size can obtain bigger, thus Obtain higher precision and efficiency when pile neutron calculation of Flux Distribution simultaneously；

2. give the special neutron transport discrete locking nub method expansion coefficient expression formula of zero cross-section situation so that Neutron transport discrete locking nub method can be used to calculate band vacuum area reactor or reactor periphery Flux Distribution.

Detailed description of the invention

Below in conjunction with detailed description of the invention, the present invention is described in further detail:

The inventive method is based on neutron transport discrete locking nub method, in order to obtain arbitrary order neutron transport in Neutron Transport Equation Discrete locking nub method arbitrary order expansion coefficient, under the framework of neutron transport discrete locking nub method, by computer programming by above Step code realizes.

The most basic relational expression used in programming is (23), (24) and (25) formula, observes these three relational expression and understands, one Aspect middle coefficient a to be obtained, b and c；On the other hand need to obtain shape such asAnd J_knExponential function and arbitrary order multinomial take advantage of Long-pending integration.

The angle cosine value μ, Yi Jijie of the size Δ x of middle coefficient a, b, c and locking nub, neutron heading and coordinate axes The neutron overall reaction cross section Σ of block_tRelevant, this tittle is according to the character of concrete problem to be calculated and combines stress and strain model and discrete Angular divisions obtains, and can regard given value when solving；

Shape is such asAnd J_knArbitrary order integration obtained by the integral expression that is above given or recurrence Relation.Owing to passing Push away relational expression and use the value of the lower-degree coefficient calculated when calculating higher order coefficient, it is possible to reduce amount of calculation, should in practice With middle use recurrence Relation, during concrete calculating, the value of exponent number N and Flux Expansion exponent number K is launched in source is set-point (convenient discussion Period, it is assumed here that N=K), during for middle coefficient a=0, directly utilize (30) and (33) formula calculating integral value；For The situation of middle coefficient a ＞ 0, uses (34)-(37) formula recurrence calculation integrated value: calculate first by (34) formulaThen make With (35) formula progressively recurrence calculationThen relational expression (31) is used to calculateThen (36) formula of use calculates J_k0(k=1,2 ..., K), finally utilize (37) formula to calculate J_kn(k=1,2 ..., K；N=1,2 ..., N), each rank middle coefficient storage these calculated is standby.

Finally use (23), (24) and (25) formula combination middle coefficient and intermediary intergal, it is thus achieved that final arbitrary order (source exhibition Open the desirable any nonnegative integer of exponent number N and Flux Expansion exponent number K) source coefficient, sediment flux ratio and the value of the coefficient of coup.

Claims (1)

1. one kind calculates the method for arbitrary order coefficient in neutron transport discrete locking nub method, it is characterised in that: comprise the steps:

The first step, by introducing middle coefficient, three key coefficients i.e. source coefficient, the flux in locking nub method discrete to neutron transport The form of coefficient and the coefficient of coup simplifies:

Rewrite virgin neutron and transport three key coefficients in discrete locking nub method, i.e. source coefficient F_xn, sediment flux ratio G_xkAnd the coefficient of coup G_xknIt is as follows,

$F_{xn}=\frac{\mathrm{\Δ}x}{2\mathrm{\μ}}\frac{2n+1}{2}\exp \left(\frac{-{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}\right)\underset{-1}{\overset{1}{\∫}}{P}_n\left(x\right)\exp \left(\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x\right)dx---\left(13\right)$

$G_{xk}=\exp \left(\frac{-{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}\right)\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x)dx---\left(14\right)$

$G_{xkn}=\frac{\mathrm{\Δ}x}{2\mathrm{\μ}}\frac{2n+1}{2}\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)\exp (-\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}x)\[\underset{-1}{\overset{x}{\∫}}{P}_n\left(x^{\′}\right)\exp \left(\frac{{\Σ}_t\mathrm{\Δ}x}{2\mathrm{\μ}}{x}^{\′}\right){\mathrm{dx}}^{\′}\]dx---\left(15\right)$

Wherein, F_xn、G_xkAnd G_xknSubscript x represent that this coefficient is relevant with the x direction in cartesian coordinate system, k and n is that non-negative is whole Number, represents neutron flux and the Legendre polynomial expansion exponent number of neutron source in locking nub respectively, is called for short and launches exponent number；In bracket X' and x represents the location variable in x direction along cartesian coordinate system, and unit is cm, is x' to distinguish in integration Subscript；Δ x represents locking nub size cm in the x-direction, and exp represents natural exponential function, and μ is the angle of neutron heading and x-axis Degree cosine value, in this formula, μ is limited on the occasion of, Σ_tFor the neutron overall reaction cross section of this locking nub, represent response probability, unit cm^-1, P_n X () is the n rank Legnedre polynomial about x, P_kX () is the k rank Legnedre polynomial about x, P_n(x') it is the n rank about x' Legnedre polynomial；

In order to simplify source coefficient, sediment flux ratio and the coefficient of coup, introduce following middle coefficient b, a and c:

$b=\frac{\mathrm{\Δ}x}{2\left|\mathrm{\μ}\right|}---\left(16\right)$

A=Σ_tb (17)

C=e^-a (18)

Wherein e is natural Exponents,

(16), (17) and (18) formula is substituted into (13), (14) and (15) formula respectively and obtains the source coefficient of following form, sediment flux ratio And the coefficient of coup:

$F_{xn}=bc(n+0.5)\underset{-1}{\overset{1}{\∫}}{P}_n\left(x\right)e^{ax}dx---\left(19\right)$

$G_{xn}=c\underset{-1}{\overset{1}{\∫}}{P}_n\left(x\right)e^{-ax}dx---\left(20\right)$

$G_{xkn}=b(n+0.5)\underset{-1}{\overset{1}{\∫}}{P}_k\left(x\right)e^{-ax}\[\underset{-1}{\overset{x}{\∫}}{P}_n\left(x^{\′}\right)e^{{\mathrm{ax}}^{\′}}{\mathrm{dx}}^{\′}\]dx---\left(21\right)$

Son transports discrete locking nub method when solving pile neutron flux in the application, and middle coefficient b, a and c are as given value, source Coefficient, sediment flux ratio and the coefficient of coup are the functions of Legendre polynomial expansion exponent number n and k, but functional relationship (19), (20) (21) containing Legnedre polynomial and the integration of exponential function in formula, direct numerical solution is time-consuming and coarse, below Step use the thought of integral transformation and Analytical Solution to obtain accurate arbitrary order source coefficient, sediment flux ratio and the coefficient of coup Value；

Second step, is split as the form of multinomial summation, by source coefficient, sediment flux ratio and the coefficient of coup by Legnedre polynomial The exponential function occurred and the integration of Legnedre polynomial product are called for short original integration, are converted into exponential function and general polynomial The integration abbreviation intermediary intergal of product:

Legnedre polynomial is written as multinomial summation form:

$P_n\left(x\right)=\underset{n^{\′}=0}{\overset{n}{\Σ}}{p}_{n^{\′}-n}{x}^{n^{\′}}---\left(22\right)$

Wherein p_n'-nFor n rank Legnedre polynomial n-th ' the coefficient of power item,

(22) formula is substituted in (19), (20) and (21) formula, obtains source coefficient, sediment flux ratio and the pass of coefficient of coup intermediary intergal It is formula:

$F_{xn}=bc(n+0.5)\underset{n^{\′}=0}{\overset{n}{\Σ}}{p}_{n^{\′}-n}{I}_{n^{\′}}^a---\left(23\right)$

$G_{xk}=c\underset{k^{\′}=0}{\overset{k}{\Σ}}{p}_{k^{\′}-k}{I}_{k^{\′}}^{-a}---\left(24\right)$

$G_{xkn}=b(n+0.5)\underset{n=0}{\overset{n}{\Σ}}\underset{k=0}{\overset{k}{\Σ}}{p}_{n^{\′}-n}{p}_{k^{\′}-k}{J}_{k^{\′}{n}^{\′}}---\left(25\right)$

Wherein intermediary intergal form is as follows:

$I_{n^{\′}}^a=\underset{-1}{\overset{1}{\∫}}{x}^{n^{\′}}{e}^{ax}dx---\left(26\right)$

$I_{k^{\′}}^{-a}=\underset{-1}{\overset{1}{\∫}}{x}^{k^{\′}}{e}^{-ax}dx---\left(27\right)$

$J_{k^{\′}{n}^{\′}}=\underset{-1}{\overset{1}{\∫}}{x}^{k^{\′}}{e}^{-ax}\left(\underset{-1}{\overset{x}{\∫}}{x}^{\′n^{\′}}{e}^{{\mathrm{ax}}^{\′}}{\mathrm{dx}}^{\′}\right)dx---\left(28\right)$

3rd step, is obtained the analytical expression of intermediary intergal, thus obtains arbitrary order source by integration by parts and mathematical induction The analytical expression of coefficient, sediment flux ratio and the coefficient of coup:

(23)-(28) formula will ask the key transformation of arbitrary order coefficient for asking shape such asAnd J_knExponential function and arbitrary order multinomial Formula integration；By mathematical integration by parts and mathematical induction, obtain the arbitrary order intermediary intergal resolution table of following form Reach formula:

$I_n^a={(-1)}^{n+1}{e}^a\underset{l=0}{\overset{n}{\Σ}}\frac{{(-1)}^{l+1}}{a^{n-l+1}}\frac{n!}{l!}+{(-1)}^{n+1}{e}^{-a}\underset{l=0}{\overset{n}{\Σ}}\frac{1}{a^{n-l+1}}\frac{n!}{l!},a>0---\left(29\right)$

$I_n^a=\left\{\begin{array}{cc}2,& n=0\\ 0,& n>0\end{array}\right.,a=0---\left(30\right)$

$I_n^{-a}={(-1)}^n{I}_n^a---\left(31\right)$

$J_{kn}={(-1)}^{n+1}\underset{l=0}{\overset{n}{\Σ}}\frac{{(-1)}^{l+1}}{a^{n-l+1}}\frac{n!}{l!}\frac{1-{(-1)}^{l+k+1}}{l+k+1}+{(-1)}^{n+1}{e}^{-a}{I}_k^{-a}\underset{l=0}{\overset{n}{\Σ}}\frac{1}{a^{n-l+1}}\frac{n!}{l!},a>0---\left(32\right)$

$J_{kn}=\left\{\begin{array}{cc}4,& k+n=0\\ 0,& k+n>0\end{array}\right.,a=0---\left(33\right)$

And obtain following recurrence Relation:

$I_0^a=(e^a-e^{-a})/a,a>0---\left(34\right)$

$I_n^a=-\frac{n}{a}{I}_{n-1}^a+(e^a+{\left(-1\right)}^{n+1}{e}^{-a})/a,a>0---\left(35\right)$

$J_{k0}=\[\frac{1-{(-1)}^{k+1}}{k+1}-e^{-a}{I}_k^{-a}\]/a,a>0---\left(36\right)$

$J_{kn}=-\frac{n}{a}{I}_{k(n-1)}+\[\frac{1-{(-1)}^{n+k+1}}{n+k+1}+{(-1)}^{n+1}{e}^{-a}{I}_k^{-a}\]/a,a>0---\left(37\right)$

(29)-(33) formula is the analytical expression of intermediary intergal, for arbitrary Legendre polynomial expansion exponent number n and k, it is possible to These intermediary intergals are tried to achieve conveniently by computer programming；The value of the intermediary intergal tried to achieve is substituted into the source system of integrated conversion In expression formula (23), (24) and (25) formula of number, sediment flux ratio and the coefficient of coup, just obtain arbitrary order source coefficient, sediment flux ratio With the value of calculation of the coefficient of coup,

Noticing that (30), (31) formula give the special circumstances of a=0, this corresponds to neutron overall reaction cross section Σ_tIt is the situation of zero, this Make neutron transport discrete locking nub method can process the Neutron Transport Equation of band vacuum material in theory.