CN107092581A - The transport equation response matrix Block Diagonalization method discussed based on symmetric group - Google Patents
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Abstract
The transport equation response matrix Block Diagonalization method discussed based on symmetric group, is comprised the following steps：1st, divided for whole reactor core domain according to the geometry of rule, obtain the locking nub of rule, based on variation Nodal method, functional is set up to neutron-transport equation, the weak form of transport equation is constructed；Discrete to the progress of neutron angular flax density using spheric harmonic function and orthogonal polynomial, spheric harmonic function together constitutes the basic function of flux with orthogonal space multinomial；2nd, according to the geometry of locking nub, the symmetric group that all symmetry changes of locking nub are constituted is determined；Basic function is projected using the projection operator of symmetric group, so as to construct the basic function of irreducible symmetrization；3rd, by the spheric harmonic function and orthogonal space multinomial in the basic function replacement step 1 of irreducible symmetrization, the matrix that discrete, new response matrix is diagonal piecemeal is carried out to variable using new basic function；The floating-point operation amount of matrix is substantially reduced using the inventive method, the calculating internal memory needed for memory response matrix is greatly reduced.
Description
Technical field
The present invention relates to nuclear reactor design and security technology area, and in particular to a kind of based on the defeated of symmetric group opinion
Transport equation response matrix Block Diagonalization method.
Background technology
Whole core transports locking nub and calculated to have great importance in the calculating of neutronics.In the solution of neutron-transport equation
In, if the response relation that neutron at locking nub inner boundary goes out incident flow can be constructed in advance, it can be passed through according to the response relation
The scan iterations of locking nub are until convergence, you can the neutron leakage of locking nub is obtained, so as to solve the neutron flux in each locking nub
Distribution.This method is referred to as response matrix method.The angle item of transport equation is usually carried out discrete using spheric harmonic function.If angle
The spheric harmonic function of N ranks before item expansion takes, and transport equation is projected in the spheric harmonic function of preceding N ranks, you can obtain (N+1
)2Individual linear algebraic equation systems, i.e. PN Approximate Equations.Because the sum and the number of spheric harmonic function of unknown quantity inside locking nub are
Positively related, therefore, the sum of unknown quantity deploys exponent number with angle and increased in square.
The structure of response matrix is usually directed to multiplication of matrices and inversion operation, these floating-point operation amounts substantially with it is unknown
Measure the proportional relation of cube of number.Therefore, when the number of unknown quantity is more, calculate internal memory and floating-point operation total amount will
It can greatly increase.
It is anti-for leakage relatively strong, the small-sized fast neutron with strong anisotropic scattering effect in the calculating of actual reactor core
Heap is answered, if expecting more accurate numerical result, higher angle exponent number is typically employed to and is approached.If in addition,
Carry out burnup, transient state etc. to calculate, the section of each locking nub is different, causes each locking nub to need individually to calculate respective response square
Battle array, this causes great challenge for calculating internal memory and computational efficiency.
The content of the invention
In order to overcome the problem of above-mentioned prior art is present, it is an object of the invention to provide a kind of based on the defeated of symmetric group opinion
Equation response matrix Block Diagonalization method is transported, this method is discussed based on symmetric group, will be defeated using the basic function of irreducible symmetrization
The solution of fortune equation is reduced to several subproblems decoupled mutually, a series of is decoupled mutually so that response matrix be decomposed into
Positioned at the sub-block of the diagonal positions of original matrix.In matrix multiplication and inversion operation, can individually it be carried out with antithetical phrase block matrix
Multiplication and inversion operation, so as to substantially reduce the floating-point operation amount of matrix.Further, since matrix has been carried out at Block Diagonalization
Reason, compared with original matrix, new nonzeros is greatly reduced, and the calculating internal memory needed for memory response matrix also drops significantly
It is low.
To achieve these goals, it is practiced this invention takes following technical scheme：
The transport equation response matrix Block Diagonalization method discussed based on symmetric group, this method is comprised the following steps：
Step one, divided for whole reactor core domain according to the geometry of rule, so that the locking nub of rule is obtained,
Based on variation Nodal method, functional is set up to neutron-transport equation, the weak form of transport equation is constructed；Use spheric harmonic function with
And orthogonal polynomial is discrete to the progress of neutron angular flax density, spheric harmonic function together constitutes flux with orthogonal space multinomial
Shown in basic function, such as formula (8)
Whereinψ(r, Ω) is even flux density；J (r, Ω) is the Lagrange multiplier of locking nub boundary；G (Ω)=
[Y0,0,Y2,-2,Y2,-1,Y2,0,Y2,1,Y2,2...]TFor the normalized spheric harmonic function of even order；F (r)=[f1,f2,f3...]TFor
It is defined on the orthogonal polynomial inside locking nub v；H (r)=[h1,h2...]TTo be defined on the orthogonal polynomial of locking nub v boundaries；The border basic function positioned at No. 1 edge surface is represented,Represent to be located at No. 2 edge surfaces
Border basic function；And ξ is respectively even neutron angular flux density expansion square and neutron-current expansion square；T is transposition
Symbol；
The flux of discrete form and Lagrange multiplier item are brought into arrange in functional and can obtain response matrix equation：
j+=Bs+Rj-
(16)
WhereinDeploy square for even neutron angular flux density；j+Deploy square for outgoing partial current density；j-For in incidence partially
Subflow density deploys square；sDeploy square for source item；R, H, C, B are four response matrixs of variation Nodal method；
Step 2, according to the geometry of the locking nub in step one, determines pair that all symmetry changes of locking nub are constituted
Claim group；The basic function in step one is projected using the projection operator of symmetric group, so as to construct irreducible symmetrization
Basic function；
Projection operatorIt is made up of symmetrical operator
In formula
n--- group's member number in order of a group number, i.e. finite group；
sα--- the dimension of the α irreducible representation matrix；
R --- group element；
PR--- with R groups of first corresponding symmetry operators；
The title of G --- group；
--- the complex conjugate of the j row jth column elements of the corresponding the α irreducible representation matrix of group element R；
I.e. acquisition basic function is projected to basic function using projection operator irreducible under different symmetry spaces
Component；If boundary face set of basis function is into basis function vectorUse projection operatorCarrying out projection to functional vector can
：
Projection operator is made up of symmetrical operator, and symmetrical operator acts on basis function vectorIt is equal to a certain matrix
URFunctional vector is multiplied by, i.e.,：
So as to：
Wherein matrix
Functional vectorFunction be linear correlation, at this moment need to matrixGaussian elimination is carried out, so that
Obtain new non-singular matrixIts size is For matrixOrder；
Obtain matrixAfterwards so as to obtain the basic function of the jth row of the α irreducible representation：
The above method is repeated, using other projection operators, the basic function of all irreducible symmetrizations is finally tried to achieve；
Step 3, the spheric harmonic function and orthogonal space in the basic function replacement step one of irreducible symmetrization is multinomial
Formula,
Wherein K (r, Ω) and H (r, Ω) is the basic function of irreducible symmetrization；Using new basic function variable is carried out from
Dissipate, new response relation can be obtained：
New response matrixFor the matrix of diagonal piecemeal.
Compared with prior art, the present invention has following outstanding advantages：
1. complicated transport issues can be decomposed into a series of subproblem of decouplings using the symmetry of locking nub, so that simple
The solution of change problem.
2. basic function is carried out after symmetrization processing, you can the response matrix of diagonal piecemeal is obtained, so as to greatly reduce
The calculating time of response matrix.
3. the number of the nonzero element of response matrix is greatly reduced, so as to largely reduce response matrix
Required calculating internal memory.
Brief description of the drawings
Fig. 1 hexagon locking nub symmetry schematic diagrames.
Fig. 2 response matrixsNonzero element position schematic diagram.
Fig. 3 is using the response matrix after the basic function of irreducible symmetrizationNonzero element position schematic diagram.
Embodiment
The present invention is described in further detail with reference to the accompanying drawings and detailed description：
The inventive method will be theoretical based on symmetric group, using the basic function of irreducible symmetrization, by the solution of transport equation
Be reduced to several subproblems decoupled mutually, thus by response matrix be decomposed into it is a series of decouple mutually be located at original matrix
The sub-block of diagonal positions.In matrix multiplication and inversion operation, multiplication and fortune of inverting can be individually carried out with antithetical phrase block matrix
Calculate, so as to substantially reduce the floating-point operation amount of matrix.Further, since matrix has carried out Block Diagonalization processing, with original matrix
Compare, new nonzeros is greatly reduced, the calculating internal memory needed for memory response matrix is greatly reduced.
Whole calculate specifically includes following steps：
Step one, based on variation Nodal method, functional is set up to neutron-transport equation, the weak solution shape of transport equation is constructed
Spheric harmonic function is respectively adopted for formula, the angle of odd even flux and space item and orthogonal polynomial progress is discrete.
The neutron-transport equation of the second order even symmetry form of single energy can be written as form：
R --- locus variable；
Ω --- angle variables；
ψg--- g can group's neutron angular flux density；
Φg--- neutron scalar flux density；
G --- can group identification；
Σt,g--- the total cross section of g energy groups；
--- the self-scattering section of g energy groups；
Sg--- the neutron-transport equation source item of g energy groups, comprising scattering source item and fission source term between group,
It is assumed here that scattering is isotropic, the odd even flux of g groups is defined as follows
ψg(r, Ω) --- the even flux density of g groups；
χg(r, Ω) --- the strange flux density of g groups；
Order
Jγ(r, Ω)=nγ·Ωχγ(r,Ω) (3)
nγ--- it is the borderline exterior normal directions of locking nub γ；
χγ--- it is the borderline strange flux of locking nub γ；
Global functional is set up to the region solved, global functional adds and form for each locking nub local functional.
Local functional is
Wherein I [ψ, J] is boundary integral, for inner boundary
For vacuum boundary
Γ in above formulaiFor inner boundary, ΓjFor vacuum boundary.
Then, the problem of solving neutron-transport equation is then converted to solve the variational problem of global functional minimum, and
The solution of variational problem is the Solution of Weak Formulation of second order parity equation.
(5) J (r, Ω) is Lagrange multiplier in formula, and it is to eliminate the constraint that boundary condition is brought that it, which is acted on, by drawing
Boundary condition is added in Ge Lang multipliers, functional, it is achieved thereby that the coupling between locking nub.
The angle item of even flux and Lagrange multiplier is launched into the form of spheric harmonic function, space expansion is orthogonal
The form of basic function：
Wherein g (Ω)=[Y0,0,Y2,-2,Y2,-1,Y2,0,Y2,1,Y2,2...]TFor the normalized spheric harmonic function of even order；f
(r)=[f1,f2,f3...]TFor the orthogonal polynomial being defined on inside locking nub v, h (r)=[h1,h2...]TTo be defined on locking nub v
The orthogonal polynomial of boundary；The border basic function positioned at No. 1 edge surface is represented,Table
Show the border basic function positioned at No. 2 edge surfaces；And ξ is respectively even neutron angular flux density expansion square and neutron current
Density deploys square；T is transposition symbol.
(8) formula is brought into the response matrix that traditional variation Nodal method is can obtain in functional (5).
Wherein
M=[M1 M2 ... Mγ ...] (10)
Functional is zero on the derivative of Flux Expansion square, can obtain even Flux Expansion square and the relation of Lagrange multiplier：
Define the shunting expansion square in γ faces：
Wherein
Retouched on the strategy for solving expansion square using four coloured chess sweeping and response matrix computational methods, response matrix is public
Formula is as follows：
j+=Bs+Rj- (16)
Wherein
B=[G+I]-1CT (18)
R=[G+I]-1[G-I] (19)
H=A-1 (22)
Step 2, according to the geometry of locking nub, determines the symmetric group that all symmetry changes of locking nub are constituted.Two dimension six
Angular all symmetry changes have 12, as shown in Figure 1.These symmetry changes include 6 rotation transformation operatorsThey represent respectively 0 degree of rotate counterclockwise, 60 degree, 120 degree, 180 degree, 240 degree, 300 degree.Its
InAlso unit operator E can be denoted as.6 reflection transformation operators are { σv1,σv2,σv3,σd1,σd2,σd3, they are on each right
Claim the reflection transformation of axle.All symmetrical operators constitute C6vGroup.Also need to consider on xoy planes for six prism locking nubs
Symmetry, therefore have 24 symmetrical operators,Structure
D in groups6h。
Basic function is projected using the projection operator of symmetric group, so as to construct the basic function of irreducible symmetrization.
It might as well assume that all symmetry changes constitute group G, group G has several projection operators, whereinFor α not
The projection operator of the jth row of reducible representation matrix.
Projection operatorIt is made up of symmetry operator
In formula
N --- group's member number in order of a group number, i.e. finite group；
sα--- the dimension of the α irreducible representation matrix；
R --- group element；
PR--- with R groups of first corresponding symmetry operators；
The title of G --- group；
--- the complex conjugate of the j row jth column elements of the corresponding the α irreducible representation matrix of group element R；
By taking two-dimensional hexagonal locking nub as an example, it can be seen from the group theory, group C6v8 projection operators are had, they are respectively
Wherein
--- irreducible representation matrix A1Projection operator；
--- irreducible representation matrix A2Projection operator；
--- irreducible representation matrix B1Projection operator；
--- irreducible representation matrix B2Projection operator；
--- irreducible representation matrix E1First row projection operator；
--- irreducible representation matrix E1Secondary series projection operator；
--- irreducible representation matrix E2First row projection operator；
--- irreducible representation matrix E2Secondary series projection operator；
The operator of right of formula is the symmetry change operator in step 2.
Using projection operator to basic function carry out projection can obtain basic function under different symmetry spaces can not
About component.If boundary face set of basis function is into basis function vectorIt might as well setUseFunctional vector is projected
It can obtain：
Projection operator is made up of symmetrical operator, and a certain symmetrical operator acts on basis function vectorIt can obtain
One group of new basic function.If basic function space is closing for symmetrical operator, then obtained new basic function is still fallen within
Function space originally.Therefore symmetrical operator acts on basis function vectorIt is equal to a certain matrix URBe multiplied by function to
Amount, i.e.,：
Wherein matrix URIt can be obtained by following formula
It can be obtained according to formula (32) and formula (34)
Wherein matrix
Functional vectorLength and original basis function vectorLength formula it is equal.And
In function only formulaThe α irreducible representation jth row component, it can be seen thatLetter
Number is linear correlation.At this moment need to matrixGaussian elimination is carried out, so as to obtain new non-singular matrixIts
Size is For matrixOrder.If rightCarry out all irreducible representation spaces and different row are asked
With can obtain：
Obtain matrixAfterwards it is hereby achieved that the basic function of the jth row of the α irreducible representation：
The above method is repeated, using other projection operators, all irreducible basis functions are finally tried to achieve.
For two-dimensional hexagonal locking nub, C can be constructed according to projection operator of the formula (24) into formula (31)6vGroup is not
Reducible basic function.And for three-dimensional six prisms locking nub, due toTherefore C can first be generated6vIrreducible basis
Function, reuses C1hProjection operator to C6vIrreducible basis function further project, can finally obtain crowd D6hIt is irreducible right
The basic function of titleization.
Step 3, by the spheric harmonic function and orthogonal polynomial in the basic function replacement formula (8) of irreducible symmetrization：
The relation of irreducible basis function and former base function is as follows：
According to the basic function of irreducible symmetrization, then new response relation formula can be obtained：
As a result of the basic function of irreducible symmetrization, the symmetry of locking nub is make use of, new response matrix is diagonal
Matrix in block form, as shown in Figure 3.
By taking the response matrix G (formula (20)) of three-dimensional hexagon locking nub as an example, according to former base function, matrix such as Fig. 2 institutes
Show.The angle that the calculating employs P3 is approximate, and the space multinomial that border goes out is deployed using 2 ranks, and the multinomial of internal volume is adopted
Deployed with 6 ranks.Matrix G has 11504 nonzero elements in Fig. 2.And according to irreducible basis function, new matrix can be obtainedThe relation of the two matrixes is：
MatrixAs shown in Figure 3.It can be seen that matrixWith 16 sub-blocks for being located at diagonal positions.
Matrix is inverted and during multiplying, can progress independent to sub-block invert multiplying, this will greatly save matrix meter
Evaluation time.Nonzeros is also greatly reduced simultaneously, is reduced from 11504 to 1454, therefore, is largely reduced meter
Internal memory needed for calculating.The comparison that TAKETA problems are calculated using the total time of different angle expansion exponent numbers is as shown in table 1.For
P7 angle expansion, calculates speed-up ratio and can reach 15 times.
The different angles of table 1 deploy comparing calculating total time for exponent number problem
Claims (1)
1. the transport equation response matrix Block Diagonalization method discussed based on symmetric group, it is characterised in that：Comprise the following steps：
Step one, divide, so as to obtain the locking nub of rule, be based on according to the geometry of rule for whole reactor core domain
Variation Nodal method, functional is set up to neutron-transport equation, constructs the weak form of transport equation；Using spheric harmonic function and just
Multinomial is handed over to carry out neutron angular flax density discrete, spheric harmonic function together constitutes the base letter of flux with orthogonal space multinomial
Shown in number, such as formula (8)
Wherein ψ (r, Ω) is even flux density；J (r, Ω) is the Lagrange multiplier of locking nub boundary；G (Ω)=[Y0,0,
Y2,-2,Y2,-1,Y2,0,Y2,1,Y2,2...]TFor the normalized spheric harmonic function of even order；F (r)=[f1,f2,f3...]TTo be defined on
Orthogonal polynomial inside locking nub v；H (r)=[h1,h2...]TTo be defined on the orthogonal polynomial of locking nub v boundaries；The border basic function positioned at No. 1 edge surface is represented,Represent to be located at No. 2 edge surfaces
Border basic function；And ξ is respectively even neutron angular flux density expansion square and neutron-current expansion square；T accords with for transposition
Number；
The flux of discrete form and Lagrange multiplier item are brought into arrange in functional and can obtain response matrix equation：
j+=Bs+Rj- (16)
WhereinDeploy square for even neutron angular flux density；j+Deploy square for outgoing partial current density；j-For incident neutron current partially
Density deploys square；S is that source item deploys square；R, H, C, B are four response matrixs of variation Nodal method；
Step 2, according to the geometry of the locking nub in step one, determines the symmetric group that all symmetry changes of locking nub are constituted；
The basic function in step one is projected using the projection operator of symmetric group, so as to construct the base letter of irreducible symmetrization
Number；
Projection operatorIt is made up of symmetrical operator
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R --- group element；
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The title of G --- group；
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN112364294A (en) * | 2020-10-27 | 2021-02-12 | 中国核动力研究设计院 | Automatic conversion method, device, equipment and medium for reactor core numerical calculation matrix |
| CN113010841A (en) * | 2021-02-08 | 2021-06-22 | 南京信息工程大学 | Any-aperture optical element surface shape reconstruction method based on numerical orthogonal transformation |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN112364294A (en) * | 2020-10-27 | 2021-02-12 | 中国核动力研究设计院 | Automatic conversion method, device, equipment and medium for reactor core numerical calculation matrix |
| CN113010841A (en) * | 2021-02-08 | 2021-06-22 | 南京信息工程大学 | Any-aperture optical element surface shape reconstruction method based on numerical orthogonal transformation |
| CN113010841B (en) * | 2021-02-08 | 2024-02-23 | 南京信息工程大学 | Surface shape reconstruction method of arbitrary aperture optical element based on numerical orthogonal transformation |
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