CN113673116A - Three-dimensional quasi-transportation acceleration method aiming at uniform geometric variable block method - Google Patents

Three-dimensional quasi-transportation acceleration method aiming at uniform geometric variable block method Download PDF

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CN113673116A
CN113673116A CN202111021903.4A CN202111021903A CN113673116A CN 113673116 A CN113673116 A CN 113673116A CN 202111021903 A CN202111021903 A CN 202111021903A CN 113673116 A CN113673116 A CN 113673116A
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殷晗
张滕飞
刘晓晶
熊进标
柴翔
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Shanghai Jiaotong University
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Abstract

A three-dimensional quasi-transport acceleration method aiming at a uniform geometric variation block method eliminates the cross derivative term of the axial direction and the radial direction in a second-order even-parity neutron transport equation through quasi-transport approximate treatment; considering that the axial nonuniformity is weak in the actual pressurized water reactor, the method adopts diffusion approximation to the angle distribution of the odd neutron angular flux density of the axial surface in the Ritz dispersion process; in the Ritz dispersion process, the number of odd neutron angular flux density angle basis functions of the radial surface is reduced by using the symmetry of an angle space. The method can greatly improve the calculation efficiency and reduce the calculation memory under the condition of not obviously influencing the calculation precision, thereby being used for accurate and efficient neutron science simulation in the design of the nuclear reactor.

Description

Three-dimensional quasi-transportation acceleration method aiming at uniform geometric variable block method
Technical Field
The invention relates to a technology in the field of nuclear engineering, in particular to a three-dimensional quasi-transportation accelerating method aiming at a uniform geometric variable block method.
Background
In the design of a reactor, in order to analyze the neutron performance and safety of the reactor, accurate and efficient neutron simulation needs to be carried out on the reactor so as to obtain the effective multiplication factor of the reactor and the neutron flux density distribution in the reactor. The effective multiplication coefficient and the neutron flux density are obtained by solving a neutron transport equation, the method widely adopted at present for solving the neutron transport equation is a variable-partition block method, the method divides a reactor area into a series of typical blocks, and the effective multiplication coefficient of the reactor and the neutron flux distribution in the reactor are finally obtained by constructing a corresponding response matrix for the typical blocks and completing the solution of the response matrix equation. However, in large-scale reactor three-dimensional neutron simulation, especially when high-order angle approximation is adopted, the scale and number of response matrixes are often large, which poses severe challenges to computational efficiency and computational memory, and greatly limits the engineering applicability of the method in reactor design.
Disclosure of Invention
The invention provides a three-dimensional quasi-transport acceleration method aiming at a uniform geometric variable block method, aiming at the problem that the calculation cost is high when the conventional uniform geometric variable block method is used for large-scale reactor three-dimensional neutron simulation, and the method can greatly improve the calculation efficiency and reduce the calculation memory under the condition of not obviously influencing the calculation precision, so that the method is used for accurate and efficient neutron simulation in nuclear reactor design.
The invention is realized by the following technical scheme:
the invention relates to a three-dimensional quasi-transportation accelerating method aiming at a uniform geometric variable block method, which comprises the following steps of:
step 1) starting from a steady-state neutron transport equation, and obtaining a second-order even-parity neutron transport equation without an axial radial cross derivative term through quasi-transport approximation.
And 2) establishing a functional containing a second-order even-parity neutron transport equation and boundary conditions by using a variational principle.
Step 3), expanding the even neutron angular flux density inside the segment in space by adopting a standard orthogonal polynomial; the angle is processed by an integral method.
And 4) performing standard orthogonal polynomial expansion on the odd neutron angular flux density of the radial surface of the segment in space, performing perfect spherical harmonic expansion on the angle, and reducing the number of the spherical harmonics by half by using the symmetry of the angular space.
And 5) the odd neutron angular flux density of the axial surface of the segment is spatially expanded by adopting a standard orthogonal polynomial, and the angle is approximated by adopting diffusion.
And 6) substituting the discrete form of the function into the segment functional, and obtaining a response matrix equation by adopting a variation method and utilizing the idea of variable replacement.
And 7) solving a response matrix equation by adopting a triple iteration method of fission source iteration, multi-group iteration and intra-group iteration to obtain the effective multiplication coefficient of the reactor and the neutron flux density distribution in the reactor.
The invention relates to a nuclear reactor neutron performance analysis system for realizing the method, which comprises the following steps: the device comprises an input information reading unit, a response matrix constructing unit, a matrix equation solving unit and a post-processing unit, wherein: the input information reading unit is used for reading the describing reactor parameter information input by a user and transmitting the describing reactor parameter information to the response matrix constructing unit, the response matrix constructing unit constructs a response matrix according to the reactor parameter information to obtain a response matrix B, R, V, C corresponding to each typical segment under each energy group, the matrix equation solving unit carries out power method iterative solution on the response matrix equation according to the constructed response matrix, and the post-processing unit outputs the effective multiplication coefficient of the reactor and the neutron flux density distribution diagram in the reactor according to the calculation result of the matrix equation solving unit to analyze the neutron performance of the nuclear reactor.
The reactor parameter information comprises: the method comprises the following steps of obtaining the geometric parameter information of a reactor, the material composition information of the reactor, the section information of a material, the space set by a user, the expansion order information of an angle and the convergence limit information set by the user.
Technical effects
According to the invention, through quasi-transport approximate treatment, the cross derivative terms of the axial direction and the radial direction in the second-order even-parity neutron transport equation are eliminated; considering that the axial nonuniformity is weak in the actual pressurized water reactor, the method adopts diffusion approximation to the angle distribution of the odd neutron angular flux density of the axial surface in the Ritz dispersion process; in the Ritz dispersion process, the number of odd neutron angular flux density angle basis functions of the radial surface is reduced by using the symmetry of an angle space.
Compared with the condition that the calculation cost is high in the process of neutron simulation in a large-scale reactor by the conventional uniform geometric variable block method, the method disclosed by the invention has the advantages that the calculation efficiency is greatly improved, the calculation memory is reduced under the condition that the calculation precision is not obviously influenced, and the method is applied to the large-scale neutron transport problem represented by the TAKEDA-3 benchmark, so that a better application effect is obtained: the calculation memory is reduced by 75%, and the calculation time is reduced by 93%.
Drawings
FIG. 1 is a schematic diagram of a TAKEDA-3 benchmark topic model in the embodiment;
FIG. 2 is a calculation efficiency and a calculation memory test result of the quasi-transport acceleration method on a TAKEDA-3 benchmark;
FIG. 3 is a flow chart of the present invention;
FIG. 4 is a flow chart of an iterative solution of response matrix equations by a power method.
Detailed Description
The embodiment relates to a three-dimensional quasi-transportation accelerating method aiming at a uniform geometric variable block method, which comprises the following steps:
step 1: solving a neutron transport equation: starting from a steady-state neutron transport equation, a leakage term in the equation is separated into an axial leakage part and a radial leakage part:
Figure BDA0003241787750000021
wherein: the subscript p denotes the radial contribution, z denotesAn axial contribution; introducing odd and even neutron angular flux density
Figure BDA0003241787750000022
Figure BDA0003241787750000031
And omitting high-order derivative terms to obtain a second-order even-parity neutron transport equation without axial and radial cross derivative terms
Figure BDA0003241787750000032
Wherein: omega is the neutron motion direction; sigmatSum ΣsRespectively a macroscopic total cross section and a macroscopic scattering cross section; psi+Is even astronomical neutron angular flux density; phi is neutron standard flux density; q is a source term, including inter-cluster scatter sources and fission sources.
Step 2: a functional comprising a second-order even-parity neutron transport equation and boundary conditions is established by utilizing a variational principle: f [ psi+，ψ-]＝∑vFv[ψ+，ψ-]，
Figure BDA0003241787750000033
Figure BDA0003241787750000034
Wherein: f is a functional of the whole problem domain and can be expressed as a functional F of each section blockvLinear superposition of (2); d Λ is the area infinitesimal of the radial surface of the segment; dA is the area infinitesimal of the axial surface; n isp、nzThe unit vectors of the outer normal directions of the radial surface and the axial surface, respectively.
And step 3: the even neutron angular flux density inside the segment is spatially expanded by adopting a standard orthogonal polynomial, and is angularly processed by adopting an integration method, which specifically comprises the following steps:
Figure BDA0003241787750000035
the discrete form of the corresponding neutron fluence density is:
Figure BDA0003241787750000036
the discrete format of the source item is:
Figure BDA0003241787750000037
wherein: f (z) and f (x, y) are orthonormal polynomial vectors, meet orthonormality and are obtained by Gramm-Schmidt orthogonalization; psi (omega), phi and q are respectively unfolding moment vectors corresponding to the functions.
And 4, step 4: the odd neutron angular flux density of the radial surface of the segment is expanded by adopting a standard orthogonal polynomial in space, is expanded by adopting a perfect spherical harmonic function in angle, and reduces the number of the spherical harmonic functions by half by utilizing the symmetry of an angular space, specifically:
Figure BDA0003241787750000038
Figure BDA0003241787750000039
y, and y; y, η x, wherein: γ is 1, 2, 3, 4 the radial surface number of the segment; y is-γ(Ω) is an odd-parity spherical harmonic vector defined on the γ plane, whose polar axis direction coincides with the outer normal direction of the γ plane; f. ofγ(η) is an orthonormal polynomial defined on the γ plane, satisfying orthonormality; chi shapeγThe moment vector is developed for the odd neutron angular flux density of the gamma plane.
And 5: the odd neutron angular flux density of the axial surface of the segment is spatially expanded by adopting a standard orthogonal polynomial, and the angle is approximate by adopting diffusion, and specifically comprises the following steps:
Figure BDA00032417877500000310
z ═ z ±, where: f. ofz′(x, y) is an orthonormal polynomial vector defined on the z' plane, satisfying orthonormality; chi shapez′The moment vector is unwrapped for the odd neutron angular flux density on the z' plane.
Step 6: substituting discrete form of function into segment functional, using variation principle, and introducing
Figure BDA00032417877500000311
To obtainResponse matrix equation: j is a function of+＝Bq+Rj-，φ＝Vq-C(j+-j-) Wherein: j is a function of+、j-Respectively an outgoing neutron flux density expansion moment and an incoming neutron flux density expansion moment,
Figure BDA00032417877500000312
and 7: solving a response matrix equation by adopting triple iteration of fission source iteration, multi-group iteration and intra-group iteration, which specifically comprises the following steps: the fission source iteration is used for converging the fission source and the characteristic value, the multi-group iteration is used for converging the neutron flux density of each energy group, the intra-group iteration converges the intra-group neutron flux density by adopting a red-black scanning method, and finally the effective multiplication coefficient of the reactor and the intra-reactor neutron flux density distribution are obtained.
As shown in FIG. 1, the TAKEDA-3 reference problem is a typical large scale reactor neutron transport reference problem, and comprises: case 1, control rod withdrawal and case 2, control rod insertion.
The method is applied to a variable block method program VITAS, and tests on the technical feasibility and performance including time calculation and memory calculation are completed by adopting a TAKEDA-3 benchmark. The calculation parameters in the specific implementation are set as follows: setting the size of a segment to be 5cm multiplied by 5cm, setting the expansion order of the internal space of the segment to be 6 orders, setting the expansion order of the surface of the segment to be 2 orders, setting the maximum red-black iteration times to be 10, setting the maximum multi-group iteration times to be 5, and respectively setting the convergence limits of the characteristic value, the fission rate and the neutron flux density to be: 1E-5, 5E-4, 5E-4.
The comparison results of the calculation time and the calculation memory under different angle expansion orders are shown in fig. 2. The results show that: compared with the existing uniform geometric variable block method, the method can reduce the calculation memory by 75 percent and the calculation time by 93 percent, and effectively solves the problem of high calculation cost in the neutron simulation of the large-scale reactor in the prior art.
Compared with the prior art, the method can obviously improve the calculation efficiency and reduce the calculation memory: the calculation memory is reduced by 75%, and the calculation time is reduced by 93%.
The foregoing embodiments may be modified in many different ways by those skilled in the art without departing from the spirit and scope of the invention, which is defined by the appended claims and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.

Claims (3)

1. A three-dimensional quasi-transport acceleration method aiming at a uniform geometric variation block method is characterized in that a second-order even-parity neutron transport equation without an axial radial cross derivative term is obtained by quasi-transport approximation from a steady-state neutron transport equation; establishing a functional containing a second-order even-parity neutron transport equation and boundary conditions by using a variational principle; the even neutron angular flux density inside the segment is spatially expanded by adopting a standard orthogonal polynomial, and is angularly processed by adopting an integration method; the odd neutron angular flux density of the radial surface of the segment is expanded by adopting a standard orthogonal polynomial in space, the complete spherical harmonic function is expanded in angle, and the number of the spherical harmonic functions is reduced by half by utilizing the symmetry of the angular space; the odd neutron angular flux density of the axial surface of the segment is spatially expanded by adopting a standard orthogonal polynomial, and the angle is approximate to the angle by adopting diffusion; substituting the discrete form of the function into the segment functional, adopting a variation method and utilizing the idea of variable replacement to obtain a response matrix equation; and solving a response matrix equation by adopting a triple iteration method of fission source iteration, multi-group iteration and intra-group iteration to obtain the effective multiplication coefficient of the reactor and the neutron flux density distribution in the reactor.
2. The three-dimensional quasi-transportation acceleration method aiming at the uniform geometric variation block method as claimed in claim 1, which is characterized by comprising:
step 1: solving a neutron transport equation: starting from a steady-state neutron transport equation, a leakage term in the equation is separated into an axial leakage part and a radial leakage part:
Figure FDA0003241787740000011
wherein: the subscript p represents the radial contribution, z represents the axial contribution; introducing odd and even neutron angular flux density
Figure FDA0003241787740000012
Figure FDA0003241787740000013
And omitting high-order derivative terms to obtain a second-order even-parity neutron transport equation without axial and radial cross derivative terms
Figure FDA0003241787740000014
Wherein: omega is the neutron motion direction; sigmatSum ΣsRespectively a macroscopic total cross section and a macroscopic scattering cross section; psi+Is even astronomical neutron angular flux density; phi is neutron standard flux density; q is a source term, including inter-cluster scattering sources and fission sources;
step 2: a functional comprising a second-order even-parity neutron transport equation and boundary conditions is established by utilizing a variational principle: f [ psi+，ψ-]＝∑vFv[ψ+，ψ-]，
Figure FDA0003241787740000015
Figure FDA0003241787740000016
Wherein: f is a functional of the whole problem domain and can be expressed as a functional F of each section blockvLinear superposition of (2); d Λ is the area infinitesimal of the radial surface of the segment; dA is the area infinitesimal of the axial surface; n isp、nzThe outer normal direction unit vectors of the radial surface and the axial surface, respectively;
and step 3: the even neutron angular flux density inside the segment is spatially expanded by adopting a standard orthogonal polynomial, and is angularly processed by adopting an integration method, which specifically comprises the following steps:
Figure FDA0003241787740000017
the discrete form of the corresponding neutron fluence density is:
Figure FDA0003241787740000018
the discrete format of the source item is:
Figure FDA0003241787740000019
wherein: f (z) and f (x, y) are orthonormal polynomial vectors, meet orthonormality and are obtained by Gramm-Schmidt orthogonalization; psi (omega), phi and q are respectively expansion moment vectors corresponding to the functions;
and 4, step 4: the odd neutron angular flux density of the radial surface of the segment is expanded by adopting a standard orthogonal polynomial in space, is expanded by adopting a perfect spherical harmonic function in angle, and reduces the number of the spherical harmonic functions by half by utilizing the symmetry of an angular space, specifically:
Figure FDA0003241787740000021
Figure FDA0003241787740000022
y, and y; y, η x, wherein: γ is 1, 2, 3, 4 the radial surface number of the segment; y is-γ(Ω) is an odd-parity spherical harmonic vector defined on the γ plane, whose polar axis direction coincides with the outer normal direction of the γ plane; f. ofγ(η) is an orthonormal polynomial defined on the γ plane, satisfying orthonormality; chi shapeγUnfolding a moment vector for the odd neutron angular flux density of the gamma plane;
and 5: the odd neutron angular flux density of the axial surface of the segment is spatially expanded by adopting a standard orthogonal polynomial, and the angle is approximate by adopting diffusion, and specifically comprises the following steps:
Figure FDA0003241787740000023
z ═ z ±, where: f. ofz′(x, y) is an orthonormal polynomial vector defined on the z' plane, satisfying orthonormality; chi shapez′Unwrapping odd neutron angular flux density on the z' -planeMoment vector quantity;
step 6: substituting discrete form of function into segment functional, using variation principle, and introducing
Figure FDA0003241787740000024
Obtaining a response matrix equation: j is a function of+＝Bq+Rj-，φ＝Vq-C(j+-j-) Wherein: j is a function of+、j-Respectively an outgoing neutron flux density expansion moment and an incoming neutron flux density expansion moment,
Figure FDA0003241787740000025
and 7: solving a response matrix equation by adopting triple iteration of fission source iteration, multi-group iteration and intra-group iteration, which specifically comprises the following steps: the fission source iteration is used for converging the fission source and the characteristic value, the multi-group iteration is used for converging the neutron flux density of each energy group, the intra-group iteration converges the intra-group neutron flux density by adopting a red-black scanning method, and finally the effective multiplication coefficient of the reactor and the intra-reactor neutron flux density distribution are obtained and are used for analyzing the neutron performance and safety of the nuclear reactor so as to ensure that the reactor design meets the design specification and safety standard.
3. A nuclear reactor neutronics performance analysis system implementing the method of claim 1 or 2, comprising: the device comprises an input information reading unit, a response matrix constructing unit, a matrix equation solving unit and a post-processing unit, wherein: the system comprises an input information reading unit, a response matrix construction unit, a matrix equation solving unit, a post-processing unit and a data processing unit, wherein the input information reading unit is used for reading descriptive reactor parameter information input by a user and transmitting the descriptive reactor parameter information to the response matrix construction unit, the response matrix construction unit constructs a response matrix according to the reactor parameter information to obtain a response matrix B, R, V, C corresponding to each typical segment under each energy group, the matrix equation solving unit carries out power method iterative solution on a response matrix equation according to the constructed response matrix, and the post-processing unit outputs an effective multiplication coefficient of a reactor and a neutron flux density distribution diagram in the reactor according to a calculation result of the matrix equation solving unit to analyze the neutron performance of the nuclear reactor;
the reactor parameter information comprises: the method comprises the following steps of obtaining the geometric parameter information of a reactor, the material composition information of the reactor, the section information of a material, the space set by a user, the expansion order information of an angle and the convergence limit information set by the user.
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