Coarse mesh diffusion coefficient calculation method based on quasi-three-dimensional Fourier analysis
By introducing periodic boundary conditions and Fourier analysis into the quasi-three-dimensional transport model and optimizing the calculation of the coarse-grid diffusion coefficient, the problems of high computational cost and poor convergence of the quasi-three-dimensional transport model were solved, efficient and accurate neutron transport calculation was achieved, and the safety and economy of the pressurized water reactor were improved.
Patent Information
- Application Number
- CN202510769316.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-10
- Publication Date
- 2025-09-16
- Estimated Expiration
- Not applicable · inactive patent
AI Technical Summary
Existing quasi-3D transport models suffer from high computational cost, low efficiency, and poor convergence during the calculation process, especially when using the coarse-grid finite difference method, which limits the engineering application of high-fidelity numerical simulation methods.
A coarse-grid diffusion coefficient calculation method based on quasi-three-dimensional Fourier analysis is adopted. By establishing a quasi-three-dimensional transport model, introducing periodic boundary conditions and internal iteration process, and combining the Fourier analysis method, the coarse-grid diffusion coefficient is corrected. Through iterative acceleration processing, the correction factor with the minimum spectral radius is obtained, and the neutron target flux density and eigenvalue are updated.
It effectively reduces the errors introduced by spatial discretization, improves the accuracy and efficiency of neutron transport calculations, enhances the stability and convergence of calculations, adapts to the speed and accuracy requirements of high-fidelity calculations, reduces design redundancy, and improves the safety and economy of pressurized water reactors.
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Figure CN120654483A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of coarse mesh diffusion coefficient calculation, and in particular to a coarse mesh diffusion coefficient calculation method based on quasi-three-dimensional Fourier analysis. Background Art
[0002] With the continuous progress of the nuclear power industry, high-fidelity numerical simulation methods have become a key technological development direction, reducing the degree of approximation in the calculation process and improving design accuracy by directly solving the entire core. This optimizes the safety margin configuration and further enhances the economic efficiency while ensuring the safety of pressurized water reactors.
[0003] Compared to traditional quasi-3D transport models, high-fidelity numerical simulation methods avoid leakage terms generated by transverse integration and eliminate the computational inconsistency between 2D and 1D equations, making the calculation process more stable. However, using only quasi-3D transport models for direct full-core solutions is computationally expensive and inefficient. To improve the computational efficiency of quasi-3D transport models, the coarse-mesh finite-difference (CMFD) method is often used for acceleration. However, this approach introduces computational convergence issues, severely restricting the engineering application of high-fidelity numerical simulation methods. To improve the convergence of CMFD in transport model acceleration, a bias-flow-based coarse-mesh finite-difference method (pCMFD) has been proposed, fixing the coarse-mesh diffusion coefficient correction factor at 0.25. Based on a one-dimensional coarse-mesh finite-difference model and a one-dimensional transport model, an optimal diffusion coefficient coarse-mesh finite-difference method (odCMFD) is established, using a one-dimensional Fourier analysis model to calculate the coarse-mesh diffusion coefficient correction factor as a function of the coarse-mesh optical thickness. However, the above-mentioned pCMFD method or odCMFD method still diverges in the accelerated quasi-3D transport model, indicating that the fixed correction factor in the pCMFD method is not universally applicable to the coarse-mesh finite difference accelerated quasi-3D transport model. The odCMFD method is only proposed for the one-dimensional Fourier analysis model and cannot reflect the convergence characteristics of the coarse-mesh finite difference model accelerated quasi-3D transport model.
[0004] Therefore, it is necessary to optimize the quasi-three-dimensional coarse-grid diffusion coefficient calculation method to improve the stability of the calculation and the convergence ability of the model iteration. Summary of the Invention
[0005] The purpose of the present invention is to provide a coarse-grid diffusion coefficient calculation method based on quasi-three-dimensional Fourier analysis, which can improve the stability of the calculation and the convergence ability of the model iteration.
[0006] The present invention is achieved through the following technical solutions:
[0007] A method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis comprises the following steps:
[0008] Obtain geometric information, material information and boundary condition information of the three-dimensional core of the pressurized water reactor;
[0009] The three-dimensional core of the pressurized water reactor is divided into several layers, several coarse meshes are divided in each layer, and several fine meshes are radially divided in each coarse mesh;
[0010] Establish a quasi-3D transport model;
[0011] Obtaining the neutron flux density of the fine mesh and the neutron flux density at the interface of the coarse mesh according to the quasi-three-dimensional transport model, and establishing a coarse mesh finite difference model;
[0012] Periodic boundary conditions are introduced to establish the relationship between the neutron angular flux density at the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh;
[0013] Based on the quasi-three-dimensional transport model with periodic boundary conditions, a quasi-three-dimensional Fourier analysis model of the first inner iteration is established and multiple inner iterations are established according to mathematical induction;
[0014] Considering the coarse-grid finite difference model for acceleration, a coarse-grid accelerated quasi-three-dimensional Fourier analysis model is established by accelerating the iteration;
[0015] Obtain the correction factor of the coarse mesh diffusion coefficient that varies with the coarse mesh optical thickness and scattering rate when the spectral radius is minimum;
[0016] The neutron flux density and eigenvalue of the fine mesh in the coarse mesh are updated according to the correction factor of the coarse mesh diffusion coefficient.
[0017] Preferably, the method for establishing the quasi-three-dimensional transport model is:
[0018] Establish a quasi-3D transport model that scans from bottom to top:
[0019]
[0020] Quasi-3D transport model scanning from top to bottom:
[0021]
[0022] Among them, i is the number of the fine mesh along the x-axis, j is the number of the fine mesh along the y-axis, iz is the number of the fine mesh or coarse mesh along the z-axis, h is the size of the fine mesh along the x-axis and y-axis, Δ z The size of the fine or coarse mesh along the z-axis, m is the number of inner iterations of the quasi-three-dimensional transport equation, M is the maximum number of inner iterations of the quasi-three-dimensional transport equation, l is the number of outer iterations of the quasi-three-dimensional transport equation, p is the angular number of the neutron angular flux density, p is the angular number of the neutron angular flux density, μ pis the cosine of the angle between direction p and the positive direction of the x-axis, η p is the cosine of the angle between direction p and the positive direction of the y-axis, ξ p is the cosine of the angle between direction p and the positive direction of the z axis, ∑ t is the total neutron cross section of the three-dimensional core of the pressurized water reactor, t is the symbol of the total neutron cross section, ∑ s is the neutron scattering cross section of the three-dimensional core of the pressurized water reactor, s is the symbol of the neutron scattering cross section, ∑ f is the neutron fission cross section of the three-dimensional core of the pressurized water reactor, f is the neutron fission cross section symbol, v is the average number of neutrons per fission, is the eigenvalue of the lth outer iteration, and are the neutron angular flux densities along direction p on the west and east sides of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the iz-th coarse mesh layer of the 3D PWR core for the l+1 / 2-th outer iteration and the m-th inner iteration, respectively. and are the neutron angular flux densities along direction p on the south and north sides of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the l+1 / 2-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, respectively. and are the neutron angular flux densities along direction p at the bottom and top surfaces of the i-th fine mesh in the x-axis direction, the j-th fine mesh in the y-axis direction, and the z-th fine mesh in the z-axis direction within the coarse mesh, for the 1+1 / 2-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, is the neutron flux density of the ith fine mesh in the x-axis direction and the jth fine mesh in the izth coarse mesh layer in the z-axis direction in the l-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, is the neutron target flux density of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the iz-th coarse mesh layer in the z-axis direction in the l-th outer iteration and the 0-th inner iteration of the three-dimensional core of the pressurized water reactor.
[0023] Preferably, the neutron target flux density The method to obtain is:
[0024] The diamond difference formula is introduced to average the neutron angular flux density in the east, west, south and north:
[0025]
[0026] Preferably, the method for establishing the coarse-grid finite difference model is:
[0027]
[0028] Among them, n is the coarse mesh number, u is the coordinate axis direction, The eigenvalue of the l+1th outer iteration, is the coarse grid neutron flux density of the nth coarse grid provided by the transport calculation at the l+1 / 2th outer iteration, and are the coarse grid neutron flux densities of the nth, n-1th and n+1th coarse grids at the l+1th outer iteration, Δ n,u is the coarse mesh size of the nth coarse mesh along the u-axis direction, and are the diffusion coefficients of the nth coarse mesh interface along the positive and negative directions of the u axis, and are the neutron flux correction coefficients of the nth coarse mesh interface along the positive and negative directions of the u axis, ∑ s is the neutron scattering cross section of the three-dimensional core of the pressurized water reactor, s is the symbol of the neutron scattering cross section, ∑ f is the neutron fission cross section of the three-dimensional core of the pressurized water reactor, f is the neutron fission cross section symbol, and v is the average number of neutrons per fission.
[0029] Preferably, the diffusion coefficient and the neutron flux correction coefficient are obtained by:
[0030]
[0031] Among them, D n,u 、D n-1,u and D n+1,u is the coarse mesh diffusion coefficient of the nth, n-1th and n+1th coarse mesh along the u-axis, θ n,u ,θ n-1,u and θ n+1,u are the correction factors of the coarse mesh diffusion coefficients of the nth, n-1th and n+1th coarse meshes along the u-axis, and are the coarse-grid neutron flux densities provided by transport calculations at the interface of the l+1 / 2th outer iteration, the nth coarse-grid, and the positive and negative directions of the u-axis, respectively. and is the coarse grid neutron flux density of the nth, n-1th and n+1th coarse grids provided by transport calculations at the l+1 / 2th outer iteration, Δ n,u 、D n-1,u and D n+1,u are the coarse mesh sizes of the nth, n-1th and n+1th coarse meshes along the u-axis respectively.
[0032] Preferably, the method for establishing the relationship between the neutron angular flux density at the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh is:
[0033]
[0034] in, and are the neutron angular flux densities of the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh, e is a natural constant, t is the coefficient of Fourier transform, Δ n,z is the coarse mesh size of the nth coarse mesh along the z-axis.
[0035] Preferably, the method of establishing the quasi-three-dimensional Fourier analysis model of the first inner iteration and establishing multiple inner iterations according to mathematical induction is:
[0036] Establish the quasi-three-dimensional Fourier analysis model of the first inner iteration
[0037]
[0038] The fine mesh neutron flux density conversion matrix of the Mth inner iteration is:
[0039]
[0040] Among them, S (l,0) is the fine-grid neutron flux density matrix of the 0th inner iteration of the lth outer iteration, is the fine-grid neutron flux density matrix of the first inner iteration of the l+1 / 2th outer iteration, U is the fine-grid neutron flux density conversion matrix of the first inner iteration, is the fine grid neutron flux density conversion matrix of the Mth inner iteration, I is the unit matrix, X s represents the scattering matrix, X a is the absorption matrix, S (l+1,M) is the fine mesh neutron flux density matrix of the Mth inner iteration of the l+1th outer iteration.
[0041] Preferably, the method for establishing a coarse-grid accelerated quasi-three-dimensional Fourier analysis model by accelerating iteration is:
[0042]
[0043] Among them, R (l+1) is the conversion matrix of the coarse-grid neutron flux obtained by solving the coarse-grid finite difference model after the l+1th outer iteration, is the conversion matrix of the sub-beacon flux density in the fine grid to obtain the sub-beacon flux in the coarse grid after the l+1 / 2th outer iteration and the Mth inner iteration, ρ(λ x ,λ y ,λ z) is the spectral radius of the quasi-3D Fourier analysis model, that is, the maximum absolute value of the eigenvalue of the conversion matrix between the fine-grid neutron flux density of the l+1th outer iteration and the lth outer iteration, max is the function for finding the maximum value, abs is the function for finding the absolute value, and eig is the function for finding the eigenvalue.
[0044] Preferably, the method for obtaining the correction factor of the coarse mesh diffusion coefficient that varies with the coarse mesh optical thickness and scattering rate when the spectral radius is minimum is:
[0045]
[0046] Among them, θ n,z Represents the correction factor for the diffusion coefficient of the coarse mesh n without considering the scattering rate in the direction of the coordinate axis u, θ n,u (u=x,y) represents the correction factor of the diffusion coefficient of the coarse mesh n without considering the scattering rate when the direction of the coordinate axis u is the x-axis or the y-axis, index is the order index of the polynomial, a index is the index-order coefficient of polynomial a, b index is the index-order coefficient of polynomial b, c index is the coefficient of the index-th order of polynomial c, d index is the index-order coefficient of polynomial d, Δ n,u is the coarse mesh size of the nth coarse mesh along the u-axis.
[0047] Preferably, the method for updating the neutron flux density and characteristic value of the fine mesh in the coarse mesh according to the correction factor of the coarse mesh diffusion coefficient is:
[0048] Calculating the coarse-mesh diffusion coefficient of the coarse-mesh finite difference model according to the correction factor of the coarse-mesh diffusion coefficient;
[0049] Solve the modified coarse-grid finite difference model and obtain the coarse-grid neutron flux density and eigenvalues;
[0050] The neutron target flux density and the characteristic value of the fine mesh in the coarse mesh are updated.
[0051] The technical solution of the present invention has at least the following advantages and beneficial effects:
[0052] The present invention establishes a quasi-three-dimensional transport model and combines it with the Fourier analysis method to correct the coarse-grid diffusion coefficient, effectively reducing the error introduced by spatial discretization and improving the accuracy of neutron transport calculations.
[0053] The present invention introduces a Fourier analysis model of the internal iterative process and performs iterative acceleration processing, which effectively reduces the calculation convergence time, improves the overall calculation efficiency, and meets the dual requirements of high-fidelity calculation for speed and accuracy;
[0054] The present invention introduces periodic boundary conditions and a fine-grid neutron flux density conversion matrix obtained through internal iteration based on mathematical induction. It also considers a coarse-grid finite difference model for acceleration. This not only significantly improves the stability of the quasi-3D transport method calculation problem, but also enables the quasi-3D transport method calculation problem to have fewer iteration steps and more efficient convergence capability.
[0055] The present invention obtains a correction factor for the coarse mesh diffusion coefficient that varies with the coarse mesh optical thickness and scattering rate when the spectral radius is minimum, so that the spectral radius of the quasi-3D transport method calculation problem reaches the theoretical minimum, improving the accuracy and reliability of the calculation based on the quasi-3D model.
[0056] The present invention has a reasonable design and can provide more reasonable and accurate diffusion parameter support, further reduce design redundancy, improve the safety and economy of pressurized water reactors, and facilitate promotion and implementation. BRIEF DESCRIPTION OF THE DRAWINGS
[0057] Figure 1 A schematic flow chart of a method for calculating a coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis provided in Example 1 of the present invention;
[0058] Figure 2 A schematic diagram of a quasi-3D model of a pressurized water reactor core provided in Example 1 of the present invention;
[0059] Figure 3 This is a schematic diagram of a three-dimensional coarse-mesh model of a pressurized water reactor core provided in Example 1 of the present invention. DETAILED DESCRIPTION
[0060] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Generally, the components of the embodiments of the present invention described and shown in the drawings herein can be arranged and designed in various different configurations.
[0061] Example 1
[0062] This embodiment provides a method for calculating the coarse mesh diffusion coefficient based on quasi-three-dimensional Fourier analysis. Figure 1 , including the following steps:
[0063] Reading the geometric information, material information and boundary condition information of the three-dimensional core of the pressurized water reactor to be simulated;
[0064] Based on geometric and material information, the three-dimensional core of a pressurized water reactor is divided into several layers, each layer is divided into several coarse meshes, and each coarse mesh is radially divided into several fine meshes.
[0065] A quasi-three-dimensional transport model is established on each fine mesh to obtain the neutron flux density in the fine mesh and the neutron flux density at the interface of the coarse mesh.
[0066] Based on the neutron flux density of the fine mesh and the neutron flux density at the interface of the coarse mesh obtained by the quasi-three-dimensional transport model, a coarse mesh finite difference model is established on each coarse mesh.
[0067] Periodic boundary conditions are introduced to establish the relationship between the neutron angular flux density at the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh;
[0068] Based on the quasi-three-dimensional transport model with periodic boundary conditions, a quasi-three-dimensional Fourier analysis model of the first inner iteration is established and multiple inner iterations are established according to mathematical induction;
[0069] Considering the coarse-grid finite difference model for acceleration, a coarse-grid accelerated quasi-three-dimensional Fourier analysis model is established by accelerating the iteration;
[0070] Obtain the correction factor of the coarse mesh diffusion coefficient that varies with the coarse mesh optical thickness and scattering rate when the spectral radius is minimum;
[0071] The neutron flux density and eigenvalue of the fine mesh in the coarse mesh are updated according to the correction factor of the coarse mesh diffusion coefficient.
[0072] This example first collects and organizes the geometric information, material properties, and boundary conditions of the three-dimensional pressurized water reactor core, primarily to provide basic data support for subsequent modeling. The three-dimensional core is then divided axially into several layers. Each layer is further divided into multiple coarse-mesh cells, and each coarse-mesh cell is further divided radially into multiple fine-mesh cells. This division scheme, taking into account the structural characteristics of the three-dimensional core, ensures that the fine-mesh layers accurately capture local variations in neutron flux while facilitating the calculation and optimization of diffusion coefficients at the coarse-mesh level.
[0073] Then, a quasi-three-dimensional neutron transport model is constructed to describe the migration and reaction behavior of neutrons in the core. Next, based on the above-mentioned quasi-3D transport model, the neutron target flux density in each fine grid is numerically solved, and the neutron flux density on the interface between coarse grids is calculated and a finite difference model is constructed at the coarse grid level. Generally speaking, the neutron angular flux density on the coarse grid surface is projected onto the normal direction of the coarse grid surface to obtain the neutron flux density model. Using the diamond difference formula, the neutron target flux density of the fine grid is obtained by averaging the neutron angular flux densities on the east, west, south and north sides. Next, periodic boundary conditions are introduced to effectively reduce the interference of end boundary effects, so that the axial flux can be assumed to vary in a periodic form in the Fourier analysis, simplifying the mathematical processing. On this basis, mathematical induction is used to establish multiple inner iterations to achieve a more general quasi-3D Fourier analysis model for subsequent acceleration and optimization. The quasi-3D Fourier analysis model of coarse grid acceleration is subsequently introduced. Finally, by analyzing the trend of the spectral radius changing with the coarse grid optical thickness and scattering rate, the factor used to correct the coarse grid diffusion coefficient is extracted and the corresponding eigenvalue update is performed. The above scheme enables the spectral radius of the quasi-three-dimensional transport method calculation problem to reach the theoretical minimum value, which not only significantly improves the stability of the quasi-three-dimensional transport method calculation problem, but also enables the quasi-three-dimensional transport method calculation problem to have fewer iteration steps and more efficient convergence ability.
[0074] Figure 2 For a pressurized water reactor core model example, Figure 3 This is a schematic diagram of a 3D coarse mesh model of a pressurized water reactor core. During actual execution, the model reads the geometry, material, and boundary condition information of the small, rod-shaped pressurized water reactor core to be simulated. Geometric information such as the cell and component dimensions, as well as material information such as the nuclear density of the fuel and moderator, are obtained. The core is then divided into 15 layers. Each layer is divided into a coarse mesh with radial dimensions of 1.26 cm × 1.26 cm and axial dimensions of 2 cm, based on the cell size. Within each coarse mesh, nine fine meshes (3 × 3 × 1) are then created. As a preferred method, the SHARK program can be used for this division.
[0075] In this embodiment, the method for establishing the quasi-three-dimensional transport model is:
[0076] Establish a quasi-3D transport model that scans from bottom to top:
[0077]
[0078] Quasi-3D transport model scanning from top to bottom:
[0079]
[0080] Among them, i is the number of the fine mesh along the x-axis, j is the number of the fine mesh along the y-axis, iz is the number of the fine mesh or coarse mesh along the z-axis, h is the size of the fine mesh along the x-axis and y-axis, Δ x The size of the fine or coarse mesh along the z-axis, m is the number of inner iterations of the quasi-three-dimensional transport equation, M is the maximum number of inner iterations of the quasi-three-dimensional transport equation, l is the number of outer iterations of the quasi-three-dimensional transport equation, p is the angular number of the neutron angular flux density, p is the angular number of the neutron angular flux density, μ p is the cosine of the angle between direction p and the positive direction of the x-axis, η p is the cosine of the angle between direction p and the positive direction of the y-axis, ξ p is the cosine of the angle between direction p and the positive direction of the z axis, ∑ t is the total neutron cross section of the three-dimensional core of the pressurized water reactor, t is the symbol of the total neutron cross section, ∑ s is the neutron scattering cross section of the three-dimensional core of the pressurized water reactor, s is the symbol of the neutron scattering cross section, ∑ f is the neutron fission cross section of the three-dimensional core of the pressurized water reactor, f is the neutron fission cross section symbol, v is the average number of neutrons per fission, is the eigenvalue of the lth outer iteration, and are the neutron angular flux densities along direction p on the west and east sides of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the iz-th coarse mesh layer of the 3D PWR core for the l+1 / 2-th outer iteration and the m-th inner iteration, respectively. and are the neutron angular flux densities along direction p on the south and north sides of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the l+1 / 2-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, respectively. and are the neutron angular flux densities along direction p at the bottom and top surfaces of the i-th fine mesh in the x-axis direction, the j-th fine mesh in the y-axis direction, and the z-th fine mesh in the z-axis direction within the coarse mesh, for the 1+1 / 2-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, is the neutron flux density of the ith fine mesh in the x-axis direction and the jth fine mesh in the izth coarse mesh layer in the z-axis direction in the l-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, is the neutron target flux density of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the iz-th coarse mesh layer in the z-axis direction in the l-th outer iteration and the 0-th inner iteration of the three-dimensional core of the pressurized water reactor.
[0081] As an implementation example, a quasi-3D transport model can be established in the SHARK program for each fine mesh divided to satisfy the previously established quasi-3D transport model. The quasi-3D transport model is solved to obtain the neutron target flux density of the fine mesh and the neutron angular flux density on the coarse mesh surface. The neutron angular flux density on the coarse mesh surface is then projected onto the normal direction of the coarse mesh surface to obtain the neutron current density. Specifically, the diamond difference formula is used to average the neutron angular flux densities on the east, west, south, and north sides to obtain the neutron target flux density of the fine mesh.
[0082] In this approach, bottom-up and top-down quasi-3D transport models are established, combined with the propagation characteristics of neutron flux in different scanning directions. This allows for highly accurate 3D core modeling and calculations. The true solution for the flux distribution can be approximated layer by layer through internal and external iterations. This quasi-3D transport model provides data support for subsequent Fourier modal analysis, demonstrating both theoretical coherence and practical value.
[0083] As a preferred solution, the neutron target flux density The method to obtain is:
[0084] The diamond difference formula is introduced to average the neutron angular flux density in the east, west, south and north:
[0085]
[0086] On this basis, the method for establishing the coarse-grid finite difference model is:
[0087]
[0088] Where n is the coarse mesh number, which can be used to represent the coarse mesh number ix along the x-axis, the coarse mesh number iy along the y-axis, or the coarse mesh number iz along the z-axis. u is the coordinate axis direction. The eigenvalue of the l+1th outer iteration, is the coarse grid neutron flux density of the nth coarse grid provided by the transport calculation at the l+1 / 2th outer iteration, and are the coarse grid neutron flux densities of the nth, n-1th and n+1th coarse grids at the l+1th outer iteration, Δ n,u is the coarse mesh size of the nth coarse mesh along the u-axis direction, and are the diffusion coefficients of the nth coarse mesh interface along the positive and negative directions of the u axis, and are the neutron flux correction coefficients of the nth coarse mesh interface along the positive and negative directions of the u axis, ∑ sis the neutron scattering cross section of the three-dimensional core of the pressurized water reactor, s is the symbol of the neutron scattering cross section, ∑ f is the neutron fission cross section of the three-dimensional core of the pressurized water reactor, f is the neutron fission cross section symbol, and v is the average number of neutrons per fission.
[0089] On the other hand, the method for obtaining the diffusion coefficient and the neutron flux correction coefficient is:
[0090]
[0091] Among them, D n,u 、D n-1,u and D n+1,u is the coarse mesh diffusion coefficient of the nth, n-1th and n+1th coarse mesh along the u-axis, θ n,u ,θ n-1,u and θ n+1,u are the correction factors of the coarse mesh diffusion coefficients of the nth, n-1th and n+1th coarse meshes along the u-axis, and are the coarse-grid neutron flux densities provided by transport calculations at the interface of the l+1 / 2th outer iteration, the nth coarse-grid, and the positive and negative directions of the u-axis, respectively. and is the coarse grid neutron flux density of the nth, n-1th and n+1th coarse grids provided by transport calculations at the l+1 / 2th outer iteration, Δ n,u 、D n-1,u and D n+1,u are the coarse mesh sizes of the nth, n-1th and n+1th coarse meshes along the u-axis respectively.
[0092] In specific implementation, the neutron flux density can be obtained by using the neutron flux density of the fine mesh and the normal direction of the coarse mesh surface. A coarse mesh finite difference model can be established in SHARK, and the coarse mesh diffusion coefficient, the diffusion coefficient of the coarse mesh interface and the neutron flux correction coefficient of the coarse mesh interface can be obtained.
[0093] The calculation method of this embodiment can effectively suppress numerical oscillations, enhance the smoothness and accuracy of the calculation, help reflect local transport nonlinear effects, and significantly improve the adaptability of coarse-grid simulations to the complex neutron behavior of three-dimensional cores. Furthermore, the introduction of a correction factor for the coarse-grid diffusion coefficient and the coarse-grid size allows the solution of this embodiment to flexibly adapt to the non-uniform grids and regions of different materials in the core, improving the versatility and physical adaptability of the model. Furthermore, during the zeroth inner iteration of the quasi-three-dimensional transport equation, only the three unknowns of the neutron angular flux density on the west, south, and bottom surfaces can be considered within a single coarse grid.
[0094] Next, the method for establishing the relationship between the neutron angular flux density at the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh is:
[0095]
[0096] in, and are the neutron angular flux densities of the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh, e is a natural constant, t is the coefficient of Fourier transform, Δ n,z is the coarse mesh size of the nth coarse mesh along the z-axis.
[0097] The above scheme mainly uses periodic boundary conditions and Fourier transform to establish the numerical relationship between the bottom and top surfaces of the coarse mesh.
[0098] In addition, the method of establishing the quasi-three-dimensional Fourier analysis model of the first inner iteration and establishing multiple inner iterations according to mathematical induction is:
[0099] Establish the quasi-three-dimensional Fourier analysis model of the first inner iteration
[0100]
[0101] The fine mesh neutron flux density conversion matrix of the Mth inner iteration is:
[0102]
[0103]
[0104] Among them, S (l,0) is the fine-grid neutron flux density matrix of the 0th inner iteration of the lth outer iteration, is the fine-grid neutron flux density matrix of the first inner iteration of the l+1 / 2th outer iteration, U is the fine-grid neutron flux density conversion matrix of the first inner iteration, is the fine grid neutron flux density conversion matrix of the Mth inner iteration, I is the unit matrix, X s represents the scattering matrix, X a is the absorption matrix, S (l+1,M) is the fine mesh neutron flux density matrix of the Mth inner iteration of the l+1th outer iteration.
[0105] Furthermore, the method for establishing a coarse-grid accelerated quasi-three-dimensional Fourier analysis model by accelerating iteration is:
[0106]
[0107] Among them, R (l+1) is the conversion matrix of the coarse-grid neutron flux obtained by solving the coarse-grid finite difference model after the l+1th outer iteration, is the conversion matrix of the sub-beacon flux density in the fine grid to obtain the sub-beacon flux in the coarse grid after the l+1 / 2th outer iteration and the Mth inner iteration, ρ(λ x ,λ y ,λ z ) is the spectral radius of the quasi-3D Fourier analysis model, that is, the maximum absolute value of the eigenvalue of the conversion matrix between the fine-grid neutron flux density of the l+1th outer iteration and the lth outer iteration, max is the function for finding the maximum value, abs is the function for finding the absolute value, and eig is the function for finding the eigenvalue.
[0108] The above scheme uses mathematical induction to construct the transfer relationship for the Mth iteration, ensuring that the model maintains a consistent mathematical structure at any iteration step. This transforms the originally complex numerical iteration process into a highly interpretable algebraic problem with a clear structure and explicit physical meaning, making it easy to analyze and implement. The introduced acceleration mechanism significantly improves the convergence speed of the entire iteration process while maintaining fine-grid accuracy, reducing computational costs.
[0109] Next, the method for obtaining the correction factor of the coarse mesh diffusion coefficient that varies with the coarse mesh optical thickness and scattering rate when the spectral radius is minimum is:
[0110]
[0111] Among them, θ n,z Represents the correction factor for the diffusion coefficient of the coarse mesh n without considering the scattering rate in the direction of the coordinate axis u, θ n,u (u=x,y) represents the correction factor of the diffusion coefficient of the coarse mesh n without considering the scattering rate when the direction of the coordinate axis u is the x-axis or the y-axis, index is the order index of the polynomial, a index is the index-order coefficient of polynomial a, b index is the index-order coefficient of polynomial b, c index is the coefficient of the index-th order of polynomial c, d index is the index-order coefficient of polynomial d, Δ n,u is the coarse mesh size of the nth coarse mesh along the u-axis.
[0112] Finally, the method for updating the neutron flux density and characteristic value of the fine mesh in the coarse mesh according to the correction factor of the coarse mesh diffusion coefficient is:
[0113] Calculating the coarse-mesh diffusion coefficient of the coarse-mesh finite difference model according to the correction factor of the coarse-mesh diffusion coefficient;
[0114] Solve the modified coarse-grid finite difference model and obtain the coarse-grid neutron flux density and eigenvalues;
[0115] The neutron target flux density and the characteristic value of the fine mesh in the coarse mesh are updated.
[0116] This embodiment uses different fitting functions to obtain correction factors in different coordinate axis directions, fully accounting for the characteristics and physical differences of meshes in different directions, improving adaptability in heterogeneous structures. Furthermore, the fitting method can dynamically adjust the diffusion coefficient in different regions and scales, effectively overcoming the problem of reduced accuracy of the original coarse-mesh model in highly absorbing or highly scattering regions. By resolving the coarse-mesh finite-difference model based on the corrected coarse-mesh diffusion coefficient, the neutron flux density and eigenvalues in the fine mesh can be updated, achieving efficient linkage between the coarse and fine meshes and improving the dynamic response and adaptability of the entire multi-scale computing system.
[0117] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Those skilled in the art will readily appreciate that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.
Claims
1. A method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis, characterized in that: The following steps are involved: Obtain geometric information, material information and boundary condition information of the three-dimensional core of the pressurized water reactor; The three-dimensional core of the pressurized water reactor is divided into several layers, several coarse meshes are divided in each layer, and several fine meshes are radially divided in each coarse mesh; A quasi-three-dimensional transport model is established on each fine mesh to obtain the neutron flux density in the fine mesh and the neutron flux density at the interface of the coarse mesh. Based on the neutron standard flux density of the fine mesh and the neutron flux density at the interface of the coarse mesh obtained from the quasi-3D transport model, a coarse mesh finite difference model is established on each coarse mesh. Periodic boundary conditions are introduced to establish the relationship between the neutron angular flux density at the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh. Based on the quasi-three-dimensional transport model with periodic boundary conditions, a quasi-three-dimensional Fourier analysis model of the first inner iteration is established and multiple inner iterations are established according to mathematical induction; Considering the coarse-grid finite difference model for acceleration, a coarse-grid accelerated quasi-three-dimensional Fourier analysis model is established by accelerating the iteration; Obtain the correction factor of the coarse mesh diffusion coefficient that varies with the coarse mesh optical thickness and scattering rate when the spectral radius is minimum; The neutron flux density and eigenvalue of the fine mesh in the coarse mesh are updated according to the correction factor of the coarse mesh diffusion coefficient.
2. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 1, characterized in that: The method for establishing the quasi-three-dimensional transport model is: Establish a quasi-3D transport model that scans from bottom to top: Quasi-3D transport model scanning from top to bottom: Among them, i is the number of the fine mesh along the x-axis, j is the number of the fine mesh along the y-axis, iz is the number of the fine mesh or coarse mesh along the z-axis, h is the size of the fine mesh along the x-axis and y-axis, Δ z The size of the fine or coarse mesh along the z-axis, m is the number of inner iterations of the quasi-three-dimensional transport equation, M is the maximum number of inner iterations of the quasi-three-dimensional transport equation, l is the number of outer iterations of the quasi-three-dimensional transport equation, p is the angular number of the neutron angular flux density, p is the angular number of the neutron angular flux density, μ p is the cosine of the angle between direction p and the positive direction of the x-axis, η p is the cosine of the angle between direction p and the positive direction of the y-axis, ξ p is the cosine of the angle between direction p and the positive direction of the z axis, ∑ t is the total neutron cross section of the three-dimensional core of the pressurized water reactor, t is the symbol of the total neutron cross section, ∑ s is the neutron scattering cross section of the three-dimensional core of the pressurized water reactor, s is the symbol of the neutron scattering cross section, ∑ f is the neutron fission cross section of the three-dimensional core of the pressurized water reactor, f is the neutron fission cross section symbol, v is the average number of neutrons per fission, is the eigenvalue of the lth outer iteration, and are the neutron angular flux densities along direction p on the west and east sides of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the iz-th coarse mesh layer of the 3D PWR core for the l+1 / 2-th outer iteration and the m-th inner iteration, respectively. and are the neutron angular flux densities along direction p on the south and north sides of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the l+1 / 2-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, respectively. and are the neutron angular flux densities along direction p at the bottom and top surfaces of the i-th fine mesh in the x-axis direction, the j-th fine mesh in the y-axis direction, and the z-th fine mesh in the z-axis direction within the coarse mesh, for the 1+1 / 2-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, is the neutron flux density of the ith fine mesh in the x-axis direction and the jth fine mesh in the izth coarse mesh layer in the z-axis direction in the l-th outer iteration and the m-th inner iteration of the three-dimensional core of the pressurized water reactor, is the neutron target flux density of the i-th fine mesh in the x-axis direction and the j-th fine mesh in the y-axis direction in the iz-th coarse mesh layer in the z-axis direction in the l-th outer iteration and the 0-th inner iteration of the three-dimensional core of the pressurized water reactor.
3. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 2, characterized in that: The neutron flux density The method to obtain is: The diamond difference formula is introduced to average the neutron angular flux density in the east, west, south and north:
4. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 1, characterized in that: The method for establishing the coarse-grid finite difference model is: Among them, n is the coarse mesh number, u is the coordinate axis direction, The eigenvalue of the l+1th outer iteration, is the coarse grid neutron flux density of the nth coarse grid provided by the transport calculation at the l+1 / 2th outer iteration, and are the coarse grid neutron flux densities of the nth, n-1th and n+1th coarse grids at the l+1th outer iteration, Δ n,u is the coarse mesh size of the nth coarse mesh along the u-axis direction, and are the diffusion coefficients of the nth coarse mesh interface along the positive and negative directions of the u axis, and are the neutron flux correction coefficients of the nth coarse mesh interface along the positive and negative directions of the u axis, ∑ s is the neutron scattering cross section of the three-dimensional core of the pressurized water reactor, s is the symbol of the neutron scattering cross section, ∑ f is the neutron fission cross section of the three-dimensional core of the pressurized water reactor, f is the neutron fission cross section symbol, and v is the average number of neutrons per fission.
5. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 4, characterized in that: The method for obtaining the diffusion coefficient and the neutron flux correction coefficient is: Among them, D n,u 、D n-1,u and D n+1,u is the coarse mesh diffusion coefficient of the nth, n-1th and n+1th coarse mesh along the u-axis, θ n,u ,θ n-1,u and θ n+1,u are the correction factors of the coarse mesh diffusion coefficients of the nth, n-1th and n+1th coarse meshes along the u-axis, and are the coarse-grid neutron flux densities provided by transport calculations at the interface of the l+1 / 2th outer iteration, the nth coarse-grid, and the positive and negative directions of the u-axis, respectively. and is the coarse grid neutron flux density of the nth, n-1th and n+1th coarse grids provided by transport calculations at the l+1 / 2th outer iteration, Δ n,u 、D n-1,u and D n+1,u are the coarse mesh sizes of the nth, n-1th and n+1th coarse meshes along the u-axis respectively.
6. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 1, characterized in that: The method for establishing the relationship between the neutron angular flux density of the bottom surface and the top surface of the fine mesh in the z-axis direction within the coarse mesh is: in, and are the neutron angular flux densities of the bottom and top surfaces of the fine mesh in the z-axis direction within the coarse mesh, e is a natural constant, t is the coefficient of Fourier transform, Δ n,z is the coarse mesh size of the nth coarse mesh along the z-axis.
7. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 1, characterized in that: The method of establishing the quasi-three-dimensional Fourier analysis model of the first inner iteration and establishing multiple inner iterations according to mathematical induction is as follows: Establish the quasi-three-dimensional Fourier analysis model of the first inner iteration The fine mesh neutron flux density conversion matrix of the Mth inner iteration is: Among them, S (l,0) is the fine-grid neutron flux density matrix of the 0th inner iteration of the lth outer iteration, is the fine-grid neutron flux density matrix of the first inner iteration of the l+1 / 2th outer iteration, U is the fine-grid neutron flux density conversion matrix of the first inner iteration, is the fine grid neutron flux density conversion matrix of the Mth inner iteration, I is the unit matrix, X s represents the scattering matrix, X a is the absorption matrix, S (l+1,M) is the fine mesh neutron flux density matrix of the Mth inner iteration of the l+1th outer iteration.
8. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 7, characterized in that: The method for establishing a coarse-grid accelerated quasi-three-dimensional Fourier analysis model by accelerating iteration is: Among them, R (l+1) is the conversion matrix of the coarse-grid neutron flux obtained by solving the coarse-grid finite difference model after the l+1th outer iteration, is the conversion matrix of the sub-beacon flux density in the fine grid to obtain the sub-beacon flux in the coarse grid after the l+1 / 2th outer iteration and the Mth inner iteration, ρ(λ x ,λ y ,λ z 0 is the spectral radius of the quasi-3D Fourier analysis model, which represents the maximum absolute value of the eigenvalue of the conversion matrix of the fine-grid neutron flux density of the l+1th outer iteration and the lth outer iteration, max is the function for finding the maximum value, abs is the function for finding the absolute value, and eig is the function for finding the eigenvalue.
9. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 7, characterized in that: The method for obtaining the correction factor of the coarse mesh diffusion coefficient that changes with the coarse mesh optical thickness and scattering rate when the spectral radius is minimum is: Among them, θ n,z Represents the correction factor for the diffusion coefficient of the coarse mesh n without considering the scattering rate in the direction of the coordinate axis u, θ n,u (u=x,y) represents the correction factor of the diffusion coefficient of the coarse mesh n without considering the scattering rate when the direction of the coordinate axis u is the x-axis or the y-axis, index is the order index of the polynomial, a index is the index-order coefficient of polynomial a, b index is the index-order coefficient of polynomial b, c index is the coefficient of the index-th order of polynomial c, d index is the index-order coefficient of polynomial d, Δ n,u is the coarse mesh size of the nth coarse mesh along the u-axis.
10. The method for calculating the coarse-grid diffusion coefficient based on quasi-three-dimensional Fourier analysis according to claim 1, characterized in that: The method for updating the neutron flux density and characteristic value of the fine mesh in the coarse mesh according to the correction factor of the coarse mesh diffusion coefficient is: Calculating the coarse-mesh diffusion coefficient of the coarse-mesh finite difference model according to the correction factor of the coarse-mesh diffusion coefficient; Solve the modified coarse-grid finite difference model and obtain the coarse-grid neutron flux density and eigenvalues; The neutron target flux density and the characteristic value of the fine mesh in the coarse mesh are updated.