Self-adaptive source iteration acceleration method based on physical-data driving

Abstract

The invention provides a self-adaptive source iteration acceleration method based on physics-data driving, and belongs to the field of reactor physical computation.The method comprises the steps that an initial threshold value and a convergence standard are set firstly, a steady-state neutron diffusion equation is discretized through a coarse mesh finite difference method, and an equation set is established; whether an acceleration strategy is executed or not is determined by judging the relation between the maximum flux relative error and a threshold value. When the conditions are met, acceleration parameters are calculated, a flux error snapshot matrix is constructed, a residual error ratio is calculated by adopting incremental orthogonalization analysis, and whether flux prediction correction is carried out or not is judged according to the residual error ratio; and after correction, continuously calculating an equation set twice, and adaptively updating a threshold value. And carrying out loop iteration until convergence conditions are met. According to the method, physical constraint and data driving advantages are fused, parameters and strategies are adjusted in a self-adaptive mode, the number of iterations is greatly reduced on the premise that the calculation precision is guaranteed, and the solving efficiency of the steady-state multi-dimensional multi-group neutron diffusion equation is remarkably improved.

Description

Technical Field

[0001] This invention belongs to the field of reactor physics computation, specifically relating to a physics-data driven adaptive source iteration acceleration method. Background Technology

[0002] In numerical calculations of reactor physics, computational accuracy and solution efficiency are core factors determining the reliability and engineering applicability of numerical simulation results. Currently, with increasingly complex reactor core geometries and a continuously increasing number of energy groups, the scale of solving multidimensional, multi-group neutron diffusion equations has increased dramatically. Traditional source iteration methods, when solving large-scale reactor core neutron diffusion problems, suffer from convergence speed limitations imposed by the scattering ratio of the problem being solved. The closer the scattering ratio is to 1, the slower the convergence speed. To achieve the preset convergence criterion, a large number of iteration steps are required, significantly increasing the overall computational cost. Therefore, it is necessary to develop a physics-data-driven adaptive source iteration acceleration method that adaptively adjusts the solution under different iteration conditions through a physically constrained incremental orthogonalization analysis method. , Construct the flux error snapshot matrix and calculate the first... n +1 step and the n The flux error term of the first step, using the first step n Step flux and error term update n Adding one flux step allows the updated flux to quickly approach the convergence value. Ultimately, this significantly reduces the number of iterations and substantially improves computational efficiency while maintaining computational accuracy. Summary of the Invention

[0003] The technical problem this invention aims to solve is that traditional source iteration methods, when solving large-scale reactor core neutron diffusion problems, require multiple solutions to the discrete equations, regardless of whether the nodal method or the finite difference method is used, to continuously reduce the relative error of flux before and after iteration. In the later stages of iteration, slow convergence is prone to occur, requiring a large number of iterations to reach the preset convergence criterion, significantly increasing the overall computational cost. Therefore, this invention proposes a physics-data-driven adaptive source iteration acceleration method based on the change in flux error between adjacent steps. This method saves a significant amount of computation time and cost while maintaining computational accuracy.

[0004] To achieve the above-mentioned objectives, this invention provides a physical-data-driven adaptive source iteration acceleration method, comprising the following steps: S1: Set the maximum relative error threshold for flux. Noise threshold Initial value, maximum relative error of flux and Convergence criteria; The steady-state neutron diffusion equation is discretized using the coarse-grid finite difference method to establish a coarse-grid finite difference equation system; S2: Calculate the coarse mesh finite difference equation system; S3: Judgment Is it less than : like Greater than or equal to Then proceed to S2; like Less than Then proceed to S4; S4: Will Assign to ; S5: Save the first n Step flux, calculate the flux error between two adjacent steps, and construct a snapshot matrix; S6: The incremental orthogonalization analysis method is used to calculate the flux error modal decay in real time, i.e., the residual ratio. r and with Compare: like r Greater than or equal to Then proceed to S2; like r Less than Then proceed to S7; S7: Extract snapshot matrix information and calculate the... n +1 step and the n The flux error term of the first step, using the first step n Step flux and flux error term update n +1 step flux; S8: Enter S2 to calculate the coarse mesh finite difference equation system twice, and update. and compare and Update : like Greater than or equal to ,Will Assign to ; like Less than ,Will Assign to ; S8: Determine the result calculated after S2 is completed. and Has the convergence criterion been met? If the current or If the convergence criterion is not met, proceed to S2; If the current and If the convergence criterion is met, proceed to S9; S9: End the iterative calculation.

[0005] Preferably, in S5, the snapshot matrix includes the following steps: S501: When Less than At that time, the flux is saved sequentially, and the flux error between adjacent steps is calculated: (1) (2) In the formula: This is a column vector representing the flux errors between two adjacent steps in different regions and energy groups. For the preservation of the first m The flux of the group For the first m Flux of group +1 for m A vector consisting of the flux errors of the group; S502: Will of m Gram-Schmidt orthogonalization of the group of column vectors: (3) In the formula: The orthogonalized first m Column vectors; S503: Calculate the residual ratio after orthogonalization r : (4) S504: When r Greater than or equal to Then, continue calculating the coarse mesh finite difference equation system, repeating steps S501 to S503 until... r Less than When the time comes, stop saving throughput; S505: Constructing the snapshot matrix: (5) In the formula: the total number of saves from the start to the end of the save process. n Group flux, This is a snapshot matrix that preserves the flux errors of all adjacent pairs of steps.

[0006] Preferably, in S7, the calculation of the first... n +1 step and the n The flux error term of the first step, using the first step n Step flux and flux error term update n +1 step flux; including the following steps: S701: Will Divide into submatrices and as follows: (6) (7) S702: The relationship between the construction formula (6) and formula (7) is as follows: (8) In the formula: for and Relationship matrix; S703: Yes Perform polar decomposition: (9) In the formula: It is an orthogonal matrix. It is a positive semi-definite matrix; S704: The relationship between equation (8) and equation (9) is as follows: (10) S705: Equation (10) multiplied by left Right multiplication for: (11) In the formula: Let K equal ; S706: The iterative scheme for the steady-state multidimensional multigroup neutron diffusion equation is: (12) In the formula: L The coefficient matrix of the vanishing terms. S To generate the term coefficient matrix, Q For external terms vector; S707: Equation (12) can be written as: (13) In the formula: B For matrix L inverse and matrix S The product of b For matrix L The inverse and the external term vector Q The product; S708: When n When approaching infinity: (14) S709: The relationship between the construct (13) and the construct (14) is as follows: (15) In the formula: For the present The flux convergence value under the given conditions; S710: Introducing a flux error term : (16) In the formula: For the present The flux convergence value and the saved first n The difference in flux per step; S711: The relationship between the construct (15) and the construct (16) is as follows: (17) S712: Introduction : (18) In the formula: for exist The projection below; S713: The relationship between the construction formulas (11), (17), and (18) is as follows: (19) S714: The flux error term is obtained through equations (18) and (19): (20) S715: Update via equation (16) and equation (20) n +1 step flux: ;(twenty one)

[0007] The present invention has the following beneficial effects: This invention discloses a physics-data-driven adaptive source iteration acceleration method. The method first sets an initial threshold and convergence criterion, then discretizes the steady-state neutron diffusion equation using a coarse-mesh finite difference method to establish a coarse-mesh finite difference equation system. Finally, it calculates the coarse-mesh finite difference equation system and determines the maximum relative error of the flux. Is it less than the set maximum relative error threshold for flux? If greater than or equal to Then calculate the next step of the coarse mesh finite difference equation system; if it is less than Then Assign to Save the first n For each step of flux, the flux error between two adjacent steps is calculated, a snapshot matrix is ​​constructed, and the modal decay of the flux error, i.e., the residual ratio, is calculated in real time using the incremental orthogonalization analysis method. r,like r Greater than or equal to Then calculate the next step of the coarse mesh finite difference equation system; if r Less than Then extract the snapshot matrix information and calculate the first snapshot matrix. n +1 step and the n The flux error term of the first step, using the first step n Step flux and flux error term update n After adding one flux step, perform two more calculations using the coarse mesh finite difference equations, and then update... ,like Greater than or equal to Then Assign to ,like Less than Then Assign to This process is repeated until the convergence condition is met. This method can significantly reduce the number of iterations while maintaining computational accuracy throughout the entire adaptive iterative process, thus significantly improving the computational efficiency of solving the multidimensional multi-group steady-state neutron diffusion equation, and providing a reference for related research. Attached Figure Description

[0008] The present invention will be further described below with reference to the accompanying drawings and embodiments. The accompanying drawings, which constitute a part of this application, are used to provide a further understanding of the present invention. The illustrative embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention.

[0009] Figure 1 This is a flowchart of a physical-data-driven adaptive source iteration acceleration method according to the present invention.

[0010] Table 1 shows the iteration count and iteration time for the IAEA benchmark problem involving a 3D quadrilateral grid, with no subdivision of the x and y axes and bisection of the z axis. The iteration count and iteration time are calculated using the physics-data-driven adaptive source iteration acceleration method, but without employing this method. The relative error with the reference solution.

[0011] Table 2 shows the number of iterations and iteration time for the 3D hexagonal VVER440 benchmark problem without using the physics-data-driven adaptive source iteration acceleration method, and the number of iterations and iteration time using the physics-data-driven adaptive source iteration acceleration method. The relative error with the reference solution. Detailed Implementation

[0012] The present invention will be further described in detail below through specific embodiments. These embodiments are intended to enable those skilled in the art to gain a more comprehensive understanding of the present invention, but do not limit the invention in any way.

[0013] Example 1: This embodiment uses the 3D IAEA and VVER440 international benchmark problems to perform physical-data driven adaptive source iteration.

[0014] like Figure 1 As shown in the figure, this embodiment provides a physical-data-driven adaptive source iteration acceleration method, which is carried out according to the following steps: The first step is to set the maximum relative error threshold for flux. Noise threshold Initial value, maximum relative error of flux and Convergence criteria; The steady-state neutron diffusion equation is discretized using the coarse-grid finite difference method to establish a coarse-grid finite difference equation system; The second step is to calculate the coarse mesh finite difference equation system; The third step is to determine... Is it less than : like Greater than or equal to Then proceed to the second step; like Less than Then proceed to step four; Fourth step, use Update ; Fifth step, when Less than At that time, the flux is saved sequentially, and the flux error between adjacent steps is calculated: (1) (2) In the formula: This is a column vector representing the flux errors between two adjacent steps in different regions and energy groups. For the preservation of the first m The flux of the group For the first m Flux of group +1 for m A vector consisting of the flux errors of the group; Step 6, of m Gram-Schmidt orthogonalization of the group of column vectors: (3) In the formula: The orthogonalized first m Column vectors; Step 8: Calculate the ratio of the orthogonalized residuals. r : (4) Step 9, Comparison r and Size: like r Greater than or equal to Then proceed to the second step; like r Less than Then proceed to step eleven; Step 10, construct the snapshot matrix: (5) In the formula: the total number of saves from the start to the end of the save process. n Group flux, This is a snapshot matrix that preserves the flux errors of all adjacent pairs of steps.

[0015] Step 11, Divide into submatrices and as follows: (6) (7) Step 12: Construct the relationship between equation (6) and equation (7) as follows: (8) In the formula: for and Relationship matrix; Step thirteen, for Perform polar decomposition: (9) In the formula: It is an orthogonal matrix. It is a positive semi-definite matrix; Step fourteen, construct the relationship between equation (8) and equation (9) as follows: (10) Step 15, multiply equation (10) on the left. Right multiplication for: (11) In the formula: Let K equal ; Step sixteen, the iterative scheme for the steady-state multidimensional multigroup neutron diffusion equation is: (12) In the formula: L The coefficient matrix of the vanishing terms. S To generate the term coefficient matrix, Q For external terms vector; Step 17, Equation (12) can be written as: (13) In the formula: B For matrix L inverse and matrix S The product of b For matrix L The inverse and the external term vector Q The product; Step 18, when n When approaching infinity: (14) Step 19: Construct the relationship between equation (13) and equation (14) as follows: (15) In the formula: For the present The flux convergence value under the given conditions; Step 20: Introduce the flux error term : (16) In the formula: For the present The flux convergence value and the saved first n The difference in flux per step; Step 21: The relationship between equation (15) and equation (16) is constructed as follows: (17) Step 22, Introduction : (18) In the formula: for exist The projection below; Step 23: Construct the relationship between equations (11), (17), and (18) as follows: (19) Step 24: Obtain the flux error term using equations (18) and (19): (20) Step 25: Update using equations (16) and (20) n +1 step flux: ;(twenty one) Step 26: Proceed to the second step, calculate the coarse mesh finite difference equations twice, and update... and compare and Update : like Greater than or equal to ,Will Assign to ; like Less than ,Will Assign to ; Step 27: Determine the result calculated after step 2. and Has the convergence criterion been met? If the current or If the convergence criterion is not met, proceed to the second step; If the current and If the convergence criterion is met, proceed to step twenty-eight. Step 28: End the iterative calculation.

[0016] Based on the above process, the dataset used in this invention is derived from numerical simulation.

[0017] Table 1

[0018] Table 1 shows the iteration count, iteration time, and other parameters for the IAEA benchmark problem involving a 3D quadrilateral with no subdivision of the coarse mesh at x and y, and bisection of z. The method is based on a physics-data driven adaptive source iteration acceleration approach. The calculation results are compared with the number of iterations and iteration time without using the physics-data-driven adaptive source iteration acceleration method. Throughout the calculation process, the results calculated using the physics-data-driven adaptive source iteration acceleration method are... It matches the reference solution well, and the efficiency improvement can reach more than 50%.

[0019] Table 2

[0020] Table 2 shows the number of iterations, iteration time, and iteration duration for the 3D hexagonal VVER440 benchmark problem using a physics-data driven adaptive source iteration acceleration method. The calculation results are compared with the number of iterations and iteration time without using the physics-data-driven adaptive source iterative acceleration method. The calculation results show that the calculation using the physics-data-driven adaptive source iterative acceleration method... It matches the reference solution well, and the efficiency is improved by nearly 50%.

[0021] Although the preferred embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many specific modifications under the guidance of the present invention without departing from the spirit of the invention and the scope of protection of the claims, and these modifications all fall within the scope of protection of the present invention.

Claims

1. A physical-data-driven adaptive source iteration acceleration method, characterized in that, Includes the following steps: S1: Set the maximum relative error threshold for flux. Noise threshold Initial value, maximum relative error of flux and Convergence criteria; The steady-state neutron diffusion equation is discretized using the coarse-grid finite difference method to establish a coarse-grid finite difference equation system; S2: Calculate the coarse mesh finite difference equation system; S3: Judgment Is it less than : like Greater than or equal to Then proceed to S2; like Less than Then proceed to S4; S4: Will Assign to ; S5: Save the first n Step flux, calculate the flux error between two adjacent steps, and construct a snapshot matrix; S6: The incremental orthogonal analysis method is used to calculate the flux error modal decay in real time, i.e., the residual ratio. r and with Compare: like r Greater than or equal to Then proceed to S2; like r Less than Then proceed to S7; S7: Extract snapshot matrix information and calculate the... n +1 step and the n The flux error term of the first step, using the first step n Step flux and flux error term update n +1 step flux; S8: Enter S2 to calculate the coarse mesh finite difference equation system twice, and update. and compare and Update : like Greater than or equal to ,Will Assign to ; like Less than ,Will Assign to ; S8: Determine the result calculated after S2 is completed. and Has the convergence criterion been met? If the current or If the convergence criterion is not met, proceed to S2; If the current and If the convergence criterion is met, proceed to S9; S9: End the iterative calculation.

2. The physical-data-driven adaptive source iteration acceleration method according to claim 1, characterized in that, In S5, the snapshot matrix includes the following steps: S501: When Less than At that time, the flux is saved sequentially, and the flux error between adjacent steps is calculated: ；（1） ；（2） In the formula: This is a column vector representing the flux errors between two adjacent steps in different regions and energy groups. For the preservation of the first m The flux of the group For the first m Flux of group +1 for m A set of vectors consisting of flux errors; S502: Will of m Gram-Schmidt orthogonalization of the group of column vectors: ；（3） In the formula: The orthogonalized first m Column vectors; S503: Calculate the residual ratio after orthogonalization r : ；（4） S504: When r Greater than or equal to Then, continue calculating the coarse mesh finite difference equation system, repeating steps S501 to S503 until... r Less than When the time comes, stop saving throughput; S505: Constructing the snapshot matrix: ；（5） In the formula: the total number of saves from the start to the end of the save process. n Group flux, This is a snapshot matrix that preserves the flux errors of all adjacent pairs of steps.

3. The physical-data-driven adaptive source iteration acceleration method according to claim 1, characterized in that, In S7, the calculation of the first n +1 step and the n The flux error term of the first step, using the first step n Step flux and flux error term update n +1 step flux; including the following steps: S701: Will Divide into submatrices and as follows: ；（6） ；（7） S702: The relationship between the construction formula (6) and formula (7) is as follows: ；（8） In the formula: for and Relationship matrix; S703: Yes Perform polar decomposition: ； （9） In the formula: It is an orthogonal matrix. It is a positive semi-definite matrix; S704: The relationship between formula (8) and formula (9) is as follows: ；（10） S705: Equation (10) multiplied by left Right multiplication for: ；（11） In the formula: Let K equal ; S706: The iterative scheme for the steady-state multidimensional multigroup neutron diffusion equation is: ； （12） In the formula: L The coefficient matrix of the vanishing terms. S To generate the term coefficient matrix, Q For external terms vector; S707: Equation (12) can be written as: ； （13） In the formula: B For matrix L inverse and matrix S The product of b For matrix L The inverse and the external term vector Q The product; S708: When n When approaching infinity: ；（14） S709: The relationship between the construct (13) and the construct (14) is as follows: ； （15） In the formula: For the present The flux convergence value under the given conditions; S710: Introducing a flux error term : ；（16） In the formula: For the present The flux convergence value and the saved first n The difference in flux per step; S711: The relationship between the construct (15) and the construct (16) is as follows: ；（17） S712: Introduction : ；（18） In the formula: for exist The projection below; S713: The relationship between the construction formulas (11), (17), and (18) is as follows: ；（19） S714: The flux error term is obtained through equations (18) and (19): ；（20） S715: Update via equation (16) and equation (20) n +1 step flux: ；（21）。

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