MMPA burnup equation set accelerated solving method based on Gaussian Seidel
By directly constructing an iterative matrix using the Gauss-Seidel iterative method to solve the MMPA burnup equation, the problems of large memory overhead and long computation time in existing technologies are solved, achieving efficient burnup calculation and improving the accuracy and efficiency of reactor physics calculations.
Patent Information
- Application Number
- CN202511355027.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-09-22
- Publication Date
- 2026-01-09
AI Technical Summary
Existing methods for solving the MMPA fuel consumption equations rely on matrix inverse operations, resulting in high memory overhead and long computation time, which limits the improvement of fuel consumption calculation efficiency. In particular, the preprocessing process in large-scale nuclide systems causes a large amount of memory overhead and computational cost.
The Gauss-Seidel iterative method is used to directly construct the iteration matrix, and the MMPA fuel consumption equation is solved by Gauss-Seidel iteration. This avoids the preprocessing of the fuel consumption matrix and only requires storing the coefficient matrix, initial vector and solution vector, which simplifies the calculation process and reduces memory overhead.
While ensuring computational accuracy, it significantly improves fuel consumption calculation efficiency, reduces hardware costs and operational burden, and enhances the engineering applicability and promotional value of the calculation.
Smart Images
Figure CN121301716A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of reactor physics calculation, specifically involving an accelerated solution method for the MMPA burnup equations based on Gauss-Seidel. Background Technology
[0002] Reactor burnup calculation is crucial for the safe and stable operation of a reactor. It optimizes core design and controls reactivity, ensuring the reactor's safety and reliability throughout its entire lifespan. In detailed calculations throughout the reactor's lifespan, burnup calculation should simultaneously consider computational accuracy and efficiency to achieve a precise characterization of various physical phenomena in the core, thereby accurately predicting key safety parameters during reactor service. Mini-Max Polynomial Approximation (MMPA) is one of the main methods for solving the burnup equations. Traditional MMPA methods typically employ Sparse Gaussian Elimination (SGE) for solving the matrix inverse, significantly increasing computational costs and offering limited improvement in burnup calculation efficiency, thus restricting its application in practical engineering calculations. To further improve the efficiency of reactor burnup calculation, research is needed on novel inverse calculation methods based on traditional methods. Therefore, it is necessary to develop an accelerated solution method for the MMPA burnup equations based on Gauss-Seidel iteration, introducing the Gauss-Seidel iterative method to solve the MMPA burnup equations. While ensuring computational accuracy, computational efficiency is significantly improved. Summary of the Invention
[0003] The technical problem this invention aims to solve is the bottleneck in existing MMPA fuel consumption equation solving techniques, which suffer from high memory overhead and long computation time due to reliance on matrix inverse operations. Conventional methods for solving matrix inverse operations require repeated preprocessing of the fuel consumption matrix. For large-scale fuel consumption systems containing hundreds to thousands of nuclides, the preprocessing process not only requires storing multiple intermediate matrices after preprocessing but also performing high-frequency matrix multiplication and inverse operations. This process incurs significant memory overhead and offers very limited efficiency improvement for the MMPA fuel consumption equation. Therefore, this invention proposes an accelerated solution method for the MMPA fuel consumption equation based on the Gauss-Seidel iterative method. This method ensures that no preprocessing of the fuel consumption matrix is required throughout the entire fuel consumption calculation process. Furthermore, the Gauss-Seidel iterative method has advantages in storage, requiring only the storage of the coefficient matrix, initial vector, and solution vectors used for iteration, without the need to store a large number of intermediate calculation results. This saves significant computation time and cost while maintaining computational accuracy.
[0004] To achieve the above-mentioned technical features, the objective of this invention is as follows: a method for accelerating the solution of the MMPA fuel consumption equation system based on Gauss-Seidel, comprising the following steps: S1: Obtain the initial nuclide density and burnup matrix under the current burnup conditions; wherein, the current burnup conditions include the current neutron flux density distribution of the reactor, the initial fuel composition, and the coolant temperature; S2: Based on the fuel consumption matrix obtained in S1, construct the MMPA fuel consumption equation; S3: Perform a formal transformation on the MMPA fuel consumption equation constructed in S2, and obtain the Gaussian Seidel form of the iterative matrix by constructing the iterative matrix; S4: Set the iteration convergence criterion, and determine the initial value of the iteration based on the initial nuclide density obtained in S1. Perform Gaussian-Seidel iteration to solve the iteration matrix obtained in step S3. If the convergence criterion is met, execute S5; otherwise, execute S4. S5: Substitute the Gauss-Seidel iteration result that satisfies the convergence criterion in step S4 back into step S3 to obtain the final nuclide density distribution result.
[0005] Preferably, in S2, constructing the MMPA fuel consumption equation includes the following steps: S201: Construct the nuclide density variation of the current system based on the fuel consumption system that needs to be calculated: (1) In the formula: express The nuclide density vector at time t; Represents the fuel consumption matrix; S202: Given an initial nuclide density vector Then, the matrix exponential solution of equation (1) is as follows: (2) S203: Equation (3) is N The solution to the fuel consumption equation is calculated using the first-order MMPA method: (3) In the formula: It is related to the fuel consumption matrix Identity matrices of the same order; , c and All coefficients are real numbers; N It is the order of the MMPA method.
[0006] Preferably, in S3, transforming the MMPA fuel consumption equation form and extracting the iteration matrix to derive the iteration matrix suitable for Gauss-Seidel iteration includes the following steps: S301: Transform equation (3) in S203 to obtain the improved MMPA-CMIA equation (4): (4) S302: Iterative format based on Gaussian Seidel construct (5): (5) In the formula: For the iterative term, the boundary conditions are as follows (6); (6) S303: The first iteration based on equations (5) and (6) is as follows: equation (7): (7) S304: Applying the Gauss-Seidel method to the solution term When the component form of the iteration is as follows: (8) In the formula: Representation matrix The diagonal elements; Representation matrix The Middle Line number Column elements; Representing vectors The nth element; Indicates the number of Gauss-Seidel iterations; Representing vectors The Middle The element of the first The result of the next iteration; For vectors The length of represents the total number of nuclides involved in the fuel consumption calculation.
[0007] Preferably, in S4, Gaussian-Seidel iterative calculations are performed by setting an iterative convergence criterion: S401: Set the iterative convergence criterion By comparing whether the difference in the 2-norm between two adjacent Gauss-Seidel iteration results satisfies the iteration convergence criterion: (9) In the formula: Let be the order of MMPA; S402: If the iterative convergence criterion is met, proceed to S5; if the iterative convergence criterion is not met, repeat S4 until the convergence criterion is met.
[0008] Preferably, in S5, the final nuclide density distribution result is obtained by substituting back into equation (5) using the Gauss-Seidel iteration result that satisfies the convergence criterion.
[0009] The present invention has the following beneficial effects: 1. The present invention provides an accelerated solution method for the MMPA fuel consumption equation system based on Gauss-Seidel. The method first constructs the MMPA fuel consumption equation and transforms its form. By constructing an iterative matrix in Gauss-Seidel form, the MMPA fuel consumption equation is solved using the Gauss-Seidel iterative method.
[0010] 2. This method ensures that no preprocessing of the fuel consumption matrix is required throughout the entire fuel consumption calculation process; it significantly reduces memory consumption, lowers hardware costs and operational burden; simplifies the calculation process, enhances engineering applicability and promotional value; and improves calculation efficiency while maintaining calculation accuracy, ensuring that the relative error of each nuclide density is within an acceptable range. This method can provide a reference for effectively improving the efficiency of high-fidelity fuel consumption calculation. Attached Figure Description
[0011] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0012] Figure 1 This is a flowchart of an accelerated solution method for the MMPA fuel consumption equation system based on Gauss-Seidel's method according to the present invention.
[0013] Figure 2 The relative deviation of nuclide density at different MMPA expansion orders is based on a database of 71 nuclides.
[0014] Figure 3 The relative deviation of nuclide density at different MMPA expansion orders is based on a database of 221 nuclides.
[0015] Figure 4 The relative deviation of nuclide density at different MMPA expansion orders is based on a database of 1487 nuclides. Detailed Implementation
[0016] 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.
[0017] This embodiment calculates the nuclide density using a light water reactor benchmark. This benchmark uses a single-group flux of 1.5 × 10⁻⁶. 14 cm -2 s -1 Irradiation for 200 days. Three different fuel consumption databases of varying sizes (71, 221, and 1487) were used.
[0018] like Figure 1As shown in the figure, this embodiment provides an accelerated solution method for the MMPA fuel consumption equation system based on Gauss-Seidel, which is carried out according to the following steps: The first step is to obtain the initial nuclide density and burnup matrix under the current burnup conditions; The second step is to construct the MMPA fuel consumption equation: (1) In the formula: It is related to the fuel consumption matrix Identity matrices of the same order; , c and All coefficients are real numbers; N It is the order of the MMPA method. express The nuclide density vector at time t; Represents the fuel consumption matrix; The third step is to transform the current MMPA fuel consumption matrix to obtain the improved MMPA-CMIA formula: (2) The fourth step is to construct the MMPA-CMIA Gauss-Seidel iterative scheme based on the Gauss-Seidel iterative method: (3) In the formula: For the iterative term, the boundary conditions are as follows (4); (4) Fifth step, construct the first iterative formula based on equations (2) and (3): (5) Step 6: Apply the Gauss-Seidel method to the solution terms. ; (6) In the formula: Representation matrix The diagonal elements; Representation matrix The Middle Line number Column elements; Representing vectors The nth element; Indicates the number of Gauss-Seidel iterations; Representing vectors The Middle The element of the first The result of the next iteration; For vectors The length of represents the total number of nuclides involved in the fuel consumption calculation; Step 7: Define the convergence criterion for iterative iteration. : (7) In the formula: Let be the order of MMPA; Step 8: Perform Gaussian-Seidel iteration calculations. If the iteration convergence criterion is met, proceed to step 9. If the iteration convergence criterion is not met, repeat step 8 until the convergence criterion is met. In the ninth step, substitute the iterative results obtained in the eighth step back into equation (3) to obtain the final nuclide density distribution: Based on the above process, the dataset used in this invention is derived from numerical simulation.
[0019] Figure 2 The figure shows the relative deviation of nuclide densities at different MMPA expansion orders based on a database of 71 nuclides (AO represents the expansion order). As can be seen from the figure, when using the Gauss-Seidel method of this invention to solve the MMPA burnup equation, the calculated relative deviation of nuclide densities is strictly controlled within 0.000001%, which fully verifies that the method has excellent computational accuracy and stability in small-scale nuclide systems.
[0020] Figure 3 The results show the relative deviations of nuclide densities at different MMPA expansion orders (AO represents the expansion order) based on a database of 221 nuclides. The results demonstrate that, even with a database of 221 nuclides, the relative deviations of nuclide densities at different expansion orders remain within 0.000001% when using the Gauss-Seidel method of this invention to solve the MMPA burnup equation. This further proves that the method can reliably guarantee computational accuracy in medium-sized nuclide systems.
[0021] Figure 4 To investigate the relative deviations of nuclide densities at different MMPA expansion orders based on a database of 1487 nuclides (AO being the expansion order), the Gauss-Seidel method was used to solve the MMPA burnup equation. The relative deviations of nuclide densities at different expansion orders remained within an acceptable range, strongly demonstrating that this method possesses both high accuracy and strong adaptability in large-scale nuclide burnup calculations.
[0022] 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 method for accelerating the solution of the MMPA fuel consumption equations based on Gauss-Seidel equations, characterized in that, Includes the following steps: S1: Obtain the initial nuclide density and burnup matrix under the current burnup conditions; wherein, the current burnup conditions include the current neutron flux density distribution of the reactor, the initial fuel composition, and the coolant temperature; S2: Based on the fuel consumption matrix obtained in S1, construct the MMPA fuel consumption equation; S3: Perform a formal transformation on the MMPA fuel consumption equation constructed in S2, and obtain the Gaussian Seidel form of the iterative matrix by constructing the iterative matrix; S4: Set the iteration convergence criterion, and determine the initial value of the iteration based on the initial nuclide density obtained in S1. Perform Gaussian-Seidel iteration to solve the iteration matrix obtained in step S3. If the convergence criterion is met, execute S5; otherwise, execute S4. S5: Substitute the Gauss-Seidel iteration result that satisfies the convergence criterion in step S4 back into step S3 to obtain the final nuclide density distribution result.
2. The accelerated solution method for the MMPA fuel consumption equations based on Gauss-Seidel as described in claim 1, characterized in that, In S2, the construction of the MMPA fuel consumption equation includes the following steps: S201: Construct the nuclide density variation of the current system based on the fuel consumption system that needs to be calculated: ； (1) In the formula: express The nuclide density vector at time t; Represents the fuel consumption matrix; S202: Given an initial nuclide density vector Then, the matrix exponential solution of equation (1) is as follows: ；(2) S203: Equation (3) is N The solution to the fuel consumption equation is calculated using the first-order MMPA method: ； (3) In the formula: It is related to the fuel consumption matrix Identity matrices of the same order; , c and All coefficients are real numbers; N It is the order of the MMPA method.
3. The accelerated solution method for the MMPA fuel consumption equations based on Gauss-Seidel as described in claim 2, characterized in that, In S3, the transformation of the MMPA fuel consumption equation and the extraction of the iteration matrix to derive the iteration matrix suitable for Gauss-Seidel iteration includes the following steps: S301: Transform equation (3) in S203 to obtain the improved MMPA-CMIA equation (4): ； (4) S302: Iterative format based on Gaussian Seidel construct (5): ； (5) In the formula: For the iterative term, the boundary conditions are as follows (6); ；(6) S303: The first iteration based on equations (5) and (6) is as follows: equation (7): ；(7) S304: Applying the Gauss-Seidel method to the solution term When the component form of the iteration is as follows: ； (8) In the formula: Representation matrix The diagonal elements; Representation matrix The Middle Line number Column elements; Representing vectors The nth element; Indicates the number of Gauss-Seidel iterations; Representing vectors The Middle The element of the first The result of the next iteration; For vectors The length of represents the total number of nuclides involved in the fuel consumption calculation.
4. The accelerated solution method for the MMPA fuel consumption equations based on Gauss-Seidel as described in claim 1, characterized in that, In S4, Gaussian-Seidel iterative calculations are performed by setting an iterative convergence criterion: S401: Set the iterative convergence criterion By comparing whether the difference in the 2-norm between two adjacent Gauss-Seidel iteration results satisfies the iteration convergence criterion: ； (9) In the formula: Let be the order of MMPA; S402: If the iterative convergence criterion is met, proceed to S5; if the iterative convergence criterion is not met, repeat S4 until the convergence criterion is met.
5. The accelerated solution method for the MMPA fuel consumption equations based on Gauss-Seidel as described in claim 1, characterized in that, In S5, the final nuclide density distribution result is obtained by substituting back into equation (5) through the Gauss-Seidel iteration result that satisfies the convergence criterion.