Self-adaptive relaxation method applied to neutronics-thermotechnical-fuel performance multi-physical coupling

Abstract

The invention discloses a self-adaptive relaxation method applied to neutronics-thermotechnical-fuel performance multi-physical coupling, and aims at solving the neutronics-thermotechnical-fuel performance analysis multi-physical coupling problem through a Picatan iteration method. According to the method, a proper relaxation factor is automatically calculated based on the results of the first two steps of Picatard iteration to be used for current iterative calculation, the relaxation factor can be dynamically adjusted along with iteration so as to get rid of dependence on artificial experience, and the iteration efficiency and calculation precision of multi-physical coupling calculation are effectively improved.

Description

Technical Field

[0001] The present invention relates to the technical field of nuclear reactor core design and fuel performance analysis, and particularly relates to an adaptive relaxation method based on iterative history and applied to multi-physics coupling of neutronics-thermohydraulics-fuel performance. Background Technique

[0002] Multi-physics coupling calculations of neutronics, thermohydraulics, and fuel performance are the core links in studying nuclear reactor design and safety. The interaction of these physical fields has an important impact on the operation efficiency and safety of the reactor core. However, traditional coupling calculation methods usually adopt the method of fixed relaxation factors, and in the face of complex multi-physics problems, there are defects such as low convergence efficiency and poor adaptability. An adaptive relaxation method for multi-physics coupling of neutronics-thermohydraulics-fuel performance based on iterative history. Summary of the Invention

[0003] In order to solve the problems existing in the above-mentioned prior art, the object of the present invention is to propose an adaptive relaxation method based on iterative history and applied to multi-physics coupling of neutronics-thermohydraulics-fuel performance. This method automatically calculates a suitable relaxation factor for the current iteration step based on the results of the previous two iterations. In this way, the relaxation factor can get rid of the dependence on manual experience and will be dynamically adjusted as the iteration progresses to adapt to the current iteration step.

[0004] In order to achieve the above object, the present invention adopts the following technical solutions:

[0005] An adaptive relaxation method applied to multi-physics coupling of neutronics-thermohydraulics-fuel performance includes the following steps:

[0006] Step 1: Numerically discretize the physical field expressions of neutronics, thermohydraulic, and fuel performance analysis, and consider the Picard iteration process. In the case of power relaxation, obtain the iterative equations of each physical field for the k-th iteration;

[0007] Step 2: Define the compression factor for the k-th iteration according to the iterative equations of each physical field obtained in Step 1, and obtain the relaxation factor for the k-th iteration;

[0008] Step 3: Calculate the power relaxation according to the power relaxation equation in the iterative equations of each physical field and the relaxation factor obtained in Step 2;

[0009] Step 4: Considering large-scale parallel computing, the computing region will be decomposed into several sub-regions. Therefore, the power vector of the computing region is written in the form of a block vector. Combining the defined compression factor, obtain the large-scale parallel compression factor;

[0010] Step 5: Based on the large-scale parallel compression factor obtained in Step 4, combined with the power relaxation equation, calculate the power field, and then perform thermohydraulic calculations to update the coolant temperature field, coolant density field, and the temperature field of the outer wall of the cladding, so as to solve and obtain the fuel temperature field through fuel performance analysis; then determine whether the multi-physics iteration converges. If it does not converge, continue the iteration.

[0011] In Step 1, the physical field expressions for neutronics, thermohydraulics, and fuel performance analysis are as follows:

[0012]

[0013] In the formula: p—the core power field / W·m -3 ; N solve ([T cool ρ cool T fuel )—the neutronics solution function; TH T (p)—the thermohydraulics solution function; FP solve ([p T solve ([p T clad_surface )—the fuel performance solution function. T T )—the fuel performance solution function. T cool —the coolant temperature / K; ρ cool —the coolant density / kg·m -3 ; T clad_surface —the temperature of the outer wall of the cladding / K; T fuel —the fuel temperature / K;

[0014] The iteration equations of each physical field for the k-th iteration are as follows:

[0015]

[0016] In the formula: —the power vector after the k-th iteration without considering relaxation / W·m -3 ; —the coolant temperature vector after the (k - 1)-th iteration / K; —the coolant density vector after the (k - 1)-th iteration / kg·m -3 ; —the fuel temperature vector after the (k - 1)-th iteration / K; p (k) —the power vector after the k-th iteration / W·m -3 ; ω—the relaxation factor; p (k-1) —the power vector after the (k - 1)-th iteration / W·m -3 ; —the coolant temperature vector after the k-th iteration / K; —the coolant density vector after the k-th iteration / kg·m-3 ; —— The temperature vector of the outer wall of the cladding after the k-th iteration / K; —— The fuel temperature vector after the k-th iteration / K.

[0017] In step 2, the compression factor of the k-th iteration is defined as follows:

[0018]

[0019] Where: —— The compression factor of the k-th iteration; θ (k) —— The vector and p (k-1) -p (k-2) The generalized included angle.

[0020] Then the relaxation factor of the k-th iteration is obtained as follows:

[0021]

[0022] In step 3, the power relaxation is calculated as:

[0023]

[0024] In step 4, the power vector in the calculation region is written in the form of a block vector as follows:

[0025] p = [p1 p2... p N T (6)

[0026] Where: p1, p2,..., p N —— The power vectors in the calculation sub-regions 1, 2,..., N.

[0027] In step 4, the large-scale parallel compression factor is obtained as follows:

[0028]

[0029] Where: —— The power vector in the sub-region n after the k-th iteration without considering relaxation; —— The power vector in the sub-region n after the k-th iteration.

[0030] Compared with the prior art, the present invention has the following outstanding advantages:

[0031] Based on the adaptive relaxation method of the iteration history, the relaxation factor can get rid of the dependence on artificial experience and will be dynamically adjusted as the iteration progresses to adapt to the current iteration step, effectively improving the iteration efficiency and calculation accuracy of multi-physical coupling calculations.​ Description of the Drawings

[0032] Figure 1 is the flowchart of the method of the present invention.

[0033] Figure 2 is the number of Picard iterations when convergence is reached at each time step in the single fuel element problem. Detailed Description of the Invention

[0034] The present invention will be further described in detail below with reference to the drawings and the detailed description of the invention.

[0035] As Figure 1 shown, an adaptive relaxation method based on the iteration history and applied to the multi-physics coupling of neutronics-thermal-hydraulics-fuel performance of the present invention comprises the following specific steps:

[0036] Step 1: Analyze the function to be solved according to neutronics, thermal-hydraulics and fuel performance, and obtain the iteration equations of each physical field after the k-th iteration.

[0037] The objective of neutronics solution is to solve the power field, the objective of thermal-hydraulics solution is to calculate the coolant temperature and density, and the outer surface temperature of the cladding, and the objective of fuel performance solution is the fuel temperature. The expressions of the physical fields to be solved are as follows:

[0038]

[0039] In the formula: p - core power field / W·m -3 ; N solve ([T cool ρ cool T fuel ) - neutronics solution function; TH T (p) - thermal-hydraulics solution function; FP solve ([p T solve ([p T clad_surface ) - fuel performance solution function. T T ) - fuel performance solution function. T cool - coolant temperature / K; ρ cool - coolant density / kg·m -3 ; T clad_surface - outer wall temperature of the cladding / K; T fuel - fuel temperature / K.

[0040] For Equation (1), perform numerical discretization on it and consider the Picard iteration process. In the case of power relaxation, given the results of each physical field after the (k - 1)-th iteration, the results of each physical field after the k-th iteration are updated by the following iteration format:

[0041]

[0042] In the formula: —— The power vector after the k-th iteration without considering relaxation / W·m -3 ; —— The coolant temperature vector after the (k - 1)-th iteration / K; —— The coolant density vector after the (k - 1)-th iteration / kg·m -3 ; —— The fuel temperature vector after the (k - 1)-th iteration / K; p (k) —— The power vector after the k-th iteration / W·m -3 ; ω—— The relaxation factor; p (k-1) —— The power vector after the (k - 1)-th iteration / W·m -3 ; —— The coolant temperature vector after the k-th iteration / K; —— The coolant density vector after the k-th iteration / kg·m -3 ; —— The temperature vector of the outer wall of the cladding after the k-th iteration / K; —— The fuel temperature vector after the k-th iteration / K.

[0043] Step 2: According to the iteration equations of each physical field obtained in Step 1, define the compression factor for the k-th iteration as:

[0044]

[0045] In the formula: —— The compression factor for the k-th iteration; θ (k) —— The vector and p (k-1) -p (k-2) The generalized included angle.

[0046] Then the relaxation factor for the k-th iteration is obtained:

[0047]

[0048] Step 3: According to the power relaxation equation in the iteration equations of each physical field and the relaxation factor obtained in Step 2, the power relaxation can be calculated as:

[0049]

[0050] Step 4: When considering large-scale parallel computing, the power vector in the computing region can be written in the form of a block vector. In large-scale parallel computing, the computing region will be decomposed into several sub-regions, and the power information in the sub-regions is stored in the memory space corresponding to the computing core responsible for that sub-region. Therefore, the power vector in the computing region can be written in the form of a block vector:

[0051] p = [p1 p2...p N T (6)

[0052] where: p1, p2,..., p N —— power vectors in sub - regions 1, 2,..., N for calculation.

[0053] Then, combining with the defined compression factor, the large - scale parallel compression factor is obtained as:

[0054]

[0055] where: —— power vector in sub - region n after the k - th iteration without considering relaxation; —— power vector in sub - region n after the k - th iteration.

[0056] Step 5: According to the large - scale parallel compression factor obtained in Step 4, combining with the power relaxation equation, the power field is obtained, and then the thermal - hydraulic calculation is carried out to update the coolant temperature field, coolant density field and the temperature field of the outer wall of the cladding, so as to solve and obtain the fuel temperature field through fuel performance analysis. Then judge whether the multi - physical iteration converges. If both the fuel temperature and the effective multiplication factor of the core converge, the iteration converges; otherwise, continue the iteration until it converges.

[0057] The advantages of the adaptive relaxation factor of the present invention are illustrated below through an example of a specific single - fuel element.

[0058] The design parameters of this single - fuel element refer to the VERA series benchmark problems released by the US CASL project. The pellet material of the fuel element is UO2, the cladding material is Zr alloy, the pellet radius is 0.4096 cm, the inner radius of the cladding is 0.418 cm, the outer radius of the cladding is 0.475 cm, the pitch of the fuel element is 1.26 cm, the height of the fuel element is 385.1 cm, the height of the active zone is 365.76 cm, the height of the gas cavity is 16.0 cm, the coolant inlet temperature is 565 K, the system pressure is 15.5 MPa, the power of the fuel element is 80 kW, the convergence limit of the calculated keff is 5 pcm, the power convergence limit is 0.01%, and the convergence limits of both the fuel temperature and the coolant temperature are 1 K.

[0059] ​The single fuel element was simulated using the non - relaxation method, the traditional relaxation method, and the adaptive relaxation method based on the iteration history of the present invention for neutronics - thermal - fuel performance multi - physical coupling. In the traditional relaxation method, the relaxation factor was taken as 0.1, 0.2, ……, 0.9 for a total of 9 cases. The convergence of each case is shown in Table 1. It can be seen that without relaxation, only the first time step can converge; for the traditional relaxation method, when the relaxation factor is taken as 0.6 - 1.0, only the first time step can converge; when the relaxation factor is taken as 0.1 - 0.5, the convergence stability is greatly improved, but in the best case of convergence stability (relaxation factor taken as 0.1), non - convergence still occurs at the 15th time step. For the adaptive relaxation method, all 17 time steps of the single fuel element can converge. The above results show that for this single fuel element, the adaptive relaxation method can convert the iterative format that cannot converge by the traditional relaxation method into a converging iterative format, with better convergence stability.

[0060] Table 1 Convergence of each case in the single fuel element problem

[0061]

[0062] Figure 2 It shows the number of Picard iterations required for convergence at each time step in the single fuel element. It can be seen that in the traditional relaxation method, when the relaxation factor is taken as 0.1 - 0.4, as the relaxation factor gradually decreases, the number of Picard iterations required for convergence gradually increases; the case where the relaxation factor is taken as 0.5 is not much different from that when it is taken as 0.4; when the relaxation factor is taken as 0.6 - 1.0, for the only converging first time step, as the relaxation factor gradually decreases, the number of Picard iterations required for convergence will decrease; generally speaking, the number of Picard iterations required for convergence is the least when the relaxation factor is taken as 0.4, that is, the convergence speed is the fastest. Comparing the adaptive relaxation method of the present invention with the traditional relaxation method with a relaxation factor of 0.1 - 0.5 in the case where both can converge, it can be found that the convergence speed of the adaptive relaxation method of the present invention is only better than the case where the relaxation factor is taken as 0.1, and is not as good as the cases where the relaxation factor is taken as 0.2 - 0.5, but for most time points, this disadvantage is not significant.

[0063] The above calculation results show that compared with the traditional relaxation method, the adaptive relaxation method based on the iteration history of the present invention shows better convergence stability without the need for artificially given relaxation factors.

[0064] The method of the present invention can mine the evolution law between multi - physical fields through the concept of data analysis and learning of the iteration history, realize the efficient optimization of the coupled system, significantly improve the calculation efficiency, and ensure the accuracy and stability of the results at the same time.

Claims

1. An adaptive relaxation method applied to the multi-physics coupling of neutronics-thermal-hydraulics-fuel performance, characterized in that: The steps are as follows: Step 1: Numerically discretize the physical field expressions of neutronics, thermal-hydraulics, and fuel performance analysis, and consider the Picard iteration process. Under the condition of power relaxation, obtain the iterative equations of each physical field for the k-th iteration; Step 2: According to the iterative equations of each physical field obtained in Step 1, define the compression factor for the k-th iteration, and obtain the relaxation factor for the k-th iteration; Step 3: Calculate the power relaxation according to the power relaxation equation in the iterative equations of each physical field and the relaxation factor obtained in Step 2; Step 4: Considering large-scale parallel computing, the computational domain will be decomposed into several sub-domains. Therefore, the power vector of the computational domain is written in the form of a block vector. Combining the defined compression factor, obtain the large-scale parallel compression factor; Step 5: According to the large-scale parallel compression factor obtained in Step 4, combine the power relaxation equation to obtain the power field, then perform thermal-hydraulic calculations to update the coolant temperature field, coolant density field, and cladding outer wall temperature field, and thus solve the fuel temperature field through fuel performance analysis; then judge whether the multi-physics iteration converges. If it does not converge, continue the iteration.

2. An adaptive relaxation method applied to the multi-physics coupling of neutronics-thermal-hydraulics-fuel performance according to claim 1, characterized in that: In Step 1, the physical field expressions of neutronics, thermal-hydraulics, and fuel performance analysis are as follows: Where: p——core power field / W·m -3 ; N solve ([T cool ρ cool T fuel ] T )——neutronics solution function; TH solve (p)——thermal hydraulic solution function; FP solve ([p T clad_surface ] T )——Fuel performance solution function. cool ——Coolant temperature / K; ρ cool ——Coolant density / kg·m -3 ; T clad_surface ——Clad outer wall temperature / K; T fuel ——fuel temperature / K; The iterative equations of each physical field for the k-th iteration obtained are as follows: Wherein: —— Power vector after the k-th iteration without considering relaxation / W·m -3 ; —— Coolant temperature vector after the (k - 1)-th iteration / K; —— Coolant density vector after the (k - 1)-th iteration / kg·m -3 ; —— Fuel temperature vector after the (k - 1)-th iteration / K; p (k) —— Power vector after the k-th iteration / W·m -3 ; ω—— Relaxation factor; p (k-1) —— Power vector after the (k - 1)-th iteration / W·m -3 ; —— Coolant temperature vector after the k-th iteration / K; —— Coolant density vector after the k-th iteration / kg·m -3 ; —— Cladding outer wall temperature vector after the k-th iteration / K; —— Fuel temperature vector after the k-th iteration / K.

3. An adaptive relaxation method applied to multi-physics coupling of neutronics-thermal-hydraulics-fuel performance according to claim 1, characterized in that: In Step 2, the compression factor for the k-th iteration is defined as follows: where: —— the compression factor for the k-th iteration; θ (k) —— the vector and p (k-1) -p (k-2) is the generalized included angle. Then the relaxation factor for the k-th iteration is obtained as follows:

4. An adaptive relaxation method applied to neutronics-thermal-fuel performance multi-physics coupling according to claim 1, characterized in that: In Step 3, the calculated power relaxation is:

5. An adaptive relaxation method applied to neutronics-thermal-hydraulics-fuel performance multi-physics coupling according to claim 1, characterized in that: In Step 4, the power vector in the computational domain is written in the form of a block vector as follows: p = [p1 p2... p N T (6)​ Where: p1, p2, ……, p N —— Power vectors in sub-regions 1, 2, ……, N for calculation.

6. An adaptive relaxation method applied to neutronics-thermal-hydraulic-fuel performance multi-physics coupling according to claim 1, characterized in that: In step 4, calculate the large-scale parallel compression factor as follows: where: —— the power vector in sub-region n after the k-th iteration without considering relaxation; —— the power vector in sub-region n after the k-th iteration.

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