High-fidelity intrusive reduced order method based on deterministic neutron transport solver

Abstract

The application discloses a high-compatibility intrusive reduced-order method based on a deterministic neutron transport solver, which combines the neutron flux field distribution calculated by the deterministic neutron transport solver with the corresponding neutron flow distribution, extracts characteristic modes through eigen-orthogonal decomposition and determines the truncation order, combines the control equation with the eigen-orthogonal basis to construct a reduced-order model, calculates the eigen-orthogonal basis coefficients under the corresponding cross-section parameters, and reconstructs the neutron flux field distribution; the application can construct a reduced-order model compatible with various neutron transport solvers without modifying the existing neutron transport solver; especially, the relationship between the established neutron flux and neutron flow is mapped to the reduced basis space, and a high-compatibility neutron transport intrusive reduced-order model is constructed based on the new control equation combined with the POD basis, which overcomes the defect that the existing full-order neutron transport equation solver needs to consume a large amount of memory, has higher calculation efficiency, and has higher universality.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of nuclear reactor core calculation method, in particular to a high compatibility intrusive reduced order method based on deterministic neutron transport solution. BACKGROUND

[0002] At present, China's nuclear energy industry is still maintaining a benign high-speed development, for example, China's self-developed third-generation reactor technology "Hualong No. 1" has been in commercial operation for many years, and the research and development of various fourth-generation reactor types is also being carried out in an orderly manner. Nuclear energy technology plays an irreplaceable role in China's energy transformation and upgrading process.

[0003] In order to improve the rationality and reliability of the nuclear reactor core design, efficient numerical simulation methods are needed to describe the neutron behavior of the nuclear reactor. For example, the neutron transport equation, as one of the key physical control equations, plays a crucial role in nuclear reactor design, safety analysis, etc. However, due to the complexity of the neutron transport equation, the process of solving the neutron transport equation often consumes a large amount of computing resources and computing time. Therefore, model reduction methods have been applied to nuclear reactor neutron calculation in recent years, aiming to map from high-dimensional space to low-dimensional space to reduce the number of variables to be solved, thereby reducing the consumption of computing resources and computing time.

[0004] The previously used neutron transport reduction method is based on the Sn neutron transport equation solver, and the specific equation can be represented as follows:

[0005]

[0006]

[0007] wherein ψ and φ represent the neutron angular flux and the neutron scalar flux, respectively, represents the spatial position, g represents the energy group index, Σ t,g represents the total cross section of the gth energy group, Σ s,g’→g represents the scattering cross section from the g'th energy group to the gth energy group, χ represents the fission spectrum, υ represents the number of neutrons released per fission, Σ f,g represents the fission cross section, G represents the total number of energy groups, L represents the total number of discrete angle directions, k eff represents the effective multiplication factor.

[0008] The neutron transport equation solver only takes the calculated neutron angular flux ψ as the sample α, that is:

[0009]

[0010] wherein N DoF represents the total number of degrees of freedom of the full order.

[0011] First, the corresponding POD base is extracted, and the neutron angular flux ψ is reconstructed in the following form:

[0012]

[0013] Then, the reconstructed form is substituted into the control equation, and it is assumed that The discrete form of the spatial derivative term is taken as an example of the first-order upwind scheme, which can be expressed in the following form:

[0014]

[0015] Where the subscripts i and j represent the discrete grid number indexes in the x and y directions, respectively.

[0016] Then, D and the neutron flux φ can also be expressed in the form of the POD base:

[0017]

[0018] On this basis, the reduced-order model is constructed:

[0019]

[0020] Where C represents a column vector composed of c m The matrix U and the matrix V can be expressed as:

[0021]

[0022] As can be seen from the above calculation process, the existing technology for the reduced-order model of the neutron transport problem uses the neutron angular flux ψ as the sample to construct the reduced-order model, which leads to a serious dependence on the full-order solver, and the reduced-order models constructed based on different full-order solvers also lack universality. At the same time, due to the large amount of data of the neutron angular flux ψ, a large amount of memory is consumed in the process of constructing the sample matrix of the neutron angular flux ψ and calculating the POD base.

[0023] Therefore, the existing technology still needs to be improved and developed. SUMMARY

[0024] To solve the above technical problems, the present application provides a high compatibility invasive reduced-order method based on deterministic neutron transport solution, which can overcome the defect of consuming a large amount of memory of the existing full-order neutron transport equation solver, has higher calculation efficiency, and has higher universality.

[0025] The technical scheme of the present application is as follows: a high compatibility invasive reduced-order method based on deterministic neutron transport solution, which is composed of the following steps:

[0026] Step S210, the neutron flux field distribution is calculated by using the deterministic neutron transport solver under different cross section parameters;

[0027] Step S220, the neutron flow distribution corresponding to different neutron flux fields is calculated;

[0028] Step S230, the two physical fields in steps S210 and S220 are combined, and the characteristic mode is extracted by intrinsic orthogonal decomposition;

[0029] Step S240, the truncation order N of the intrinsic orthogonal basis is determined P ;

[0030] Step S250, the reduced order model is constructed by combining the control equation and the intrinsic orthogonal basis;

[0031] Step S260, according to the reduced order model, different cross section parameters are input to calculate the intrinsic orthogonal basis coefficients c m ;

[0032] Step S270, the neutron flux field distribution is reconstructed.

[0033] The high compatibility invasive reduced order method based on deterministic neutron transport solution, wherein the deterministic neutron transport solver of step S210 adopts the steady-state neutron transport fission source equation under two-dimensional multi-group approximation, and the equation is expressed as follows:

[0034]

[0035] Wherein, ψ and φ represent the neutron angular flux and the neutron flux respectively, represents the spatial position, represents the angle direction, g represents the energy group index, Σ t,g represents the total cross section of the gth energy group, Σ s,g’→g represents the scattering cross section from the g'th energy group to the gth energy group, χ represents the fission spectrum, υ represents the number of neutrons released per fission, Σ f,g represents the fission cross section, G represents the total number of energy groups, k eff represents the effective multiplication factor.

[0036] The high compatibility invasive reduced order method based on deterministic neutron transport solution, wherein the equation in step S210 is integrated in the angle direction to obtain the following new control equation form:

[0037]

[0038] Wherein, J g represents the neutron flow, which can be defined as:

[0039]

[0040] When the neutron flux φ is calculated by the deterministic neutron transport solver, the neutron flow J is calculated by the following formula g :

[0041]

[0042] The high compatibility intrusive reduced order method based on deterministic neutron transport solution, wherein step S230 takes the neutron flux φ g and the neutron flow J g as samples α, and is defined as:

[0043]

[0044] Where N DoF represents the total number of degrees of freedom of the full order;

[0045] Then the samples under different parameter variation states are constructed into a sample matrix A:

[0046] Where N S represents the number of samples.

[0047] The high compatibility intrusive reduced order method based on deterministic neutron transport solution, wherein step S240 performs singular value decomposition (SVD) on the sample matrix, including the following three steps:

[0048] S241, calculate the square matrix S corresponding to the sample matrix: S = A T A;

[0049] S242, calculate the eigenvalues {λ1,…,λ m ,…,λ NS} and

[0050] eigenvectors {X1,…,X m ,…,X NS} of the square matrix S, so as to form the POD basis P:

[0051]

[0052] S243, calculate the truncation order N P of the POD basis, and the calculation formula is as follows:

[0053]

[0054] The high compatibility intrusive reduced order method based on deterministic neutron transport solution, wherein step S250, according to the definition of the above sample, the POD basis is divided into the following two parts:

[0055]

[0056] wherein P m,φ,g and P m,J,g represent the neutron flux and the neutron flow POD basis of the g-th energy group, respectively; thus, the neutron flux φ g and the neutron flow J g are expressed as follows:

[0057]

[0058] wherein c m represents the coefficient corresponding to the POD basis, that is, the variable to be solved in the reduced-order model;

[0059] Combining the above equation with the new control equation, we have:

[0060]

[0061] The high-compatibility intrusive reduced-order method based on the deterministic neutron transport solution, wherein step S260 defines the residual function E as follows:

[0062]

[0063] By the least square method, c m is obtained, which makes the residual function E reach the minimum value.

[0064] The high-compatibility intrusive reduced-order method based on the deterministic neutron transport solution, wherein step S270 represents the reduced-order model equation constructed in step S250 as follows:

[0065]

[0066] wherein C represents a column vector composed of c m , and the matrix U and the matrix V can be expressed as:

[0067]

[0068] The high-compatibility intrusive reduced-order method based on the deterministic neutron transport solution, wherein the deterministic neutron transport solver in step S210 adopts the finite difference method, the finite element method, the finite volume method, or the method of characteristics to construct the full-order model.

[0069] The high-compatibility intrusive reduced-order method based on the deterministic neutron transport solution, wherein step S250 adopts the dynamic mode decomposition method, the principal component analysis method, the autoencoder method, or the locally linear embedding method when constructing the reduced-order model.

[0070] The application provides a high-compatibility intrusive reduced-order method based on a deterministic neutron transport solution, which is based on model reduction and can construct a reduced-order model compatible with various neutron transport solvers without modifying an existing neutron transport solver; especially, a relationship between neutron flux g and neutron flow J g is mapped to a reduced basis space, and a high-compatibility neutron transport intrusive reduced-order model can be constructed based on a new control equation combined with a POD basis, which overcomes the defect that an existing full-order neutron transport equation solver needs to consume a large amount of memory, has higher calculation efficiency, and has higher universality. BRIEF DESCRIPTION OF DRAWINGS

[0071] The drawings described herein are for illustrative purposes only, and are not intended to limit the scope of the present application in any way; the shapes and scales of the components in the drawings are only illustrative, and are used to help understand the present application, and are not specific limitations on the shapes and scales of the components; those skilled in the art can select various possible shapes and scales according to specific circumstances to implement the present application under the guidance of the present application.

[0072] Figure 1 is a flow chart of the high-compatibility intrusive reduced-order method based on the deterministic neutron transport solution of the application;

[0073] Figure 2 is a schematic diagram of a two-dimensional TWIGL example used in an embodiment of the application. DETAILED DESCRIPTION

[0074] The specific embodiments and examples of the application will be described in detail below with reference to the drawings, and the specific embodiments described are only used to explain the application and are not used to limit the specific embodiments of the application.

[0075] The high-compatibility intrusive reduced-order method based on the deterministic neutron transport solution of the application aims to solve the compatibility problem of the neutron transport reduced-order model for various deterministic methods, and the entire technical solution is divided into three parts, including neutron flow calculation based on the neutron transport equation, eigenvalue orthogonal decomposition, and construction of a reduced-order model, the first two parts belong to the offline stage, and the last part belongs to the online stage.

[0076] As shown in Figure 1 , the high-compatibility intrusive reduced-order method based on the deterministic neutron transport solution of the application includes the following steps:

[0077] Step S210, under different cross-section parameters, the neutron flux field distribution is calculated by using a deterministic neutron transport solver;

[0078] Step S220, calculate the neutron flux distribution corresponding to different neutron flux fields;

[0079] Step S230, combine the two physical fields in step S210 and step S220, and extract the characteristic mode through the intrinsic orthogonal decomposition;

[0080] Step S240, determine the truncation order N of the intrinsic orthogonal basis P ;

[0081] Step S250, combine the control equation and the intrinsic orthogonal basis to construct the reduced order model;

[0082] Step S260, according to the reduced order model, input different cross-section parameters, and calculate the intrinsic orthogonal basis coefficients c m under the corresponding cross-section parameters.

[0083] Step S270, reconstruct the neutron flux field distribution.

[0084] Specifically, step S210 takes the steady-state neutron transport fission source equation under two-dimensional multi-group approximation as an example, which can be specifically expressed as:

[0085]

[0086] Where ψ and φ represent the neutron angular flux and the neutron flux, respectively, represents the spatial position, represents the angle direction, g represents the energy group index, Σ t,g represents the total cross-section of the gth energy group, Σ s,g’→g represents the scattering cross-section from the g'th energy group to the gth energy group, χ represents the fission spectrum, υ represents the number of neutrons released per fission, Σ f,g represents the fission cross-section, G represents the total number of energy groups, k eff represents the effective multiplication factor.

[0087] Specifically, step S220 integrates the above equation in the angle direction to obtain the following new control equation form:

[0088]

[0089] Where J g represents the neutron flux, which can be defined as:

[0090]

[0091] In actual application, after calculating the neutron flux φ through the deterministic neutron transport solver, the neutron flux J g is calculated through the following formula: , instead of calculating through the above definition:

[0092]

[0093] Thus, the scheme can obtain the physical quantity required by the new control equation without any modification to the existing deterministic neutron transport solver, and prepare for subsequent reduced-order model construction.

[0094] Specifically, step S230 takes the above neutron flux φ g and neutron flow J g as the sample α, which can be specifically defined as:

[0095]

[0096] Where N DoF represents the total number of degrees of freedom of the full order;

[0097] Then, the samples under different parameter variation states are constructed into a sample matrix A:

[0098] Where N S represents the number of samples.

[0099] Specifically, step S240 performs singular value decomposition (SVD) on the sample matrix, which specifically includes the following three steps:

[0100] S241, calculate the square matrix S corresponding to the sample matrix: S = A T A;

[0101] S242, calculate the eigenvalues {λ1,…,λ m ,…,λ NS} and

[0102] eigenvectors {X1,…,X m ,…,X NS} of the square matrix S, to form the POD basis P:

[0103]

[0104] S243, calculate the truncation order N P of the POD basis, which is specifically calculated as follows:

[0105]

[0106] Specifically, step S250 can divide the POD basis according to the above sample definition method into the following two parts:

[0107]

[0108] Where P m,φ,g and Pm,J,g respectively represent the neutron flux and the neutron current POD basis of the gth energy group; thus, the neutron flux φ g and the neutron current J g can be expressed in the following reconstruction form:

[0109]

[0110] where c m represents the coefficient corresponding to the POD basis, that is, the variable to be solved in the reduced-order model;

[0111] By combining the above equation with the new control equation, the following equation can be obtained:

[0112]

[0113] Specifically, the residual function E is defined in step S260 as follows:

[0114]

[0115] By the least square method, c m can be calculated when the residual function E reaches the minimum value, so as to reconstruct the neutron flux distribution.

[0116] Specifically, the reduced-order model equation constructed in step S250 is expressed in step S270 as follows:

[0117]

[0118] where C represents a column vector composed of c m , and the matrix U and the matrix V can be expressed as:

[0119]

[0120] A two-dimensional TWIGL example is verified and analyzed to prove the feasibility and computational efficiency of the scheme.

[0121] A two-dimensional TWIGL example is verified and analyzed to prove the feasibility and computational efficiency of the scheme. Figure 2 The two-dimensional TWIGL example is shown in the following table, and the entire research area is a square of 80 cm x 80 cm, the left boundary and the lower boundary are reflective boundaries, the right boundary and the upper boundary are vacuum boundaries, and the entire research area includes three materials, and the cross sections of each material are as shown in the following table:

[0122]

[0123] In the offline stage, the finite difference method (FDM) and the finite element method (FEM) are respectively used as full-order neutron transport solvers to solve the neutron flux φ of material 1 under different cross section parameters, and the corresponding neutron current J is calculated by the neutron current calculation method proposed in the scheme.g The specific cross-section parameter value range is shown in the following table:

[0124]

[0125] In the online phase, the calculation of the test example is performed, and the specific values of the various changes in the cross-section of material 1 are as follows:

[0126]

[0127] The following two tables respectively show the comparison of the reduced order results of the samples based on FDM and FEM as full-order neutron transport solvers:

[0128]

[0129] Meanwhile, the calculation time of different methods is shown in the following table:

[0130]

[0131] From the above table results, it can be seen that the reduced order model proposed by the present application has good calculation accuracy, and the relative error of the maximum neutron flux is only 0.27%, and also has excellent calculation efficiency, compared with the traditional full-order neutron transport solver, the acceleration ratio of the present application can reach more than 10000 times.

[0132] For different types of full-order models, i.e., neutron transport high-fidelity deterministic algorithms, including but not limited to finite difference method, finite element method, finite volume method or characteristic line method, the above method can be used to construct the corresponding reduced order model.

[0133] In addition, other model reduction methods can also be used to construct the above reduced order model, including linear model reduction methods: dynamic mode decomposition (Dynamic Mode Decomposition) method, principle component analysis (Principle Component Analysis) method; and nonlinear model reduction methods: auto-encoder (Auto-Encoder) method, linear locally embedding (Linear Locally Embedding) method.

[0134] It should be noted that the present application is based on the high compatibility invasive reduced order method of deterministic neutron transport solution, which is applied to the real-time calculation of the neutron field of the nuclear reactor, and the focus is on establishing the neutron transport calculation results, i.e., the neutron flux φ g and the neutron flow J gThe relationship between the two is mapped to the reduced basis space, and a high compatibility neutron transport intrusive reduced order model can be constructed based on the new control equation combined with the POD basis.

[0135] The contents not described in detail in the specification are all the prior art known to those skilled in the art.

[0136] It should be understood that the above description is only the preferred embodiment of the present application, and is not intended to limit the technical solutions of the present application. Those skilled in the art can make additions, substitutions, transformations or improvements to the above description within the spirit and principles of the present application, and all these additions, substitutions, transformations or improvements should belong to the protection scope of the appended claims of the present application.

Claims

1. A high-consistency intrusive reduced order method based on deterministic neutron transport solvers, characterized in that, Comprise the following steps: Step S210, under different cross-section parameters, the neutron flux field distribution is calculated by using the deterministic neutron transport solver; Step S220, the neutron flow distribution corresponding to different neutron flux fields is calculated; Step S230, combine the two physical fields in step S210 and step S220, and extract the characteristic mode through the intrinsic orthogonal decomposition; Step S240, determining the truncation order N of the eigen-orthogonal basis P ; Step S250, combine the control equation with the intrinsic orthogonal basis to construct the reduced order model; Step S260, according to the reduced order model, input different cross-section parameters, and calculate the intrinsic orthogonal basis coefficients c under the corresponding cross-section parameters m ; Step S270, reconstruct the neutron flux field distribution.

2. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 1, wherein, The deterministic neutron transport solver of step S210 adopts the steady-state neutron transport fission source equation under two-dimensional multi-group approximation, and its equation is expressed as follows: where ψ and φ represent the angular flux and the scalar flux of neutrons, respectively, denotes the spatial position, denotes the angular direction, g denotes the energy group index, t,g denotes the total cross section of the gth energy group, s,g’→g denotes the scattering cross section from the g'th energy group to the gth energy group, χ denotes the fission spectrum, υ denotes the number of neutrons released per fission, Σ f,g denotes the fission cross section, G represents the total number of energy groups, k eff denotes the effective multiplication factor.

3. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 2, wherein, Step S220 integrates the equation in step S210 in the angle direction to obtain the following new control equation form: where J g represents the neutron flux, which can be defined as: When the neutron flux φ is calculated by the determinist neutron transport solver, the neutron current J is calculated by the following equation g :

4. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 3, wherein, Step S230 calculates the neutron fluence φ g and the neutron flux J g When taken as a sample α, and defined as: where N DoF represents the total number of degrees of freedom of the full order; Then the samples under different parameter changes are constructed into a sample matrix A: where N S represents the number of samples.

5. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 4, wherein, Step S240 singular value decomposition (SVD) is performed on the sample matrix, including the following three steps: S241. Calculate the square matrix S corresponding to the sample matrix: S = A T A; S242、computing eigenvalues of a square matrix S and Feature vector POD base P is thus constituted: S243、Calculate the truncation order N of the POD basis P The calculation formula is as follows:

6. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 5, wherein, Step S250, according to the definition of the above sample, the POD basis is divided into the following two parts: where P m,φ,g and P m,J,g represent the neutron fluence and the neutron current POD basis of the gth energy group, respectively; thus, the reconstructed neutron fluence φ g and the neutron current J g are expressed as follows: where c m denotes the coefficient corresponding to the POD basis, i.e. the variable to be solved in the reduced model; Combine the above formula with the new control equation, then:

7. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 6, wherein, Step S260 defines the residual function E as follows: By least square method, the c which can make the residual function E reach the minimum value is calculated m .

8. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 7, wherein, Step S270 represents the reduced order model equation constructed in step S250 as: where C represents a c m The matrix U and the matrix V can be represented as:

9. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 1, wherein: The deterministic neutron transport solver in step S210 adopts finite difference method, finite element method, finite volume method or characteristic line method to construct the full order model.

10. The high-fidelity intrusive reduced order method based on deterministic neutron transport solvers of claim 1, wherein: Step S250 adopts dynamic mode decomposition method, principal component analysis method, autoencoder method or local linear embedding method when constructing the reduced order model.

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