High-compatibility intrusive order reduction method for neutron transport solution based on deterministic theory

Abstract

The invention discloses a high-compatibility intrusive order reduction method based on deterministic theory neutron transport solution, which comprises the following steps of: combining neutron standard flux field distribution calculated by using a deterministic theory neutron transport solver with corresponding neutron current distribution, extracting a characteristic mode through intrinsic orthogonal decomposition, and determining a truncation order of the characteristic mode; constructing a reduced-order model by combining the control equation and the intrinsic orthogonal basis, and calculating to obtain an intrinsic orthogonal basis coefficient under the corresponding section parameter so as to reconstruct neutron standard flux field distribution; according to the method, under the condition that an existing neutron transport solver is not modified, the reduced-order model compatible with various neutron transport solvers can be constructed; particularly, the established relationship between the neutron standard flux and the neutron current is mapped to a reduced base space, and a neutron transport intrusive reduced-order model with high compatibility is constructed based on a new control equation in combination with a POD base, so that the defect that an existing full-order neutron transport equation solver needs to consume a large amount of memory is overcome; and the method has higher calculation efficiency and higher universality.

Description

Technical Field

[0001] The invention relates to the field of calculation methods for nuclear reactor cores, and in particular to a highly compatible intrusive order reduction method based on deterministic neutron transport solutions. Background Art

[0003] At present, China's nuclear energy industry is still maintaining healthy and rapid development. For example, China's independently developed third-generation reactor technology "Hualong One" has been put into commercial operation for many years, and the research and development of various types of fourth-generation reactor types are also proceeding in an orderly manner. The development of nuclear energy technology plays an irreplaceable role in China's energy transformation and upgrading process.

[0004] In order to improve the rationality and reliability of nuclear reactor core design, efficient numerical simulation methods are needed to describe the neutron behavior of nuclear reactors. For example, the neutron transport equation is one of the key physical control equations, and its numerical solution plays a vital 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 lot of computing resources and computing time. Therefore, model reduction methods have been applied to nuclear reactor neutron calculations in recent years, aiming to reduce the variables to be solved by mapping high-dimensional space to low-dimensional space, thereby reducing the consumption of computing resources and computing time.

[0005] The neutron transport reduction method used in the past has a full-order model based on the Sn-based neutron transport equation solver. The specific equation can be expressed as follows:

[0006]

[0007]

[0008] Where ψ and φ represent the neutron angular flux and neutron scalar flux, respectively. represents the spatial position, g represents the energy group index, Σ t,g represents the total cross section of the g-th 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 energy spectrum, υ represents the number of neutrons released in each 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 Indicates effective proliferation factor.

[0009] The neutron transport equation solver only uses the calculated neutron angular flux ψ as the sample α, that is:

[0010]

[0011] Among them, N DoF represents the total number of degrees of freedom of the full order.

[0012] First, extract the corresponding POD basis and reconstruct the neutron angular flux ψ into the following form:

[0013]

[0014] Substitute this reconstructed form into the control equation and assume is the discrete form of the spatial partial derivative, taking the first-order upwind format as an example, It can be expressed as follows:

[0015]

[0016] The subscripts i and j represent the discrete grid number indexes in the x-direction and y-direction, respectively.

[0017] Then, D and the neutron scalar flux φ can also be expressed in POD basis as:

[0018]

[0019] On this basis, the reduced-order model is constructed:

[0020]

[0021] Among them, C represents c m The column vectors composed of the matrix U and matrix V can be expressed as:

[0022]

[0023] It is not difficult to see from the above calculation process that the reduced-order models of the existing technologies for neutron transport problems all use the neutron angular flux ψ as a sample to construct the reduced-order model, which leads to a serious reliance of the reduced-order model 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 huge amount of data of the neutron angular flux ψ, a large amount of memory is required in the process of constructing the sample matrix of the neutron angular flux ψ and calculating the POD basis.

[0024] Therefore, the prior art still needs to be improved and developed. Summary of the invention

[0025] In order to solve the above technical problems, the present invention provides a highly compatible intrusive order reduction method based on deterministic neutron transport solution, which can overcome the defect of existing full-order neutron transport equation solvers that consume a large amount of memory, and has higher computational efficiency and higher versatility.

[0026] The technical solution of the present invention is as follows: a highly compatible intrusive order reduction method based on deterministic neutron transport solution, which consists of the following steps:

[0027] Step S210, using a deterministic neutron transport solver to calculate the neutron flux field distribution under different cross-sectional parameters;

[0028] Step S220, calculating the neutron flux distribution corresponding to different neutron target flux fields;

[0029] Step S230, combining the two physical fields in step S210 and step S220, and extracting characteristic modes through intrinsic orthogonal decomposition;

[0030] Step S240: Determine the truncation order N of the eigenorthogonal basis P ;

[0031] Step S250, combining the control equation with the intrinsic orthogonal basis to construct a reduced-order model;

[0032] Step S260: According to the reduced-order model, different cross-sectional parameters are input to calculate the intrinsic orthogonal basis coefficients c under the corresponding cross-sectional parameters. m ;

[0033] Step S270: reconstruct the neutron target flux field distribution.

[0034] The highly compatible intrusive order reduction 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 the two-dimensional multi-group approximation, and its equation is expressed as follows:

[0035]

[0036] Where ψ and φ represent the neutron angular flux and neutron scalar flux, respectively. Indicates the spatial position, represents the angle direction, g represents the energy group index, Σ t,g represents the total cross section of the g-th 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 energy spectrum, υ represents the number of neutrons released in each fission, Σ f,g represents the fission cross section, G represents the total number of energy groups, k eff Indicates effective proliferation factor.

[0037] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein step S220 integrates the equation in step S210 in the angular direction to obtain the following new governing equation form:

[0038]

[0039] Among them, J g represents the neutron flux and can be defined as:

[0040]

[0041] After the neutron flux φ is calculated by the deterministic neutron transport solver, the neutron flux J is calculated by the following formula: g :

[0042]

[0043] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein step S230 converts the neutron flux φ g and neutron flux J g As sample α, it is defined as:

[0044]

[0045] Among them, N DoF represents the total number of degrees of freedom of the full order;

[0046] Then construct the samples under different parameter change states into sample matrix A:

[0047] Among them, N S Represents the sample size.

[0048] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein step S240 performs singular value decomposition (SVD) on the sample matrix, includes the following three steps:

[0049] S241. Calculate the square matrix S corresponding to the sample matrix: S = A T A;

[0050] S242, calculate the eigenvalues ​​of the square matrix S {λ1,…,λ m ,…,λ NS}and

[0051] The eigenvector {X1,…,X m ,…,X NS}, thus forming a POD basis P:

[0052]

[0053] S243. Calculate the truncation order N of the POD basis P , the calculation formula is as follows:

[0054]

[0055] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein step S250 is divided into the following two parts according to the definition of the above sample:

[0056]

[0057] Among them, P m,φ,g and P m,J,g denote the neutron flux and neutron flux POD basis of the g-th energy group respectively; thus, the neutron flux φ is reconstructed g and neutron flux J g It is expressed as:

[0058]

[0059] Among them, c m represents the coefficient corresponding to the POD basis, which is the variable to be determined in the reduced-order model;

[0060] Combining the above formula with the new control equation, we get:

[0061]

[0062] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein step S260 defines the residual function E as follows:

[0063]

[0064] By using the least squares method, we can calculate the c that can make the residual function E reach the minimum value. m .

[0065] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein step S270 expresses the reduced order model equation formed in step S250 as:

[0066]

[0067] Among them, C represents c m The column vectors composed of the matrix U and matrix V can be expressed as:

[0068]

[0069] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein: the deterministic neutron transport solver in step S210 uses a finite difference method, a finite element method, a finite volume method or a characteristic line method to construct a full-order model.

[0070] The highly compatible intrusive order reduction method based on deterministic neutron transport solution, wherein: step S250 adopts a dynamic mode decomposition method, a principal component analysis method, an autoencoder method or a local linear embedding method when constructing a reduced-order model.

[0071] The invention provides a highly compatible intrusive order reduction method based on deterministic neutron transport solution. Based on model order reduction, it is possible to construct a reduced-order model compatible with a variety of neutron transport solvers without modifying the existing neutron transport solver. In particular, the established neutron scalar flux φ g and neutron flux J g The relationship between them is mapped to the reduced basis space. At the same time, based on the new control equation combined with the POD basis, a highly compatible neutron transport intrusive reduced-order model can be constructed, which overcomes the defect that the existing full-order neutron transport equation solver consumes a lot of memory, has higher computational efficiency and is more versatile. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] The drawings described herein are only for explanation purposes and are not intended to limit the scope of the present invention in any way; the shapes and proportional dimensions of the components in the drawings are only schematic and are used to help understand the present invention, but are not intended to specifically limit the shapes and proportional dimensions of the components of the present invention; those skilled in the art can select various possible shapes and proportional dimensions to implement the present invention according to the specific circumstances under the guidance of the present invention.

[0073] Figure 1 It is a flowchart of the high-compatibility intrusive order reduction method based on deterministic neutron transport solution of the present invention;

[0074] Figure 2 It is a schematic diagram of a two-dimensional TWIGL example used in an embodiment of the present invention. DETAILED DESCRIPTION

[0075] The specific implementation modes and embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings. The described specific embodiments are only used to explain the present invention and are not used to limit the specific implementation modes of the present invention.

[0076] The present invention is based on a highly compatible intrusive order reduction method for solving deterministic neutron transport, aiming to solve the compatibility problem of the neutron transport reduced-order model with various deterministic methods. The whole technical solution is divided into three parts, including neutron flux calculation based on the neutron transport equation, intrinsic orthogonal decomposition and construction of the reduced-order model. The first two parts belong to the offline stage, and the last part belongs to the online stage.

[0077] like Figure 1 As shown, the present invention is based on a highly compatible intrusive order reduction method for solving deterministic neutron transport, which generally includes the following steps:

[0078] Step S210, using a deterministic neutron transport solver to calculate the neutron flux field distribution under different cross-sectional parameters;

[0079] Step S220, calculating the neutron flux distribution corresponding to different neutron target flux fields;

[0080] Step S230, combining the two physical fields in step S210 and step S220, and extracting characteristic modes through intrinsic orthogonal decomposition;

[0081] Step S240: Determine the truncation order N of the eigenorthogonal basis P ;

[0082] Step S250, combining the control equation with the intrinsic orthogonal basis to construct a reduced-order model;

[0083] Step S260: According to the reduced-order model, different cross-sectional parameters are input to calculate the intrinsic orthogonal basis coefficients c under the corresponding cross-sectional parameters. m ;

[0084] Step S270: reconstruct the neutron target flux field distribution.

[0085] Specifically, step S210 takes the steady-state neutron transport fission source equation under the two-dimensional multi-group approximation adopted by the deterministic neutron transport solver as an example, and the equation can be specifically expressed as:

[0086]

[0087] Where ψ and φ represent the neutron angular flux and neutron scalar flux, respectively. Indicates the spatial position, represents the angle direction, g represents the energy group index, Σ t,g represents the total cross section of the g-th 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 energy spectrum, υ represents the number of neutrons released in each fission, Σ f,g represents the fission cross section, G represents the total number of energy groups, k eff Indicates effective proliferation factor.

[0088] Specifically, step S220 integrates the above equation in the angular direction to obtain the following new control equation form:

[0089]

[0090] Among them, J g represents the neutron flux and can be defined as:

[0091]

[0092] In practical applications, after the neutron flux φ is calculated by the deterministic neutron transport solver, the neutron flux J is calculated by the following formula: g , instead of calculating it through the above definition:

[0093]

[0094] Therefore, this scheme can obtain the physical quantities required for the new control equations without making any changes to the existing deterministic neutron transport solver, thus preparing for the subsequent reduced-order model construction.

[0095] Specifically, step S230 converts the above neutron flux φ g and neutron flux J g As sample α, it can be defined as:

[0096]

[0097] Among them, N DoF represents the total number of degrees of freedom of the full order;

[0098] Then construct the samples under different parameter change states into sample matrix A:

[0099] Among them, N S Represents the sample size.

[0100] Specifically, step S240 performs singular value decomposition (SVD) on the sample matrix, which specifically includes the following three steps:

[0101] S241. Calculate the square matrix S corresponding to the sample matrix: S = A T A;

[0102] S242, calculate the eigenvalues ​​of the square matrix S {λ1,…,λ m ,…,λ NS}and

[0103] The eigenvector {X1,…,X m ,…,X NS}, thus forming a POD basis P:

[0104]

[0105] S243. Calculate the truncation order N of the POD basis P , the specific calculation formula is as follows:

[0106]

[0107] Specifically, according to the definition of the above sample, the POD base of step S250 can be divided into the following two parts:

[0108]

[0109] Among them, P m,φ,g and Pm,J,g denote the neutron flux and neutron flux POD basis of the g-th energy group respectively; hence, the neutron flux φ g and neutron flux J g It can be expressed as the following refactored form:

[0110]

[0111] Among them, c m represents the coefficient corresponding to the POD basis, which is the variable to be determined in the reduced-order model;

[0112] Combining the above formula with the new control equation, we can get:

[0113]

[0114] Specifically, step S260 defines the residual function E as follows:

[0115]

[0116] By using the least squares method, we can calculate the c that can make the residual function E reach the minimum value m , thereby reconstructing the neutron flux distribution.

[0117] Specifically, step S270 expresses the reduced-order model equation constructed in step S250 as:

[0118]

[0119] Among them, C represents c m The column vectors composed of the matrix U and matrix V can be expressed as:

[0120]

[0121] The feasibility and computational efficiency of this scheme are demonstrated through verification and analysis of a two-dimensional TWIGL example.

[0122] The schematic diagram of the two-dimensional TWIGL example is as follows Figure 2 As shown, the entire research area is a 80cm×80cm square, the left and lower boundaries are reflection boundaries, the right and upper boundaries are vacuum boundaries, and the entire research area includes three materials. The various cross-sections of each material are shown in the following table:

[0123]

[0124] In the offline stage, the finite difference method (FDM) and the finite element method (FEM) are used as full-order neutron transport solvers to solve the neutron flux φ of material 1 under different cross-sectional parameters, and the corresponding neutron flux J is calculated by the neutron flux calculation method proposed in this scheme.g The specific range of cross-sectional parameters is shown in the following table:

[0125]

[0126] In the online stage, by calculating the test case, the specific values ​​of various variable cross sections of material 1 are:

[0127]

[0128] The following two tables compare the reduced-order results of samples obtained using FDM and FEM as full-order neutron transport solvers:

[0129]

[0130] At the same time, the calculation time of different methods is given as shown in the following table:

[0131]

[0132] From the results in the above table, it can be seen that the reduced-order model proposed in the present invention has good calculation accuracy, and the relative error of the maximum neutron standard flux is only 0.27%. At the same time, it also has excellent calculation efficiency. Compared with the traditional full-order neutron transport solver, the speedup ratio of the present invention can reach more than 10,000 times.

[0133] For different types of full-order models, i.e., high-fidelity deterministic algorithms for neutron transport, 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 corresponding reduced-order models.

[0134] In addition, other model order reduction methods can also be used to construct the above-mentioned reduced-order model, including linear model order reduction methods: Dynamic Mode Decomposition method, Principle Component Analysis method; and nonlinear model order reduction methods: Auto-Encoder method, Linear Locally Embedding method.

[0135] It should be noted that the present invention is based on a highly compatible intrusive order reduction method for solving deterministic neutron transport, and is applied to real-time calculation of the neutron field in a nuclear reactor. The focus is on establishing the neutron transport calculation results, i.e., the neutron standard flux φ g and neutron flux J gThe relationship between them is mapped to the reduced basis space, and at the same time, a highly compatible neutron transport intrusive reduced-order model can be constructed based on the new control equation combined with the POD basis; therefore, the present invention does not implement simple functions with complex steps, nor does it use conventional or simple features for combination or stacking, so it conforms to the common sense of technical improvement and is practical.

[0136] The contents not described in detail in this specification belong to the prior art known to ordinary technicians in this field.

[0137] It should be understood that the above description is only a preferred embodiment of the present invention and is not sufficient to limit the technical solution of the present invention. For ordinary technicians in this field, within the spirit and principles of the present invention, additions, substitutions, changes or improvements can be made according to the above description, and all these additions, substitutions, changes or improved technical solutions should fall within the scope of protection of the claims attached to the present invention.

Claims

1. A highly compatible intrusive order reduction method based on deterministic neutron transport solution, characterized in that: It consists of the following steps: Step S210, using a deterministic neutron transport solver to calculate the neutron flux field distribution under different cross-sectional parameters; Step S220, calculating the neutron flux distribution corresponding to different neutron target flux fields; Step S230, combining the two physical fields in step S210 and step S220, and extracting characteristic modes through intrinsic orthogonal decomposition; Step S240: Determine the truncation order N of the eigenorthogonal basis P ; Step S250, combining the control equation with the intrinsic orthogonal basis to construct a reduced-order model; Step S260: According to the reduced-order model, different cross-sectional parameters are input to calculate the intrinsic orthogonal basis coefficients c under the corresponding cross-sectional parameters. m ; Step S270: reconstruct the neutron target flux field distribution.

2. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 1 is characterized in that: The deterministic neutron transport solver of step S210 adopts the steady-state neutron transport fission source equation under the two-dimensional multi-group approximation, and its equation is expressed as follows: Where ψ and φ represent the neutron angular flux and neutron scalar flux, respectively. Indicates the spatial position, represents the angle direction, g represents the energy group index, Σ t,g represents the total cross section of the g-th 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 energy spectrum, υ represents the number of neutrons released in each fission, Σ f,g represents the fission cross section, G represents the total number of energy groups, k eff Indicates effective proliferation factor.

3. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 2 is characterized in that: Step S220 integrates the equation in step S210 in the angular direction to obtain the following new control equation form: Among them, J g represents the neutron flux and can be defined as: After the neutron flux φ is calculated by the deterministic neutron transport solver, the neutron flux J is calculated by the following formula: g :

4. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 3 is characterized in that: Step S230: neutron flux φ g and neutron flux J g As sample α, it is defined as: Among them, N DoF represents the total number of degrees of freedom of the full order; Then construct the samples under different parameter change states into sample matrix A: Among them, N S Represents the sample size.

5. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 4 is characterized in that: Step S240 performs singular value decomposition (SVD) 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, calculate the eigenvalues ​​of the square matrix S {λ1,…,λ m ,…,λ NS }and The eigenvector {X1,…,X m ,…,X NS }, thus forming a POD basis P: S243. Calculate the truncation order N of the POD basis P , the calculation formula is as follows: N P =arg min{I(N P ):I(N P )≥1-μ}, 6. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 5, characterized in that: In step S250, according to the definition of the above sample, the POD base is divided into the following two parts: Among them, P m,φ,g and P m,J,g denote the neutron flux and neutron flux POD basis of the g-th energy group respectively; thus, the neutron flux φ is reconstructed g and neutron flux J g It is expressed as: Among them, c m represents the coefficient corresponding to the POD basis, which is the variable to be determined in the reduced-order model; Combining the above formula with the new control equation, we get:

7. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 6 is characterized in that: Step S260 defines the residual function E as follows: By using the least squares method, we can calculate the c that can make the residual function E reach the minimum value. m .

8. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 7 is characterized in that: Step S270 expresses the reduced-order model equation constructed in step S250 as: Among them, C represents c m The column vectors composed of the matrix U and matrix V can be expressed as:

9. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 1, characterized in that: The deterministic neutron transport solver in step S210 constructs a full-order model using a finite difference method, a finite element method, a finite volume method or a characteristic line method.

10. The highly compatible intrusive order reduction method based on deterministic neutron transport solution according to claim 1, characterized in that: Step S250 adopts a dynamic mode decomposition method, a principal component analysis method, an autoencoder method or a local linear embedding method when constructing a reduced-order model.

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