Pressure vessel lower chamber nonlinear model order reduction method based on POD and secondary manifold

Abstract

The invention discloses a pressure vessel lower chamber nonlinear model order reduction method based on POD and secondary manifold. The method comprises the steps that sample points of sampling parameters are determined, a lower chamber full-order fluid mechanics CFD model corresponding to the sample points is operated, internal velocity field and temperature field data of a lower chamber are obtained, and a snapshot matrix of the velocity field and a snapshot matrix of the temperature field are constructed; performing singular value decomposition on the snapshot matrixes of the velocity field and the temperature field to obtain a left singular value matrix, a diagonal singular value matrix and a right singular value matrix; determining an intrinsic orthogonal decomposition (POD) modal matrix and an intrinsic orthogonal decomposition (POD) coefficient matrix according to the singular value proportion; determining a mapping relation between the sampling parameter and the POD coefficient; and respectively calculating the POD error matrix, a Kronerkel matrix of the POD coefficient matrix and a quadratic manifold modal matrix to obtain a nonlinear reduced-order model of the lower chamber of the pressure vessel. The method can be used for rapidly obtaining field distribution data of internal flow parameters of the lower chamber.

Description

Technical Field

[0001] The present invention relates to the technical field of nuclear reactor engineering, and particularly to a method for reducing the order of a nonlinear model of the lower plenum of a pressure vessel based on POD and quadratic manifolds. Background Art

[0002] There are complex flow mixing and flow direction transformation phenomena in the lower plenum of the reactor pressure vessel. Currently, computational fluid dynamics (CFD) programs are often used to simulate the lower plenum to obtain the field distribution of the internal flow parameters of the lower plenum. However, the CFD program calculation requires a very high cost and the calculation speed is very slow. For application scenarios that require fast simulation, such as parametric studies, online simulations, and digital twins, etc., the advantages of the CFD program cannot be exerted. Therefore, a reduced-order model of the lower plenum CFD model can be constructed. Once the reduced-order model is constructed, only matrix algebraic operations are required to obtain the field distribution data of the internal flow parameters of the lower plenum, and the calculation speed is greatly improved. Currently, the proper orthogonal decomposition (POD) method is often used to construct a parametric reduced-order model, but the POD method belongs to a linear reduced-order method. For physical phenomena dominated by transport, more POD modes are required to ensure the accuracy of the reduced-order model. Summary of the Invention

[0003] To solve the above technical problems existing in the prior art, the present invention proposes a method for reducing the order of a nonlinear model of the lower plenum of a pressure vessel based on POD and quadratic manifolds. This method combines POD and quadratic manifolds to construct a non-intrusive nonlinear reduced-order model of the lower plenum of a pressure vessel, which can be used to quickly obtain the field distribution data of the internal flow parameters of the lower plenum.

[0004] On the one hand, to achieve the above object, the present invention provides a method for reducing the order of a nonlinear model of the lower plenum of a pressure vessel based on POD and quadratic manifolds, including:

[0005] Determine the sample points of the sampling parameters, run the full-order fluid mechanics CFD model of the lower plenum corresponding to the sample points, obtain the internal velocity field and temperature field data of the lower plenum, and construct a snapshot matrix of the velocity field and a snapshot matrix of the temperature field;

[0006] Perform singular value decomposition on the snapshot matrix of the velocity field and the snapshot matrix of the temperature field respectively to obtain a left singular value matrix, a diagonal singular value matrix, and a right singular value matrix;

[0007] Determine the proper orthogonal decomposition POD mode matrix and the proper orthogonal decomposition POD coefficient matrix according to the singular value ratio;

[0008] Determine the mapping relationship between the sampling parameters and the POD coefficients;

[0009] Calculate the POD error matrix, the Kronecker matrix of the POD coefficient matrix, and the quadratic manifold mode matrix respectively, obtain the nonlinear reduced-order model of the lower chamber of the pressure vessel, and obtain the distribution data of the velocity field and temperature field inside the lower chamber through the nonlinear reduced-order model.

[0010] Preferably, the full-order computational fluid dynamics (CFD) model of the lower chamber is the Reynolds-averaged equations:

[0011]

[0012] where ρ is the density, t is the time, u is the velocity, p is the pressure, g is the acceleration due to gravity, r is the position vector of the control volume, k is the turbulent kinetic energy, μ eff is the equivalent dynamic viscosity, h is the specific enthalpy, K is the kinetic energy of unit mass motion, α eff is the equivalent thermal diffusivity.

[0013] Preferably, determining the proper orthogonal decomposition (POD) mode matrix and the POD coefficient matrix according to the singular value ratio includes:

[0014] Determine the number of POD modes, take the first r columns of the left singular matrix as the POD mode matrix, and obtain the POD coefficient matrix through the POD mode matrix.

[0015] Preferably, the method for determining the number of POD modes is:

[0016]

[0017] where r is the number of POD modes, σ is the diagonal element of the singular value matrix E, is a constant set by the user;

[0018] The obtained POD coefficient matrix is:

[0019] C = Z T Y;

[0020] where Z is the POD mode matrix, Y is the snapshot matrix, and T is the matrix transpose symbol.

[0021] Preferably, the mapping relationship between the sampling parameters and the POD coefficients is obtained by training a BP neural network.

[0022] Preferably, calculate the POD error matrix as:

[0023] Epsi = Y - Z * C;

[0024] where Epsi is the POD error matrix, Y is the snapshot matrix, Z is the POD mode matrix, and C is the POD coefficient matrix.

[0025] Preferably, obtaining the Kronecker matrix of the POD coefficient matrix includes:

[0026] Performing a Kronecker product operation on each column element of the POD coefficient matrix C with itself, and retaining the independent elements to obtain the Kronecker matrix W of the POD coefficient matrix.

[0027] Preferably, the method for calculating the quadratic manifold modal matrix is:

[0028] M = Epsi * W T (WW T + γI) -1 ;

[0029] where γ is the regularization factor, I is the identity matrix, Epsi is the POD error matrix, W is the Kronecker matrix of the POD coefficient matrix, and W T is the transpose matrix of the Kronecker matrix of the POD coefficient matrix, and M is the quadratic manifold modal matrix.

[0030] Preferably, the non - linear reduced - order model is:

[0031] Yr = Z * C + M * W;

[0032] where Yr is the non - linear reduced - order model, Z is the POD modal matrix, C is the POD coefficient matrix, and W is the Kronecker matrix of the POD coefficient matrix.

[0033] On the other hand, to achieve the above - mentioned purpose, the present invention also provides a non - linear model reduction system for the lower chamber of a pressure vessel based on POD and quadratic manifold, including a POD module and a quadratic manifold module;

[0034] Among them, the POD module is used to determine the sample points of the sampling parameters, run the full - order fluid mechanics CFD model of the lower chamber corresponding to the sample points, obtain the internal velocity field and temperature field data of the lower chamber, and construct the snapshot matrix of the velocity field and the snapshot matrix of the temperature field; perform singular value decomposition on the snapshot matrix of the velocity field and the snapshot matrix of the temperature field respectively to obtain the left singular matrix, diagonal singular value matrix, and right singular value matrix; determine the proper orthogonal decomposition POD modal matrix and the proper orthogonal decomposition POD coefficient matrix according to the singular value proportion; determine the mapping relationship between the sampling parameters and the POD coefficients;

[0035] The quadratic manifold module is used to calculate the POD error matrix, the Kronecker matrix of the POD coefficient matrix, and the quadratic manifold modal matrix respectively, and obtain the non - linear reduced - order model.

[0036] Compared with the prior art, the present invention has the following advantages and technical effects:

[0037] The present invention constructs a non-intrusive non-linear reduced-order model for the lower chamber of a pressure vessel by combining POD and quadratic manifolds, which can be used to quickly obtain the field distribution data of the flow parameters inside the lower chamber. Compared with the linear reduced-order model of the lower chamber constructed by the POD method, the present invention improves the accuracy of the reduced-order model by introducing a quadratic manifold module. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The drawings forming a part of this application are used to provide a further understanding of this application. The schematic embodiments of this application and their descriptions are used to explain this application and do not constitute an improper limitation of this application. In the drawings:

[0039] Figure 1 It is a schematic flow chart of a method for reducing the order of a non-linear model of the lower chamber of a pressure vessel based on POD and quadratic manifolds according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0040] It should be noted that, without conflict, the embodiments in this application and the features in the embodiments can be combined with each other. The following will refer to the drawings and combine the embodiments to detail this application.

[0041] It should be noted that the steps shown in the flowchart of the drawings can be executed in a computer system such as a set of computer-executable instructions, and although the logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in a different order than here.

[0042] This embodiment proposes a method for reducing the order of a non-linear model of the lower chamber of a pressure vessel based on POD and quadratic manifolds, as Figure 1 , including:

[0043] Determine the sample points of the sampling parameters, run the full-order fluid dynamics CFD model of the lower chamber corresponding to the sample points, obtain the velocity field and temperature field data inside the lower chamber, and construct a snapshot matrix of the velocity field and temperature field;

[0044] Perform singular value decomposition on the snapshot matrix to obtain a left singular matrix, a diagonal singular value matrix, and a right singular value matrix;

[0045] Determine the proper orthogonal decomposition POD mode matrix and the proper orthogonal decomposition POD coefficient matrix according to the singular value ratio;

[0046] Determine the mapping relationship between the sampling parameters and the POD coefficients;

[0047] Calculate the POD error matrix, the Kronecker matrix of the POD coefficient matrix, and the quadratic manifold mode matrix respectively to obtain a non-linear reduced-order model.

[0048] In this embodiment, a non-invasive nonlinear reduced-order model of the lower chamber of a pressure vessel is constructed by combining POD and quadratic manifolds, which can be used to quickly obtain the field distribution data of the flow parameters inside the lower chamber. The reduced-order model obtained in this embodiment is used to quickly obtain the field distribution data of the flow parameters (velocity and temperature) inside the lower chamber, and can be used in some application scenarios that require quick calculation results, such as parametric studies, online simulations, and digital twins.

[0049] Further, in this embodiment, based on the uniform sampling method, the sample points of the sampling parameter μ are determined, and the full-order CFD model of the lower chamber corresponding to the sample points is run to obtain the velocity field and temperature field data inside the lower chamber. Then, the snapshot matrix of the velocity field and the snapshot matrix of the temperature field are constructed respectively, where the full-order CFD model uses the Reynolds-averaged Navier-Stokes equations:

[0050]

[0051] In the formula, ρ is the density, t is the time, u is the velocity, p is the pressure, g is the acceleration due to gravity, r is the control volume radius vector, k is the turbulent kinetic energy, μ eff is the equivalent dynamic viscosity, h is the specific enthalpy, K is the kinetic energy of unit mass motion, α eff is the equivalent thermal diffusivity.

[0052] The form of the snapshot matrix Y is:

[0053]

[0054] In the formula, y is the velocity or temperature, N C is the number of grids of the full-order CFD model, N S is the number of sample points.

[0055] Further, the snapshot matrix of the velocity field and the snapshot matrix of the temperature field are respectively subjected to singular value decomposition to obtain the left singular value matrix U, the diagonal singular value matrix E, and the right singular value matrix V;

[0056] According to the singular value ratio, the POD mode matrix Z and the POD coefficient matrix C are determined, specifically:

[0057] First, according to the formula the number of POD modes r is determined, where σ is the diagonal element of the diagonal singular value matrix E, is a constant set by the user, which is taken as 0.995 in this embodiment;

[0058] Then, the first r columns of the left singular value matrix U are taken as the POD mode matrix Z;

[0059] Finally, the POD coefficient matrix C is obtained through the formula C = Z T Y.

[0060] Furthermore, the mapping relationship between the sampling parameters and the proper orthogonal decomposition (POD) coefficients is obtained by training a BP neural network.

[0061] Specifically, the mapping relationship c = c(μ) between the parameter μ and the POD coefficient c is determined, where the mapping relationship is obtained by training a BP neural network. The input of the network is the parameter μ, and the output is the POD coefficient.

[0062] The POD coefficient c is a row in the POD coefficient matrix. Each row of the POD coefficient matrix C is a POD coefficient, and each column of the POD coefficient matrix C is the value of different POD coefficients under the parameter μ.

[0063] Furthermore, the POD error matrix is calculated as:

[0064] Epsi = Y - Z * C;

[0065] In the formula, Epsi is the POD error matrix, Y is the snapshot matrix, Z is the POD mode matrix, and C is the POD coefficient matrix.

[0066] Obtaining the Kronecker matrix of the POD coefficient matrix includes:

[0067] After performing the Kronecker product operation on each column element of the POD coefficient matrix C with itself, the independent elements are retained to obtain the Kronecker matrix W of the POD coefficient matrix.

[0068] Specifically, the Kronecker product is defined as follows: For two column vectors a ∈ R m and b ∈ R n , their Kronecker inner product is a column vector of size (m·n) × 1:

[0069]

[0070] Furthermore, the method for calculating the secondary manifold mode matrix is:

[0071] M = Epsi * W T (WW T + γI) -1 ;

[0072] In the formula, γ is the regularization factor, I is the identity matrix, Epsi is the POD error matrix, W is the Kronecker matrix of the POD coefficient matrix, W T is the transpose matrix of the Kronecker matrix of the POD coefficient matrix, and M is the secondary manifold mode matrix.

[0073] The nonlinear reduced-order model is:

[0074] Yr = Z * C + M * W;

[0075] Wherein, Yr is a non - linear reduced - order model, Z is a POD modal matrix, C is a POD coefficient matrix, and W is a Kronecker matrix of the POD coefficient matrix.

[0076] This embodiment also provides a non - linear model reduction system for the lower chamber of a pressure vessel based on POD and quadratic manifold, including a POD module and a quadratic manifold module:

[0077] The POD module is used to determine the sample points of the sampling parameters, run the full - order fluid dynamics CFD model of the lower chamber corresponding to the sample points, obtain the internal velocity field and temperature field data of the lower chamber, and construct a snapshot matrix of the velocity field and a snapshot matrix of the temperature field; perform singular value decomposition on the snapshot matrix of the velocity field and the snapshot matrix of the temperature field respectively to obtain a left singular matrix, a diagonal singular value matrix, and a right singular value matrix; determine the proper orthogonal decomposition (POD) modal matrix and the proper orthogonal decomposition (POD) coefficient matrix according to the singular value ratio; determine the mapping relationship between the sampling parameters and the POD coefficients;

[0078] The quadratic manifold module is used to calculate the POD error matrix, the Kronecker matrix of the POD coefficient matrix, and the quadratic manifold modal matrix respectively to obtain a non - linear reduced - order model.

[0079] The present invention combines POD and quadratic manifold to construct a non - intrusive non - linear reduced - order model for the lower chamber of a pressure vessel, which can be used to quickly obtain the field distribution data of the internal flow parameters of the lower chamber. Compared with the linear reduced - order model of the lower chamber constructed by the POD method, the present invention improves the accuracy of the reduced - order model by introducing a quadratic manifold module.

[0080] The above is only a preferred specific embodiment of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed in the present application should be covered within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold, characterized in that: include: Determine the sample points of the sampling parameters, run the full-order fluid mechanics CFD model of the lower chamber corresponding to the sample points, obtain the velocity field and temperature field data inside the lower chamber, and construct a snapshot matrix of the velocity field and a snapshot matrix of the temperature field; Performing singular value decomposition on the snapshot matrix of the velocity field and the snapshot matrix of the temperature field to obtain a left singular value matrix, a diagonal singular value matrix and a right singular value matrix; Determine the intrinsic orthogonal decomposition POD mode matrix and the intrinsic orthogonal decomposition POD coefficient matrix according to the proportion of singular values; Determine a mapping relationship between the sampling parameters and the POD coefficients; The POD error matrix, the Kronecker matrix of the POD coefficient matrix and the secondary manifold modal matrix are calculated respectively to obtain a nonlinear reduced-order model of the lower chamber of the pressure vessel, and the distribution data of the velocity field and the temperature field inside the lower chamber are obtained through the nonlinear reduced-order model.

2. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 1 is characterized in that: The full-order fluid mechanics CFD model of the lower chamber is the Reynolds average equations: Where ρ is density, t is time, u is velocity, p is pressure, g is gravitational acceleration, r is the control body radius, k is turbulent kinetic energy, μ is eff is the equivalent dynamic viscosity, h is the specific enthalpy, K is the kinetic energy per unit mass, α eff is the equivalent thermal diffusion coefficient.

3. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 1 is characterized in that: Determining the intrinsic orthogonal decomposition POD mode matrix and the intrinsic orthogonal decomposition POD coefficient matrix according to the singular value proportion includes: Determine the number of POD modes, take the first r columns of the left singular matrix as the POD mode matrix, and obtain the POD coefficient matrix through the POD mode matrix.

4. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 3 is characterized in that: The method for determining the number of POD modes is: Where r is the number of POD modes, σ is the diagonal element of the singular value matrix E, Constants set by the user; The POD coefficient matrix is ​​obtained as: C=Z T Y; Where Z is the POD modal matrix, Y is the snapshot matrix, and T is the matrix transpose symbol.

5. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 1, characterized in that: The mapping relationship between the sampling parameters and the intrinsic orthogonal decomposition POD coefficients is obtained by BP neural network training.

6. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 1, characterized in that: The POD error matrix is ​​calculated as: Epsi=YZ*C; Where Epsi is the POD error matrix, Y is the snapshot matrix, Z is the POD modal matrix, and C is the POD coefficient matrix.

7. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 1, characterized in that: Obtaining the Kronnicke matrix of the POD coefficient matrix includes: After performing a Kronecker product operation on each column element of the POD coefficient matrix C and itself, the independent elements are retained to obtain the Kronecker matrix W of the POD coefficient matrix.

8. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 1, characterized in that: The method for calculating the secondary manifold modal matrix is: M=Epsi*W T (WW T +γI) -1 ; Where γ is the regularization factor, I is the identity matrix, Epsi is the POD error matrix, W is the Kronnick matrix of the POD coefficient matrix, and W T is the transposed matrix of the Kronnicke matrix of the POD coefficient matrix, and M is the secondary manifold modal matrix.

9. The nonlinear model reduction method for the lower chamber of a pressure vessel based on POD and secondary manifold according to claim 8, characterized in that: The nonlinear reduced-order model is: Yr=Z*C+M*W; Where Yr is the nonlinear reduced-order model, Z is the POD modal matrix, C is the POD coefficient matrix, and W is the Kronnicke matrix of the POD coefficient matrix.

10. A nonlinear model reduction system for the lower chamber of a pressure vessel based on POD and secondary manifold, characterized in that: Includes POD module and secondary manifold module; The POD module is used to determine the sample points of the sampling parameters, run the full-order fluid mechanics CFD model of the lower chamber corresponding to the sample points, obtain the velocity field and temperature field data inside the lower chamber, and construct the snapshot matrix of the velocity field and the snapshot matrix of the temperature field; perform singular value decomposition on the snapshot matrix of the velocity field and the snapshot matrix of the temperature field to obtain a left singular matrix, a diagonal singular value matrix and a right singular value matrix; determine the intrinsic orthogonal decomposition POD mode matrix and the intrinsic orthogonal decomposition POD coefficient matrix according to the proportion of singular values; Determine a mapping relationship between the sampling parameters and the POD coefficients; The secondary manifold module is used to respectively calculate the POD error matrix, the Kronnicke matrix of the POD coefficient matrix, and the secondary manifold modal matrix to obtain a nonlinear reduced-order model.

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