POD secondary manifold lower chamber flow field nonlinear model order reduction method based on anchor point optimization

Abstract

The invention relates to a POD secondary manifold lower chamber flow field nonlinear model order reduction method based on anchor point optimization, and the method comprises the steps: obtaining a two-dimensional lower chamber computational fluid mechanics model as a full-order model, and building a snapshot matrix according to the full-order model; carrying out POD order reduction on the snapshot matrix, and calculating a POD reconstruction error matrix and a secondary manifold coefficient matrix of the snapshot matrix; determining an anchor point, and calculating a quadratic manifold modal matrix at the anchor point according to the POD reconstruction error matrix and the quadratic manifold coefficient matrix; according to the secondary manifold modal matrix at each anchor point, obtaining a secondary manifold modal matrix at the test point; and according to the quadratic manifold modal matrix at the test point, calculating an approximate snapshot of the POD quadratic manifold nonlinear reduced-order model at the test point. According to the method, the representative anchor points are introduced into the parameter sampling space, so that self-adaptive calculation of the secondary manifold modal matrix is realized, and the approximation precision of the reduced-order model is improved.

Description

Technical Field

[0001] This invention relates to the field of thermal-hydraulic model order reduction technology, and in particular to a method for order reduction of a nonlinear model of the chamber flow field under a POD quadratic manifold based on anchor point optimization. Background Technology

[0002] The flow within the lower chamber of a nuclear reactor pressure vessel exhibits complex phenomena such as abrupt changes in flow direction, strong backflow, and large-scale vortices, leading to uneven flow distribution at the lower chamber outlet. Simulating the flow phenomena within the lower chamber using a full-order model based on the finite volume method is computationally expensive and unsuitable for applications requiring repeated solutions. To improve computational efficiency, model reduction methods based on Proper Orthogonal Decomposition (POD) are widely adopted, allowing for approximate reconstruction of the full-order model's response characteristics at a lower dimension. However, the POD method is essentially a linear approximation; for convection-dominated physical phenomena, it typically requires preserving a large number of modes to achieve the target accuracy. The quadratic manifold method alleviates these problems by introducing a second-order extension term, but the calculation of this extension term depends on the selection of a regularization parameter. It is difficult to select a unique regularization parameter within the global parameter sampling space to ensure optimal accuracy for the reduced-order model at all parameter points, increasing the difficulty of constructing a nonlinear reduced-order model using the POD quadratic manifold. Summary of the Invention

[0003] The purpose of this invention is to provide a method for reducing the order of a nonlinear model of a cavity flow field under a POD quadratic manifold based on anchor point optimization. By introducing representative anchor points in the parameter sampling space, the method enables adaptive calculation of the mode matrix of the quadratic manifold. This method is suitable for constructing a nonlinear reduced-order model of the cavity flow field under a high-dimensional system, thereby improving the approximate accuracy of the reduced-order model.

[0004] To achieve the above objectives, the present invention provides the following solution:

[0005] A method for order reduction of the nonlinear model of the chamber flow field under a POD quadratic manifold based on anchor point optimization includes:

[0006] A two-dimensional computational fluid dynamics model of the lower chamber is obtained as a full-order model, and a snapshot matrix is ​​constructed based on the full-order model.

[0007] The snapshot matrix is ​​subjected to POD reduction, and the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix are calculated.

[0008] Determine the anchor point, and calculate the quadratic manifold mode matrix at the anchor point based on the POD reconstruction error matrix and the quadratic manifold coefficient matrix;

[0009] Based on the secondary manifold mode matrix at each anchor point, obtain the secondary manifold mode matrix at the test point;

[0010] Based on the modal matrix of the quadratic manifold at the test point, calculate the approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the test point.

[0011] Optionally, constructing the snapshot matrix includes:

[0012] The sampling parameters are sampled within the sampling space, and a snapshot of the lower chamber velocity amplitude field is obtained through the full-order model to construct the snapshot matrix Y∈R. n×m =[y1 … y m ], where the sampling parameters are the mass flow rates of the two inlets of the lower chamber, n is the full-order model dimension, m is the number of snapshots, and y is the snapshot of the velocity amplitude field of the lower chamber.

[0013] Optionally, the snapshot matrix is ​​subjected to POD reduction, and the calculation of the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix includes:

[0014] Singular value decomposition is performed on the snapshot matrix, and the first r columns of the left singular vector matrix are selected as the POD mode matrix, where each column of the POD mode matrix is ​​a POD mode, and r is the truncation rank determined according to the proportion of the diagonal elements of the singular value matrix.

[0015] Based on the POD mode matrix, calculate the POD coefficient matrix, wherein each row of the POD coefficient matrix represents a POD coefficient;

[0016] Based on the POD mode matrix and the POD coefficient matrix, calculate the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix.

[0017] Optionally, the POD reconstruction error matrix of the snapshot matrix is:

[0018] E∈R n×m =YZ*C;

[0019] The quadratic manifold coefficient matrix is:

[0020]

[0021] Where E is the POD reconstruction error matrix, n is the full-order model dimension, m is the number of snapshots, Y is the snapshot matrix, Z is the POD modality matrix, C is the POD coefficient matrix, and W is the quadratic manifold coefficient matrix. This represents the Kronikel product of the column vectors of the POD coefficient matrix C with itself. After the operation, only independent elements are retained, and r is the truncated rank.

[0022] Optionally, calculating the modal matrix of the quadratic manifold at the anchor point includes:

[0023] Vp (γ p )∈R n×0.5r(r+1) =EW T (WW T +γ p I);

[0024] Among them, V p (γ p ) represents the modal matrix of the quadratic manifold, with the superscript T indicating transpose, I representing the identity matrix, and γ... p The optimal regularization factor at the anchor point is solved by minimizing the problem.

[0025] Optionally, after calculating the modal matrix of the quadratic manifold at the anchor point, the following is included:

[0026] Calculate the approximate snapshot error of the POD linear reduced-order model at the anchor point.

[0027]

[0028] Calculate the approximate snapshot error of the nonlinear reduced-order model of the POD quadratic manifold at the anchor point.

[0029]

[0030] Where n is the full-order model dimension, y p For anchor point μ p A snapshot of the full-order model, where r is the truncation rank. Anchor point μ is obtained based on the mapping relationship between sampling parameter μ and POD coefficient. p POD coefficient values ​​at each column For a POD mode, This is an approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the anchor point;

[0031] Determine the approximate snapshot error of the linear reduced-order model at the anchor point. Is it less than or equal to the approximate snapshot error of the nonlinear reduced-order model? If so, set the corresponding quadratic manifold mode matrix at the anchor point to a zero matrix, that is, do not introduce a quadratic manifold.

[0032] Optionally, based on the secondary manifold mode matrix at each anchor point, the secondary manifold mode matrix at the test point is obtained, including:

[0033] The quadratic manifold mode matrix at each anchor point is weighted and averaged to obtain the quadratic manifold mode matrix at the test point. The weighting coefficient of each anchor point is calculated based on the Euclidean distance between the test point and the anchor point in the parameter space.

[0034] Optionally, calculating the approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the test point includes:

[0035]

[0036] in, This is an approximate snapshot of the reduced-order nonlinear model of the POD quadratic manifold at the test point, μ test V is the test point. test Let w(μ) be the modal matrix of the quadratic manifold at the test point. test ) represents the corresponding quadratic manifold coefficient, and r is the cutoff rank. The test point μ is obtained based on the mapping relationship between the sampling parameter μ and the POD coefficient. test POD coefficient values ​​at each column This is a POD mode.

[0037] The beneficial effects of this invention are as follows: 1. This invention introduces an anchor point mechanism to perform local optimization of regularization parameters at representative locations in the parameter sampling space, effectively avoiding the problem of excessive error fluctuations caused by globally fixed regularization parameters in traditional methods. By weighted averaging the modal matrices of the quadratic manifold at the anchor points, local adaptive adjustment of the quadratic manifold terms is achieved, thereby improving the generalization ability and approximation accuracy of the reduced-order model in the entire parameter space.

[0038] 2. This invention enhances the robustness of the model by comparing the errors of linear and nonlinear reduced-order models at anchor points and updating the quadratic manifold mode matrix based on the errors, thus avoiding unnecessary nonlinear compensation. Attached Figure Description

[0039] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0040] Figure 1 This is a flowchart illustrating a method for reducing the order of a nonlinear model of the flow field in a cavity under a POD quadratic manifold based on anchor point optimization, according to an embodiment of the present invention.

[0041] Figure 2 This is a schematic diagram of the geometry and boundary conditions of the full-order model in an embodiment of the present invention;

[0042] Figure 3 This is a distribution diagram of snapshot points, anchor points, and test points in the parameter space according to an embodiment of the present invention;

[0043] Figure 4 This is a root mean square error diagram of the test points in an embodiment of the present invention. Detailed Implementation

[0044] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0045] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0046] like Figure 1 As shown, this embodiment provides a method for order reduction of the nonlinear model of the chamber flow field under a POD quadratic manifold based on anchor point optimization, including:

[0047] A two-dimensional computational fluid dynamics model of the lower chamber is obtained as a full-order model, and a snapshot matrix is ​​constructed based on the full-order model.

[0048] The snapshot matrix is ​​reduced to POD order, and the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix are calculated.

[0049] Determine the anchor point, and calculate the quadratic manifold mode matrix at the anchor point based on the POD reconstruction error matrix and the quadratic manifold coefficient matrix;

[0050] Based on the secondary manifold mode matrix at each anchor point, obtain the secondary manifold mode matrix at the test point;

[0051] Based on the modal matrix of the quadratic manifold at the test point, calculate the approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the test point.

[0052] Furthermore, constructing the snapshot matrix includes: sampling the sampling parameters in the sampling space, obtaining a snapshot of the lower chamber velocity amplitude field through a full-order model, and constructing a snapshot matrix Y∈R. n×m =[y1 … y m ], where the sampling parameters are the mass flow rates of the two inlets of the lower chamber, n is the full-order model dimension, m is the number of snapshots, and y is the snapshot of the velocity amplitude field of the lower chamber.

[0053] In this embodiment, constructing the snapshot matrix Y includes: selecting Figure 2 The two-dimensional lower chamber computational fluid dynamics model shown is a full-order model. Steady-state numerical solutions to the Navier-Stokes equations are obtained using the Reynolds-averaged method to obtain data such as velocity, pressure, and temperature fields. Within the sampling space, the sampling parameter μ is sampled, and the corresponding velocity amplitude field snapshot y is obtained through the full-order model. Then, a snapshot matrix Y∈R is constructed. n×m =[y1 … y m], where n = 47418 is the full-order model dimension and m = 68 is the number of snapshots.

[0054] Furthermore, the snapshot matrix is ​​reduced to its POD order to obtain the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix, including:

[0055] Perform singular value decomposition on the snapshot matrix, and select the first r columns of the left singular vector matrix as the POD mode matrix. Each column of the POD mode matrix is ​​a POD mode, and r is the truncation rank determined according to the proportion of the diagonal elements of the singular value matrix.

[0056] Based on the POD modality matrix, calculate the POD coefficient matrix and determine the mapping relationship between each POD coefficient and the sampling parameters. Each row of the POD coefficient matrix represents a POD coefficient.

[0057] Based on the POD mode matrix and the POD coefficient matrix, calculate the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix.

[0058] Specifically, perform singular value decomposition on matrix Y: Y = UΣV T According to the singular value matrix Σ∈R m×m The proportion of diagonal elements σ Determine the truncation rank r, where The threshold constant set by the user is taken in this embodiment.

[0059] Take the left singular vector matrix U∈R n×m The first r columns are the POD mode matrix Z∈R n×r Each column For a POD mode, calculate the corresponding POD coefficient matrix C∈R. r×m =Z T Y, each row Each column represents a POD coefficient, with each column showing the POD coefficient value corresponding to different snapshot sampling parameters μ.

[0060] A mapping relationship between the sampling parameter μ and each POD coefficient is established by polynomial fitting.

[0061] Furthermore, the POD reconstruction error matrix of the snapshot matrix is:

[0062] E∈R n×m =YZ*C;

[0063] The quadratic manifold coefficient matrix is:

[0064]

[0065] Where E is the POD reconstruction error matrix, n is the full-order model dimension, m is the number of snapshots, Y is the snapshot matrix, Z is the POD modality matrix, C is the POD coefficient matrix, and W is the quadratic manifold coefficient matrix. This represents the Kronikel product of the column vectors of the POD coefficient matrix C with itself. After the operation, only independent elements are retained, and r is the truncated rank.

[0066] Furthermore, calculating the modal matrix of the quadratic manifold at the anchor point includes:

[0067] V p (γ p )∈R n×0.5r(r+1) =EW T (WW T +γ p I);

[0068] Among them, V p (γ p ) represents the modal matrix of the quadratic manifold, with the superscript T indicating transpose, I representing the identity matrix, and γ... p The optimal regularization factor at the anchor point is determined by solving a minimization problem using the function fminbnd in the open-source program Octave.

[0069]

[0070] Where, γ min =0 and γ max =10 10 The range of values ​​for the regularization factor γ set by the user, y p This is a snapshot of the full-order model at the anchor point. This is an approximate snapshot of the reduced-order nonlinear model of the POD quadratic manifold at the anchor point, z i For the i-th POD mode, The anchor point μ is obtained based on the mapping relationship between the POD coefficient and the sampling parameters. p The POD coefficient value at location w(μ) p ) represents the corresponding quadratic manifold coefficient:

[0071]

[0072] Furthermore, after calculating the modal matrix of the quadratic manifold at the anchor point, the following is included:

[0073] Calculate the approximate snapshot error of the POD linear reduced-order model at the anchor point.

[0074]

[0075] Calculate the approximate snapshot error of the nonlinear reduced-order model of the POD quadratic manifold at the anchor point.

[0076]

[0077] Where n is the full-order model dimension, y p For anchor point μ p A snapshot of the full-order model, where r is the truncation rank. Anchor point μ is obtained based on the mapping relationship between sampling parameter μ and POD coefficient. p POD coefficient values ​​at each column For a POD mode, This is an approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the anchor point;

[0078] Determine the approximate snapshot error of the linear reduced-order model at the anchor point. Is it less than or equal to the approximate snapshot error of the nonlinear reduced-order model? If so, set the corresponding quadratic manifold mode matrix at the anchor point to a zero matrix, that is, do not introduce a quadratic manifold.

[0079] Furthermore, based on the secondary manifold mode matrix at each anchor point, the secondary manifold mode matrix at the test point is obtained, including:

[0080] The quadratic manifold mode matrix at each anchor point is weighted and averaged to obtain the quadratic manifold mode matrix at the test point. The weighting coefficient of each anchor point is calculated based on the Euclidean distance between the test point and the anchor point in the parameter space. The weight is defined as the proportion of the reciprocal of the distance between the anchor point and the test point to the sum of the reciprocals of the distances from all anchor points to the test point.

[0081] Furthermore, the approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the test point includes:

[0082]

[0083] in, This is an approximate snapshot of the reduced-order nonlinear model of the POD quadratic manifold at the test point, μ test V is the test point. test Let w(μ) be the modal matrix of the quadratic manifold at the test point. test ) represents the corresponding quadratic manifold coefficient. The test point μ is obtained based on the mapping relationship between the sampling parameter μ and the POD coefficient. test POD coefficient values ​​at each column This is a POD mode.

[0084] In this embodiment, the distribution of snapshot points, anchor points, and test points in the parameter space is as follows: Figure 3 As shown, the root mean square error of the nonlinear reduced-order model constructed based on the method proposed in this invention at the test points is as follows: Figure 4As shown, the root mean square error obtained by the POD linear order reduction model method is given. It can be seen that the method proposed in this embodiment is superior to the POD method.

[0085] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for order reduction of the nonlinear model of the cavity flow field under a POD quadratic manifold based on anchor point optimization, characterized in that, include: A two-dimensional computational fluid dynamics model of the lower chamber is obtained as a full-order model, and a snapshot matrix is ​​constructed based on the full-order model. The snapshot matrix is ​​subjected to POD reduction, and the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix are calculated. Determine the anchor point, and calculate the quadratic manifold mode matrix at the anchor point based on the POD reconstruction error matrix and the quadratic manifold coefficient matrix; Based on the secondary manifold mode matrix at each anchor point, obtain the secondary manifold mode matrix at the test point; Based on the modal matrix of the quadratic manifold at the test point, calculate the approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the test point.

2. The method for order reduction of the nonlinear model of the cavity flow field under the POD quadratic manifold based on anchor point optimization according to claim 1, characterized in that, Constructing the snapshot matrix includes: The sampling parameters are sampled within the sampling space, and a snapshot of the lower chamber velocity amplitude field is obtained through the full-order model to construct the snapshot matrix Y∈R. n×m =[y1 … y m ], where the sampling parameters are the mass flow rates of the two inlets of the lower chamber, n is the full-order model dimension, m is the number of snapshots, and y is the snapshot of the velocity amplitude field of the lower chamber.

3. The method for order reduction of the nonlinear model of the cavity flow field under the POD quadratic manifold based on anchor point optimization according to claim 1, characterized in that, The snapshot matrix is ​​subjected to POD reduction, and the calculation of the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix includes: Singular value decomposition is performed on the snapshot matrix, and the first r columns of the left singular vector matrix are selected as the POD mode matrix, where each column of the POD mode matrix is ​​a POD mode, and r is the truncation rank determined according to the proportion of the diagonal elements of the singular value matrix. Based on the POD mode matrix, calculate the POD coefficient matrix, wherein each row of the POD coefficient matrix represents a POD coefficient; Based on the POD mode matrix and the POD coefficient matrix, calculate the POD reconstruction error matrix and the quadratic manifold coefficient matrix of the snapshot matrix.

4. The method for order reduction of the nonlinear model of the cavity flow field under the POD quadratic manifold based on anchor point optimization according to claim 3, characterized in that, The POD reconstruction error matrix of the snapshot matrix is: E∈R n×m =Y-Z*C; The quadratic manifold coefficient matrix is: Where E is the POD reconstruction error matrix, n is the full-order model dimension, m is the number of snapshots, Y is the snapshot matrix, Z is the POD modality matrix, C is the POD coefficient matrix, and W is the quadratic manifold coefficient matrix. This represents the Kronikel product of the column vectors of the POD coefficient matrix C with itself. After the operation, only independent elements are retained, and r is the truncated rank.

5. The method for order reduction of the nonlinear model of the cavity flow field under the POD quadratic manifold based on anchor point optimization according to claim 4, characterized in that, The calculation of the modal matrix of the quadratic manifold at the anchor point includes: V p (c p )∈R n×0.5r(r+1) =EW T (WW T +g p I); Among them, V p (γ p ) represents the modal matrix of the quadratic manifold, with the superscript T indicating transpose, I representing the identity matrix, and γ... p The optimal regularization factor at the anchor point is solved by minimizing the problem.

6. The method for order reduction of the nonlinear model of the cavity flow field under the POD quadratic manifold based on anchor point optimization according to claim 5, characterized in that, After calculating the modal matrix of the quadratic manifold at the anchor point, the following is included: Calculate the approximate snapshot error of the POD linear reduced-order model at the anchor point. Calculate the approximate snapshot error of the nonlinear reduced-order model of the POD quadratic manifold at the anchor point. : Where n is the full-order model dimension, y p For anchor point μ p A snapshot of the full-order model, where r is the truncation rank. Anchor point μ is obtained based on the mapping relationship between sampling parameter μ and POD coefficient. p The POD coefficient value at each column For a POD mode, This is an approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the anchor point; Determine the approximate snapshot error of the linear reduced-order model at the anchor point. Is it less than or equal to the approximate snapshot error of the nonlinear reduced-order model? If so, set the corresponding quadratic manifold mode matrix at the anchor point to a zero matrix, that is, do not introduce a quadratic manifold.

7. The method for order reduction of the nonlinear model of the cavity flow field under the POD quadratic manifold based on anchor point optimization according to claim 1, characterized in that, Based on the secondary manifold mode matrices at each anchor point, the secondary manifold mode matrices at the test points are obtained, including: The quadratic manifold mode matrix at each anchor point is weighted and averaged to obtain the quadratic manifold mode matrix at the test point. The weighting coefficient of each anchor point is calculated based on the Euclidean distance between the test point and the anchor point in the parameter space.

8. The method for order reduction of the nonlinear model of the cavity flow field under the POD quadratic manifold based on anchor point optimization according to claim 1, characterized in that, The calculation of the approximate snapshot of the nonlinear reduced-order model of the POD quadratic manifold at the test point includes: in, This is an approximate snapshot of the reduced-order nonlinear model of the POD quadratic manifold at the test point, μ test V is the test point. test Let w(μ) be the modal matrix of the quadratic manifold at the test point. test ) represents the corresponding quadratic manifold coefficient, and r is the cutoff rank. The test point μ is obtained based on the mapping relationship between the sampling parameter μ and the POD coefficient. test The POD coefficient value at each column This is a POD mode.

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