Sparse grid and POD combined pressure vessel lower chamber model order reduction method

Abstract

The invention discloses a pressure vessel lower chamber model order reduction method combining sparse grids and POD, and the method comprises the steps: adding one to the grade of the sparse grids after iteration initialization, and obtaining test grid points according to important grid points; executing the lower chamber full-order computational fluid mechanics model corresponding to the test grid points to obtain a first snapshot; reconstructing a second snapshot by using the mapping relation and the POD modal matrix; calculating a root-mean-square error of the two snapshots, and if the root-mean-square error is smaller than a threshold value, completing construction of a reduced-order model Otherwise, the grid points with the errors meeting the requirements are stored in a non-important grid point set, the grid points which do not meet the requirements are marked as important grid points, and the snapshot matrix is updated. Performing singular value decomposition on the snapshot matrix, updating the mapping relation, judging whether the maximum sparse grid grade is reached or not, and continuing iteration if the maximum sparse grid grade is not reached. According to the method, the field distribution of the thermal hydraulic parameters in the lower chamber can be quickly obtained through matrix algebraic operation, and the method can be used for application scenes such as parameterization analysis, design optimization and digital twinning.

Description

Technical Field

[0001] The present invention belongs to the technical field of nuclear reactor engineering, and in particular relates to a pressure vessel lower chamber model reduction method combining sparse grids and POD. Background Art

[0002] When the coolant in the reactor's primary circuit system flows through the lower chamber of the reactor pressure vessel, there are complex flow turbulence and flow direction changes. This phenomenon directly determines the coolant flow rate flowing through the reactor core rod bundle channel, which in turn significantly affects the calculation of neutron dynamics and the safety of nuclear power system operation. In order to obtain the three-dimensional flow distribution of the coolant inside the lower chamber, a computational fluid dynamics (CFD) program can be used for simulation. The CFD program is based on the three-dimensional Navier-Stokes equations and can achieve high-precision three-dimensional scale simulation of complex flow phenomena. However, the CFD program requires high resources for calculation and cannot achieve fast calculation. For application scenarios such as parametric analysis, design optimization and digital twins, its high computational cost and long calculation cycle limit its actual engineering application. Summary of the Invention

[0003] To address the aforementioned technical issues, the present invention provides a method for reducing the order of the lower chamber model of a pressure vessel by combining sparse grids and POD. This method combines sparse grid theory with the POD method to construct a non-invasive reduced-order model of the lower chamber of a pressure vessel. The sparse grid theory addresses the issues of snapshot sample selection and POD coefficient fitting during the reduced-order model construction process. Based on the reduced-order model of the lower chamber, the field distribution of the thermal-hydraulic parameters within the lower chamber can be rapidly obtained. This method can be used in applications such as parametric analysis, design optimization, and digital twins, demonstrating practical engineering applications.

[0004] To achieve the above objectives, the present invention provides a method for reducing the order of the lower chamber model of a pressure vessel by combining sparse grids and POD, comprising:

[0005] After iterative initialization, the sparse grid level is increased by one;

[0006] Acquire test grid points based on the important grid points obtained by initialization; wherein the test grid points are grid points included in the set of non-important grid points removed;

[0007] executing a full-order computational fluid dynamics model of the lower chamber corresponding to the test grid point to obtain a first snapshot of the corresponding velocity field, temperature field, and pressure field;

[0008] Reconstructing a second snapshot of the velocity field, temperature field, and pressure field corresponding to the test grid point according to a mapping relationship between the grid point position and the POD coefficient and the POD modal matrix Z;

[0009] Calculate the root mean square error between the first snapshot and the second snapshot, and determine whether the root mean square error is less than a preset threshold. If so, complete the reduced-order model construction; otherwise, store the descendants of the grid points whose error meets the requirement into a set of non-significant grid points, up to the maximum sparse grid level; wherein the descendants include all direct descendants of the grid point; record the grid points whose error does not meet the requirement as significant grid points, and append the corresponding snapshots to the snapshot matrix;

[0010] Performing singular value decomposition on the snapshot matrix to obtain a left singular vector matrix, a diagonal singular value matrix, and a right singular vector matrix, and determining the POD modal matrix Z and the POD coefficient matrix C according to the singular value proportions;

[0011] Update the mapping relationship between the grid point position and the POD coefficient to determine whether the maximum sparse grid level is reached. If so, complete the reduced-order model construction; otherwise, continue to iterate and increase the sparse grid level by one.

[0012] Preferably, the iterative initialization process includes:

[0013] Determine the relationship between the number of sparse grid points m, the position of the grid points x, and the basis function a(x) at the grid points as a function of the grid level L;

[0014] Execute the full-order computational fluid dynamics model of the lower chamber corresponding to the level 1 grid point X1, and obtain the distribution data of the velocity field, temperature field, and pressure field inside the lower chamber as the initial snapshot matrix Y through the Reynolds-averaged Navier-Stokes equations;

[0015] Perform singular value decomposition on the initial snapshot matrix Y to obtain a left singular vector matrix U, a diagonal singular value matrix E, and a right singular vector matrix V, and determine the POD modal matrix Z and the POD coefficient matrix C according to the singular value proportions;

[0016] Determine the mapping relationship c=c(x) between the grid point position x and the POD coefficient matrix c, and initialize the important grid point Mx=X1.

[0017] Preferably, the formula expression of the change relationship includes:

[0018]

[0019] Preferably, for the case of a multi-dimensional parameter space, the formula expression of the change relationship can be expanded according to the tensor product.

[0020] Preferably, the formula expression of the Reynolds-averaged Navier-Stokes equations is:

[0021]

[0022] Preferably, the formula expression of the initial snapshot matrix Y is:

[0023]

[0024] Where y is a parameter among velocity, temperature and pressure, that is, the reduced-order model is constructed for different thermal hydraulic parameters, N C is the number of grids in the full-order computational fluid dynamics model, N S is the number of grid points X1.

[0025] Preferably, the process of determining the POD modal matrix Z and the POD coefficient matrix C according to the singular value proportion includes:

[0026] First, according to the formula Determine the number of POD modes r, where σ is the diagonal element of the singular value matrix E, Constants set by the user;

[0027] Then, take the first r columns of the left singular vector matrix U as the POD modal matrix Z;

[0028] Finally, through the formula C = Z T Y obtains the POD coefficient matrix C.

[0029] Preferably, the mapping relationship c=c(x) between the grid point position x and the POD coefficient c is expressed as follows:

[0030]

[0031] Where, The number of new grid points added for level i, is a grid point The value of the POD coefficient at The mapping relationship of level i-1 at the grid point The POD coefficient value at is a grid point The basis function at the position, in the case of multi-dimensionality, the basis function in the form of a tensor basis is used, c L (x) is the final mapping relationship.

[0032] Preferably, the process of obtaining the test grid points according to the important grid points obtained by initialization includes:

[0033] Obtain the test grid point Tx according to the important grid point Mx obtained by initialization: Tx = Ψ(Mx);

[0034] Where Ψ(x) is used to obtain the direct descendants {y} of the grid point x and is defined as:

[0035]

[0036] Where d is the dimension of the multidimensional parameter space, b(y) is the direct parent of point y, and is defined as the grid point closest to y among the newly added grid points in the previous grid level of y.

[0037] Preferably, the process of reconstructing the second snapshot of the velocity field, temperature field, and pressure field corresponding to the test grid point according to the mapping relationship between the grid point position and the POD coefficient and the POD modal matrix Z includes:

[0038] According to the mapping relationship c=c(x) between the grid point position and the POD coefficient, the POD coefficient column vector C corresponding to the test grid point Tx is obtained. Tx , then the POD coefficient column vector C Tx The second snapshot is obtained by multiplying the POD modal matrix Z on the left; the formula is: Z×C Tx .

[0039] Compared with the prior art, the present invention has the following advantages and technical effects:

[0040] This paper combines sparse grid theory and the POD method to construct a non-invasive reduced-order model for the lower chamber of a pressure vessel. The sparse grid theory addresses the issues of snapshot sample selection and POD coefficient fitting during the reduced-order model construction process. The reduced-order model of the lower chamber constructed using the method described in this invention can rapidly determine the field distribution of thermal-hydraulic parameters within the lower chamber through matrix algebraic operations. This model can be used in applications such as parametric analysis, design optimization, and digital twins, demonstrating practical engineering applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0042] Figure 1 Schematic diagram of a method flow in an embodiment of the present invention. DETAILED DESCRIPTION

[0043] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0044] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0045] like Figure 1As shown, this embodiment provides a method for reducing the order of the lower chamber model of a pressure vessel by combining sparse grids and POD, including the following steps:

[0046] S01: Iterative initialization, including:

[0047] S01-1: Determine the relationship between the number of sparse grid points m, the grid point location x, and the basis function a(x) at the grid point as a function of the grid level L, specifically:

[0048]

[0049] For the case of multi-dimensional parameter space, the above formula can be expanded according to the tensor product;

[0050] S01-2: Execute the full-order computational fluid dynamics model of the lower chamber corresponding to the level 1 grid point X1, specifically through the Reynolds-averaged Navier-Stokes equations:

[0051]

[0052] The distribution data of the velocity field, temperature field, and pressure field inside the lower chamber are obtained as the initial snapshot matrix Y:

[0053]

[0054] Where y is a parameter among velocity, temperature and pressure, that is, the reduced-order model is constructed for different thermal hydraulic parameters, N C is the number of grids in the full-order computational fluid dynamics model, N S is the number of grid points X1;

[0055] S01-3: Perform singular value decomposition on the initial snapshot matrix Y to obtain the left singular vector matrix U, the diagonal singular value matrix E, and the right singular vector matrix V. The POD modal matrix Z and the POD coefficient matrix C are determined based on the proportion of singular values. Specifically:

[0056] First, according to the formula Determine the number of POD modes r, where σ is the diagonal element of the singular value matrix E, is a constant set by the user, which is 0.995. Then, the first r columns of the left singular vector matrix U are taken as the POD modal matrix Z. Finally, the formula C = Z T Y obtains the POD coefficient matrix C;

[0057] S01-4: Determine the mapping relationship c=c(x) between the grid point position x and the POD coefficient matrix c. The specific formula is:

[0058]

[0059] Where, The number of new grid points added for level i, is a grid point The value of the POD coefficient at The mapping relationship of level i-1 at the grid point The POD coefficient value at is a grid point The basis function at the position, in the case of multi-dimensionality, the basis function in the form of a tensor basis is required, c L (x) is the final mapping relationship;

[0060] S01-5: Initialize important grid points Mx=X1;

[0061] S02: Sparse grid level plus one: L = L + 1;

[0062] S03: Obtain the test grid point Tx according to Mx: Tx = Ψ(Mx), where Ψ(x) is used to obtain the direct descendant {y} of the grid point x, which is defined as:

[0063]

[0064] Where d is the dimension of the multidimensional parameter space, b(y) is the direct parent of point y, defined as the grid point closest to y among the newly added grid points in the previous grid level. The test grid points Tx are the ones that are included in the set of non-significant grid points γ.

[0065] S04: Execute the full-order computational fluid dynamics model of the lower chamber corresponding to Tx to obtain the corresponding velocity field, temperature field and pressure field snapshot {y1};

[0066] S05: Reconstruct the velocity field, temperature field and pressure field snapshot {y2} corresponding to Tx according to the mapping relationship c=c(x) and the POD modal matrix Z. The specific steps are: Obtain the POD coefficient column vector C corresponding to the test grid point Tx according to the mapping relationship Tx , then multiply the POD modal matrix Z on the left, that is, Z×C Tx ;

[0067] S06: Calculate the root mean square error between {y1} and {y2};

[0068] S07: Determine whether all the errors corresponding to Tx obtained in S06 are less than 0.01. If so, complete the construction of the reduced-order model. Otherwise, execute S08.

[0069] S08: Store the descendants of the grid points whose errors meet the requirements into γ. The descendants include all direct descendants of the grid points up to the maximum sparse grid level.

[0070] S09: The grid points whose errors do not meet the requirements are recorded as important grid points Mx, and the corresponding snapshots are appended to the snapshot matrix Y;

[0071] S10: Perform singular value decomposition on Y to obtain the left singular vector matrix U, the diagonal singular value matrix E, and the right singular vector matrix V. Determine the POD modal matrix Z and the POD coefficient matrix C based on the proportion of singular values, as in S01-3;

[0072] S11: Update the mapping relationship c=c(x) between the grid point position x and the POD coefficient c, same as S01-4;

[0073] S12: Determine whether the maximum sparse grid level 8 is reached. If so, complete the reduced-order model construction; otherwise, execute S02.

[0074] The above are merely preferred embodiments of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in this application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. A pressure vessel lower chamber model reduction method combining sparse grid and POD, characterized in that: include: After iterative initialization, the sparse grid level is increased by one; Acquire test grid points based on the important grid points obtained by initialization; wherein the test grid points are grid points included in the set of non-important grid points removed; executing a full-order computational fluid dynamics model of the lower chamber corresponding to the test grid point to obtain a first snapshot of the corresponding velocity field, temperature field, and pressure field; Reconstructing a second snapshot of the velocity field, temperature field, and pressure field corresponding to the test grid point according to a mapping relationship between the grid point position and the POD coefficient and the POD modal matrix Z; Calculate the root mean square error between the first snapshot and the second snapshot, and determine whether the root mean square error is less than a preset threshold. If so, complete the reduced-order model construction; otherwise, store the descendants of the grid points whose error meets the requirement into a set of non-significant grid points, up to the maximum sparse grid level; wherein the descendants include all direct descendants of the grid point; record the grid points whose error does not meet the requirement as significant grid points, and append the corresponding snapshots to the snapshot matrix; Performing singular value decomposition on the snapshot matrix to obtain a left singular vector matrix, a diagonal singular value matrix, and a right singular vector matrix, and determining the POD modal matrix Z and the POD coefficient matrix C according to the singular value proportions; Update the mapping relationship between the grid point position and the POD coefficient to determine whether the maximum sparse grid level is reached. If so, complete the reduced-order model construction; otherwise, continue to iterate and increase the sparse grid level by one.

2. The method according to claim 1, characterized in that The iterative initialization process includes: Determine the relationship between the number of sparse grid points m, the position of the grid points x, and the basis function a(x) at the grid points as a function of the grid level L; Execute the full-order computational fluid dynamics model of the lower chamber corresponding to the level 1 grid point X1, and obtain the distribution data of the velocity field, temperature field, and pressure field inside the lower chamber as the initial snapshot matrix Y through the Reynolds-averaged Navier-Stokes equations; Perform singular value decomposition on the initial snapshot matrix Y to obtain a left singular vector matrix U, a diagonal singular value matrix E, and a right singular vector matrix V, and determine the POD modal matrix Z and the POD coefficient matrix C according to the singular value proportions; Determine the mapping relationship c=c(x) between the grid point position x and the POD coefficient matrix c, and initialize the important grid point Mx=X1.

3. The method according to claim 2, characterized in that The formula expression of the change relationship includes:

4. The method according to claim 2, characterized in that For the case of multi-dimensional parameter space, the formula expression of the change relationship can be expanded according to the tensor product.

5. The method according to claim 2, characterized in that The formula expression of the Reynolds-averaged Navier-Stokes equations is:

6. The method according to claim 2, characterized in that The formula expression of the initial snapshot matrix Y is: Where y is a parameter among velocity, temperature and pressure, that is, the reduced-order model is constructed for different thermal hydraulic parameters, N C is the number of grids in the full-order computational fluid dynamics model, N S is the number of grid points X1.

7. The method according to claim 2, characterized in that The process of determining the POD modal matrix Z and POD coefficient matrix C based on the proportion of singular values includes: First, according to the formula Determine the number of POD modes r, where σ is the diagonal element of the singular value matrix E, Constants set by the user; Then, take the first r columns of the left singular vector matrix U as the POD modal matrix Z; Finally, through the formula C = Z T Y obtains the POD coefficient matrix C.

8. The method according to claim 2, characterized in that The mapping relationship c=c(x) between the grid point position x and the POD coefficient c is expressed as follows: Where, The number of new grid points added for level i, is a grid point The value of the POD coefficient at The mapping relationship of level i-1 at the grid point The POD coefficient value at is the grid point The basis function at the position, in the case of multi-dimensionality, the basis function in the form of a tensor basis is used, c L (x) is the final mapping relationship.

9. The method according to claim 1, characterized in that The process of obtaining test grid points based on the important grid points obtained by initialization includes: Obtain the test grid point Tx according to the important grid point Mx obtained by initialization: Tx = Ψ(Mx); Where Ψ(x) is used to obtain the direct descendants {y} of the grid point x and is defined as: Where d is the dimension of the multidimensional parameter space, b(y) is the direct parent of point y, and is defined as the grid point closest to y among the newly added grid points in the previous grid level of y.

10. The method according to claim 1, characterized in that The process of reconstructing the second snapshot of the velocity field, temperature field, and pressure field corresponding to the test grid point according to the mapping relationship between the grid point position and the POD coefficient and the POD modal matrix Z includes: According to the mapping relationship c=c(x) between the grid point position and the POD coefficient, the POD coefficient column vector C corresponding to the test grid point Tx is obtained. Tx , then the POD coefficient column vector C Tx The second snapshot is obtained by multiplying the POD modal matrix Z on the left; the formula is: Z×C Tx .

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