A time super-resolution flow field prediction method, program, device and storage medium based on DMD and Schur-Pade combination

Abstract

This invention belongs to the field of flow field super-resolution prediction technology, specifically involving a time-based super-resolution flow field prediction method, program, device, and storage medium based on a combination of DMD and Schur-Pade methods. This invention is based on the fundamental idea of ​​treating the evolution of complex flow fields as a linear dynamic system using the DMD method. It combines this with the Schur-Pade method for solving real powers of a square matrix, employing four techniques—Schur decomposition, square root matrix decomposition, Pade approximation, and explicit correction of diagonal elements—to perform real-power operations on the flow field transfer matrix. Then, based on the flow field prediction formula obtained from the DMD method, it achieves super-resolution prediction of the flow field at any given time. This invention, through the combination of the DMD and Schur-Pade methods, forms a stable, efficient, and high-precision time-based super-resolution flow field prediction method, avoiding the problems of high randomness, long training time, and difficult parameter tuning associated with machine learning-based flow field prediction methods.

Description

Technical Field

[0001] This invention belongs to the field of flow field super-resolution prediction technology, specifically relating to a time super-resolution flow field prediction method, program, device and storage medium based on a combination of DMD and Schur-Pade. Background Technology

[0002] Currently, solving the Navier-Stokes (NS) equations using computational fluid dynamics (CFD) methods to achieve high-fidelity numerical simulations of flow fields has become an important tool for analyzing complex fluid flow phenomena. However, even the Reynolds-averaged NS method requires a very high computational cost to obtain flow field results with high spatiotemporal resolution. Therefore, researchers have been actively exploring simpler and more efficient methods to obtain high-fidelity flow field data.

[0003] Super-resolution flow field prediction is one solution to this problem. It can transform low-resolution CFD data into high-resolution data while significantly reducing computational costs and time. With the development of big data technologies such as cloud computing and machine learning, data-driven methods have shown great potential in this field. Flow field reduction models, as one of the main data-driven methods, can achieve rapid capture and prediction of flow field information while maintaining accuracy.

[0004] Existing research indicates that reduced-order models such as Orthogonal Eigenvalue Decomposition (POD) and Dynamic Mode Decomposition (DMD) can conveniently achieve flow field prediction at the same resolution. However, current methods rarely employ reduced-order models like POD and DMD for super-resolution flow field prediction from low to high resolution; instead, they primarily rely on deep learning models, such as U-net networks and diffusion models. These methods excel at generating high-resolution flow field information from low-resolution input, accurately reproducing flow field details. However, deep learning models like neural networks often exhibit relative error-level randomness during training, posing challenges to model stability and reliability. Furthermore, obtaining deep learning models with smaller errors often requires a large amount of data samples and training time, resulting in significant costs and hindering practical applications in various engineering fields, making it less suitable for flow mechanism analysis and hydrodynamic performance optimization of structures. Therefore, it is necessary to simplify the evolution of nonlinear complex flow fields by referencing the linearization concept of dynamic systems in DMD, and propose an efficient and stable time-based super-resolution flow field prediction method without employing neural network algorithms. Summary of the Invention

[0005] The purpose of this invention is to provide a time super-resolution flow field prediction method, program, device and storage medium based on the combination of DMD and Schur-Pade, which can realize flow field prediction at any time by using flow field information at typical times within a time interval.

[0006] A temporal super-resolution flow field prediction method based on a combination of DMD and Schur-Pade includes the following steps:

[0007] Step 1: Obtain the initial flow field data vector x1 and the time t to be predicted for the target, and obtain the flow field data of the target within the time interval T through numerical simulation.

[0008] Step 2: Divide the time interval T evenly to obtain M typical moments, and establish the correlation matrix Y = [x1, x2, ..., x] for the flow field data at all typical moments. M ], element x j This represents the flow field data vector of the target at the j-th typical time, where the first typical time is the initial time.

[0009] Step 3: Use the DMD method to obtain the transfer matrix A corresponding to matrix Y;

[0010] Step 4: Perform Schur decomposition on the transfer matrix A to obtain the decomposed matrices T and Q;

[0011] Step 5: Initialize the number of iterations k = 1, and set the convergence parameter θ;

[0012] Step 6: Perform k square root operations on matrix T to obtain the square root matrix. Calculate the residual matrix X and its F-norm ||X|| F , I represents the identity matrix;

[0013] Step 7: If ||X|| F If ≤θ, then output the residual matrix X and the current iteration number k; otherwise, let k = k + 1 and return to step 6.

[0014] Step 8: Set the Pade approximation parameters m and α, and obtain (IX) based on the Pade approximation. α Approximate matrix r m (X);

[0015] Step 9: Approximate matrix r m (X) is squared k times to obtain matrix V, and the elements of matrix V are further modified to obtain matrix W;

[0016] Step 10: Calculate the super-resolution prediction result x of the flow field at the target time t. t_pre =QWQ * x1.

[0017] Furthermore, in step 3, the transfer matrix A corresponding to matrix Y is obtained using the DMD method, specifically as follows:

[0018] Based on the correlation matrix Y, flow field data from the first typical time point to the (M-1)th typical time point are selected and denoted as X1=[x1,x2,…,x M-1 Select the flow field data from the second typical time point to the Mth typical time point, denoted as X2=[x2,x3,…x M Based on the DMD method, X1 and X2 approximately satisfy a linear transformation relationship, i.e., X2 = AX1. The transfer matrix A is obtained by solving for the generalized inverse of the matrix, i.e. This represents the generalized inverse of X1.

[0019] Furthermore, in step 4, the transfer matrix A is decomposed using Schur decomposition, specifically as follows:

[0020] A = QTQ *

[0021] Where Q is a unitary matrix; Q * Let Q be the conjugate transpose of Q; T is an upper triangular matrix, and its main diagonal elements are the eigenvalues ​​of A.

[0022] Furthermore, in step 6, the matrix T is subjected to the kth square root operation, specifically as follows:

[0023] For an upper triangular matrix T, the square root matrix T 1 / 2 =U is also an upper triangular matrix, and the diagonal elements u of U are... ii satisfy Pick The off-diagonal element of U, u ij Satisfy U 2 =T, then we have:

[0024]

[0025] When u ii +u ij When not equal to 0, there is

[0026] When u ii +u ij When it equals 0, u is undetermined. ij and with ||U 2 -T|| F The objective is to minimize the value of T, and an approximate solution is obtained based on an optimization algorithm. The method for solving the square root matrix is ​​repeated k times to finally obtain the 2^T value of T. k Root matrix

[0027] Furthermore, the method for setting parameter α in step 8 is as follows:

[0028]

[0029] in,

[0030] Furthermore, in step 8, (IX) is obtained based on Pade approximation. α Approximate matrix r m (X), specifically:

[0031] r m (X)=I+c1X[I+c2X[…I+c 2m-1 X[I+c 2m X] -1 …] -1 ] -1

[0032] in,

[0033] Furthermore, in step 9, the elements of matrix V are further modified to obtain matrix W, specifically as follows:

[0034] Replace the elements on the main diagonal and the first upper diagonal of matrix V. The elements on the main diagonal of the resulting matrix W are:

[0035]

[0036] The elements on the first upper diagonal of matrix W are:

[0037]

[0038] Where i and j satisfy i = 1, 2, ..., n-1 and j = i + 1;

[0039] Matrix W is:

[0040]

[0041] Among them, v ij Let t be the element in the i-th row and j-th column of matrix V. ij Let be the element in the i-th row and j-th column of matrix T.

[0042] A computer device / apparatus / system includes a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the above-described time super-resolution flow field prediction method based on the combination of DMD and Schur-Pade.

[0043] A computer-readable storage medium having a computer program / instructions stored thereon, which, when executed by a processor, implements the steps of the above-described time super-resolution flow field prediction method based on the combination of DMD and Schur-Pade.

[0044] A computer program product includes a computer program / instructions that, when executed by a processor, implement the steps of the aforementioned time-super-resolution flow field prediction method based on a combination of DMD and Schur-Pade.

[0045] The beneficial effects of this invention are as follows:

[0046] This invention is based on the fundamental idea of ​​treating the evolution of complex flow fields as a linear dynamic system using the DMD method. It combines this with the Schur-Pade method for solving real powers of a square matrix, employing four techniques—Schur decomposition, square root matrix decomposition, Pade approximation, and explicit correction of diagonal elements—to perform real power operations on the flow field transfer matrix. Then, based on the flow field prediction formula obtained from the DMD method, it achieves super-resolution prediction of the flow field at any given time. This invention, by combining the DMD and Schur-Pade methods, forms a stable, efficient, and high-precision time-based super-resolution flow field prediction method, avoiding the problems of high randomness, long training time, and difficult parameter tuning inherent in machine learning-based flow field prediction methods. Attached Figure Description

[0047] Figure 1 This is a flowchart of the present invention.

[0048] Figure 2 This is a CFD simulation wave field cloud diagram of a trimaran hull at a typical moment in an embodiment of the present invention.

[0049] Figure 3 This is a transfer matrix cloud diagram of the wave field evolution of a trimaran hull in an embodiment of the present invention.

[0050] Figure 4 This is a time-super-resolution wave field cloud map of a trimaran hull at the time to be predicted, as shown in an embodiment of the present invention.

[0051] Figure 5 This is a CFD simulation wave field cloud map of a trimaran hull at the predicted time in an embodiment of the present invention.

[0052] Figure 6 This is an example of an error cloud map between the time super-resolution predicted wave field and the CFD simulated wave field at the predicted time for a trimaran hull in an embodiment of the present invention. Detailed Implementation

[0053] The present invention will now be further described with reference to the accompanying drawings.

[0054] This invention proposes a time-based super-resolution flow field prediction method based on a combination of DMD and Schur-Pade. The invention utilizes the fundamental idea of ​​treating complex flow fields as linear dynamic systems based on the DMD method, and combines this with the Schur-Pade method, which solves for real powers of a square matrix, to achieve super-resolution prediction of the flow field at any given time.

[0055] A temporal super-resolution flow field prediction method based on a combination of DMD and Schur-Pade includes the following steps:

[0056] Step 1: Obtain the initial flow field data vector x1 and the time t to be predicted for the research object (e.g., ship, airfoil, etc.), and obtain the flow field data of the target within the time interval T through numerical simulation.

[0057] Step 2: Divide the time interval T evenly to obtain M typical moments, and establish the correlation matrix Y = [x1, x2, ..., x] for the flow field data at all typical moments. M ], element x j (j=1,2,…,M) represents the flow field data vector of the target at the j-th typical time, where the first typical time is the initial time;

[0058] Step 3: Use the DMD method to obtain the transfer matrix A corresponding to matrix Y;

[0059] Based on the correlation matrix Y, flow field data from the first typical time point to the (M-1)th typical time point are selected and denoted as X1=[x1,x2,…,x M-1 Select the flow field data from the second typical time point to the Mth typical time point, denoted as X2=[x2,x3,…x M Based on the DMD method, X1 and X2 approximately satisfy a linear transformation relationship, i.e., X2 = AX1, where the matrix A corresponding to the linear transformation is called the transfer matrix, which can be obtained by solving the generalized inverse of the matrix. This represents the generalized inverse of X1.

[0060] Step 4: Perform Schur decomposition on the transfer matrix A to obtain the decomposed matrices T and Q;

[0061] A = QTQ *

[0062] Where Q is a unitary matrix, Q * Let Q be the conjugate transpose of Q, and T be an upper triangular matrix whose main diagonal elements are the eigenvalues ​​of A.

[0063] Step 5: Initialize the number of iterations k = 1, and set the convergence parameter θ;

[0064] Step 6: Perform k square root operations on matrix T to obtain the square root matrix. Calculate the residual matrix X and its F-norm ||X|| F , I represents the identity matrix;

[0065] For an upper triangular matrix T, the square root matrix T 1 / 2 =U is also an upper triangular matrix, and the diagonal elements u of U are... ii satisfy Pick The off-diagonal element of U, u ij Satisfy U 2 =T, then we have:

[0066]

[0067] When u ii +u ij When not equal to 0, there is

[0068] When u ii +u ij When it equals 0, u is undetermined. ij and with ||U 2 -T|| F The goal is to minimize the value, and an approximate solution is obtained based on an optimization algorithm.

[0069] Repeating the above method for solving the square root matrix k times will eventually yield the 2^T of T. k Root matrix Finally, through the formula Obtain the residual matrix X, and calculate the F-norm ||X|| of the residual matrix X. F .

[0070] Step 7: If ||X|| F If ≤θ, then output the residual matrix X and the current iteration number k; otherwise, let k = k + 1 and return to step 6.

[0071] Step 8: Set the Pade approximation parameters m and α, and obtain (IX) based on the Pade approximation. α Approximate matrix r m (X);

[0072] The method for setting α is as follows:

[0073]

[0074] in,

[0075] The Pade approximation method is as follows:

[0076] For the function (1-x) αAccording to the Pade approximation method, when the value of |x| is small, r can be used. m (x) versus (1-x) α To approximate r m The specific form of (x) is:

[0077]

[0078] in j = 1, 2, ..., m.

[0079] Extending the above approximation method to matrices, that is, for matrix functions (IX) α When ||X|| F When the value is small, r can be used. m (X) versus (IX) α An approximate representation can be given as follows:

[0080] r m (X)=I+c1X[I+c2X[…I+c 2m-1 X[I+c 2m X] -1 …] -1 ] -1

[0081] Step 9: Approximate matrix r m (X) is squared k times to obtain matrix V, and the elements of matrix V are further modified to obtain matrix W;

[0082] For the exponentiation of an upper triangular matrix, since the elements on its diagonal and the elements on the first upper diagonal can have explicit expressions, in order to reduce the error caused by the Pade approximation, the elements on the main diagonal and the first upper diagonal of matrix V can be replaced, and the matrix after replacement is denoted as W.

[0083] The elements on the main diagonal of matrix W can be represented as:

[0084]

[0085] Where w ii (i = 1, 2, ..., n) are the main diagonal elements of matrix W, t ii (j=1,2,…,n) are the main diagonal elements of matrix T.

[0086] The elements on the first upper diagonal of matrix W can be represented as:

[0087]

[0088] Where w ij Let t be the element in the i-th row and j-th column of matrix W.ij Let i be the element in the i-th row and j-th column of matrix T, and let i and j satisfy i = 1, 2, ..., n-1 and j = i + 1.

[0089] Thus, the replaced matrix W can be obtained as follows:

[0090]

[0091] Among them, v ij Let be the element in the i-th row and j-th column of matrix V.

[0092] Step 10: Calculate the super-resolution prediction result x of the flow field at the target time t. t_pre =QWQ * x1. Where x t_pre This represents the flow field data vector obtained at the predicted time t.

[0093] This invention consists of a DMD calculation module for solving the transfer matrix, a Schur decomposition module for obtaining the upper triangular matrix, a square root calculation module for obtaining the residual matrix, a Pade approximation module for obtaining the approximate matrix, and a correction module for replacing the diagonal elements of matrix exponentiation.

[0094] Example 1:

[0095] In one specific embodiment of the present invention,

[0096] A trimaran hull type was selected as the sample hull, and the wave field around the trimaran was calculated using CFD numerical simulation (the main parameters are: incident wave is a first-order Stokes wave, amplitude is 0.02 m, wavelength is 4 m, waterline length of the hull is 4 m, and ship speed is 1.88 m / s). For a time interval T = 0.8 s, nine typical time points were evenly divided, and the flow field information for each typical time point is as follows: Figure 2 As shown, a correlation matrix for the flow field data at all typical moments is established.

[0097] The time interval Δt = 0.1s can be obtained. Table 1 shows the time t to be predicted and the corresponding calculated parameter α. All the times to be predicted are outside the typical times. The number of iterations is initialized to k = 1, and the convergence parameter θ = 0.01 and the Pade approximation parameter m = 7 are set.

[0098] Table 1. Predicted time t and corresponding parameter α

[0099] t 0.01s 0.02s 0.03s 0.04s 0.05s 0.06s 0.07s 0.08s 0.09s α 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

[0100] Based on the correlation matrix of the flow field data at all typical time points, the transfer matrix A (dimension 19950×19950) of the flow field data at different time points is obtained using the DMD method. The transfer matrix contour plot is shown below. Figure 3 As shown.

[0101] Performing Schur decomposition on the transfer matrix A yields decomposed matrices T and Q. Through continuous iteration and checking for convergence criteria, it is found that when k=3, the square root matrix obtained is T. 1 / 8 At this point, the F-norm ||X|| obtained by calculating the residual matrix X F Satisfying the convergence criterion ||X|| F ≤0.01.

[0102] Based on the parameter α in Table 1 and the obtained final residual matrix X, (IX) is obtained using Pade approximation. α The approximate matrix r7(X) can be expressed in the following form:

[0103] r7(X)=I+c1X[I+c2X[…I+c 13 X[I+c 14 X] -1 …] -1 ] -1

[0104] in j = 1, 2, ..., 7.

[0105] The approximate matrix r7(X) is squared three times to obtain matrix V, and the elements of matrix V are further modified to obtain matrix W. The initial flow field data vector x1 is obtained from the correlation matrix Y, and the calculated matrix Q is used to calculate the flow field data vector x1 using the formula x1. t_pre =QWQ * x1 obtains the super-resolution prediction results of the flow field at time t, such as Figure 4 As shown.

[0106] from Figure 5 and Figure 6 It can be seen that the predicted wave field of the trimaran at any given time (all outside the typical timeframe) obtained by this system is basically consistent with the simulation results obtained through further CFD calculations, indicating that the overall prediction error is small. Therefore, the computational and storage costs of numerical simulation can be greatly reduced, enabling time-resolution prediction of the flow field of the trimaran moving in waves.

[0107] This invention considers the flow field information of the research object at M typical moments uniformly distributed within a time interval T, and realizes time super-resolution prediction of the flow field at any moment. Based on the DMD method, it uses Schur decomposition, square root matrix solving, Pade approximation, and diagonal element correction to calculate the real power of the transfer matrix, resulting in a stable, efficient, and high-precision flow field time super-resolution prediction method. This avoids the problems of strong randomness, long training time, and difficult parameter tuning caused by machine learning-based flow field prediction methods.

[0108] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A time-based super-resolution flow field prediction method based on a combination of DMD and Schur-Pade, characterized in that, Includes the following steps: Step 1: Obtain the initial flow field data vector x1 and the time t to be predicted for the target, and obtain the flow field data of the target within the time interval T through numerical simulation. Step 2: Divide the time interval T evenly to obtain M typical moments, and establish the correlation matrix Y = [x1, x2, ..., x] for the flow field data at all typical moments. M ], element x j This represents the flow field data vector of the target at the j-th typical time, where the first typical time is the initial time. Step 3: Use the DMD method to obtain the transfer matrix A corresponding to matrix Y; Based on the correlation matrix Y, flow field data from the first typical time point to the (M-1)th typical time point are selected and denoted as X1=[x1,x2,…,x M-1 Select the flow field data from the second typical time point to the Mth typical time point, denoted as X2=[x2,x3,…x M Based on the DMD method, X1 and X2 approximately satisfy a linear transformation relationship, i.e., X2 = AX1. The transfer matrix A is obtained by solving for the generalized inverse of the matrix, i.e. Represents the generalized inverse of X1; Step 4: Perform Schur decomposition on the transfer matrix A to obtain the decomposed matrices T and Q; A=QTQ * Where Q is a unitary matrix; Q * Let Q be the conjugate transpose of Q; T is an upper triangular matrix, and its main diagonal elements are the eigenvalues ​​of A; Step 5: Initialize the number of iterations k = 1, and set the convergence parameter θ; Step 6: Perform k square root operations on matrix T to obtain the square root matrix. Calculate the residual matrix X and its F-norm ||X|| F , I represents the identity matrix; Step 7: If ||X|| F If ≤θ, then output the residual matrix X and the current iteration number k; otherwise, let k = k + 1 and return to step 6. Step 8: Set the Pade approximation parameters m and α, and obtain (IX) based on the Pade approximation. α Approximate matrix r m (X); Step 9: Approximate matrix r m (X) is squared k times to obtain matrix V, and the elements of matrix V are further modified to obtain matrix W; Step 10: Calculate the super-resolution prediction result x of the flow field at the target time t. t_pre =QWQ * x1.

2. The time-super-resolution flow field prediction method based on the combination of DMD and Schur-Pade as described in claim 1, characterized in that: In step 6, the matrix T is subjected to k square root operations, specifically as follows: For an upper triangular matrix T, the square root matrix T 1 / 2 =U is also an upper triangular matrix, and the diagonal elements u of U are... ii satisfy Pick The off-diagonal element of U, u ij Satisfy U 2 =T, then we have: When u ii +u ij When not equal to 0, there is When u ii +u ij When u equals 0, it is undetermined. ij and with ||U 2 -T|| F The objective is to minimize the value, and an approximate solution is found based on an optimization algorithm. Repeat the above method for solving the square root matrix k times to finally obtain the 2^T of T. k Root matrix 3. The time-super-resolution flow field prediction method based on the combination of DMD and Schur-Pade as described in claim 1, characterized in that: The method for setting parameter α in step 8 is as follows: in, 4. The time-super-resolution flow field prediction method based on the combination of DMD and Schur-Pade as described in claim 1, characterized in that: In step 8, (IX) is obtained based on the Pade approximation. α Approximate matrix r m (X), specifically: r m (X)=I+c1X[I+c2X[…I+c 2m-1 X[I+c 2m X] -1 …] -1 ] -1 in, 5. The time-super-resolution flow field prediction method based on the combination of DMD and Schur-Pade as described in claim 1, characterized in that: In step 9, the elements of matrix V are further modified to obtain matrix W, specifically as follows: Replace the elements on the main diagonal and the first upper diagonal of matrix V. The elements on the main diagonal of the resulting matrix W are: The elements on the first upper diagonal of matrix W are: Where i and j satisfy i = 1, 2, ..., n-1 and j = i + 1; Matrix W is: Among them, v ij Let t be the element in the i-th row and j-th column of matrix V. ij Let be the element in the i-th row and j-th column of matrix T.

6. A computer device, comprising a memory, a processor, and a computer program stored in the memory, characterized in that: The processor executes the computer program to implement the steps of the method according to any one of claims 1 to 5.

7. A computer-readable storage medium having a computer program stored thereon, characterized in that: When executed by a processor, the computer program implements the steps of the method according to any one of claims 1 to 5.

8. A computer program product comprising computer instructions, characterized in that: When executed by a processor, the computer instructions implement the steps of the method according to any one of claims 1 to 5.

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