Time super-resolution flow field forecasting method based on combination of DMD and Schur-Pade, program, equipment and storage medium

Abstract

The invention belongs to the technical field of flow field super-resolution forecasting, and particularly relates to a time super-resolution flow field forecasting method, program and device based on combination of a DMD and Schur-Pade and a storage medium. According to the method, on the basis of a DMD method, complex flow field evolution is regarded as a basic thought of a linear power system, a Schur-Pade method for solving square matrix real-number power is combined, and real-number power operation of a flow field transfer matrix is achieved by means of four technical means of Schur decomposition, square root matrix decomposition, Pade approximation and diagonal element explicit correction. And realizing super-resolution forecasting of the flow field at any moment according to a flow field forecasting formula obtained by a DMD (Digital Micromirror Device) method. The DMD method and the Schur-Pade method are combined to form a stable, efficient and high-precision time super-resolution flow field forecasting method, and the problems of high randomness, long training time, difficulty in parameter tuning and the like caused by a flow field forecasting method based on machine learning are solved.

Description

Technical Field

[0001] The present invention belongs to the technical field of flow field super-resolution prediction, and particularly relates to a time super-resolution flow field prediction method, program, device and storage medium based on the combination of DMD and Schur-Pade. Background Technique

[0002] Currently, based on the computational fluid dynamics (CFD) method to solve the Navier-Stokes (NS) equations and achieve high-fidelity numerical simulation of the flow field has become an important means to analyze complex fluid flow phenomena. However, to obtain flow field results with high spatio-temporal resolution, even the computational cost required by the Reynolds-averaged NS method is extremely high. Therefore, researchers have been actively exploring simpler and more effective methods to obtain high-fidelity flow field data.

[0003] Flow field super-resolution prediction is one of the solutions to this problem. It can convert low-resolution CFD data into high-resolution data, while significantly reducing the computational cost 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. As one of the main data-driven methods, the reduced-order model can quickly capture and predict flow field information while ensuring accuracy.

[0004] According to the investigation of existing research, it is found that the reduced-order models such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) can more conveniently achieve flow field prediction at the same resolution; however, currently, POD, DMD and other reduced-order models are less used to achieve flow field super-resolution prediction from low resolution to high resolution, but mainly based on deep learning models, such as U-net network, diffusion model, etc. These methods perform well in generating high-resolution flow field information from low-resolution inputs and can relatively accurately restore flow field details. However, in the training process of deep learning models such as neural networks, there is often randomness at the relative error level, which poses certain challenges to the stability and reliability of the model. In addition, to obtain a deep learning model with small errors often requires a large number of data samples and training time, with extremely high costs, and it is not convenient for practical applications in various engineering fields to more conveniently conduct flow mechanism analysis and optimize the hydrodynamic performance design of structures, etc. To sum up, it is necessary to simplify the evolution of the non-linear complex flow field by referring to the linearization idea of the dynamic system of DMD, etc., and propose an efficient and stable time super-resolution flow field prediction method without using neural network algorithms. Summary of the Invention

[0005] The purpose of the present 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 the flow field prediction at any time through the flow field information at typical moments within a time interval.

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

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

[0008] Step 2: Uniformly divide the time interval T to obtain M typical moments, and establish a correlation matrix Y = [x1, x2,..., x M of the flow field data at all typical moments, where the element x j represents the flow field data vector of the target at the jth typical moment, and the first typical moment is the initial moment;

[0009] Step 3: Use the DMD method to obtain the transfer matrix A corresponding to the 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 iteration number k = 1 and set the convergence parameter θ;

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

[0013] Step 7: If ||X|| F ≤θ, then output the calculated 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 the approximate matrix r α of (I - X) m (X) based on Pade approximation;

[0015] Step 9: Perform k square calculations on the approximate matrix r m (X) to obtain the matrix V, and further correct the elements of the matrix V to obtain the matrix W;

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

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

[0018] According to the correlation matrix Y, the flow field data from the 1st typical moment to the (M - 1)th typical moment are selected and denoted as X1 = [x1, x2, …, x M-1 , and the flow field data from the 2nd typical moment to the Mth typical moment are selected and denoted as X2 = [x2, x3, …, x M ; Based on the DMD method, X1 and X2 approximately satisfy a linear transformation relationship, that is, X2 = AX1, and the transfer matrix A is obtained by solving the generalized inverse of the matrix, that is denotes the generalized inverse of X1.

[0019] Further, in step 4, the Schur decomposition of the transfer matrix A is performed, specifically as follows:

[0020] A = QTQ *

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

[0022] Further, in step 6, the k - th square root operation is performed on matrix T, specifically as follows:

[0023] For the upper triangular matrix T, the square root matrix T 1 / 2 = U is also an upper triangular matrix, and the diagonal elements u ii satisfy Take The off - diagonal elements u ij of U satisfy U 2 = T, then there is:

[0024]

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

[0026] When u ii + u ij is equal to 0, u ij is to be determined, and with ||U 2 - T|| F minimized as the goal, approximate solution is carried out based on the optimization algorithm; The above method for solving the square root matrix is repeated k times, and finally the 2 k - th root matrix of T

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

[0028]

[0029] Wherein,

[0030] Further, the approximate matrix r α (X) of (I - X) obtained based on Pade approximation in step 8 is specifically: m (X) = I + c1X[I + c2X[…I + c

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

[0032] Wherein,

[0033] Further, in step 9, the elements of matrix V are further corrected to obtain matrix W, specifically:

[0034] The elements on the main diagonal and the first superdiagonal of matrix V are replaced. The elements on the main diagonal in the replaced matrix W are:

[0035]

[0036] The elements on the first superdiagonal in matrix W are:

[0037]

[0038] Wherein, i and j satisfy i = 1, 2, …, n - 1 and j = i + 1;

[0039] Matrix W is:

[0040]

[0041] Wherein, v ij is the element in the i-th row and j-th column of matrix V, and t ij is the element in the i-th row and j-th column of matrix T.

[0042] A computer device / system, including 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 time super-resolution flow field prediction method based on the combination of DMD and Schur-Pade.

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

[0044] A computer program product, including a computer program / instructions, and when the computer program / instructions are executed by a processor, the steps of the above-mentioned time super-resolution flow field prediction method based on the combination of DMD and Schur-Pade are implemented.

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

[0046] Based on the basic idea of regarding the complex flow field evolution as a linear dynamic system by the DMD method, the present invention combines the Schur-Pade method for solving the real power of a square matrix, and realizes the real power operation of the flow field transfer matrix by means of four technical means: Schur decomposition, square root matrix decomposition, Pade approximation, and explicit correction of diagonal elements. Furthermore, according to the flow field prediction formula obtained by the DMD method, the super-resolution prediction of the flow field at any time is realized. By combining the DMD method with the Schur-Pade method, the present invention forms a stable, efficient, and high-precision time super-resolution flow field prediction method, avoiding the problems of strong randomness, long training time, and difficult parameter tuning brought by the flow field prediction method based on machine learning. Description of the Drawings

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

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

[0049] Figure 3 It is a transfer matrix cloud map of the wave field evolution of a trimaran hull form in the embodiment of the present invention.

[0050] Figure 4 It is a time super-resolution prediction wave field cloud map at a moment to be predicted of a trimaran hull form in the embodiment of the present invention.

[0051] Figure 5 It is a CFD simulation wave field cloud map at a moment to be predicted of a trimaran hull form in the embodiment of the present invention.

[0052] Figure 6 It is an error cloud map between the time super-resolution prediction wave field and the CFD simulation wave field at a moment to be predicted of a trimaran hull form in the embodiment of the present invention. Detailed Embodiment

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

[0054] The present invention designs a time super-resolution flow field prediction method based on the combination of DMD and Schur-Pade. Based on the basic idea of regarding a complex flow field as a linear dynamic system by the DMD method and combining with the Schur-Pade method for solving the real power of a square matrix, the super-resolution prediction of the flow field at any time is realized.

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

[0056] Step 1: Obtain the flow field data vector x1 at the initial moment of the research object (such as a ship, airfoil, etc.) and the moment t to be predicted, and obtain the flow field data of the target within the time interval T through numerical simulation methods;

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

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

[0059] According to the correlation matrix Y, select the flow field data from the 1st typical moment to the (M - 1)-th typical moment, denoted as X1 = [x1, x2,..., x M-1 , and select the flow field data from the 2nd typical moment to the M-th typical moment, denoted as X2 = [x2, x3,...x M ; Based on the DMD method, X1 and X2 approximately satisfy the linear transformation relationship, that is, X2 = AX1, where the linear transformation corresponding matrix A is called the transfer matrix and can be obtained by solving the generalized inverse of the matrix, that is 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 * is the conjugate transpose matrix of Q, and T is an upper triangular matrix and its main diagonal elements are the eigenvalues of A.

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

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

[0065] For the upper triangular matrix T, the square root matrix T 1 / 2 = U is also an upper triangular matrix, and the diagonal element u of U ii satisfies Take The off-diagonal element u of U ij satisfies U 2 = T, then there is:

[0066]

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

[0068] When u ii + u ij is equal to 0, determine u ij , and take ||U 2 - T|| F as the minimum target, and perform approximate solution based on the optimization algorithm.

[0069] Repeat the above method for solving the square root matrix k times, and finally the 2 k th root matrix of T can be obtained Finally, obtain the residual matrix X through the formula and calculate the Frobenius norm ||X|| of the residual matrix X F .

[0070] Step 7: If ||X|| F ≤ 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 the approximate matrix r α of (I - X) m (X);

[0072] The setting method of α is:

[0073]

[0074] Among them,

[0075] The Pade approximation method is as follows:

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

[0077]

[0078] where j = 1, 2, …, m.

[0079] Extend the above approximation method to matrices, that is, for the matrix function (I - X) α , when the value of ||X|| F is small, r m (X) can be used to approximate (I - X) α , and the specific form can be expressed as:

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

[0081] Step 9: Perform k - th power calculation on the approximate matrix r m (X) to obtain the matrix V, and further correct the elements of the matrix V to obtain the matrix W;

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

[0083] For the elements on the main diagonal of the matrix W, it can be expressed as:

[0084]

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

[0086] For the elements of the first super - diagonal of the matrix W, it can be expressed as:

[0087]

[0088] where w ij is the element in the i - th row and j - th column of the matrix W, and t​ij is the element in the \(i\)-th row and \(j\)-th column of matrix \(T\), and \(i\) and \(j\) satisfy \(i = 1, 2, \cdots, n - 1\) and \(j = i + 1\).

[0089] In this way, the replaced matrix \(W\) can be obtained as

[0090]

[0091] where \(v\) ij is 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 prediction time \(t\) t_pre \(= QWQ\) * \(x_1\). Where \(x\) t_pre represents the flow field data vector at time \(t\) obtained by prediction.

[0093] The present 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 the matrix power.

[0094] Example 1:

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

[0096] Select a certain trimaran hull form as the sample hull form, and calculate the surrounding wave field of the trimaran through CFD numerical simulation (where the main parameters are: the incident wave is a first-order Stokes wave, the wave amplitude is 0.02 m, the wavelength is 4 m, the waterline length of the ship's main body is 4 m, and the ship speed is 1.88 m / s). For a time interval \(T = 0.8\ s\), 9 typical moments are evenly divided, and the flow field information at each typical moment is as Figure 2 shown, and a correlation matrix regarding the flow field data at all typical moments is established.

[0097] The time interval \(\Delta t = 0.1\ s\) can be obtained. Table 1 shows the prediction time \(t\) and the corresponding calculated parameter \(\alpha\), where each prediction time is outside the typical moments; initialize the iteration number \(k = 1\), and set the convergence parameter \(\theta = 0.01\) and the Pade approximation parameter \(m = 7\).

[0098] Table 1 Prediction time \(t\) and corresponding parameter \(\alpha\)

[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 moments, use the DMD method to obtain the transfer matrix \(A\) (dimension \(19950\times19950\)) of the flow field data at different moments. The transfer matrix contour plot is asFigure 3 as shown

[0101] Perform Schur decomposition on the transfer matrix A to obtain the decomposed matrices T and Q. By continuously iterating and judging whether the convergence criterion is satisfied, it is found that when k = 3, the square root matrix obtained is T 1 / 8 , and at this time, the F-norm ||X|| of the residual matrix X is calculated F satisfies the convergence criterion ||X|| F ≤0.01

[0102] Based on the parameters α in Table 1 and the obtained final residual matrix X, an approximate matrix r7(X) of (I - X) is obtained based on Pade approximation. The specific form of r7(X) can be expressed as: α r7(X) = I + c1X[I + c2X[…I + c

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

[0104] where j = 1, 2, …, 7

[0105] Perform 3 square calculations on the approximate matrix r7(X) to obtain the matrix V, and further correct the elements of the matrix V to obtain the matrix W. Obtain the initial moment flow field data vector x1 according to the relevant matrix Y, and according to the calculated matrix Q, use the formula x t_pre = QWQ * x1 to obtain the time super-resolution prediction result of the flow field at time t, as Figure 4 shown

[0106] From Figure 5 and Figure 6 it can be seen that by this system, the predicted wave field of the trimaran at any moment (all outside the typical moments) is obtained. Compared with the simulation results further calculated by CFD, the two are basically the same, that is, the overall prediction error is small. Therefore, the numerical simulation calculation and storage costs can be greatly reduced, and the time super-resolution prediction of the flow field of the trimaran in waves can be realized

[0107] ​The present invention considers the flow field information at M typical moments uniformly distributed within a time interval T of the research object, and realizes the time super-resolution prediction of the flow field at any moment; on the basis of the DMD method, technical means such as Schur decomposition, square root matrix solution, Pade approximation, and diagonal element correction are adopted to realize the calculation of the real power of the transfer matrix, and a stable, efficient, and high-precision flow field time super-resolution prediction method is obtained, avoiding the problems of strong randomness, long training time, and difficult parameter tuning brought by the flow field prediction method based on machine learning.

[0108] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A temporal super-resolution flow field prediction method based on the combination of DMD and Schur-Pade, characterized in that: The following steps are involved: Step 1: Obtain the flow field data vector x1 at the initial time of the target and the time to be predicted t, and obtain the flow field data of the target in the time interval T through the numerical simulation method; Step 2: Divide the time interval T evenly to obtain M typical moments, and establish the correlation matrix Y = [x1, x2, ..., x M ], element x j represents the flow field data vector of the target at the jth typical moment, and the first typical moment is the initial moment; Step 3: Use the DMD method to obtain the transfer matrix A corresponding to the matrix Y; Step 4: Perform Schur decomposition on the transfer matrix A to obtain the decomposed matrices T and Q; Step 5: Initialize the number of iterations k = 1 and set the convergence parameter θ; Step 6: Perform k square root operations on the matrix T to obtain the square root matrix T 1 / 2k ; Calculate the residual matrix X and its F-norm ||X|| F , I is the unit matrix; Step 7: If ||X|| F ≤θ, then output the residual matrix X and the current iteration number k; otherwise, set 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 The approximate matrix r m (X); Step 9: Approximate the matrix r m (X) Perform k-times square calculations to obtain a matrix V, and further modify the elements of the matrix V to obtain a matrix W; Step 10: Calculate the super-resolution prediction result x of the flow field at the target prediction time t t_pre =QWQ * x1.

2. The method for predicting flow field with temporal super-resolution based on the combination of DMD and Schur-Pade according to claim 1, characterized in that: In step 3, the DMD method is used to obtain the transfer matrix A corresponding to the matrix Y, specifically: According to the correlation matrix Y, the flow field data from the first typical moment to the (M-1)th typical moment are selected, denoted as X1 = [x1, x2, …, x M-1 ], select the flow field data from the second typical moment to the Mth typical moment, recorded as X2 = [x2, x3, ... x M ]; Based on the DMD method, X1 and X2 approximately satisfy the linear transformation relationship, that is, X2 = AX1, and the transfer matrix A is obtained by solving the generalized inverse of the matrix, that is, represents the generalized inverse of X1.

3. The temporal super-resolution flow field prediction method based on the combination of DMD and Schur-Pade according to claim 1, characterized in that: In step 4, the transfer matrix A is subjected to Schur decomposition, specifically: A=QTQ * Among them, Q is a unitary matrix; Q * is the conjugate transposed matrix of Q; T is an upper triangular matrix whose main diagonal elements are the eigenvalues ​​of A.

4. The method for predicting flow field with temporal super-resolution based on the combination of DMD and Schur-Pade according to claim 1, characterized in that: In step 6, k square root operations are performed on the matrix T, specifically: 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 elements u of U ij Meet U 2 =T, then: When u ii +u ij When it is not equal to 0, there is When u ii +u ij When it is equal to 0, u is to be determined ij , and with ||U 2 -T|| F The goal is to minimize the error and to obtain an approximate solution based on the optimization algorithm. Repeat the above method of solving the square root matrix k times and finally get T2 k Power Root Matrix 5. The method for predicting flow field with temporal super-resolution based on the combination of DMD and Schur-Pade according to claim 1, characterized in that: The method for setting the parameter α in step 8 is: in, 6. The method for predicting flow field with temporal super-resolution based on the combination of DMD and Schur-Pade according to claim 1, characterized in that: In step 8, based on the Pade approximation, (IX) is obtained α The 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, 7. The method for predicting flow field with temporal super-resolution based on the combination of DMD and Schur-Pade according to claim 1, characterized in that: In step 9, the elements of the matrix V are further modified to obtain the matrix W, which is specifically: Replace the elements of the main diagonal and the first upper diagonal of the matrix V. The elements on the main diagonal of the replaced matrix W are: The elements of the first upper diagonal of the matrix W are: Wherein, i and j satisfy i=1,2,…,n-1 and j=i+1; The matrix W is: Among them, v ij is the element in the i-th row and j-th column of the matrix V, t ij is the element in the i-th row and j-th column of matrix T.

8. A computer device / equipment / system 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 7.

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

10. A computer program product comprising a computer program / instructions, characterized in that: When the computer program / instructions are executed by a processor, the steps of the method according to any one of claims 1 to 7 are implemented.

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