A dynamic mode decomposition flow field prediction method based on nonlinear data dimensionality reduction

Abstract

The present invention belongs to the technical field of computer-aided fluid mechanics, and proposes a dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction, which combines orthogonal decomposition and multi-layer perceptron autoencoder. First, the modal coefficients in the unsteady flow field are extracted through two-step POD to form low-dimensional features, and then the low-dimensional features are subjected to nonlinear feature learning through the multi-layer perceptron autoencoder, so as to better capture the complex dynamic behavior of the unsteady flow field. The unsteady flow field prediction method of the present invention can not only greatly reduce the computational complexity, but also retain the key features in the unsteady flow field. In addition, the present invention also combines dynamic modal decomposition with the random forest algorithm to further improve the prediction accuracy of the time evolution of the modal coefficients, which is particularly suitable for the rapid prediction and reconstruction of high-dimensional nonlinear flow fields.

Description

Technical Field

[0001] The invention relates to the technical field of computer-aided fluid mechanics, and in particular to a dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction. Background Art

[0002] In the field of fluid mechanics, computational fluid dynamics (CFD) is a core tool for fluid mechanics research and is widely used in aerospace, automobile manufacturing, environmental engineering and other fields to simulate complex flow phenomena. CFD solves the control equations of fluid motion through numerical methods and can provide accurate flow field data for engineering applications. However, despite the high accuracy and wide applicability of CFD, its high computational cost limits its application in real-time simulation and multi-scenario optimization. Especially in unsteady flow fields with complex geometries, CFD calculations often require millions or even hundreds of millions of grid points, resulting in huge overhead in computing resources and time. Therefore, how to reduce the computing cost while ensuring accuracy has become the focus of researchers.

[0003] In order to meet these challenges, order reduction modeling technology has emerged. Proper orthogonal decomposition (POD) is a common order reduction technology. By performing singular value decomposition (SVD) on flow field data, it can map high-dimensional data to a low-dimensional feature space, thereby reducing the complexity of the data. However, the traditional POD method has certain limitations in dealing with nonlinear or complex dynamic behaviors. Especially when dealing with unsteady flow fields, the POD method may not be able to effectively capture the time-varying characteristics and nonlinear relationships of the flow field, resulting in insufficient accuracy and generalization of the model. In recent years, researchers have introduced multilayer perceptron (MLP) autoencoder (AE) as a nonlinear dimensionality reduction tool. Autoencoder is an unsupervised learning model that performs nonlinear mapping and feature extraction on data through encoder and decoder structures. Although autoencoders perform well in processing nonlinear data, it is still a challenge to effectively combine POD and autoencoders to obtain more accurate models when facing complex unsteady flow fields.

[0004] In addition, dynamic modal decomposition (DMD), as a technique for extracting dynamic system modes, can further improve the accuracy of flow field modeling. However, the existing DMD method also has certain limitations when dealing with high-dimensional data, especially when dealing with short-term and long-term dynamic changes. How to optimize DMD analysis to adapt to different time scales still needs further research.

[0005] In summary, the existing unsteady flow prediction methods have problems such as insufficient accuracy, high computational complexity, and difficulty in dealing with nonlinear dynamic behaviors in terms of reduced-order modeling. Therefore, an innovative method is urgently needed to combine the advantages of POD and autoencoder, while introducing DMD analysis to improve the accuracy, efficiency, and adaptability of the prediction model. Summary of the invention

[0006] In view of the above shortcomings in the prior art, the present invention proposes a dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction. This method combines the dimensionality reduction capability of POD and the nonlinear feature extraction capability of MLP autoencoder, and realizes the modeling and prediction of unsteady flow field through reasonable step design.

[0007] The technical solution of the present invention is as follows:

[0008] A dynamic mode decomposition flow field prediction method based on nonlinear data dimensionality reduction comprises the following steps:

[0009] Step S1: obtaining the original flow field parameter matrix from the unsteady flow field;

[0010] Step S2: performing the first step POD decomposition on the original flow field parameter matrix based on the time direction to obtain the time modal matrix of each characteristic dimension, and extracting the first POD modal matrix and the first POD modal coefficient matrix based on the cumulative energy criterion;

[0011] Step S3: performing a second POD decomposition on the first POD modal matrix based on the characteristic dimension direction, and extracting a second POD modal matrix and a second POD modal coefficient matrix based on a cumulative energy criterion;

[0012] Step S4: establishing a low-dimensional unsteady flow field prediction model, including a multi-layer perceptron autoencoder, a random forest model and a dynamic mode decomposition model;

[0013] Step S5: training the low-dimensional unsteady flow field prediction model with the second POD modal coefficient matrix as input and the future time flow field parameters as output;

[0014] Step S6: Apply the trained low-dimensional unsteady flow field prediction model to realize unsteady flow field prediction.

[0015] Furthermore, the step S2 specifically includes:

[0016] Step S2-1: performing singular value decomposition on the original flow field parameter matrix based on the time direction to obtain a time mode matrix of each characteristic dimension, wherein each column of the time mode matrix is ​​a time mode;

[0017] Step S2-2: The temporal modal matrix of each feature dimension is processed as follows:

[0018] Sort the time modes by energy from large to small, and extract the time mode, ensuring that the proportion of retained energy is greater than a first threshold;

[0019] Step S2-3: horizontally merge the processed time modal matrices of each feature dimension to obtain a first POD modal matrix and its corresponding first POD modal coefficient matrix.

[0020] Furthermore, the step S3 specifically includes:

[0021] The first POD modal matrix is ​​decomposed by singular value based on the characteristic dimension direction, and the time modes in the first POD modal matrix after singular value decomposition are sorted from large to small according to energy, and the previous time modes, ensuring that the retained energy ratio is greater than the second threshold, that is, obtaining the second POD modal matrix and its corresponding second POD modal coefficient matrix.

[0022] Furthermore, the multilayer perceptron autoencoder performs nonlinear mapping on the second POD modal coefficient matrix, divides the second POD modal coefficient matrix into three parts according to energy size, and uses them as input feature variables of three autoencoders with different structures to extract the first low-dimensional features.

[0023] Furthermore, the three autoencoders with different structures are , and ,in:

[0024] The input layer, bottleneck layer and output layer all contain the same number of neurons, and the bottleneck layer uses a linear mapping function;

[0025] The input layer and output layer contain the same number of neurons, which is greater than the number of neurons in the bottleneck layer. The input layer and the bottleneck layer use linear rectification activation function, and the output layer uses linear activation function.

[0026] The input layer and output layer contain the same number of neurons, which is greater than the number of neurons in the bottleneck layer. The input layer and bottleneck layer use linear rectification activation function, and the output layer uses linear activation function.

[0027] Furthermore, the random forest model interpolates the first low-dimensional feature in the feature dimension direction, and obtains a low-dimensional feature of the desired feature dimension after filtering.

[0028] Furthermore, the dynamic modal decomposition model performs dynamic modal decomposition analysis on the low-dimensional features of the prediction parameters to obtain a low-dimensional time modal coefficient matrix.

[0029] Furthermore, the dynamic modal decomposition analysis includes a window size of s Windowed DMD analysis and the cutoff dimension r The Optimized DMD analysis of Windowed DMD is used to process the dynamic evolution within the preset time range, and the Optimized DMD analysis is used to process the dynamic evolution outside the preset time range.

[0030] Furthermore, the multi-layer perceptron autoencoder decodes the low-dimensional time modal coefficients to obtain a POD modal coefficient matrix, and the POD modal matrix is ​​combined to achieve reconstruction of the unsteady flow field.

[0031] Furthermore, the characteristic dimension is the spanwise position, Reynolds number or geometric shape parameter of a plane in space. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. The features and advantages of the present invention can be more clearly understood by referring to the drawings. The drawings are schematic and should not be understood as limiting the present invention in any way. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0033] Figure 1 The present invention is a flowchart of a dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction.

[0034] Figure 2 It is a schematic diagram of the principle of the low-dimensional unsteady flow field prediction model of the present invention.

[0035] Figure 3 This is a schematic diagram of the composition of three autoencoders with different structures.

[0036] Figure 4 This is a result diagram of POD coefficients of various orders corresponding to the plane z*=0.125 of the reconstructed test set of the multilayer perceptron autoencoder in Example 1.

[0037] Figure 5 This is the comparison of the average reconstruction errors of POD-AE and POD in Example 1 on the test set plane (ac) and the training set plane z*=0.12 (d).

[0038] Figure 6 This is the comparison of the evolution errors of POD-AE, POD-AE-RF and POD-AE-RF-DMD in Example 1 on the test set plane (ac) and the training set plane z*=0.12 (d). DETAILED DESCRIPTION

[0039] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present invention and the features in the embodiments can be combined with each other without conflict.

[0040] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

[0041] This embodiment takes the flow around a three-dimensional airfoil as an example to specifically illustrate the dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction of the present invention:

[0042] In this example, the unsteady flow field pressure data of the three-dimensional airfoil flow is used as the original flow field parameter matrix. As the characteristic dimension direction, the unsteady flow law of different planes is studied. In the flow field, a total of 51 planes are extracted from the spanwise coordinates 0 to 0.25 at a uniform interval of 0.005. The training set contains a total of 48 planes, and the test set includes three planes. =[0.06,0.125,0.19], the amount of grid intercepted on each plane is The 1110 time snapshot samples of the sub-area are used as the original flow field parameter matrix ,in x Indicates the flow velocity value, represents the spanwise position, t Indicates time.

[0043] like Figure 1-2 As shown, the present embodiment provides a dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction, which is specifically performed in the following steps:

[0044] The first step is to perform the first POD decomposition, perform singular value decomposition on the original flow field parameter matrix based on the time direction, obtain the time mode matrix of each spanwise position, sort the time modes in the time mode matrix of each spanwise position from large to small according to the energy, and extract the previous time modes, ensuring that the retained energy ratio is greater than the first threshold, and then merging horizontally to obtain the first POD mode matrix and the first POD mode coefficient matrix;

[0045]

[0046] in, is the original flow field parameter matrix, is the first dimension reduction threshold , is the first POD modal matrix, is the first POD modal coefficient matrix.

[0047] The design of the first threshold needs to strike a balance between computational efficiency and model accuracy.

[0048] The second step is to perform the second POD decomposition, perform singular value decomposition on the first POD modal matrix, sort the time modes in the first POD modal matrix after singular value decomposition from large to small energy, and extract the previous time modes, ensuring that the retained energy ratio is greater than the second threshold, that is, obtaining the second POD modal matrix and the second POD modal coefficient matrix.

[0049]

[0050] in, is the second dimension reduction threshold, is the second POD modal matrix, is the second POD modal coefficient matrix.

[0051] The design of the second threshold needs to strike a balance between computational efficiency and model accuracy.

[0052] In the third step, the multilayer perceptron autoencoder performs nonlinear mapping on the second POD modal coefficient matrix, divides the second POD modal coefficient matrix into three parts according to energy size, and uses them as input feature variables of three autoencoders with different structures to extract the first low-dimensional features.

[0053] like Figure 3 As shown, the three autoencoders with different structures are , and .in, Responsible for reconstructing the contribution to the unsteady flow field The input layer, bottleneck layer and output layer all contain the same number of neurons. In order to avoid unnecessary distortion of modal coefficients by nonlinear reconstruction, the bottleneck layer adopts a linear mapping function. The encoding and decoding process is:

[0054]

[0055] in, for The encoder mapping function converts the modal coefficients Mapping to bottleneck layer features , for The encoder mapping function transforms the bottleneck layer features Mapping back to the reconstructed modal coefficients .

[0056] To reconstruct the Step to The order modal coefficients, both the input layer and the output layer contain neurons, and the bottleneck layer contains neurons (of which ), realize the dimensionality reduction of modal coefficients, the input layer and bottleneck layer use linear rectification activation function, the output layer uses linear activation function, and the error measure of modal coefficient reconstruction is mean square error (MSE). The encoding and decoding process is:

[0057]

[0058] in, for The encoder mapping function converts the modal coefficients Mapping to bottleneck layer features , for The encoder mapping function transforms the bottleneck layer features Mapping back to the reconstructed modal coefficients .

[0059] For reconstruction Step to The order modal coefficients, both the input layer and the output layer contain neurons, and the bottleneck layer contains neurons (of which , is the number of the first low-dimensional features), and a large dimension reduction is performed. The activation function and error measurement criteria are the same as same. The encoding and decoding process is:

[0060]

[0061] in, for The encoder mapping function converts the modal coefficients Mapping to bottleneck layer features , for The encoder mapping function transforms the bottleneck layer features Mapping back to the reconstructed modal coefficients .

[0062] Figure 4 Shows the training completed and The reconstruction results of the POD modal coefficients of each order corresponding to the test set plane z*=0.125. It can be seen that The reconstructed results are in good agreement with the initial POD modal coefficients both within and outside the time range of the training set; Reconstruction error relative Although it is relatively large, it can basically reconstruct the changing trend of the high-order POD modal coefficients and has the ability to reconstruct outside the time range of the training set. Figure 5 The comparison results of the average reconstruction errors of POD-AE (a method combining POD and AE, abbreviated as POD-AE) and POD in the test set plane (ac) and the training set plane z*=0.12 (d). It can be seen that the flow field further reconstructed by the coefficients output by the autoencoder has higher accuracy and stronger ability to capture the structure existing in the flow field, whether in the test set plane or the training set plane. In the result of plane z*=0.06 in (a), AE effectively improves the problem of insufficient generalization ability that may occur in the POD method, and also proves that in this problem, the POD-AE dimensionality reduction method is superior to the traditional POD method in terms of dimensionality reduction ability.

[0063] The fourth step is to use the random forest (RF) model to interpolate the first low-dimensional feature in the spanwise direction. The data label vector of each first low-dimensional feature is and order i, the data label vector As input features, the test set =[0.06,0.125,0.19] as the output variable. After the training is completed, the output variable is filtered by a Gaussian filter. The kernel size of the Gaussian filter is selected according to the noise spectrum characteristics and is usually set to a high-frequency noise suppression range below a specific frequency to ensure that the smoothing of the signal does not affect the dynamic characteristics of the unsteady flow field.

[0064] The fifth step is to use the dynamic modal decomposition model to perform dynamic modal decomposition analysis on the low-dimensional features of the test set to obtain its evolution process in the future, that is, the low-dimensional time modal coefficient matrix. Specifically, it includes Windowed DMD analysis with a window size of s and Optimized DMD analysis with a truncation dimension of r.

[0065] Windowed DMD analysis is used for short-term dynamic evolution within the time range of the training set. It divides the time series into multiple windows, each with a length of s, and performs DMD decomposition on the low-dimensional features of the test set in each time window to extract local low-dimensional time modal coefficients to capture the short-term flow characteristics in the training set. Optimized DMD is used for the long-term evolution after the last time window of the training set. It adopts the optimized DMD analysis method with a truncation dimension of r. By extracting r stable non-zero modal coefficients from the low-dimensional features of the test set, it can be effectively used to predict the long-term dynamic changes of unsteady flow fields outside the time range of the training set.

[0066] In the sixth step, the low-dimensional time modal coefficient matrix obtained in the fifth step is decoded through a multi-layer perceptron autoencoder and converted into a POD modal coefficient matrix. The POD modal matrix is ​​combined to realize the reconstruction of the unsteady flow field, and the evolution of the unsteady flow field outside the time range of the training set is predicted to ensure the generalization ability of the low-dimensional unsteady flow field prediction model in time extrapolation. Figure 6 The figure shows the evolution error comparison of POD-AE, POD-AE-RF (a method combining POD, AE and RF, abbreviated as POD-AE-RF) and the dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction (POD-AE-RF-DMD) of the present invention on the test set plane (ac) and the training set plane z*=0.12 (d). It can be clearly seen that whether in the test set plane or the training set plane, the algorithm with DMD improvement has higher evolution accuracy outside the training set time range, which more intuitively proves the effectiveness of the POD-AE-RF-DMD algorithm.

[0067] In the present invention, unless otherwise clearly specified and limited, the terms "installed", "connected", "connected", "fixed" and the like should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium, it can be the internal connection of two elements or the interaction relationship between two elements. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0068] In the present invention, unless otherwise clearly specified and limited, a first feature being "above" or "below" a second feature may include that the first and second features are in direct contact, or may include that the first and second features are not in direct contact but are in contact through another feature between them. Moreover, a first feature being "above", "above" and "above" a second feature includes that the first feature is directly above and obliquely above the second feature, or simply indicates that the first feature is higher in level than the second feature. A first feature being "below", "below" and "below" a second feature includes that the first feature is directly below and obliquely below the second feature, or simply indicates that the first feature is lower in level than the second feature.

[0069] In the present invention, the terms "first", "second", "third", "fourth" are used for descriptive purposes only and cannot be understood as indicating or implying relative importance. The term "plurality" refers to two or more, unless otherwise clearly defined.

[0070] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A dynamic mode decomposition flow field prediction method based on nonlinear data dimensionality reduction, characterized in that: The following steps are involved: Step S1: obtaining the original flow field parameter matrix from the unsteady flow field; Step S2: performing the first step POD decomposition on the original flow field parameter matrix based on the time direction to obtain the time modal matrix of each characteristic dimension, and extracting the first POD modal matrix and the first POD modal coefficient matrix based on the cumulative energy criterion; Step S3: performing a second POD decomposition on the first POD modal matrix based on the characteristic dimension direction, and extracting a second POD modal matrix and a second POD modal coefficient matrix based on a cumulative energy criterion; Step S4: establishing a low-dimensional unsteady flow field prediction model, including a multi-layer perceptron autoencoder, a random forest model and a dynamic mode decomposition model; Step S5: training the low-dimensional unsteady flow field prediction model with the second POD modal coefficient matrix as input and the future time flow field parameters as output; Step S6: Apply the trained low-dimensional unsteady flow field prediction model to realize unsteady flow field prediction; The multilayer perceptron autoencoder performs nonlinear mapping on the second POD modal coefficient matrix, divides the second POD modal coefficient matrix into three parts according to energy size, and respectively serves as input feature variables of three autoencoders with different structures to extract the first low-dimensional feature; The three different structure autoencoders are , and ,in: The input layer, bottleneck layer and output layer all contain the same number of neurons, and the bottleneck layer uses a linear mapping function; The input layer and output layer contain the same number of neurons, which is greater than the number of neurons in the bottleneck layer. The input layer and the bottleneck layer use linear rectification activation function, and the output layer uses linear activation function. The input layer and output layer contain the same number of neurons, which is greater than the number of neurons in the bottleneck layer. The input layer and bottleneck layer use linear rectification activation function, and the output layer uses linear activation function.

2. The dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction according to claim 1 is characterized in that: The step S2 specifically includes: Step S2-1: performing singular value decomposition on the original flow field parameter matrix based on the time direction to obtain a time mode matrix of each characteristic dimension, wherein each column of the time mode matrix is ​​a time mode; Step S2-2: The temporal modal matrix of each feature dimension is processed as follows: Sort the time modes by energy from large to small, and extract the time mode, ensuring that the proportion of retained energy is greater than a first threshold; Step S2-3: horizontally merge the processed time modal matrices of each characteristic dimension to obtain a first POD modal matrix and its corresponding first POD modal coefficient matrix.

3. The dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction according to claim 2 is characterized in that: The step S3 specifically includes: The first POD modal matrix is ​​decomposed by singular value based on the characteristic dimension direction, and the time modes in the first POD modal matrix after singular value decomposition are sorted from large to small according to energy, and the previous time modes, ensuring that the retained energy ratio is greater than the second threshold, that is, obtaining the second POD modal matrix and its corresponding second POD modal coefficient matrix.

4. The dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction according to claim 3 is characterized in that: The random forest model interpolates the first low-dimensional feature in the feature dimension direction, and obtains a low-dimensional feature of the desired feature dimension after filtering.

5. The dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction according to claim 4 is characterized in that: The dynamic modal decomposition model performs dynamic modal decomposition analysis on the low-dimensional characteristics of the prediction parameters to obtain a low-dimensional time modal coefficient matrix.

6. The dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction according to claim 5, characterized in that: The dynamic modal decomposition analysis includes a window size of s Windowed DMD analysis and the cutoff dimension r The Optimized DMD analysis is used to process the dynamic evolution within the preset time range, and the OptimizedDMD analysis is used to process the dynamic evolution outside the preset time range.

7. The dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction according to claim 6 is characterized in that: The multi-layer perceptron autoencoder decodes the low-dimensional time modal coefficients to obtain the POD modal coefficient matrix, and the POD modal matrix is ​​combined to realize the reconstruction of the unsteady flow field.

8. The dynamic modal decomposition flow field prediction method based on nonlinear data dimensionality reduction according to claim 7 is characterized in that: The characteristic dimension is the spanwise position, Reynolds number or geometric shape parameter of the plane in space.

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