A flow field data smoothing method based on augmented Lagrange constraint and DMD decomposition

Abstract

The present application relates to the technical field of fluid mechanics measurement, solves the technical problems that space smoothing and time fidelity cannot be considered in traditional flow field data smoothing method, and error accumulation and phase drift are caused by step-by-step serial processing, and particularly relates to a flow field data smoothing method based on augmented Lagrange constraint and DMD decomposition, obtains particle image velocimetry time series flow field data to be processed, rearranges the time series flow field data based on spatial discrete points, constructs a time series flow field snapshot matrix according to time sampling sequence, and constructs a unified constraint objective function for flow field optimization. The present application can effectively eliminate random noise and abnormal values in original PIV data which do not conform to physical evolution law, so that the final output smoothed flow field has strict time sequence consistency and physical interpretability while maintaining high signal-to-noise ratio, can more truly reflect the dynamic characteristics of fluid, and realizes deep fusion of flow field physical mechanism and observation data.

Description

Technical Field

[0001] This invention relates to the field of fluid dynamics measurement technology, and in particular to a smoothing method for flow field data based on augmented Lagrangian constraints and DMD decomposition. Background Technology

[0002] As particle image velocimetry technology advances towards higher frequency and higher spatiotemporal resolution, experimental fluid mechanics researchers generally face bottlenecks in data quality. Due to limitations imposed by high-frequency laser irradiation and high-speed imaging equipment, the signal-to-noise ratio of a single frame image decreases significantly, and the calculated velocity field often contains random high-frequency noise and local outlier vectors. To improve flow field data quality, existing technologies typically employ the following post-processing methods: First, spatial filtering methods, such as median filtering, Gaussian filtering, and total variation TV denoising, can smooth spatial gradients, but they can easily lead to over-smoothing, thus smoothing out key flow structures with large gradient characteristics, such as shear layers and vortex cores, causing spatial feature distortion. Second, temporal filtering methods, such as moving average and POD low-rank reconstruction, can suppress temporal jitter, but they can easily introduce phase drift, leading to distortion of dynamic characteristics such as vortex shedding frequency and turbulent energy spectrum. Moreover, the POD method often ignores the predictable dynamic characteristics of the flow field, resulting in larger errors in the processing of non-stationary processes. Third, a step-by-step processing method of first spatial filtering and then temporal correction cannot coordinate spatial and temporal constraints within the same mathematical framework, resulting in the inability to simultaneously ensure the fidelity of spatial gradients and temporal evolution. This leads to unpredictable systematic errors introduced into subsequent advanced analyses such as pressure reconstruction and eigenfrequency extraction. Summary of the Invention

[0003] To address the shortcomings of existing technologies, this invention provides a flow field data smoothing method based on augmented Lagrangian constraints and DMD decomposition. This method solves the technical problems of traditional flow field data smoothing methods, which cannot simultaneously achieve spatial smoothing and temporal fidelity, as well as the error accumulation and phase drift caused by step-by-step serial processing. It achieves the goal of effectively removing high-frequency noise while maintaining the consistency between the edge features of the flow field spatial structure and the dynamic phase of the temporal evolution.

[0004] To address the aforementioned technical problems, this invention provides the following technical solution: a flow field data smoothing method based on augmented Lagrangian constraints and DMD decomposition, comprising the following steps: S1. Obtain the time-series flow field data of particle image velocimetry to be processed, rearrange the time-series flow field data into vectors based on spatial discrete points, and construct the time-series flow field snapshot matrix according to the time sampling order; S2. Construct a unified constraint objective function for flow field optimization. The unified constraint objective function includes a total variation regularization term for constraining the sparsity of the spatial gradient of the flow field, a dynamic mode decomposition dynamic constraint term for constraining the linear dynamic evolution law of the flow field, and a data fidelity term for maintaining consistency with the time-series flow field data. S3. Perform variable splitting on the unified constraint objective function, introduce a first auxiliary variable to characterize the spatial gradient of the flow field and a second auxiliary variable to characterize the predicted value of the flow field temporal evolution, and combine them with the preset Lagrange multipliers and quadratic penalty terms to construct an augmented Lagrange function. S4. The augmented Lagrangian function is solved iteratively using the alternating direction multiplier method. In each iteration, the variable update step and the multiplier update step are executed sequentially until the preset convergence condition is met, and the smoothed flow field data is output.

[0005] Furthermore, step S1 specifically includes the following steps: S11. Obtain the flow field velocity field data at T consecutive sampling times within the experimental observation area, where the flow field at each time time contains the velocity values ​​of N spatial discrete points; S12. Rearrange the two-dimensional or three-dimensional flow field velocity data at the t-th sampling time into a one-dimensional column vector according to the preset spatial index order to obtain the instantaneous flow field velocity vector. S13. Arrange the instantaneous flow field velocity vectors corresponding to all T sampling times as column vectors according to their time sequence to construct a time-series flow field snapshot matrix.

[0006] Furthermore, step S2 specifically includes the following steps: S21. Construct a constraint term for the spatial gradient of the flow field velocity variable using the L1 norm, so as to preserve the spatial edge characteristics of the flow field while removing noise. S22. Use the L2 norm to construct residual constraint terms between the flow field velocity variables and the time-series flow field snapshot matrix to prevent the optimization results from deviating from the original observation data; S23. Introduce a linear evolution operator and use the L2 norm to construct the dynamic residual constraint terms between the current flow field after linear evolution and the flow field at the next time step; S24. Combine the preset weight coefficients to construct a unified constraint objective function.

[0007] Furthermore, the expression for the unified constraint objective function is as follows: in, This represents the unified constraint objective function. This represents the optimized time-series flow field data to be solved. Represents the time-series flow field snapshot matrix. The spatial gradient represents the velocity variable in the flow field. This represents the linear evolution matrix obtained from dynamic mode decomposition. This represents the instantaneous flow field velocity vector of u at the t-th sampling time. This represents the instantaneous flow field velocity vector of u at the (t+1)th sampling time. This represents the total number of sampling frames for the flow field data. The weight coefficients of the total variation regularization term are represented. This represents the weighting coefficient of the data fidelity item. The weighting coefficients represent the dynamic evolution constraint terms. Describing the L1 norm, This represents the L2 norm.

[0008] Furthermore, step S3 specifically includes the following steps: S31. Introduce the first auxiliary variable and the second auxiliary variable respectively, and establish their spatial gradient constraint relationship and temporal evolution constraint relationship with the flow field velocity variable; S32. For spatial gradient constraint relationship and temporal evolution constraint relationship, respectively introduce the first Lagrange multiplier and the first penalty parameter corresponding to the first auxiliary variable, and the second Lagrange multiplier and the second penalty parameter corresponding to the second auxiliary variable; S33. Replace the total variation regularization term in the unified constraint objective function with the L1 norm term about the first auxiliary variable, replace the dynamic mode decomposition dynamic constraint term with the residual term about the second auxiliary variable and the linear evolution result of the flow field, and add the linear multiplier term and quadratic penalty term corresponding to the constraint conditions to construct the augmented Lagrangian function.

[0009] Furthermore, the expression for the augmented Lagrange function is: in, This represents the augmented Lagrange function. Indicates the first auxiliary variable. This represents the value of the second auxiliary variable at time t. Represents the first Lagrange multiplier. Indicates the second Lagrange multiplier. Indicates the first penalty parameter. This represents the second penalty parameter.

[0010] Furthermore, the variable update step includes a flow field velocity update sub-step, a spatial gradient update sub-step, and a dynamic evolution update sub-step. Specifically, step S4 includes the following steps: S41. Fixing the first auxiliary variable, the second auxiliary variable, and all Lagrange multipliers, the spatial difference operation in the solution process of the flow field velocity variable is transformed into a frequency domain multiplication operation using the fast Fourier transform. The flow field velocity variable in the current (k+1)th iteration is calculated using the following formula: In the above formula, This represents the flow field velocity variable calculated in the (k+1)th iteration. This represents the Fast Fourier Transform operator. This represents the inverse Fast Fourier Transform operator. Represents the time-series flow field snapshot matrix. This represents the first auxiliary variable obtained in the k-th iteration. This represents the second auxiliary variable obtained from the k-th iteration. This represents the first Lagrange multiplier corresponding to the spatial gradient constraint, calculated in the k-th iteration. This represents the second Lagrange multiplier corresponding to the temporal evolution constraint, calculated in the k-th iteration. This represents the first penalty parameter corresponding to the spatial gradient constraint. This represents the second penalty parameter corresponding to the temporal evolution constraint. This represents the weighting coefficient of the data fidelity item. Represents the discrete divergence operator. Represents the square of the discrete wavenumber in the frequency domain; S42. With the flow field velocity variable fixed, solve the spatial gradient minimization subproblem under L1 regularization using the soft threshold operator. Calculate the first auxiliary variable for the (k+1)th iteration using the following formula: In the above formula, This represents the first auxiliary variable (characterizing the spatial gradient) obtained in the (k+1)th iteration. Represents the spatial gradient operator. This represents the soft threshold operator. This represents the shrinkage threshold of the soft threshold operator; S43. With the flow field velocity variable fixed, the time-series flow field is reconstructed using singular value decomposition with low-rank truncation. The second auxiliary variable for the current (k+1)th iteration is calculated using the following formula: In the above formula, This represents the second auxiliary variable calculated in the (k+1)th iteration. This represents the low-rank truncation reconstruction operator. Indicates a timing misalignment operation; S44. Based on the constraint residuals of the current iteration, calculate the first and second Lagrange multipliers used for dual compensation of the constraint residuals; S45. Determine whether the spatial gradient constraint residual and the temporal evolution constraint residual satisfy the corresponding preset convergence conditions; If not, proceed to the next iteration; If so, stop the iteration and output the current flow field velocity variable as the smoothed flow field data.

[0011] Furthermore, the formulas for calculating the first and second Lagrange multipliers are as follows: In the above formula, Let the first Lagrange multiplier in the (k+1)th iteration be denoted as . This represents the second Lagrange multiplier in the (k+1)th iteration. This represents the spatial gradient constraint residual in the current iteration round. This represents the temporal evolution constraint residual of the current iteration round.

[0012] By employing the above technical solution, the present invention provides a flow field data smoothing method based on augmented Lagrangian constraints and DMD decomposition, which has at least the following beneficial effects: 1. This invention introduces dynamic mode decomposition as a dynamic constraint term to construct a unified constraint objective function, which can force the optimized flow field data to meet the linear evolution law of fluid mechanics. By introducing a second auxiliary variable and performing low-rank SVD truncation during the optimization process, random noise and outliers in the original PIV data that do not conform to the physical evolution law can be effectively removed. This makes the final output smooth flow field maintain a high signal-to-noise ratio while having strict temporal consistency and physical interpretability, and can more realistically reflect the dynamic characteristics of the fluid, realizing the deep integration of the physical mechanism of the flow field and the observation data.

[0013] 2. This invention utilizes a first auxiliary variable and a first Lagrange multiplier to transform the complex total variational regularization problem into a linear subproblem. It also creatively uses the Fast Fourier Transform to transform large-scale difference operations in the spatial domain into dot multiplication and division operations in the frequency domain, thereby realizing the closed-loop analytical solution of the flow field velocity variable. This significantly reduces the computational complexity of the algorithm, enabling rapid processing of long-term flow field data with high spatial resolution, greatly improving computational efficiency, and meeting the real-time processing requirements of high-resolution flow fields.

[0014] 3. This invention integrates a total variational regularization term into the objective function and uses the soft threshold operator in the augmented Lagrange multiplier method to constrain the sparsity of the spatial gradient, giving it anisotropic diffusion characteristics. This can smooth the uniform flow field region and effectively solve the problem of oversmoothing in traditional flow field denoising methods, which easily blurs the fine structure of the flow field and causes the loss of important physical features.

[0015] 4. This invention adopts the augmented Lagrange multiplier method framework and establishes a residual feedback mechanism for spatial gradient constraints and temporal evolution constraints by introducing and dynamically updating the Lagrange multiplier matrix. It uses multiplier terms to perform dual compensation for the constraint residuals of each iteration, which can force auxiliary variables and flow field variables to strictly satisfy the equality constraints when the penalty parameter value is small. This not only eliminates the systematic calculation bias caused by variable splitting, but also gives the algorithm extremely strong robustness, enabling it to stably converge to the global optimum when facing PIV raw data with different signal-to-noise ratios and different flow types. Attached Figure Description

[0016] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart of the flow field data smoothing method of the present invention. Detailed Implementation

[0017] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. This will allow for a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects, and to facilitate its implementation.

[0018] Traditional methods for improving flow field data quality often suffer from drawbacks, failing to simultaneously achieve spatial smoothing and temporal fidelity, leading to error accumulation and phase drift, which in turn affects the physical accuracy of the denoising results. To achieve a deep fusion of flow field physical mechanisms and observational data, and significantly improve the physical accuracy of the denoising results, this invention proposes a flow field data smoothing method based on augmented Lagrangian constraints and DMD decomposition, such as... Figure 1 As shown, the method includes the following steps: S1. Obtain the time-series flow field data of particle image velocimetry through the experimental observation platform. Rearrange the time-series flow field data into vectorized arrays based on spatial discrete points, and construct a time-series flow field snapshot matrix according to the time sampling order. This improves the temporal continuity and evolution smoothness of the denoising results, while also increasing computational efficiency. The specific steps include: S11. Obtain flow field velocity field data at T consecutive sampling times within the experimental observation area. The flow field at each time time contains velocity values ​​at N spatial discrete points. The flow field data usually comes from a particle image velocimetry system. During the experiment, a high-speed camera is used to capture images of the motion of tracer particles in the fluid at a preset sampling frequency, and a series of two-dimensional or three-dimensional velocity vector fields at consecutive times are calculated through a cross-correlation algorithm.

[0019] S12. The two-dimensional or three-dimensional flow field velocity data at the t-th sampling time are rearranged into a one-dimensional column vector according to the preset spatial index order to obtain the instantaneous flow field velocity vector. If the flow field contains two components in the x and y directions, the preferred rearrangement method is to first arrange the x-direction components of all grid points into a column, then arrange the y-direction components of all grid points into a column, and then splice the two together to form a high-dimensional column vector of length N.

[0020] S13. Arrange the instantaneous flow field velocity vectors corresponding to all T sampling times as column vectors according to the chronological order to construct a time-series flow field snapshot matrix. By constructing this time-series flow field snapshot matrix, the complex spatiotemporal flow field data is transformed into a standard linear algebra matrix form, providing the necessary data structure foundation for subsequent extraction of linear evolution calculations using dynamic mode decomposition and overall solution using fast Fourier transform.

[0021] S2. To establish a mathematical optimization model that simultaneously considers spatial smoothness, data fidelity, and temporal dynamics, a unified constraint objective function for flow field optimization is constructed. This unified constraint objective function includes a total variational regularization term to constrain the sparsity of the flow field's spatial gradient, a dynamic mode decomposition dynamic constraint term to constrain the linear dynamic evolution of the flow field, and a data fidelity term to maintain consistency with the temporal flow field data. This accurately preserves the edge and step characteristics of the flow field, improving the temporal consistency and physical realism of the results. Specifically, the following steps are included: S21. Use the L1 norm to construct a constraint term for the spatial gradient of the velocity variable in the flow field, so as to maintain the spatial edge characteristics of the flow field while removing noise. The L1 norm has sparsity-induced properties, which allows the gradient to take large values ​​at a few locations (i.e., at the edge) and zero values ​​in flat regions. This enables the optimized flow field to achieve anisotropic diffusion, that is, while smoothing out random high-frequency noise, it accurately preserves the edge sharpness and geometric features of the flow field.

[0022] S22. To ensure data consistency of the algorithm and make the final output smooth flow field the best approximation of the original measurement in a statistical sense, the L2 norm is used to construct a residual constraint term between the flow field velocity variable and the time-series flow field snapshot matrix to prevent the optimization result from deviating from the original observation data.

[0023] S23. Introducing a linear evolution operator, using the L2 norm to construct the dynamic residual constraint term between the current flow field after linear evolution and the flow field at the next time step, effectively eliminating non-physical modal noise that does not conform to the laws of fluid dynamics evolution, and significantly improving the temporal consistency of the results.

[0024] S24. Combining the preset weighting coefficients, the total variation regularization term, data fidelity term, and dynamic constraint term are combined to obtain the final unified constraint objective function, as shown in the following expression: in, This represents the unified constraint objective function, used to characterize the total cost of flow field optimization. This represents the optimized time-series flow field data to be solved. Represents the time-series flow field snapshot matrix. The spatial gradient represents the velocity variable in the flow field. This represents the linear evolution matrix obtained from dynamic mode decomposition, used to characterize the linear dynamic evolution of the flow field from time t to time t+1. This represents the instantaneous flow field velocity vector of u at the t-th sampling time. This represents the instantaneous flow field velocity vector of u at the (t+1)th sampling time. This represents the total number of sampling frames for the flow field data. These represent the weighting coefficients of the total variation regularization term, used to adjust the strength of spatial smoothing. The weighting coefficients for data fidelity terms are used to adjust how closely the results approximate the original data. These represent the weighting coefficients of the dynamic evolution constraint terms, used to adjust the strength of temporal evolution consistency. This represents the L1 norm, used to constrain the sparsity of gradients. This represents the L2 norm (Euclidean norm), whose squared form is used to calculate the energy residual.

[0025] In practical applications, users can adjust the weighting coefficients according to the flow field characteristics: if the original data has extremely high noise, the weighting coefficients can be increased appropriately. and This enhances smoothness and physical constraint; if the flow field contains a large number of subtle high-frequency physical structures, the strength can be appropriately increased. This reduces the loss of detail caused by smoothing.

[0026] S3. Variable splitting is performed on the unified constraint objective function. A first auxiliary variable characterizing the spatial gradient of the flow field and a second auxiliary variable characterizing the predicted temporal evolution of the flow field are introduced. Combined with pre-defined Lagrange multipliers and quadratic penalty terms, an augmented Lagrange function is constructed. This decouples the complex optimization problem, improves the robustness and convergence stability of the algorithm, and precisely controls the separation of the physical characteristics of the flow field. Specifically, the steps include: S31. To address the difficulty in solving the problem caused by the non-differentiability of the total variational regularization term at the origin, a first auxiliary variable is introduced to proxy the spatial gradient, establishing its spatial gradient constraint relationship with the flow field velocity variable: To decouple the time-dimensional dependency, a second auxiliary variable is introduced to proxy the time evolution of the flow field, and its temporal evolution constraint relationship with the flow field velocity variable is established: , where t is the time index. By introducing independent auxiliary variables for spatial gradient and temporal evolution respectively, the drawback of accidentally erasing spatial details in order to smooth temporal noise in traditional methods is avoided, and the spatiotemporal characteristics of the flow field are refined and reconstructed.

[0027] S32. To ensure the numerical stability of the algorithm while satisfying the hard constraints, a first Lagrange multiplier and a first penalty parameter are introduced for the spatial gradient constraint relationship, corresponding to the first auxiliary variable. The Lagrange multiplier serves as the dual variable, used to dynamically accumulate constraint errors during iteration. The first penalty parameter, as a positive real scalar, determines the basic penalty strength for constraint violations. A larger penalty parameter... Values ​​that can accelerate convergence can lead to numerical rigidity; smaller values... The values ​​are the opposite; for the temporal evolution constraint relationship, a second Lagrange multiplier and a second penalty parameter corresponding to the second auxiliary variable are introduced.

[0028] S33. Replace the total variation regularization term in the unified constraint objective function with the L1 norm term about the first auxiliary variable, while retaining the data fidelity term; The dynamic mode decomposition dynamic constraint terms are replaced with residual terms relating to the second auxiliary variable and the linear evolution of the flow field, and linear multiplier terms and quadratic penalty terms corresponding to the constraint conditions are added; The expression for the constructed augmented Lagrange function is: in, This represents the augmented Lagrange function. This represents the first auxiliary variable, used to decouple the non-smooth gradient operator in the total variation regularization term. This represents the value of the second auxiliary variable at time t, used to decouple temporal correlations in dynamic constraints. Let represent the first Lagrange multiplier, used to dynamically correct the residuals of the spatial gradient constraints during iteration. This represents the second Lagrange multiplier, used to dynamically correct the residuals of the dynamic evolution constraints during the iteration process. This represents the first penalty parameter, used to adjust the penalty weight of the spatial gradient constraint error term in the augmented Lagrangian function. This represents the second penalty parameter, used to adjust the penalty weight of the temporal evolution constraint error term in the augmented Lagrangian function.

[0029] S4. To address the challenges of large-scale flow field optimization under multiple constraints, an alternating direction multiplier method is employed to iteratively solve the augmented Lagrangian function. In each iteration, variable update steps and multiplier update steps, including flow field velocity update, spatial gradient update, and dynamic evolution update sub-steps, are executed sequentially until the preset convergence condition is met. Smoothed flow field data is then output, ensuring that the algorithm can stably converge to the global optimum even when processing high-dimensional flow field data with millions of grid points, avoiding getting trapped in local optima. Specifically, the steps include: S41. Fixing the first auxiliary variable, the second auxiliary variable, and all Lagrange multipliers, the spatial difference operation in the solution process of the flow field velocity variable is transformed into a frequency domain multiplication operation using the Fast Fourier Transform (FFT). This calculates the flow field velocity variable for the current (k+1)th iteration, enabling real-time or near-real-time processing of long-term flow field data with high spatial resolution, significantly improving data processing speed. The calculation formula is as follows: In the above formula, This represents the flow field velocity variable calculated in the (k+1)th iteration. This represents the Fast Fourier Transform operator, used to transform spatial domain data to the frequency domain. This represents the inverse Fast Fourier transform operator, used to restore frequency domain calculation results to the spatial domain. Represents the time-series flow field snapshot matrix. This represents the first auxiliary variable (characterizing the spatial gradient) obtained in the k-th iteration. This represents the second auxiliary variable (characterizing the time-series evolution prediction value) obtained from the k-th iteration. This represents the first Lagrange multiplier corresponding to the spatial gradient constraint, calculated in the k-th iteration. This represents the second Lagrange multiplier corresponding to the temporal evolution constraint, calculated in the k-th iteration. This represents the first penalty parameter corresponding to the spatial gradient constraint. This represents the second penalty parameter corresponding to the temporal evolution constraint. This represents the weighting coefficient of the data fidelity item. This represents the discrete divergence operator (i.e., the transpose of the spatial gradient operator), used in frequency domain computation to project the auxiliary gradient field back into the velocity field. It represents the discrete wavenumber square in the frequency domain (representing the wavenumber vector), which corresponds to the eigenvalue of the Laplacian operator in the spatial domain in the frequency domain, and plays a role in high-frequency suppression and regularization.

[0030] S42. With the flow field velocity variable fixed, the spatial gradient minimization subproblem under L1 regularization is solved using the soft threshold operator. The first auxiliary variable in the (k+1)th iteration is calculated, achieving anisotropic smoothing of the flow field space. This ensures that the output flow field data is clean and clearly preserves the physical boundaries. The calculation formula is as follows: In the above formula, This represents the first auxiliary variable (characterizing the spatial gradient) obtained in the (k+1)th iteration. Represents the spatial gradient operator. The soft threshold operator is mathematically defined as follows: ,in For symbolic functions, It is a function with maximum value. The shrinkage threshold of the soft threshold operator is expressed by the following formula: ,in The weighting coefficients are the total variation regularization terms.

[0031] S43. With the flow field velocity variable fixed, singular value decomposition (SVD) is used to reconstruct the time-series flow field using a low-rank truncation method. The second auxiliary variable in the current (k+1)th iteration is calculated, effectively filtering out artifact noise that may be statistically reasonable but is unreasonable in terms of fluid dynamics evolution. The calculation formula is as follows: In the above formula, This represents the second auxiliary variable (characterizing the predicted value of the temporal evolution of the flow field) obtained from the (k+1)th iteration. The low-rank truncation reconstruction operator is calculated as follows: The input matrix is ​​subjected to singular value decomposition, retaining only the first r largest singular values ​​and their corresponding singular vectors. The matrix is ​​then reconstructed using the retained components. This indicates a timing misalignment operation, referring to the operation performed on the flow field velocity variable. Extract snapshot data from time 2 to time T and construct a time series matrix to align with the second auxiliary variable.

[0032] S44. Based on the constraint residuals of the current iteration, calculate the first and second Lagrange multipliers used for dual compensation of the constraint residuals. Apply dynamic weighted penalties to the constraint residuals that were not completely eliminated in the current iteration, thereby forcing the auxiliary variables to further approximate the transformed values ​​of the flow field velocity variables in the next iteration, until strict equality constraint closure is achieved. The calculation formula is as follows: In the above formula, Let the first Lagrange multiplier in the (k+1)th iteration be denoted as . It represents the second Lagrange multiplier in the (k+1)th iteration.

[0033] The formulas for calculating the spatial gradient constraint residual and the temporal evolution constraint residual are as follows: In the above formula, This represents the spatial gradient constraint residual in the current iteration round. This represents the temporal evolution constraint residual of the current iteration round.

[0034] S45. The preset convergence condition includes any one of the following two cases: Residual compliance condition: The maximum value of the L2 norm of the spatial gradient constraint residual and the L2 norm of the temporal evolution constraint residual calculated in the current iteration is less than or equal to the preset convergence error threshold. The number of iterations has reached the preset maximum number of iterations.

[0035] If the maximum value of the L2 norm of the spatial gradient constraint residual and the L2 norm of the temporal evolution constraint residual in the current iteration is less than or equal to a preset convergence error threshold, or if the current iteration number reaches a preset maximum iteration number, then the preset convergence condition is satisfied, the iteration stops, and the flow field velocity variable of the current iteration is output as smoothed flow field data. Otherwise, the preset convergence condition is not satisfied, the iteration number is increased by one, and the variable update step is returned to be executed. This eliminates the approximation error caused by the splitting of dependent variables, ensures that the final output flow field data strictly satisfies the definition of spatial gradient and the temporal evolution law, and guarantees the mathematical accuracy and physical rigor of the results.

[0036] This flow field data smoothing method first acquires the time-series flow field data of particle image velocimetry to be processed, rearranges it based on spatial discrete points using vectorization, and constructs a time-series flow field snapshot matrix. Then, it constructs a unified constraint objective function that integrates a total variational regularization term for constraining spatial gradient sparsity, a dynamic mode decomposition dynamic constraint term for constraining linear dynamic evolution, and a data fidelity term. Next, it performs variable splitting on the objective function, and constructs an augmented Lagrangian function by introducing first and second auxiliary variables representing the spatial gradient and time-series evolution prediction values ​​of the flow field, respectively, along with corresponding Lagrange multipliers and quadratic penalty terms. Finally, it iteratively solves the augmented Lagrangian function using the alternating direction multiplier method, alternately performing variable update and multiplier update steps during iteration until the convergence condition is met, thereby outputting the smoothed flow field data.

[0037] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0038] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. Since the above embodiments are substantially similar to the method embodiments, their descriptions are relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

[0039] The above embodiments provide a detailed description of the present invention. Specific examples have been used to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of the present invention. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A flow field data smoothing method based on augmented Lagrangian constraints and DMD decomposition, characterized in that, The method includes the following steps: S1. Obtain the time-series flow field data of particle image velocimetry to be processed, rearrange the time-series flow field data into vectors based on spatial discrete points, and construct the time-series flow field snapshot matrix according to the time sampling order; S2. Construct a unified constraint objective function for flow field optimization, which includes a total variation regularization term to constrain the sparsity of the spatial gradient of the flow field, a dynamic mode decomposition dynamic constraint term to constrain the linear dynamic evolution of the flow field, and a data fidelity term to maintain consistency with the time-series flow field data. S3. Perform variable splitting on the unified constraint objective function, introduce a first auxiliary variable to characterize the spatial gradient of the flow field and a second auxiliary variable to characterize the predicted value of the flow field temporal evolution, and combine them with the preset Lagrange multipliers and quadratic penalty terms to construct an augmented Lagrange function. S4. The augmented Lagrangian function is solved iteratively using the alternating direction multiplier method. In each iteration, the variable update step and the multiplier update step are executed sequentially until the preset convergence condition is met, and the smoothed flow field data is output.

2. The flow field data smoothing method according to claim 1, characterized in that, Step S1 specifically includes the following steps: S11. Obtain the flow field velocity field data at T consecutive sampling times within the experimental observation area, where the flow field at each time time contains the velocity values ​​of N spatial discrete points; S12. Rearrange the two-dimensional or three-dimensional flow field velocity data at the t-th sampling time into a one-dimensional column vector according to the preset spatial index order to obtain the instantaneous flow field velocity vector. S13. Arrange the instantaneous flow field velocity vectors corresponding to all T sampling times as column vectors according to their time sequence to construct a time-series flow field snapshot matrix.

3. The flow field data smoothing method according to claim 1, characterized in that, Step S2 specifically includes the following steps: S21. Construct a constraint term for the spatial gradient of the flow field velocity variable using the L1 norm, so as to preserve the spatial edge characteristics of the flow field while removing noise. S22. Use the L2 norm to construct residual constraint terms between the flow field velocity variables and the time-series flow field snapshot matrix to prevent the optimization results from deviating from the original observation data; S23. Introduce a linear evolution operator and use the L2 norm to construct the dynamic residual constraint terms between the current flow field after linear evolution and the flow field at the next time step; S24. Combine the preset weight coefficients to construct a unified constraint objective function.

4. The flow field data smoothing method according to claim 3, characterized in that, The expression for the unified constraint objective function is as follows: ; in, This represents the unified constraint objective function. This represents the optimized time-series flow field data to be solved. Represents the time-series flow field snapshot matrix. The spatial gradient represents the velocity variable in the flow field. This represents the linear evolution matrix obtained from dynamic mode decomposition. This represents the instantaneous flow field velocity vector of u at the t-th sampling time. This represents the instantaneous flow field velocity vector of u at the (t+1)th sampling time. This represents the total number of sampling frames for the flow field data. The weight coefficients represent the total variation regularization term. This represents the weighting coefficient of the data fidelity item. The weighting coefficients represent the dynamic evolution constraint terms. Describing the L1 norm, This represents the L2 norm.

5. The flow field data smoothing method according to claim 4, characterized in that, Step S3 specifically includes the following steps: S31. Introduce the first auxiliary variable and the second auxiliary variable respectively, and establish their spatial gradient constraint relationship and temporal evolution constraint relationship with the flow field velocity variable; S32. For spatial gradient constraint relationship and temporal evolution constraint relationship, respectively introduce the first Lagrange multiplier and the first penalty parameter corresponding to the first auxiliary variable, and the second Lagrange multiplier and the second penalty parameter corresponding to the second auxiliary variable; S33. Replace the total variation regularization term in the unified constraint objective function with the L1 norm term about the first auxiliary variable, replace the dynamic mode decomposition dynamic constraint term with the residual term about the second auxiliary variable and the linear evolution result of the flow field, and add the linear multiplier term and quadratic penalty term corresponding to the constraint conditions to construct the augmented Lagrangian function.

6. The flow field data smoothing method according to claim 5, characterized in that, The expression for the augmented Lagrange function is: ; in, This represents the augmented Lagrange function. Indicates the first auxiliary variable. This represents the value of the second auxiliary variable at time t. Represents the first Lagrange multiplier. Indicates the second Lagrange multiplier. Indicates the first penalty parameter. This represents the second penalty parameter.

7. The flow field data smoothing method according to claim 1, characterized in that, The variable update step includes a flow field velocity update sub-step, a spatial gradient update sub-step, and a dynamic evolution update sub-step. In step S4, the specific steps are as follows: S41. Fixing the first auxiliary variable, the second auxiliary variable, and all Lagrange multipliers, the spatial difference operation in the solution process of the flow field velocity variable is transformed into a frequency domain multiplication operation using the fast Fourier transform. The flow field velocity variable in the current (k+1)th iteration is calculated using the following formula: ; In the above formula, This represents the flow field velocity variable calculated in the (k+1)th iteration. This represents the Fast Fourier Transform operator. This represents the inverse Fast Fourier Transform operator. Represents the time-series flow field snapshot matrix. This represents the first auxiliary variable obtained in the k-th iteration. This represents the second auxiliary variable obtained from the k-th iteration. This represents the first Lagrange multiplier corresponding to the spatial gradient constraint, calculated in the k-th iteration. This represents the second Lagrange multiplier corresponding to the temporal evolution constraint, calculated in the k-th iteration. This represents the first penalty parameter corresponding to the spatial gradient constraint. This represents the second penalty parameter corresponding to the temporal evolution constraint. This represents the weighting coefficient of the data fidelity item. Represents the discrete divergence operator. Represents the square of the discrete wavenumber in the frequency domain; S42. With the flow field velocity variable fixed, solve the spatial gradient minimization subproblem under L1 regularization using the soft threshold operator. Calculate the first auxiliary variable for the (k+1)th iteration using the following formula: ; In the above formula, This represents the first auxiliary variable (characterizing the spatial gradient) obtained in the (k+1)th iteration. Represents the spatial gradient operator, This represents the soft threshold operator. This represents the shrinkage threshold of the soft threshold operator; S43. With the flow field velocity variable fixed, the time-series flow field is reconstructed using singular value decomposition with low-rank truncation. The second auxiliary variable for the current (k+1)th iteration is calculated using the following formula: ; In the above formula, This represents the second auxiliary variable calculated in the (k+1)th iteration. This represents the low-rank truncation reconstruction operator. Indicates a timing misalignment operation; S44. Based on the constraint residuals of the current iteration, calculate the first and second Lagrange multipliers used for dual compensation of the constraint residuals; S45. Determine whether the spatial gradient constraint residual and the temporal evolution constraint residual satisfy the corresponding preset convergence conditions; If not, proceed to the next iteration; If so, stop the iteration and output the current flow field velocity variable as the smoothed flow field data.

8. The flow field data smoothing method according to claim 7, characterized in that, The formulas for calculating the first and second Lagrange multipliers are as follows: ； ； In the above formula, Let the first Lagrange multiplier in the (k+1)th iteration be denoted as . Denotes the second Lagrange multiplier in the (k+1)th iteration. This represents the spatial gradient constraint residual in the current iteration round. This represents the temporal evolution constraint residual of the current iteration round.

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