Improved DMD (Digital Micromirror Device) method suitable for internal flow analysis of rotating machinery

Abstract

The invention discloses an improved DMD (Digital Mode Decomposition) method suitable for internal flow analysis of a rotating machine, and particularly relates to an improved DMD (Digital Mode Decomposition) method based on an energy truncation SVET (Singular Value Element Truncation), which is particularly suitable for the internal flow analysis of the rotating machine. A traditional DMD method is easy to be interfered by low frequency during analysis of internal flow of a rotating machine with relatively low dominant frequency, and a real dominant mode is difficult to accurately capture, so that the analysis precision is reduced. According to the method, the energy truncation strategy is introduced, the reasonable truncation point is determined according to the energy proportion of the singular value, the low-energy low-frequency interference mode is eliminated, and the data matrix dimension is effectively reduced, so that the dominant mode of the internal flow of the rotating machine is accurately captured. According to the method, the identification capability of the real flow mode in the rotating machine is remarkably enhanced, the analysis precision and efficiency of the flow characteristics in the rotating machine are improved, and reliable theoretical support and technical guidance are provided for design optimization of the rotating machine.

Description

Technical Field

[0001] The present invention belongs to the technical field of fluid mechanics and engineering applications, and in particular relates to an improved dynamic modal decomposition method, which is particularly suitable for analyzing the internal flow field of rotating machinery. Background Art

[0002] Rotating machinery plays an important role in numerous fields, and its internal flow fields exhibit unsteady and complex characteristics. While dynamic modal decomposition (DMD) methods are widely used for flow analysis, traditional methods suffer from significant low-frequency interference when identifying the internal flow fields of rotating machinery with relatively low dominant frequencies. This makes it difficult to identify the dominant mode, reduces reconstruction errors, and impacts analysis accuracy and the optimal design of rotating machinery. Summary of the Invention

[0003] The present invention discloses an improved DMD method suitable for internal flow analysis of rotating machinery, aiming to improve the accuracy and stability of unsteady flow field data analysis, and is particularly suitable for main mode identification and flow field reconstruction of complex flow systems in low main frequency rotating machinery.

[0004] The DMD method is a typical data-driven decomposition technique. Its basic assumption is that the system can be described by a linear dynamic model. Modal features are extracted by constructing linear mappings between state matrices. Traditional DMD methods have two approaches for dealing with high-dimensional, low-frequency-dominated flow systems. One approach is to retain only the first r orders of singular value decomposition results, disregarding the energy distribution, regardless of whether the first r singular values ​​contribute 60% or 99%. The other approach is to retain all singular values ​​while introducing redundant modes. Both approaches result in increased reconstruction errors and inaccurate identification of primary modes.

[0005] To address these issues, the present invention proposes a singular value energy accumulation truncated SVET strategy to improve the modal extraction and flow field reconstruction process of the DMD method. This method achieves technical improvements through the following core steps.

[0006] An improved DMD method suitable for internal flow analysis of rotating machinery includes the following steps:

[0007] (1) A total of m snapshots of the flow field inside the rotating machinery are collected at a sampling time interval Δt, and the flow field state matrix X is constructed;

[0008] (2) Divide the flow field state matrix X into a high-dimensional matrix X D With X' D ; Derive a mapping matrix A D , so that the matrix X D and X' D There is a linear relationship;

[0009] (3) For the high-dimensional data matrix X DPerform singular value decomposition to obtain several singular values ​​μ i ;

[0010] (4) Construct the energy function E based on the cumulative energy ratio of the square of the singular value T (z), setting an energy retention threshold γ, selecting the main mode associated with the first z singular values ​​whose retention energy is not less than the threshold as the final singular value cutoff point; used to construct a low-dimensional matrix to eliminate low-energy interference modes and achieve data dimensionality reduction processing;

[0011] (5) Reconstruct the mapping matrix after energy truncation, calculate the DMD eigenvalues ​​and modes, and reconstruct and output the flow field characteristics.

[0012] The unsteady flow system inside the rotating machinery is approximated as a linear system, and its behavior can be described by a linear differential equation:

[0013]

[0014] Where x(t) is the system state vector, A is the system matrix, b(t) is the external excitation, and t is time. In the absence of external force input, the system solution is:

[0015] x(t)=e At x(0) (2)

[0016] Among them, x(0) is the initial state, e At Used to describe changes in state over time.

[0017] In step (1), m snapshots of an unsteady flow system inside a rotating machinery (such as a centrifugal pump, compressor, or turbine) are obtained by numerical simulation or experimental means with a sampling time interval of Δt. The sampling time interval Δt satisfies the following relationship:

[0018]

[0019] in,

[0020] λ is the rotation angle step of the rotating component (preferably λ=4);

[0021] θ is the current angular step of the rotating part of the rotating machinery;

[0022] ω is the angular velocity of the rotating parts of the rotating machinery;

[0023] The snapshot selection time range is limited to the last five cycles after the system stabilizes, namely:

[0024] t range ∈[t end -5T,t end ]

[0025] The number of snapshots m and the total sampling time T t and the time interval Δt satisfy the following relationship:

[0026]

[0027]

[0028] Where m is the total number of snapshots collected; k is the number of retention cycles (preferably k=5); N is the impeller speed;

[0029] Combine the total number of snapshots collected into a set of flow field state matrices X:

[0030] X=[x1,x2,...,x m ] (6)

[0031] Each column of X represents a snapshot of the flow field at a different moment.

[0032] In step (2), the flow field snapshots are arranged in time to construct two high-dimensional state matrices:

[0033] X D =[x1,x2,...,x m-1 ] (7)

[0034] X' D =[x2,x3,...,x m ] (8)

[0035] Based on the linear principle of equations (1) and (2), combined with equations (7) and (8), a mapping matrix A can be derived: D , so that the matrix X D and X' D There is a linear relationship, expressed as:

[0036] X' D =A D X D (9)

[0037] In step (3), the matrix X D Perform singular value decomposition:

[0038] X D =UΣV * (10)

[0039] Among them, U is the left singular matrix, which contains the spatial characteristics of the flow field, Σ is the diagonal matrix containing the singular values, V * is the conjugate transpose of the right singular matrix, describing the time characteristics.

[0040] in,

[0041] In step (4), the cumulative energy ratio of the first z singular values ​​is calculated:

[0042]

[0043] Among them, μ i is the i-th singular value, r is X D The matrix rank of is obtained by singular value decomposition.

[0044] Set the energy retention threshold γ. The energy retention threshold γ and the impeller speed N satisfy the following relationship:

[0045]

[0046] Wherein, N is the rotating speed of the impeller of the rotating machinery.

[0047] Starting from z=1, the cumulative calculation is continued until the

[0048] E T (z)≥γ (14)

[0049] The minimum integer z at this time is the final singular value truncation point.

[0050] In step (5), the first z singular values ​​and corresponding vectors are retained after energy truncation, and the dimensionality reduction expression is obtained:

[0051]

[0052] In the formula, the variables with a tilde ~ correspond to the data after truncation according to the energy retention threshold.

[0053] Furthermore, a truncated mapping matrix can be derived So that the matrix and There is a linear relationship:

[0054]

[0055] Using the singular value decomposition results after energy truncation, the dimension-reduced linear mapping matrix of the system is calculated according to formula (15) and formula (16):

[0056]

[0057] Among them, the matrix Characterize the system dynamics evolution after dimensionality reduction.

[0058] right Perform eigenvalue decomposition:

[0059] Among them, λ is the eigenvalue and ω is the corresponding eigenvector.

[0060] Calculate the DMD spatial mode φ:

[0061] Among them, according to the eigenvalue λ, the growth rate σ of the mode is calculated:

[0062] According to the eigenvalue λ, the frequency f of the mode is calculated:

[0063] Calculate the initial modal contribution coefficient b, which is the projection of the first snapshot on each mode:

[0064] b=φ + x1 (22)

[0065] Among them, φ + represents the pseudo-inverse of φ.

[0066] Construct the Vandermonde time evolution matrix V f :

[0067]

[0068] Realize the reconstruction expression of each dominant mode over time t

[0069]

[0070] in, Each column of represents the reconstructed state at time t.

[0071] The present invention has the following beneficial effects:

[0072] (1) Effectively suppress low-frequency redundant modal interference and improve the accuracy of main mode extraction;

[0073] (2) Significantly reduce the flow field reconstruction error, which is reduced by more than 50% compared with the traditional DMD method;

[0074] (3) Adapt to the modeling requirements of low-frequency complex systems, especially for non-periodic vortex structure systems such as rotating machinery;

[0075] (4) Improve computational efficiency and physical interpretability, retain physically meaningful main modes and speed up the numerical solution process.

[0076] (5) While maintaining the excellent data-driven characteristics of DMD, the present invention enhances its applicability and stability in complex engineering flow field analysis, and has important engineering value and theoretical significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] Figure 1This is a flow chart of an improved DMD method suitable for rotating machinery flow analysis provided by the present invention.

[0078] Figure 2 The figure shows a comparison of the reconstruction error inside the pump impeller using the traditional DMD method and the improved DMD method.

[0079] Figure 3 The relationship between the growth rate and frequency in the impeller under the improved DMD method is shown as an example.

[0080] Figure 4 The relationship between amplitude and frequency in the impeller under the improved DMD method is shown as an example.

[0081] Figure 5 The traditional DMD method is used as an example to capture the energy distribution at different frequencies within the impeller.

[0082] Figure 6 The improved DMD method used as an example captures the energy distribution at different frequencies within the impeller.

[0083] Figure 7 The spatial distribution cloud diagram of the first-order pressure mode of the improved DMD method is shown as an example. DETAILED DESCRIPTION

[0084] The present invention will be described in further detail below through specific examples. The following examples are provided to enable those skilled in the art to more fully understand the present invention, but are not intended to limit the present invention in any way.

[0085] This embodiment uses the pressure field dataset of the first-stage impeller of a deep-sea mining pump as an example to illustrate the implementation process of the present invention:

[0086] like Figure 1 As shown, this embodiment provides an improved DMD method suitable for internal flow analysis of rotating machinery, which mainly includes the following steps:

[0087] (1) A total of m snapshots of the flow field inside the rotating machinery are collected at a sampling time interval Δt, and the flow field state matrix X is constructed;

[0088] (2) Divide the flow field state matrix X into a high-dimensional matrix X D With X' D ;

[0089] (3) For the high-dimensional data matrix X D Perform singular value decomposition to obtain several singular values ​​μ i ;

[0090] (4) Set the energy retention threshold γ and construct the energy function E based on the cumulative energy ratio of the square of the singular value T(z), selecting the modes corresponding to the first z singular values ​​whose energy is not less than the threshold, and using them to construct a low-dimensional feature subspace to eliminate low-energy interference modes and reduce data dimensionality;

[0091] (5) Reconstruct the truncated mapping matrix, calculate the DMD eigenvalues ​​and modes, and reconstruct and output the flow field characteristics.

[0092] CFX was used to calculate the internal flow of a deep-sea mining pump. Following the method described in step (1), flow field data were saved every 4° as a sampling interval. A total of 90 flow field data points were saved for each impeller rotation. The impeller flow field data for the last five impeller rotation cycles were statistically analyzed, resulting in a total of 450 pressure field snapshots. The pressure field snapshots were exported as matrices and imported into MATLAB for data processing.

[0093] Every 4° is a time interval, and 450 column vectors are formed. Each column is the pressure field at a different moment, and the row vector is the pressure data value corresponding to different position points in the pressure field.

[0094] The flow field is arranged according to time, and the matrix is ​​obtained:

[0095] X=[x1,x2,...,x m ] (6)

[0096] Construct two high-dimensional state matrices:

[0097] X D =[x1,x2,...,x m-1 ] (7)

[0098] X' D =[x2,x3,...,x m ] (8)

[0099] Where m is the total number of snapshots, x i Represents the pressure field dataset corresponding to the i-th moment;

[0100] Formula (7) and formula (8) are approximated as a linear system to form a relationship.

[0101] X' D =A D X D (9)

[0102] For matrix X D Perform singular value decomposition:

[0103] X D =UΣV * (10)

[0104] Among them, U is the left singular matrix, which contains the spatial characteristics of the flow field, Σ is the diagonal matrix containing the singular values, V * is the conjugate transpose of the right singular matrix, describing the time characteristics.

[0105] in,

[0106] Calculate the cumulative energy ratio of the first z singular values:

[0107]

[0108] Among them, μ i is the i-th singular value, r is the matrix rank, obtained by singular value decomposition.

[0109] Set the energy retention threshold γ. The energy retention threshold γ and the impeller speed N satisfy the following relationship:

[0110]

[0111] N is the speed of the rotating mechanical impeller.

[0112] Start calculating cumulatively from z=1 until you find:

[0113] E T (z)≥γ (14)

[0114] Determine the final singular value cutoff point z as 3.

[0115] After energy truncation, the first z singular values ​​and corresponding vectors are retained to obtain the dimensionality reduction expression:

[0116]

[0117] In the formula, the variables with a tilde ~ correspond to the data after truncation according to the energy retention threshold.

[0118] A truncated mapping matrix can be derived So that the matrix and There is a linear relationship:

[0119]

[0120] Using the truncated singular value decomposition results, the system's dimensionality reduction linear mapping matrix is ​​calculated according to formula (15) and formula (16):

[0121]

[0122] matrix Characterize the system dynamics evolution after dimensionality reduction.

[0123] right Perform eigenvalue decomposition:

[0124]

[0125] λ is the eigenvalue and ω is the corresponding eigenvector.

[0126] Calculate the DMD mode φ:

[0127]

[0128] Each column of φ represents a spatial mode of each order, Figure 7 The spatial distribution cloud diagram of the first-order pressure mode. The first-order pressure mode consists of four pairs of alternating positive and negative pressures, corresponding to the main frequency of the pressure pulsation in the impeller (the guide vane frequency f gv ).

[0129] Based on the eigenvalue λ, calculate the mode growth rate σ:

[0130]

[0131] According to the eigenvalue λ, the frequency f of the mode is calculated:

[0132]

[0133] According to the growth rate and frequency, the relationship between the growth rate and frequency can be realized, such as Figure 2 Mode 0 represents the average mode, reflecting the steady-state part of the flow field that does not change with time, so its growth rate is 0; Mode 1 and Mode 2 have relatively small growth rates, corresponding to the first and second mode of DMD analysis. The initial modal contribution coefficient b is calculated, that is, the projection of the first snapshot on each mode:

[0134] b=φ + x1 (22)

[0135] φ + represents the pseudo-inverse of φ.

[0136] The energy matrix of each mode can be expressed as E K :

[0137] E K =||φ|| 2 |b| 2 (26)

[0138] According to the frequency corresponding to the energy matrix, the energy corresponding to each mode can be realized to observe the dominant mode with the largest energy.

[0139] Figure 5The relationship diagram between mode and energy obtained by the traditional DMD method is shown in the figure. The second-order mode (2f gv ) is disturbed by the low-frequency mode, and the energy occupied by the low frequency in the energy sorting process is significantly higher than that of the second-order mode; Figure 6 The improved DMD method accurately captures the first-order mode with the largest energy (f gv ) and the second-order mode.

[0140] The amplitude is defined by b of the DMD mode, using the formula:

[0141] α=|b| (27)

[0142] According to the amplitude and frequency, the amplitude corresponding to different modes can be plotted to observe the mode with the largest amplitude, such as Figure 3 As shown in Figure 1, Mode 1 and Mode 2 correspond to the first-order mode and the second-order mode of DMD analysis, respectively, and their corresponding energies are the largest.

[0143] Construct the Vandermonde time evolution matrix V f :

[0144]

[0145] Realize the reconstruction expression of each dominant mode over time:

[0146]

[0147] The time evolution characteristics of the i-th mode are obtained, and its expression is:

[0148]

[0149] The matrix structure is:

[0150]

[0151] Each represents the contribution of the i-th spatial mode at the t-th time point.

[0152] The reconstruction error is used to measure the difference between the flow field reconstructed by the DMD method and the original flow field. The smaller the error, the higher the accuracy of the reconstructed flow field. The expression is as follows:

[0153]

[0154] Figure 4The reconstruction errors of the traditional DMD method and the improved DMD method under different cycles are as follows. When the number of cycles is 5, the reconstruction error of the traditional DMD method is greater than 13%. The reconstruction error of the improved DMD method is less than 0.7% under different cycle numbers. The traditional DMD method has a large error in the process of reconstructing the flow field characteristics inside the deep-sea mining pump impeller, especially when the number of cycles is 5. The improved DMD method significantly improves the reconstruction accuracy.

[0155] Although the preferred embodiments of the present invention have been described in detail above with reference to the accompanying drawings, it should be understood that the contents are merely for illustrating the principles and technical implementations of the present invention and do not constitute a limitation on the scope of protection of the present invention. Those skilled in the art may, without departing from the basic concept and core technical solution of the present invention, make various equivalent variations, substitutions, or improvements based on the technical teachings of the present invention, and such variations or improvements shall also be within the scope defined by the claims of the present invention.

Claims

1. An improved DMD method suitable for internal flow analysis of rotating machinery, characterized in that: The steps include: (1) A total of m snapshots of the flow field inside the rotating machinery are collected at a sampling time interval Δt, and the flow field state matrix X is constructed; (2) Divide the flow field state matrix X into a high-dimensional matrix X D With X' D ; Derive a mapping matrix A D , so that the matrix X D and X' D There is a linear relationship; (3) For the high-dimensional data matrix X D Perform singular value decomposition to obtain several singular values ​​μ i ; (4) Construct the energy function E based on the cumulative energy ratio of the square of the singular value T (z), setting an energy retention threshold γ, selecting the main mode associated with the first z singular values ​​whose retention energy is not less than the threshold as the final singular value cutoff point; used to construct a low-dimensional matrix to eliminate low-energy interference modes and achieve data dimensionality reduction processing; (5) Reconstruct the mapping matrix after energy truncation, calculate the DMD eigenvalues ​​and modes, and reconstruct and output the flow field characteristics.

2. The improved DMD method for internal flow analysis of rotating machinery as claimed in claim 1, characterized in that: In step (1), m snapshots with a sampling time interval of Δt in the unsteady flow system inside the rotating machinery are obtained by numerical simulation or experimental means. The sampling time interval Δt satisfies the following relationship: in, λ is the rotation angle step of the rotating component (preferably λ=4); θ is the current angular step of the rotating part of the rotating machinery; ω is the angular velocity of the rotating parts of the rotating machinery; The snapshot selection time range is limited to the last five cycles after the system stabilizes, namely: t range ∈[t end -5T,t end ] The number of snapshots m and the total sampling time T t and the time interval Δt satisfy the following relationship: Where m is the total number of snapshots collected; k is the number of retention cycles; N is the impeller speed; Combine the total number of snapshots collected into a set of flow field state matrices X: X=[x1,x2,...,x m ] (6) Each column of X represents a snapshot of the flow field at a different moment.

3. The improved DMD method for internal flow analysis of rotating machinery as claimed in claim 2, characterized in that: In step (1), the rotating machinery includes a centrifugal pump, a compressor or a turbine; and the number of cycles k is retained as 5.

4. The improved DMD method for internal flow analysis of rotating machinery as claimed in claim 1, characterized in that: In step (2), The unsteady flow system inside the rotating machinery is approximated as a linear system, and its behavior can be described by a linear differential equation: Where x(t) is the system state vector, A is the system matrix, b(t) is the external excitation, and t is time. In the absence of external force input, the system solution is: x(t)=e At x(0) (2) Among them, x(0) is the initial state, e At Used to describe changes in state over time.

5. The improved DMD method for internal flow analysis of rotating machinery as claimed in claim 2 or 4, characterized in that: In step (2), the flow field snapshots are arranged in time to construct two high-dimensional state matrices: X D =[x1,x2,...,x m-1 ] (7) X' D =[x2,x3,...,x m ] (8) Based on the linear principle of equations (1) and (2), combined with equations (7) and (8), a mapping matrix A can be derived: D , so that the matrix X D and X' D There is a linear relationship, expressed as: X' D =A D X D (9)。 6. The improved DMD method for internal flow analysis of rotating machinery as claimed in claim 1, characterized in that: In step (3), the matrix X D Perform singular value decomposition: X D =UΣV * (10) Among them, U is the left singular matrix, which contains the spatial characteristics of the flow field, Σ is the diagonal matrix containing the singular values, V * is the conjugate transpose of the right singular matrix, describing the time characteristics; in, 7. The improved DMD method for internal flow analysis of rotating machinery as claimed in claim 1, characterized in that: In step (4), the cumulative energy ratio of the first z singular values ​​is calculated: Among them, μ i is the i-th singular value, r is X D The matrix rank of is obtained by singular value decomposition; Set the energy retention threshold γ. The energy retention threshold γ and the impeller speed N satisfy the following relationship: Wherein, N is the rotating machine impeller speed; Starting from z=1, the cumulative calculation is continued until the E T (z)≥γ (14) The minimum integer z at this time is the final singular value truncation point.

8. The improved DMD method for internal flow analysis of rotating machinery as claimed in claim 1, characterized in that: In step (5), the first z singular values ​​and corresponding vectors are retained after energy truncation, and the dimensionality reduction expression is obtained: In the formula, the variables with tilde ~ correspond to the data after truncation according to the energy retention threshold; Furthermore, a truncated mapping matrix can be derived So that the matrix and There is a linear relationship: Using the singular value decomposition results after energy truncation, the dimension-reduced linear mapping matrix of the system is calculated according to formula (15) and formula (16): Among them, the matrix Characterize the system dynamics evolution after dimensionality reduction; right Perform eigenvalue decomposition: Among them, λ is the eigenvalue and ω is the corresponding eigenvector; Calculate the DMD spatial mode φ: Among them, according to the eigenvalue λ, the growth rate σ of the mode is calculated: According to the eigenvalue λ, the frequency f of the mode is calculated: Calculate the initial modal contribution coefficient b, which is the projection of the first snapshot on each mode: b=φ + x1 (22) Among them, φ + represents the pseudo-inverse of φ; Construct the Vandermonde time evolution matrix V f : Realize the reconstruction expression of each dominant mode over time t in, Each column of represents the reconstructed state at time t.

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