CN110826197A - Wind speed field simulation method based on improved Cholesky decomposition closed solution - Google Patents
Wind speed field simulation method based on improved Cholesky decomposition closed solution Download PDFInfo
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Abstract
The invention discloses a wind speed field simulation method based on an improved Cholesky decomposition closed solution, which comprises the following steps of firstly, calculating to obtain a corresponding power spectrum matrix according to a given power spectrum of a horizontal stationary wind speed field simulation point; converting Cholesky decomposition of the power spectrum matrix into decomposition of a coherent matrix to obtain a closed solution expression; then, obtaining a simulation sample by a closed solution expression and a stable simulation formula; then, the closed solution is popularized from a horizontal wind speed field to a vertical wind speed field and an inclined wind speed field; finally, the improved closed solution is further applied to the simulation of the non-stationary wind speed field. The invention can consider the simulation points randomly distributed along the horizontal axis, the self-spectrums of all the points can be different, and the traveling wave effect can be considered; under the condition of extremely small approximation, the closed solution is also expanded to vertical and oblique wind speed fields, and has deeper significance for simulation calculation in practice. Furthermore, the method can also be used for non-stationary wind velocity field simulation and random vibration analysis based on the decomposition of the coherence matrix.
Description
Technical Field
The invention relates to the technical field of random signal simulation, in particular to a wind speed field simulation method based on an improved Cholesky decomposition closed solution.
Background
Simulations of wind velocity fields have been widely applied in wind-resistant design of structures, especially when structural and aerodynamic nonlinearities are considered. Among the various simulation methods, the Spectral Representation (SRM) is the most popular method in wind field simulation because of its accuracy and immediacy. Yang realizes the superposition of trigonometric function terms by introducing Fast Fourier Transform (FFT), and remarkably improves the Simulation efficiency of a steady wind field (Yang, J.N (1972) 'Simulation of random evolution processes,' J.Sound VIb.,21(1), 73-85.). In addition, the application of FFT has also been extended to non-stationary wind field simulations (Zhao, n., and Huang, G. (2017). "Fast correlation of multivariate statistical process and entity application to environment windows." j.wind.end.ind.aerodyne., 170, 118. 127.).
Despite these advances, there remains a need to improve simulation efficiency, particularly for large structures. For example, the main spans of a completed mingshi strait bridge and a planned mexican strait bridge are 1991 and 3300 meters, respectively; the length of a multi-span transmission line can usually reach several kilometers. In these cases, hundreds or even more simulation points are often needed to simulate the wind speed field in the time domain response analysis.
To further speed up the simulation, most efforts have focused on the Cholesky decomposition of the spectral matrix. Yang et al (Yang, W.W., Chang, T.Y.P., and Chang, C.C. (1997) "An effective with field modulation technique for bridges." J.Wind.Eng.Ind.Aedodyn., 67, 697-. Although the closed-form formulation of Cholesky decomposition is very attractive, it suffers from one major limitation: the simulation points must be evenly distributed along the horizontal axis.
Disclosure of Invention
In view of the above problems, the present invention aims to provide a wind speed field simulation method based on an improved Cholesky decomposition closed solution, which can be used for simulating a wind speed field on a horizontal axis, a vertical axis or even an inclined axis, and has important engineering application value. The technical scheme is as follows:
a wind speed field simulation method based on an improved Cholesky decomposition closed solution comprises the following steps:
step 1: calculating according to the given power spectrum of the simulation point of the horizontal stationary wind speed field to obtain a corresponding power spectrum matrix;
step 2: converting Cholesky decomposition of the power spectrum matrix into decomposition of a coherent matrix to obtain a closed solution expression;
and step 3: obtaining a simulation sample by a closed solution expression and a stable simulation formula;
and 4, step 4: the closed solution is popularized from a horizontal wind speed field to a vertical and inclined wind speed field;
and 5: the improved closed-loop solution is further applied to the simulation of non-stationary wind speed fields.
Further, the power spectrum matrix in step 1 is represented as:
S(ω)＝D(ω)Γ(ω)DT(ω) (1)
wherein
Wherein D (ω) represents a diagonal matrix associated with the self-spectrum; sjj(ω) represents the self-power spectral function of the jth analog point; Γ (ω) denotes a coherence matrix, which is a non-negative Hermite matrix, γjk(ω) represents the coherence function between the simulation points j and k; j is 1,2, …, n; k is 1,2, …, n.
Further, the step 2 specifically includes:
decomposing the coherence matrix Γ (ω) into
Γ(ω)＝B(ω)BT*(ω) (4)
Where B (ω) is a lower triangular matrix, represented by
In the formula, βjk(ω) represents the corresponding element of the lower triangular matrix B (ω); j is 1,2, …, n; j is more than or equal to k;
accordingly, the following relationship is obtained
H(ω)＝D(ω)B(ω) (6)
The above process replaces the decomposition of the power spectrum matrix with the decomposition of the coherence matrix, as shown in formula (4); then the auto-PSD for each simulation point may be different when applying the improved closing solution;
considering the traveling wave effect, a generalized coherence function is as follows
In the formula vappIs the wave velocity; cyIs a horizontal exponential decay factor; y isjIs a j point horizontal coordinate value; y iskIs a horizontal coordinate value of the k point; omega is the circular frequency;
is the average wind speed value;
for the simulation points arbitrarily distributed along the horizontal axis, Cholesky decomposition of the coherence matrix composed of coherence functions is calculated by the following closed equation
Further, the step 3 specifically includes:
and (3) substituting corresponding elements into the following formula according to the closed formula obtained in the step (2) to obtain a simulation sample:
in the formula, N is the number of discrete points; omegalIs the l-th discrete frequency, △ omega is the frequency step, thetajkIs a phase angle; phi is aklIs a random phase angle.
Further, the step 4 specifically includes:
the coherence function of the vertical wind velocity field and the corresponding closed solution are expressed as follows:
in the formula, CzIs a vertical exponential decay factor; z is a radical ofjIs the j point height value; z is a radical ofkIs the k-point height value, α is the power exponent relating to the roughness of the ground, U (10) is the average wind speed at 10m height;
is gammak,k-1(ω) conjugation;
the tilted wind velocity field coherence function is expressed as follows:
where κ is the slope. The closed solution corresponding to the coherent matrix decomposition is also equation (12).
Further, the step 5 specifically includes:
by performing Cholesky decomposition on the evolved power spectrum matrix, the corresponding simulation formula of the non-stationary wind speed field along any axis is obtained as follows:
in the formula (I), the compound is shown in the specification,
is the jth non-stationary random process; | Hjk(ω, t) | and θjk(ω, t) are each HjkThe mode and complex phase angle of (ω, t); hjk(ω, t) is the element of the decomposed matrix; hjk(ω, t) is calculated by the following formula:
in the formula, Sjj(ω, t) j ═ 1,2, …, n is
βjk(ω) can be determined from the modified closed solution equation previously described.
The invention has the beneficial effects that: the high-efficiency simulation method of the wind field has important significance in structural wind vibration response analysis; can be used to simulate wind speed fields in horizontal, vertical and even oblique axes; the simulation points can be distributed at will, the self-spectrums of different points can be different, and the traveling wave effect can be considered; the invention can also accelerate the simulation of non-stationary wind speed fields. The proposed method has been shown to perform well by simulations of two stationary wind speed fields in horizontal and vertical directions.
Drawings
FIG. 1 is a schematic diagram of a simulated point distribution of the present invention.
FIG. 2 is a schematic diagram of a correlation function of a sample of a simulated sample obtained by an example; the correlation function of the simulated wind speed at points 1, 2.
FIG. 3 is a schematic diagram of a correlation function of a sample of a simulated sample obtained by an example; the correlation function of the simulated wind speed at points 1 and 51.
FIG. 4 is a comparison graph of the power spectra of the sample samples obtained by the calculation; (a) point 1 and (b) PSD of simulated wind speed at point 51.
Detailed Description
The invention is described in further detail below with reference to the figures and specific embodiments. The invention discloses a wind speed field simulation method based on an improved Cholesky decomposition closed solution. The method mainly comprises the following steps:
step 1: and calculating to obtain a corresponding power spectrum matrix according to the given power spectrum of the horizontal stationary wind speed field simulation point.
S(ω)＝D(ω)Γ(ω)DT(ω) (17)
Wherein
Step 2: converting Cholesky decomposition of the power spectrum matrix into decomposition of a coherent matrix to obtain a closed solution expression;
the coherence matrix Γ (ω) is also a non-negative Hermitian matrix that can be decomposed into
Γ(ω)＝B(ω)BT*(ω) (20)
Where B (ω) is a lower triangular matrix, represented by
Accordingly, the following relationship can be obtained
H(ω)＝D(ω)B(ω) (22)
It can be seen that the decomposition of the spectral matrix can be replaced by the decomposition of the coherence matrix, as shown in equation (20). Thus, the auto-PSD for each analog point may be different.
To account for the traveling wave effect, a generalized coherence function is given as follows
In the formula, vappIs the wave velocity.
For the simulation points arbitrarily distributed along the horizontal axis, it can be shown that Cholesky decomposition of the coherence matrix composed of coherence functions can be calculated with the following closed formula
And step 3: obtaining a simulation sample by a closed solution expression and a stable simulation formula;
according to the obtained closed formula, substituting the corresponding elements into the following formula to obtain a simulation sample:
and 4, step 4: the closed solution is popularized from a horizontal wind speed field to a vertical and inclined wind speed field;
when generalizing the horizontal wind speed field to the vertical wind speed field and the oblique wind speed field, the coherence function and the closed solution of the vertical wind speed field are expressed as follows:
the tilted wind velocity field coherence function is expressed as follows:
the remaining calculations are similar to the horizontal wind velocity field.
And 5: the improved closed-loop solution is further applied to the simulation of non-stationary wind speed fields.
When a stationary wind speed field is generalized to a non-stationary wind speed field, similar to stationary simulation, by performing Cholesky decomposition on the evolved power spectrum matrix, the corresponding simulation formula of the non-stationary wind speed field along any axis can be obtained as:
in the formula
Is the jth non-stationary random process; | Hjk(ω, t) | and θjk(ω, t) are each HjkThe mode and complex phase angle of (ω, t); hjkAnd (ω, t) is the element of the decomposed matrix. Obviously, Hjk(ω, t) can be explicitly calculated using the proposed method:
in the formula Sjj(ω, t) j ═ 1,2, …, n is
From EPSD, βjk(ω) is determined by a closed solution formula.
To verify the efficiency and accuracy of the enhanced closed-solution algorithm of the proposed Cholesky decomposition, a horizontal stationary wind speed field was simulated. Through numerical calculation, the conclusion that the method performs well can be obtained.
It is assumed that a large-span suspension bridge deck wind speed field with a main span of 1000m and two side spans of 500m needs to be simulated. For the sake of simplicity, only the longitudinal wind speed component perpendicular to the bridge axis is considered. The distances between two adjacent simulation points of the side span and the main span are set to be 10m and 20m respectively. Thus, a wind speed field with 151 simulation points will be simulated, as shown in FIG. 1.
The bilateral spectrum of kalman was used as the target power spectrum for the main cross simulation point, given below
The height of the bridge surface from the ground is z 60 m; the average wind speed of the bridge deck is U (z) 40 m/s; tangential velocity u ═ kU (z)/ln (z/z)0) And k is 0.4 and the roughness of the ground is z0＝0.01m；The power spectrum of the edge-crossing simulation point is assumed to be 0.8 times the above karman spectrum. In the coherence function C y10 and vapp10m/s, is used to characterize the correlation of each simulation point.
To simulate the ergodic wind speed field, the time and frequency parameters in this simulation are given as: cut-off frequency omega u4 pi rad/s, 2048 discrete frequency points, 0.25s time step △ t, and 154624s period.
Fig. 2 shows the estimated and target autocorrelation functions for the 1, 2-point simulated wind speeds, and the cross-correlation functions and target cross-correlation functions for the 1 and 51-point simulated wind speeds. The result shows that the estimated autocorrelation function of the simulated wind speed has better consistency with the target autocorrelation function, and the cross-correlation function also has better consistency. In addition, the estimated PSD of the simulated wind speed for point 1 and point 51 is also compared to the target PSD, as shown in FIG. 3. Satisfactory consistency can be observed. Therefore, the proposed method has a good performance for the simulation of horizontal wind velocity fields.
Claims (6)
1. A wind speed field simulation method based on an improved Cholesky decomposition closed solution is characterized by comprising the following steps of:
step 1: calculating according to the given power spectrum of the simulation point of the horizontal stationary wind speed field to obtain a corresponding power spectrum matrix;
step 2: converting Cholesky decomposition of the power spectrum matrix into decomposition of a coherent matrix to obtain a closed solution expression;
and step 3: obtaining a simulation sample by a closed solution expression and a stable simulation formula;
and 4, step 4: the closed solution is popularized from a horizontal wind speed field to a vertical and inclined wind speed field;
and 5: the improved closed-loop solution is further applied to the simulation of non-stationary wind speed fields.
2. The method for simulating a wind speed field based on the improved Cholesky decomposition closed solution as claimed in claim 1, wherein the power spectrum matrix in step 1 is represented as:
S(ω)＝D(ω)Γ(ω)DT(ω) (1)
wherein
Wherein D (ω) represents a diagonal matrix associated with the self-spectrum; sjj(ω) represents the self-power spectral function of the jth analog point;
Γ (ω) denotes a coherence matrix, which is a non-negative Hermite matrix, γjk(ω) is a coherence function between the simulation points j and k;
j＝1,2,…,n；k＝1,2,…,n。
3. the wind speed field simulation method based on the improved Cholesky decomposition closed solution according to claim 2, wherein the step 2 specifically comprises:
decomposing the coherence matrix Γ (ω) into
Γ(ω)＝B(ω)BT*(ω) (4)
Where B (ω) is a lower triangular matrix, represented by
In the formula, βjk(ω) represents the corresponding element of the lower triangular matrix B (ω); j is 1,2, …, n; j is more than or equal to k;
accordingly, the following relationship is obtained
H(ω)＝D(ω)B(ω) (6)
Replacing the decomposition of the power spectrum matrix with the decomposition of the coherence matrix, as shown in formula (4); then the self-power spectrum of each simulation point may be different when applying the improved closed solution;
considering the traveling wave effect, a generalized coherence function is as follows
In the formula vappIs the wave velocity; cyIs a horizontal exponential decay factor; y isjIs a j point horizontal coordinate value; y iskIs a horizontal coordinate value of the k point;
omega is the circular frequency;
is the average wind speed value;
for the simulation points arbitrarily distributed along the horizontal axis, Cholesky decomposition of the coherence matrix composed of coherence functions is calculated by the following closed equation
4. The wind speed field simulation method based on the improved Cholesky decomposition closed solution according to claim 3, wherein the step 3 specifically comprises:
and (3) substituting corresponding elements into the following formula according to the closed formula obtained in the step (2) to obtain a simulation sample:
in the formula, N is the number of discrete points; omegalIs the l-th discrete frequency, △ omega is the frequency step, thetajkIs a phase angle; phi is aklIs a random phase angle.
5. The wind speed field simulation method based on the improved Cholesky decomposition closed solution according to claim 4, wherein the step 4 specifically comprises:
the coherence function of the vertical wind velocity field and the corresponding closed solution are expressed as follows:
in the formula, CzIs a vertical exponential decay factor; z is a radical ofjIs the j point height value; z is a radical ofkIs the k-point height value, α is the power exponent relating to the roughness of the ground, U (10) is the average wind speed at 10m height;
is gammak,k-1(ω) conjugation;
the tilted wind velocity field coherence function is expressed as follows:
where κ is the slope.
6. The wind speed field simulation method based on the improved Cholesky decomposition closed solution according to claim 5, wherein the step 5 specifically comprises:
by performing Cholesky decomposition on the evolved power spectrum matrix, the corresponding simulation formula of the non-stationary wind speed field along any axis is obtained as follows:
in the formula (I), the compound is shown in the specification,is the jth non-stationary random process; | Hjk(ω, t) | and θjk(ω, t) are each HjkThe mode and complex phase angle of (ω, t); hjk(ω, t) is the element of the decomposed matrix; hjk(ω, t) is calculated by the following formula:
in the formula, Sjj(ω, t) j ═ 1,2, …, n is
βjk(ω) can be determined from the modified closed solution equation previously described.
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