Wind Velocity Field Simulation of the Large-Span Spatial Structure Based on the Wavelet Decomposition Method

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Introduction
Wind load is an important design load for large-span spatial structures. With the development of numerical simulation theory, computer simulation of wind velocity and wind load is increasingly applied. To study the wind load of long-span spatial structures, it is necessary to simulate the wind speed time history at a large number of points and consider the temporal and spatial correlation characteristics of the wind. At present, there are two main models widely used in wind feld simulation: harmonic superposition method [1] and linear flter method [2]. Te former is computationally intensive, especially when there are many simulated points, which is quite time-consuming. However, the AR model in the latter method simulates the stochastic process directly from the perspective of time domain, so it has less computation and faster simulation speed, especially in the wind feld simulation of large complex structures, which is more widely used. AR models can be classifed into two categories: AR models of scalar processes [3] and AR models of vector processes [4,5]. Te AR model of vector process is very commonly used in multivariate time series analysis due to its stationarity and simplicity of parameter interpolation, and the fuctuation of natural wind can be approximated as a stable Gaussian random process experienced by various states. If the wind feld is regarded as the sum of random wind waves at discrete spatial points, it can be treated as a one-dimensional multivariable random process. However, the disadvantage of the vector AR model is that it is not suitable for random feld simulation of nonuniform grids. Moreover, these processes are difcult to strengthen the local solution of random feld samples, and wavelet transform is an appropriate method that has a nonuniform grid and can detect the local similarity of time series [6]. Wavelet decomposition is widely used in signal feature extraction in various felds [7][8][9][10]. Because of its good localization characteristics in both the time domain and frequency domain, it has broad application prospects in wind engineering applications [11].
At present, there are few studies about the specifc application of wavelet analysis in wind feld simulation. Zeldin and Spanos [12] simulated one-dimensional and multidimensional random felds with Daubechies orthogonal wavelets based on the principle of linear estimation. Chen and Wang [6] simulated the fuctuating wind velocity time series at a point with wavelet analysis and compared them with the measured results. Gao et al. [13] proposed an ultrashort-term wind speed prediction method based on neural network and wavelet analysis. Zhang et al. [14] proposed to use discrete biorthogonal wavelets with a distance-dependent threshold to denoise wind speed that has large dynamic range along the data profle, and wavelet analysis is adopted to improve the accuracy of the wind velocity derived from lidar backscattering. Cava et al. [15] applied the Euler autocorrelation function and Morlet continuous wavelet transform to study the turbulence of the boundary layer at night under low wind conditions. Zheng [16] considered the diference and predictability of the frequency series obtained after the decomposition of the actual wind speed series and proposed a combined forecasting method of ultrashort-term wind speed mixed model based on wavelet decomposition to improve the prediction accuracy. Chandra et al. [17] and Doucoure et al. [18] also applied wavelet analysis to wind speed prediction. Based on the wavelet extension of autoregressive coefcients, the estimation method of vector process AR model is presented in this paper so as to simulate the wind speed feld of longspan roof.
In this study, a new method for simulating the fuctuating wind feld of large-span structures is given by combining the vector process AR model in the linear flter method and applying the wavelet decomposition method. In this method, the autoregressive coefcients of the vector process AR model is spatially extended by wavelet, and the least square method is applied to estimate the autoregressive coefcients of the AR model, and the realization steps of the wind feld simulation of the large-span structure are given. Te method is applied to the simulation of wind velocity feld of a large-span structure, and the simulation results are compared with the target value.

Methodology
Te methodology is developed as follows.

Teory of Wavelet Decomposition.
Wavelet is a special kind of waveform with fast decay or a small area with a fnite length and a mean value of 0. In the early 20th century, the wavelet was frst discovered and used by Haar, and this wavelet was also named as Haar wavelet. In the 1980s, French scholar Morlet proposed wavelet transform, a theoretical method of time-frequency analysis based on the Fourier transform [19]. French scientist Meyer successfully constructed a smooth function with a certain attenuation and constructed the normal orthogonal basis of the L 2 (R) space through scaling and translation. Daubechies, Coifman, and Wickerhauser have also made outstanding contributions to the development of wavelet theory and engineering applications. Wavelet transform not only has the nature of frequency analysis but also can represent the time of occurrence, which is convenient for analyzing and determining the phenomenon of time. Wavelet transform has been widely used in many felds [20] such as signal processing, pattern recognition, and image processing.
Wavelet transform is developed from Fourier transform. Compared with Fourier transform, wavelet transform emphasizes the local characterization of time and frequency. Fourier transform can only get the spectrogram of the signal but not the time of appearance, and even the same spectrogram may appear for two diferent nonstationary sequences. Te wavelet transform can change adaptively, and the variable time window enables the wavelet transform to extract efective information from the signal and clearly shows the trend component and detail component of the signal, which is more advantageous than the Fourier transform [21].
Wavelet transform can be divided into three wavelet transforms: continuous wavelet transform, discrete wavelet transform, and dyadic wavelet transform [22].
If ψ(t) ∈ L 2 (R), its Fourier transform ψ(ω) satisfes the following formula: where C ψ is bounded and ψ is called a base wavelet or mother wavelet. Equation (1) is called the admissible condition of wavelet function. After the mother wavelet is stretched and translated, the wavelet sequence can be obtained as where a is the expansion factor and b is the translation factor. Te shape of the time-frequency window is determined by the value of the scaling factor a, and the position of the window is determined by the translation factor b. When both a and b take values continuously, ψ a,b (t) is called a continuous wavelet function. For ∀f(t) ∈ L 2 (R), when the function f(t) is expanded under the continuous wavelet base ψ a,b (t), the expression is Equation (3) is called continuous wavelet transform (CWT) on mother wavelet ψ.
Because the continuous wavelet transform needs to calculate the integral in the application, it is inconvenient to process digital signals, so it is mainly used for theoretical analysis and demonstration. In practical problems, the discrete form is usually used, namely, discrete wavelet transform (DWT). DWT can be obtained by discretizing the expansion factor a and the translation factor b in CWT. Usually, a � a 0 m and b � nb 0 a 0 m ; then, Equation (4) can be obtained as Te wavelet function in equation (4) is a discrete wavelet. Te discrete wavelet transform is Taking a 0 � 2 and b 0 � 1, the dyadic wavelet can be obtained as follows: In practical applications, it is usually necessary to construct a wavelet function with orthogonality to make the calculation of wavelet transform more efective:

Wind Load Simulation of the Vector AR Model.
Te AR model is commonly used in wind load simulation. Here is only a brief introduction to the equation of vector process AR model, which lays the foundation for the subsequent analysis using the wavelet method. N related fuctuating wind velocity time series v(t) � [v 1 (t), v 2 (t), · · · , v n (t)] T can be generated by the following equation: where p is the regression order of the AR model, ψ k is the N × N order autoregressive coefcient matrix, is an independent random process with undetermined zero mean and covariance.
From the nature of the stationary random process, the regular equation of the AR model can be obtained as where ψ � [I, ψ 1 , · · · ψ p ] T is an (p + 1)M × M order matrix, I is an M-order unit matrix, O p is a pM × M order matrix with all zero elements, and R is a (p + 1)M × (p + 1)M order autocorrelation matrix. Te correlation function R(kΔt) is a square matrix of order M × M, k � 0, · · · p, determined by the Wiener-Khintchine equation: where f is the frequency of fuctuating wind velocity and S ij (f) can be determined by the autospectral density function S ii (f) and coherence function r ij (f): Te large-span spatial structure has a relatively small span and little change in node height. Davenport wind velocity spectrum can be adopted [23] as where k is the surface resistance coefcient and V 10 is the average wind velocity at a height of 10 m.
Considering the spatial correlation of wind velocity time series through r ij (f) [23], its three-dimensional expression is where C x , C y , C z , respectively, represent the attenuation coefcients of left and right, up and down, and front and back of any two points in space, which are determined by experiment or actual measurement and V(z i ), V(z j ), respectively, represent the average wind velocity at the ith point and the jth point.

AR Model Parameter Estimation Based on the Wavelet
Method. Te wavelet decomposition method is used to spatially expand the autoregressive coefcients of the above AR model. Te wavelet expansion method is adopted to expand the vector process AR model coefcients in sufcient space. For any function f(x) ∈ L 2 , it can be extended to where ψ j,k (x) is the mother wavelet and β j,k � 〈f(x); ψ j,k 〉.
Regarding the above multidimensional AR model as a function of time and belonging to L 2 , here, we refer to the representation method in [24], and the model can be expressed as In the equation,

Mathematical Problems in Engineering
where A (l) j,k (l � 1, 2 · · · , j � 1, 2 · · · , k � 1, 2 · · ·) is the matrix containing wavelet expansion coefcients, u j,k is the intercept vector, A (l) j,k and u j,k contain the coefcients to be estimated, η t is the independent zero mean variable, and ζ t is the truncation error. Te above vectors are all functions of time.
Te reliability and applicability of the time-varying model mainly depends on the accuracy of its parameter estimation. Tis study adopts the least square method to determine the autoregressive coefcient of the AR model; that is, the coefcients contained in A (l) j,k and u j,k . Te commonly used estimation method for time-varying models is based on adaptive flter and window estimation, but this method is not accurate enough for the simulation of shortterm time series because the window length should not be too large at this time. At this time, the method of function expansion in enough space can be used. Tis study adopts the expansion method based on wavelet.
First, assuming that the covariance matrix (t) is known, equation (15) can be rewritten as where ⊗ represents the Kronecker product, Θ is the column vector is the 1 × 2 J order vector containing the wavelet function, Γ(t) � [c −1,0 (t), c 0,0 (t), · · · , c J,2 J−1 (t)], and Η is the matrix containing all wavelet expansion coefcients. Ten, the p-order wavelet vector AR model can be expressed as where λ is the generalized least squares estimation coefcient, and its value is determined by the following equation: Among them,

Implementation Steps of Wind Field Simulation for Large-Span Structures.
In the previous steps, it is assumed that the covariance matrix (t) is known, but the covariance matrix is usually unknown in the actual use of the least square method. For this reason, we assume that η t has zero mean and time-varying variance σ 2 (t); then, we obtain Te reasonable estimation of σ 2 (t) is the squared residual [25], so the estimation of the covariance σ 2 lm (t) of the two time series x lt and x mt that changes with time is obtained by the following wavelet expansion: where the coefcients v j,k and c j,k can be obtained by the classical wavelet smoothing of the squared residual [26], from which the estimation of the covariance matrix (t) is obtained.
Te implementation steps of simulating spatial wind feld based on wavelet method combined with the AR model are summarized as follows: (1) Preliminarily, we assume that the covariance matrix � I and estimate the generalized least squares estimation coefcients by equations (19) to (24) (2) We apply equations (26)∼ (27) to obtain estimates of variance and covariance (3) apply the covariance matrix obtained above and then use the least square method introduced above to estimate the autoregressive coefcients A (l) j,k and u j,k of the AR model (4) return to step 2 until the result converges

Case Analysis
Te case analysis is explained in the following sections.

Simulation Results.
Te method mentioned above is applied to simulate the fuctuating wind feld of a large-span rectangular fat roof structure. According to the above steps, a calculation program is compiled in MATLAB language. Trough the analysis of the example, the simulation accuracy of the method in this paper is verifed, and the accuracy and simulation efciency of the vector process AR method are compared. Tere are 444 nodes on the roof, and the geometric dimensions of the roof are shown in Figure 1. Te height of the roof above the ground is 20 m, the landform type is Class B, and the ground roughness k � 0.03. In the simulation, the regression order p � 4, the time step Δt � 0.1s, and the simulation time t � 500s. Wind velocity is taken as V 10 � 30m/s, Davenport spectrum is taken as the target spectrum, db3 wavelet is adopted, and the function expansion of four wavelets (24 coefcients) is considered, and the wind velocity time series of all nodes on the structure considering time and space correlation is calculated. Here, only the horizontally correlated wind feld is simulated, and the simulation of the vertical correlated wind feld is the same. Only the simulated wind velocity time series of representative nodes A, D, G, and J are given here, as shown in  In order to verify the correctness of the proposed method, the comparison results of the simulated wind velocity time-series power spectrum of nodes A and G and the target spectrum are given, as shown in Figure 6, and the results show that the two methods are in good agreement. In order to verify the statistical characteristics of the sample, the wind velocity time-series autocorrelation and cross-correlation functions of nodes D and J are obtained through time average and compared with the corresponding objective correlation functions. Te results show that the two are in good agreement, as shown in Figures 7 and 8.
At the same time, in order to verify the correctness of the AR model parameters estimated by the least square method in this paper, the estimated values of the parameters are compared with the theoretical values. Here, only the estimated average and theoretical values of the coefcients a 11 and a 22 in the matrix A (l) j,k and the coefcient u 11 in the vector u j,k are given. Te comparison result is shown in Figure 9.
In order to further illustrate the accuracy and efciency of the method proposed in this study, the comparison of the wind velocity time-series and the target spectrum at nodes A and G using the AR model is also shown in Figure 10. Table 1 shows the deviation statistics of the nodes A-J using the vector AR model and the method in this study to simulate the correlation function and the target value and the simulated wind speed mean square value and the target value. Te total time spent using the vector AR model is 12684 s, and the total time using the method proposed in this study is 10247 s.

Results' Analysis.
From the comparison between the simulated wind velocity time-series power spectrum obtained by the vector AR model and the target spectrum in Figure 10, it can be seen that the diference between the simulated wind velocity power spectrum value and the target wind spectrum value at the corresponding elevation is small, and the high frequency part is in good agreement. While the simulated wind velocity power spectrum has errors in the low frequency part compared with the target spectrum, the calculation and simulation found that the wind velocity time series has higher accuracy in the time domain, but there is information loss in the frequency domain. Te simulated wind velocity time series power spectrum in Figure 6 adopts the method proposed in this paper which is in good agreement with the target wind spectrum value. It can be seen that the proposed method reduces the loss of information in the time-frequency domain of wind velocity time series analysis and can more accurately simulate the wind velocity time series of a large-span structure. Tis can also be confrmed from the consistency between the simulation correlation function and the objective correlation function in Figures 7 and 8.
It can be seen from the comparison between the estimated average value of autoregressive coefcient and the theoretical value in Figure 9 that the estimated value is very consistent with the theoretical value, which proves the correctness of the wavelet spread function adopted in this paper and the least square method adopted in parameter estimation.
It can be seen from Table 1 that the correlation function deviation and wind speed mean square error deviation using this method have been greatly improved, and the correlation function deviation is 50% higher than the accuracy of the vector process AR model, and the wind speed mean square error deviation accuracy is improved by 69.4% on average,   Mathematical Problems in Engineering indicating that the simulation accuracy of the proposed method is higher. In addition, from the perspective of the total simulation time, the method proposed in this paper requires less time than the AR model simulation, and the running time is reduced by about 20%, which shows that the efciency of this method is higher.

Conclusions
In this study, a new method for simulating the wind feld of long-span structures is presented by combining the vector process AR model of linear flter method and wavelet decomposition method. In this method, the autoregressive function of the AR model is extended by the wavelet function in sufcient space, and the least square method is applied to estimate the parameters of the AR model. Te results of the numerical example in this study prove that the proposed method can reduce the information loss of wind speed time history analysis in the frequency domain and accurately simulate the wind speed time history of long-span spatial structure and has high computational efciency. Te specifc conclusions are as follows: (1) Te AR model based on the wavelet method can reduce the information loss of wind velocity timeseries analysis in the frequency domain, can better preserve the integrity of the simulated wind velocity time series in the frequency domain, and improve the accuracy of the wind feld simulation (2) Te method of simulating spatial wind feld proposed in this study has higher accuracy and higher computational efciency for the simulation of shortterm time series (3) Te expansion of the wavelet function in sufcient space can overcome the shortcomings of the timevarying model based on adaptive flters and window estimation methods that the window length should not be too large, which is an efective means to simulate short-term time series

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.