Simulation method of completely non-stationary wind field for mountainous bridge

Abstract

The application particularly relates to a simulation method of a completely non-stationary wind field of a mountain bridge, which comprises the following steps: obtaining an evolution power spectrum function of a completely non-stationary wind field of a bridge to be simulated; determining representative interpolation nodes based on the evolution power spectrum function; the interpolation nodes comprise time domain interpolation nodes and frequency domain interpolation nodes; performing Cholesky decomposition on the interpolation nodes to obtain corresponding node Cholesky decomposition values; then further decomposing the node Cholesky decomposition values into a series of time and frequency function products by a non-negative matrix decomposition method; applying a Hermite interpolation method to establish global time and frequency interpolation functions; and generating a simulation high-efficiency calculation expression of the corresponding completely non-stationary wind field of the bridge based on the global time and frequency interpolation functions, so as to realize the simulation of the completely non-stationary wind field of the mountain bridge. The simulation method in the application can adapt to the completely non-stationary wind field, can take into account the time-varying frequency characteristics and spatial correlation of the wind speed field simulation, and can improve the practicability and efficiency of the bridge wind field simulation.

Description

Technical Field

[0001] This invention relates to the field of bridge wind field simulation technology, specifically to a method for simulating a completely non-stationary wind field on a bridge in a mountainous area. Background Technology

[0002] The substructure of long-span bridges in mountainous areas will be subjected to complex, spatiotemporally varying random environmental loads during construction and operation, among which wind load is one of the most significant. Extreme winds (such as typhoons and downbursts) are highly non-stationary and typically pose a significant threat to the wind resistance safety of long-span bridges in mountainous areas. Therefore, accurately predicting the structural response caused by these special wind events is particularly important during the design, construction, and operation phases; and accurately and effectively simulating the time history of non-stationary wind fields is a crucial step in estimating the structural response.

[0003] To address the issues of low matrix decomposition efficiency and high computational complexity in existing technologies, Chinese Patent Publication No. CN110196960A discloses "An Efficient Simulation Method for Eccentric Wind Fields Based on Two-Dimensional FFT," which includes: decomposing a given power spectrum matrix to obtain a wind field simulation formula in summative form; using FFT to calculate the summation in the frequency dimension; and using FFT to calculate the summation in the spatial dimension to complete the efficient simulation of the wind field. This existing wind field simulation method applies FFT technology to both frequency and spatial dimensions, improving simulation efficiency; and because it does not introduce any assumptions, it is applicable to the simulation of arbitrary ergodic multivariate stochastic processes.

[0004] The applicant discovered that, compared to existing methods, the Evolutionary Power Spectral Density (EPSD) function can better characterize non-stationary wind fields. Furthermore, one of the most commonly used methods for simulating non-stationary stochastic processes is the Classical Spectral Representation Method (SRM), which is suitable for simulating wind fields with various spectral characteristics and features mature theory and high application accuracy. However, the Classical Spectral Representation Method suffers from low computational efficiency when dealing with a large number of simulation points or long time histories, resulting in poor simulation results for wind fields on mountain bridges. In addition, existing non-stationary wind field simulations mainly focus on wind fields with time-invariant coherence. However, field measurements show that extreme winds exhibit significant time-varying frequency characteristics and spatial correlation, i.e., completely non-stationary wind fields. This makes it difficult for existing simulation methods to effectively guarantee simulation efficiency and results when facing completely non-stationary wind fields. Therefore, designing a simulation method for mountain bridge wind fields that can adapt to completely non-stationary wind fields is an urgent technical problem to be solved. Summary of the Invention

[0005] In view of the shortcomings of the prior art, the technical problem to be solved by the present invention is: how to provide a method for simulating wind fields of bridges in mountainous areas that can adapt to completely non-stationary wind fields, so as to effectively take into account the time-varying frequency characteristics and spatial correlation of wind speed field simulation, thereby improving the practicality and efficiency of wind field simulation of bridges in mountainous areas.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0007] A method for simulating a completely non-stationary wind field on a bridge in a mountainous area includes the following steps:

[0008] S1: Obtain the evolution power spectrum function of the completely non-stationary wind field of the bridge to be simulated;

[0009] S2: Determine representative interpolation nodes based on the evolutionary power spectrum function; interpolation nodes include time-domain interpolation nodes and frequency-domain interpolation nodes;

[0010] S3: Perform Cholesky decomposition on the interpolation nodes to obtain the corresponding node Cholesky decomposition values; then further decompose the node Cholesky decomposition values ​​into a series of time and frequency function products using the non-negative matrix factorization method;

[0011] S4: Apply Hermite interpolation to establish global time and frequency interpolation functions;

[0012] S5: Generate efficient calculation expressions for simulating the completely non-stationary wind field of bridges based on global time and frequency interpolation functions, so as to realize the simulation of the completely non-stationary wind field of bridges in mountainous areas.

[0013] Preferably, in step S1, the evolution power spectrum function of the completely non-stationary wind field of the bridge is obtained by numerical simulation.

[0014] Preferably, in step S2, representative interpolation nodes are determined by an interpolation scheme in which the interpolation nodes are uniformly distributed in the time direction and non-uniformly distributed in the frequency direction.

[0015] Preferably, representative interpolation nodes are determined through the following steps:

[0016] S201: Generate time-domain interpolation nodes in Cartesian coordinates using the following formula, with a uniform distribution.

[0017]

[0018]

[0019] In the formula: Δt It N represents the time-domain interpolation interval. rtIndicates the number of interpolation nodes in the time domain. Indicates the lower limit value in the time domain. Indicates the upper limit of the time domain;

[0020] S202: Frequency domain interpolation nodes are generated in a uniform distribution in logarithmic coordinates using the following formula.

[0021]

[0022]

[0023]

[0024] In the formula: θ represents the parameter controlling the distribution of different frequency regions; Δω Iω N represents the frequency domain interpolation interval; rω Indicates the number of frequency domain interpolation nodes; Indicates the lower limit value in the frequency domain; Iω represents the upper limit of the frequency domain; e represents the number of frequency interpolation points; θ(Iω-1) It represents the exponential function of the natural constant e.

[0025] Preferably, in step S3, Cholesky decomposition is performed on the interpolation nodes using the following formula:

[0026]

[0027] In the formula: This represents the spectral value of the power spectrum at the point of time-frequency difference. This represents the Cholesky decomposition value of the corresponding interpolation node; The power spectral density is the decomposed power spectrum at the time-frequency interpolation point; the superscript T indicates the matrix transpose operation.

[0028] Preferably, in step S3, the Cholesky decomposition value of the node is performed through the following steps. Further breakdown:

[0029] S301: Construct the optimization function for nonnegative matrix decomposition using the following formula:

[0030]

[0031] In the formula: H represents the nonnegative matrix to be decomposed, W represents the fundamental matrix, V represents the coefficient characteristic matrix, ||·|| F Denotes the Frobenius norm;

[0032] S302: The optimization function in step S301 is transformed into two sub-problems using the following formula:

[0033]

[0034]

[0035] In the formula: V p W represents the value of the p-th iteration step. p+1 This represents the value at the (p+1)th iteration step;

[0036] S303: Solve the two subproblems in step S302 using the monotonic projection Barzilai-Borwein method in each iteration;

[0037] S304: Cholesky decomposition value of each interpolation node. Repeat steps S301 to S303 to decompose it into:

[0038]

[0039]

[0040] In the formula: Represents the time component of spectral decomposition; Represents the frequency components of the spectral decomposition;

[0041] S305: Calculated using the following formula element value

[0042]

[0043] In the formula: and They represent and The elements; r represents the rank of the matrix and r << N.

[0044] Preferably, in step S4, a global time and frequency interpolation function is established through the following steps:

[0045] S401: Establish the time-domain and frequency-domain interpolation functions using the following formulas:

[0046]

[0047] S402: The target time function is approximated at all required time and frequency domain interpolation points using the following formula. and frequency function

[0048]

[0049] In the formula: and These are the interpolation approximation expressions for the target time and frequency functions, respectively;

[0050] S403: The decomposed spectral value H is approximately calculated using the following formula. jk (ω,t):

[0051]

[0052] Preferably, in step S4, the fast Fourier transform method is used to accelerate the simulation process of a completely non-stationary wind field.

[0053] The preferred efficient calculation expression for simulating a completely non-stationary wind field on a bridge is as follows:

[0054]

[0055]

[0056] In the formula: ω u Indicates the upper limit of the cutoff frequency; N represents the number of discrete frequency points; φ kl Let represent independent random phase angles, uniformly distributed in [0, 2π]; M represents the number of time divisions, satisfying M ≥ 2N; Re is the operation of taking the real part of a complex function; x j (t) represents the simulated wind speed time history at the j-th location; ω l t represents frequency.

[0057] Compared with existing technologies, the method for simulating completely non-stationary wind fields on mountain bridges in this invention has the following advantages:

[0058] In this invention, by combining interpolation nodes, Cholesky decomposition, nonnegative matrix decomposition, global time and frequency interpolation function establishment, and Fast Fourier Transform (FFT), the wind field simulation method can adapt to completely non-stationary wind fields. It effectively balances the time-varying frequency characteristics and spatial correlation of wind speed field simulation, thereby improving the practicality and efficiency of wind field simulation for mountain bridges. Simultaneously, this invention employs an improved interpolation scheme with uniformly distributed interpolation nodes in the time direction and non-uniformly distributed in the frequency direction. The relationship between the number of frequency interpolation nodes and their desired non-uniform distribution is expressed explicitly, significantly reducing computational errors and workload, thus improving interpolation accuracy and model practicality. Furthermore, this invention effectively decouples the Cholesky decomposition results into a series of time and frequency function products, easily integrating FFT techniques into the classic spectral decomposition method. A small number of FFT operations are sufficient to replace the cumbersome trigonometric term summation, further simplifying the simulation process. Finally, this invention does not require the establishment of complex time-frequency interpolation functions, but instead uses a simple Hermite interpolation function to approximate the decoupled form of the decomposed spectrum, avoiding a large number of Cholesky decompositions and the resulting storage requirements, thus making it easier to implement and apply. Attached Figure Description

[0059] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0060] Figure 1 This is a logic block diagram of the method for simulating a completely non-stationary wind field on a bridge in a mountainous area, as shown in the embodiment.

[0061] Figure 2 This is a schematic diagram of the wind speed field simulation points for a long-span suspension bridge in the embodiment;

[0062] Figure 3 This is a schematic diagram illustrating the impact of the number of interpolation nodes on modeling error in the embodiment.

[0063] Figure 4 This is a schematic diagram illustrating the decoupling error of the improved NMF under different size parameters in the embodiments;

[0064] Figure 5 In the example, H 11 A diagram comparing the target value and approximate value of (ω,t);

[0065] Figure 6 In the example, H 11 A schematic diagram of the interpolation approximation results for (ω,t);

[0066] Figure 7 These are schematic diagrams of three typical simulation results obtained using the technology of the present invention in the embodiments;

[0067] Figure 8 This is a schematic diagram showing the comparison between the evolutionary power spectrum and the correlation function and the target value in the embodiment;

[0068] Figure 9 This is a schematic diagram illustrating how the computation time changes as the number of simulation points increases, as shown in the example. Detailed Implementation

[0069] The following detailed explanation illustrates the specific implementation methods:

[0070] Example:

[0071] In their practical research, the applicant of this invention discovered that the main factors affecting the computational efficiency of classical spectral representation are spectral matrix decomposition and the summation of triangular terms.

[0072] In fact, spectral matrix decomposition can be achieved through Cholesky decomposition or orthogonal decomposition. The former is more widely used because it directly provides an accurate decomposition without any iterative operations. Generally, the computational burden of Cholesky decomposition increases gradually with the number of simulation points and frequency discrete points. To address this issue, researchers have made efforts such as closed-form formulas and interpolation methods. For example, a closed-form formula for Cholesky decomposition has been proposed based on the assumption that the simulation locations are uniformly distributed and have the same height. However, such an explicit expression is often unavailable in practice. Fortunately, interpolation schemes offer a good compromise between these approaches. Instead of performing Cholesky decomposition at all discrete frequencies, interpolation schemes only require decomposition at a few representative frequencies, and the decomposition results at the remaining frequencies can be approximated using interpolation techniques. Clearly, this approach can significantly reduce the computational cost and storage space of Cholesky decomposition, thus holding great potential for practical applications.

[0073] Besides efforts to improve the efficiency of spectral matrix decomposition, improvements to harmonic superposition can further reduce computational costs and accelerate the simulation process, typically achieved by introducing the Fast Fourier Transform (FFT). FFT techniques can be directly applied to classical spectral representations to accelerate the simulation of stationary scenes. However, in non-stationary simulations, this approach is limited by the time-varying properties of the decomposed matrix, as the EPSD matrix is ​​a coupling function relative to time and frequency. To effectively utilize the Fast Fourier Transform, several mathematical tools with time-frequency interpretation capabilities have been used to transform the aforementioned coupling function into a sum of products of time-frequency functions, such as polynomial expansion, wavelet decomposition, POD, and Non-negative Matrix Factorization (NMF). NMF, due to its strict non-negativity constraint, allows the resulting low-rank matrix to convey more physical meaning. However, this method may only converge to a local optimum at a slow pace, and its accuracy and efficiency are closely related to the choice of initial conditions. Although wavelet-based methods have been used to set the initial matrix, it is difficult to determine suitable wavelet basis functions and guarantee the non-negativity of all matrix elements.

[0074] More importantly, existing non-stationary wind field simulations primarily focus on wind fields with time-invariant coherence. However, field measurements show that extreme winds exhibit significant time-varying frequency characteristics and spatial correlation, indicating completely non-stationary wind fields. In this case, the Cholesky decomposition needs to be performed at every time and frequency discrete point, resulting in a decomposition workload that is the square of the workload under time-invariant conditions, and placing enormous demands on memory, especially for scenarios with a large number of simulation points.

[0075] The above analysis shows that wind field simulations for bridges in mountainous areas mainly focus on non-stationary wind fields, and techniques such as Cholesky decomposition, non-negative matrix decomposition, and fast Fourier transform are used to improve simulation efficiency. However, field measurements show that bridges in mountainous areas are situated in complex, completely non-stationary wind fields, exhibiting significant time-varying frequency characteristics and spatial correlations. This renders existing simulation methods ineffective in improving efficiency.

[0076] To this end, the applicant designed the following method for simulating completely non-stationary wind fields on mountain bridges.

[0077] like Figure 1 As shown, a method for simulating a completely non-stationary wind field on a bridge in a mountainous area includes the following steps:

[0078] S1: Obtain the evolution power spectrum function of the completely non-stationary wind field of the bridge to be simulated. Specifically, the bridge evolution power spectrum function can be obtained through a data acquisition system installed at the bridge site, and the evolution power spectrum function of the completely non-stationary wind field of the bridge can be obtained through numerical simulation.

[0079] S2: Representative interpolation nodes are determined based on the evolved power spectrum function; the interpolation nodes include time-domain interpolation nodes and frequency-domain interpolation nodes. Specifically, a representative interpolation node is determined using an interpolation scheme in which the interpolation nodes are uniformly distributed in the time direction and non-uniformly distributed in the frequency direction.

[0080] S3: Perform Cholesky decomposition on the interpolation nodes to obtain the corresponding node Cholesky decomposition values; then, further decompose the node Cholesky decomposition values ​​into a series of time and frequency function products using the non-negative matrix decomposition method.

[0081] S4: Use the Hermite interpolation method to establish global time and frequency interpolation functions.

[0082] S5: Based on global time and frequency interpolation functions, an efficient calculation expression for simulating the completely non-stationary wind field of a bridge is generated to achieve the simulation of the bridge's completely non-stationary wind field. Specifically, the Fast Fourier Transform method is used to accelerate the simulation process of the completely non-stationary wind field. The efficient calculation expression for simulating the completely non-stationary wind field of a bridge is as follows:

[0083]

[0084]

[0085] In the formula: ω u Indicates the upper limit of the cutoff frequency; N represents the number of discrete frequency points; φ kl Let represent independent random phase angles, uniformly distributed in [0, 2π]; M represents the number of time divisions, satisfying M ≥ 2N; Re is the operation of taking the real part of a complex function; x j (t) represents the simulated wind speed time history at the j-th location; ω l t represents frequency.

[0086] In this invention, by combining interpolation nodes, Cholesky decomposition, nonnegative matrix decomposition, global time and frequency interpolation function establishment, and Fast Fourier Transform (FFT), the wind field simulation method can adapt to completely non-stationary wind fields. It effectively balances the time-varying frequency characteristics and spatial correlation of wind speed field simulation, thereby improving the practicality and efficiency of bridge wind field simulation. Simultaneously, this invention employs an improved interpolation scheme where interpolation nodes are uniformly distributed in the time direction and non-uniformly distributed in the frequency direction. The relationship between the number of frequency interpolation nodes and their desired non-uniform distribution is expressed explicitly, significantly reducing computational errors and workload, thus improving interpolation accuracy and model practicality. Furthermore, this invention effectively decouples the Cholesky decomposition results into a series of time and frequency function products, easily integrating FFT techniques into the classic spectral decomposition method. A small number of FFT operations are sufficient to replace the cumbersome trigonometric summation, further simplifying the simulation process. Finally, this invention does not require the establishment of complex time-frequency interpolation functions, but instead uses a simple Hermite interpolation function to approximate the decoupled form of the decomposed spectrum, avoiding a large number of Cholesky decompositions and the resulting storage requirements, thus making it easier to implement and apply.

[0087] In practice, Cholesky decomposition is performed on the interpolation nodes using the following formula:

[0088]

[0089] In the formula: This represents the spectral value of the power spectrum at the point of time-frequency difference. This represents the Cholesky decomposition value of the corresponding interpolation node; The power spectral density is the decomposed power spectrum at the time-frequency interpolation point; the superscript T indicates the matrix transpose operation.

[0090] Representative interpolation nodes are determined through the following steps:

[0091] S201: Generate time-domain interpolation nodes in Cartesian coordinates using the following formula, with a uniform distribution.

[0092]

[0093]

[0094] In the formula: Δt It N represents the time-domain interpolation interval. rt Indicates the number of interpolation nodes in the time domain. Indicates the lower limit value in the time domain. Indicates the upper limit of the time domain;

[0095] S202: Frequency domain interpolation nodes are generated in a uniform distribution in logarithmic coordinates using the following formula.

[0096]

[0097]

[0098]

[0099] In the formula: θ represents the parameter controlling the distribution of different frequency regions; Δω Iω N represents the frequency domain interpolation interval; rω Indicates the number of frequency domain interpolation nodes; Indicates the lower limit value in the frequency domain; Iω represents the upper limit of the frequency domain; e represents the number of frequency interpolation points; θ(Iω-1) It represents the exponential function of the natural constant e.

[0100] In this invention, an improved interpolation scheme is adopted, in which interpolation nodes are uniformly distributed in the time direction and non-uniformly distributed in the frequency direction. The relationship between the number of frequency interpolation nodes and their desired non-uniform distribution is expressed explicitly, which significantly reduces computational errors and workload, thereby improving interpolation accuracy and model practicality. Simultaneously, this invention effectively decouples Cholesky's decomposition results into a series of products of time and frequency functions, easily integrating Fast Fourier Transform (FFT) techniques into classical spectral decomposition methods. A small number of FFT operations are sufficient to replace the cumbersome summation of trigonometric terms, further simplifying the simulation process.

[0101] In the specific implementation process, the Cholesky decomposition value of the node is processed through the following steps. Further breakdown:

[0102] S301: Construct the optimization function for nonnegative matrix decomposition using the following formula:

[0103]

[0104] In the formula: H represents the nonnegative matrix to be decomposed, W represents the fundamental matrix, V represents the coefficient characteristic matrix, ||·|| F Denotes the Frobenius norm;

[0105] S302: The optimization function in step S301 is transformed into two sub-problems using the following formula:

[0106]

[0107]

[0108] In the formula: V p W represents the value of the p-th iteration step. p+1 This represents the value at the (p+1)th iteration step;

[0109] S303: Solve the two subproblems in step S302 using the monotonic projection Barzilai-Borwein method in each iteration;

[0110] S304: Cholesky decomposition value of each interpolation node. Repeat steps S301 to S303 to decompose it into:

[0111]

[0112]

[0113] In the formula: Represents the time component of spectral decomposition; Represents the frequency components of the spectral decomposition;

[0114] S305: Calculated using the following formula element value

[0115]

[0116] In the formula: and They represent and The elements; r represents the rank of the matrix and r << N.

[0117] In this invention, the Cholesky decomposition result can be effectively decoupled into a sum of products of time and frequency functions. The fast Fourier transform technique can be easily assembled into the classic spectral decomposition method. A small number of fast Fourier transform operations are sufficient to replace the cumbersome summation of trigonometric terms, thereby further simplifying the simulation process.

[0118] In the specific implementation process, the global time and frequency interpolation functions are established through the following steps:

[0119] S401: Establish the time-domain and frequency-domain interpolation functions using the following formulas:

[0120]

[0121] S402: The target time function is approximated at all required time and frequency domain interpolation points using the following formula. and frequency function

[0122]

[0123] In the formula: and These are the interpolation approximation expressions for the target time and frequency functions, respectively;

[0124] S403: The decomposed spectral value H is approximately calculated using the following formula. jk (ω,t):

[0125]

[0126] In this invention, instead of establishing a complex time-frequency interpolation function, a simple Hermite interpolation function is used to approximate the decoupled form of the decomposed spectrum, avoiding a large number of Cholesky decompositions and the resulting storage requirements, thus making it easier to implement and apply.

[0127] To further illustrate the advantages of the bridge completely non-stationary wind field simulation method in this patent, the following experiment is also disclosed in this embodiment.

[0128] like Figure 2 As shown, a simulation experiment was conducted on a completely non-stationary wind field of a long-span suspension bridge, with simulation points evenly distributed on the horizontal bridge surface. Specifically, a case study of 100 simulation points will be used for further explanation.

[0129] The extended two-sided Kaimal spectrum was used to describe longitudinal wind fluctuations at bridge deck height, i.e.:

[0130]

[0131] In the formula, z represents the height of the bridge deck; u * (z,t) and U(z,t) represent the corresponding frictional wind speed and time-varying average wind speed, respectively, given by the following formula:

[0132]

[0133] U(z,t)=A(t)U(z)

[0134] In the formula, Karman constant k′=0.4, surface roughness coefficient z0=0.01m, U(z) represents the average wind speed at the bridge deck height; A(t) is a three-parameter time modulation function, and at t max The maximum value is obtained at β0 / λ, that is:

[0135]

[0136] For simplicity, the self-spectrum of all simulated points on the bridge surface is assumed to be identical; then, the spatial correlation between the j-th and k-th simulated points is characterized by the Davenport coherence function, which has an extended time-frequency expression given by the following equation:

[0137]

[0138] In the formula, C x It is the attenuation coefficient in the horizontal direction; Δ jk This represents the horizontal distance between the j-th and k-th simulation points.

[0139] Clearly, this coherence function can be viewed as a deterministic time-frequency bivariate function, which makes the simulated wind field completely non-stationary. Table 1 summarizes the basic parameters of the special wind field to be simulated.

[0140] Table 1 Simulation parameters of the power spectrum of a completely non-stationary wind field evolution

[0141]

[0142]

[0143] Taking the evolutionary power spectrum to be simulated in Table 1 as an example, the number of time-domain interpolation nodes N needs to be determined first. rt The number of frequency domain interpolation nodes N rω Generally speaking, assuming a high level of approximation in the interpolation, the fewer the number of time and frequency interpolation nodes, the better the interpolation performance. Figure 3 This demonstrates the sensitivity of interpolation accuracy to the two key parameters mentioned above. From Figure 3 It can be observed that the error increases with parameter N. rt and N rω The error decreases rapidly as the value increases, until it exceeds a certain value, at which point the error becomes relatively small and decays slowly. If the allowable error for interpolation is within 1%, then N... rt =20 and N rω =15 (i.e., E = 0.8 × 10) -2 This provides the highest computational efficiency, requiring only 300 Cholesky decompositions. Therefore, this embodiment sets N... rt =20, N rω =15, θ=0.4903.

[0144] Taking the data shown in Table 1 as an example, after performing Cholesky decomposition on the evolutionary power spectrum at the interpolation nodes, the key issue is determining the size parameter r. A larger value means higher decoupling accuracy, but lower simulation efficiency. Typically, r ≤ 9 is sufficient to provide a satisfactory approximation. To explore suitable size parameters, an error function is defined to measure the difference between the decomposition spectrum obtained by the improved NMF approximation and the decomposition spectrum obtained by direct Cholesky decomposition, i.e.:

[0145]

[0146] In the formula, and These are the approximate spectrum and the target decomposition spectrum, respectively.

[0147] Therefore, a smaller size, which translates to a simpler simulation process and potentially higher accuracy, is better.

[0148] Figure 4 The decoupling error of the improved NMF is shown under different size parameters. From Figure 4 It can be observed that the error decreases rapidly with increasing dimension, and then decreases by 10 in the range r≥4. -4 The order of magnitude has remained almost stable.

[0149] Therefore, the calculation error is E = 3.6 × 10⁻⁶. -4 When r = 4, the superiority of the improved NMF is sufficiently demonstrated, and it will be used as the optimal size parameter thereafter.

[0150] Figure 5 (b) shows the decomposed spectrum through the improved NMF approximation. For comparison, Figure 5 (a) shows the target decomposition spectrum calculated from the EPSD (Evolutionary Power Spectral Density) matrix. The comparison shows that the improved NMF performs very well in matching the decomposed spectrum.

[0151] This invention can effectively process Cholesky decomposition spectra, thus easily integrating Fast Fourier Transform (FFT) techniques into classical spectral decomposition methods. A small number of FFT operations are sufficient to replace the cumbersome summation of trigonometric terms, further simplifying the simulation process.

[0152] Taking the data shown in Table 1 as an example, Hermite interpolation is performed on the time-domain and frequency-domain functions of the improved NMF decomposition. Figure 6 (a) shows typical results. Compare them with... Figure 5 The comparison with the target values ​​in (a) shows that the approximation is quite accurate, with no significant difference (E < 1%). Furthermore, Figure 6(b) and (c) show a comparison of them in typical time and frequency slices. The results show that the approximate results agree well with the target values, further confirming the reliability of the improved interpolation scheme.

[0153] Based on this invention, simulation Figure 2 The completely non-stationary wind field of the entire bridge (i.e., 100 simulated points) is shown. Figure 7 Figures (a), 7(b), and 7(c) present the simulated wind fluctuation time histories for three typical simulation points (points 54, 58, and 100).

[0154] To illustrate the practicality of the simulated wind field, 5000 samples were generated to statistically calculate the average EPSD and correlation function. The simulation performance was then evaluated by comparing these results with a target. As a typical example, in... Figure 8 Tables (a), (b), (c), and (d) provide detailed comparisons between the estimated statistical properties at points 54 and 58 and their target results. Clearly, the estimated EPSD and correlation function are consistent with their respective targets. This phenomenon indicates a high fidelity in the simulated wind field, further validating the practicality of this method in simulating completely non-stationary wind fields.

[0155] To demonstrate the superiority of the proposed method, the computational efficiency of two recently developed methods is compared using the methods presented in Table 2. According to Zhao's method, when the number of simulation points exceeds 22, Zhao's method may encounter severe computational challenges and lead to termination (i.e., insufficient computer memory). In contrast, Tao's method is computationally efficient. Table 3 shows a comparison of its simulation efficiency with the proposed method for a single sample of the vector process under the same simulation parameters.

[0156] Table 2 Comparison of computational complexity of related methods

[0157]

[0158] Table 3. Efficiency comparison of different methods (100 simulation points; unit: seconds)

[0159]

[0160] Table 3 summarizes the computational cost of the numerical results using 100 simulation points as an example. The results show that the proposed method is significantly superior to Tao's method in application, with the latter requiring 32 times the computational cost. Theoretically, they only require 300 Cholesky decompositions, thus occupying the same time in this part. However, due to the use of complex 2D interpolation techniques, Tao's method may suffer computational efficiency losses in the interpolation approximation of the decomposed spectrum, incurring a greater computational burden than 1D interpolation techniques. Furthermore, Tao's method has low computational efficiency for harmonic superposition, requiring as many as 10,746,400 Fast Fourier Transform operations, while the proposed method only requires 20,200. Undoubtedly, the proposed method is significantly superior to Tao's method in terms of total computational cost, interpolation time, and the time consumed by Fast Fourier Transform operations.

[0161] In addition, it is necessary to investigate the impact of the number of simulation points on simulation efficiency, such as Figure 9 As shown, the time consumed by each process gradually increases with the number of simulation points, but the total time cost is acceptable. Unlike other fast Fourier transform-assisted methods whose computational efficiency is mainly determined by the Cholesky decomposition and fast Fourier transform, the efficiency of this invention is independent of the number of Cholesky decompositions and mainly relies on the simple Herimte interpolation technique. Therefore, this invention may be very effective in practice. Taking 500 simulation points as an example, its total time cost is only 501s, while Tao's method can reach as high as 20,000s.

[0162] The above discussion demonstrates the practicality and efficiency of this invention in simulating completely non-stationary wind fields. While maintaining satisfactory simulation accuracy, this invention not only significantly reduces the computational burden associated with spectral matrix decomposition but also greatly accelerates the summation of trigonometric terms. Its computational efficiency is primarily controlled by the simple Hermite interpolation technique. Therefore, this invention significantly extends the performance of classical spectral representation, enabling rapid simulation of completely non-stationary wind fields and possessing broad application prospects.

[0163] In summary, this invention employs an improved interpolation scheme in which interpolation nodes are uniformly distributed in the time direction and non-uniformly distributed in the frequency direction. The relationship between the number of frequency interpolation nodes and their desired non-uniform distribution is expressed by an explicit formula. This can significantly reduce computational errors and workload, and further improve interpolation accuracy and the practicality of the model.

[0164] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described with reference to preferred embodiments, those skilled in the art should understand that various changes in form and detail can be made without departing from the spirit and scope of the invention as defined in the appended claims. Furthermore, common knowledge such as specific structures and characteristics known in the embodiments is not described in detail here. Finally, the scope of protection claimed by the present invention should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.

Claims

1. A method for simulating a completely non-stationary wind field on a bridge in a mountainous area, characterized in that, Includes the following steps: S1: Obtain the evolution power spectrum function of the completely non-stationary wind field of the bridge to be simulated; S2: Determine representative interpolation nodes based on the evolved power spectrum function; interpolation nodes include time-domain interpolation nodes and frequency-domain interpolation nodes; among them, a scheme in which interpolation nodes are uniformly distributed in the time direction and non-uniformly distributed in the frequency direction is used to determine representative interpolation nodes; representative interpolation nodes are determined through the following steps: S201: Generate time-domain interpolation nodes in Cartesian coordinates using the following formula, with a uniform distribution. In the formula: Δt It N represents the time-domain interpolation interval. rt Indicates the number of interpolation nodes in the time domain. Indicates the lower limit value in the time domain. Indicates the upper limit of the time domain; S202: Frequency domain interpolation nodes are generated in a uniform distribution in logarithmic coordinates using the following formula. In the formula: θ represents the parameter controlling the distribution of different frequency regions; N represents the number of discrete frequency points; Δω Iω N represents the frequency domain interpolation interval; rω Indicates the number of frequency domain interpolation nodes; Indicates the lower limit value in the frequency domain; Iω represents the upper limit of the frequency domain; e represents the number of frequency interpolation points; θ(Iω-1) The exponential function representing the natural constant e; S3: Perform Cholesky decomposition on the interpolation nodes to obtain the corresponding node Cholesky decomposition values; then further decompose the node Cholesky decomposition values ​​into a series of time and frequency function products using the non-negative matrix factorization method; S4: Apply Hermite interpolation to establish global time and frequency interpolation functions; S5: Generate efficient calculation expressions for simulating the completely non-stationary wind field of bridges based on global time and frequency interpolation functions, so as to realize the simulation of the completely non-stationary wind field of bridges in mountainous areas.

2. The simulation method for completely non-stationary wind fields on mountain bridges as described in claim 1, characterized in that: In step S1, the evolution power spectrum function of the completely non-stationary wind field of the bridge is obtained through numerical simulation.

3. The simulation method for completely non-stationary wind fields on mountain bridges as described in claim 1, characterized in that, In step S3, Cholesky decomposition is performed on the interpolation nodes using the following formula: In the formula: This represents the spectral value of the power spectrum at the point of time-frequency difference. This represents the Cholesky decomposition value of the corresponding interpolation node; The power spectral density is the decomposed power spectrum at the time-frequency interpolation point; the superscript T indicates the matrix transpose operation.

4. The simulation method for completely non-stationary wind fields on mountain bridges as described in claim 1, characterized in that, In step S3, the Cholesky decomposition value of the node is performed through the following steps. Further breakdown: S301: Construct the optimization function for nonnegative matrix decomposition using the following formula: stW≥0,V≥0 In the formula: H represents the nonnegative matrix to be decomposed, W represents the fundamental matrix, V represents the coefficient characteristic matrix, ||·|| F Denotes the Frobenius norm; S302: The optimization function in step S301 is transformed into two sub-problems using the following formula: In the formula: V p W represents the value of the p-th iteration step. p+1 This represents the value at the (p+1)th iteration step; S303: Solve the two subproblems in step S302 using the monotonic projection Barzilai-Borwein method in each iteration; S304: Cholesky decomposition value of each interpolation node. Repeat steps S301 to S303 to decompose it into: In the formula: Represents the time component of spectral decomposition; Represents the frequency components of the spectral decomposition; S305: Calculated using the following formula element value In the formula: and They represent and The elements; r represents the rank of the matrix and r << N.

5. The simulation method for completely non-stationary wind fields on mountain bridges as described in claim 1, characterized in that, In step S4, the global time and frequency interpolation functions are established through the following steps: S401: Establish the time-domain and frequency-domain interpolation functions using the following formulas: S402: The target time function is approximated at all required time and frequency domain interpolation points using the following formula. and frequency function In the formula: and These are the interpolation approximation expressions for the target time and frequency functions, respectively; S403: The decomposed spectral value H is approximately calculated using the following formula. jk (ω,t):

6. The simulation method for completely non-stationary wind fields on mountain bridges as described in claim 1, characterized in that, In step S4, the Fast Fourier Transform method is used to accelerate the simulation process of a completely non-stationary wind field.

7. The simulation method for completely non-stationary wind fields on mountain bridges as described in claim 6, characterized in that, The efficient calculation expression for simulating a completely non-stationary wind field on a bridge is as follows: Give = oh u / N;ω l =lΔω,l=1,2,...,N; t=p′Δt; Δt=2π / (MΔω), p′=1,2,...,M; In the formula: ω u Indicates the upper limit of the cutoff frequency; N represents the number of discrete frequency points; φ kl Let represent independent random phase angles, uniformly distributed in [0, 2π]; M represents the number of time divisions, satisfying M ≥ 2N; Re is the operation of taking the real part of a complex function; x j (t) represents the simulated wind speed time history at the j-th location; ω l t represents frequency.

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