CN115526125B - Numerical cutoff-based efficient simulation method for random wind speed field - Google Patents
Numerical cutoff-based efficient simulation method for random wind speed field Download PDFInfo
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Abstract
The invention discloses a numerical cutoff-based random wind speed field efficient simulation method, and relates to the technical field of civil engineering wind resistance design. Selecting a wind spectrum model and a coherence function model which are commonly used for describing a wind field; fast Cholesky decomposition is carried out on a coherent matrix formed by coherent functions; rounding the Cholesky decomposition result of the coherence matrix by using a reasonable threshold; exchanging the inner layer summation sequence and the outer layer summation sequence of the simulation formula, substituting the rounded coherent matrix Cholesky decomposition result into the inner layer summation result and calculating the inner layer summation result; for a stable wind speed field, performing high-efficiency outer layer summation calculation by directly utilizing fast Fourier transformation; for non-stationary wind speed field, the binary function needs to be decoupled by a time-frequency decoupling toolAnd then calculating the outer layer summation by adopting fast Fourier transform. The method and the device solve the problems that in the multi-point wind field simulation process, the calculated amount of double summation is overlarge and long calculation time is needed in harmonic synthesis calculation, and when a proper threshold value is taken for cutting, the requirement of simulation precision can be effectively met.
Description
Technical Field
The invention relates to the technical field of civil engineering wind resistance design, in particular to a numerical cutoff-based random wind speed field efficient simulation method.
Background
Spectral Representation Methods (SRMs) have been widely used for simulation of non-uniform random wind velocity fields for large span bridges. However, a significant amount of computational resources are typically required in this simulation process, particularly as the number of simulation points and simulation samples continue to increase, the simulation efficiency decreases dramatically.
Specifically, the number of orders and trigonometric summations of Cholesky decomposition increases as the number of analog points increases. At the same time, the addition of the analog time and the number of samples increases the summation of the trigonometric functions. There are many ways currently available to accelerate the computation in Cholesky decomposition for this problem. For example, a closed-form solution formula of spectral matrix decomposition is directly introduced into simulation of wind speed fields, simulation calculation of wind speed fields uniformly distributed in the horizontal direction is performed, and a modified Cholesky decomposition closed-form solution can be performed to simulate wind speed fields distributed at any position. In addition, interpolation may be used to reduce the Cholesky decomposition step. However, with a large number of simulation points and a large number of simulation samples, the computer memory and time required for summing calculation are quite large, which results in low simulation efficiency of the wind speed field of the large-span bridge.
To solve this problem, the present study proposes a numerical truncation method. By setting a proper threshold value, the Cholesky decomposition closed solution is truncated by using the threshold value, so that the summation process is simplified, and the calculated amount is effectively reduced.
Disclosure of Invention
The invention aims to provide a numerical cutoff-based random wind speed field efficient simulation method which can provide input for power time course analysis of wind resistance designs of large structures such as bridges, high-rise buildings and large roofs.
The technical aim of the invention is realized by the following technical scheme: a method for efficiently simulating a random wind speed field based on numerical cutoff comprises the following steps:
s1: selecting a wind spectrum model and a coherence function model which are commonly used for describing a wind field;
s2: fast Cholesky decomposition is carried out on a coherent matrix formed by coherent functions;
s3: rounding the Cholesky decomposition result of the coherence matrix by using a reasonable threshold;
s4: exchanging the inner layer summation sequence and the outer layer summation sequence of the simulation formula, substituting the rounded coherent matrix Cholesky decomposition result into the inner layer summation result and calculating the inner layer summation result;
s5: for a stable wind speed field, performing high-efficiency outer layer summation calculation by directly utilizing fast Fourier transformation; for non-stationary wind speed field, the binary function needs to be decoupled by a time-frequency decoupling toolAnd then calculating the outer layer summation by adopting fast Fourier transform.
The invention is further provided with: in the step S1, a stable wind spectrum model S describing a spatial n-point wind speed field is selected jj (omega) or non-stationary wind spectrum model S jj (ω, t) and a Davenport coherence function model γ jk (ω) the expression is as follows:
where j, k=1, 2, n; omega and t respectively represent frequencyAnd time; c (C) y And C z Respectively representing the exponential decay coefficients in the horizontal direction and the vertical direction; y is j And y is k The points respectively represent the horizontal coordinates of the j-th point and the k-th point; z j And z k Respectively representing the vertical coordinates of the jth point and the kth point; u (z) j ) And U (z) k ) Respectively represent z j And z k Is set in the wind turbine.
The invention is further provided with: u (z) j ) And U (z) k ) Respectively represent z j And z k They satisfy the following formula:
wherein α is an index related to surface roughness, and is specified as α=0.12 to 0.3.
The invention is further provided with: in the step S2, wind field simulation points are arbitrarily distributed along the horizontal axis, and a Cholesky decomposition closed solution of the coherence matrix formed by the coherence functions may be expressed as:
for an arbitrary distribution of analog points along the vertical axis, the Cholesky decomposition closed solution of the coherence matrix consisting of coherence functions can be approximated as:
for other arbitrarily distributed analog points, interpolation can be used to quickly complete Cholesky decomposition of the coherence matrix. First, determining a frequency interpolation pointThen calculate the frequency interpolation point by Davenport coherent function model>Coherence function matrix->I.e.
Re-pairingPerforming Cholesky decomposition to obtain +.>Finally use +.>Interpolation results in B (ω) at other frequency points, i.e
The invention is further provided with: in step S3, a suitable threshold value ε is set, when β jk When (ω) is smaller than the threshold value, the rounding can be 0, and the specific expression is:
where the threshold is typically set to a small value. On the one hand, the truncation errors can be reduced, especially when the frequency ω is small, ensuring the accuracy of the simulation results, and on the other hand, β for most frequencies jk (ω), a smaller threshold epsilon may already ensure that most elements are truncated.
The invention is further provided with: in the step S4, the double summation is divided into two layers of inner and outer summation, that is, the summation of the space dimension k is first performed, and then the summation of the frequency dimension l is performed, where the inner summation is:
wherein i represents an imaginary unit; e represents an exponential function; omega l =lΔω, Δω is the frequency step, Δω=ω up /N(ω up Is the upper cutoff frequency, N is the total number of frequencies);is at a random phase angle (0, 2 pi) uniformly distributed;
stationary wind speed field x j (t) non-stationary wind speed fieldIs a classical simulation formula:
can be simplified into the following form:
the invention is further provided with: in the step S5, for the simulation of the steady wind speed field, the calculation of the above-mentioned simulation formula (12) can directly use the fast fourier transform technology, so that the calculation amount of the outer layer summation can be greatly reduced, and the simulation efficiency is improved.
For non-stationary wind speed field simulations, binary functions with respect to time and frequency in the simulation formulaDecoupling is performed. It can be decomposed into a series of sums of time and frequency function products as follows:
in the middle ofIs the q-th order primary coordinate; />Is the q-th order feature vector; />Representing the number of low-order terms comprising most of the energy, typically +.>The accuracy requirement can be met. By decoupling, equation (13) can be rewritten as:
equation (15) above can be efficiently calculated using fast fourier transform techniques to accomplish efficient simulation of non-stationary wind velocity fields.
In summary, the invention has the following beneficial effects:
1. the method can efficiently simulate a stable or non-stable wind speed field. The Cholesky decomposition result of the coherence matrix and the numerical truncation operation are used for combination to simplify the double summation process in the simulation. By setting a proper threshold value, the Cholesky decomposition result of the coherent matrix can be truncated under different frequencies, and the workload can be reduced because truncated elements are eliminated in the summation calculation, so that the efficiency of the whole simulation process is greatly improved under the condition of ensuring enough calculation accuracy.
2. The method can simulate the wind speed field of the large-span bridge structure and provide the input wind speed time course for the large-span bridge structure.
Drawings
FIG. 1 is a flow chart in an embodiment of the invention;
FIG. 2 is a plot of verification of the point 1 autocorrelation function of an simulated wind speed field in an embodiment of the present invention;
FIG. 3 is a graph of verification of the 1 st and 5 th point cross-correlation functions of an simulated wind speed field in an embodiment of the invention.
Detailed Description
The invention is described in further detail below with reference to fig. 1-3.
Examples: a method for efficiently simulating a random wind speed field based on numerical cutoff is shown in figures 1-3, and specifically comprises the following steps:
s1: and selecting a wind spectrum model and a coherence function model which are used for describing a wind field. Selecting a stable wind spectrum model S for describing a spatial n-point wind speed field jj (omega) or non-stationary wind spectrum model S jj (ω, t) and a Davenport coherence function model γ jk (ω) the expression is as follows:
where j, k=1, 2, n; ω and t represent frequency and time, respectively; c (C) y And C z Respectively representing the exponential decay coefficients in the horizontal direction and the vertical direction; y is j And y is k The points respectively represent the horizontal coordinates of the j-th point and the k-th point; z j And z k Respectively representing the vertical coordinates of the jth point and the kth point; u (z) j ) And U (z) k ) Respectively represent z j And z k Are satisfied by the average wind speed ofThe following formula:
wherein α is an index related to surface roughness, and is specified as α=0.12 to 0.3.
S2: and (3) performing fast Cholesky decomposition on a coherence matrix formed by the coherence functions. If wind field simulation points are arbitrarily distributed along the horizontal axis, cholesky decomposition closed solution of the coherence matrix composed of the coherence function (1) can be expressed as:
for an arbitrary distribution of analog points along the vertical axis, the Cholesky decomposition closed solution of the coherence matrix consisting of the coherence function (1) can be approximated as:
for other arbitrarily distributed analog points, interpolation can be used to quickly complete Cholesky decomposition of the coherence matrix. First, determining a frequency interpolation pointThen calculate the frequency interpolation point by Davenport coherent function model>Coherence function matrix->I.e.
Re-pairingPerforming Cholesky decomposition to obtain +.>Finally use +.>Interpolation results in B (ω) at other frequency points.
S3: the Cholesky decomposition result of the coherence matrix is rounded with a reasonable threshold. Setting a proper threshold epsilon, when beta jk When (ω) is smaller than the threshold value, the rounding can be 0, and the specific expression is:
where the threshold is typically set to a small value. On the one hand, the truncation errors can be reduced, especially when the frequency ω is small, ensuring the accuracy of the simulation results, and on the other hand, β for most frequencies jk (ω), a smaller threshold epsilon may already ensure that most elements are truncated.
It is apparent from the closed-form solutions formulas (3) and (4) that the closed-form solution is represented by the Davenport coherence function. As can be seen from equation (1), the value of the Davenport coherence function is mainly affected by two key parameters, namely, the distance between two analog points and the frequency ω. Taking the horizontal closed-form solution as an example, when k=1, the closed-form solution decreases rapidly with increasing frequency until convergence to 0. When k is more than or equal to 2, the closed solution shows a trend of increasing and then decreasing along with the increase of the frequency,and finally converges to 0. In general, the closed-form solution converges to 0 as the frequency increases. In fact, when the closure solution is reduced to some extent, its contribution to the simulation is negligible. Thus, for each closure solution beta jk (ω) are truncated in frequency using equation (8), respectively. As the frequency increases, more and more elements are rounded. In the simulation calculation, the 0 element does not participate in the summation calculation of the simulation formula. Therefore, the calculated amount is reduced, the calculated time is shortened, and the simulation efficiency is improved.
S4: exchanging the inner layer summation sequence and the outer layer summation sequence of the simulation formula, substituting the rounded coherent matrix Cholesky decomposition result into the inner layer summation result and calculating the inner layer summation result; from the spectral representation-based simulation formulas (10) and (11), it is known that both the simulation of a stationary wind speed field and a non-stationary wind speed field requires a double summation with respect to the frequency dimension l and the spatial dimension k. To simplify the calculation, the double summation can be divided into an inner layer summation and an outer layer summation, namely, the summation of the space dimension k is performed first, and then the summation of the frequency dimension l is performed. The inner layer summation can be written as:
wherein i represents an imaginary unit; e represents an exponential function; omega l =lΔω, Δω is the frequency step, Δω=ω up /N(ω up Is the upper cutoff frequency, N is the total number of frequencies);is at a random phase angle of (0, 2 pi) uniform distribution.
Then the wind velocity field x is stabilized j (t) non-stationary wind speed fieldIs a classical simulation formula:
can be simplified into the following form:
s5: for a stable wind speed field, performing high-efficiency outer layer summation calculation by directly utilizing fast Fourier transformation; for non-stationary wind speed field, the binary function needs to be decoupled by a time-frequency decoupling toolAnd then calculating the outer layer summation by adopting fast Fourier transform. For the simulation of the stable wind speed field, the calculation of the simulation formula (12) can directly use the fast Fourier transform technology, so that the calculation amount of outer layer summation can be greatly reduced, and the simulation efficiency is improved.
For non-stationary wind speed field simulation, first, a binary function with respect to time and frequency in the simulation formulaDecoupling is performed. It can be decomposed into a series of sums of time and frequency function products as follows:
in the middle ofIs the q-th order primary coordinate; />Is the q-th order feature vector; />Representing the number of low-order terms comprising most of the energy, typically +.>The accuracy requirement can be met. By decoupling, equation (13) can be rewritten as:
equation (15) can be efficiently calculated using fast fourier transform techniques to accomplish efficient simulation of non-stationary wind velocity fields.
The invention is further illustrated by the following example of simulating a horizontally distributed non-stationary wind velocity field:
(1) The simulation points are evenly distributed with 120 points along the horizontal straight line direction of 1200 m.
(2) Assume that the Kametal evolution power spectral density function of each point is:
wherein the height of the simulation point is z j ＝60m，u * Can be expressed in the following form:
in the ground roughness z 0 =0.01, karman constant is k=0.4. The time-varying wind speed for this altitude is expressed as:
wherein the average wind speed at the altitude is U (z j )＝40m/s，d (t) is a time modulation function, which can be expressed as:
wherein t is max ＝β 0 /λ，The parameter in the time modulation function is set to t max ＝600，β 0 ＝2。
The coherence function uses a Davenport coherence function model, which can be expressed in the following form:
wherein C is y Taking C as an exponential decay coefficient in the horizontal direction y ＝8，y j And y is k The horizontal coordinate positions of j and k points are respectively represented.
The Cholesky decomposition closed-loop solution of the coherence function of the horizontally distributed wind field can be expressed as:
(4) Setting a threshold epsilon=0.01, and obtaining a closed solution after interception according to the following interception formula
(5) By eigen-orthogonal decomposition (POD)(j=1, 2, …, n) into a small number of principal coordinates +.>And feature vector->The specific method comprises the following steps:
to be continuously functionalThe value is expressed as a matrix of N x M-dimensional constants in a time domain and a frequency domain, namely:
each column of the constant matrix is regarded as a column vector, and M column vectors are taken as a total, and the constant matrix is decomposed by the following eigenvectors:
RΦ q ＝λ q Φ q ,q＝1,2,…,N (24)
find a set of optimal orthonormal basis Φ= [ Φ ] 1 ,Φ 2 ,…,Φ N ] N×N The projection of the column vector will be maximized thereon, where Φ q Is the q-th feature vector; lambda (lambda) q Is the q-th eigenvalue; r is the correlation matrix for these N column vectors, which can be calculated by:
after obtaining N feature vectors, the projection of each column vector, i.e., the primary coordinates, is determined by the following equation:
wherein a is q Is the q-th projection vector;
the eigenvalues are reorganized in descending order, retaining lower order eigenvalues containing more energy, and L is approximately expressed as:
wherein, the liquid crystal display device comprises a liquid crystal display device,the number of low-order items selected to meet the precision requirement is taken +.>Whereby each of(j=1, 2, …, n) with its main coordinates +.>And feature vector->Expressed as:
wherein, the liquid crystal display device comprises a liquid crystal display device,and->Respectively is vector a q And phi is q A corresponding discrete function.
(6) A random simulation formula based on a spectral representation method utilizes Fast Fourier Transform (FFT) to generate a spatial multipoint wind speed field sample, and the specific method comprises the following steps:
based on the spectral representation method, the simulation formula of the non-stationary wind field can be expressed as:
where N is the number of frequency steps, 1024 is taken, Δω=ω u N is the frequency increment, ω l LΔω is the frequency point coordinates, t is the time coordinates, time interval Δt=1s; omega l Is the first frequency point;is a set of random phase angles, at [0,2 pi ]]Evenly distributed over the range, a defined amount for a particular sample.
The simulation formula after exchanging the summation sequence can be expressed as:
wherein the internal summing portion is denoted as:
taking the threshold value as epsilon=0.01, and carrying the POD decomposition result after the truncation treatment, the summation formula can be expressed as:
then go onAnd generating a wind speed field sample by FFT operation.
By the method, the wind field time course of 120 points in the non-stationary horizontal wind speed field can be simulated, the autocorrelation function and the cross correlation function are calculated by using 2000 samples, then the wind field time course is compared with a given target value, as shown in fig. 2 and 3, verification results of the autocorrelation function and the cross correlation function of the simulated wind speed time course are given, and the simulated value is consistent with the target value, so that the rationality of the simulation method can be illustrated. Further, by comparing the simulation efficiency of the method with that of the non-cutting method, the result shows that the simulation efficiency of the method is greatly improved as the simulation of 2000 samples only takes 332.2 seconds, while the non-cutting method takes 3528.9 seconds.
In the embodiment of the invention, the method provided by the invention is adopted to simulate the non-stationary wind speed field distributed along the horizontal direction, the Cholesky decomposition result of the coherence matrix is truncated, the summation calculation in a simulation formula is greatly simplified, and the FFT is utilized to accelerate the summation operation. The efficiency of the overall wind speed field simulation process is greatly improved while maintaining sufficient accuracy. The method can provide an input wind load time course for time course analysis of the wind-resistant design of the large-span bridge.
The present embodiment is only for explanation of the present invention and is not to be construed as limiting the present invention, and modifications to the present embodiment, which may not creatively contribute to the present invention as required by those skilled in the art after reading the present specification, are all protected by patent laws within the scope of claims of the present invention.
Claims (5)
1. A high-efficiency simulation method of a random wind speed field based on numerical value cutoff is characterized by comprising the following steps: the method comprises the following steps:
s1: selecting a wind spectrum model and a coherence function model which are commonly used for describing a wind field;
s2: fast Cholesky decomposition is carried out on a coherent matrix formed by coherent functions;
s3: rounding the Cholesky decomposition result of the coherence matrix by using a threshold; setting a threshold epsilon, and randomly distributing wind field simulation points along a horizontal axis, wherein Cholesky decomposition closed solution of a coherent matrix formed by coherent functions is beta jk (omega), when beta jk (omega) is less than the threshold, i.e. rounded to 0, withThe volume expression is:
s4: exchanging the inner layer summation sequence and the outer layer summation sequence of the simulation formula, substituting the rounded coherent matrix Cholesky decomposition result into the inner layer summation result and calculating the inner layer summation result; the double summation is divided into an inner layer summation and an outer layer summation, namely, the summation of the space dimension k is firstly carried out, and then the summation of the frequency dimension l is carried out, wherein the inner layer summation is as follows:
wherein i represents an imaginary unit; e represents an exponential function; omega l =lΔω, Δω is the frequency step, Δω=ω up /N，ω up Is the upper cutoff frequency, N is the total number of frequencies;is at a random phase angle (0, 2 pi) uniformly distributed;
stationary wind speed field x j (t) non-stationary wind speed fieldIs a classical simulation formula:
is simplified into the following form:
s5: for a stable wind speed field, performing high-efficiency outer layer summation calculation by directly utilizing fast Fourier transformation; for non-stationary wind speed field, the binary function needs to be decoupled by a time-frequency decoupling toolAnd then calculating the outer summation by adopting fast Fourier transform, wherein omega and t respectively represent frequency and time.
2. The method for efficiently simulating the random wind speed field based on numerical cutoff according to claim 1, wherein the method is characterized by comprising the following steps of: in the step S1, a stable wind spectrum model S describing a spatial n-point wind speed field is selected jj (omega) or non-stationary wind spectrum model S jj (ω, t) and a Davenport coherence function model γ jk (ω) the expression is as follows:
where j, k=1, 2, n; ω and t represent frequency and time, respectively; c (C) y And C z Respectively representing the exponential decay coefficients in the horizontal direction and the vertical direction; y is j And y is k Respectively representing horizontal coordinates of a j-th point and a k-th point; z j And z k Respectively representing the vertical coordinates of the jth point and the kth point; u (z) j ) And U (z) k ) Respectively represent z j And z k Is set in the wind turbine.
3. The method for efficiently simulating the random wind speed field based on numerical cutoff according to claim 2, wherein the method is characterized by comprising the following steps of: u (z) j ) And U (z) k ) Respectively represent z j And z k They satisfy the following formula:
wherein α is an index related to surface roughness, and α=0.12 to 0.3.
4. The method for efficiently simulating the random wind speed field based on numerical cutoff according to claim 2, wherein the method is characterized by comprising the following steps of: in the step S2, wind field simulation points are arbitrarily distributed along the horizontal axis, and Cholesky decomposition closed solution of the coherence matrix formed by the coherence function (1) is expressed as:
for analog points arbitrarily distributed along the vertical axis, the Cholesky decomposition closed solution of the coherence matrix consisting of the coherence function (1) is approximately expressed as:
for other arbitrarily distributed simulation points, rapidly completing Cholesky decomposition of the coherent matrix by adopting an interpolation method; first, determining a frequency interpolation pointThen calculate the frequency interpolation point by Davenport coherent function model>Coherence function matrix->I.e.
Re-pairingPerforming Cholesky decomposition to obtain +.>Finally use +.>Interpolation results in B (ω) at other frequency points, i.e
5. The method for efficiently simulating the random wind speed field based on numerical cutoff according to claim 1, wherein the method is characterized by comprising the following steps of: in the step S5, for the simulation of the steady wind speed field, the calculation of the above-mentioned simulation formula (12) directly uses the fast fourier transform technique, so as to reduce the calculation amount of the outer layer summation and improve the simulation efficiency;
for non-stationary wind speed field simulations, binary functions with respect to time and frequency in the simulation formulaDecoupling is performed and decomposed into a series of sums of time and frequency function products, as follows:
in the middle ofIs the q-th order primary coordinate; />Is the q-th order feature vector; />Representing the number of low-order terms comprising most of the energy,/->The precision requirement is met; by decoupling, equation (13) is rewritten as:
the above formula (15) is calculated efficiently by using a fast fourier transform technique, and the efficient simulation of the non-stationary wind speed field is completed.
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