基于特征正交分解的桥梁风场随机模拟
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摘要
现代桥梁正在向大跨方向发展,桥梁结构越来越轻柔,其抖振性能必须引起重视。为便于考虑各种非线性因素的影响,抖振响应计算多采用基于Monte Carlo随机模拟的时域分析方法。而作为时域方法的输入,桥梁脉动风速场的随机模拟是首先要解决的问题。目前的模拟方法中,原型谱表示法相对于线性滤波法,具有高精度和鲁棒性等特征,在工程实际中已经得到了广泛的应用。特征正交分解(POD)型谱表示法的核心是以对功率谱矩阵的特征值分解取代原型谱表示法中的Cholesky分解,物理意义明确且可通过模态截断节省计算量。本文的研究主要在POD型谱表示法的精度、误差、运算速度、加速算法等以前研究较少涉及的方面开展,希望能给出在工程实际运用时风场随机模拟方法的合理选择指引。主要工作包括以下方面:
(1)研究了用POD型谱表示法来精确的模拟一维多变量风场。给出了仅随机相位的特征正交分解型谱表示法模拟公式,研究了运用模态截断所带来风场模拟的误差和一维复杂风场特征正交分解的特性。使用Matlab和Fortran两类语言编制程序,对比分析了两类谱表示法的计算效率。研究表明,POD型谱表示法具有非常明确的物理意义,模态截断给模拟带来的误差可以得到有效控制,但POD型谱表示法在计算效率方面并无明显的优势。提出了一种改进型的POD型谱表示法,分离了自谱和相干函数的影响,有更明确的物理意义。给出了谱表示法的统一表达式,以及FFT用于加速谱表示法的算法。讨论了风场模拟结果各项矩统计量的时域估计方法。给出了POD型谱表示法模拟具有桥塔风效应风场的简化算法。
(2)对两类谱表示法在理论上和数值上进行了精细化的误差分析。以模拟三变量风场为例,详细推导了一个风场样本的均值、相关函数、功率谱函数、根方差等概率统计量的时域估计表达式。推导得到了各统计量时域估计值的偏度误差和随机误差的解析解,然后将三变量风场的误差计算结果推广至一般情况,对比分析了两类谱表示法的模拟随机误差。POD型谱表示法的总体相对随机误差和根方差相对随机误差要小于原型谱表示法,且在空间分布是均匀的,而原型谱表示法误差随点号递增,这是POD型谱表示法的一个重大优点。考察了降低随机误差的可行方法,且详细讨论了利用双索引频率和POD型谱表示法模拟各态历经性风场的新算法。建议使用多次模拟取平均值的方法来降低随机误差。
(3)研究了提高谱表示法计算效率的方法。逐一讨论了利用快速Hartley变换(FHT)、连续线状结构风场POD的半解析解和POD频率插值三类算法来加速谱表示法时,它们各自的实用算法,及各类算法对计算效率的提高和对模拟精度的影响。研究结果表明,使用FHT提高计算效率远不如采用高优化的FFT算法有效;对点数较多的桥面风场,可以利用连续线状结构风场POD的半解析解加速;应当优先选择对数等距布置基点、基点数量为总频率点数的1/16左右、特征值线性插值、特征向量近似插值的改进POD型插值模拟算法。
(4)研究了三维风场模拟中的若干具体问题。描述了一般三维风场相干模型,推导了双特征正交分解法的简化模拟公式,可有效的节省计算量。并将简化的模拟算法应用于四渡河大跨悬索桥的三维风场模拟。
最后,对计算效率、随机误差、物理意义、截断误差、加速算法等5个影响选择因素的综合对比分析结论表明,POD型谱表示法比原型谱表示法更值得选择。
In China, it is worth noting that more and more long-span bridges are now being constructed planned, or will be planned. The structural typology of these bridges, even more dominated by growing slenderness, lightness, and elegance, is putting wind effects, especially the buffeting responses of the structures, into a pronominent role. It is very convenient to take into account manifold nonlinear effects, if the buffeting responses of bridges are evaluated in time domain. When using the time domain methods, namely, the Monte Carlo simulation-based methods, the stochastic wind velocity fields on bridges must be simulated first of all.
Here study on the simulation methods will be addressed. Due to its higher accuracy and robustness, the spectral representation method is more popular in the simulation of wind fields, than the linear filtering (AR, MA, or ARMA) method. Actually, it has become the dominant class of simulation methods. Two types of spectral representation methods exist. One is the original spectral representation method, involving the Cholesky decomposition of spectral density matrixes. The other, the newer one, is named the proper orthogonal decomposition(POD)-based spectral representation method, in which the eigen decomposition of the spectral matrixes is taken for instead of the Cholesky decomposition.
From the physical viewpoint, the mechanism of the latter method can be described more meaningfully. And less computational cost will be consumed if the modal truncation technique is embedded into the POD-based method. In order to provide a guide of the choice between the two types of spectral representation methods, the truncation accuracy, the stochastic error assessment, the computation efficiency and the improvement approaches of the POD-based method deserve special attention. Consequently, study on the following aspects will be included in this dissertation.
(1) Concerning how to simulate 1d-nV(1 dimensional, n variate) stochastic wind velocity fields more accurate by using the POD-based spectral representation method. Simulation formula of POD-based spectral representation method is given at first. Then discussion on the effects on the simulation accuracy, induced by involving the modal truncation technique, is carried out. Some characteristics of POD of a 1d-nV complex wind field are also investigated. The computational efficiency of the two types of methods, programmed in both Matlab and Fortran 90 language, is compared. It is shown that the POD-based method is more meaningful in physical sense, but it consumes more computational cost even if the modal truncation technique is embedded in. The error induced by this technique can be controlled effictively. In addition, an alternative form of POD, CPOD, is presented, whose physical mechanism can be interpreted more clearly than POD, due to the separation of the auto/cross effect of/between the components of the field. The CPOD-based spectral representation method is also derived. Thus, a unified formula for POD-based, CPOD-based and original methods is proposed, with the approach of incorporating the FFT (Fast Fourier Transform) algorithm into the formula, to speed-up computation of the spectral representation methods. The methods, which can be used to estimate the statistical moments of the simulated wind fields, are also provided. Furthermore, the POD-based method can be employed to simulate the wind fields including wind passage effects.
(2) Concerning the detailed error assessment of the two types of spectral representation methods, both analytical and numerical. A one dimensional tri-variate (1d-3V) stationary vector process is considered firstly, as the example case. The temporal estimations of the mean values, correlation functions, power spectra and standard deviations of the generated process are derived over the period of the simulated process. By calculating the mathematic expectations and standard deviations of the temporal estimations, the bias errors and stochastic errors of the first and second order statistical moments of the sample process are obtained in close-form. Therefore, a series of formulas for error assessment of the two types of methods are derived, respectively, by extending the results of 1d-3V process into the general 1d-nV process. Then the stochastic errors of the two types of methods are analyzed numerically, for comparison. It is manifested that the POD-based method is superior to the original method, because the total stochastic errors and the relative stochastic errors of the standard deviations of the processes generated by the former are less in sum. Also these errors generated by the POD-based method are distributed uniformly in space, according to the order of point numbers. In contrast, the errors of the processes generated by the original method are not uniform, but increasing according to the order of point numbers. Further, discussion on some possible methodologies for reducing stochastic errors is carried out. In particular, simulation of ergodic wind fields is studied in detail, by using the POD-based method incorporated with the double-index frequency series technique. It is suggested that taking averages of buffeting responses due to several simulated samples of the same wind field could produce less stochastic errors.
(3) Concerning the improvement approaches for speeding up the computation of the POD-based spectral representation method. Three approaches are considered, including utilizing the fast Hartley transform(FHT) algorithm, utilizing the close-form solutions of POD of the linear continuous wind fields and utilizing interpolation of POD eignvectors and eignvalues in frequency domain. The practical procedures, efficiency, and effects on accuracy of the three improvement approaches are investigated. Firstly, it is concluded that some optimized algorithms of FFT are far more efficient than the only FHT algorithm. Secondly, the close-form solutions of POD can be efficient, only when the simulation of wind fields on bridge decks is needed, in which an amount of points are included. Thirdly, the CPOD-based method is suitable for combination with the interpolation approach. The amount of the interpolation nodes can be equal to 1/16 of the total number of the sampling discrete frequency series or so. These nodes are distributed linearly if the frequency series are taken with logarithmic scale. Furthermore, the CPOD eignvectors and eignvalues can be piecewise interpolated linearly and roundly, respectively, accounting for higher accuracy.
(4) Concerning the simulation of 3-dimentional wind fields on complex bridge structures. The generalization of auto/cross power spectral density functions of complex 3-dimentional wind fields is described. Then the double POD(DPOD)-based spectral representation method for the simulation of 3-dimentional wind fields are derived, in order to make the simulation more efficient. A series of simplified formulas of the DPOD-based method are obtained, programmed, and then used in the simulation of the 3-dimentional wind velocity field on the Sidu River long-span suspension bridge.
Finally, based on the comparison of the two types of spectral representation method in five aspects, that is, the computation efficiency, the stochastic error, the mechanism in the physical sense, the truncation accuracy, and the improvement approaches, it can be concluded that, when an engineer or a researcher need to simulate some wind velocity fields on bridges, he or she should prefer to the POD-based method firstly.
(1)研究了用POD型谱表示法来精确的模拟一维多变量风场。给出了仅随机相位的特征正交分解型谱表示法模拟公式,研究了运用模态截断所带来风场模拟的误差和一维复杂风场特征正交分解的特性。使用Matlab和Fortran两类语言编制程序,对比分析了两类谱表示法的计算效率。研究表明,POD型谱表示法具有非常明确的物理意义,模态截断给模拟带来的误差可以得到有效控制,但POD型谱表示法在计算效率方面并无明显的优势。提出了一种改进型的POD型谱表示法,分离了自谱和相干函数的影响,有更明确的物理意义。给出了谱表示法的统一表达式,以及FFT用于加速谱表示法的算法。讨论了风场模拟结果各项矩统计量的时域估计方法。给出了POD型谱表示法模拟具有桥塔风效应风场的简化算法。
(2)对两类谱表示法在理论上和数值上进行了精细化的误差分析。以模拟三变量风场为例,详细推导了一个风场样本的均值、相关函数、功率谱函数、根方差等概率统计量的时域估计表达式。推导得到了各统计量时域估计值的偏度误差和随机误差的解析解,然后将三变量风场的误差计算结果推广至一般情况,对比分析了两类谱表示法的模拟随机误差。POD型谱表示法的总体相对随机误差和根方差相对随机误差要小于原型谱表示法,且在空间分布是均匀的,而原型谱表示法误差随点号递增,这是POD型谱表示法的一个重大优点。考察了降低随机误差的可行方法,且详细讨论了利用双索引频率和POD型谱表示法模拟各态历经性风场的新算法。建议使用多次模拟取平均值的方法来降低随机误差。
(3)研究了提高谱表示法计算效率的方法。逐一讨论了利用快速Hartley变换(FHT)、连续线状结构风场POD的半解析解和POD频率插值三类算法来加速谱表示法时,它们各自的实用算法,及各类算法对计算效率的提高和对模拟精度的影响。研究结果表明,使用FHT提高计算效率远不如采用高优化的FFT算法有效;对点数较多的桥面风场,可以利用连续线状结构风场POD的半解析解加速;应当优先选择对数等距布置基点、基点数量为总频率点数的1/16左右、特征值线性插值、特征向量近似插值的改进POD型插值模拟算法。
(4)研究了三维风场模拟中的若干具体问题。描述了一般三维风场相干模型,推导了双特征正交分解法的简化模拟公式,可有效的节省计算量。并将简化的模拟算法应用于四渡河大跨悬索桥的三维风场模拟。
最后,对计算效率、随机误差、物理意义、截断误差、加速算法等5个影响选择因素的综合对比分析结论表明,POD型谱表示法比原型谱表示法更值得选择。
In China, it is worth noting that more and more long-span bridges are now being constructed planned, or will be planned. The structural typology of these bridges, even more dominated by growing slenderness, lightness, and elegance, is putting wind effects, especially the buffeting responses of the structures, into a pronominent role. It is very convenient to take into account manifold nonlinear effects, if the buffeting responses of bridges are evaluated in time domain. When using the time domain methods, namely, the Monte Carlo simulation-based methods, the stochastic wind velocity fields on bridges must be simulated first of all.
Here study on the simulation methods will be addressed. Due to its higher accuracy and robustness, the spectral representation method is more popular in the simulation of wind fields, than the linear filtering (AR, MA, or ARMA) method. Actually, it has become the dominant class of simulation methods. Two types of spectral representation methods exist. One is the original spectral representation method, involving the Cholesky decomposition of spectral density matrixes. The other, the newer one, is named the proper orthogonal decomposition(POD)-based spectral representation method, in which the eigen decomposition of the spectral matrixes is taken for instead of the Cholesky decomposition.
From the physical viewpoint, the mechanism of the latter method can be described more meaningfully. And less computational cost will be consumed if the modal truncation technique is embedded into the POD-based method. In order to provide a guide of the choice between the two types of spectral representation methods, the truncation accuracy, the stochastic error assessment, the computation efficiency and the improvement approaches of the POD-based method deserve special attention. Consequently, study on the following aspects will be included in this dissertation.
(1) Concerning how to simulate 1d-nV(1 dimensional, n variate) stochastic wind velocity fields more accurate by using the POD-based spectral representation method. Simulation formula of POD-based spectral representation method is given at first. Then discussion on the effects on the simulation accuracy, induced by involving the modal truncation technique, is carried out. Some characteristics of POD of a 1d-nV complex wind field are also investigated. The computational efficiency of the two types of methods, programmed in both Matlab and Fortran 90 language, is compared. It is shown that the POD-based method is more meaningful in physical sense, but it consumes more computational cost even if the modal truncation technique is embedded in. The error induced by this technique can be controlled effictively. In addition, an alternative form of POD, CPOD, is presented, whose physical mechanism can be interpreted more clearly than POD, due to the separation of the auto/cross effect of/between the components of the field. The CPOD-based spectral representation method is also derived. Thus, a unified formula for POD-based, CPOD-based and original methods is proposed, with the approach of incorporating the FFT (Fast Fourier Transform) algorithm into the formula, to speed-up computation of the spectral representation methods. The methods, which can be used to estimate the statistical moments of the simulated wind fields, are also provided. Furthermore, the POD-based method can be employed to simulate the wind fields including wind passage effects.
(2) Concerning the detailed error assessment of the two types of spectral representation methods, both analytical and numerical. A one dimensional tri-variate (1d-3V) stationary vector process is considered firstly, as the example case. The temporal estimations of the mean values, correlation functions, power spectra and standard deviations of the generated process are derived over the period of the simulated process. By calculating the mathematic expectations and standard deviations of the temporal estimations, the bias errors and stochastic errors of the first and second order statistical moments of the sample process are obtained in close-form. Therefore, a series of formulas for error assessment of the two types of methods are derived, respectively, by extending the results of 1d-3V process into the general 1d-nV process. Then the stochastic errors of the two types of methods are analyzed numerically, for comparison. It is manifested that the POD-based method is superior to the original method, because the total stochastic errors and the relative stochastic errors of the standard deviations of the processes generated by the former are less in sum. Also these errors generated by the POD-based method are distributed uniformly in space, according to the order of point numbers. In contrast, the errors of the processes generated by the original method are not uniform, but increasing according to the order of point numbers. Further, discussion on some possible methodologies for reducing stochastic errors is carried out. In particular, simulation of ergodic wind fields is studied in detail, by using the POD-based method incorporated with the double-index frequency series technique. It is suggested that taking averages of buffeting responses due to several simulated samples of the same wind field could produce less stochastic errors.
(3) Concerning the improvement approaches for speeding up the computation of the POD-based spectral representation method. Three approaches are considered, including utilizing the fast Hartley transform(FHT) algorithm, utilizing the close-form solutions of POD of the linear continuous wind fields and utilizing interpolation of POD eignvectors and eignvalues in frequency domain. The practical procedures, efficiency, and effects on accuracy of the three improvement approaches are investigated. Firstly, it is concluded that some optimized algorithms of FFT are far more efficient than the only FHT algorithm. Secondly, the close-form solutions of POD can be efficient, only when the simulation of wind fields on bridge decks is needed, in which an amount of points are included. Thirdly, the CPOD-based method is suitable for combination with the interpolation approach. The amount of the interpolation nodes can be equal to 1/16 of the total number of the sampling discrete frequency series or so. These nodes are distributed linearly if the frequency series are taken with logarithmic scale. Furthermore, the CPOD eignvectors and eignvalues can be piecewise interpolated linearly and roundly, respectively, accounting for higher accuracy.
(4) Concerning the simulation of 3-dimentional wind fields on complex bridge structures. The generalization of auto/cross power spectral density functions of complex 3-dimentional wind fields is described. Then the double POD(DPOD)-based spectral representation method for the simulation of 3-dimentional wind fields are derived, in order to make the simulation more efficient. A series of simplified formulas of the DPOD-based method are obtained, programmed, and then used in the simulation of the 3-dimentional wind velocity field on the Sidu River long-span suspension bridge.
Finally, based on the comparison of the two types of spectral representation method in five aspects, that is, the computation efficiency, the stochastic error, the mechanism in the physical sense, the truncation accuracy, and the improvement approaches, it can be concluded that, when an engineer or a researcher need to simulate some wind velocity fields on bridges, he or she should prefer to the POD-based method firstly.
引文
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