Simulation Of Stochastic Wind Fields On Long-Span Bridges Based On Proper Orthogonal Decomposition

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.