二维空间脉动风场波数-频率联合功率谱表达的FFT模拟

针对二维空间纵向脉动风场模拟问题,采用多维随机场理论,基于Davenport脉动风速谱和相干函数模型,导出了二维空间均匀脉动风场的波数-频率联合功率谱。通过谐波叠加直接获得二维空间均匀脉动风场,避免了经典谱表达方法在脉动风场模拟时的空间离散和互功率谱矩阵的Cholesky分解或本征正交分解(POD)。为了进一步提高风场模拟的效率,在数值程序中引入了快速Fourier变换(FFT)技术。最后,对风力机桨叶所在平面进行了二维空间脉动风场模拟,验证了该方法的准确性和高效性。     

与反应谱相容的多点完全非平稳地震动随机过程的快速模拟

为了精确评估结构地震响应的概率特性,地震动随机过程的模拟需要考虑时间变异性(频率和强度非平稳)、空间变异性以及与反应谱的相容性。在经典的多点完全非平稳随机过程的模拟方法中,由于频率与时间变量不可分离,演化功率谱矩阵分解效率较低。为了加快谱矩阵的分解,提出了新Cholesky分解方法。该方法的核心是将演化谱矩阵分离为相位和模矩阵,而模矩阵进一步被转化为与时间不相关的延迟相干矩阵。通过与时间相关的演化谱矩阵相比,延迟相干矩阵仅与频率相关,这样就显著提高了矩阵分解的效率;此外,延迟相干矩阵更适合采用插值技术。最后,将新Cholesky分解方法和插值技术应用到生成与反应谱相容的随机方法中。结果表明:新Cholesky分解与插值能够高效地模拟多点完全非平稳并且与反应谱相容的地震动样本;线性插值与三次样条插值均可达到良好的分辨率,少量的插值点即可满足精度的要求。     

大跨度结构随机脉动风场的快速模拟方法

针对传统谐波合成法计算量巨大、内存耗费多的缺点,研究了空间各点的分解功率谱密度函数曲线随频率变化的特点。在此基础上,提出采用三次均匀B样条插值方法来拟合分解谱密度函数曲线,引入矩阵Cholesky分解的优化递归算法来加速矩阵分解速度,同时利用FFT技术来减少谐波项合成所需要的时间。经过上述的改进,可以大大提高双索引频率下谐波合成法的计算效率。最后利用两个算例表明,这种方法可以高效、准确地模拟出适合大跨度结构的随机脉动风场。     

基于相干函数矩阵的风场本征正交分解

提出了基于相干函数矩阵的风场本征正交分解(CPOD,Coherency matrix-based POD)。首先推导了基于功率谱函数矩阵的本征正交分解。通过引入功率谱矩阵的预分解,将推导过程中的功率谱矩阵特征值分解改进为相干函数矩阵的特征值分解,得到了CPOD的表达式。对CPOD作频域离散化,得到了基于CPOD的谱表示法随机风场模拟公式。以模拟某18点脉动风速场为算例,对比分析了CPOD与SPT振型的特征并验证了所提模拟算法的有效性。结果表明,CPOD的显式分离特性使得它具有比POD更加明确的物理意义,并适用于风场特征分析。     

具有桥塔风效应的桥梁风场数值模拟

将特征正交分解型谱表示法运用于模拟具有桥塔风效应的桥梁风场中。首先介绍桥塔风效应和桥梁风场的概率描述,然后结合模态截断技术,介绍特征正交分解（Proper Orthogonal Decomposition,POD）型谱表示法,该方法是对常用的原型谱表示法的继承和提高,且物理概念更加清晰。通过引入对风速谱矩阵的显式预分解,推导模拟具有桥塔风效应的桥梁风场的简化计算公式,将对付目标功率谱矩阵的特征值分解运算简化为对实矩阵的运算。该方法可用FFT加速,相对于原有的模拟方法具有较高的计算效率。最后,以模拟龙潭河特大桥施工最大双悬臂阶段的脉动风速场为算例,解释了脉动风速过程特征正交分解模态的物理意义,说明该方法的可靠性。在算例中,观察到复杂相关结构下,特征正交分解发生振型交换的现象,并分析其原因。     

基于Hermite插值的简化风场模拟

针对传统Deodatis谐波合成法的模拟效率受Cholesky分解次数制约的问题,通过对互谱密度矩阵分解引入Hermite插值,推导了基于Hermite插值的简化风场模拟方法,将传统谐波合成法中的Cholesky分解次数由n×N次缩减为n×2k次（2k < N）,从而大幅度提升了传统谐波合成法的计算效率。以某大跨度三塔悬索桥主梁风场模拟为例,分别基于传统Deodatis法、三次Lagrange插值法、Hermite插值法模拟了时长为4096 s的脉动风速时程,三者在模拟耗时与模拟精度方面的对比表明:Hermite插值法与Lagrange插值法均能显著提高传统谐波合成法的模拟效率;Hermite插值法的模拟效率略低于三次Lagrange插值法,但其对H矩阵的模拟精度明显高出一个层次,因而Hermite插值法在风场模拟中表现更优。采用基于Hermite插值的简化方法,模拟脉动风速的功率谱与相关函数均能与目标值吻合较好,表明所模拟的脉动风速仍具有较高的保真度。在此基础上,通过插值间距的优化分析给出了插值间距的建议取值区间。     

基于协方差本征变换与谐波合成法的随机风场模拟

为解决谐波合成法模拟多变量随机脉动风场计算效率低下,内存易超限的问题,提出了协方差本征变换与谐波合成法相结合的风场模拟新方法。在谐波合成法和协方差本征变换的基础上,采用基于方差比的模态截断准则缩阶互谱密度矩阵,以减少矩阵的Cholesky分解次数,同时运用FFT技术来提高谐波项的合成速度,并给出了该方法的误差控制指标。通过算例表明上述方法可有效减少整个模拟过程的计算时间,采用方差比模态截断准则可保证所有节点位置的样本方差具有较高精度,模拟样本的功率谱密度、相关函数与目标值较为符合,并且具有较好的随机性。证实了所提出方法可以高效、准确地的模拟出随机脉动风场。     

使用特征正交分解型谱表示法的汽车受路面激励随机模拟

将特征正交分解型谱表示法用于模拟汽车受路面激励。首先给出了路面不平度对汽车输入的位移随机激励的谱描述。基于路面激励的功率谱矩阵,结合特征正交分解（POD,Proper Orthogonal Decomposition,）型谱表示法的模拟表达式,给出了路面激励的显式POD分解,定义了汽车的“路面激励模态”,推导了路面对汽车输入激励随机模拟的简化计算公式。该方法可用FFT来减少计算量以提高计算速度。它由于完全消除掉了原型谱表示法的Cholesky分解过程而具有较高的计算效率和更明确的物理意义。最后,通过对一个四轮轿车在国标GB7031—87中的A级路面不平度下受到的位移随机激励进行模拟,说明了该方法的有效性。     

大跨桥梁非平稳台风风场数值模拟与验证

为实现大跨桥梁非平稳台风风场模拟,系统提出大跨桥梁台风风场模拟验证方法,并基于实测台风数据对进化谱作为风场验证风谱的适用性进行检验。首先提出台风风场模拟方法,基于进化谱理念,获取时变功率谱,采用谐波合成法,对非平稳台风风场进行模拟;其次基于Kaimal风谱和进化谱理论,推导非均匀调制函数,获取非均匀调制进化谱,验证模拟台风风场的合理性;再次基于实测脉动风场特性理论和经验模态分解(EMD)方法,构建非平稳风速模型,获取实测功率谱,对进化谱作为风场验证风谱的适用性进行检验;最后应用提出的台风风场模拟和验证系统方法对某沿海斜拉桥台风风场进行模拟和验证。研究结果表明:实测功率谱与进化谱吻合较好,进化谱适用于台风模拟风场的验证;应用提出的台风风场模拟方法对台风风场进行模拟,模拟风场的功率谱与进化谱吻合非常好,模拟方法合理有效。     

脉动风速连续随机场的降维模拟

基于标准正交随机变量的波数谱表示,通过定义标准正交随机变量集的随机函数形式,建立了连续时空随机场模拟的波数谱-随机函数方法。同时,引入快速傅里叶变换（FFT）的算法,极大地提高了波数谱-随机函数方法的模拟效率。在波数谱-随机函数模拟方法中,仅需两个基本随机变量即可在概率密度层次上描述时空随机场的概率特性,并利用数论方法选取基本随机变量的代表性点集,实现对连续时空随机场模拟的降维表达。数值算例表明,当模拟相同数量的样本时,综合考虑模拟的效率和精度两方面,该文方法与传统的波数谱表示方法不分伯仲,但该文方法所需的基本随机变量最少,生成的代表性样本数量少且构成一个完备的概率集,从而可结合概率密度演化理论实现结构随机动力反应及动力可靠度的精细化分析。最后,结合Kaimal风速谱及Davenport空间相干函数模型,模拟了水平向脉动风速连续随机场,验证了该文方法的有效性和优越性。     

超高层建筑脉动风速场模拟的改进谐波合成法

针对传统的谐波合成法模拟超高层建筑脉动风速场存在计算量大的问题,提出了考虑互谱密度矩阵为复数矩阵的一般情况,使用插值精度较高的样条插值技术来减少Cholesky谱矩阵分解的次数,再使用FFT技术进一步加快模拟计算速度的改进谐波合成法.为了阐明改进谐波合成法具有较好的模拟精度和较快的计算速度, 运用该方法模拟了一栋225 m高的超高层建筑脉动风速场,并与仅使用FFT技术的谐波合成法即传统谐波合成法的模拟结果进行了对比分析.模拟结果的对比分析表明:尽管改进后的谐波合成法对谱分解矩阵采用了插值近似措施,但模拟的随机风速样本仍具有很好的精度,计算效率比传统的谐波合成法有了较大的提高.     

An efficient space–time based simulation approach of wind velocity field with embedded conditional interpolation for unevenly spaced locations

The classical spectral representation method (SRM) has been extensively used in the simulation of multivariate stationary Gaussian random processes. Due to the application of fast Fourier transform (FFT), the simulation is usually efficient. However, for processes with a large number of simulation points, it becomes necessary to enhance the simulation efficiency. One example is the wind velocity field along a large-span bridge, where hundreds of wind velocity fluctuations are required. In the case of bridges built over a homogeneous terrain such as coastal area or flat plain, the wind velocity field can be modeled as a multivariate homogeneous random process i.e., the auto power spectral densities (PSDs) at evenly-spaced simulation points are same and the cross PSD is a function of separation distance between two simulation points. Furthermore, in some applications, additional simulation points need to be included to a set of uniformly distributed points in order to make the wind velocity field consistent to the structural dynamic analysis requirement.In this paper, a hybrid approach of space–time random-field based SRM and proper orthogonal decomposition (POD)-based interpolation is developed for simulating the above wind velocity process. In this approach, the random-field based SRM is used to simulate the multivariate homogeneous random process composed of a set of uniformly distributed simulation locations while POD-based interpolation is used to conditionally generate the wind velocities at a few unevenly distributed points using the previously simulated wind velocities. The idea of the former is based on transforming the simulation of the homogeneous random process into that of the corresponding space–time random field where the phase angle is assumed to be zero and the coherence function must be an even function in terms of separation distance. Through this procedure, customary requirement for spectral matrix decomposition is eliminated and application of two dimensional FFT can improve the simulation efficiency dramatically. The shortcomings of this method include a slight approximation regarding the simulated sample and the non-ergodicity for the correlation function. The numerical example of a homogeneous wind velocity field along a bridge deck shows that the proposed random field-based method is very efficient in terms of accuracy and efficiency when the number of simulation locations is large and the POD-based interpolation also has good performance.     

New formulation of Cholesky decomposition and applications in stochastic simulation

Monte Carlo simulation plays a significant role in the mechanical and structural analysis due to its versatility and accuracy. Classical spectral representation method is based on the direct decomposition of the power spectral density (PSD) or evolutionary power spectral density (EPSD) matrix through Cholesky decomposition. This direct decomposition of complex matrix usually results in large computational time and storage memory.In this study, a new formulation of the Cholesky decomposition for the EPSD/PSD matrix and corresponding simulation scheme are presented. The key idea to this approach is to separate the phase from the complex EPSD/PSD matrix. The derived real modulus matrix evidently expedites decomposition compared to the direct Cholesky decomposition of the complex EPSD/PSD matrix. In the proposed simulation scheme, the separated phase can be easily assembled. The modulus of EPSD/PSD matrix could be further decomposed into the modulus of coherence matrix (or lagged coherence matrix), which describes the basic coherence structure of stochastic process. The lagged coherence matrix is independence of time and thus remarkably improves the Cholesky decomposition efficiency.The application of the proposed schemes to Gaussian stochastic simulations is presented. Firstly, the previous closed-form wind speed simulation algorithm for equally-spaced locations is extended to a more general situation. Secondly, the proposed approach facilitates the application of interpolation technique in stochastic simulation. The application of interpolation techniques in the wind field simulation is studied as an example.     

大跨平屋盖风荷载特性及风压预测研究

在大气边界层风洞中开展了大跨平屋盖结构刚性模型试验,获得了屋盖表面测点的风压时程,分析了典型风向下屋盖表面平均风压与脉动风压特性。结合本征正交分解技术(POD)与BP神经网络法,提出了一种可用于大跨结构进行空间插值的机器学习法—POD-BPNN法,实现了对风压的高效预测。预测的平均风压系数、脉动风压系数、脉动风压的时域与频域特性均与风洞试验值相吻合。表明运用POD-BPNN方法预测大跨结构表面风压是可行的。     

An improved approximation for the spectral representation method in the simulation of spatially varying ground motions

The spectral representation method (SRM), based on the Cholesky decomposition of either cross spectral density matrix or lagged coherency matrix, is widely used in the simulation of spatially varying ground motions. In this study, the SRM, based on the decomposition of lagged coherency matrix, is modified to apply to the common case which the auto spectral densities of simulation points are not the same. When using interpolation approximation approach to improve the efficiency, the SRM based on the decomposition of lagged coherency matrix exhibits much higher accuracy than the SRM based on the decomposition of cross spectral density matrix, because the elements of lower triangular matrix obtained by the Cholesky decomposition of lagged coherency matrix vary slowly with the frequency. Therefore, the SRM, based on the decomposition of lagged coherency matrix, is generally suitable for the combination with the interpolation approximation approach.     

An efficient simulation approach for multivariate nonstationary process: Hybrid of wavelet and spectral representation method

Currently, the classical spectral representation method (SRM) for nonstationary process simulation is widely used in the engineering community. Although this scheme has the higher accuracy, the time-dependent spectra results in unavailability of fast Fourier transform (FFT) and thus the simulation efficiency is lower. On the other hand, the approach based on stochastic decomposition can apply FFT in the simulation. However, the algorithm including the fitting procedure is relatively complicated and thus limits its use in practice.In this paper, the hybrid efficient simulation method is proposed for the vector-valued nonstationary process, which contains the spectra decomposition via wavelets and SRM. This method can take advantage of FFT and is also straightforward to engineering application. Numerical examples are employed to evaluate the proposed method. Results show that the method performs fairly well for the scalar process and vector-valued process with real coherence function. In the case of complex coherence function, the majority of the phase in the coherence function cannot be remained in the simulation. In addition, the validity of proper orthogonal decomposition (POD) in nonstationary process simulation via the decomposition of the time-dependent nonstationary spectra is studied. Analysis shows that the direct use of POD in nonstationary spectra decomposition may not be useful in nonstationary process simulations.     

基于双POD模型的空间相关三维随机风场数值模拟

POD法提供了一种高效、准确的风荷载模拟方法,通过对风场的功率谱密度矩阵进行Schur分解,得到一系列的特征值和特征向量,选取主要的几阶特征模态进行计算就可以得到比较精确的结果。该文讨论了具有空间相关性三维随机风场的数学模型,利用双POD模型和蒙特卡罗模拟法,详细描述了空间相关三维风场的数值模拟方法。通过大跨越输电塔三维风场的数值模拟研究表明,模拟的顺风向、横风向、竖直向风速的功率谱密度函数与理论值较为符合,并且具有较好的随机性。证实了该文提出的方法是一种高效、准确的结构三维风场模拟方法,并可应用于大跨空间结构、高层建筑及大跨度桥梁等结构之中。