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。     

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.     

考虑相位角的脉动风场模拟

为克服考虑相位角后风谱密度矩阵可能不正定而无法进行Cholesky分解的困难,提出了一种基于插值技术的改进风场模拟方法。首先依据风谱密度矩阵的正定性或不正定性将模拟风频范围划分为正定区间和不正定区间。在正定区间内,可直接将风谱密度矩阵进行Cholesky分解。而在不正定区间内,可采用插值技术获得风谱密度矩阵的分解式。然后运用谐波合成法模拟出全部风频范围内的脉动风速。数值算例表明:该方法具有较高的精度,且模拟结果能较好地符合自然风的基本特性。     

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.     

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。     

基于波数-频率联合演变功率谱的一维空间非均匀脉动风场模拟

在高层建筑、大跨桥梁和大型风力发电系统等大型工程中,风荷载对结构的安全性至关重要、甚至是起控制作用的要素。因此,脉动风速场的模拟具有重要意义。谱表达方法得到了广泛的应用。但经典的谱表达方法需要对互功率谱矩阵的逐个频率点进行Cholesky分解或本征正交分解（POD）,当模拟的空间点数较多时,分解效率非常低下、甚至可能出现矩阵奇异而难以实现。基于波数-频率联合功率谱或联合演变谱,不需要Cholesky分解或POD,可以方便地实现均匀或非均匀脉动风场的模拟。当模拟点为等间距点时,该方法能够使用FFT技术提高模拟效率,而当模拟点为非等间距点、不能使用FFT技术时,模拟效率依然有待提高。鉴于此,该文引入“结构化”非均匀离散方法和“舍选法”思想,建议了二维波数-频率域的非均匀离散策略,显著地提高了模拟效率。以某桥塔的一维空间非均匀脉动风场的数值模拟为例,验证了该方法的有效性。     

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.     

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。     

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.     

Dimension reduction model for two-spatial dimensional stochastic wind field: Hybrid approach of spectral decomposition and wavenumber spectral representation

A renewed methodology for simulating two-spatial dimensional stochastic wind field is addressed in the present study. First, the concept of cross wavenumber spectral density (WSD) function is defined on the basis of power spectral density (PSD) function and spatial coherence function to characterize the spatial variability of the stochastic wind field in the two-spatial dimensions. Then, the hybrid approach of spectral representation and wavenumber spectral representation and that of proper orthogonal decomposition and wavenumber spectral representation are respectively derived from the Cholesky decomposition and eigen decomposition of the constructed WSD matrices. Immediately following that, the uniform hybrid expression of spectral decomposition and wavenumber spectral representation is obtained, which integrates the advantages of both the discrete and continuous methods of one-spatial dimensional stochastic field, allowing for reflecting the spatial characteristics of large-scale structures. Moreover, the dimension reduction model for two-spatial dimensional stochastic wind field is established via adopting random functions correlating the high-dimensional orthogonal random variables with merely 3 elementary random variables, such that this explicitly describes the probability information of stochastic wind field in probability density level. Finally, the numerical investigations of the two-spatial dimensional stochastic wind fields respectively acting on a long-span suspension bridge and a super high-rise building are implemented embedded in the FFT algorithm. The validity and engineering applicability of the proposed method are thus fully verified, providing a potentially effective approach for refined wind-resistance dynamic reliability analysis of large-scale complex engineering structures.     

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。     

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。     

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.     

Nonstationary response of nonlinear systems using equivalent linearization with a compact analytical form of the excitation process

A new method based on equivalent linearization approaches is presented for estimating the nonstationary response of a class of nonlinear multi-degree-of-freedom systems subjected to nonstationary excitations. The highly efficient method is based on creating a compact analytical approximation of measured nonstationary excitation process data through use of a two-stage decomposition procedure. The analytic data condensation of the excitation process is performed in two stages; (1) by performing the Karhunen–Loeve spectral decomposition on the covariance matrix of the input random process to obtain the dominant eigenvectors, and (2) by fitting these eigenvectors with orthogonal polynomials to produce a truncated series of analytically approximated eigenvectors. The efficiency and accuracy of the method is demonstrated through simulation with synthetically generated excitation data as well as measured data from a real-world physical process. Although the decomposition procedure used can characterize very general input processes, because the equivalent linearization technique requires the Gaussian assumption of the response process, the constraint on applying this approach is similar to the constraints on all other equivalent linearization techniques. However, the additional freedom gained from being able to work with data-based nonstationary random processes is a significant addition to this area of research.     

Shape-dependent regularization for the retrieval of atmospheric state parameter profiles

We present an adaptive regularization approach to retrieve vertical state parameter profiles from limb-sounding measurements with high accuracy. This is accomplished by introducing a dedicated regularization functional based on a reasonable assumption of the profile characteristics. The approach results in shape-dependent weighting during least-squares computations and relies on a Cholesky decomposition of a preselected L(T)L matrix. Our method is compared with established regularization functionals such as optimal estimation and Tikhonov with respect to errors and achievable height resolution. The results show an improved height resolution of the retrieved profiles together with a reduction of absolute and relative errors obtained by test-bed simulations.     

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.     

土木工程下击暴流风速数值模拟的研究

雷暴天气下击暴流之类的高强风对诸如输电塔这样的晶格状结构具有很大的破坏性。因此,建立可靠合理的下击暴流风速模型来分析这些结构在这种极端风荷载作用下的动力响应是很有必要的。通过运用一些时频分析工具可以发现:下击暴流风速表现出明显的非平稳性。而且,下击暴流非平稳风速时程可以分解为随时间变化的平均风速即时变平均风速与具有一定相关性的随机脉动成分。基于Holmes平均风速模型和Vicroy风速竖向分布模型,使用Deo-datis提出的均匀调制非平稳随机场的模拟方法,数值模拟了雷暴天气下击暴流行进路线上某一固定位置处的竖向分布风速场。在随机脉动成分的数值模拟过程中,引入了三次样条函数插值技术,以减少Cholesky分解的次数,进一步提高了数值模拟雷暴天气下击暴流风速的效率。     

Spatial propagation of coherency matrix in polarization optics

In this work, the question of the coherency matrix propagation of a light beam is addressed by means of the analysis of interpolation processes between two physical situations. These physical situations are defined according to the second order statistical properties of the underlying process. Two states of a light beam or the path in a medium to go from a physical situation at distance z(1) to another one at distance z(2) is related to the correlation between both these physical situations. Equivalence classes are derived from the definition of a group action on the set of coherency matrices. The geodesic curves on each equivalence class define the process of interpolation. The general solution is derived as a symbolic equation, and the solution is explicitly developed for the situation of uncorrelated statistical processes. Interpolating coherency matrix in this particular case describes the propagation of a light beam into a uniform nondepolarizing medium characterized by a differential Jones matrix determined by the far points of the interpolation curve up to a unitary matrix.     

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。     

On the Cholesky algorithm with shifts for the eigensolution of real symmetric matrices

This paper presented an efficient computational algorithm for the eigensolution for real symmetric positive definite matrices. The algorithm is based on the Cholesky decomposition and will be referred to as the Cholesky algorithm. The efficiency of the algorithm is due to the combination of several iteration cycles into one, as well as to a judicious choice of shifts. The latter is accomplished by combining an approach based on Sturm's theorem for the initial stages of the process with another approach based on Gersch-gorin's theorem for the subsequent and final stages of the process. The efficiency of the proposed algorithm was tested against that of the QL method and was found to be superior. Moreover, its superiority increases with an increase in the order of the eigenvalue problem. For an eigenvalue problem of order 100, the proposed algorithm produced a solution in about one-half the time required by the QL method.