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.     

An efficient ergodic simulation of multivariate stochastic processes with spectral representation

A simulation formula to generate stationary multivariate stochastic processes is derived from the Fourier-Stieltjes integral of spectral representation. It is proved that the proposed algorithm generates ergodic sample functions in the mean value and in the correlation when the sample length is equal to one period (the generated sample functions are periodic). The algorithm is very efficient computationally since it takes advantage of the fast Fourier transform technique. The simulation of longitudinal wind velocity fluctuations and the simulation of longitudinal and vertical wind fluctuating components on a bridge deck are performed. It has been noted that there are good agreements between the temporal and target auto-/cross-correlation functions of simulated wind velocities.     

Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure

The direct methods for the solution of systems of linear equations with a symmetric positive‐semidefinite (SPS) matrix A usually comprise the Cholesky decomposition of a nonsingular diagonal block A _???? of A and effective evaluation of the action of a generalized inverse of the corresponding Schur complement. In this note we deal with both problems, paying special attention to the stiffness matrices of floating structures without mechanisms. We present a procedure which first identifies a well‐conditioned positive‐definite diagonal block A _???? of A , then decomposes A _???? by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complement S of A _????. The Schur complement S is typically very small, so the generalized inverse can be effectively evaluated by the singular value decomposition (SVD). If the rank of A or a lower bound on the nonzero eigenvalues of A are known, then the SVD can be implemented without any ‘epsilon’. Moreover, if the kernel of A is known, then the SVD can be replaced by effective regularization. The results of numerical experiments show that the proposed method is useful for effective implementation of the FETI‐based domain decomposition methods. Copyright © 2011 John Wiley & Sons, Ltd.     

A novel mean square formulation of stochastic nonlinear dynamic systems based on Adomian decomposition

An Adomian decomposition based mathematical framework to derive the mean square responses of nonlinear structural systems subjected to stochastic excitation is presented. The exact mean square response estimation of certain class of nonlinear stochastic systems is achieved using Fokker–Planck–Kolmogorov (FPK) equations resulting in analytical expressions or using Monte Carlo simulations. However, for most of the nonlinear systems, the response estimation using Monte Carlo simulations is computationally expensive, and, also, obtaining solution of FPK equation is mathematically exhaustive owing to the requirement to solve a stochastic partial differential equation. In this context, the present work proposes an Adomian decomposition based formalism to derive semi-analytical expressions for the second order response statistics. Further, a derivative matching based moment approximation technique is employed to reduce the higher order moments in nonlinear systems into functions of lower order moments without resorting to any sort of linearization. Three case studies consisting of Duffing oscillator with negative stiffness, Rayleigh Van-der Pol oscillator and a Pendulum tuned mass damper inerter system with linear auxiliary spring–damper arrangement subjected to white noise excitation are undertaken. The accuracy of the closed form expressions derived using the proposed framework is established by comparing the mean square responses of the systems with the exact solutions. The results demonstrate the robustness of the proposed framework for accurate statistical analysis of nonlinear systems under stochastic excitation.     

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.     

Spectral density estimation of stochastic vector processes

A spectral density matrix estimator for stationary stochastic vector processes is studied. As the duration of the analyzed data tends to infinity, the probability distribution for this estimator at each frequency approaches a complex Wishart distribution with mean equal to an aliased version of the power spectral density at that frequency. It is shown that the spectral density matrix estimators corresponding to different frequencies are asymptotically statistically independent. These properties hold for general stationary vector processes, not only Gaussian processes, and they allow efficient calculation of updated probabilities when formulating a Bayesian model updating problem in the frequency domain using response data. A three-degree-of-freedom Duffing oscillator is used to verify the results.     

A stochastic model for localized disturbances and its applications

A stochastic model for local disturbances, particularly for a temporal harmonic with random modulations in amplitude and/or phase, is proposed in this paper. Results for the second moment responses of a linear single-degree-of-freedom system to this type of stochastic loading demonstrate a significant change in response characteristics due to a small uncertainty. A local phenomenon may last much longer and resonance may be smeared to a broad range. Integrated with wavelet transform, the proposed approach may be used to model a random process with non-stationary frequency content. Especially, it can be effectively used for Monte Carlo simulation to generate large size of samples that have similar characteristics in time and frequency domains as a pre-selected mother sample has. The technique has a great potential for the case where uncertainty study is warranted but the available samples are limited.     

On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: An alternative perspective

An approximate formula which utilizes the concept of conditional power spectral density (PSD) has been employed by several investigators to determine the response PSD of stochastically excited nonlinear systems in numerous applications. However, its derivation has been treated to date in a rather heuristic, even “unnatural” manner, and its mathematical legitimacy has been based on loosely supported arguments. In this paper, a perspective on the veracity of the formula is provided by utilizing spectral representations both for the excitation and for the response processes of the nonlinear system; this is done in conjunction with a stochastic averaging treatment of the problem. Then, the orthogonality properties of the monochromatic functions which are involved in the representations are utilized. Further, not only stationarity but ergodicity of the system response are invoked. In this context, the nonlinear response PSD is construed as a sum of the PSDs which correspond to equivalent response amplitude dependent linear systems. Next, relying on classical excitation-response PSD relationships for these linear systems leads, readily, to the derivation of the formula for the determination of the PSD of the nonlinear system. Related numerical results are also included.     

An RKPM-based formulation of the generalized probability density evolution equation for stochastic dynamic systems

In the stochastic dynamic analysis, the probability density evolution method (PDEM) provides an optional way to capture the complete probability distribution of the stochastic response of general nonlinear systems. In the PDEM, the key point is to solve the generalized probability density evolution equation (GDEE), which governs the evolution of the joint probability density function (PDF) of the response and the randomness. In this paper, a new numerical method based on the reproducing kernel particle method (RKPM) is proposed. The GDEE can be approximated through the RKPM. By some particles in the response domain, the instantaneous PDF and its partial derivative with respect to response are smoothly expressed. Then, the approximated GDEE can be discretized directly at the collocation points in the response domain. At the same time, discretization in the time domain is achieved by the difference scheme. Therefore, the RKPM-based formulation to obtain the numerical solution of GDEE is formed. The implementation procedure of the proposed method is given in detail. The accuracy and efficiency of this method are illustrated with some numerical examples. Some details of parameter analysis are also discussed.     

A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems

In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.     

Domain decomposition of stochastic PDEs: Theoretical formulations

We present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations. The methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space. The probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes. The spatial decomposition is achieved through a Schur‐complement‐based domain decomposition. The methodology aims to exploit the full potential of high‐performance computing platforms by reducing discretization errors with high‐resolution numerical model in conjunction to giving due regards to uncertainty in the system. The mathematical formulation is numerically validated with an example of waves in random media. Copyright © 2008 John Wiley & Sons, Ltd.     

局部均值分解与经验模式分解的对比研究

摘要：介绍了一种新的非平稳信号分析方法——局部均值分解（Local mean decomposition,简称LMD）。 LMD方法可以自适应地将任何一个复杂信号分解为若干个具有一定物理意义的PF（Product function）分量之和,其中每个PF分量为一个包络信号和一个纯调频信号的乘积,从而获得原始信号完整的时频分布。本文首先介绍了LMD方法,然后将LMD方法对仿真信号进行了分析,取得了满意的效果,最后将其和经验模式分解EMD（Empirical mode decomposition）方法进行了对比,结果表明在端点效应、迭代次数等方面LMD方法要优于EMD方法。
     

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。     

基于本征正交分解的谱表示法模拟风场的误差

推导了本征正交分解(Proper Orthogonal Decomposition,POD)型谱表示法模拟所得平稳正态脉动风场的偏度误差和随机误差.从POD型谱表示法的模拟公式出发,推导了Ⅳ变量风场模拟结果序列的样本均值、相关函数、功率谱函数和根方差等前二阶矩统计特征的时域估计表达式;并证明了时域估计相关函数是正态过程,功率谱函数为非正态随机过程.进一步,计算上述样本时域估计二阶矩特征的均值和根方差,即得到了POD型谱表示法模拟所得风场的各统计量时域估计的偏度误差和随机误差,并以此给出了误差计算的通式.算例中统计误差和理论误差值的对比验证了所推导的解析解.     

Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes

A random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen–Loeve (K–L) series expansion is based on the eigen‐decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non‐stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K–L expansion are investigated by comparing the second‐order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen‐solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K–L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K–L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K–L expansion method is that it can be easily generalized to simulate non‐stationary processes with little additional effort. Copyright © 2001 John Wiley & Sons, Ltd.     

New concepts and their applications in underwater acoustic warfare simulation system

An underwater acoustic warfare simulation system (UAWSS) with a structure of high level architecture (HLA) is studied based upon a previous research project. With the experience and lessons learned, some new concepts are adopted in the implementation of UAWSS according to the essence of simulation and the objective of the system, among which are simulation synthetic environment, signal processing at other simulation nodes, decomposition of underwater sound channel, channel varying law and rules on system and parts evaluation, etc. Applications of these new ideas show that they are effective.     

The method of separation for evolutionary spectral density estimation of multi-variate and multi-dimensional non-stationary stochastic processes

The method of separation can be used as a non-parametric estimation technique, especially suitable for evolutionary spectral density functions of uniformly modulated and strongly narrow-band stochastic processes. The paper at hand provides a consistent derivation of method of separation based spectrum estimation for the general multi-variate and multi-dimensional case. The validity of the method is demonstrated by benchmark tests with uniformly modulated spectra, for which convergence to the analytical solution is demonstrated. The key advantage of the method of separation is the minimization of spectral dispersion due to optimum time- or space–frequency localization. This is illustrated by the calibration of multi-dimensional and multi-variate geometric imperfection models from strongly narrow-band measurements in I-beams and cylindrical shells. Finally, the application of the method of separation based estimates for the stochastic buckling analysis of the example structures is briefly discussed.     

分解炉内煤粉燃烧和CaCO3分解流场的数值模拟

针对国内某大型水泥高性能分解炉，基于Fluent软件，采用有限速率／涡耗散模型模拟了炉内煤粉燃烧和生料分解的湍动多相流场，给出了炉内速度矢量、温度、压力和组分分布，结果显示。炉内流动趋势合理，可为分解炉的研究提供参考。     

Gaussian process emulators for the stochastic finite element method

This paper explores a method to reduce the computational cost of stochastic finite element codes. The method, known as Gaussian process emulation, consists of building a statistical approximation to the output of such codes based on few training runs. The incorporation of emulation is explored for two aspects of the stochastic finite element problem. First, it is applied to approximating realizations of random fields discretized via the Karhunen–Loève expansion. Numerical results of emulating realizations of Gaussian and lognormal homogeneous two‐dimensional random fields are presented. Second, it is coupled with the polynomial chaos expansion and the partitioned Cholesky decomposition in order to compute the response of the typical sparse linear system that arises due to the discretization of the partial differential equations that govern the response of a stochastic finite element problem. The advantages and challenges of adopting the proposed coupling are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

A warranty forecasting model based on piecewise statistical distributions and stochastic simulation

This paper presents a warranty forecasting method based on stochastic simulation of expected product warranty returns. This methodology is presented in the context of a high-volume product industry and has a specific application to automotive electronics. The warranty prediction model is based on a piecewise application of Weibull and exponential distributions, having three parameters, which are the characteristic life and shape parameter of the Weibull distribution and the time coordinate of the junction point of the two distributions. This time coordinate is the point at which the reliability ‘bathtub’ curve exhibits a transition between early life and constant hazard rate behavior. The values of the parameters are obtained from the optimum fitting of data on past warranty claims for similar products. Based on the analysis of past warranty returns it is established that even though the warranty distribution parameters vary visibly between product lines they stay approximately consistent within the same product family, which increases the overall accuracy of the simulation-based warranty forecasting technique. The method is demonstrated using a case study of automotive electronics warranty returns. The approach developed and demonstrated in this paper represents a balance between correctly modeling the failure rate trend changes and practicality for use by reliability analysis organizations.