Response analysis under stochastic loading in presence of structural uncertainties

The response of a structure with random parameters (e.g. stiffness, or damping coefficients) is investigated, when it is subjected to a random load. The structural behaviour can be either linear or non‐linear, but the forcing process has been assumed to be second‐order Gaussian. A modified series expansion is used to determine response statistics, in the various cases of a Duffing oscillator with random stiffness and damping coefficients being subjected to a deterministic sinusoidal force, a random amplitude sinusoidal force, a Kanai–Taijimi‐type earthquake. Copyright © 1999 John Wiley & Sons, Ltd.     

A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

Solution of linear dynamic systems with uncertain properties by stochastic reduced order models

A novel method, referred to as the stochastic reduced order model (SROM) method, is proposed for finding statistics of the state of linear dynamic systems with random properties subjected to random noise. The method is conceptually simple, accurate, computationally efficient, and non-intrusive in the sense that it uses existing solvers for deterministic differential equations to find state properties.Bounds are developed on the discrepancy between the exact and the SROM solutions under some assumptions on system properties. The bounds show that the SROM solutions converge to the exact solutions as the SROM representation of the vector of random system parameters is refined. Numerical examples are presented to illustrate the implementation of the SROM method and demonstrate its accuracy and efficiency.     

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.     

Control Method for Multiple‐Input Multiple‐Output Non‐Gaussian Random Vibration Test

A control method for multi‐input multi‐output non‐Gaussian random vibration test based on an improved zero‐memory nonlinear transformation and an inverse system method is proposed. Compared with the classic zero‐memory nonlinear transformation method, the improved one can overcome the defect of the dynamic range loss. The inverse system method is put forward in order to control the kurtoses and the spectra for multi‐input multi‐output non‐Gaussian random vibration test simultaneously. The main idea of inverse system method is to generate the Gaussian reference response signals first from the reference spectra, and the improved zero‐memory nonlinear transformation method is utilized to obtain the non‐Gaussian reference response signals with the reference kurtoses, then the continuous and stationary coupled driving signals can be derived from the relationship between the inputs and outputs of the test system. Thus, the difficulty in generation of driving signals in multi‐input multi‐output non‐Gaussian random vibration test can be overcome. The matrix power control algorithm is introduced for the spectrum control, and a kurtosis control algorithm is set up similarly. A simulation example and an experimental test are provided in the paper, and the results illustrate the effectiveness and feasibility of the proposed control method. Copyright © 2017 John Wiley & Sons, Ltd.     

Linear systems driven by martingale noise

A method is developed for calculating second moment properties and moments of order three and higher of the state X of a linear filter driven by martingale noise. The martingale noise is interpreted as the formal derivative of a square integrable martingale with continuous samples. The Gaussian white noise is an example of a martingale noise. It is shown that the differential equations of the mean and correlation functions of the state X developed in the paper resemble the corresponding equations of the classical linear random vibration and coincide with these equations if the input is a Gaussian white noise. The moment equations are derived by (1) the Itô formula for semimartingales and (2) the classical Itô formula applied to a diffusion process whose coordinates include X. An advantage of the second method is use of more familiar concepts. However, this method requires to calculate unnecessary moments and can be applied only for a class of martingale noise processes. Examples are presented to illustrate and evaluate the two methods for calculating moments of X and demonstrate the use of these methods in linear random vibration.     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

This paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

随机动力学响应的高斯相关过程模拟

动力学响应是描述系统振动状态、评估其性能、用于控制等的重要变量,复杂系统的不确定性、随机激励样本的不可测量性等导致传统随机响应时程与统计分析的计算困难,因此需要发展基于系统响应观测的、直接的随机过程概率模型与评估新方法。近年来,人工智能与数据处理技术等领域发展的无确定性系统模型的、直接随机过程概率模型,及其概率评估、系统状态预测等方法为动力学响应的概率分析提供了新思路,特别是具有很好普适性与可分析性的高斯相关过程已具有较完整的理论方法。鉴于此,本文提出针对动力学系统响应的、直接的随机过程概率模型与评估方法,并作探索性研究。先基于高斯白噪声激励动力学系统响应的统计特性分析,说明系统响应的高斯随机过程特性、响应在时间维度上的相关性、及其协方差随时间差的指数衰减特性等;再给出该系统响应的高斯相关过程概率建模与评估方法,包括由响应协方差计算,高斯过程协方差或核函数的拟合,到高斯相关过程概率模型的确定,响应样本过程的直接生成,及其统计评估等,并给出高斯相关过程的贝叶斯更新与系统状态预测有关基本公式。数值结果表明该高斯相关过程的概率建模与响应评估方法的可行性与有效性。     

A modal correction method for non-stationary random vibrations of linear systems

The computational effort in determining the dynamic response of linear systems is usually reduced by adopting the well-known modal analysis along with modal truncation of higher modes. However, in the case in which the contribution of higher modes is not negligible, modal correction methods have been introduced to improve the accuracy of the dynamic response, for both deterministic and stochastic input. In the latter case the random response is usually corrected via various methods determined as rough extensions of methods originally proposed for deterministic input. Consequently the efficiency of the correction methods is not suitable, from both theoretical and computational points of view. In this paper, a new approach to cope with the non-stationary response of linear systems is presented. The proposed modal correction method provides a correction term determined as a pseudo-stationary contribution of the equation governing either first-order or second-order statistics. Owing to the fact that no truncation criteria are well established for random vibration study, the proposed modal correction method offers a suitable vehicle for determining very accurately the stochastic response of MDOF linear systems under Gaussian stationary and non stationary excitation as evidenced in the numerical applications.     

Synthesizing nonstationary,non‐Gaussian random vibrations

This paper presents a novel technique by which non‐Gaussian vibrations are synthesized by generating a sequence of random Gaussian processes of varying root mean square (rms) levels and durations. The technique makes use of previous research by the authors which shows that non‐Gaussian vibrations can be decomposed into a sequence of Gaussian processes. Synthesis is achieved by first computing a modulation function which is produced from the rms and the segment length distribution functions, both of which were developed in previous research. This is achieved by first generating a sequence of uniformly distributed random numbers scaled to the range of segment length, which itself is a function of the desired total duration of the synthesized process. In order to transform a uniformly distributed random variable into any arbitrary non‐uniform distribution, the cumulative distribution function is established and used as a transfer function applied to the uniformly distributed random variable. This modulation function is applied to a Gaussian random signal itself generated by a standard laboratory random vibration controller (RVC) by means of a purposed‐designed variable gain amplifier system. In order to counteract the feedback function of the RVC, a second variable gain amplifier is introduced into the system in order to attenuate the feedback signal in inverse proportion to the gain applied to the command signal. This result is a nonstationary, non‐Gaussian random signal that statistically conforms to the desired PSD as well as the RMS distribution function. Copyright © 2010 John Wiley & Sons, Ltd.     

On the dual iterative stochastic perturbation‐based finite element method in solid mechanics with Gaussian uncertainties

The main idea is a dual mathematical formulation and computational implementation of the iterative stochastic perturbation‐based finite element method for both linear and nonlinear problems in solid mechanics. A general‐order Taylor expansion with random coefficients serves here for the iterative determination of the basic probabilistic characteristics, where linearization procedure widely applicable in stochastic perturbation technique is replaced with the iterative one. The expected values and, in turn, the variances are derived first, and then, they are substituted into the equations for higher central probabilistic moments and additional probabilistic characteristics. The additional formulas for up to the fourth‐order probabilistic characteristics are derived thanks to the 10th‐order Taylor expansion. Computational implementation of this idea in the stochastic finite element method is provided by using the direct differentiation method and, independently, the response function method with polynomial basis. Numerical experiments include the simple tension of the elastic bar, nonlinear elasto‐plastic analysis of the aluminum 2D truss, and solution to the homogenization problem of periodic fiber‐reinforced composite with random elastic properties. The expected values, coefficients of variation, skewness, and kurtosis of the structural response determined via this iterative scheme are contrasted with these estimated by the Monte Carlo simulation as well as with the results of the semi‐analytical probabilistic technique following the response function method itself. Although the entire methodology is illustrated here by using the Gaussian variables where all odd‐order terms simply vanish, it can be extended towards non‐Gaussian processes as well and completed with all the perturbation orders. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.     

Linear systems with fractional Brownian motion and Gaussian noise

Methods are presented for calculating the evolution in time of the second moment properties of the output of linear systems subjected to fractional Brownian motion and fractional Gaussian noise, defined as the formal derivative of fractional Brownian motion. The study also examines whether the output of linear systems to fractional Brownian motion and fractional Gaussian noise exhibits long range dependence. Numerical examples are presented to illustrate the calculation of output statistics for some linear systems with fractional Brownian motion and fractional Gaussian noise input, and show that output of linear systems to these input processes may not have long memory.     

Chaotic descent method and fractal conjecture

Very often, when dealing with computational methods in engineering analysis, the final state depends so sensitively on the system's precise initial conditions that the behaviour becomes unpredictable and cannot be distinguished from a random process. This outcome is rooted in an intricate phenomenon labelled ‘chaos’, which is a synonym for unpredictable events in nature. In contrast, chaos is a deterministic feature that can be utilized for problems of finding global solutions in both non‐linear systems of equations as well as optimization. The focus of this paper is an attempt to utilize computational instabilities in solving systems of non‐linear equations and optimization theory that resulted in development of a new method, chaotic descent. The method is based on descending to global minima via regions that are the source of computational chaos. Also, one very important conjecture is presented that in the future might lead the way towards direct solving of the systems of simultaneous non‐linear equations for all the solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

About stability of stochastic elastic and viscoelastic systems

On examples of elastic and viscoelastic systems it is shown that unstable deterministic parametric systems can be stabilized by the addition of a random noise to the deterministic parametric load. The noise is assumed in the form of a Gaussian stationary process with an implicit periodicity. The almost sure stability and stability with respect to statistical moments of the first and second-order is considered. The analysis is built on the basis of the calculation of the top Liapunov exponent.     

A non‐stationary covariance‐based Kriging method for metamodelling in engineering design

Metamodels are widely used to facilitate the analysis and optimization of engineering systems that involve computationally expensive simulations. Kriging is a metamodelling technique that is well known for its ability to build surrogate models of responses with non‐linear behaviour. However, the assumption of a stationary covariance structure underlying Kriging does not hold in situations where the level of smoothness of a response varies significantly. Although non‐stationary Gaussian process models have been studied for years in statistics and geostatistics communities, this has largely been for physical experimental data in relatively low dimensions. In this paper, the non‐stationary covariance structure is incorporated into Kriging modelling for computer simulations. To represent the non‐stationary covariance structure, we adopt a non‐linear mapping approach based on parameterized density functions. To avoid over‐parameterizing for the high dimension problems typical of engineering design, we propose a modified version of the non‐linear map approach, with a sparser, yet flexible, parameterization. The effectiveness of the proposed method is demonstrated through both mathematical and engineering examples. The robustness of the method is verified by testing multiple functions under various sampling settings. We also demonstrate that our method is effective in quantifying prediction uncertainty associated with the use of metamodels. Copyright © 2006 John Wiley & Sons, Ltd.     

Closed-form solutions for the time-variant spectral characteristics of non-stationary random processes

Spectral characteristics are important quantities in describing stationary and non-stationary random processes. In this paper, the spectral characteristics for complex-valued random processes are evaluated and closed-form solutions for the time-variant statistics of the response of linear single-degree-of-freedom (SDOF) and both classically and non-classically damped multi-degree-of-freedom (MDOF) systems subjected to modulated Gaussian colored noise are obtained. The time-variant central frequency and bandwidth parameter of the response processes of linear SDOF and MDOF elastic systems subjected to Gaussian colored noise excitation are computed exactly in closed-form. These quantities are useful in problems which require the use of complex modal analysis, such as random vibrations of non-classically damped MDOF linear structures, and in structural reliability applications. Monte Carlo simulation has been used to confirm the validity of the proposed solutions.     

A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation

A method for computing the lower-order moments of response of randomly excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficients and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases there is a significant reduction in the number of equations to be solved. The method is illustrated for a five-story shear-frame structure with nonlinear interstory restoring forces and random damping and stiffness properties. The results of the proposed method are compared to those estimated by extensive Monte-Carlo simulation.