Structural Uncertainty Analysis Methods Involving Random Processes Or Fields

Uncertain factors are widely encountered in practical engineering problems,in which the probabilistic tool is the most commonly used measure to quantify the uncertainty.On this basis,some uncertain problems are put forward,such as reliability analysis,uncertainty propagation and uncertainty identification.According to whether the frequency,time,temperature,space and other deterministic variables are considered or not,uncertain problems are divided into conventional ones and ones with deterministic variables.As methods for conventional uncertain problems only concentrate on the calculation about random variables,and much attention has been paid to them,they have gradually matured.Uncertain analysis with deterministic variables usually involves the simulation of random processes or random fields,on whose basis a series of uncertain problems are raised,such as reliability analysis and uncertainty propagation involving random processes/fields and identification of random processes/fields.Since random processes/fields are much more complex than random variables,uncertain analysis involving them is more difficult to solve compared with the conventional one,and there are still a bunch of key problems need to be addressed.Therefore,this paper mainly concentrates on the uncertain problems involving deterministic variables,and three common deterministic variables in engineering problems are chosen as frequency,time and space.Then considering the uncertainty varying with these deterministic variables,corresponding studies are carried out on uncertainty analysis involving random processes/fields.According to the above frame,the main work completed in this paper is as follows:(1)Considering uncertain parameters varying with frequency and frequency dependent performance indicators in electromagnetic field,reliability of performance indicator in frequency domain is valuable to be studied.As the designed frequency domain usually extends to both sides from the center frequency,and the reliability of each required frequency band needs to be solved,the idea of frequency domain discretization is suitable to be adopted to transform the reliability problem of frequency domain into a system reliability problem at discrete frequency points.Following this idea,a reliability analysis method based on discretization of frequency domain is proposed.The method firstly discretizes the frequency-variant performance function into a series of functions at discrete frequency points,and then transforms the reliability problem in frequency domain into a series system reliability problem of these discrete functions.Secondly,the univariate dimension reduction method is introduced to solve the probability distribution functions and correlation coefficients of these functions in the system.Finally,according to the above calculation results,the series system reliability can be readily solved,and reliability corresponding to different failure thresholds can also be obtained.In this study,Monte Carlo simulation is adopted to demonstrate the validity of the method.Three examples are investigated to demonstrate the accuracy and efficiency of the proposed method.(2)Aimed at solving the time-variant reliability of arbitrary time interval and arbitrary failure threshold,the time-evolution process of extreme-value event is proposed.In this method,the random process in a time-variant function is firstly expanded by an improved orthogonal series expansion method.Second,we introduce the idea of extreme-value event to describe the time-variant reliability problem.And by discretizing the time domain,we can obtain a series of extreme-value events.The moments of extreme-value event in every discrete time interval will be solved by the integration of BFGS method and univariate dimension reduction method.Third,a time-dependent polynomial chaos expansion method is proposed to simulate the time-evolution process of extreme-value events,and it will be simulated as a function in terms of a standard normal variable and time.Finally,Monte Carlo simulation is adopted to sample the standard normal variable to obtain the time-variant reliability of arbitrary failure threshold and time interval.Four examples are investigated to demonstrate the effectiveness of the proposed method.(3)Given that time-variant reliability problems only concern about a small part of the information in the output stochastic process of a time-variant function with uncertainty,and can’t consider its whole uncertain characteristics such as the higher order statistical moments and autocorrelation function,which are necessary to conduct further analysis and applications,the time-variant uncertainty propagation analysis is defined to solve the output stochastic process of a time-variant function with uncertainty.And a time-variant uncertainty propagation analysis method is constructed with the combination of an extended orthogonal series expansion method(extended OSE)and sparse grid numerical integration(SGNI).The SGNI serving as a classical uncertainty propagation method is utilized here to solve the moments and autocorrelation function at discrete time points of the output stochastic process.And the extended OSE is used to simulate it based on the results from SGNI.By extended OSE,a non-Gaussian stochastic process is represented as the sum of orthogonal time functions with random coefficients,and these coefficients can be directly obtained by discretization of the target process.For these coefficients are correlated and non-Gaussian,the correlated polynomial chaos expansion method(c-PCE)is presented to represent them in terms of correlated standard Gaussian variables,and then the principal component analysis is adopted to transform them into independent ones with dimension reduction.Finally we can obtain an explicit expression to represent the non-Gaussian process whatever it is stationary or non-stationary.Three illustrative examples are used to verify the performance of the extended OSE.In addition,two engineering problems are investigated to demonstrate the effectiveness of the time-variant uncertainty propagation method.(4)In most developed methods involving estimation of random modal parameters for linear structures,the modal parameters are usually seen as independent random variables,and only the first two order moments of them are considered.However,neglect of higher order moments of the random modal parameters and the correlation among them will cause errors.Concerning this problem,the extended orthogonal series expansion method is applied to simulate the random fields of mode shapes in terms of discrete random mode shapes,which can be solved by stochastic finite element methods,while given their non-Gaussianity,the Monte Carlo simulation based stochastic finite element method is required to ensure the accuracy of higher order of moments.Based on the solved discrete random mode shapes,and referring to the above mentioned simulation of random process by extended OSE,the simulated random fields of mode shapes can be obtained.Then discrete random mode shapes and random natural frequencies are treated as correlated random variables which can be simulated in terms of independent Gaussian variables by the combination of c-PCE and orthogonal transformation.Based on the samples of simulated random modal parameters,the random dynamic responses under different inputs can be calculated by using the mode superposition method.Compared with the results by Monte Carlo simulation based stochastic finite element method,the proposed method is of good simulation accuracy.(5)The developed identification methods of Gaussian fields by hierarchical Bayesian framework are usually based on the Karhunen-Loeve(K-L)expansion simulating Gaussian fields.However the introduction of K-L expansion not only leads to identification bias of hyper parameters,but also causes uncertainty propagation of model making computation prohibitive.Besides the developed methods are only verified for stationary Gaussian fields.Aiming at the above problems,a new framework for Gaussian fields identification based on hierarchical Bayesian theorem is constructed,where the improved OSE is applied to simulate a Gaussian field into multiple correlated Gaussian variables,which can be readily obtained by discretizing itself at integral points.In the proposed framework,the hyper parameters to be identified are mean,standard deviation and correlation length of autocorrelation function of a Gaussian field,and the model parameters are discrete Gaussian variables.Since there is an iteration process for selection of integer parameters(the number of integral points and orthogonal functions)in the simulation of Gaussian fields by improved OSE,the model selection method is introduced to the identification process of model parameters.Given different models of a Gaussian field(determined by different integer parameters),the identification process is conducted.And the most suitable integer parameters are chosen according to the identification results based on a strategy of model selection.Then by utilizing them with corresponding hyper parameters,the identification of Gaussian fields can be realized.The proposed framework is applied to two dynamic examples.According to the random dynamic responses as observed data,it is used to identify a non-stationary Gaussian process load,and a stationary Gaussian field of Young’s modulus of a planar structure,respectively,which demonstrates the effectiveness of the framework.