Efficient Uncertainty Propagation Numerical Analysis Methods For Probability-box Model

There are various kinds of uncertainties in the whole life cycle from design,production to scrap of engineering products.In general,these uncertainties are small in most cases,but their spread and amplification may cause large fluctuation of product peformance,poor reliability,and even cause catastrophic accidents.Therefore,throughout the whole life cycle of the product,from design to use,it is of great significance to adopt uncertainty propagation analysis to ensure the reliability,safety and even economy of products.Actually,according to the different generation mechanisms and physical meanings,uncertainties can be divided into two different categories: aleatory uncertainty and epistemic uncertainty.Aleatory uncertainty,known as statistical uncertainty or objective uncertainty,which refers to the r andomness existing in nature or physical phenomena,can not be controlled or reduced by any designer.The analysis methods of aleatory uncertainty include probability theory,mathematical statistics and random process,whose theory and application research es are relatively mature and perfect.Epistemic uncertainty,generally called subjective uncertainty,is caused by insufficient subjective knowledge,data or incomplete information of the designer.The uncertainty analysis methods corresponding to epistemic uncertainty are non-probabilistic theories,which include fuzzy set,interval analysis theory and evidence theory etc.In fact,both aleatory uncertainty and epistemic uncertainty play an important role in the field of uncertainty analysis,and have obtained a lot of research achievements on theories.But in practice,there exists multiple uncertainty models rather than a single uncertainty model in practical engineering.Thus a new model is needed to integrate the existing multiple uncertainty models.Probabiligy box(p-box)is the model that can describe both aleatory uncertainty and epistemic uncertainty problems simultaneously,in gereral,the existing Dempster-Shafer structures,probability distribution,hybrid probability and interval model,etc.can be transformed into probability box.More importantly,probability box can be regarded as the mixture of probability theory and interval theory,which can be easily understood and accepted by engineers who are familiar with probability theory and interval theory.Due to this characteristic,the uncertainty analysis of p-box is getting more and more domestic and international attention in recent years,and some progressive achievements have been made.However,the research of uncertainty analysis based on probability box is still in the primary stage.A series of key scientific issues are required to be solved.Especailly,the large-scale computing and complex correlation problems hinder the applicability of probability box in practical engineering problems.Therefore,to deal with such challenges,the following aspects are carried out in this paper:(1)For the low computational efficiency of traditional sampling method,an efficient uncertainty propagation method based on univariate dimension reduction method(UDRM)and Johnson distribution is constructed for parameterized p-box inputs.Firstly,an optimization method based on univariate dimension reduction method is constructed to calculate the bounds on statistical moments of response function.Then,the Johnson distribution function is utilized for fitting the possible distribution functions of response.Finally,the moment matching method is presented based on the interval-valued moments,by which the probability bounds of the response p-box can be successfully obtained,and the uncertainty propagation analysis based on parameterized p-box is completed at the same time.(2)For the response function with strong interaction terms problems,a p-box uncertainty propagation analysis method based on sparse grid numerical integration and saddlepoint approximation is presented.Firstly,an optimization method based on the sparse grid numerical integration is presented to calculate the bounds on the statistical moments of the response function and the cumulants of the cumulant generating function,respectively.Then,an optimization strategy based on the saddlepoint approximation is proposed to calculate the probability bounds of reponse.The proposed method has high computational accuracy for high dimension and strong interaction problems,and can obtain a good tail probability distribution information.(3)For parameterized p-boxes variables,an uncertainty propagation analysis method based on sparse-decomposition-based polynomial chaos expansion is developed.Firstly,a sparse-decomposition-based polynomial chaos expansion(PCE)method is presented to process the aleatory uncertainty,in which a basis selec tion strategy that based on sparse decomposition is devised for automatically detecting the significant basis set of PCE.Then,to deal with the epistemic uncertainty,the coefficients of the sparse-decomposition-based PCE are treated as quadratic polynomial functions of the intervalvalued distribution parameters.Finally,the interval-valued mean,interval-valued standard deviation and probability bounds of the response function can be successfully obtained.The proposed method provides a new solution for the uncertainty propagation of probability boxes.(4)An uncertainty propagation method for correlated probability-boxes is given for input variables with complex nonlinear dependence.The parameterized p-box model is constructed first based on limited experimental data.Then,the AIC criterion is adopted to select the optimal copula function,by which the joint cumulative distribution function is acquired for the correlated input variables.Then the correlated variables are further transformed into independent normal variables through rosenblatt transformation.Finally,the bounds on statistic momens of the response function can be obtained based on the sparse grid numerical integration.(5)For the multiple failure modes problem,an efficient system reliability analysis method for structures with probability boxes uncertainties is proposed.Firstly,the minimum reliability index of each failure mode is obtained based on an efficient solution method.Then the system reliability model under multiple failure modes with probability box uncertainties is constructed.Considering the dependence between different failure modes of systems,a correlation coefficient matrix is obtained by the linear correlati on calculated method.Finally,the maximum failure probabilities are calculated for series and parallel system,respecively.The presented method has high computational efficiency and accuracy,which can meet the practical engineering requirements.