Researches On Forward And Inverse Propagations Of The Structural Random Uncertainties

Uncertainty widely exists in practical engineering,and that the accurate evaluation of the influence of the uncertain parameters on structural response has become an essential part of system health monitoring and product reliability analysis in engineering.On the one hand,it is necessary to analyze the influence of uncertain factors on system performance to judge whether the system performance can meet the design requirements and to further ensure the stability of system performance and the reliability of service.On the other hand,the obtained system performance information in return evaluates the system parameters and optimizes the system parameters to further improve the system performance.This makes the uncertainty propagation technology has important practical significance and broad applications in the field of theoretical research and engineering.At present,lots of scholars have done a lot of researches on the structural random uncertainty quantification,but most of them only focus on the unimodal uncertainty propagation of known probability distributions.However,in practical engineering problems,it often occurs that the probability distributions of random variables may be arbitrary,and there exist multimodal distribution propagation and multi-level interdisciplinary propagation.The research on these problems belongs to the primary stage,and the related uncertainty theory is not perfect.Based on these cases,this paper focuses on the complex uncertainty propagation under arbitrary probability distribution,multimodal distribution and multi-level uncertainty propagation,and carries out the following works:(1)In order to measure arbitrary probability distribution,a unified framework based on derivative ?-PDF is proposed.Through strict mathematical proof,the fitting ability of derivative ?-PDF is derived and expressed by a closed fitting region.For any probabaility distributions in the fitting domain,a single derivative ?-PDF can fit them accurately.For the unimodal probability distribution outside the fitting region and multimodal probability distributions,a derivative ?-PDF mixed model(?MM)is presented.At the same time,in order to realize the parameter estimation of ?MM more simply and conveniently,a pseudo expectation maximization method proved strictly is proposed,and it can match the first four statistical moments of variable.In a word,the derivative ?-PDF and ?MM effectively solve the fitting of random varialbes inside and outside the fitting region,and perfects the general measurement framework of arbitrary probability distributions.(2)In order to the forward uncertainty propagation problem of arbitrary probability distributions in the fitting domain,a high-precision propagation method based on fractional moments is developed.The method is a double-layer structure.The inner layer uses the dimension reduction integration method to calculate the forward uncertainty propagation under the arbitrary distribution,and obtains the entropy value of the probability density function of structural response.The outer layer takes the entropy value as the optimization objective function,and optimizes the fractional order by using the optimization algorithm.Until the objective function converges,the optimal fractional order is obtained,and simultaneously the fractional moment and probability density function of the response can be outputed.Because the error of fractional moments is smaller than that of integer moments,the probability density function of response can be estimated more accurately,so as to realize the high-precision forward uncertainty propagation under arbitrary probability distributions.(3)Aiming at the complex forward propagation problems of random uncertainty,such as probability distributions outside the fitting region,multimodal probability distributions and multi-level uncertainty propagation,a general framework based on?MM and polynomial chaos expansion method is proposed.In this method,?MM with strong fitting ability is used to represent arbitrary random variables.Then,the multimodal uncertainty propagation is equivalent to multiple unimodal uncertainty propagations,which are sovled by polynomial chaos expansion method to obtain the statistical moment of structural response.For multi-level uncertainty propagation,Gaussian Copula is adopted to measure the uncertainty and correlation of the intermediate variables,and then combines the choleskey factorization decorrelation to decouple the correction and polynomial chaos expansion are used to solve the latter uncertainty propagation,so as to complete the multi-level uncertainty propagation analysis eventually.This method not only can solve the complex uncertainty propagation efficiently and accurately,but also can get the accurate PDFof the structural response through the maximum entropy principle with only the first four statistical moments,which improves the efficiency of forward uncertainty propagation.(4)Aiming at the inverse uncertainty propagation under arbitrary probability distributions,a method based on ?MM and optimization algorithm is proposed.The inverse uncertainty propagation is transformed into forward uncertainty propagation and structural parameter optimization problem.The forward uncertainty propagation uses?MM to measure the unknown input variables,and combines the dimension integration method to solve it,so as to obtain the statistical moment of the structural response.In the parameter optimization problem,the optimization objective function is constructed by calculating the residual of the calculated statistical moment and measured statistical moment,and the distribution parameters of ?MM are optimized until the objective function converges,thus realizing the unimodal reverse uncertainty propagation.Particularly,when there is multimodal probability distribution,the multimodal inverse uncertainty quatification is transformed into the several unimodal uncertainty propagations,and the final result is the weight superposition of the result of the unimodal uncertainty propagations.In a word,the inverse uncertainty propagation problem is transformed into the parameter optimization problem of forward uncertainty propagation by the optimization algorithm,so as to obtain a set of optimal parameters and to realize the inverse uncertainty propagation under arbitrary probability distributions.