Research On The Method Of Uncertainty Quantification And Propagation Analysis Based On Probability

In practical engineering problems,there are many inevitable uncertainty factors due to the complexity of structures;dispersion of materials;as well as errors in manufacturing,installation,and measurement.The coupling of multiple uncertainties often results in a considerable fluctuation of the structures or the product performance,sometimes even leads to a failure.In general,the uncertainty can be described by the interval quantification or the probability quantification.However,the latter one is employed in this paper because of a relative perfect theory and relative precise uncertainty quantification.In terms of the uncertainty analysis,the influence of the parameter uncertainty on the system response is of great concern to many researchers,which can be known as uncertainty propagation(UP).Concentrating on the UP problems has become a key issue for the reliability design of products or a system.In recent years,a type of an efficient numerical method,namely,the sparse grid method,has been introduced into the UP field and gets a relative good effect.Nevertheless,with regard to the complex practical engineering problems,it is imprecise for calculating the high order moments in UP problems with the sparse grid method.Furthermore,when it comes to the input random variables which have various types of distributions,it is difficult to solve the statistics characteristics by sparse grid,thereby limiting the versatility of the method to some extent.Aiming at improving the accuracy of the high order moments and making the sparse grid suitable to different types of distributions of the input random variables,this study tries to make a meaningful exploration on its algorithm and practical value.The research content of this paper is as follows:(1)Proposing an uncertainty propagation method based on an extended sparse grid technique,which can ensure that the integration points in each variable can be transformed the extended integration points.Moreover,it can improve the accuracy of the system response’s high order moments,thereby providing an efficient computational tool for the complex engineering problems.The proposed method introduces the extended Gauss integration into the UP field,and uses Rosenblatt transformation to alter the integration points of arbitrary variables to the extended Gauss-Hermite points.Different from the traditional transformation method,the proposed transformation is based on the extended Gauss points,so it can enhance the algebraic accuracy of integration points.Furthermore,it can also provide sparse grid the univariate integration points,which can efficiently solve the UP problem.(2)As for the different types of distributions in unimodal variable(included direct orthogonal polynomials,non-included direct orthogonal polynomials,large amounts of data),this paper proposed an UP method based on the unknown distribution in order to unite the solution of sparse grid method.Using λ-PDF and its extended probability density function(PDF)to fit the distributions of the input random variables,and solving the undetermined coefficients in the fitting function by a gradient-based optimization algorithm,thereby constructing the PDF of unimodal variable.Based on the Gegenbauer orthogonal polynomials,the Gauss integration points and its weights can be efficiently solved,which can be used in the sparse grid method.Finally,the maximum entropy principle can be employed to fit the PDF of the system response.(3)With regard to the problem which has multimodal distribution in the input random variable,this paper proposed an UP method based on the multimodal distribution.First,the mixture model is specified as the second-order extended function of λ-PDF,and the determined parameter can be solved by EM algorithm.Furthermore,the standard moment based quadrature rule is introduced into the UP problem,and then it can be used to solve the univariate integration points and its weights in the mixture model.Finally,the sparse grid method and the maximum entropy principle are used to obtain the statistics moments as well as the PDF.The proposed method can fit the curve of multimodal distribution well,so it can efficiently apply to the UP problems with multimodal variables in practical engineering.