Research On The Method Of Uncertainty Analysis Based On Polynomial Chaos Expansion

In practical engineering problems,uncertainties are widespread because of the influence of many factors,such as different material properties,geometric parameters of structures,process conditions and manufacturing errors.Ignoring the impact of these uncertainties may lead to loss of accuracy,performance degradation,structural failure and even casualties in the actual engineering design.For these engineering problems,in the design stage,we should fully consider the influence of uncertainty on the structure and reserve a certain safety margin,and then carry out the corresponding structural uncertainty analysis and structural reliability optimization design.This is of great value to improve the safety,reliability and economy.According to the probabilistic distribution of uncertain variables,this paper studies three important problems based on the theory of polynomial chaos expansion,namely the difficulty of selecting orthogonal polynomial bases under different types of single-peak distribution,the difficulty of selecting orthogonal polynomial bases without corresponding polynomial bases under multi-peak distribution and the difficulty of polynomial chaos expansion under arbitrary peak distribution.The following three tasks have been completed:(1)Aiming at the difficulty of choosing orthogonal polynomial bases under different single-peak distributions,a method based on chaos expansion of Gegenbauer polynomial is studied.In this method,a series of uncertain variables of single-peak distribution are fitted by ?-PDF and Gegenbauer polynomials are used as the orthogonal bases of polynomial chaos expansion,which effectively solves the difficulty of orthogonal base selection under different types of single-peak distribution,and provides a new idea for solving uncertain propagation with single-peak distribution.(2)A polynomial chaos expansion method based on Gauss mixture model is studied to solve the difficulty that there is no corresponding orthogonal polynomial bases to solve multi-peak distribution problems.The method uses the Gauss mixture model to model the uncertain variable with multi-peak distribution and constructs the orthogonal bases of the model based on the recurrence relations of the orthogonal polynomial.It effectively solves the problem that there is no corresponding orthogonal polynomial dealing with the multi-peak distribution,and provides an effective solution to multi-peak distribution problems in practical engineering.(3)To handle the problem of polynomial chaos expansion under arbitrary-peak distribution,a polynomial chaos expansion method is studied based on ?-PDF mixture model.In this method,the uncertain variable with arbitrary peak distribution can be modeled by using the ?-PDF mixed model,and then the orthogonal polynomial bases can be directly constructed according to its moment information.This method effectively solves the problem with mixing single-peak distribution and multi-peak distribution,which provides theoretical support for complex engineering problems.