Efficient Response Surface Method For Structural Stochastic Analysisi Of Non-Gaussian Space

In actual engineering,uncertainty factors such as material parameters and external loads can affect the output response of the structure,and the effect of uncertainty factors must be adequately considered to gaurantee the reliability of the structure,and the common method of structural stochastic analysis is Response Surface Method(RSM),which uses a known function with certain characteristics to construct a quantitative relationship between the basic variables and the response quantity.The implicit functional relationship is made explicit,which has the characteristics of simple principle and practical convenience.However,the traditional response surface method has the problems of low computational accuracy and efficiency in solving structural multi-objective non-Gaussian random variable problems.Therefore,in this paper,the response surface method for structural stochastic analysis is systematically studied,and an efficient response surface method for nonGaussian space is established based on the traditional response surface method.The main research contents of this paper are as follows:(1)For the existing common methods of structural reliability analysis:Monte Carlo Simulation(MCS),Hermite Response Surface Method(HRSM)and Full-space Response Surface Method(FRSM),the basic principle,the selection of matching points and calculation steps are studied in detail;The structural reliability indexes and statistical eigenvalues are calculated by these three methods,and the results of MCS are used as the benchmark solution for comparison and analysis.It is shown that FRSM can effectively solve the problem of repeatedly constructing the response surface of HRSM for solving the multi-objective response of the structure,and it has good full domain capability,providing a conceptual basis for the following research on response surface method in non-Gaussian space.(2)In order to solve the problem of low computational accuracy and efficiency when the traditional response surface method is applied to nonGaussian space,the Generalized-chaos Response Surface Method(GRSM)is proposed.Firstly,constructing mixed generalized chaotic polynomials based on different distribution types of random variables,whereby the explicit expressions of the target response quantities are established;Then,the roots of the one-dimensional generalized chaos polynomials are orthogonally combined to construct the response surface collocation points,and the optimal probability collocation points with higher contribution are selected based on the principle of full rank of the coefficient matrix.The linear equations are then used to determine the coefficients of the generalized chaotic polynomials and to establish the generalized-chaos response surface expressions.Finally,by using the MCS results as the benchmark and comparing with the current HRSM,it is verified that the GRSM established in this paper can achieve higher computational accuracy with fewer collocation points and lower response surface expansion order,and significantly improve the computational efficiency when solving non-Gaussian spatial nonlinear stochastic problems.It is shown that the GRSM effectively overcomes the shortcomings of HRSM which is affected by nonlinearity and type of stochastic distribution.(3)In order to solve the problem of complicated computational paths when FRSM is applied to non-Gaussian space,generalized chaos theory is introduced into FRSM,and the Improved Full-space Response Surface Method(IFRSM)is proposed.The Krylov basis vector is constructed in the non-Gaussian probability space,and the target response is expanded by the basis vector,and then the roots of the generalized chaos polynomial corresponding to the distribution type of the random variables are directly used to construct the matching points,and then the optimal matching points are selected based on the principle of linear irrelevance of the combined row vectors,and the coefficients to be determined by the expansion are used to establish the improved full-space response surface expression.Finally,the accuracy and efficiency of IFRSM are verified through the analysis of arithmetic cases.It is shown that,for solving the multi-objective response of the structure,IFRSM applies the pending coefficients of the response surface calculated by using only one degree of freedom of the structure to the whole structure,which has good full domain in the structure and probability space,and effectively overcomes the drawback of GRSM that needs to reconstruct the response surface.