Research On High-dimensional Surrogate Model And Structural Uncertainty Based On Sparse Polynomial Chaos Expansion

In recent years,with the ever-increasing of engineering structural is becoming more and more large and complicated.Surrogate model(also known as metamodel)has been widely used to approximately replace the actual engineering structure for uncertainty analysis.The traditional technology of surrogate model has good accuracy,efficiency and robustness in solving low-dimensional nonlinear problems.However,with the increase of the dimensionality of computer models for engineering simulations,the computational cost of using the traditional surrogate model increases exponentially,and its approximation ability will be greatly affected or even a feasible computer model can not be obtained.Because of its rigorous theoretical basis and wide applicability,polynomial chaos expansion(PCE)has rapidly become one of the popular methods in the field of uncertainty analysis.This thesis aims to tackle the"curse of dimensionality"encountered by classic polynomial chaos expansion in modeling high-dimensional problems,as well as its limitations in global sensitivity analysis and structural reliability analysis.Firstly,since high dimensional model representation(HDMR)can decompose high-dimensional problems into the sum of low-dimensional component functions,it is proposed to solve high-dimensional problems efficiently by combining polynomial chaos expansion with high dimensional model representation.Secondly,the greedy coordinate descent method based on Bregman iteration is integrated into polynomial chaos expansion,which provides a new idea for the modeling and uncertainty analysis of large-scale complex structures.The main contributions of this thesis are as follows:(1)The polynomial chaos expansion is introduced into the high dimensional model representation,and a new high-dimensional surrogate model,named adaptive PCE-HDMR,is proposed by using intelligent dividing rectangles(DIRECT)sampling.An analytical function is used to illustrate the modeling principles and procedures of the algorithm.(2)A comprehensive comparison between the proposed PCE-HDMR and other well-established Cut-HDMRs is made on fourteen representative mathematical functions and five engineering examples with a wide scope of dimensionalities.The results show that the proposed PCE-HDMR has much superior accuracy and robustness in terms of both global and local error metrics while requiring fewer number of samples,and its superiority becomes more significant for polynomial-like functions,higher-dimensional problems,and relatively larger PCE degrees.(3)A novel methodology for developing sparse PCE is proposed by making use of the efficiency of greedy coordinate descent(GCD)in sparsity exploitation and the capability of Bregman iteration in accuracy enhancement.By minimizing an objective function composed of the?₁ norm(sparsity)of the polynomial chaos(PC)coefficients and regularized?₂ norm of the approximation fitness,the proposed algorithm screens the significant basis polynomials and builds an optimal sparse PCE with model evaluations much fewer than unknown coefficients.To validate the effectiveness of the developed algorithm,three benchmark examples are investigated for global sensitivity analysis(GSA).A detailed comparison is made with the well-established orthogonal matching pursuit(OMP),least angle regression(LAR)and two adaptive algorithms.Results show that the proposed method is superior to the benchmark methods in terms of accuracy while maintaining a better balance among accuracy,complexity and computational efficiency.(4)The sparse PCE built by greedy coordinate descent method and Bregman iteration is applied to the field of uncertainty analysis.Global sensitivity analysis,structural reliability analysis and mean variance and probability density function evaluation of plane truss structure and space truss structure are carried out.Compared to the Monte Carlo method,the results show that the proposed method can efficiently and accurately perform structural uncertainty analysis.The research results of this thesis will improve the modeling efficiency and optimal design ability of major engineering structures in China to a certain extent,and provide theoretical basis and technical support for the uncertainty analysis of structural parameters.