Efficient Stochastic Uncertainty Analysis Algorithms Based On High-dimensional Model Representation

Uncertainties are quite common in practical engineering,e.g.,the randomness of structural size,material properties and external loads.These uncertainties will spread to the response through the structure,and lead to the variability of the response described by a statistical feature or distribution function.Therefore,accurately analyzing uncertainties of structural responses is of great significance to evaluate the performance of the structure.Recently,uncertainty analysis methods based on probability theory and mathematical statistics theory,such as reliability analysis methods and sensitivity analysis methods,are widely used in practical engineering.The key of these uncertainty analysis methods lies in the calculation of statistical moment of the response and the metamodel technique.With the increasing complexity and scale of modern engineering structures,the strong correlation and high dimension of input random variables are frequently encountered,which poses a great challenge to the accuracy and efficiency of statistical moment estimation and metamodel technique.When the correlation of input variables is considered,the accuracy and efficiency of traditional statistical moment estimation methods may be hardly guaranteed simultaneously.For complex high-dimensional problems,current metamodel methods are usually front with problems of curse of dimensionality or overfitting.In order to solve these problems,this dissertation carried out in-depth research on the high-dimensional model representation method:(1)Based on the dimension-reduction method(the reference high-dimensional model representation),a statistical moment estimation method for stochastic systems with correlated variables is proposed;(2)Based on the analysis of variance high-dimensional model representation method,two efficient metamodel methods are proposed.The main contents of this dissertation are as follows:1.An uncertainty analysis method based on hybrid dimension-reduction method is proposed for the stochastic systems with correlated input random variables.Firstly,the hybrid dimension-reduction method based on the Nataf transform is proposed by combining the univariate dimension reduction method with the bivariate dimension reduction method,which greatly improves the calculation accuracy of statistical moment and has satisfactory efficiency.Then,the maximum entropy method is used for reliability analysis.The results show that the calculation of failure probability is highly accurate under the premise of accurate statistical moment estimation,when medium and low-level reliability analysis is required.2.An accurate uncertainty analysis method is proposed for problems without probability distributions of the input random variables.Firstly,we proposed an adaptive polynomial representation method to recover the probabilistic information of random variables with the aid of statistical moments or data of input variables,and a distribution-free copula method to transform the correlated variables into independent standard normal probability space.Then,the hybrid dimension reduction method combined with the distribution-free copula method is used to calculate the statistical moments,which only needs the statistical information of random variables.Based on the accurate statistical moment estimation,an improved fractional moment-based maximum entropy method is proposed,which is of good performance even for high-level reliability analysis problem.3.For solving high-dimensional uncertainty analysis problems,an efficient metamodel is proposed,which can overcome the difficulty of the curse of the dimensionlity caused byhigh-dimensional input random variables.Firstly,the original function is approximately expressed by univariate dimension reduction method,and then the component function is fitted by orthogonal polynomials.Therefore,the high-dimensional problem can be transformed into a series of one-dimensional problems.Based on the adaptive polynomial representation method of random variables,a strategy for updating polynomial bases based on information entropy is proposed.Then,an adaptive polynomial dimension decomposition metamodel based on univariate dimension reduction method is established.The results show that the proposed method solves the curse of dimensionality with high efficiency,and works well for the problems which are difficult for common polynomial chao expansion methods.4.For solving problems of complex high-dimensional uncertainty analysis,an adaptive sparse polynomial dimension decomposition metamodel based on the Bayesian theory is proposed.Firstly,in order to overcome the shortcomings of traditional Bayesian lasso algorithm,the analytical expression of the posterior prediction distribution is derived,and an efficient iterative algorithm of maximum a posteriori estimation is proposed.In order to reduce the scale of candidate bases and improve the modeling accuracy,a model selection criterion for model accuracy is proposed based on the Bayesian model averaging.Based on the cross-entropy,we proposed a low fidelity prior model to provide reasonable prior information for Bayesian model averaging,which is composed of the polynomial dimension decompositions of each univariate component function.Then,add the polynomial bases to the low fidelity model,select the bases by the proposed model selection criteria,perform the regression analysis by using the analytical Bayesian lasso algorithm,and finaly obtain the sparse polynomial dimension decomposition metamodel with high fidelity.5.The adaptive sparse polynomial dimension decomposition metamodel based on the Bayesian theory is applied to the uncertainty analysis of dynamic responses of the underwater vehicle structure.The uncertainty analysis of structural dynamic response is essentially a time-variant problem.The extremum method transforms the original time-variant problem into a non-time-variant problem by quantifying the uncertainty of the extremum of the time-history curve,and then evaluates the performance of the structure conservatively.Firstly,internal force responses of each time node are fitted by the proposed metamodel method,and then extreme value of predicted responses are obtained by maximizing the predicted time-histories.Based on the predicted extreme responeses,the uncertainty analysis can be performed.The results show that the proposed metamodel can accurately quantify the uncertainty of dynamic responses of the underwater vehicle structure,and the single internal force response can not fully reflect the bearing capacity of the structure.The extreme value of equivalent bending moment combinging the axial force,bending moment and shear force can well reflect the transverse and longitudinal coupling effect of the structure,and is helpful to reduce uncertainty analysis errors.