A Regular Vine Copula-based Reliability Analysis Method For Multivariate Structural Variables

With the gradual improvement of the industrial level,the reliability analysis and design have played an important role in the complex engineering problems.In complex engineering problems,the correlation between design variables is widespread,and it directly affects the reliability results.In recent years,Copula functions,the new correlation analysis tool with high flexibility and multi-variables distribution,have been widely used in the field of structural reliability analysis.Copula functions can separate the marginal distribution functions and correlation structure of multi-variables.It can not only deal with a variety of correlation problems,but also does not depend on a specific distribution.However,Copula function can only construct the correlation structure between bivariate variables,and it is difficult to expand to high-dimensional cases.Therefore,in order to solve the problem of multiple correlation,the Regular Vine Copula models based on graph theory have been proposed.Based on the marginal distribution functions of variables and multiple Pair-Copula constructions,the Regular Vine Copula models can accurately construct the joint distribution functions.However,most of the current research concentrates on a subclass of the Regular Vine Copula models,namely the Drawable Vine(D-Vine)Copula model.In the D-Vine Copula model,the whole design variables are sequenced,which directly reduces the flexibility of the Regular Vine Copula models.Meanwhile,for different problems,only using D-Vine Copula model to build the correlation structure will cause the error results.Therefore,it is of great significance to build a correlation model with higher flexibility and better sample fitness for the reliability analysis of multiple structural variables.Based on the existing correlation analysis methods,the paper explores the application of Regular Vine Copula models in the reliability analysis of multiple structural variables.The main research contents are as follows:(1)Aiming at the problem that the conventional copula functions cannot accurately describe the asymmetric tail negative correlations,a reliability analysis method of multivariate structural variables based on the rotated copula functions is proposed.Firstly,the rotated copula functions are introduced into the set of candidate Pair-Copula Constructions(PCC).Then,the AIC information criterion and MLE(Maximum Likelihood Estimation)are used to select the optimal Pair-Copula Constructions and approximate the corresponding correlation parameters.Finally,the correlation structure of multivariate variables is constructed accurately.Moreover,in the numerical examples,the results of the algorithm are compared with the results of the Monto Carlo method,and the significant impact of rotated copula functions on the reliability results is further analyzed.(2)In view of the multi-source and high-dimensional correlations between variables,we determine the applicable Regular Vine Copula models by derivation.And proposes six multiple structure reliability analysis algorithms based on the Regular Vine Copula models,which are CVC-FORM,DVC-FORM,B0VC-FORM,B1VC-FORM,B2VC-FORM and B3VC-FORM algorithms,respectively.The above six algorithms convert the joint probability density function(PDF)into the product of multiple bivariate Pair-Copula Constructions and marginal PDF according to a certain logical structure.After the joint PDF is constructed,the reliability of multivariate structural variables is solved by the improved First Order Reliability Method(FORM).Finally,in the numerical examples,six RVC-FORM algorithms are compared with Monto Carlo method.The accuracy and robustness of the proposed algorithms are verified,and the sample fitness of several Regular Vine Copula models is further analyzed.(3)In view of the complex engineering problems which the performance functions are implicit and difficult to obtain,this paper proposes an RVC-FORM algorithm based on the local encryption approximation model technology.Firstly,the approximate limit state function of complex engineering problems is constructed by using the local encryption approximate model technology.Secondly,the reliability of complex engineering problems with multiple correlation variables is solved based on six RVC-FORM algorithms.Finally,the accuracy and robustness of the algorithm are verified by two numerical examples and an engineering example of commercial vehicle occupant living space analysis.