Global Sensitivity Analysis For Structure System With Multivariate Outputs

In the design of aircraft structure and mechanism system,on the one hand it is necessary to execute global sensitivity analysis by considering the uncertainty of input variables,such as the geometrical dimension and performance parameters of the material,constraint and load in the surrounding environment,and the errors resulting from the instrument measurement,etc,and on the other hand more and more mathematical models encountered in engineering structure system are involved with the multivariate outputs.In this situation,to improve the performance of the structure and mechanism system under uncertainty circumstances,this paper proposes three global sensitivity analysis methods based on decomposition of the covariance matrix of the multivariate output responses,multidimensional spatial distance and the joint entropy of the multivariate output responses.The detailed contents are summarized as follows:1.The covariance matrix of the multivariate output responses has emerged to characterize the uncertainty of structure system and a set of generalized global sensitivity indices based on the decomposition of the covariance matrix was proposed for structure system with multivariate outputs.In consideration of the imperfections in the main effects of Sobol' global sensitivity analysis and the dimensional influence of each output response,this paper defines a set of new global sensitivity indices based on the decomposition of the covariance matrix using the multivariate non-dimensional outputs.Numerical example and industrial design case illustrate that the new indices can synthetically measure the uncertainty effect on the multivariate output responses induced by the corresponding input random variable expediently and contain global sensitivity analysis information of each output effectively.The new indices are calculated by using a surrogate model which is based on a multiplicative version of the dimensional reduction method.The new algorithm can greatly reduce the number of model calls without decreasing its accuracy compared with the Monte Carlo simulation.2.Since the distance metrics between the multivariate data can be used to describe the differences among samples of the multivariate output responses,the multidimensional spatial distance can be used to characterize the uncertainty of structure system with multivariate outputs.The global sensitivity indices based on Euclidean distance and weighted Mahalanobis distance which are commonly used in the clustering analysis and discriminant analysis are defined to synthetically measure the uncertainty effect on the multivariate outputs induced by the corresponding input random variable expediently.The geometric and physical properties of the new indices are analyzed as well.Furthermore,the sparse grid integration(SGI),which is adept in solving high-dimensional integration problems,is used to calculate the new global sensitivity indices based on multidimensional spatial distance.The established SGI-based method can improve the computational efficiency of the new global sensitivity analysis indices considerably in case of acceptable accuracy.3.Since the joint probability density function(PDF)can perfectly describe the uncertainty information of the multivariate output responses,which contains the complicated correlation between multivariate outputs.The joint entropy based on the joint PDF of multivariate output responses can be utilized to describe the uncertainty of structure system with multivariate outputs according to the information entropy.Then a new global sensitivity analysis method based on the joint entropy of multivariate output responses is proposed,which can indicate the importance of input variables through the effects of the input variables on the joint entropy of multivariate output responses.At the same time,the mathematical properties of the proposed global sensitivity indices are discussed,as well as the relationship between the new indices and the relative entropy.The double-loop Monte Carlo simulation and single-loop Monte Carlo simulation methods are used to calculate the proposed new global sensitivity indices and the joint PDF is estimated by the multivariate kernel density estimation method with Gaussian kernel.