【作者】 王飞；

【导师】 吕震宙；

【作者基本信息】 西北工业大学 ， 飞行器设计， 2016， 硕士

【摘要】 在飞行器结构、机构或系统的设计过程中,一方面,需要考虑材料的几何尺寸和性能参数以及在所处环境中受到的约束和外载荷等不确定性因素的影响开展全局灵敏度分析;另一方面,工程上越来越需要考虑多个结构性能同时进行设计。为提高结构系统在不确定性条件下的多维输出性能,在已有的研究基础上,本文从多维输出响应的协方差矩阵分解,多维空间距离度量和多维随机变量的联合熵三个角度出发对多维输出情况下的结构系统开展全局灵敏度分析的有关研究:(1)利用多维输出响应间的协方差矩阵表示结构系统的整体不确定性,再根据协方差矩阵的分解来定义多维输出情况下的广义全局灵敏度指标。针对基于方差的全局灵敏度分析主效应的不足以及各个输出响应间量纲差异的影响两个方面进一步完善,从而提出一种使用无量纲化模型的基于协方差矩阵分解的全局灵敏度分析新指标。数值与工程算例的分析说明了新指标可以方便地综合衡量输入随机变量的变异性对多维输出结构系统变异性影响的重要程度,而且能够有效地保留各个输出响应提供的全局灵敏度信息。同时,应用一种乘法降维模型来代替原始功能函数去求解新指标,并利用蒙特卡洛数字模拟方法作对照说明新算法在保证指标求解精度的同时节约了模型运算成本。(2)多元数据之间的空间距离度量可以描述多维输出响应不同样本间的差异性。根据这一特点采用空间距离来表征多维输出情况下结构系统的整体不确定性。具体把聚类分析和判别分析中常用的欧氏距离和加权马氏距离引入到结构系统不确定性的表示中,从而开展多维输出情况下的重要性分析和定义输入随机变量对多维输出影响的全局灵敏度分析新指标,并针对所提出的新指标的数学性质和物理意义进行分析讨论。由于稀疏网格积分求解高维问题的优势,据此建立基于空间距离的全局灵敏度分析新指标的求解算法,可以在保证计算精度的同时大幅度提高多维输出结构系统的全局灵敏度分析效率。(3)多维输出响应的不确定性信息完整描述是其联合概率密度函数,其包含了各输出响应间的复杂相关关系。多维输出响应联合熵直接是在联合概率密度函数的基础上定义的,因而用信息熵来表示结构系统的整体不确定性更加合理。基于多维随机变量联合熵提出一种新的全局灵敏度分析方法,用消除输入变量的不确定性后对多维输出响应联合熵的影响来表征输入随机变量的重要性程度,并就所提全局灵敏度指标的有关数学性质以及它同相关熵的联系展开讨论。利用双层蒙特卡洛法和单层蒙特卡洛法求解所提出的输入随机变量基于联合熵的新全局灵敏度指标,而联合概率密度函数则可以利用基于高斯核的多维核密度估计算法求解。更多还原

【Abstract】 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.更多还原