【作者】 欧阳衡；

【导师】 韩旭； 刘杰；

【作者基本信息】 湖南大学 ， 机械工程， 2020， 博士

【摘要】 实际工程中普遍存在着不确定性,准确地研究结构不确定性参数的度量模型及其传播与反求规律是结构优化设计与分析的重要基础。传统的不确定性度量、传播与反求方法大多数都是基于概率模型开展的,即先采用概率模型对结构不确定性参数进行建模,再利用概率密度或统计矩实现不确定性传播与反求。一般来说,构建准确的概率模型需要抽取大量的结构参数样本,但由于实验条件和外界环境的限制,实际问题中往往只能收集少量的样本信息,无形地限制了概率模型的适应性。此外,结构几何参数、材料参数和外部条件等不确定因素之间通常是具有相关性的,且相关性的存在会对结构不确定性参数的传播与反求结果产生一定的影响。然而,基于概率模型的结构不确定性传播与反求方法难以确定传播响应与待反求参数的概率密度类型,无法准确地描述其联合概率密度函数。近年来,随着非概率凸模型的出现及日益完善,特别是针对小样本下度量结构参数不确定性与相关性的椭球凸模型,使得基于非概率模型的结构不确定性传播与反问题研究受到了广泛关注。目前,基于非概率椭球模型的结构不确定性传播与反问题中仍有诸多问题亟待解决,包括结构参数不确定性与相关性序列传播、相关参数的全局敏感性分析、反求过程中最优测点选择和结构参数不确定性与相关性序列反求等。为此,本文针对上述问题逐个开展研究工作,力求在结构不确定性传播与反求方面做出一些有效的尝试和探索,并有序地建立一套考虑非概率相关性分析的结构不确定性传播与反求方法体系。本文的研究工作是基于结构输入参数端和输出响应端等两个层面开展的。首先,为了准确地度量结构输出响应不确定性边界,推导了结构参数到响应的非概率相关性传播公式;其次,为了实现结构相关参数的重要性排序,确定主要的反求参数,研究了结构相关参数的全局敏感性问题;再次,为了保证结构不确定性反问题中测量信息的有效性及反求结果的准确性,研究了结构不确定性参数识别的最优测点布局问题;最后,为了准确地度量结构参数具有相关性的不确定性边界,在上述研究的基础上,研究了结构参数不确定性与相关性的解耦和反求问题。基于上述思路,本文所开展和完成的主要工作如下:(1)提出了基于非概率相关分析的结构不确定性传播方法,实现了结构参数不确定性与相关性的共同传播。针对少量样本下结构参数不确定性与相关性的度量问题,采用椭球凸模型对其进行建模,准确地度量其不确定性边界。将考虑非概率相关性的结构不确定性传播问题分解为区间传播问题和非概率相关系数传播问题。针对区间传播问题,发展了椭球约束下的子区间分解方法,高效地获取了结构响应的区间。针对非概率相关系数传播问题,推导了从结构参数空间到响应空间的非概率相关性传播公式,且可根据实际精度要求将结构模型扩展至n维泰勒展开,解析地确定结构响应的非概率相关系数矩阵,并建立其椭球凸模型。(2)提出了考虑非概率相关性的结构不确定性参数全局敏感性分析方法,实现了非概率相关参数的重要性排序。采用椭球凸模型对相关参数的不确定性边界进行度量,并将非概率方差定义为非概率敏感性指标。推导了从不确定结构参数到结构响应的非概率方差传播公式,并基于方差分解的思想将结构输入参数对输出响应方差的影响分解成了方差贡献和协方差贡献。在椭球模型的基础上,定义了结构输入参数对输出响应方差的总贡献率、独立贡献率和相关贡献率等评价指标,从而准确地评价各输入参数对输出响应的敏感程度。(3)提出了基于最大独立均方差准则的结构最优测点布局方法,保证了测量信息的有效性和不确定性反求过程的稳定性。将结构不确定性参数识别的最优测点布局问题转换为均值已知和均值未知的结构参数不确定性传播问题。针对均值已知的结构参数不确定性传播问题,建立了最优测点布局的最大独立方差准则,并发展了基于蒙特卡洛模拟和正交匹配追踪的最优测点布局方法。同时,为了传感器布置效率,提出了基于降维积分和正交匹配追踪的最优测点布局方法。针对均值未知的结构参数不确定性传播问题,建立了最优测点布局的最大独立均方差准则,并发展了基于双层降维积分和正交匹配追踪的最优测点布局方法,从而确定最优测点的位置及数量。(4)提出了结构不确定性参数识别的序列区间与相关性反求方法,实现了结构参数区间和相关系数矩阵的反求。考虑结构测量响应的不确定性和相关性,采用椭球凸模型对其不确定性边界进行了量化。同时,将考虑相关性的不确定性反求问题分解为区间反求问题和相关性反求问题。针对结构参数的区间反求问题,分情况讨论了结构测量响应区间与计算响应区间的区间接近程度,定义了误差区间,并采用子区间分解分析法获取结构计算响应的区间。针对结构参数的相关性反求问题,采用非概率相关性传播公式确定结构计算响应的相关系数矩阵。利用优化算法循环减小测量响应与计算响应之间的区间误差和相关系数误差,获取结构参数的反求区间与相关系数矩阵,并建立其相应的椭球凸模型。更多还原

【Abstract】 Uncertainties widely exist in practical engineering problems.It is an important basis for structural optimization design and analysis to accurately investigate the uncertainty quantification,uncertain forward propagation and uncertain inverse propagation problems.Most of the traditional methods for the above problems were proposed based on the probabilistic model,in which the probability model was applied to describe the uncertainties in structural parameters and responses,and then used the probability density functions or statistical moments to realize the uncertain forward and inverse propagation processes.Generally,constructing an accurate probability model needs to extract a large number of samples according to the known probability density functions.However,due to the limitations of experimental environment and economic conditions,only a few samples can be obtained in practical engineering,which limites the adaptability of the traditional methods.Besides,in practical engineering problems,the uncertainties such as material properties,geometric dimensions and boundary conditions are usually correlated,which may have a great influence on the uncertainty analysis results.For this reason,it is difficult to determine the probability density types and to accurately describe the joint probability density functions of the propagated results,which makes the uncertain forward and inverse propagation analysis methods based on probabilistic model difficult to be widely applied.In recent years,with the emergence and improvement of the non-probabilistic convex model,especially for the ellipsoidal convex model which measures the uncertainties and correlations of the structural parameters under the limited samples,the studies about non-probability uncertain forward and inverse propagations have been widely concerned.At present,there are still many problems in uncertain inverse propagation process should be solved in practical engineering,including the correlation propagation,the global sensitivity analysis of the input correlated parameters,the selection of the optimal sensor positions and the decoupling problem of the non-probabilistic correlation coefficient matrix and intervals.Thus,this paper studies the above problems one by one,and tries to make some effective attempts and explorations in the aspect of structural uncertainty propagation and computational inverse problems.Then,a set of structural inverse method system considering non-probabilistic correlation analysis is established.The research idea of this paper is carried from two aspects of the input parameters and output responses.Firstly,to establish an accurate and compact uncertainty boundary of sturctural responses,the relationship bewteen the non-probability correlation coefficients of structural parameters and responses is discussed,and the correlation propagation equations are newly derived.Secondly,due to the data analysis is usually heavy and the structural model is commonly complex,the importance of the structural parameters should be sorted,so as to ignore the parameters that have little impact on the structural responses,and focus on the parameters that have significant impact on the structural responses.Therefore,the global sensitivity method considering the correlated parameters is studied.Thirdly,in order to overcome the ill-posed problem in the uncertian inverse problem,the optimal sensor placement for stable identification of structural parameter is investigated.Importantly,the coupling problem of uncertainties and correlations of structural parameters to be identified in uncertain inverse problem is also considered,and a sequence interval and correlation inverse strategy is propsoed.Based on the above ideas,the main works of this paper are listed as follows:(1)A correlation propagation method for uncertainty analysis of structures based on the non-probabilistic ellipsoidal model is proposed to propagate the uncertainties and correlations of the structural parameters simultaneously.Aiming at the quantification problem of the uncertainties and correlations of structural parameters with a limited sample,the ellipsoidal convex model is adopted to model their uncertainty boundary.The propagation problem considering correlations and uncertainties is decomposed into an interval propagation problem and a correlation coefficient propagation problem.For the interval propagation problem,a subinterval decomposition analysis method based on the ellipsoidal convex model is developed to evaluate the intervals of the structural responses with a low computational cost.For the correlation propagation problem,the non-probabilistic correlation propagation equations from uncertain parameters to structural responses are newly derived for predicting the correlation coefficient matrix of the responses,which can be expected to extend the n-order Taylor expansion according to the accuracy requirement.(2)A global sensitivity analysis method considering correlation for uncertain structure is proposed to determine the important uncertain parameters.The ellipsoidal convex model is used to quantify the uncertainty boundary of the correlated parameters,and the non-probabilistic variance is defined as a sensitivity index.Importantly,the non-probabilistic variance propagation equations from uncertain structural parameters to responses are derived,and the influence of input parameters on the non-probabilistic variance of output response is decomposed into the independent contribution and correlated contribution.Simultaneously,the total contribution rate,independent contribution rate and correlation contribution rate are defined to accurately estimate the sensitivity of each parameter to the response.(3)An optimal sensor placement method based on maximum independent mean-variance criterion is proposed to ensure the validity of measurement information and the stability of uncertain inverse process.The optimal sensor placement problem for structural parameter identification is transformed into an uncertainty propagation problem,and the maximum independent variance and maximum independent mean-variance criteria are established.The Monte Carlo simulation and the dimension reduction integral method are used to evaluate the maximum independent variance of structural response respectively,and an orthogonal matching pursuit algorithm is developed to determine the positions and the number of sensors.Especially,to ensure the accuracy and stability of the identification results,two-layers Monte Carlo simulation and dimension reduction integral method are developed to achieve the maximum independent mean-variance of the structural responses.(4)A sequence interval and correlation inverse strategy for structural parameter identification is proposed to realize the interval and correlation inverse processes of structural parameters.In practical engineering,considering the non-probabilistic uncertainties and correlations of measured responses,the ellipsoidal convex model is used to quantify their uncertainty boundary.At the same time,the uncertain inverse problem considering correlations is decomposed into the interval inverse problem and correlation inverse problem.For the interval inverse problem,the subinterval decomposition analysis method is adopted to achieve the intervals of the calculated structural responses.For the correlation inverse problem,the correlation coefficient matrix of the structural responses is determined by correlation propagation equations.In this way,the intervals and correlation coefficient matrix of the structural parameters are obtained by using the optimization methods with the help of the measured responses and the calculated responses,so that the uncertainty boundary of the structural parameters can be established by using the ellipsoidal convex model.更多还原