考虑认知不确定性的结构可靠性分析方法研究
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摘要
实际工程问题中广泛存在着与几何尺寸、材料属性、边界条件等相关的不确定性,采用合理有效的理论与方法度量、传播和控制这些不确定性对于提高产品或结构的安全性能具有极其重要的意义。不确定性可分为随机不确定性和认知不确定性两大类,随机不确定性建模通常需要大量的样本信息以构造不确定性参数的精确概率分布,且不能随着认识水平的增加而消除；而认知不确定性则往往是由于样本信息匮乏无法构建精确的概率分布,且会随着认识水平的增加而逐渐消除。现代产品和结构的设计、制造、服役及老化等全生命周期普遍存在认知不确定性,仅仅采用传统的随机建模、分析与设计将无法对认知不确定性下结构的性能做出客观有效的评估,甚至可能导致不可靠的设计。
目前,以概率论这一个统一完善的理论体系的支撑,随机不确定性结构响应与可靠性分析在理论方法与工程应用均发展的较为成熟。相对而言,认知不确定性的建模与分析手段则存在多种理论体系并存的状况,这就使得认知不确定性的建模与分析在一定程度上较随机不确定性的处理方法更为复杂。尽管认知不确定性结构响应与可靠性分析得到了较为迅速地发展,但整体而言该领域的研究依然处于初步阶段,还有诸多关键问题亟待解决。为此,本文将从认知不确定性建模较为方便的两类方法即凸模型和证据理论入手,着重针对基于证据理论的结构不确定性分析、考虑区间与随机场混合不确定性结构分析、基于凸模型的高效非概率可靠性分析和基于证据理论的可靠性分析等方面展开研究。基于此思路,本论文开展和完成了如下的研究工作：
(1)针对证据不确定性下结构响应问题,将证据理论直接与有限元方法结合,从而发展出基于证据理论的不确定性结构分析方法。通过引入概率论中的矩概念,并针对证据变量和证据函数发展出相应的证变量矩和证据函数矩；通过区间运算计算证据变量矩和证据函数矩；将矩概念分别与静力学有限元方法和动力学有限元方法结合发展出结构静动态响应分析方法来计算不确定性结构的响应区间矩。此外,在每一焦元上采用区间结构分析来高效求解结构的近似响应边界。
(2)针对带有区间相关长度的随机场结构响应分析问题,发展出一种区间与随机场混合的不确定结构分析方法。实验数据匮乏情况下的随机场相关模型的相关长度采用区间进行度量,基于这种混合模型将能更为合理的描述不确定参数的空间变异特性；基于Karhunen-Loeve级数展式离散带有区间相关长度的随机场,并获得均值单元刚度矩阵和区间加权单元刚度矩阵；基于带有区间权系数的多项式混沌展式近似结构位移响应；通过Element-by-Element(EBE)技术将单元刚度矩阵组装为整体刚度矩阵,以消除单元刚度矩阵中区间元素耦合对计算精度的影响；基于Galerkin法构造出一区间线性方程组,通过求解获取相应的区间权系数,进而获得结构响应的区间矩。
(3)针对具有黑盒子型极限状态函数的椭球非概率可靠性分析问题,借鉴传统的概率可靠性分析中的经典响应面法思想,发展出了一种基于序列迭代响应面的椭球非概率可靠性分析方法。基于相关性分析技术构建多维椭球模型,从而使多维椭球模型的构建更为方便经济；通过不带交叉项二次多项式构建真实极限状态函数的代理模型,并采用迭代策略对采样中心、设计空间和代理模型进行更新直至逼近于真实的极限状态函数。
(4)提出了一种基于局部加密代理模型的椭球非概率可靠性分析方法。通过在第一迭代步采用不带交叉项二次多项式构建极限状态函数的代理模型,并求解获取近似设计点以作为整体设计域的采样中心；在随后的迭代步,采用径向基函数构建极限状态函数的代理模型,并将该迭代步获得的近似设计点加入先前的样本集实现代理模型的逐步更新。该方法着重于提高极限状态函数在设计点附近区域的近似精度,所需的样本大大减少,因而能进一步提高可靠性分析的效率。
(5)针对证据理论在描述不确定性变量的不连续性而带来的计算极端耗时问题,分别基于不带交叉项二次多项式响应面、径向基函数和结合移动最小二乘法的高维代理模型发展出三种基于代理模型的证据可靠性分析方法。相比于概率可靠性分析和椭球非概率可靠性分析,证据可靠性分析则需要对穿过辨识框架的极限状态函数而非对设计点局部区域具有较高的近似精度。通过引入一统计量指标,在不同的样本点数、阈值设置、基本概率分配数及不同问题规模等多个方面对三种方法的优势和劣势进行系统比较研究以考察不同代理模型方法在证据可靠性分析中的适用性。此外,还针对工程中常见的低失效概率问题进一步研究了三种方法的优缺点。
(6)针对证据不确定性下的系统可靠度问题,分别针对串联系统和并联系统发展出高效的结构系统可靠度分析方法。通过引入系统可靠度的概念,给出串并联系统下证据失效概率信度和似真度的定义。针对串联系统和并联系统分别构造出相应的两层优化格式以判断焦元与系统失效域的相对位置,从而高效确定属于失效概率信度和似真度的焦元集合。对于极大极小和极小极大优化问题,将其转换为一单层优化问题并结合泰勒展式及单纯形方法实现高效判断焦元与系统失效域的相对位置；对于极小极小和极大极大优化问题,通过区间分析方法来高效求解每一焦元上极值以判断焦元与系统失效域的相对位置。
Uncertainties associated with geometric tolerances, material properties, boundary conditions widely exist in practical engineering problems. Quantifying, propagating and managing the concerned uncertainty based on related uncertainty theory have become extremely important to improve the performances of products or structures. Uncertainty can be categorized into stochastic and epistemic types. Stochastic uncertainty quantification always requires a large amount of experimental samples to construct the precise probability distributions of uncertain parameters, and cann't be reducible as the state of knowledge increases. While epistemic uncertainty arises from lack of knowledge without sufficient sample information to construct the corresponding precise probability distributions, and can be reducible as the state of knowledge increases. Actually, epistemic uncertainty widely exists during the design, manufacturing, service and aging stages of the whole life cycle of current products or structures. Structural analysis and design by only using stocastic modeling may not provide effective assessments of structural performances under epistemic uncertainties, or even will lead to unreliable designs.
Comparatively, structural stochastic response analysis and reliability theory has exhibited full developments presently in support of a unified systematic framework of probability theory. In contrast, there co-exist several theories to characterize the epistemic uncertainty, which makes epistemic uncertainty quantification more complicated. Several progresses have been made in the field of structural response and reliability analysis theories and algorithms under epistemic uncertainty in recent years, however, it seems that it is still in its preliminary stages and several related important issues have not been well solved. Thus, this dissertation concentrates on two types of relatively convinent uncertainty modeling approaches, namely convex sets and evidence theory, and conducts systematic studies at structural response analysis algorithm based on evidence theory, structural response analysis under convex sets and probability hybrid uncertainty model, non-probability reliability analysis methods by using ellipsoid-type convex model, and evidence-theory-based structural reliability analysis. As a result, the following studies are carried out in this dissertation:
(1) A structural uncertainty analysis method is developed to calculate the structural static and dynamic response under epistemic uncertainties by directly integrating evidence theory with finite element method. By introducing the moment concept in probability theory, the moments of evidence variables and associated functions are developed to describe their distributions. The static and dynamic response analysis method is developed by integrating the moment concept with static and dynamic finite element method to compute the respone moments. Besides, the interval structural analysis technique is used to efficiently calculate the approximate response for each focal element.
(2) A structural uncertainty analysis method with interval and stochastic field hybrid model is proposed to conduct structural response analysis under stochastic field with interval correlation length. The correlation length of the stochastic correlation model is quantified by using interval due to lack of sufficient experimental data, and this proposed hybrid model can more reasonably describe the spatial variability characteristics of the uncertain parameters. The stochastic field with interval correlation length is discretized based on Karhunen-Loeve series expansion, and the mean element stiffness matrix and the interval weighted element stiffness matrix can be further obtained. The structural displacement response is approximated by using polynomial chaos expansion with interval weights. The Element-by-Element technique is used to assemble the corresponding global stiffness matrix to avoid the couping effects of the element stiffness matrices. An extended-order system of interval linear equations is formulated, and is further solved to obtain the corresponding interval weights, and the interval moments of the structural response can be achieved.
(3) For ellipsoid-model-based non-probabilistic reliability problems with black-box limit-state equations, a sequential response surface method is developed to conduct reliability analysis by referring to the basic ideas of conventional response surface method in probability-based reliability analysis. The correlation analysis technique is used to construct the multi-dimensional ellipsoid model, which will be more convenient and economic to quantify uncertain parameters. A quadratic polynomial without cross terms is adopted to approximate the limit-state function, and an iterative strategy is employed to update sampling center, design space and metamodel until the convergence criteria is satisfied.
(4) A local-densifying metamodeling approach is suggested to conduct ellipsoid-model-based non-probabilistic reliability analysis. A quadratic polynomial without cross terms is used to approximate the limit-state function in the first iterative step, and the approximate design point is located based on the created metamodel as the sampling center of the design space. In subsequent iterative steps, radial basis function is used to construct the metamodel of the limit-state function, and the obtained approximate design point is added into the previous sampling point sets to update the metamodel. The proposed method can gradually improve the approximation accuracy of the limit-state function at the local region nearby the design point, and hence much less samples are required comprared with the sequential response surface method, and whereby improve the reliability analysis solution efficiency.
(5) For the extremely computational cost problems due to the discontinuous nature of epistemic uncertainty quantification based on evidence theory, three types of evidence-theory-based reliability analysis method are developed by intergating the quadratic polynomial without cross terms, radial basis function and high-dimensional model representation in combination with moving least square method to overcome the low-efficiency problem, respectively. Compared to probability-based and convex-model-based reliability analysis, evidence-theory-based reliability analysis requires relatively high approximation accuracy of the limit-state function across the frame of discernment. The proposed three metamodel-based approaches are systematically compared to test the applicability of different metamodeling techniques in evidence-theory-based reliability analysis under different sampling points, threshold setting, basic probability assignments, and different problem scale by introducing a statistical measure. Besides, their advantages and disadvantages are further investigated for low-failure-probability problems.
(6) A structural system reliability analysis method is developed to deal with series and parallel system under epistemic uncertainty by extending evidence theory to system reliability problems. By introducing the concept of system reliabillity, the definitions of failure probability belief and plausibility for series and parallel systems are provided. By constructing two-level optimization formulations to locate the relative position between the focal element and system failure region, and whereby efficiently determine the focal element sets belonging to failure belief or plausibility. The first-order Taylor expansion and simplex method are employed to determine the relative positions for the max-min and min-max optimization problems, while the interval analysis method is used for min-min and max-max optimization problems.
目前,以概率论这一个统一完善的理论体系的支撑,随机不确定性结构响应与可靠性分析在理论方法与工程应用均发展的较为成熟。相对而言,认知不确定性的建模与分析手段则存在多种理论体系并存的状况,这就使得认知不确定性的建模与分析在一定程度上较随机不确定性的处理方法更为复杂。尽管认知不确定性结构响应与可靠性分析得到了较为迅速地发展,但整体而言该领域的研究依然处于初步阶段,还有诸多关键问题亟待解决。为此,本文将从认知不确定性建模较为方便的两类方法即凸模型和证据理论入手,着重针对基于证据理论的结构不确定性分析、考虑区间与随机场混合不确定性结构分析、基于凸模型的高效非概率可靠性分析和基于证据理论的可靠性分析等方面展开研究。基于此思路,本论文开展和完成了如下的研究工作：
(1)针对证据不确定性下结构响应问题,将证据理论直接与有限元方法结合,从而发展出基于证据理论的不确定性结构分析方法。通过引入概率论中的矩概念,并针对证据变量和证据函数发展出相应的证变量矩和证据函数矩；通过区间运算计算证据变量矩和证据函数矩；将矩概念分别与静力学有限元方法和动力学有限元方法结合发展出结构静动态响应分析方法来计算不确定性结构的响应区间矩。此外,在每一焦元上采用区间结构分析来高效求解结构的近似响应边界。
(2)针对带有区间相关长度的随机场结构响应分析问题,发展出一种区间与随机场混合的不确定结构分析方法。实验数据匮乏情况下的随机场相关模型的相关长度采用区间进行度量,基于这种混合模型将能更为合理的描述不确定参数的空间变异特性；基于Karhunen-Loeve级数展式离散带有区间相关长度的随机场,并获得均值单元刚度矩阵和区间加权单元刚度矩阵；基于带有区间权系数的多项式混沌展式近似结构位移响应；通过Element-by-Element(EBE)技术将单元刚度矩阵组装为整体刚度矩阵,以消除单元刚度矩阵中区间元素耦合对计算精度的影响；基于Galerkin法构造出一区间线性方程组,通过求解获取相应的区间权系数,进而获得结构响应的区间矩。
(3)针对具有黑盒子型极限状态函数的椭球非概率可靠性分析问题,借鉴传统的概率可靠性分析中的经典响应面法思想,发展出了一种基于序列迭代响应面的椭球非概率可靠性分析方法。基于相关性分析技术构建多维椭球模型,从而使多维椭球模型的构建更为方便经济；通过不带交叉项二次多项式构建真实极限状态函数的代理模型,并采用迭代策略对采样中心、设计空间和代理模型进行更新直至逼近于真实的极限状态函数。
(4)提出了一种基于局部加密代理模型的椭球非概率可靠性分析方法。通过在第一迭代步采用不带交叉项二次多项式构建极限状态函数的代理模型,并求解获取近似设计点以作为整体设计域的采样中心；在随后的迭代步,采用径向基函数构建极限状态函数的代理模型,并将该迭代步获得的近似设计点加入先前的样本集实现代理模型的逐步更新。该方法着重于提高极限状态函数在设计点附近区域的近似精度,所需的样本大大减少,因而能进一步提高可靠性分析的效率。
(5)针对证据理论在描述不确定性变量的不连续性而带来的计算极端耗时问题,分别基于不带交叉项二次多项式响应面、径向基函数和结合移动最小二乘法的高维代理模型发展出三种基于代理模型的证据可靠性分析方法。相比于概率可靠性分析和椭球非概率可靠性分析,证据可靠性分析则需要对穿过辨识框架的极限状态函数而非对设计点局部区域具有较高的近似精度。通过引入一统计量指标,在不同的样本点数、阈值设置、基本概率分配数及不同问题规模等多个方面对三种方法的优势和劣势进行系统比较研究以考察不同代理模型方法在证据可靠性分析中的适用性。此外,还针对工程中常见的低失效概率问题进一步研究了三种方法的优缺点。
(6)针对证据不确定性下的系统可靠度问题,分别针对串联系统和并联系统发展出高效的结构系统可靠度分析方法。通过引入系统可靠度的概念,给出串并联系统下证据失效概率信度和似真度的定义。针对串联系统和并联系统分别构造出相应的两层优化格式以判断焦元与系统失效域的相对位置,从而高效确定属于失效概率信度和似真度的焦元集合。对于极大极小和极小极大优化问题,将其转换为一单层优化问题并结合泰勒展式及单纯形方法实现高效判断焦元与系统失效域的相对位置；对于极小极小和极大极大优化问题,通过区间分析方法来高效求解每一焦元上极值以判断焦元与系统失效域的相对位置。
Uncertainties associated with geometric tolerances, material properties, boundary conditions widely exist in practical engineering problems. Quantifying, propagating and managing the concerned uncertainty based on related uncertainty theory have become extremely important to improve the performances of products or structures. Uncertainty can be categorized into stochastic and epistemic types. Stochastic uncertainty quantification always requires a large amount of experimental samples to construct the precise probability distributions of uncertain parameters, and cann't be reducible as the state of knowledge increases. While epistemic uncertainty arises from lack of knowledge without sufficient sample information to construct the corresponding precise probability distributions, and can be reducible as the state of knowledge increases. Actually, epistemic uncertainty widely exists during the design, manufacturing, service and aging stages of the whole life cycle of current products or structures. Structural analysis and design by only using stocastic modeling may not provide effective assessments of structural performances under epistemic uncertainties, or even will lead to unreliable designs.
Comparatively, structural stochastic response analysis and reliability theory has exhibited full developments presently in support of a unified systematic framework of probability theory. In contrast, there co-exist several theories to characterize the epistemic uncertainty, which makes epistemic uncertainty quantification more complicated. Several progresses have been made in the field of structural response and reliability analysis theories and algorithms under epistemic uncertainty in recent years, however, it seems that it is still in its preliminary stages and several related important issues have not been well solved. Thus, this dissertation concentrates on two types of relatively convinent uncertainty modeling approaches, namely convex sets and evidence theory, and conducts systematic studies at structural response analysis algorithm based on evidence theory, structural response analysis under convex sets and probability hybrid uncertainty model, non-probability reliability analysis methods by using ellipsoid-type convex model, and evidence-theory-based structural reliability analysis. As a result, the following studies are carried out in this dissertation:
(1) A structural uncertainty analysis method is developed to calculate the structural static and dynamic response under epistemic uncertainties by directly integrating evidence theory with finite element method. By introducing the moment concept in probability theory, the moments of evidence variables and associated functions are developed to describe their distributions. The static and dynamic response analysis method is developed by integrating the moment concept with static and dynamic finite element method to compute the respone moments. Besides, the interval structural analysis technique is used to efficiently calculate the approximate response for each focal element.
(2) A structural uncertainty analysis method with interval and stochastic field hybrid model is proposed to conduct structural response analysis under stochastic field with interval correlation length. The correlation length of the stochastic correlation model is quantified by using interval due to lack of sufficient experimental data, and this proposed hybrid model can more reasonably describe the spatial variability characteristics of the uncertain parameters. The stochastic field with interval correlation length is discretized based on Karhunen-Loeve series expansion, and the mean element stiffness matrix and the interval weighted element stiffness matrix can be further obtained. The structural displacement response is approximated by using polynomial chaos expansion with interval weights. The Element-by-Element technique is used to assemble the corresponding global stiffness matrix to avoid the couping effects of the element stiffness matrices. An extended-order system of interval linear equations is formulated, and is further solved to obtain the corresponding interval weights, and the interval moments of the structural response can be achieved.
(3) For ellipsoid-model-based non-probabilistic reliability problems with black-box limit-state equations, a sequential response surface method is developed to conduct reliability analysis by referring to the basic ideas of conventional response surface method in probability-based reliability analysis. The correlation analysis technique is used to construct the multi-dimensional ellipsoid model, which will be more convenient and economic to quantify uncertain parameters. A quadratic polynomial without cross terms is adopted to approximate the limit-state function, and an iterative strategy is employed to update sampling center, design space and metamodel until the convergence criteria is satisfied.
(4) A local-densifying metamodeling approach is suggested to conduct ellipsoid-model-based non-probabilistic reliability analysis. A quadratic polynomial without cross terms is used to approximate the limit-state function in the first iterative step, and the approximate design point is located based on the created metamodel as the sampling center of the design space. In subsequent iterative steps, radial basis function is used to construct the metamodel of the limit-state function, and the obtained approximate design point is added into the previous sampling point sets to update the metamodel. The proposed method can gradually improve the approximation accuracy of the limit-state function at the local region nearby the design point, and hence much less samples are required comprared with the sequential response surface method, and whereby improve the reliability analysis solution efficiency.
(5) For the extremely computational cost problems due to the discontinuous nature of epistemic uncertainty quantification based on evidence theory, three types of evidence-theory-based reliability analysis method are developed by intergating the quadratic polynomial without cross terms, radial basis function and high-dimensional model representation in combination with moving least square method to overcome the low-efficiency problem, respectively. Compared to probability-based and convex-model-based reliability analysis, evidence-theory-based reliability analysis requires relatively high approximation accuracy of the limit-state function across the frame of discernment. The proposed three metamodel-based approaches are systematically compared to test the applicability of different metamodeling techniques in evidence-theory-based reliability analysis under different sampling points, threshold setting, basic probability assignments, and different problem scale by introducing a statistical measure. Besides, their advantages and disadvantages are further investigated for low-failure-probability problems.
(6) A structural system reliability analysis method is developed to deal with series and parallel system under epistemic uncertainty by extending evidence theory to system reliability problems. By introducing the concept of system reliabillity, the definitions of failure probability belief and plausibility for series and parallel systems are provided. By constructing two-level optimization formulations to locate the relative position between the focal element and system failure region, and whereby efficiently determine the focal element sets belonging to failure belief or plausibility. The first-order Taylor expansion and simplex method are employed to determine the relative positions for the max-min and min-max optimization problems, while the interval analysis method is used for min-min and max-max optimization problems.
引文
[1]邱志平,王晓军.不确定性结构力学问题的集合理论凸方法.北京：科学出版社,2008.
[2]赵国藩,金伟良,贡金鑫.结构可靠度理论.北京：中国建筑工业出版社,2000.
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