力学进展  2019 , 49 (1): 201909-201909 https://doi.org/10.6052/1000-0992-18-004

力学

有限元模型修正研究进展:从线性到非线性

张皓1, 李东升2, 李宏男1

1 大连理工大学土木工程学院, 海岸和近海工程国家重点实验室, 大连 116024
2 汕头大学土木与环境工程系,广东省结构安全与监测工程技术研究中心, 广东 汕头515063

Recent progress on finite element model updating: From linearity to nonlinearity

ZHANG Hao1, LI Dongsheng2, LI Hongnan1

1 School of Civil Engineering, State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024,China
2 Guangdong Engineering Center for Structure Safety and Health Monitoring,Department of Civil and Environmental Engineering,Shantou University,Guangdong, 515063, China

中图分类号:  O322

文献标识码:  A

通讯作者:  †E-mail: lids@stu.edu.cn; dsli@dlut.edu.cn†E-mail: lids@stu.edu.cn; dsli@dlut.edu.cn

收稿日期: 2018-03-15

接受日期:  2018-07-6

网络出版日期:  2019-01-15

版权声明:  2019 中国力学学会 This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

基金资助:  国家重点基础研究发展计划(973计划) (2015CB057704)、国家自然科学基金(51121005, 51578107,51778103)、中央高校基本科研业务费(DUT18LAB07)和汕头大学科研启动基金(NTF18012)资助项目.

作者简介:

李东升, 博士, 汕头大学教授, 博士生导师.


李宏男, 博士, 大连理工大学教授, 博士生导师.


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摘要

有限元分析在实际工程中得到了广泛应用.然而有限元模型由于受到网格划分、边界条件和材料物理参数不确定性等的影响,与真实结构有差异. 因此须通过试验数据加以修正,使其尽可能接近实际结构,以保证之后的结构动力模拟分析和监测等具有实际意义. 经过多年发展,有限元模型修正技术已经能够成功应用于一些实际工程,但现代工程技术的进步对有限元模型修正提出了更高要求,修正后的有限元模型不仅要有较高的精确度,还需要为后续应用给出具有指导意义的置信度.而现有的有限元模型修正、确认方法多基于结构线性的假设,而未能考虑实际结构中广泛存在的非线性.因此本文以土木工程结构模型修正的一些研究成果为例,通过对传统有限元模型修正的发展历程进行全面回顾;总结评述传统有限元修正技术的主要方法,以及包括有限元模型确认在内的最新研究进展;重点探讨有限元模型修正技术向非线性发展的技术路线和目前主要研究成果,展望其未来发展方向, 并提出值得研究的问题.

关键词: 有限元模型修正 ; 有限元建模 ; 非线性有限元模型修正 ; 模型确认 ; 灵敏度

Abstract

Finite element (FE) analysis is extensively applied in practical engineering. However, FE model usually differs much from actual engineering structures due to modeling errors caused by meshing scales, boundary conditions, and material properties. Therefore an FE model has to be modified or updated by test data to make the FE model as close as possible to the real structure so that the updated model can be convincingly used for structural simulation, dynamic analysis, health monitoring or other engineering applications. Over the years, although FE model updating has been applied to many engineering applications successfully, the development of modern technology has put forward higher requirements on updated models, not only higher accuracy level but also higher confidence level. However, current methods are mostly confined to the linear structure hypothesis, which disagrees with the actual situations in many cases. Based on the background, this paper comprehensively reviews traditional FE model updating techniques with civil engineering structure as an example. Furthermore, influential model updating methods and their progress are surveyed and critically commented including recent progress of FE model validation. Especially, the evolution from linear to nonlinear FE model updating technology and essential research findings are summarized, then attractive perspectives are forecasted, and several promising issues are sketched out.

Keywords: finite element model updating ; finite element modelling ; nonlinear finite element model updating ; model validation ; sensitivity

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张皓, 李东升, 李宏男. 有限元模型修正研究进展:从线性到非线性[J]. 力学进展, 2019, 49(1): 201909-201909 https://doi.org/10.6052/1000-0992-18-004

ZHANG Hao, LI Dongsheng, LI Hongnan. Recent progress on finite element model updating: From linearity to nonlinearity[J]. Advances in Mechanics, 2019, 49(1): 201909-201909 https://doi.org/10.6052/1000-0992-18-004

1 引言

有限单元法是工程分析应用最为广泛的数值方法,目前已成为计算机辅助设计和计算机辅助制造的重要组成部分(王勖成2003),通过有限元分析软件建立有限元模型进行结构设计、分析、监测等在工程界已基本得到普及和认可.有限元模拟分析目前已经成为与理论分析、试验研究鼎足而立的重要科学研究手段之一,特别是对于大型结构、复杂结构,有限元模拟分析提供了设计方案可行性的理论保证, 更显得尤为关键.但是, 有限元模型难免会引入误差, 主要包括:模型结构误差、参数误差和阶次误差(Brownjohn et al. 1992,Mottershead & Friswell 1993). 如果误差超过规定的阈值,则有限元模型将不能反映实际结构特性------有限元模型修正技术应运而生.由于建模时缺乏先验信息, 通常认为实测数据更为准确可靠,因此有限元模型修正技术的思想是根据静动力试验数据,对有限元模型进行修正, 最大限度缩小模拟结果与实测数据之间的差距,使有限元模型能够更可靠地反映实际结构特性(宗周红和任伟新 2012).初始有限元模型建立后,对连续结构离散化造成的阶次误差可以通过单元网格细化来控制;多数有限元模型修正方法实际上是对刚度、质量、材料属性、几何尺寸等模型参数的修正,以此降低参数误差(李辉 和丁桦 2005).有限元模型修正技术最早出现在航空领域. 1958年,Gravitz通过飞机地面振动测试数据修正飞机结构柔度矩阵,被认为是有限元模型修正技术的早期探索(吴晓菊 2009). 经过多年的发展,有限元模型修正已经成为一项比较成熟的工程技术,并成功应用于诸多实际工程(Brughmans et al. 1991, 1993, Schedlinski2000, Venture et al. 2001, Goge 2003). 以土木工程领域为例,又以在桥梁工程中的应用最为成熟,已经能够完成特大跨桥梁有限元模型修正的问题(范立础 等 2000, Wang etal. 2005, Xu et al. 2011, 郁胜 等 2014). 另外,虽然目前主流大型有限元分析软件的模型修正功能尚不完善,但也有如Dynamic Design Solution (DDS)公司的FEMtools,西门子(SIEMENS)公司的LMS Gateway, Simcenter 3D等软件可供使用.工程实用软件的开发将极大地推动有限元模型修正技术的发展,因此也是这一领域的热点发展方向之一. 然而,尽管在工程上已经有一些成功应用, 这项技术还远远谈不上完善,现代工程技术的飞速发展已经对有限元模型修正提出了更高要求.实践表明,修正后的模型仍然不能保证对结构在真实环境中的响应做出准确预测,因此需要进行模型确认(张令弥 2002),并给出模型置信度以指导后续其他应用. 事实上,受到噪声、摩擦等因素影响,无论在试验数据还是模型参数中均广泛存在不确定性,因此基于确定性的有限元模型修正只是一个特例,有限元模型确认具有更一般的意义,是有限元模型修正在统计学上的发展(郭勤涛和张令弥 2005,宗周红和任伟新 2012).模型确认这一概念首先在美国能源部所属的洛斯$\cdot$阿拉莫斯国家实验室(Los Alamos National Laboratory,LANL)、桑迪亚国家实验室(Sandia National Laboratory,SNL)和劳伦斯$\cdot $利弗莫尔国家实验室(Lawrence Livermore NationalLaboratory, LLNL)负责的专项研究计划中提出(Martin 2000),并引起学术界的广泛关注. 目前,有限元模型确认的总体技术路线已经基本形成,相关研究仍在不断深入(郭勤涛 等 2006a, 宗周红和牛杰 等 2012).

另外, 除了通过网格细化控制阶次误差,以及通过传统有限元模型修正方法控制参数误差之外,初始有限元模型的另一误差来源------模型结构误差,在传统的有限元模型修正技术中并不涉及. 但对实际结构的简化,对结构非线性的线性化已经成为了传统有限元模型修正技术提升的瓶颈,特别是对于具有明显非线性特征的结构,传统基于线性的有限元模型修正从理论上也不再合理. 事实上,非线性有限元模型修正问题早已引起研究人员的注意. Schmidt(1994)对非线性有限元模型修正问题进行了较早的实践,通过拟合时程响应数据,对连接结构缝隙、摩擦的局部非线性进行了模型修正研究.另外还有一些综述文章展望或探讨过涉及非线性的模型修正问题(Mottershead et al. 2011, 宗周红 等 2012), 但相关研究成果并不丰富.原因是传统线性结构的模态分析理论在非线性结构中并不适用,而非线性动力系统理论的建立远不如线性成熟,且其中往往蕴含着如分岔、分形、混沌等极为复杂的现象(Kerschen et al.2006), 因此非线性有限元模型修正是更具挑战的问题.近年来非线性科学、非线性系统识别以及时--频分析技术等的发展和应用,为有限元模型修正技术向非线性发展提供了理论支持.非线性有限元模型修正能够增加模型的复杂程度, 降低模型的不确定性,使其更接近实际结构, 从根本上提升修正结果的置信度,是基于线性结构传统方法质的飞跃.国内外学者已经进行了一些有价值的探索,非线性有限元模型修正将逐渐得到更广泛的关注.

发展有限元模型修正技术, 提升有限元模型模拟实际结构的能力,能够进一步发挥有限元模拟分析方法的优势, 对结构设计、优化, 结构振动控制,结构健康监测, 结构性能评估、预测等工程应用,以及解决碰撞、爆炸等工程难题具有重要意义; 同时, 也能够缩减不必要的试验研究,减少资源消耗, 节约成本, 缩短研发周期, 提高企业技术水平和市场竞争力,促进相关行业绿色发展和转型升级.本文立足于对传统有限元模型修正基本方法、理论的总结和评述,以其在土木工程领域的应用为例, 全面回顾有限元模型修正技术的发展历程,同时讨论其最新发展方向;重点探讨有限元模型修正技术向非线性的发展现状并进行展望, 供同行研究人员参考.

2 线性有限元模型修正及其发展

2.1 传统有限元模型修正方法

有限元模型修正技术包含了有限元建模、试验模态分析、模型缩聚扩阶、灵敏度分析、优化算法等丰富的内容,图1所示为目前最常用的基于灵敏度的参数型修正方法技术路线.Berman和Flannelly (1971)Berman(1984)的研究对有限元模型修正进行了最早的论述并指出,由于模态不完备等原因,通过试验数据建立接近实际结构的理论模型是比较困难的,因此提倡使用模型修正技术建立具有物理意义的模型.Friswell和Mottershead (1995)的专著《Finite Element Model Updatingin StructuralDynamics》对有限元模型修正的技术路线、修正过程中可能遇到的问题和具体解决方法进行了较为全面的论述,将有限元模型修正方法分为基于模态数据的直接法、基于模态数据的迭代法和基于频域数据的方法.经过多年的发展, 已经涌现出了大量有限元模型修正方法,相关文献浩如烟海, 方法的分类也有所不同. 综合来看,可以根据修正对象的不同分为矩阵型修正方法和参数型修正方法.修正过程中使用的实测数据可以是动力试验时域、频域数据或者静力试验数据,其中静力试验数据受噪声影响较小, 有研究表明,联合使用静、动力试验数据进行有限元模型修正也可以取得较好的效果(宗周红和 任伟新 2012), 但基于动力的有限元模型修正仍为本领域研究的主流,因此本文主要探讨基于动力的有限元模型修正方法.对于传统有限元模型修正及其相关问题,国内学者已经进行了一些重要的讨论、总结和评述. 比如,魏来生(1998)介绍了5种修正方法,反映出矩阵型方法到参数型方法的发展和过渡;朱安文等(2002)除了综述矩阵型方法到元素型方法的发展过程,还探讨了模型缩聚、灵敏度分析和模型修正有效性检验的问题;宋汉文等(2003)探讨了有限元模型修正中的模型降阶、参数化以及反问题不适定性的处理方法等,并展望了基于统计理论的修正方法;李辉和丁桦(2005)综述了矩阵型方法、参数型方法和神经网络法,并指出各种方法的不足, 提出值得研究的问题;郭勤涛等(2006a)重点探讨了有限元模型修正向统计意义下的有限元模型确认的发展;杨智春等(2009)总结了有限元模型修正中基于模态参数、频响函数和动力响应的各种目标函数及其解法.这些综述文章反映了这一技术领域的发展历程, 内容上也各有侧重,但均未对非线性有限元模型修正进行综述或展望.为了反映有限元模型修正技术完整的发展脉络,进而引出非线性有限元模型修正的讨论,本文简要而全面地总结评述传统有限元模型修正方法,但不延展到基于有限元模型修正技术的损伤识别、参数识别等其他应用.

图 1   传统有限元模型修正方法的技术路线

   

2.1.1 矩阵型方法

Berman (1979)Berman和Nagy (1983)以及Baruch (1978,1982)等在基于模态数据的直接法方面较早地开展了相关研究工作,他们运用模态正交性条件作为约束,通过拉格朗日乘子法使矩阵修正量范数最小化, 直接修正质量和刚度矩阵.使用这种方法修正后的模型能够精确``复制''试验测试数据,但是破坏了质量、刚度矩阵原有的带状、稀疏特征. 针对这一问题, Kabe(1985)提出了一种保持矩阵带状、稀疏特征的刚度矩阵修正方法,其思想是通过引入约束条件, 使修正过程只改变矩阵中非零元素,原有零元素不变. 修正过程从矩阵整体转变到矩阵中的元素,这是方法上的一个较大进步. 后续又有较多研究成果涌现,包括我国学者早期的相关研究工作也是围绕着直接修正刚度、质量矩阵元素展开的(张德文和 魏阜旋 1999). Roy (1990)通过对比多种修正方法,提出考虑模型物理意义、修正后矩阵元素关联性、修正结果真实性的有限元模型修正方法评价标准.从这个角度来看,直接修正质量、刚度矩阵的方法虽然可以通过施加相应约束保证其带状、稀疏特征,但这类矩阵型方法在修正过程和修正结果上仍普遍缺乏物理意义,修正后的模型只在数学结果上与实际结构相近而不具备实际工程意义,因而逐渐退出历史舞台(李伟明 2011).此类方法的技术细节已有大量相关文章介绍, 不再赘述.

2.1.2 参数型方法

参数型方法不仅能够保证修正过程、修正结果的物理意义,还能够保证矩阵原有带状、稀疏特性以及各元素之间原有的连接意义,是矩阵型方法的重要发展, 国内外学者进行了较为深入的研究.参数型方法中所使用的参数可以是子矩阵参数,也可以是具有实际物理意义的弹性模量、密度、几何尺寸等物理参数.

子矩阵参数最早由Natke (1988)所提倡.通过子矩阵参数将修正后的质量、刚度矩阵$\pmb M ^* $, $\pmb K^*$表示为

$$\left. \begin{array}{l} \pmb M ^* = \pmb M_0 + \sum\limits_{n = 1}^{N} \alpha _{n} \pmb M _{ n}^{e} \\ \pmb K ^* = \pmb K_0 + \sum\limits_{ n = 1 }^{ N} \beta _{ n} \pmb K _{ n} ^{ e} \\ \end{array}\right\}(1)$$

其中, $ \pmb M_0$, $\pmb K_{0}$为修正前的质量、刚度矩阵; 每个求和单项$\pmb M _{n}^{e} $, $\pmb K _{n}^{ e} $可以是单元矩阵或多个单元的装配矩阵; $\alpha _n $,$\beta _n $即为子结构参数.可将修正后的质量、刚度矩阵代入特征方程或正交性条件,构造关于子结构参数的方程并求解, 便达到模型修正的目的.

Friswell (1990)较早地将质量、刚度矩阵表示为关于待修正物理参数$p$的一阶Taylor展开式

$$\left.\begin{array}{l} \pmb M = \pmb M _0 + \sum\limits_{i = 1}^l \delta p_i \dfrac{\partial \pmb M }{\partial p_i } \\ \pmb K = \pmb K _0 + \sum\limits_{i = 1}^l \delta p_i \dfrac{\partial \pmb K }{\partial p_i } \\ \end{array}\right\}(2)$$

然后可以将其代入特征方程或正交性条件,由同阶项相等构造方程组求解参数.相比于子矩阵参数代表单元内部物理参数的综合作用,这种修正参数表达方式是子矩阵参数的进步.

实际上,更为常用的是构造结构特征量残差目标函数并运用优化算法的有限元模型修正方法.这类方法可以表示为

$$ J\left( {\Delta \pmb p} \right) = \pmb \varepsilon ^{\rm T}\pmb W \pmb \varepsilon(3)$$

其中, 残差$ \pmb \varepsilon $表示为

$$\pmb \varepsilon = \Delta \pmb f - \pmb S \Delta \pmb p(4)$$

$ \Delta \pmb p$为待修正参数改变量, $ \Delta \pmb f$为参数改变时结构特征量的变化量,这里的结构特征量可以是特征值、特征向量、模态保证准则(modal assurance criteria,MAC)、频响函数、反共振频率、静力位移、静力应变等,也可以是它们的组合. 待修正参数与特征量残差之间通过灵敏度矩阵$\pmb S$联系起来, 它是一个关于待修正参数的雅可比矩阵

$$ \pmb S = \left[ \begin{array}{ccc} {\dfrac{\partial f_1 }{\partial p_1 }} \quad & \cdots \quad & {\dfrac{\partial f_1 }{\partial p_n }} \\ \vdots \quad & \ddots \quad & \vdots \\ {\dfrac{\partial f_m }{\partial p_1 }} \quad & \cdots \quad & {\dfrac{\partial f_m }{\partial p_n }} \\ \end{array} \right](5)$$

1969年, Fox和Kapoor为快速考察设计变量对动力响应的影响, 计算了设计参数改变时特征值$\lambda $和特征向量$ \pmb \varphi$的``变化率'', 即灵敏度, 得到了工程界的广泛重视. 国内外众多学者对基于灵敏度的有限元模型修正方法进行了最为广泛的研究, 基于灵敏度的有限元模型修正方法也是目前应用最为成熟的方法. Mottershead等(2011)对基于灵敏度的有限元模型修正方法进行了系统、全面的讨论, 并以一个直升机有限元模型修正的工程实例说明该方法的基本流程. 值得注意的是, 文中使用的是Fox和Kapoor提出的特征值和特征向量灵敏度计算方法. 特征向量灵敏度的计算相对复杂, 因此涌现出许多改进计算的研究成果, 其中Nelson (1976)提出的方法因其不需要测得全部模态数据、较为适于实际应用而得到了广泛的关注. 且Sutter等(1989)在对比了4种方法的基础上认为, Nelson法特别是针对多个设计变量的情况下, 在计算效率和结果准确性上具有一定优势. Nelson法第$r$阶特征向量灵敏度$\dfrac{\partial \pmb \varphi^{\left( r \right)}} {\partial p_i}$的计算方法是将其表达为通解与特解和的形式

$$\dfrac{\partial \pmb \varphi ^{\left( r \right)} }{\partial p_i } = C^{\left( r \right)} \pmb \varphi^{\left( r \right)} + \pmb V ^{\left( r \right)}(6)$$

$$\pmb A ^{\left( r \right)} = \pmb K - \lambda ^{\left( r \right)}\pmb M(7)$$

$$\pmb F ^{\left( r \right)} = - \left[ {\frac{\partial \pmb K}{\partial p_i } - \lambda ^{\left( r \right)}\frac{\partial \pmb M}{\partial p_i } - \pmb M \left( {\pmb \varphi ^{\left( r\right)}} \right)^{\rm T}\left( {\frac{\partial \pmb K }{\partial p_i } - \lambda ^{\left( r \right)}\frac{\partial \pmb M}{\partial p_i }} \right)\pmb \varphi ^{\left( r \right)} }\right]\pmb \varphi ^{\left( r \right)}(8)$$

由式

$$\pmb A ^{\left( r \right)}\pmb V ^{\left( r \right)} =\pmb F ^{\left( r \right)}(9)$$

求出特解$\pmb V ^{\left( r \right)}$, 然后由式

$$C^{\left( r \right)} = - \dfrac{1}{2}\left(\pmb \varphi^{\left( r \right)}\right)^{\rm T}\dfrac{\partial \pmb M}{\partial p}\pmb \varphi ^{\left( r \right)} - \left( {\pmb \varphi ^{\left( r \right)}} \right)^{\rm T}\pmb M\pmb V ^{\left(r \right)}(10)$$

求$C^{\left( r \right)}$后代入式(6)即可求得特征向量灵敏度.各种特征量灵敏度的计算在相关文章中已有较多讨论, 不再赘述(崔飞 等2003, 戴航 和 袁爱民 2011). 另外, 式(3)中的$\pmb W$为权值矩阵,Friswell和Mottershead (1995)对权值选择问题进行了较为详细的讨论,认为选用测量量方差的倒数能够反映测量量的准确程度,对相对准确的量分配更大的权值.但目前权值的选择往往还是更多地依赖研究人员的经验,其中固有频率的测量值较易获得且较为可靠,在构造目标函数时通常不可缺少, 且联合多种特征量时,应对其分配更高的权重(张令弥 和 何柏庆 1995, 杨智春 等 2009).

在实际测试过程中, 受采样频率以及频带范围内模态数量的限制,且高阶模态数据易受噪声影响产生较大误差,所获得的结构信息是不完备的,因此使用较多的数据修正较少的参数是比较理想的情况.修正参数的选择是此类方法中极为重要的技术环节.常用的参数选择方法是工程经验与灵敏度分析结合的方法,根据经验确定大致范围后选择灵敏度较高的参数,这样既可以避免灵敏度分析选择灵敏度较高但实际修正并不改变的参数,也可以避免经验选择灵敏度较小的参数造成的灵敏度矩阵奇异(宗周红和任伟新 2012). 另外, 如果结构已经产生损伤,则需要使用损伤定位方法确定修正参数(Friswell & Mottershead1995). 值得注意的是, 参数型方法通常可以根据实际需要,多次重复修正过程进行迭代, 以期提高修正结果的精确度,但是迭代过程的收敛性缺乏相关研究证明.

试验模态分析过程造成的误差有时可能会大于模型参数误差(D'Ambrogio etal. 1993), 所以有学者开始研究直接使用频率响应数据进行模型修正.频响数据能够避免模态分析带来的误差, 且具有互易性,各测点间能够相互校核从而获得更多准确的数据,因此具有广阔的应用前景. Natke (1988)最早进行了尝试; 随后Lin和Ewins(1994)提出了基于频响函数灵敏度的方法;Zang等(2001)集中研究了有限元模型修正的频域准则,以该准则评价修正后模型, 同时也可作为修正目标函数; 徐张明等(2002,2003)提出了结合模型缩聚和频响函数灵敏度的有限元模型修正方法,以及引入频响函数相关性灵敏度的方法, 通过数值算例验证,均取得了较好的效果; 此外,还形成了方程误差法(或称输入误差法)和输出误差法两种主流方法(Cottinet al. 1984, Fritzen 1986, Fritzen & Zhu 1991).但这类方法仍有频率点选择、阻尼阵修正等问题需要解决,目前仍有一批国内外学者致力于此领域研究并取得了一些成果, 如Grafe(1999)提出了考虑黏滞阻尼的大型结构模型修正方法; Kwon和Lin(2004)提出了结合灵敏度分析与测量值依赖度指标(measurementdependency index, MDI)的频段选择新方法; Sipple和Sanayei(2014)用频响函数的数值灵敏度替代解析灵敏度,提出一种能够同时修正质量、刚度、阻尼矩阵的修正方法;刘宇飞(2015)发展了两类灵敏度指标,提出了修正前参数和频段选择的改进方法等, 相关研究仍在不断深入.

另外, 如飞机、卫星等复杂结构或微尺度结构,进行传统模态试验会存在一些限制,而且这些结构也有如火箭发射、多级火箭分离界面激振、飞机起降等模拟实际工作状态振动情况的需求,因此地面振动试验在航空航天工业中更为常用. 在土木工程中,振动台试验也是研究结构抗震的重要手段.此时应用基于基础激励数据的有限元模型修正,与上述基于模态数据、频响数据的方法不甚相同(Beliveau et al. 1986).Lin和Zhu (2007)首先提出了基于基础激励的有限元模型修正方法,并通过悬臂梁模型和桁架模型进行验证,这种方法本质上是修正了子矩阵参数;王泽宇等(2010)运用动刚度对设计参数的一阶Taylor展开式,将上述方法发展为具有物理意义的设计参数的修正;刘荣贺和于开平(2013)结合改进缩聚系统法(improved reduction system,IRS) (O'Callahan 1989), 并利用数值差分计算灵敏度,提出了基于基础激励的两步修正策略, 提高了原有方法的运算效率;Yuan和Yu (2015)将此方法发展并应用到考虑阻尼的结构,并通过卫星结构模型验证, 取得了良好的效果.

建立目标函数之后, 有限元模型修正问题转化为优化问题.随着计算机科学的发展,一些如模拟退火、遗传算法、粒子群算法等性能优秀、适于并行运算的优化算法近年来备受关注,并已被应用于有限元模型修正研究(Levin & Lieven 1998a, 余岭等 2006, Marwala 2010, Perera et al. 2010, Jafarkhani & Masri 2011, 于开平 和 刘荣贺 2013, Shabbir & Omenzetter2015, Astroza et al. 2016).实际工程的有限元模型修正运算量通常较大, 且一般为非线性优化问题,合适的优化算法不仅能提高运算效率, 而且能收敛到全局最优解,对于有限元模型修正方法的实际应用具有重要意义,研究者需根据个人经验和研究条件选用最优方案. 但是,笔者认为选用不同的优化方法并不是对有限元模型修正理论的发展,有限元模型修正理论的发展还是要依靠引入新的结构特征量,或者建立更具实际意义、能够更全面地代表结构特性的目标函数.

基于灵敏度的参数型方法虽然具有明确的物理意义, 但是由于常常要经过迭代优化,多次调用有限元模型, 计算量巨大. 其计算量主要来源于灵敏度矩阵的计算,且当参数选择不当时, 如参数灵敏度较小或几个参数影响相近等情况,亦或数据被噪声污染时, 灵敏度矩阵在求解过程中易出现病态,导致算法失效或求解错误(Friswell et al. 2001). 因此在实际应用中,特别是针对待修正参数较多的大规模、复杂结构仍有一些限制.

2.2 传统有限元模型修正方法的发展

针对传统参数型方法存在的计算量大、求解过程易出现病态的问题,有学者提出采用缩聚模型或代理模型(surrogate model)的方法,以及不基于灵敏度的参数型方法等开始受到广泛关注,有限元模型修正技术近年来取得了长足的进步. 模型缩聚的方法主要包括:静缩聚法(Guyan 1965)、动缩聚法(Paz1984)、IRS法、等效系统缩聚法(system equivalent reduction expansionprocess, SEREP) (O'Callahan et al. 1989)等.Papadimitriou和Papadioti(2013)提出的结合模态综合法进行模型修正的思路;Weng等(2011)使用基于子结构的方法,以及侯吉林(2010)提出的约束子结构修正方法也取得了较好的效果,对降低模型修正运算规模很有帮助.这些方法关键在于对原有结构特性矩阵的处理过程, 因此不多做介绍.交叉模型交叉模态法(cross-model cross-mode method,CMCM)也是一种参数型方法, 但其不需要进行迭代运算,且不需要计算灵敏度, 较好地避免了传统方法的问题;使用如神经网络、响应面等代理模型的方法, 不仅能够降低计算负担,而且不需计算灵敏度, 也体现出了实际应用方面的优势.

此外, 由于实际结构是广泛存在不确定性的,而传统有限元模型修正均为基于确定性的, 与实际情况并不相符. 因此,向不确定性的发展也是传统有限元模型修正技术重要的发展方向.Collins等(1974)结合经典统计思想提出了基于灵敏度的最小方差法,并探讨了有限元建模和动力试验过程中的不确定性,较早地运用统计理论解决有限元模型修正问题. 另外有学者注意到,贝叶斯统计推断中由先验信息和样本信息得出后验信息,然后推断未知参数的思想, 恰与有限元模型修正技术的思想类似,于是将贝叶斯统计理论应用到有限元模型修正过程中.Alvin等(1998)提出一种考虑误差和不确定性的建模框架,并基于贝叶斯统计讨论了模型中误差和不确定的估计和传递;Beck和Katafygiotis (1998)首先提出贝叶斯统计模型修正的技术框架;华宏星和傅志方(1998)从无条件极大似然估计的角度,推导了基于贝叶斯框架的模型修正公式,并讨论了其估计的无偏性、鲁棒性等问题.随后一些学者针对后验分布积分计算的困难, 发展了一系列新方法(Beck &Au 2002, Ching & Chen 2007, Cheung &Beck2009); 袁昭旭和于开平(2017)在贝叶斯修正框架下,提出了针对高温环境结构的有限元模型修正方法.目前贝叶斯统计方法已经广泛用于体系更为成熟的有限元模型确认中.除使用确定性模型和随机模型以外, 还可以使用模糊模型处理现实问题.华宏星和傅志方(1997)根据有限元模型修正中存在模糊的参数选择、模糊的修正目标等特征,提出根据修正参数及其权值模糊选择的有限元模型修正方法,且数值算例取得了理想的效果; Liu和Duan(2012)将模糊有限元理论引入到有限元模型修正中,合理地考虑了测量模态参数的不确定性对修正结果的影响; Erdogan和Bakir(2013)结合模糊数学理论和遗传算法、粒子群算法,提出基于模糊有限元模型修正的损伤识别方法,并与基于贝叶斯统计的蒙特卡洛模拟(Monte Carlosimulation)方法结果对比, 取得了较好的效果.传统方法向不确定性的发展, 特别是其中形成的有限元模型确认技术,大有替代传统方法而成为具有普遍意义方法的趋势. 以下主要介绍CMCM,神经网络法和响应面法, 并探讨有限元模型确认的发展历程和研究现状.

2.2.1 CMCM

CMCM由Hu等(2007)首先提出, 是一种不基于灵敏度的方法.这种方法与子矩阵法有些类似, 同样将修正后的质量、刚度矩阵表示为式(1) 形式并代入特征方程中,但实测数据与模拟数据的交叉使用形成了一种新方法. 即, 式 (1) 中$\pmb M _0 $, $\pmb K _0 $表示修正前的质量、刚度矩阵,则由初始有限元模型特征向量、特征值数据$\pmb \varphi _i^a $,$\lambda _i^a $和实测特征向量、特征值数据$\pmb \varphi _j^t $,$\lambda _j^t $构成模型修正前后的特征方程

$$\pmb K _0 \pmb \varphi _i^a = \lambda _i^a \pmb M _0 \pmb \varphi _i^a(11)$$

$$\pmb K ^ * \pmb \varphi _j^t = \lambda _j^t \pmb M ^ * \pmb \varphi _j^t(12)$$

将$\left(\pmb \varphi _i^a\right) ^{\rm T}$左乘式 (12),$\left(\pmb \varphi _j^t\right) ^{\rm T}$左乘式 (11),由于$\left(\pmb \varphi _j^t\right) ^{\rm T}\pmb K _0 \pmb \varphi _i^a $和$\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb K ^ * \pmb \varphi _j^t $是标量, 对调$\pmb \varphi _i^a $和$\pmb \varphi _j^t$后做两特征方程变换后的比值, 得到

$$\dfrac{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb K^ * \pmb \varphi _j^t }{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb K _0 \pmb \varphi _j^t } = \dfrac{\lambda _j^t }{\lambda_i^a }\dfrac{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb M ^ * \pmb \varphi _j^t }{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb M_0 \pmb \varphi _j^t }(13)$$

将式 (1) 代入式 (13)并进一步化简得到

$$ 1 + \sum\limits_{n = 1}^N {\alpha _n \frac{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb K _n^e \pmb \varphi _j^t}{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb K _0 \pmb \varphi_j^t }} = \frac{\lambda _j^t }{\lambda _i^a }\left( {1 + \sum\limits_{n = 1}^N {\beta _n \frac{\left(\pmb \varphi_i^a\right) ^{\rm T}\pmb M _n^e \pmb \varphi _j^t }{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb M _0 \pmb \varphi _j^t}} }\right)(14)$$

引入新符号简单表示为

$$\sum\limits_{n = 1}^N {\alpha _n C_{r,n} } + \sum\limits_{n = 1}^N {\beta _n E_{r,n} } = f_r(15)$$

其中, $C_{ij,n} = \dfrac{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb K _n^e \pmb \varphi _j^t }{\left(\pmb \varphi _i^a\right)^{\rm T}\pmb K _0 \pmb \varphi _j^t }$, $D_{ij,n} = \dfrac{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb M _n^e \pmb \varphi _j^t }{\left(\pmb \varphi _i^a\right) ^{\rm T}\pmb M _0 \pmb \varphi _j^t }$, $E_{r,n} = - \dfrac{\lambda _j^t }{\lambda_i^a }D_{ij,n}$, $f_r = \dfrac{\lambda _j^t}{\lambda _i^a } - 1$.至此有限元模型修正问题归结为解超定方程组的问题.这种方法不需要进行实测模态与模型模态的匹配, 且不要求全部模态数据,不需计算灵敏度, 不需迭代计算, 便于工程应用.但是当实测数据比较准确的时候,在做特征方程的比值时可能出现分母接近零的情况而使算法失效;另外子矩阵参数是整体参数,可以表达单元之间的连接意义和多个物理参数的综合作用,无法进行一个单元内某个特定物理参数的修正.

2.2.2 神经网络法

有限元模型修正是一种比较典型的反问题.反问题一般都是非线性且不适定的, 求解难度较正问题大得多.传统方法无论如何选择待修正参数也无法改变修正过程反问题的本质,因此有学者研究使用神经网络方法,利用其非线性映射能力强、鲁棒性较好的特点,将有限元模型修正的反问题转化为正问题求解(费庆国 和 张令弥 2004).经过多年的发展, 已经产生种类繁多的神经网络,并在多个领域得到了有效的应用. 用于有限元模型修正领域的主要有:径向基函数(RBF)神经网络(Levin & Lieven 1998b, Atalla & Inman 1998)和反向传播(BP)神经网络(徐宜桂 等 2000, 李林 等2006). 值得注意的是,神经网络方法也能用于非线性结构的有限元模型修正(费庆国 等 2005).具体做法是: 首先,通过工程经验与灵敏度分析结合的方法选择待修正参数作为输出参数,选择结构特征量作为输入参数;扰动待修正参数并通过有限元模型得到相应结构特征量,生成神经网络的训练样本数据; 然后, 确定神经网络的类型、结构,利用样本数据训练神经网络, 直至收敛;将实测数据得到的结构特征量输入神经网络,得到的参数即为修正后的参数; 最后,将修正后参数代入有限元模型重新生成结构特征量与实测数值比较,如果误差较大, 需要补充训练样本并重新训练神经网络,如此重复直至满足精度要求.神经网络取代灵敏度重新定义了结构特征量与待修正参数之间的关系,将有限元模型修正转变为正问题求解, 是有限元模型修正技术的进步.但是, 神经网络的泛化能力对修正结果的影响较大,修正后模型的外推是否合理需要进一步确认; 另外,参数较多时神经网络所需收敛时间较长,输入的训练数据越充分神经网络通常收敛越快,但是建立大量的训练数据也需要多次使用有限元模型计算,从而造成一定的计算负担.

2.2.3 响应面法

传统参数型有限元模型修正方法的迭代过程需要多次重复调用有限元模型,造成极大的计算量.通过简化有限元模型或选用更优的网格划分方案可以在一定程度上提高运算效率.响应面法是一种类似的思路,它以显式的响应面函数拟合结构特征量与参数之间复杂的隐式关系,建立一个替代有限元模型的代理模型进行优化迭代过程(蒋寅军 2011).响应面法最早由Box和Wilson (1992)提出并应用于化学工业,经过多年发展已成功应用于机械工程、土木工程等多个领域.郭勤涛等(2006b)较早地探讨了响应面法在有限元模型修正领域的应用价值.任伟新和陈华斌(2008,2010)较早地将响应面法用于解决大型土木工程结构有限元模型修正问题,且对实际桥梁进行有限元模型修正研究取得了良好的效果.响应面法有限元模型修正的具体做法是(蒋寅军 2011, 宗周红和任伟新2012): 建立初始有限元模型后, 首先根据经验确定模型参数设计空间,选择合适的试验设计方法生成样本点, 常用的试验设计方法有:全因子设计、中心复合设计、正交设计、均匀设计等,然后代入初始有限元模型生成样本数据; 通过对样本数据进行方差分析,选择对结构响应影响明显的参数; 选择响应面模型并拟合模型参数,常用的响应面模型包括:多项式模型、径向基函数模型、克里格(Kriging)模型等;确定的响应面模型需运用均方误差、平均误差、复相关系数等统计指标进行模型验证,如果达到精度要求,则可将响应面模型替代有限元模型进行模型修正优化过程;否则需要返回修改响应面模型直至满足精度要求.响应面法有限元模型修正一般流程如 图2所示.

图 2   响应面法有限元模型修正流程图

   

有研究表明, 相比于神经网络法, 响应面法建模工作量要小得多; 而且,通过统计方差分析选择影响显著的参数, 从而降低优化求解问题的维度,也是神经网络法所不具备的能力(方圣恩 2010). 另外,响应面法处理多参数影响问题体现出了明显的优越性,且回避了传统方法可能出现的不适定问题,因此在解决大型结构的有限元模型修正问题上具有广阔的前景,同时其中包含的模型确认思想也符合模型修正技术发展的方向.但是响应面法各环节方法的选择较依赖研究人员的经验,且存在最终得到的有限元模型``外推''不可靠、修正结果不唯一,生成样本数据计算量较大等问题. Chakraborty和Sen(2014)针对最小二乘法求响应面待定参数产生的误差提出使用滑动最小二乘法;Gou等(2016)基于正交设计和线性响应面提出连续选择法(successiveselection method)快速选择和分析实验设计样本点, 减轻计算负担;Jin和Jung (2016)提出序列代理建模(sequential surrogatemodeling)降低对研究者经验的依赖, 降低计算量等,这种方法仍在不断发展中.

2.2.4 有限元模型确认

有限元模型确认是有限元模型修正从确定性到不确定性的发展.以土木工程领域为例, 近年来结构健康监测技术的发展,要求所建立的有限元模型不仅能够指导结构设计, 识别结构损伤,还应当能够预测结构未来工作状态(Farrar & Worden 2012).这就要求有限元模型不仅能够准确、全面地反映结构特性,而且要保证模型修正结果具有一定置信度水平以便指导后续进一步的应用;而基于确定性的传统有限元模型修正技术显得力不从心,其修正结果实际上是不确定性模型的一个特例.基于不确定性的结构健康监测研究,有望成为解决土木工程健康监测问题的一般方法(Housner et al. 1997).有限元模型确认是其中重要的研究方向,通过对结构系统从构件到整体的分层建模和确认试验,对系统中的不确定性进行量化和传递分析,以及对有限元模型在设计空间预报精度进行评价和确认,能够建立更为可靠的有限元模型并从统计理论给出模型置信度(宗周红和任伟新2012).

有限元模型确认的技术路线主要由美国一些研究机构的研究报告、研究计划提出(朱跃2010), 包括美国能源部提出的加速战略计算首创计划(AcceleratedStrategic Computing Initiative Program),及与之配套的计算机仿真系统置信度评估方法------模型验证和模型确认(modalverification and modal validation)计划形成的ASCI V & V计划,由SNL制定的ASCI V & V指南(Martin 2000);美国航空航天学会(American Institute of Aeronautics andAstronautics, AIAA)计算流体力学委员会(Computational Fluid DynamicsCommittee)提出的计算流体动力学模拟的验证和确认指南(AIAA Guide forthe Verification and Validation of Computational Fluid DynamicsSimulations), 以及随后美国机械工程师协会(American Society ofMechanical Engineers,ASME)成立专门的委员会并发布的计算固体力学验证和确认指南(Guide forVerification and Validation in Computational Solid Mechanics)(Schwer 2009)等, 形成的技术路线如 图3所示.其中``验证''(Verification)部分是指对计算模型与数学模型之间程序代码和算法的验证,不是本文讨论范围;``确认''(Validation)部分是指试验结果与模拟结果之间的确认,即本文所讨论的有限元模型确认. 另外,这个技术路线包含了一种分层确认的思想,这种思想将复杂系统分为全系统、子系统、部件、单元四个层次,可以降低系统的复杂程度. 且系统分层研究表明,由于层次越低影响因素越少,通常只能在系统的部件和单元层次获得比较准确的试验数据,所以模型确认主要在这两个层次进行(朱跃 2010).

图 3   模型验证和确认技术路线(Schwer 2007)

   

有限元模型确认被系统提出之后, 更多学者将目光投向了这一研究领域.Hemez和Doebling (2000a, 2000b), Hanson和Hemez(2001)介绍了LANL的一些研究工作,对有限元模确认进行了较为全面的论述和展望;张令弥(2002)在回顾模型确认发展历程的基础上,讨论了模型确认以及确认试验的技术路线,并对该领域的研究方向进行了展望; Rebba等(2003,2006)运用贝叶斯方法讨论了模型确认、误差传递的问题,并探讨运用贝叶斯网络(Bayesnetwork)实现子模型到整体模型确认信息传递, 以及模型预测能力的量化;Link和Friswell (2003)针对著名的GARTEUR基准模型提出模型确认的三级确认准则,被广泛应用;Oden等(2003)指出力学问题中计算模型的不确定性建模是未来发展方向,且所建立的不确定性模型应当能够对力学问题提供可靠的分析和预测;Chen等(2004)提出利用响应面法与蒙特卡洛模拟的方法研究了金属板翻边工艺的模型确认问题;费庆国等(2004)运用Link和Friswell提出的三级确认准则,同样对GARTEUR基准模型进行了模型确认研究;郭勤涛等(2006a)在回顾传统有限元模型修正技术的基础上,全面综述了有限元模型修正向模型确认的发展方向及其现状,并对薄板结构和车架模型进行了实例分析. 2006年,为进一步推动模型确认研究的发展, SNL发布了模型确认的挑战问题,包含热传导, 静力学和动力学3个子问题,国内外学者针对此开展了广泛的研究和讨论(Red-House & Paez 2008, Hills et al. 2008), 对模型确认的发展起到了一定的推动作用.Paez和Red-Horse(2008)重点总结介绍了结构动力学挑战问题的6项研究成果, 包括:Ghanem等(2008)提出结合使用K-L过程(Karhunen-Lo$\bar{\rm e}$ve procedure)和多项式混沌展开(polynomial chaos expansion, PCE)建立随机参数模型和认知不确定性模型,并研究了不确定性传递和该挑战问题的模型确认; Hasselman和Lloyd (2008)利用摄动方法建立线性随机模型,然后按照NASA的确认准则完成挑战问题的模型确认;Horta等(2008)对奇异值分解(singular value decomposition,SVD)、核密度估计(kernel density estimate,KDE)等概率建模方法进行对比分析和总结; McFarland和Mahadevan (2008)首先用主成分分析(principal component analysis,PCA)和马尔科夫链蒙特卡洛方法(Markov chain Monte Carlo,MCMC)对模态参数进行核密度估计建立分布模型,然后利用蒙特卡洛模拟研究不确定性的传递; Rutherford (2008)采用响应主成分分析的正交向量作基函数,以其线性组合表示模拟响应, 并由拉丁超立方采样(Latin hypercube sample)及核密度估计构造参数分布, 与试验数据对比进行模型校准、确认;Zang等(2008)直接识别质量、刚度、阻尼系数并将线性与非线性系数分离建立均值模型,通过蒙特卡洛模拟进行模型确认研究,同时探讨了非线性与不确定性的关系及非线性对模型确认的影响.此外还有如Jiang和Mahadevan (2011)提出小波谱(wavelet spectrum)分析的模型确认方法, 并通过该挑战问题进行验证;张保强等(2011)使用核密度估计和核主元分析(kernel principal component analysis, KPCA)方法对该挑战问题的研究等. Oberkampf和Roy(2010)的专著对模拟研究中模型代码验证、模型确认与预测问题进行了系统的总结和论述.丁继峰等(2010)探讨了在有限元模型修正之前对模型进行确认、重构对提高最终修正结果精度的意义;朱跃(2010)采用支持向量机(support vector machine,SVM)响应面研究了复杂系统的模型确认问题;随后如毕司峰和邓忠民(2013)张冬冬和郭勤涛(2013)陈志国等(2013)也采用蒙特卡洛模拟与响应面结合的方法研究了模型确认问题,取得了较好的效果. 有限元模型确认仍在不断发展中,其主要的研究方向包括: 不确定性模型的建立; 不确定性模型选择;代理模型的建立; 包含更全面结构信息的评价准则的提出;以及与模型修正技术结合的结构安全评估预警系统的建立等.

3 非线性有限元模型修正发展现状

从线性到非线性是传统有限元模型修正技术的重要发展方向,逐渐得到国内外学者的广泛关注. 非线性与不确定性不同,非线性是结构中固有的、确定的性质,一味对结构非线性进行近似和简化会增加结构的不确定性,即使应用基于统计理论的模型修正、确认方法,也无法从根本上改善模型的置信度. 以土木工程结构为例,非线性是广泛存在的, 除材料非线性外,在结构伸缩缝、连接节点等位置均表现出非线性特征,一些新型支撑、隔震减振支座结构,摆、索结构以及损伤后的结构整体也会表现出明显的非线性.不同类型、不同材料的土木工程结构,如多层、超高层、大跨空间结构、大跨桥梁、地下结构等各自的非线性特性超出本文探讨范围,不进行分别讨论, 但不同类型的结构之间存在共性的非线性问题,如不合理地加以考虑, 一味以线性模型进行模拟和修正,可能造成结构设计缺陷、健康监测失效等工程问题,威胁人类生命财产安全(Worden & Tomlinson 2000). 另外,土木工程结构工作状态受环境因素影响较为明显,特别是在发展可靠的结构健康监测系统的道路上,首先应当确定环境因素造成的影响,而温度、湿度、风等环境因素对结构的影响也是非线性的. 目前,无论是传统有限元模型修正, 还是其向不确定性发展的有限元模型确认,以及考虑环境因素影响的有限元模型修正研究, 均为基于结构线性假设,运用固有频率、振型等线性结构特征量的方法;除了将环境影响因素去除的方法以外,有学者采用建立环境因素与结构动力特性之间关系模型的方式考虑环境因素影响,并将此关系模型参数引入模型修正过程(何成 等2013, Zhou & Song 2016), 但其模型也多为线性,尚未见针对非线性结构考虑环境因素的有限元模型修正相关研究.

基于结构线性假设的传统有限元模型修正方法的局限性, 已经引起国内外学者的关注.主要有以下研究成果:

(1) LANL的Hemez和Doebling课题组研究成果. Hemez和Doebling (2001)对非线性模型修正进行了较早的论述, 通过LANL的五项试验结果,说明发展非线性模型修正的必要性以及所面临的挑战;Beardsley等(1999)利用主成分分析结合响应面的方法,研究了某聚合物材料本构模型非线性修正问题,并与基于线性方法的修正结果对比, 说明了非线性修正方法的优势;Schultze等(2001)利用冲击试验响应峰值加速度作为特征量,结合方差分析方法, 建立待修正参数与特征量之间的响应面模型,研究了材料非线性模型修正问题.

(2)美国塔夫茨大学(Tufts University) Moaveni课题组研究成果.Moaveni等(2010,2013)先后进行了7层钢筋混凝土剪力墙结构模型和3层钢筋混凝土框架模型振动台试验,并通过线性有限元模型修正进行损伤识别, 结果表明,随着损伤程度的提高, 基于线性的方法无法很好地代表结构状态;Asgarieh等(2014,2017)随后对这两个试验进行了非线性有限元模型修正的研究,选择材料非线性本构模型参数作为待修正参数,运用时变模态参数反映结构非线性特征,以此构造目标函数并利用模拟退火法进行优化, 取得了较好的结果.

(3)英国帝国理工学院 (Imperial College London) Ewins课题组研究成果.Ewins等(2015)提出了一种非线性有限元模型修正和确认的``三阶段10步骤''技术路线,如图4所示, 即首先建立线性模型, 通过传统方法进行修正;然后通过对结构非线性的初步判断,开展针对性的动力试验对结构非线性进行探测、定位、描述和量化;文章首次提出了模型``升级''(upgrading)的概念,强调应当提升单元阶次及模型复杂程度,更准确地还原结构固有的非线性特性;最后针对``升级''后的非线性模型进行进一步的修正和确认. 随后,Ewins课题组Carri等(2017)按照所提出的技术路线对机翼结构进行了

图 4   Ewins等提出的非线性有限元模型修正技术路线(Ewins et al. 2015)

   

非线性识别、模型修正试验研究, 运用反步法(reverse path method)描述非线性, 并通过曲线拟合对非线性进行量化,进一步强调在结构承受较大幅度激励时``升级''模型的必要性,同时也指出目前用于结构非线性识别的方法均有不足,用于非线性模型修正和确认的方法更需要进一步研究. 事实上,在SNL提出的挑战问题中, 其子结构的连接处存在弱非线性,Ewins课题组的Zang等(2008)已经对其进行了探讨并提出了将模型非线性分离,先进行线性结构模型确认再探究非线性影响的思想. 另外, Carri和Di Maio (2016)还对一个哑铃模型非线性连接进行了试验研究,运用反步法进行非线性定位、描述、量化,并通过修正后模型的频响函数进行了验证.虽然该项研究取得了较好的结果,但是文中也指出了其方法无法修正精细化、大规模模型的问题,以及现有商业软件的不足.

(4)我国学者研究成果. 费庆国等(2005)使用径向基神经网络方法,通过均匀试验设计生成参数样本点训练神经网络,对一非线性梁的材料非线性模型参数进行修正. Wang等(2015,2016)通过解析模式分解(analytical mode decomposition,AMD)和希尔伯特变换(Hilbert transform,HT)提取响应主成分的瞬时频率和幅值, 并以此建立残差目标函数,然后利用模拟退火优化算法完成非线性修正,并通过剪切型结构的数值算例和变压器结构振动台试验验证了该方法的有效性;袁平平等(2016)利用类似方法对三层框架梁柱连接节点弯矩--转角关系模型进行非线性修正;另外,Yuan等(2016)还进行了使用静力数据进行非线性连接模型修正的研究,借助灵敏度分析选择连接参数, 通过挠度值构造目标函数,运用模拟退火方法进行优化,并通过悬臂梁和钢桁架桥模型进行了数值算例验证.

(5)其他学者研究成果. Meyer和Link(2003)将局部非线性抽象为两自由度系统, 采用谐波平衡法(harmonic balance method)将其线性化并变换到频域,以此修正系统质量、刚度、阻尼矩阵. Li等(2017)直接使用时程数据,对智利一座桥梁的隔震橡胶支座进行了部件层次和系统层次的修正,但其巨大的计算量限制了其进一步推广应用.Lenaerts等(2001)运用本征正交模态(proper orthogonal mode,POM)通过模型修正手段进行非线性参数识别,并通过局部非线性梁的数值算例进行方法验证,但应用于复杂实际结构还有一定困难; Kerschen 和Golinval (2004, 2005)先后利用与神经网络结合的非线性主成分分析方法(non-linear principal component analysis)和条件反步法(conditioned reverse path method)处理非线性系统模型修正问题, 但应用于实际结构尚需进一步研究.da Silva等(2009)对4种用于非线性模型修正的方法进行了比较研究.包括谐波平衡法、本构方程误差法(constitutive equation error)、恢复力面法(restoring force surface)和K-L (Karhunen-Loève)分解法. Bussetta等(2017a,2017b)从目标函数和优化算法两个角度对非线性修正方法进行了综述;并运用前三阶Volterra级数,对一杆件预加轴力试验进行了非线性有限元模型修正和确认,取得了一定的效果; Song等(2012)利用瞬时刚度作为特征量,采用线性有限元模型修正的方法,进行了混凝土剪力墙材料非线性模型的修正研究,并利用试验数据进行了模型确认; 另外, Song和Dyke (2013a,2013b)利用无迹卡尔曼滤波(unscented Kalman filter,UKF)开发了一种试验平台, 可以对材料非线性滞回模型进行实时修正,并对结构状态做出评估.Isasa等(2011)提出了一种结合多谐波平衡法与扩展本构关系误差法(extended constitutive relation error, ECRE)的局部非线性修正方法,并应用于悬臂梁数值模型对该方法进行了验证,但实际应用效果仍需进一步研究. Shahidi和Pakzad (2013)运用响应面法,对非线性材料本构模型修正进行了初步研究. Kurt等(2015,2016)提出了一种``数据驱动''的非线性有限元模型修正策略,通过对在不同激励水平下响应信号的小波谱进行叠加,得到频率--能量的经验关系并绘制频率-能量图,以此描述结构的非线性特性. 这种方法不需要事先对非线性模型做出假设,仅通过响应数据进行模型修正过程.但是这种方法需要较多的研究人员的经验, 计算量也比较大,修正结果需要通过图像对比来说明而不能量化,应用于实际结构还需要进一步研究.Chen等(2016)运用谐波平衡法计算非线性模型FRF,与试验FRF构成目标函数, 通过拉丁超立方采样方法选定初值,对目标函数进行优化以完成模型修正,并通过悬臂梁数值算例进行方法验证. Canbaloğlu和Özgüven (2016)运用其所提出的拟导纳差异法(pseudo receptance difference,PRD)从结构非线性FRF中提取线性部分FRF,然后由线性修正方法对其进行修正,并通过模拟和试验说明该方法对提高模型精度具有一定效果,但是其对强非线性的适用性还需要进一步研究.Wang等(2018)运用频响数据首先在低幅激励作用下进行线性修正,然后利用主谐波数据进行非线性修正,应用于非线性固支梁试验的非线性模型修正取得了良好的效果.Ebrahimian等(2017)运用最大似然估计法估计非线性材料本构模型的时变参数,结合克拉默--拉奥下界(Cramer-Rao lower bound,CRLB)理论估计测量过程不确定性, 提出了一种非线性模型修正框架,并通过桥墩结构和框架结构两个数值算例验证了该框架的有效性.但这种方法理论复杂, 不考虑建模误差且认为误差全部来源于测量误差,应用于实际工程仍需进一步研究验证.

相关研究从研究对象上来看主要包括材料非线性本构模型的修正和非线性连接模型的修正;从非线性程度上来看, 主要有弱非线性和强非线性之分;从研究角度也有从理论模型角度、概率模型角度以及从能量角度的不同;所使用的数据也有时域和频域的区别. 如果根据先验知识预先描述非线性,非线性模型修正转变为非线性识别, 如果是非线性形式未知的黑箱模型,主要使用反步法, 结合工程研究经验估计或试错确定非线性模型; 对弱非线性情况,有学者采用线性化的方法然后利用传统方法进行修正,对强非线性则主要通过结构瞬时特征量建立目标函数进行处理. 总之,非线性模型修正的研究成果远不如传统有限元模型修正方法丰富,非线性有限元模型修正的研究仍处于起步阶段, 相关研究仍需不断深入.

4 讨论及展望

4.1 讨论

有限元模型修正技术对进一步发挥有限元模拟分析在工业工程领域的作用具有重要意义.至今, 这一技术已经能够应用于实际工程并仍在不断发展中.本文首先对传统有限元模型修正方法进行总结评述,然后介绍了针对传统方法不足之处发展起来的一些新方法,最后重点探讨有限元模型修正向非线性的发展方向, 并介绍了目前相关的研究成果.

传统有限元模型修正方法按照模型修正数据处理的对象可分为矩阵型方法和参数型方法.其中, 矩阵型方法由于普遍缺乏物理意义而缺乏实际工程应用价值;参数型方法逐步发展并成为主流,又以基于灵敏度的参数法在实际工程中的应用最为广泛,理论研究也相对成熟. 但这类方法的缺点在于灵敏度矩阵的计算,不仅计算量大, 且容易带来求解问题的病态导致方法失效; 另外,在应用于大型结构时, 由于迭代运算需要反复调用有限元模型,造成巨大的计算量, 也限制了其更广泛的应用.针对传统有限元模型修正方法易造成问题病态和计算量大的缺点,目前的研究发展方向主要是发展不基于灵敏度的方法、发展能够实现快速运算的代理模型,以及一些结合应用新型优化算法的方法.包括交叉模型交叉模态法、神经网络法、响应面法以及结合模拟退火、遗传算法、粒子群算法等优化算法的方法.其中, 响应面法得到了广泛关注,特别是其中所包含的统计思想符合有限元模型确认的发展方向,在大型结构的有限元模型修正过程中也体现出了修正多个参数的能力.但是, 神经网络法和响应面法的成功依赖于样本空间的选择,不同的选择产生不同的修正结果,选择不当也可能会造成代理模型泛化能力不足,从而导致有限元模型修正结果外推不可靠;增加数据量能够增加模型的可靠性, 但又会带来较大的计算量. 所以,通过响应面法得到的修正模型必须经过模型确认过程.有限元模型确认是传统有限元模型修正方法在统计理论上的发展,从理论上具有更一般的意义,且它能够从理论上探究复杂结构不确定性的传递,量化评价修正后模型的不确定性, 给出指导工程应用的置信度,对于修正后有限元模型进一步应用于结构损伤识别、状态评估、性能预测等具有重要实际意义.

以上方法均基于结构线性假设, 但结构的非线性是无法回避的,如不加以合理考虑, 无论使用何种传统方法修正模型,必然会遭遇精度瓶颈; 若采用基于统计理论的方法,将非线性归为不确定性, 从理论上并不合理, 无法反应结构本质,且最终得到的模型置信度也会遇到瓶颈,所以应当从理论角度对结构中的非线性加以考虑.非线性有限元模型修正近年来已经涌现出一些研究成果,对于弱非线性情况, 一般将其线性化后使用传统有限元模型修正方法,强非线性情况多使用瞬时结构特征量.国际著名动力学专家Ewins提出先进行传统有限元模型修正建立线性模型,然后通过特定的非线性判断、定位、描述、量化技术``升级''模型,最后对非线性模型进行修正、确认这一技术路线,为非线性模型修正的发展提供了较为明确的发展方向,但相关研究仍需不断丰富和深入.

4.2 展望

非线性有限元模型修正的研究工作仍在起步阶段,距离实际工程应用更是还有很长的路要走. 基于目前研究现状,未来有限元模型修正向非线性的发展方向包括:

(1)完善模型修正过程进入不同阶段的定量判断准则. 现有研究均认为,结构的非线性特征常常依赖于振动幅度, 因此在低幅振动情况下,结构可视为线性, 或视为弱非线性作线性化处理.但当振动幅度达到一定程度后, 结构将表现出明显的非线性特征,主要表现在结构响应时程曲线畸变、频响函数产生其他频率成份,以及系统叠加性、频响函数互易性的丧失,甚至出现混沌振动等复杂现象等, 导致线性结构特征量无法代表结构特性,因此仍然按照传统方法进行模型修正将无法得到准确结果,运用模型修正进行参数识别、损伤识别的结果也将失去实际意义. 问题是,到底在激励幅度达到什么程度, 应当开始进行非线性模型修正过程,尚缺乏定量判断准则, 目前大多通过观察响应曲线变化,根据研究者的经验确定. 这样的做法造成的后果是,如果对非线性模型修正开始的``时机''把握不好,可能造成线性模型的不准确,这种误差在大幅度激励下非线性模型修正过程中将被放大,影响最终修正结果. 另外, 非线性意味着更为复杂的计算,确定在线性、弱非线性、强非线性各阶段之间合理的量化判断准则,有助于工程师和研究人员根据实际情况选择合适的方法,避免不必要的复杂计算.

(2)丰富代表结构非线性的特性指标. 如果结构表现出明显的非线性特征,以往在线性结构中所使用的振型、模态等特征量将不再适用. 所以,目前使用将系统线性化的方法处理非线性问题的研究,一般只能用于``弱非线性''结构.类似传统方法中使用固有频率、振型、反共振频率等特征量建立目标函数的思想,在非线性有限元模型修正过程中,也需要能够全面代表结构非线性特性的特性指标.现有的用于非线性模型修正的结构特性指标主要是瞬时量,包括瞬时频率、瞬时幅值、瞬时非线性模态等,然而结构非线性行为极为复杂,已有的瞬时特征量并不能完整地代表结构非线性特性.丰富、拓展代表结构非线性的特性指标,甚至建立特性指标体系是该研究领域的核心问题.而一旦能够找到合适的特性指标构建合理的目标函数,接下来的非线性有限元模型修正过程就可以转化为优化过程.

(3)发展结构非线性有效的描述和量化方法.在进行非线性有限元模型修正的过程中,常常首先需要对结构中的非线性进行识别, 包括判断非线性是否存在,确定非线性的位置, 描述非线性的特性, 量化非线性的模型参数. 其中,非线性的判断和定位尚有一些有效方法,主要通过观察时程响应曲线、频响函数曲线的畸变判断非线性是否存在,利用激励信号与响应信号之间的相关性判断非线性的位置. 但Ewins指出,目前缺乏能够进行非线性定位的仪器, 限制了实际工程应用. 更重要的是,结构中非线性的特性往往是未知的,能否正确描述、量化结构非线性将在很大程度上影响后续的非线性修正过程.针对非线性材料模型常常根据经验选取已有的本构模型进行参数识别;针对黑箱连接模型主要通过反步法, 在一定分析基础上的经验试错确定;量化过程主要通过多项式拟合. 这一过程会产生较大的计算量,即使使用代理模型或者缩聚、简化模型,应用于大型结构仍然会造成较大计算负担, 限制了实际工程应用.由于先验信息的缺乏, 无法完全确定未知的非线性关系,需要进一步研究有效的方法, 拓宽研究思路, 增加约束条件,降低对研究经验的依赖, 使非线性模型的建立以及修正结果更可靠.

李东升, 博士, 汕头大学教授, 博士生导师.先后于上海交通大学、大连理工大学和德国锡根大学获得学士、硕士和博士学位.2006年、2009年、2014年分别在意大利帕维亚大学、美国休斯顿大学、美国伊利诺伊大学芝加哥分校进行访问研究.2002年起在大连理工大学工作, 2018年作为引进人才到汕头大学工作至今,目前主要从事结构动力学和结构健康监测研究. 研究方向包括:力学与结构动力学基本问题;结构计算模型校验与验证;结构健康监测理论与关键技术;工程结构损伤识别方法与应用.主持国家自然科学基金青年基金项目一项, 面上基金两项,参加国家自然科学基金, 国家自然科学基金创新群体,国家重点基础研究发展计划(973计划)等多项科研课题.在国内外学术期刊和国际会议上发表论文50余篇, 出版英文专著一部. Advances in Civil Engineering等期刊编委,高等学校博士学科点专项科研基金网络评审专家,国家自然科学基金通讯评委.

李宏男, 博士, 大连理工大学教授, 博士生导师.教育部``长江学者奖励计划''特聘教授, 国家杰出青年基金获得者,国家级有突出贡献的中青年科技专家, 享受国务院政府特殊津贴,国家自然科学基金委员会学科评议组成员,国务院学位委员会学科评议组成员,国家科技支撑计划重点项目总体技术专家组组长,国家自然科学基金委员会创新研究群体负责人;兼任美国土木工程师学会高等材料与结构委员会副主席,国际智能基础设施健康监测学会理事,国际结构控制与监测学会中国分会副主席,中国土木工程学会和中国振动工程学会理事,中国振动工程学会抗振控制分会和随机振动分会副理事长等20余个学术团体的常务理事、理事和委员.担任国际期刊An International Journal, Structural Monitoring and Maintenance主编, 国际期刊ASCE, Journal of Aerospace Engineering, An International Journal, Smart Structures and Systems, International Journal of Distributed Sensor Networks客座主编,《建筑结构学报》、《振动工程学报》等刊物编委.主要从事工程结构的研究与教学工作,在地震动理论、结构多维抗震设计理论及其应用、结构多维随机振动理论、大跨越输电塔体系的抗震计算理论、结构减震控制与健康监测等多个领域内的前沿课题上取得了一系列的研究成果.主持国家基金重大计划重点项目、集成项目,国家基础研究与发展计划(973)课题,国家自然科学基金委员会重大国际合作项目,国家科技支撑计划重点项目等30余项国家级课题. 发表SCI论文230余篇;论文及著作被他人引用8000余次;获国家专利30余项;作为负责人获国家科技进步二等奖2项, 国家技术发明奖二等奖1项,省部级科技进步一等奖6项.

致 谢

The authors have declared that no competing interests exist.


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. 航空学报, 34: 2757-2767

DOI      URL      [本文引用: 1]      摘要

A stochastic model updating (SMU) method using distance discrimination analysis and random sampling technique is proposed and subsequently applied to the updating process of the GARTEUR benchmark structure. In contrast to the traditional deterministic model updating procedure in which parameters are calibrated by sensitivity and optimization analysis, the proposed SMU method takes into consideration uncertainties which are general in the modeling as well as test processes. Uncertainty propagation is performed by Monte Carlo sampling method in which a large scale stochastic sampling process is proposed to describe uncertainties from parameters to features. Distance discrimination analysis is presented to quantify the degree of similarity and dissimilarity between analytical and test data. Input parameters are calibrated to the test data through an iterative procedure integrating the above uncertainty propagation and quantification methods. In order to reduce calculation cost, a metamodel is constructed using radial basis function with an acceptable precision. The relative PCL program of MSC.Patran is employed to submit multiple finite element (FE) analyses and to extract information for subsequent analysis. An application is performed on the GARTEUR structure and the updating results are assessed by the widely accepted 3-steps validation criteria. The updating and validation results show the proposed SMU method is valid and effective in engineering application.

(Bi S F, Deng Z M.2013.

Stochastic model updating and validation of the GARTEUR structure based on random sampling and distance discrimination

. Acta Aeronautica et Astronautica Sinica, 34: 2757-2767).

DOI      URL      [本文引用: 1]      摘要

A stochastic model updating (SMU) method using distance discrimination analysis and random sampling technique is proposed and subsequently applied to the updating process of the GARTEUR benchmark structure. In contrast to the traditional deterministic model updating procedure in which parameters are calibrated by sensitivity and optimization analysis, the proposed SMU method takes into consideration uncertainties which are general in the modeling as well as test processes. Uncertainty propagation is performed by Monte Carlo sampling method in which a large scale stochastic sampling process is proposed to describe uncertainties from parameters to features. Distance discrimination analysis is presented to quantify the degree of similarity and dissimilarity between analytical and test data. Input parameters are calibrated to the test data through an iterative procedure integrating the above uncertainty propagation and quantification methods. In order to reduce calculation cost, a metamodel is constructed using radial basis function with an acceptable precision. The relative PCL program of MSC.Patran is employed to submit multiple finite element (FE) analyses and to extract information for subsequent analysis. An application is performed on the GARTEUR structure and the updating results are assessed by the widely accepted 3-steps validation criteria. The updating and validation results show the proposed SMU method is valid and effective in engineering application.
[2] 崔飞, 杨党旗, 高岩. 2003.

用参数识别技术进行桥梁结构损伤识别

. 中国铁道科学, 24: 37-40

DOI      URL      [本文引用: 1]      摘要

以有限元为工具,把桥梁结构模型中各单元的等效面积、惯性矩以及板壳单元的厚度作为识别参数,建立识别参数对于各种量测的灵敏度矩阵.测取结构某些部位的位移与应变,以此为基准与原先结构的分析结果进行比较,建立综合误差向量.通过优化方法调整当前计算模型的参数,使结构响应与相应的试验值最大程度地吻合,从而得到结构参数变化的信息.以此为基础实现桥梁结构的损伤判别.

(Cui F, Yang D Q, Gao Y.2003.

Bridge structural damage assessment by parameter identification

. China Railway Science, 24: 37-40).

DOI      URL      [本文引用: 1]      摘要

以有限元为工具,把桥梁结构模型中各单元的等效面积、惯性矩以及板壳单元的厚度作为识别参数,建立识别参数对于各种量测的灵敏度矩阵.测取结构某些部位的位移与应变,以此为基准与原先结构的分析结果进行比较,建立综合误差向量.通过优化方法调整当前计算模型的参数,使结构响应与相应的试验值最大程度地吻合,从而得到结构参数变化的信息.以此为基础实现桥梁结构的损伤判别.
[3] 陈志国, 邓忠民, 毕司峰. 2013.

基于Monte Carlo法的结构动力学模型确认

. 振动与冲击, 32: 76-81

DOI      URL      [本文引用: 1]      摘要

提出一种蒙特卡洛模拟(MCS)与多元回归分析相结合的有限元模型确认方法。采用基于动态响应面法生成的代理模型改进直接MCS,用一种逐步迭代的方式改进代理模型保证模型确认的精度。传统欧氏距离/马氏距离从不同方面描述了两点间距离,然而如果在距离准则中能够同时考虑两者则会更全面。在相关性分析过程中,综合欧氏/马氏距离的特点,采用了一种欧氏/马氏距离相结合的不确定量化方法(距离判别方法),并给出了基于此指标的迭代收敛判断准则。算例仿真结果表明:所提出的有限元模型确认方法和此距离判别方法的使用,能比较大地降低MCS的计算量并能得到满意的模型确认结果,迭代收敛的稳定性和精度都有了提高。

(Chen Z G, Deng Z M, Bi S F.2013.

Structural dynamics model validation based on monte carlo method

. Journal of Vibration and Shock, 32: 76-81).

DOI      URL      [本文引用: 1]      摘要

提出一种蒙特卡洛模拟(MCS)与多元回归分析相结合的有限元模型确认方法。采用基于动态响应面法生成的代理模型改进直接MCS,用一种逐步迭代的方式改进代理模型保证模型确认的精度。传统欧氏距离/马氏距离从不同方面描述了两点间距离,然而如果在距离准则中能够同时考虑两者则会更全面。在相关性分析过程中,综合欧氏/马氏距离的特点,采用了一种欧氏/马氏距离相结合的不确定量化方法(距离判别方法),并给出了基于此指标的迭代收敛判断准则。算例仿真结果表明:所提出的有限元模型确认方法和此距离判别方法的使用,能比较大地降低MCS的计算量并能得到满意的模型确认结果,迭代收敛的稳定性和精度都有了提高。
[4] 戴航, 袁爱民. 2011. 基于灵敏度分析的结构模型修正. 北京: 科学出版社

[本文引用: 1]     

(Dai H, Yuan A M.2011. Structural model updating based on sensitivity analysis. Beijing: Science Press).

[本文引用: 1]     

[5] 丁继锋, 马兴瑞, 韩增尧, 庞世伟. 2010.

结构动力学模型修正的三步策略及其实践

. 航空学报, 31: 546-552

URL      [本文引用: 1]      摘要

首先构建了由模型结构调整、模型参数修正以及模型确认组成的三步模型修正策略。该方法优于传统模型修正方法的是:在模型修正之初基于误差定位、灵敏度分析以及工程经验进行的模型结构调整可以给出一个适于参数修正的初始有限元模型,从而保证了模型修正的成功。然后,采用三步法针对国际上模型修正的标准考题——GARTEUR19结构动力学模型进行修正,详尽论述了模型结构调整、参数修正以及模型确认的过程,并将修正结果与国外同行的研究结果进行了对比,综合精度与国际先进水平相当,从而验证了三步模型修正策略的有效性。

(Ding J F, Ma X R, Han Z Y, Pang S W.2010.

Three-step model updating method in structure dynamics and its application

. Acta Aeronautica et Astronautica Sinica, 31: 546-552).

URL      [本文引用: 1]      摘要

首先构建了由模型结构调整、模型参数修正以及模型确认组成的三步模型修正策略。该方法优于传统模型修正方法的是:在模型修正之初基于误差定位、灵敏度分析以及工程经验进行的模型结构调整可以给出一个适于参数修正的初始有限元模型,从而保证了模型修正的成功。然后,采用三步法针对国际上模型修正的标准考题——GARTEUR19结构动力学模型进行修正,详尽论述了模型结构调整、参数修正以及模型确认的过程,并将修正结果与国外同行的研究结果进行了对比,综合精度与国际先进水平相当,从而验证了三步模型修正策略的有效性。
[6] 范立础, 袁万城, 张启伟. 2000.

悬索桥结构基于敏感性分析的动力有限元模型修正

. 土木工程学报, 33: 9-14

DOI      URL      [本文引用: 1]      摘要

提出一种悬索桥结构基于特征值敏感性分析的有限元模型修正方法。由结构特征值关于模型参数的 灵敏度矩阵建立理论/试验频率差与模型参数摄动之间的关系式。根据待修正参数的不确定性,将参数摄动限制于预设的范围,并将参数修正转化为不等式约束优化 问题迭代求解。通过一1/150比例的悬索桥模型的试验分析,验证了这一模型修正方法的可行性。

(Fan L C, Yuan W C, Zhang Q W.2000.

Sensitivity-based FE model updating of a suspension bridge

. China Civil Engineering Journal, 33: 9-14).

DOI      URL      [本文引用: 1]      摘要

提出一种悬索桥结构基于特征值敏感性分析的有限元模型修正方法。由结构特征值关于模型参数的 灵敏度矩阵建立理论/试验频率差与模型参数摄动之间的关系式。根据待修正参数的不确定性,将参数摄动限制于预设的范围,并将参数修正转化为不等式约束优化 问题迭代求解。通过一1/150比例的悬索桥模型的试验分析,验证了这一模型修正方法的可行性。
[7] 费庆国, 李爱群, 张令弥. 2005.

基于神经网络的非线性结构有限元模型修正研究

. 宇航学报, 26: 267-269, 281

DOI      URL      [本文引用: 2]      摘要

现有的动态有限元模型修正方法几乎都是建立在线性假设基础之上,修正中利用固有频率等线性系统特征量.工程中,真实的结构振动系统都是非线性的.虽然在许多情况下,线性化假设获得的结果能够较为准确地反映真实系统的特性.但是,在结构的非线性特征较为明显时,必须考虑非线性因素,这时,现有的模型修正方法将不再适用.现以非线性梁为研究对象,采用基于神经网络的修正方法探索了非线性结构的有限元模型修正问题.仿真研究中利用有限元分析的响应数据训练神经网络.修正结果表明,包括非线性弹簧刚度系数在内的三个设计参数修正后误差均在1%以内.说明基于神经网络的有限元模型修正方法适用于解决非线性结构的有限元模型修正问题.

(Fei Q G, Li A Q, Zhang L M. 2005.

Study on finite element model updating of nonlinear structures using neural network

. Acta Aeronautica et Astronautica Sinica, 26: 267-269, 281).

DOI      URL      [本文引用: 2]      摘要

现有的动态有限元模型修正方法几乎都是建立在线性假设基础之上,修正中利用固有频率等线性系统特征量.工程中,真实的结构振动系统都是非线性的.虽然在许多情况下,线性化假设获得的结果能够较为准确地反映真实系统的特性.但是,在结构的非线性特征较为明显时,必须考虑非线性因素,这时,现有的模型修正方法将不再适用.现以非线性梁为研究对象,采用基于神经网络的修正方法探索了非线性结构的有限元模型修正问题.仿真研究中利用有限元分析的响应数据训练神经网络.修正结果表明,包括非线性弹簧刚度系数在内的三个设计参数修正后误差均在1%以内.说明基于神经网络的有限元模型修正方法适用于解决非线性结构的有限元模型修正问题.
[8] 费庆国, 张令弥. 2004.

基于径向基神经网络的有限元模型修正研究

. 南京航空航天大学学报, 36: 748-752

DOI      URL      [本文引用: 2]      摘要

设计参数型有限元模型修正属于结构动力学反问题,其理论基础是将结构的特征量视为设计参数的函数,然后依据特征量对设计参数的一阶导数信息进行迭代求解.本文提出了一种基于径向基神经网络的有限元模型修正方法,把模型修正归结为正问题进行研究.首先将特征量视为自变量,设计参数视为因变量,以径向基神经网络逼近两者之间的非线性映射关系,然后利用神经网络的泛化特性直接求解设计参数的目标值.不但无需迭代求解,而且避开了反问题所面临的复杂的非线性优化计算.GARTEUR飞机模型仿真研究的结果表明,修正后设计参数误差在2%以内,模态频率误差在1%以内.

(Fei Q G, Zhang L M.2004.

Finite element model updating using radial basis function neutral network

. Journal of Nanjing University of Aeronautics and Austronautics, 36: 748-752).

DOI      URL      [本文引用: 2]      摘要

设计参数型有限元模型修正属于结构动力学反问题,其理论基础是将结构的特征量视为设计参数的函数,然后依据特征量对设计参数的一阶导数信息进行迭代求解.本文提出了一种基于径向基神经网络的有限元模型修正方法,把模型修正归结为正问题进行研究.首先将特征量视为自变量,设计参数视为因变量,以径向基神经网络逼近两者之间的非线性映射关系,然后利用神经网络的泛化特性直接求解设计参数的目标值.不但无需迭代求解,而且避开了反问题所面临的复杂的非线性优化计算.GARTEUR飞机模型仿真研究的结果表明,修正后设计参数误差在2%以内,模态频率误差在1%以内.
[9] 费庆国, 张令弥, 郭勤涛. 2004.

GARTEUR有限元模型修正与确认研究

. 航空学报, 25: 372-375

DOI      URL      摘要

Parameter selection and quality validation are of great importance in finite element model updating. This paper presents some results which demonstrate the relationship between parameter selection and updated model's quality through simulation cases. Three quality levels with corresponding validation criteria are employed with an emphasis on updated model's prediction ability. Results of updating based on experimental modal test data are shown as an application example. An aircraft test structure, GARTEUR, which is generally utilized in Europe, is employed in both the simulation case and the experimental case. Sensitivity-based model updating approach is applied.

(Fei Q G, Zhang L M, Guo Q T.2004.

Case study of FE model updating and validation via an aircraft model structure

. Acta Aeronautica et Astronautica Sinica, 25: 372-375).

DOI      URL      摘要

Parameter selection and quality validation are of great importance in finite element model updating. This paper presents some results which demonstrate the relationship between parameter selection and updated model's quality through simulation cases. Three quality levels with corresponding validation criteria are employed with an emphasis on updated model's prediction ability. Results of updating based on experimental modal test data are shown as an application example. An aircraft test structure, GARTEUR, which is generally utilized in Europe, is employed in both the simulation case and the experimental case. Sensitivity-based model updating approach is applied.
[10] 费庆国, 张令弥, 李爱群, 王彤. 2005.

基于不同残差的动态有限元模型修正的比较研究

. 振动与冲击, 24: 24-26,135

DOI      URL      摘要

基于模态试验结果的设计参数型动态有限元模型修正方法,较多地采用模态频率、反共振频率、模态振型以及振型相关系数等参数.分析了以上四种参数用于模型修正的优缺点,采用欧洲航空科技组织的基准模型-GARTEUR飞机模型,通过数值仿真对利用模态频率、模态频率加反共振频率、模态频率加振型以及模态频率加振型相关系数等不同残差的修正进行了比较研究.结果表明,模态频率与反共振频率是优选的两种参数.

(Fei Q G, Zhang L M, Li A Q, Wang T.2005.

Evaluation of FE model updating using four kinds of residues

. Journal of Vibration and Shock, 24: 24-26,135).

DOI      URL      摘要

基于模态试验结果的设计参数型动态有限元模型修正方法,较多地采用模态频率、反共振频率、模态振型以及振型相关系数等参数.分析了以上四种参数用于模型修正的优缺点,采用欧洲航空科技组织的基准模型-GARTEUR飞机模型,通过数值仿真对利用模态频率、模态频率加反共振频率、模态频率加振型以及模态频率加振型相关系数等不同残差的修正进行了比较研究.结果表明,模态频率与反共振频率是优选的两种参数.
[11] 方圣恩. 2010.

基于有限元模型修正的结构损伤识别方法研究. [博士论文]

. 长沙: 中南大学

DOI      URL      [本文引用: 1]      摘要

尽管有限元模型修正技术在过去20年间得到了广泛的研究和应用,但目前为止,仍有一些问题没有得到良好的解决。比如在修正复杂工程结构时,往往由于修正参数过多,容易导致病态的优化问题、优化过程收敛速度慢和计算量大等等。为此,本文提出了几种新的模型修正方法和算法,用于改善模型修正过程的优化性能并减少计算花费,最终更好地实现损伤识别的目的。文中一共提出了三种方法和算法:损伤参数化方法,基于人工智能算法的多目标优化算法和基于响应面方法的模型修正技术。前两种方法或算法仍属于传统的模型修正技术范畴;而第三种方法由于在优化迭代过程中采用了响应面模型来替代传统的有限元模型,因此可以看作是对传统方法的较大改进。首先,在损伤参数化方法上,本文建立了一种二维损伤函数,以参数化二维平面有限元模型的损伤分布,最终达到减少未知变量数目和构建状态良好的优化问题的目的。具体来说,可以将每个单元或子结构的校正系数通过全局损伤函数联系起来,使得优化过程中所要调整的参数变为全局损伤函数的系数,从而大幅降低待修正参数的数目。其次,在人工智能算法的应用上,本文将最近发展的粒子群优化算法和遗传算法相结合,形成了一种新的多目标优化算法,用于在多目标优化问题中实现快速搜索全局极值点的目的。具体应用时,粒子群优化易于编程实现且收敛速度快,但由于其单点中心的特性,使得它难以应用在损伤识别问题上,因为此类问题的优化搜索过程往往是在多约束解空间中进行的,需要的是具有良好全局搜索能力的算法;另一方面,遗传算法善于搜索全局极值点,但其在复杂问题上的搜索效率通常不高,需要相对长的时间以获取高质量的Pareto解前沿。因此,本文中将上述两种人工智能算法相结合,以期在多目标优化问题中获得快速、准确的收敛性能。此外,由文献查新结果可知,本文所提出的算法还是首次应用在基于多目标模型修正的损伤识别问题上。再次,基于对传统模型修正方法的改进目的,本文还提出了一种新的模型修正方法,即在反问题优化过程中采用响应面模型来替代有限元模型,以求在大大提高修正效率的同时,仍然能保证模型预测的精度。和传统的有限元模型修正方法相比,基于响应面模型的修正迭代过程无需构建灵敏度矩阵和计算有限元模型,且该方法易于编程实现、计算效率高;和基于神经网络的模型修正方法相比,响应面方法不仅在样本数上少很多,同时还可以提供显式的数学多项式表达式,有利于和损伤识别系统的其它模块进行接口。同时,本文还基于统计方差分析理论来量化待修正参数的重要性,为参数筛选提供定量依据,从而避免了传统经验判断的主观性和基于灵敏度分析方法的局限性。 此外,本文还首次提出了一种基于响应信号统计特征的新参数,即“功率模态振型”,用于工程结构的早期损伤检测。功率模态振型拥有和传统意义上的模态振型相似的几何形状,但前者是基于不同的原理来构造的,其间并没有利用到模态参数提取技术,为的是在复杂模态情况下可以避免模态参数的提取过程,并增强损伤指标对随机噪声的鲁棒性。同时,由功率模态振型可以派生出“功率模态振型曲率”和“功率柔度”两个参数,并由此定义2个损伤指标用于损伤定位的目的。 最后,在方法的验证上,本文采用了不同的数值和试验结构,包括一榀试验钢筋混凝土框架和一座实桥。研究结果证实了本文的方法在损伤识别上可行性和可靠性,并揭示了它们在相关领域的进一步潜在应用。

(Fang S E.2010.

Studies on structural damage detection by finite element model updating. [PhD Thesis]

. Changsha: Central South University).

DOI      URL      [本文引用: 1]      摘要

尽管有限元模型修正技术在过去20年间得到了广泛的研究和应用,但目前为止,仍有一些问题没有得到良好的解决。比如在修正复杂工程结构时,往往由于修正参数过多,容易导致病态的优化问题、优化过程收敛速度慢和计算量大等等。为此,本文提出了几种新的模型修正方法和算法,用于改善模型修正过程的优化性能并减少计算花费,最终更好地实现损伤识别的目的。文中一共提出了三种方法和算法:损伤参数化方法,基于人工智能算法的多目标优化算法和基于响应面方法的模型修正技术。前两种方法或算法仍属于传统的模型修正技术范畴;而第三种方法由于在优化迭代过程中采用了响应面模型来替代传统的有限元模型,因此可以看作是对传统方法的较大改进。首先,在损伤参数化方法上,本文建立了一种二维损伤函数,以参数化二维平面有限元模型的损伤分布,最终达到减少未知变量数目和构建状态良好的优化问题的目的。具体来说,可以将每个单元或子结构的校正系数通过全局损伤函数联系起来,使得优化过程中所要调整的参数变为全局损伤函数的系数,从而大幅降低待修正参数的数目。其次,在人工智能算法的应用上,本文将最近发展的粒子群优化算法和遗传算法相结合,形成了一种新的多目标优化算法,用于在多目标优化问题中实现快速搜索全局极值点的目的。具体应用时,粒子群优化易于编程实现且收敛速度快,但由于其单点中心的特性,使得它难以应用在损伤识别问题上,因为此类问题的优化搜索过程往往是在多约束解空间中进行的,需要的是具有良好全局搜索能力的算法;另一方面,遗传算法善于搜索全局极值点,但其在复杂问题上的搜索效率通常不高,需要相对长的时间以获取高质量的Pareto解前沿。因此,本文中将上述两种人工智能算法相结合,以期在多目标优化问题中获得快速、准确的收敛性能。此外,由文献查新结果可知,本文所提出的算法还是首次应用在基于多目标模型修正的损伤识别问题上。再次,基于对传统模型修正方法的改进目的,本文还提出了一种新的模型修正方法,即在反问题优化过程中采用响应面模型来替代有限元模型,以求在大大提高修正效率的同时,仍然能保证模型预测的精度。和传统的有限元模型修正方法相比,基于响应面模型的修正迭代过程无需构建灵敏度矩阵和计算有限元模型,且该方法易于编程实现、计算效率高;和基于神经网络的模型修正方法相比,响应面方法不仅在样本数上少很多,同时还可以提供显式的数学多项式表达式,有利于和损伤识别系统的其它模块进行接口。同时,本文还基于统计方差分析理论来量化待修正参数的重要性,为参数筛选提供定量依据,从而避免了传统经验判断的主观性和基于灵敏度分析方法的局限性。 此外,本文还首次提出了一种基于响应信号统计特征的新参数,即“功率模态振型”,用于工程结构的早期损伤检测。功率模态振型拥有和传统意义上的模态振型相似的几何形状,但前者是基于不同的原理来构造的,其间并没有利用到模态参数提取技术,为的是在复杂模态情况下可以避免模态参数的提取过程,并增强损伤指标对随机噪声的鲁棒性。同时,由功率模态振型可以派生出“功率模态振型曲率”和“功率柔度”两个参数,并由此定义2个损伤指标用于损伤定位的目的。 最后,在方法的验证上,本文采用了不同的数值和试验结构,包括一榀试验钢筋混凝土框架和一座实桥。研究结果证实了本文的方法在损伤识别上可行性和可靠性,并揭示了它们在相关领域的进一步潜在应用。
[12] 郭勤涛, 张令弥. 2005.

结构动力学有限元模型确认方法研究

. 应用力学学报, 22: 572-578, 678

DOI      URL      [本文引用: 1]      摘要

随着结构动力学求解问题的复杂化,有限元分析方法越来越起着关键作用。由于许多结构系统本身存在不确定性因素,试验数据存在随机误差,而计算的三类误差也会包含着不确定性误差,如何用有限的试验来修正和检验计算模型,最后得到具有一定置信度的有限元模型,即模型确认,在工程领域越来越得到关注。在与模型修正比较的基础上,详细讨论了模型确认的主要研究内容,并结合两个应用实例,讨论模型确认的总体思路与实现方法。

(Guo Qintao, Zhang Lingmi.2005.

Fnite element model validation in structural dynamics

. Chinese Journal of Applied Mechanics, 22: 572-578, 678).

DOI      URL      [本文引用: 1]      摘要

随着结构动力学求解问题的复杂化,有限元分析方法越来越起着关键作用。由于许多结构系统本身存在不确定性因素,试验数据存在随机误差,而计算的三类误差也会包含着不确定性误差,如何用有限的试验来修正和检验计算模型,最后得到具有一定置信度的有限元模型,即模型确认,在工程领域越来越得到关注。在与模型修正比较的基础上,详细讨论了模型确认的主要研究内容,并结合两个应用实例,讨论模型确认的总体思路与实现方法。
[13] 郭勤涛, 张令弥, 费庆国. 2006

a. 结构动力学有限元模型修正的发展------模型确认

. 力学进展, 36: 36-42

DOI      URL      [本文引用: 3]      摘要

结构动力学有限元建模精度问题是力学分支中受到广泛关注的研究课题之一,以提高建模精度为目标的有限元模型修正技术的发展日臻完善,并已逐步在工业界得到应用,然而模型修正技术未能全而解决建模精度中存在的问题。近几年以美国三大国家实验室为代表的科技人员提出了模型确认技术,以希望全面解决结构动力学建模精度问题.本文以模型修正为基础,讨论模型确认的相关问题以及与模型修正的关系。

(Guo Q T, Zhang L M, Fei Q G.2006

a. From FE model updating to model validation: advances in modeling of dynamic structures

. Advances in Mechanics, 36: 36-42).

DOI      URL      [本文引用: 3]      摘要

结构动力学有限元建模精度问题是力学分支中受到广泛关注的研究课题之一,以提高建模精度为目标的有限元模型修正技术的发展日臻完善,并已逐步在工业界得到应用,然而模型修正技术未能全而解决建模精度中存在的问题。近几年以美国三大国家实验室为代表的科技人员提出了模型确认技术,以希望全面解决结构动力学建模精度问题.本文以模型修正为基础,讨论模型确认的相关问题以及与模型修正的关系。
[14] 郭勤涛, 张令弥, 费庆国. 2006

b. 用于确定性计算仿真的响应面法及其试验设计研究

. 航空学报, 27: 55-61

DOI      URL      [本文引用: 1]      摘要

Deterministic computer simulation is widely used in science and technology. In order to illustrate the relationship between parameters and response features, Design of Experiment (DOE) based Response Surface(RS)method can be employed. This method can deal with large scale models used in design optimization, model updating and model validation. In this article, some modern DOE are introduced and developed to construct high order RS models. The proposed method is verified by studying several typical nonlinear test problems and FEA examples. The number of factors can be more than 16, and the order of the RS model can be more than 15.

(Guo Q T, Zhang L M, Fei Q G.2006

b. Response surface method and its experimental design for deterministric computer simulation

. Acta Aeronautica et Astronautica Sinica, 27: 55-61).

DOI      URL      [本文引用: 1]      摘要

Deterministic computer simulation is widely used in science and technology. In order to illustrate the relationship between parameters and response features, Design of Experiment (DOE) based Response Surface(RS)method can be employed. This method can deal with large scale models used in design optimization, model updating and model validation. In this article, some modern DOE are introduced and developed to construct high order RS models. The proposed method is verified by studying several typical nonlinear test problems and FEA examples. The number of factors can be more than 16, and the order of the RS model can be more than 15.
[15] 何成, 何欢, 陈国平. 2012.

基于分级思想的高温环境结构动力学模型修正

. 振动与冲击, 31: 145-150

URL      [本文引用: 1]      摘要

研究基于全局近似函数的高温环境结构动力学模型的修正问题.首先,对影响高温环境下的结构动力学模型的动特性因素进行分析,确定了在高温环境下结构表面温度分布和结构动响应之间近似满足独立关系,并在此基础上提出了一种分级修正思想,给出了分级修正基本流程.然后,以热参数为修正参数,构造了温度分布残差关于修正参数的全局近似函数,将模型修正问题转化为优化问题,对结构温度分布进行第一级模型修正.随后,在温度分布修正结果的基础上,考虑结构间隙对边界条件的影响,以边界接触刚度为修正参数,以高温环境下的结构固有频率为修正目标,对结构动力学模型进行第二级修正.最后,以高温环境下的机翼模型为研究对象,通过算例分析验证了本文提出的分级修正方法的有效性.

(He C, He H, Chen G P.2012.

Structural dynamic model updating in high-temperature environment based on a two stage modification

. Journal of Vibration and Shock, 31: 145-150).

URL      [本文引用: 1]      摘要

研究基于全局近似函数的高温环境结构动力学模型的修正问题.首先,对影响高温环境下的结构动力学模型的动特性因素进行分析,确定了在高温环境下结构表面温度分布和结构动响应之间近似满足独立关系,并在此基础上提出了一种分级修正思想,给出了分级修正基本流程.然后,以热参数为修正参数,构造了温度分布残差关于修正参数的全局近似函数,将模型修正问题转化为优化问题,对结构温度分布进行第一级模型修正.随后,在温度分布修正结果的基础上,考虑结构间隙对边界条件的影响,以边界接触刚度为修正参数,以高温环境下的结构固有频率为修正目标,对结构动力学模型进行第二级修正.最后,以高温环境下的机翼模型为研究对象,通过算例分析验证了本文提出的分级修正方法的有效性.
[16] 华宏星, 傅志方. 1997.

模糊数学在有限元模型修正中的应用

. 振动工程学报, 10: 42-47

URL      [本文引用: 1]      摘要

有限元模型修正的目标是使计算得到的结构动力学特性和实验结果相接近,其解往往不是唯一的,这是一个模糊的因具体问题而异的目标。修正过程中设计变量的选取,约束条件等在很大程度上也是模糊的。本文将模糊数学运用于基于灵敏度分析的有限元模型修正中,利用具有学习能力的算法,使有限元模型修正能合理地解决实际工程问题。

(Hua H X, Fu Z F.1997.

Application of Fuzzy Theory to Finite Element Model Updating

. Journal of Vibration Engineering, 10: 42-47).

URL      [本文引用: 1]      摘要

有限元模型修正的目标是使计算得到的结构动力学特性和实验结果相接近,其解往往不是唯一的,这是一个模糊的因具体问题而异的目标。修正过程中设计变量的选取,约束条件等在很大程度上也是模糊的。本文将模糊数学运用于基于灵敏度分析的有限元模型修正中,利用具有学习能力的算法,使有限元模型修正能合理地解决实际工程问题。
[17] 华宏星, 傅志方. 1998.

有限元模型修正中的BAYES方法的几点讨论

. 振动工程学报, 11: 113-118

DOI      URL      [本文引用: 1]      摘要

BAYES方程是有限元模型动力修正中常用的方法。本文从不同的途径推导了该统计估算公式,并讨论了它的有效性,无偏性,学习能力,自适应性和鲁棒性,以及和最小二乘法,卡尔门滤波器等算法的一些不同之处。该方法可以作为一个多目标,多设计变量的优化方法。

(Hua H X, Fu Z F.1998.

Discussion on the Bayes estimator used in finite element model updating

. Journal of Vibration Engineering, 11: 113-118).

DOI      URL      [本文引用: 1]      摘要

BAYES方程是有限元模型动力修正中常用的方法。本文从不同的途径推导了该统计估算公式,并讨论了它的有效性,无偏性,学习能力,自适应性和鲁棒性,以及和最小二乘法,卡尔门滤波器等算法的一些不同之处。该方法可以作为一个多目标,多设计变量的优化方法。
[18] 侯吉林. 2010.

约束子结构模型修正方法. [博士论文]

. 哈尔滨: 哈尔滨工业大学

[本文引用: 1]     

(Hou J L.2010.

Model updating based on substructure isolation methods. [PhD Thesis]

. Harbin: Harbin Institute of Technology).

[本文引用: 1]     

[19] 蒋寅军. 2011.

基于响应面方法的复杂结构模型修正方法研究. [博士论文]

. 武汉: 武汉大学

[本文引用: 2]     

(Jiang Y J.

Research on dynamic model updating technology for complex structures based on response surface methodology. [PhD Thesis]

.Wuhan: Wuhan University).

[本文引用: 2]     

[20] 李辉, 丁桦. 2005.

结构动力模型修正方法研究进展

. 力学进展, 35: 170-180

DOI      URL      [本文引用: 2]      摘要

叙述了结构动力模型修正方法的一般原理及与其密切相关的模型缩聚和模态扩展方法,并且挑选其中具有代表性的文献,介绍和比较了3种主要的修正方法,即传统的动力模型修正方法,包括矩阵型修正方法和参数型修正方法,和最近兴起的基于神经网络的模型修正方法,重点分析了这些方法的优点和不足之处,力图能使读者对于这一研究领域的发展有一个脉络清晰的了解.最后,就目前研究中尚未解决的问题作了一些探讨.

(Li H, Ding H.2005.

Progress in model updating for structural dynamics

. Advances in Mechanics, 35: 170-180).

DOI      URL      [本文引用: 2]      摘要

叙述了结构动力模型修正方法的一般原理及与其密切相关的模型缩聚和模态扩展方法,并且挑选其中具有代表性的文献,介绍和比较了3种主要的修正方法,即传统的动力模型修正方法,包括矩阵型修正方法和参数型修正方法,和最近兴起的基于神经网络的模型修正方法,重点分析了这些方法的优点和不足之处,力图能使读者对于这一研究领域的发展有一个脉络清晰的了解.最后,就目前研究中尚未解决的问题作了一些探讨.
[21] 李林, 朱宏平, 洪可柱. 2006.

基于神经网络方法的框架结构损伤检测的试验研究

. 振动与冲击, 25: 107-109, 121, 170

DOI      URL      [本文引用: 1]      摘要

首先建立了三层试验框架结构的 有限元模型,利用未损伤状态的动态测量数据,采用神经网络方法分步对原结构的有限元模型进行了修正。然后,依据修正的有限元模型,运用神经网络方法对各种 实际损伤状况进行了损伤诊断。比较了仅以三阶频率作为神经网络输入向量和三阶频率及一阶振型组合作为网络输入向量对网络训练和损伤检测结果的影响。研究表 明,神经网络的输入数据越充分,网络训练的收敛速度越快;利用三阶固有频率能够对该模型结构的各种损伤进行诊断,获得满意层间刚度识别的结果。

(Li L, Zhu H P, Hong K Z.2006.

Experimental study on damage detection of frame structure based on artificial neural network algorithm

. Journal of Vibration and Shock, 25: 107-109, 121, 170).

DOI      URL      [本文引用: 1]      摘要

首先建立了三层试验框架结构的 有限元模型,利用未损伤状态的动态测量数据,采用神经网络方法分步对原结构的有限元模型进行了修正。然后,依据修正的有限元模型,运用神经网络方法对各种 实际损伤状况进行了损伤诊断。比较了仅以三阶频率作为神经网络输入向量和三阶频率及一阶振型组合作为网络输入向量对网络训练和损伤检测结果的影响。研究表 明,神经网络的输入数据越充分,网络训练的收敛速度越快;利用三阶固有频率能够对该模型结构的各种损伤进行诊断,获得满意层间刚度识别的结果。
[22] 李伟明. 2011.

有限元模型修正方法及自由度匹配迭代技术研究. [博士论文]

. 上海: 上海交通大学

URL      [本文引用: 1]      摘要

本文对有限元模型修正领域中的修正方法及自由度匹配的迭代技术进行了深入的研究。 在工程中,通常使用有限元建模的方式来掌握结构的动力学特性。现阶段,航空、航天、大型桥梁以及其它工程结构对有限元模型的精度及可靠性提出了更高的要求。然而,结构的连接部位、边界条件以及结构阻尼的建模存在着很大的难度,其中的众多参数具有一定的不确定性,使得对复杂结构建立的有限元模型往往不精确。因此需要利用有限元模型修正技术使用实验测得的数据对有限元模型进行修正,提高有限元模型的精度及可靠性。 传统的基于灵敏度分析的模型修正方法及修正参数的选取方法尽管应用范围广泛,但是其往往只注重数学意义上的修正,存在着诸如修正参数有限、计算耗时、灵敏度大的待修正参数有时并不是真正存在建模误差的参数等不足之处。因此,有必要找到一种计算效率高,不依赖灵敏度分析,能够同时修正大量待修正参数,并且希望在修正过程中自动找到真正存在误差的参数的模型修正方法。研究发现正交模型正交模态(CMCM)法满足以上的修正需求,然而CMCM法的修正结果仅为某一单元的整体修正因子,对于某一单元存在多个待修正参数的情况则无法分别计算出各个参数的修正量,并且在修正过程中存在着分母为零或者逼近零的危险,这给修正工作带来了困难。本文在保持原有CMCM法优越性的基础上,通过公式重新推导、变形,得到了改进的正交模型正交模态法即ICMCM法。ICMCM法解决了CMCM法存在的病态问题,并且其修正对象可以是任意的材料以及几何参数,增加了待修正参数的数量。并在ICMCM法的基础上进行扩展,推导得到了含有阻尼形式的ICMCM法,扩宽了ICMCM法的应用范围。利用ICMCM法的思想,将其应用到频响型模型修正方法中,获得了新的频响型模型修正方法——正交模型正交频响函数(CMCFRF)法。通过算例证明了ICMCM法以及CMCFRF法的有效性及优越性。 另外,在使用传统的模型修正与自由度匹配相结合的迭代方法时,仅仅使用有限元模型的缩聚/扩展转换矩阵代替实验模型的缩聚/扩展转换矩阵,未对上述替代的误差进行额外的处理,因此造成使用传统的模型修正与自由度匹配相结合的迭代方法时模型修正的计算效率低,迭代收敛速度慢。针对此问题,本文进行了深入的研究并在传统迭代方法的基础提出了一种新的模型修正与模型缩聚结合应用的迭代方法即误差循环迭代缩聚(ECIMR)法以及新的模型修正与模态扩展结合应用的迭代方法即误差循环迭代扩展(ECIME)法。上述两种新的方法分别在传统迭代方法的基础上添加了误差修正项,大大加快了自由度匹配后模型修正的迭代收敛速度并减少了计算时间。ECIMR法在推导过程中未限制所使用的模型修正与模型缩聚的方法,ECIME法并未限制所使用的模型修正方法,因此二者分别具有一定的通用性。在修正过程中使用新的迭代方法较传统的迭代法具有更快的迭代收敛速度以及计算效率,这对于工程中大型、复杂结构的模型修正问题具有重要的工程应用价值,使用新的方法可以节省大量的计算时间。通过ICMCM法以及CMCFRF法证明了上述两种迭代方法的通用性以及有效性。另外,使用ECIMR法进行迭代后,在相同的迭代次数下,随着模型缩聚精度的提高,模型修正的精度也将随之提高。 为了能够进一步验证本文所提出的模型修正方法对复杂结构尤其是航天对象的模型修正能力,对评价模型修正技术的基准模型即GARTEUR benchmark模型以及某型号的火箭舱段模型进行了模型修正,并获得了非常理想的模型修正结果。修正结果表明模态扩展方法更加适合ICMCM法,此时修正的方程数量远大于使用模型缩聚方法,进而可以修正数量较大的参数,模型修正结果也远好于使用模型缩聚方法。使用ICMCM法所能修正的参数数量远大于基于灵敏度分析的模型修正方法所能修正的参数数量,尽可能包含所有真正存在建模误差的参数,并利用修正方程自动的对真正存在建模误差的参数进行修正,使得修正结果更加的符合结构真实的物理意义。

(Li Weiming.2011.

Study on finite element model updating method and iterative technique for degree of freedom matching. [PhD Thesis]

. Shanghai: Shanghai Jiao Tong University).

URL      [本文引用: 1]      摘要

本文对有限元模型修正领域中的修正方法及自由度匹配的迭代技术进行了深入的研究。 在工程中,通常使用有限元建模的方式来掌握结构的动力学特性。现阶段,航空、航天、大型桥梁以及其它工程结构对有限元模型的精度及可靠性提出了更高的要求。然而,结构的连接部位、边界条件以及结构阻尼的建模存在着很大的难度,其中的众多参数具有一定的不确定性,使得对复杂结构建立的有限元模型往往不精确。因此需要利用有限元模型修正技术使用实验测得的数据对有限元模型进行修正,提高有限元模型的精度及可靠性。 传统的基于灵敏度分析的模型修正方法及修正参数的选取方法尽管应用范围广泛,但是其往往只注重数学意义上的修正,存在着诸如修正参数有限、计算耗时、灵敏度大的待修正参数有时并不是真正存在建模误差的参数等不足之处。因此,有必要找到一种计算效率高,不依赖灵敏度分析,能够同时修正大量待修正参数,并且希望在修正过程中自动找到真正存在误差的参数的模型修正方法。研究发现正交模型正交模态(CMCM)法满足以上的修正需求,然而CMCM法的修正结果仅为某一单元的整体修正因子,对于某一单元存在多个待修正参数的情况则无法分别计算出各个参数的修正量,并且在修正过程中存在着分母为零或者逼近零的危险,这给修正工作带来了困难。本文在保持原有CMCM法优越性的基础上,通过公式重新推导、变形,得到了改进的正交模型正交模态法即ICMCM法。ICMCM法解决了CMCM法存在的病态问题,并且其修正对象可以是任意的材料以及几何参数,增加了待修正参数的数量。并在ICMCM法的基础上进行扩展,推导得到了含有阻尼形式的ICMCM法,扩宽了ICMCM法的应用范围。利用ICMCM法的思想,将其应用到频响型模型修正方法中,获得了新的频响型模型修正方法——正交模型正交频响函数(CMCFRF)法。通过算例证明了ICMCM法以及CMCFRF法的有效性及优越性。 另外,在使用传统的模型修正与自由度匹配相结合的迭代方法时,仅仅使用有限元模型的缩聚/扩展转换矩阵代替实验模型的缩聚/扩展转换矩阵,未对上述替代的误差进行额外的处理,因此造成使用传统的模型修正与自由度匹配相结合的迭代方法时模型修正的计算效率低,迭代收敛速度慢。针对此问题,本文进行了深入的研究并在传统迭代方法的基础提出了一种新的模型修正与模型缩聚结合应用的迭代方法即误差循环迭代缩聚(ECIMR)法以及新的模型修正与模态扩展结合应用的迭代方法即误差循环迭代扩展(ECIME)法。上述两种新的方法分别在传统迭代方法的基础上添加了误差修正项,大大加快了自由度匹配后模型修正的迭代收敛速度并减少了计算时间。ECIMR法在推导过程中未限制所使用的模型修正与模型缩聚的方法,ECIME法并未限制所使用的模型修正方法,因此二者分别具有一定的通用性。在修正过程中使用新的迭代方法较传统的迭代法具有更快的迭代收敛速度以及计算效率,这对于工程中大型、复杂结构的模型修正问题具有重要的工程应用价值,使用新的方法可以节省大量的计算时间。通过ICMCM法以及CMCFRF法证明了上述两种迭代方法的通用性以及有效性。另外,使用ECIMR法进行迭代后,在相同的迭代次数下,随着模型缩聚精度的提高,模型修正的精度也将随之提高。 为了能够进一步验证本文所提出的模型修正方法对复杂结构尤其是航天对象的模型修正能力,对评价模型修正技术的基准模型即GARTEUR benchmark模型以及某型号的火箭舱段模型进行了模型修正,并获得了非常理想的模型修正结果。修正结果表明模态扩展方法更加适合ICMCM法,此时修正的方程数量远大于使用模型缩聚方法,进而可以修正数量较大的参数,模型修正结果也远好于使用模型缩聚方法。使用ICMCM法所能修正的参数数量远大于基于灵敏度分析的模型修正方法所能修正的参数数量,尽可能包含所有真正存在建模误差的参数,并利用修正方程自动的对真正存在建模误差的参数进行修正,使得修正结果更加的符合结构真实的物理意义。
[23] 刘荣贺, 于开平. 2013.

基于基础激励试验数据的卫星结构有限元模型修正

. 宇航学报, 34: 1187-1194

DOI      URL      [本文引用: 1]      摘要

结合航天工程中卫星结构普遍进 行的振动台振动试验,提出了一种基于基础激励试验数据的设计参数型有限元模型修正方法,用数值差分代替灵敏度分析,提高了计算效率,整个修正步骤更加适合 工程应用。然后利用该算法对某型号卫星整星结构进行了修正,并提出了两步修正策略。修正后,有限元模型的加速度响应计算结果更加接近于试验测量值,均方根 误差也比修正前有了不同程度的降低,证实了该算法对卫星结构的修正是行之有效的。

(Liu R H, Yu K P.2013.

Model updating for a satellite structure based on basic excitation test data

. Journal of Astronautics, 34: 1187-1194).

DOI      URL      [本文引用: 1]      摘要

结合航天工程中卫星结构普遍进 行的振动台振动试验,提出了一种基于基础激励试验数据的设计参数型有限元模型修正方法,用数值差分代替灵敏度分析,提高了计算效率,整个修正步骤更加适合 工程应用。然后利用该算法对某型号卫星整星结构进行了修正,并提出了两步修正策略。修正后,有限元模型的加速度响应计算结果更加接近于试验测量值,均方根 误差也比修正前有了不同程度的降低,证实了该算法对卫星结构的修正是行之有效的。
[24] 刘宇飞. 2015.

基于模型修正与图像处理的多尺度结构损伤识别. [博士论文]

. 北京: 清华大学

URL      [本文引用: 1]      摘要

结构的损伤识别在既有结构的评价与维护中具有重要意义。论文结合结构动力指标与裂缝指标,利用基于频响函数灵敏度的模型修正方法进行结构整体损伤识别,利用基于图像处理与三维重建的裂缝识别技术进行结构局部损伤识别,形成由整体到局部、由宏观到微观的多尺度损伤识别方法。论文的主要研究成果有:(1)针对由整体到局部的多尺度损伤识别思路,论文利用模型修正的方法实现对整体结构的损伤定位及损伤程度初判;在此基础上,利用图像处理等方法实现局部裂缝的定量识别,从而实现了两种损伤识别方法在逻辑与操作上的衔接。(2)在基于频响函数灵敏度的模型修正中,针对参数敏感性分析、灵敏度方程组病态性、构造灵敏度方程的频响函数频率点选取、阻尼处理等问题展开系统讨论,给出了两类灵敏度指标,提出了综合考虑指标敏感性与灵敏度方程组病态性的频带选择指标,并结合数值算例对指标予以说明、验证。(3)在基于频响函数灵敏度的模型修正中,为处理数据不完备情况,提出了整体有限元模型缩聚、局部子结构频响函数扩展的两步自适应修正方法,实现了在测试数据严重不完备的情况下,对局部损伤的准确识别。通过四边固支有限元板的数值算例,验证了在测试数据严重不完备(约测得6%自由度)并存在噪声干扰的情况下,该损伤识别方法的有效性及准确性。(4)在基于图像处理的裂缝识别中,针对混凝土表面裂缝图像中可能存在的强噪声,提出了自适应的降噪算法。该方法结合背景减除损伤及Niblack’s二值化操作,实现在图像处理过程中维持裂缝形态不变的前提下,最优地消除噪声。同时,针对混凝土结构中常见的复杂裂缝形态,提出了复杂裂缝的分解编号、单独处理的方法,准确地完成裂缝宽度与长度的参数计算。利用多组试验对该算法的有效性及精度进行了验证。(5)在混凝土结构表面裂缝识别研究中,针对传统图像的识别方法中存在的问题,结合三维重建技术,提出了三维条件下的裂缝识别技术:通过裂缝体表面三维重建、二维图像裂缝参数识别、裂缝体三维并行投影等步骤,可实现裂缝图像的倾斜拍摄、多表面贯穿裂缝识别以及大面积裂缝的合并、集成识别。对该方法进行的试验验证表明其具有较强的应用价值。

(Liu Yunfei.2015.

Multi-scale structural damage assessment based on model updating and image processing. [PhD Thesis]

. Beijing: Tsinghua University).

URL      [本文引用: 1]      摘要

结构的损伤识别在既有结构的评价与维护中具有重要意义。论文结合结构动力指标与裂缝指标,利用基于频响函数灵敏度的模型修正方法进行结构整体损伤识别,利用基于图像处理与三维重建的裂缝识别技术进行结构局部损伤识别,形成由整体到局部、由宏观到微观的多尺度损伤识别方法。论文的主要研究成果有:(1)针对由整体到局部的多尺度损伤识别思路,论文利用模型修正的方法实现对整体结构的损伤定位及损伤程度初判;在此基础上,利用图像处理等方法实现局部裂缝的定量识别,从而实现了两种损伤识别方法在逻辑与操作上的衔接。(2)在基于频响函数灵敏度的模型修正中,针对参数敏感性分析、灵敏度方程组病态性、构造灵敏度方程的频响函数频率点选取、阻尼处理等问题展开系统讨论,给出了两类灵敏度指标,提出了综合考虑指标敏感性与灵敏度方程组病态性的频带选择指标,并结合数值算例对指标予以说明、验证。(3)在基于频响函数灵敏度的模型修正中,为处理数据不完备情况,提出了整体有限元模型缩聚、局部子结构频响函数扩展的两步自适应修正方法,实现了在测试数据严重不完备的情况下,对局部损伤的准确识别。通过四边固支有限元板的数值算例,验证了在测试数据严重不完备(约测得6%自由度)并存在噪声干扰的情况下,该损伤识别方法的有效性及准确性。(4)在基于图像处理的裂缝识别中,针对混凝土表面裂缝图像中可能存在的强噪声,提出了自适应的降噪算法。该方法结合背景减除损伤及Niblack’s二值化操作,实现在图像处理过程中维持裂缝形态不变的前提下,最优地消除噪声。同时,针对混凝土结构中常见的复杂裂缝形态,提出了复杂裂缝的分解编号、单独处理的方法,准确地完成裂缝宽度与长度的参数计算。利用多组试验对该算法的有效性及精度进行了验证。(5)在混凝土结构表面裂缝识别研究中,针对传统图像的识别方法中存在的问题,结合三维重建技术,提出了三维条件下的裂缝识别技术:通过裂缝体表面三维重建、二维图像裂缝参数识别、裂缝体三维并行投影等步骤,可实现裂缝图像的倾斜拍摄、多表面贯穿裂缝识别以及大面积裂缝的合并、集成识别。对该方法进行的试验验证表明其具有较强的应用价值。
[25] 任伟新, 陈华斌. 2008.

基于响应面的桥梁有限元模型修正

. 土木工程学报, 41: 73-78

DOI      URL      [本文引用: 1]      摘要

采用试验设计和回归分析方法,以显式的响应面模型逼近特征量与设计参数间复杂的隐式函数关系,得到简化的结构模型(Meta—model),给出有限元模型修正过程。针对复杂的土木工程结构,讨论样本选择、修正参数选取以及如何从众多因素中较合理地建立结构的响应面模型。用数值模拟算例和六跨连续梁桥环境振动试验结果,实现基于响应面模型的土木工程结构有限元模型修正,并与传统的基于灵敏度方法直接对结构有限元模型修正结果进行比较。结果表明,基于响应面方法的有限元模型修正和验证,能显著提高修正的效率,修正过程计算简洁、迭代收敛快,避开每次迭代都需要进行有限元计算,易于工程实际应用。

(Ren W X, Chen H B.2008.

Response-surface based on finite element model updating of bridge structures

. China Civil Engineering Journal, 41: 73-78).

DOI      URL      [本文引用: 1]      摘要

采用试验设计和回归分析方法,以显式的响应面模型逼近特征量与设计参数间复杂的隐式函数关系,得到简化的结构模型(Meta—model),给出有限元模型修正过程。针对复杂的土木工程结构,讨论样本选择、修正参数选取以及如何从众多因素中较合理地建立结构的响应面模型。用数值模拟算例和六跨连续梁桥环境振动试验结果,实现基于响应面模型的土木工程结构有限元模型修正,并与传统的基于灵敏度方法直接对结构有限元模型修正结果进行比较。结果表明,基于响应面方法的有限元模型修正和验证,能显著提高修正的效率,修正过程计算简洁、迭代收敛快,避开每次迭代都需要进行有限元计算,易于工程实际应用。
[26] 宋汉文, 王丽炜, 王文亮. 2003.

有限元模型修正中若干重要问题

. 振动与冲击, 22: 70-73, 81, 112

DOI      URL      [本文引用: 1]      摘要

本文论述线性常系数系统的有限元模型修正中的若干重要问题。这些问题里 :模态形状的映射 ,模型降阶 ,参数化和正则化 ,以及有限元模型修正中的贝叶斯概率方法

(Song H W, Wang L W, Wang W L.2003.

Several important problems for updating finite element model

. Journal of Vibration and Shock, 22: 70-73, 81, 112).

DOI      URL      [本文引用: 1]      摘要

本文论述线性常系数系统的有限元模型修正中的若干重要问题。这些问题里 :模态形状的映射 ,模型降阶 ,参数化和正则化 ,以及有限元模型修正中的贝叶斯概率方法
[27] 魏来生. 1998.

结构有限元动态模型修正方法综述

. 振动与冲击, 17: 46-49, 84, 92

URL      [本文引用: 1]      摘要

本文简要介绍和归纳了五种常用的动态模型修正方法对每种方法,都给出它的基本思路,目标函数和修正模型的求解过程,以及各阶段的主要计算公式,基于对各种方法的深入剖析,并结合作者本人的一些实际经验和应用体会,此文还揭示了各种修正方法之间的在联系及实际使用中应注意的问题。

(Wei Laisheng.1998.

A review on the updating methods of finite element model

. Journal of Vibration and Shock, 17: 46-49, 84, 92).

URL      [本文引用: 1]      摘要

本文简要介绍和归纳了五种常用的动态模型修正方法对每种方法,都给出它的基本思路,目标函数和修正模型的求解过程,以及各阶段的主要计算公式,基于对各种方法的深入剖析,并结合作者本人的一些实际经验和应用体会,此文还揭示了各种修正方法之间的在联系及实际使用中应注意的问题。
[28] 王勖成. 2003. 有限单元法. 北京: 清华大学出版社

[本文引用: 1]     

(Wang X C.2003. Finite element method. Beijing: Tsinghua University Press).

[本文引用: 1]     

[29] 王泽宇, 刘闯, 冯咬齐. 2010.

基于振动试验数据的有限元模型修正技术研究

. 航天器环境工程, 27: 472-476, 405

DOI      URL      [本文引用: 1]      摘要

文章针对航天器结构的有限元模型修正问题,提出利用振动试验数据对模型进行修正的方法。详细阐述了基于基础激励传递特性的模型修正方法的理论公式,对此方法的程序实现思路及修正程序与有限元分析软件NASTRAN的接口方式进行了详细的介绍。最后利用复杂桁架模型进行了仿真分析,初步验证了该方法的可行性。

(Wang Z Y, Liu C, Feng Y Q.2010.

A method to update FEA analytical model by using the shaker vibration test data

. Spacecraft Environment Engineering, 27: 472-476, 405).

DOI      URL      [本文引用: 1]      摘要

文章针对航天器结构的有限元模型修正问题,提出利用振动试验数据对模型进行修正的方法。详细阐述了基于基础激励传递特性的模型修正方法的理论公式,对此方法的程序实现思路及修正程序与有限元分析软件NASTRAN的接口方式进行了详细的介绍。最后利用复杂桁架模型进行了仿真分析,初步验证了该方法的可行性。
[30] 吴晓菊. 2009.

结构有限元模型修正综述

. 特种结构, 26: 39-45

DOI      URL      [本文引用: 1]      摘要

结构有限元分析模型计算的结构响应与实测响应之间不可避免地存在一定偏差。利用结构现场实测的响应信息修正其结构有限元分析模型,使得修正后结构有限元模型计算的响应值与试验值趋于一致,此过程即为结构有限元模型修正。本文对结构动力模型修正和静力模型修正的一般理论及其进展进行了综述,讨论了有限元模型修正中的若干重要技术问题。

(Wu X J.2009.

A review of finite element model updating

. Special Structures, 26: 39-45).

DOI      URL      [本文引用: 1]      摘要

结构有限元分析模型计算的结构响应与实测响应之间不可避免地存在一定偏差。利用结构现场实测的响应信息修正其结构有限元分析模型,使得修正后结构有限元模型计算的响应值与试验值趋于一致,此过程即为结构有限元模型修正。本文对结构动力模型修正和静力模型修正的一般理论及其进展进行了综述,讨论了有限元模型修正中的若干重要技术问题。
[31] 徐宜桂, 周轶尘, 王志华. 2000.

用神经网络方法修正悬索桥动力模型

. 振动工程学报, 13: 46-52

DOI      URL      [本文引用: 1]      摘要

提出了一种基于神经网络的结构动力模型修正方法。讨论了该法实施过程中的若干具体技术问题 ,如网络学习算法、训练样本生成、结构修正参数确定、边界条件的模拟以及结构振型变化的有效表征等。最后对某悬索桥模型进行了动力模型修正 ,数值结果表明该法是实用可行的

(Xu Y J, Zhou T C, Wang Z H.2000.

Updating of dynamic model suspension bridge using adaptive neural network method

. Journal of Vibration Engineering, 13: 46-52).

DOI      URL      [本文引用: 1]      摘要

提出了一种基于神经网络的结构动力模型修正方法。讨论了该法实施过程中的若干具体技术问题 ,如网络学习算法、训练样本生成、结构修正参数确定、边界条件的模拟以及结构振型变化的有效表征等。最后对某悬索桥模型进行了动力模型修正 ,数值结果表明该法是实用可行的
[32] 徐张明, 高天明, 沈荣瀛, 华宏星. 2002.

一种改进的利用频响函数进行有限元模型修正的方法

. 振动与冲击, 21: 45-47, 94

DOI      URL      [本文引用: 1]      摘要

在对机械结构的动态特性进行准确而可靠的预测时 ,有限元模型的设计参数的修正是很重要的。利用试验测试和预测的有限元模型计算得到的频响函数 (FRF) ,在结构动力缩聚技术的基础上 ,推导出了一种改进的基于频响函数的灵敏度分析的修正方程。数值实例研究结果表明该方法利用不完备的测量数据 ,也可在很宽的频率范围内 ,同时对多个参数进行修正 ,有限元模型修正解与真实结构参数完全吻合。本文的方法可适用于大型复杂结构的模型修正

(Xu Z M, Gao T M, Shen R Y, Hua H X.2002.

Improved finite element model updating method based on frequency response functions

. Journal of Vibration and Shock, 21: 45-47, 94).

DOI      URL      [本文引用: 1]      摘要

在对机械结构的动态特性进行准确而可靠的预测时 ,有限元模型的设计参数的修正是很重要的。利用试验测试和预测的有限元模型计算得到的频响函数 (FRF) ,在结构动力缩聚技术的基础上 ,推导出了一种改进的基于频响函数的灵敏度分析的修正方程。数值实例研究结果表明该方法利用不完备的测量数据 ,也可在很宽的频率范围内 ,同时对多个参数进行修正 ,有限元模型修正解与真实结构参数完全吻合。本文的方法可适用于大型复杂结构的模型修正
[33] 徐张明, 沈荣瀛, 华宏星. 2003.

基于频响函数相关性的灵敏度分析的有限元模型修正

. 机械强度, 25: 5-8

DOI      URL      [本文引用: 1]      摘要

有限元模型的修正对机械结构的动态特性进行准确而可靠的预测是很重要的.利用试验测试和预测的有限元模型计算得到的频响函数(FRF),引入两种频响函数相关性的判定标准,提出基于频响相关函数的灵敏度分析的修正方程.数值实例研究结果表明,该方法利用少量的测量数据,即使测试数据含附加噪声,也可在很宽的频率范围内得到接近真实结构的有限元模型修正解.本文的方法可适用于大型复杂结构的模型修正.

(Xu Z M, Shen R Y, Hua H X.2003.

Updating finite element model by the sensitivity analysis of FRF correlation functions

. Journal of Mechanical Strength, 25: 5-8).

DOI      URL      [本文引用: 1]      摘要

有限元模型的修正对机械结构的动态特性进行准确而可靠的预测是很重要的.利用试验测试和预测的有限元模型计算得到的频响函数(FRF),引入两种频响函数相关性的判定标准,提出基于频响相关函数的灵敏度分析的修正方程.数值实例研究结果表明,该方法利用少量的测量数据,即使测试数据含附加噪声,也可在很宽的频率范围内得到接近真实结构的有限元模型修正解.本文的方法可适用于大型复杂结构的模型修正.
[34] 于开平, 刘荣贺. 2013.

多族群粒子群优化算法飞行器结构模型修正

. 振动与冲击, 32: 79-83, 99

DOI      URL      [本文引用: 1]      摘要

针对粒子群优化(PSO)算法极易陷入局部最优的缺陷,提出了一种多族群粒子群优化算法(MRPSO),该算法具有较强的全局搜索能力,能极大地降低搜索陷入局部最优的概率.并将该算法引入到有限元模型修正中,对某型号飞行器结构进行了优化修正,修正后结构的固有频率都有了非常明显的改善,证实了MRPSO算法的有效性及工程应用价值.

(Yu K P, Liu R H.2013.

Model updating of a spacecraft structure based on MRPSO

. Journal of Vibration and Shock, 32: 79-83, 99).

DOI      URL      [本文引用: 1]      摘要

针对粒子群优化(PSO)算法极易陷入局部最优的缺陷,提出了一种多族群粒子群优化算法(MRPSO),该算法具有较强的全局搜索能力,能极大地降低搜索陷入局部最优的概率.并将该算法引入到有限元模型修正中,对某型号飞行器结构进行了优化修正,修正后结构的固有频率都有了非常明显的改善,证实了MRPSO算法的有效性及工程应用价值.
[35] 余岭, 万祖勇, 朱宏平, 徐德毅. 2006.

基于POS算法的结构模型修正与损伤检测

. 振动与冲击, 25: 37-39,53,190

DOI      URL      [本文引用: 1]      摘要

结构模型修正与损伤检测是结构健康监测过程中必须解决的多学科研究课题,常常转化为求解约束 优化问题。介绍粒子群优化(PSO)算法,并在此基础上利用带惯性权重因子的全局版PSO算法对结构模型修正和损伤检测等约束优化问题进行研究。通过两层 刚架单损伤和多损伤数值仿真以及三层建筑框架结构四种损伤试验研究,结果表明PSO算法对结构模型修正能够起到非常好的效果,采用PSO算法对结构损伤进 行检测不仅能够准确定位结构损伤而且能够有效识别损伤程度。由此可见,PSO算法应用于该领域的效果是显而易见的。

(Yu Ling, Wan Zuyong, Zhu Hongping, Xu D Y.2006.

Structural model updating and damage detection through particle swarm optimization

. Journal of Vibration and Shock, 25: 37-39,53,190).

DOI      URL      [本文引用: 1]      摘要

结构模型修正与损伤检测是结构健康监测过程中必须解决的多学科研究课题,常常转化为求解约束 优化问题。介绍粒子群优化(PSO)算法,并在此基础上利用带惯性权重因子的全局版PSO算法对结构模型修正和损伤检测等约束优化问题进行研究。通过两层 刚架单损伤和多损伤数值仿真以及三层建筑框架结构四种损伤试验研究,结果表明PSO算法对结构模型修正能够起到非常好的效果,采用PSO算法对结构损伤进 行检测不仅能够准确定位结构损伤而且能够有效识别损伤程度。由此可见,PSO算法应用于该领域的效果是显而易见的。
[36] 袁平平, 王佐才, 任伟新. 2016.

基于动力响应主分量瞬时频率和幅值的非线性模型修正

. 振动工程学报, 29: 887-893

DOI      URL      [本文引用: 2]      摘要

提出了一种基于动力响应主分量的瞬时频率和幅值的非线性模型修正方法。首先通过解析模式分解和希尔伯特变换提取结构动力响应主分量的瞬时频率和瞬时幅值;然后选取瞬时幅值和瞬时频率慢变部分的有限多个时间点值来反映结构的非线性特性;最后采用响应面模型,并基于试验数据和有限元模型数据之间瞬时幅值和频率的残差建立目标函数,进行结构的非线性模型修正。通过三层框架的数值模拟分析,其结果表明该方法能精确有效地修正非线性结构模型。

(Yuan Pingping, Wang Zuocai, Ren Weixin.2016.

Nonlinear model updating based on instantaneous frequencies and amplitudes of principle dynamic response components

. Journal of Vibration Engineering, 29: 887-893).

DOI      URL      [本文引用: 2]      摘要

提出了一种基于动力响应主分量的瞬时频率和幅值的非线性模型修正方法。首先通过解析模式分解和希尔伯特变换提取结构动力响应主分量的瞬时频率和瞬时幅值;然后选取瞬时幅值和瞬时频率慢变部分的有限多个时间点值来反映结构的非线性特性;最后采用响应面模型,并基于试验数据和有限元模型数据之间瞬时幅值和频率的残差建立目标函数,进行结构的非线性模型修正。通过三层框架的数值模拟分析,其结果表明该方法能精确有效地修正非线性结构模型。
[37] 郁胜, 周林仁, 欧进萍. 2014.

基于径向基函数响应面方法的超大跨悬索桥有限元模型修正

. 铁道科学与工程学报, 11: 1-9

DOI      URL      [本文引用: 1]      摘要

基于径向基函数响应面建模和遗传算法相结合的有限元模型修正方法,得到能反映结构真实状态的 超大悬索桥有限元分析模型.首先利用灵敏度分析方法选取待修正参数和可用的特征量信息,以中心复合试验设计方法构造不同摄动水平下的待修正参数样本,通过 有限元模型的静动力分析计算不同参数水平下的特征量样本;然后,以待修正参数样本和特征量样本为结构系统输入和输出,建立能逼近大型结构系统设计参数与特 征量之间复杂隐式函数关系的径向基响应面模型;最后,基于建立的响应面模型,采用遗传优化算法对结构有限元模型进行修正.以国内湖南某超大跨钢桁架悬索桥 为对象,采用高斯径向基函数响应面模型,基于成桥静动力试验监测数据,对该桥的三维有限元模型进行修正.研究结果表明:修正后的有限元模型能够更真实地反 映结构的物理状态,较好地体现了该桥梁结构的真实静动力特性.

(Yu Sheng, Zhou Linren, Ou Jinping.2014.

Finite element model updating of large suspension bridge based on radial basis function response surface

. Journal of Railway Science and Engineering, 11: 1-9).

DOI      URL      [本文引用: 1]      摘要

基于径向基函数响应面建模和遗传算法相结合的有限元模型修正方法,得到能反映结构真实状态的 超大悬索桥有限元分析模型.首先利用灵敏度分析方法选取待修正参数和可用的特征量信息,以中心复合试验设计方法构造不同摄动水平下的待修正参数样本,通过 有限元模型的静动力分析计算不同参数水平下的特征量样本;然后,以待修正参数样本和特征量样本为结构系统输入和输出,建立能逼近大型结构系统设计参数与特 征量之间复杂隐式函数关系的径向基响应面模型;最后,基于建立的响应面模型,采用遗传优化算法对结构有限元模型进行修正.以国内湖南某超大跨钢桁架悬索桥 为对象,采用高斯径向基函数响应面模型,基于成桥静动力试验监测数据,对该桥的三维有限元模型进行修正.研究结果表明:修正后的有限元模型能够更真实地反 映结构的物理状态,较好地体现了该桥梁结构的真实静动力特性.
[38] 杨智春, 王乐, 李斌, 刘江华. 2009.

结构动力学有限元模型修正的目标函数及算法

. 应用力学学报, 26: 288-296, 408

URL      [本文引用: 2]      摘要

结构动力学有限元模型修正是结构动力学领域的一个热点问题,回顾了结构动力学有限元模型修正 研究的发展历史和现状,简要评述了结构动力学有限元模型修正所使用的设计变量,着重阐述了各种结构动力学有限元模型修正方法中所使用的目标函数及修正算 法,讨论了工程结构动力学有限元模型修正的一些策略,最后对结构动力学有限元模型修正技术的发展进行了总结和展望。

(Yang Zhichun, Wang Le, Li Bin, Liu J H.2009.

Objective functions and algorithms in structural dynamic finite element model updating

. Chinese Journal of Applied Mechanics, 26: 288-296, 408).

URL      [本文引用: 2]      摘要

结构动力学有限元模型修正是结构动力学领域的一个热点问题,回顾了结构动力学有限元模型修正 研究的发展历史和现状,简要评述了结构动力学有限元模型修正所使用的设计变量,着重阐述了各种结构动力学有限元模型修正方法中所使用的目标函数及修正算 法,讨论了工程结构动力学有限元模型修正的一些策略,最后对结构动力学有限元模型修正技术的发展进行了总结和展望。
[39] 袁昭旭, 于开平. 2017.

高温环境下结构动力学模型修正方法研究

. 振动与冲击, 36: 171-180

DOI      URL      [本文引用: 1]      摘要

提出了一种新的基于贝叶斯参数估计的高温环境动力学模型修正方法,利用此方法对一复合材料层合板模型进行了修正.该方法的主旨在于,利用高温环境下的模态频率试验数据修正有限元模型.该方法的特点在于,可以通过参数和响应的离散度来描述其不确定性,从而更加合理地解决实际问题.在研究中,进行了高温环境下复合材料结构的动力学试验,通过试验得到其高温环境下的模态频率.以此为基础,对复合材料有限元模型进行修正.在修正前,对模型进行了灵敏度分析,探讨了参数与高温环境动力学响应之间的关系.根据试验所得数据和对待修正参数的预估设置其对应的离散度.利用自编译程序进行修正计算.通过修正,模型的温度一固有频率对应关系得到了明显的改善,证明了方法的有效性,可以适用于工程实际问题.

(Yuan Z X, Yu K P.2017.

Structural dynamic model updating under high temperature environment

. Journal of Vibration and Shock, 36: 171-180).

DOI      URL      [本文引用: 1]      摘要

提出了一种新的基于贝叶斯参数估计的高温环境动力学模型修正方法,利用此方法对一复合材料层合板模型进行了修正.该方法的主旨在于,利用高温环境下的模态频率试验数据修正有限元模型.该方法的特点在于,可以通过参数和响应的离散度来描述其不确定性,从而更加合理地解决实际问题.在研究中,进行了高温环境下复合材料结构的动力学试验,通过试验得到其高温环境下的模态频率.以此为基础,对复合材料有限元模型进行修正.在修正前,对模型进行了灵敏度分析,探讨了参数与高温环境动力学响应之间的关系.根据试验所得数据和对待修正参数的预估设置其对应的离散度.利用自编译程序进行修正计算.通过修正,模型的温度一固有频率对应关系得到了明显的改善,证明了方法的有效性,可以适用于工程实际问题.
[40] 朱安文, 曲广吉, 高耀南, 魏震松. 2002.

结构动力模型修正技术的发展

. 力学进展, 32: 337-348

DOI      URL      [本文引用: 1]      摘要

在计算机技术飞速发展之前,为了了解航天器在极端载荷情况下的力学行为,通常采用足尺结构星试验的办法.结构星试验方法用于结构的分析和力学行为预测,存在着耗资较大和周期较长的不足.随着数值分析技术的发展,用有限元分析结合模型修正技术代替大型试验已经成为可能.本文评述了自上个世纪70年代末期以来结构动力模型修正技术的发展,包括早期直接对总体矩阵的修正技术,还有从90年代初期发展起来的对矩阵元素或设计参数进行修正的技术.总结了元素型修正方法中的迭代法、优化法以及摄动法,其中包括考虑了试验误差的统计算法和大型复杂结构修正的遗传算法.介绍了模型修正技术中的自由度匹配技术、灵敏度分析技术和对模型修正方法的有效性进行检验的一些经验标准,归纳了目前模型修正技术还需要解决的一些关键技术问题.

(Zhu A W, Qu G J, Gao Y N, Wei Z S.2002.

A survey of the modifying techniques of structure dynamic models

. Advances in Mechanics, 32: 337-348).

DOI      URL      [本文引用: 1]      摘要

在计算机技术飞速发展之前,为了了解航天器在极端载荷情况下的力学行为,通常采用足尺结构星试验的办法.结构星试验方法用于结构的分析和力学行为预测,存在着耗资较大和周期较长的不足.随着数值分析技术的发展,用有限元分析结合模型修正技术代替大型试验已经成为可能.本文评述了自上个世纪70年代末期以来结构动力模型修正技术的发展,包括早期直接对总体矩阵的修正技术,还有从90年代初期发展起来的对矩阵元素或设计参数进行修正的技术.总结了元素型修正方法中的迭代法、优化法以及摄动法,其中包括考虑了试验误差的统计算法和大型复杂结构修正的遗传算法.介绍了模型修正技术中的自由度匹配技术、灵敏度分析技术和对模型修正方法的有效性进行检验的一些经验标准,归纳了目前模型修正技术还需要解决的一些关键技术问题.
[41] 张保强, 陈国平, 郭勤涛. 2011.

结构动力学模型确认问题的核密度估计方法

. 机械工程学报, 47: 29-36, 43

DOI      URL      [本文引用: 1]      摘要

将核密度估计方法成功用于解决结构动力学模型确认的挑战问题,进一步明确模型确认在结构动力学中的实施过程。在美国圣地亚国家实验室提出的结构动力学模型确认挑战问题中,由于子结构与整体结构的弱非线性连接以及子结构个体差异引起的统计特性并不满足标准的概率分布,因此采用核密度估计方法建立子结构的概率模型,并使用核主元分析进行降维处理来提高核密度估计的计算效率;在子结构概率模型的基础上,使用校准试验数据对模型的准确度进行定性验证,同时使用确认试验数据对模型的精度进行定量评估;最后把确认过的子结构模型用于整体认证结构的评估以及最后目标模型的预测中,得到了与其他研究者相一致的结果。研究表明核密度估计方法是一种解决结构动力学模型确认问题的有效方法。

(Zhang B Q, Chen G P, Guo Q T.2011.

Structural dynamic model validation problem solution using kernel density estimation method

. Journal of Vibration Engineering, 47: 29-36, 43).

DOI      URL      [本文引用: 1]      摘要

将核密度估计方法成功用于解决结构动力学模型确认的挑战问题,进一步明确模型确认在结构动力学中的实施过程。在美国圣地亚国家实验室提出的结构动力学模型确认挑战问题中,由于子结构与整体结构的弱非线性连接以及子结构个体差异引起的统计特性并不满足标准的概率分布,因此采用核密度估计方法建立子结构的概率模型,并使用核主元分析进行降维处理来提高核密度估计的计算效率;在子结构概率模型的基础上,使用校准试验数据对模型的准确度进行定性验证,同时使用确认试验数据对模型的精度进行定量评估;最后把确认过的子结构模型用于整体认证结构的评估以及最后目标模型的预测中,得到了与其他研究者相一致的结果。研究表明核密度估计方法是一种解决结构动力学模型确认问题的有效方法。
[42] 张冬冬, 郭勤涛. 2013.

Kriging响应面代理模型在有限元模型确认中的应用

. 振动与冲击, 32: 187-191

DOI      URL      [本文引用: 1]      摘要

结合Kriging理论,实现Kriging响应面代理模型在有限元模型确认过程中响应预测的应用.讨论模型验证与确认的基本思想,初步提出有限元模型确认流程;以Garteur benchmark飞机结构瞬态响应仿真为例,建立加速度响应最大值Kriging响应面,通过蒙特卡洛方法,实现有限元模型参数不确定性正向传递;采用核密度估计建立加速度响应最大值概率分布,计算响应量置信区间上下限.结果表明,Kriging响应面能准确对有限元模型响应进行预测,可为有限元模型确认过程提供很大便利.

(Zhang D D, Guo Q T.

Application of Kriging response surface in finite element model validation

. Journal of Vibration and Shock, 32: 187-191).

DOI      URL      [本文引用: 1]      摘要

结合Kriging理论,实现Kriging响应面代理模型在有限元模型确认过程中响应预测的应用.讨论模型验证与确认的基本思想,初步提出有限元模型确认流程;以Garteur benchmark飞机结构瞬态响应仿真为例,建立加速度响应最大值Kriging响应面,通过蒙特卡洛方法,实现有限元模型参数不确定性正向传递;采用核密度估计建立加速度响应最大值概率分布,计算响应量置信区间上下限.结果表明,Kriging响应面能准确对有限元模型响应进行预测,可为有限元模型确认过程提供很大便利.
[43] 张德文, 魏阜旋. 1999. 模型修正与破损诊断. 北京: 科学出版社

[本文引用: 1]     

(Zhang D W, Wei F X.1999. Model updating and damage detection. Beijing: Science Press).

[本文引用: 1]     

[44] 张令弥. 2002.

计算仿真与模型确认及在结构环境与强度中的应用

. 强度与环境, 29: 42-47, 56

DOI      URL      [本文引用: 2]      摘要

本文对计算仿真、模型确认和相关的确认试验作了综合评述,强调了其核心思想和主要方法,并通过两个武器装备研制计划实例,表明计算仿真和模型确认在大型复杂结构和复杂环境下的强度研究中的应用.

(Zhang L M.2002.

Computer simulation and model validation with application to strength and environment engineering

. Structure and Environment Engineering, 29: 42-47, 56)

DOI      URL      [本文引用: 2]      摘要

本文对计算仿真、模型确认和相关的确认试验作了综合评述,强调了其核心思想和主要方法,并通过两个武器装备研制计划实例,表明计算仿真和模型确认在大型复杂结构和复杂环境下的强度研究中的应用.
[45] 张令弥, 何柏庆. 1995.

利用试验数据的结构动力学数学模型修正统一方法

. 南京航空航天大学学报, 27: 33-41

URL      [本文引用: 1]      摘要

近20年来提出了一系列利用试验数据的结构动力学数学模型修正方法,用统一的观点来考查和比较各种不同的模型修正方法显得十分重要.本文提出一种统一方法,将数学模型修正表述为推广的最小二乘或贝叶斯系统识别问题,可通过优化方法求解。其中残差定义为由数学模型计算的动态参量和相应测试量、或其组合量之差.选择不同的残差量,如特征值、特征向量、特征方程、正交性条件、系统输入(力)、输出(响应)、频率响应等,可导出各种设计参数型数学模型修正方法.最后对由统一方法导出的各种数学模型修正算法进行了分析、比较和讨论.用统一方法推导的各种方法不仅涵盖了现有的主要设计参数型模型修正方法,而且还演绎出一些新的算法.

(Zhang L M, He B Q.1995.

Analytical models updating based on parameter correction using dynamic test data

. Journal of Nanjing University of Aeronautics &Austronautics, 27: 33-41).

URL      [本文引用: 1]      摘要

近20年来提出了一系列利用试验数据的结构动力学数学模型修正方法,用统一的观点来考查和比较各种不同的模型修正方法显得十分重要.本文提出一种统一方法,将数学模型修正表述为推广的最小二乘或贝叶斯系统识别问题,可通过优化方法求解。其中残差定义为由数学模型计算的动态参量和相应测试量、或其组合量之差.选择不同的残差量,如特征值、特征向量、特征方程、正交性条件、系统输入(力)、输出(响应)、频率响应等,可导出各种设计参数型数学模型修正方法.最后对由统一方法导出的各种数学模型修正算法进行了分析、比较和讨论.用统一方法推导的各种方法不仅涵盖了现有的主要设计参数型模型修正方法,而且还演绎出一些新的算法.
[46] 朱跃. 2010.

基于分层思想的复杂机械结构模型修正及确认技术研究. [博士论文]

. 南京: 南京航空航天大学

DOI      URL      [本文引用: 3]      摘要

基于传统灵敏度方法的模型修正技术,结合模型确认中分层思想,提出了分层模型修正技术,针对具有多个典型螺栓连接的框架结构,研究了模型确认的整体技术路线,开展模型确认关键环节——有限元模型不确定性量化及传递中若干关键问题的研究,为提高结构动力学领域中有限元模型的响应预报精度提供可靠的建模方法。 本文研究的主要内容如下: (1)针对复杂工程结构中连接多,连接参数变化较大,修正时目标难以收敛问题,提出分层模型修正技术修正复杂工程结构,首先研究目前有限元建模中螺栓模拟的方法,对DMIG和CHEXA单元进行了比较研究,展示了能够充分体现复杂机械结构特点的多个螺栓连接框架结构,结合同一层次模型确认广义数学模型,建立了分层修正的数学模型,基于螺栓连接框架结构探讨了分层模型修正方法和思路。 (2)针对模型确认中有限元模型不确定性量化及传递中关键问题——不确定性建模,首先从不确定性分类、不确定性参数分类开始,建立了不确定性参数与模型输出的广义数学模型;不考虑模型形式误差,针对误差参数的不确定性建模——响应面建模进行研究,提出引入统计学习理论中支持向量机建立响应面,避免常用多项式模型形式难以选择弊端,通过仿真算例和Garteur飞机模型展开支持向量回归机在不确定性建模中的应用研究;基于分层思想对复杂工程结构不确定性建模建立了数学框架,并针对螺栓连接框架结构建立子结构、零件级构件响应面;特别是对螺栓连接子结构中连接参数变化范围较大的特点,提出模糊分类结合支持向量机建立响应面,提出了样本简约原则,即用较少的样本点可建立高精度响应面模型。 (3)针对模型确认中有限元模型不确定性量化及传递中关键问题——有限元模型不确定性反向传递,采用正问题方法解决该问题,把支持向量机和联合概率密度计算方法相结合,把多输出问题转化为单输出问题,用成熟的单输出支持向量机建立实验概率密度样本和误差参数概率密度样本之间的映射关系,采用交叉选择方法获得高精度响应面,以螺栓连接框架结构为研究对象,用实验值的统计特性估计误差参数的统计特性。 (4)针对模型确认中有限元模型不确定性量化及传递中关键问题——有限元模型不确定性正向传递及确认比较,归纳了该问题中四个方面研究内容,主要研究了实验值置信度极限的模型确认准则,提出支持向量机建立响应预测误差和工况变化直接的映射关系,结合蒙特卡洛仿真,针对螺栓连接框架结构的子结构研究整个确认域中响应预测误差;在确认域中一点——即特定工况下,研究子结构连接不确定向整体结构传递方法,借鉴了多学科优化设计领域优化方法,不需做整体结构实验,基于灵敏度方法可把子结构不确定性传递至整体结构。 最后总结了本文各章内容,强调了本文的创新点和贡献之处,提出了三个层次的模型确认技术路线,并对下一步研究工作进行展望。

(Zhu Yue.2010.

Study on model updating and validation technology of complex mechanical structure based on hierarchical method. [PhD Thesis]

. Nanjing: Nanjing University of Aeronautics and Austronautics).

DOI      URL      [本文引用: 3]      摘要

基于传统灵敏度方法的模型修正技术,结合模型确认中分层思想,提出了分层模型修正技术,针对具有多个典型螺栓连接的框架结构,研究了模型确认的整体技术路线,开展模型确认关键环节——有限元模型不确定性量化及传递中若干关键问题的研究,为提高结构动力学领域中有限元模型的响应预报精度提供可靠的建模方法。 本文研究的主要内容如下: (1)针对复杂工程结构中连接多,连接参数变化较大,修正时目标难以收敛问题,提出分层模型修正技术修正复杂工程结构,首先研究目前有限元建模中螺栓模拟的方法,对DMIG和CHEXA单元进行了比较研究,展示了能够充分体现复杂机械结构特点的多个螺栓连接框架结构,结合同一层次模型确认广义数学模型,建立了分层修正的数学模型,基于螺栓连接框架结构探讨了分层模型修正方法和思路。 (2)针对模型确认中有限元模型不确定性量化及传递中关键问题——不确定性建模,首先从不确定性分类、不确定性参数分类开始,建立了不确定性参数与模型输出的广义数学模型;不考虑模型形式误差,针对误差参数的不确定性建模——响应面建模进行研究,提出引入统计学习理论中支持向量机建立响应面,避免常用多项式模型形式难以选择弊端,通过仿真算例和Garteur飞机模型展开支持向量回归机在不确定性建模中的应用研究;基于分层思想对复杂工程结构不确定性建模建立了数学框架,并针对螺栓连接框架结构建立子结构、零件级构件响应面;特别是对螺栓连接子结构中连接参数变化范围较大的特点,提出模糊分类结合支持向量机建立响应面,提出了样本简约原则,即用较少的样本点可建立高精度响应面模型。 (3)针对模型确认中有限元模型不确定性量化及传递中关键问题——有限元模型不确定性反向传递,采用正问题方法解决该问题,把支持向量机和联合概率密度计算方法相结合,把多输出问题转化为单输出问题,用成熟的单输出支持向量机建立实验概率密度样本和误差参数概率密度样本之间的映射关系,采用交叉选择方法获得高精度响应面,以螺栓连接框架结构为研究对象,用实验值的统计特性估计误差参数的统计特性。 (4)针对模型确认中有限元模型不确定性量化及传递中关键问题——有限元模型不确定性正向传递及确认比较,归纳了该问题中四个方面研究内容,主要研究了实验值置信度极限的模型确认准则,提出支持向量机建立响应预测误差和工况变化直接的映射关系,结合蒙特卡洛仿真,针对螺栓连接框架结构的子结构研究整个确认域中响应预测误差;在确认域中一点——即特定工况下,研究子结构连接不确定向整体结构传递方法,借鉴了多学科优化设计领域优化方法,不需做整体结构实验,基于灵敏度方法可把子结构不确定性传递至整体结构。 最后总结了本文各章内容,强调了本文的创新点和贡献之处,提出了三个层次的模型确认技术路线,并对下一步研究工作进行展望。
[47] 宗周红, 牛杰, 王浩. 2012.

基于模型确认的结构概率损伤识别方法研究进展

. 土木工程学报, 45: 121-130

URL      [本文引用: 2]      摘要

基于静动力特性改变的结构损伤识别方法在过去几十年中得到了长足发展。相对于确定性的损伤识别,基于概率统计理论框架的损伤识别方法能够更好地反应损伤问题的不确定性本质,是一种非常有发展潜力的损伤识别方法,有望解决当前大多数损伤识别方法对于测量误差和噪声敏感的问题,从统计意义上实现复杂工程结构健康监测早期损伤的诊断。在回顾结构确定性损伤识别方法的基础上,主要介绍不确定性损伤识别方法和有限元模型确认的研究进展,分析比较各种方法的优点和不足之处;最后对有待进一步研究的问题和基于模型确认的概率损伤识别方法发展趋势进行展望。

(Zong Z H, Niu J, Wang H.2012.

A review of structural damage identification methods based on the finite element model validation

. China Civil Engineering Journal, 45: 121-130).

URL      [本文引用: 2]      摘要

基于静动力特性改变的结构损伤识别方法在过去几十年中得到了长足发展。相对于确定性的损伤识别,基于概率统计理论框架的损伤识别方法能够更好地反应损伤问题的不确定性本质,是一种非常有发展潜力的损伤识别方法,有望解决当前大多数损伤识别方法对于测量误差和噪声敏感的问题,从统计意义上实现复杂工程结构健康监测早期损伤的诊断。在回顾结构确定性损伤识别方法的基础上,主要介绍不确定性损伤识别方法和有限元模型确认的研究进展,分析比较各种方法的优点和不足之处;最后对有待进一步研究的问题和基于模型确认的概率损伤识别方法发展趋势进行展望。
[48] 宗周红, 任伟新. 2012. 桥梁有限元模型修正和模型确认. 北京: 人民交通出版社

[本文引用: 6]     

(Zong Z H, Ren W X.2012. Finite element model updating and model validation of bridge structures. Beijing: China Communications Press).

[本文引用: 6]     

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//16th International Modal Analysis Conference, 1998, Santa Babara, USA, 2: 1191-1198.

URL      [本文引用: 1]      摘要

We present an overview of new research efforts underway at Sandia National Laboratories to understand the sources of uncertainty and error in computational structural dynamics and other physics simulations, and to quantify their effects on predictive accuracy. In order to establish confidence in computational simulations as these simulations move further from the established experimental database, a new approach to modeling and simulation validation is needed. In particular, when simulations are used to qualify the safety and reliability of systems, we believe that validation should be based upon a comprehensive quantification of uncertainties and errors from all phases of the modeling and simulation process. Uncertainty and error quantification is a two-step process, the first step being the identification of all uncertainty and error sources in each phase of modeling and simulation. The second step is the assessment and propagation of the most significant uncertainties and errors through the phases of the modeling and simulation process to the predicted response quantities. This paper outlines the phases of modeling and simulation, the distinction between uncertainty and error, and a categorization of uncertainty and error sources in each phase of modeling and simulation. We also address the question of how uncertainties in the form or structure of the model might be assessed using multiple models. Examples from linear structural dynamics are given to illustrate these concepts.
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Nonlinear model calibration of a shear wall building using time and frequency data features

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DOI      URL      [本文引用: 1]      摘要

61Nonlinear models of a shear wall structure are identified.61Effects of modeling errors and data features used in the identification are studied.61Certain data features provide better identification results.61For large modeling errors, addition of data does not significantly improve the results.
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DOI      URL      [本文引用: 1]      摘要

61A methodology is proposed for nonlinear FE model calibration of civil structures.61Parameters of material models are updated to match time-varying modal parameters.61Performance of method is evaluated through applications on a large-scale structure.61Updated models can accurately predict the response time histories.61Updated models are validated by predicting the response to other input excitations.
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DOI      URL      [本文引用: 1]      摘要

This paper proposes a method for finite element (FE) model updating of civil structures. The method is a hybrid global optimization algorithm combining simulated annealing (SA) with the unscented Kalman filter (UKF). The objective function in the optimization problem can be defined in the modal, time, or frequency domains. The algorithm improves the accuracy, convergence rate, and computational cost of the SA algorithm by local improvements of the accepted candidates though the UKF. The proposed methodology is validated using a mathematical function and numerically simulated response data from linear and nonlinear FE models of realistic three-dimensional structures.
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Key parameters in dynamic systems often change during their life cycle due to repair and replacement of parts or environmental changes. This paper presents a new approach to account for these changes by updating the system models. Current iterative methods developed to solve the model updating problem rely on minimisation techniques to find the set of model parameters that yield the best match between experimental and analytical responses. These minimisation procedures require considerable computation time, making the existing techniques infeasible for some applications, such as in an adaptive control scheme, correcting the model parameters as the system changes. The proposed approach uses frequency domain data and a neural network to estimate the updated parameters quickly, yielding a model representative of the measured data. Besides control-related applications, this may also be of use for manufacturing systems, where parameters change during operation requiring repeated updates of the nominal model. Numerical simulations and experimental results show that the neural network updating method (NNUM) has good accuracy and generalisation properties, and it is therefore a suitable alternative for the solution of the model updating problem of this class of systems
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Abstract Previously proposed stiffness and flexibility matrices are corrected optimally by using vibration tests. The two different solutions obtained can be used to check the measurements and the computations. The corrected matrices can be used for further dynamic computations.
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Current software packages used for modal parameter identification are based on measured frequency response of either a fixed structure having no rigid body modes or a free structure with flexible supports. Procedures are now available to extract complex modes with the usual assumptions that the mass, stiffness and damping matrices are symmetric. In the case of base excitation, the equations of structural dynamics involve relative displacement with respect to the base, rather than with respect to the inertial frame of reference. Measurements, usually in the form of acceleration, are, however, for the total response. In this paper, a procedure is outlined for obtaining modal information from total acceleration measurements for the case of base excitation using current software capabilities. The frequency response of the acceleration measurements must be modified algebraically before parameter estimation is performed. Once this is done, however, the modal testing procedure remains the same as for the other experimental setups.
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This paper is a survey of the literature about the nonlinear updating process. It is focused on the computation of the difference between the numerical model and the reference data as well as the algo
[69] Bussetta P, Shiki S B, da Silva S.2017

b. Updating of a nonlinear finite element model using discrete-time Volterra series

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61A new approach for model updating of nonlinear structures is proposed.61A method is developed to calculate linear FRFs from measured nonlinear FRFs.61The method identifies nonlinearities in the system simultaneously.61The model updating approach proposed is verified via example applications.61The accuracy of the approach is demonstrated with a real experimental case study.
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DOI      URL      [本文引用: 1]      摘要

61Outlined a pragmatic and data-driven approach to the modal testing of nonlinear structures.61Studied a reference test-structure (Wing-Pylon) following the mentioned approach.61Addressed the identification of the nonlinear elements that act over the Underlying Linear Model.61Augmented (Upgrade) and corrected (Update) the ULM, accommodating those elements.61Checked the validity of the upgraded model across structural modifications.
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61Moving least-squares method (MLSM) based response surface method (RSM) for model parameter updating.61Comparative assessment between the MLSM based and the conventional least-squares method (LSM) based approach.61The MLSM based RSM identifies better than the LSM based approach.
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The applications of radioguided surgery, an approach to oncologic surgery involving a multidisciplinary team, are expanding at a rapid pace. The technique of radioguided occult lesion (ROLL) was originally introduced in the mid- 90s for applications in breast surgery, and later adapted also to other lesions such as (during either open or laparoscopic surgery) and colonic lesions. Concerning the latter, in particular, the technique called radioguided occult colonic lesion identification (ROCLI) consists of identifying, with the aid of intraoperative gammaprobe counts, small lesions that may escape colic intraoperative palpation, after prior tagging of the lesions performed endoscopically through - or intra-lesional injection of -99m-labeled albumin macroaggregates (99mTc- ), a particulate radiopharmaceutical (25-100 m) that does not migrate from the site of interstitial administration. Since September 2001, ROCLI has been employed in 12 patients, using a collimated gamma- probe measuring 11 mm in external diameter (Scintiprobe MR100 .Hi.Tech.). All patients underwent preoperative colonoscopy in order to inject 0.2 mL of a 99mTc-suspension (10-20 MBq) into the submucosa or intra-perilesionally; such tagging required only a few minutes. Eight of the 12 patients were then submitted to open laparotomy, while laparoscopic access was utilized in the remaining 4 patients. In all 12 patients, of the lesion with the ROCLI technique was technically feasible, safe, efficient and highly accurate, enabling quick detection of the lesion during surgery, with a 100% success rate. No complications occurred, and there was no risk of contamination by ionizing radiation.
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Bayesian model updating using hybrid Monte Carlo simulation with application to structural dynamic models with many uncertain parameters

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Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging

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The results of this simulation study deal with the signal-to-noise ratio of the measured responses and with the choice of the residuals. In order to obtain reliable estimates for the improved computational model it is necessary to use mean values obtained by repeated estimations using different measurement data sets.
[80] D'Ambrogio W, Fregolent A, Salvini P.1993.

Updatability conditions of nonconservative FE models with noise on incomplete input-output data

//1st International Conference on Structural Dynamics Modelling: Test, Analysis, Correlation. 29-38.

[本文引用: 1]     

[81] Ebrahimian H, Astroza R, Conte J P, et al.2017.

Nonlinear finite element model updating for damage identification of civil structures using batch Bayesian estimation

. Mechanical Systems and Signal Processing, 84: 194-222.

DOI      URL      摘要

61This paper proposes a new framework for nonlinear system and damage identification of civil structures based on nonlinear finite element (FE) model updating using the Batch Bayesian estimation approach.61The Batch Bayesian approach leads to an extended maximum likelihood (ML) estimation method to estimate jointly the unknown FE model parameters and measurement noise variances.61The parameter estimation uncertainties are quantified using the Cramer–Rao lower bound theorem by computing the (asymptotically exact) Fisher Information matrix.61Two validation studies using numerically simulated measurement data are provided to investigate the performance and accuracy of the proposed framework.
[82] Erdogan Y S, Bakir P G.2013.

Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic

. Engineering Applications of Artificial Intelligence, 26: 357-367.

DOI      URL      [本文引用: 1]      摘要

A fuzzy finite element model updating (FFEMU) method is presented in this study for the damage detection problem. The uncertainty caused by the measurement noise in modal parameters is described by fuzzy numbers. Inverse analysis is formulated as a constrained optimization problem at each a-cut level. Membership functions of each updating parameter which correspond to reduction in bending stiffness of the finite elements is determined by minimizing an objective function using a hybrid version of genetic algorithms (GA) and particle swarm optimization method (PSO) which is very efficient in terms of accuracy and robustness. Practical evaluation of the approximate bounds of the interval modal parameters in FFEMU iterations is addressed. A probabilistic analysis is performed using Monte Carlo simulation (MCS) and the results are compared with presented FFEMU method. It is apparent from numerical simulations that the proposed method is well capable in finding the membership functions of the updating parameters within reasonable accuracy. It is also shown that the results obtained by FFEMU are in good agreement with the MCS results while FFEMU is not as computationally expensive as the MCS method. Nevertheless, the proposed FFEMU do not required derivatives of the objective function like existing methods except in the deterministic case. (C) 2012 Elsevier Ltd. All rights reserved.
[83] Ewins D J, Weekes B, D Carri A.2015.

Modal testing for model validation of structures with discrete nonlinearities

. Phil Trans R Soc A, 373: 20140410.

DOI      URL      PMID      [本文引用: 2]      摘要

Model validation using data from modal tests is now widely practiced in many industries for advanced structural dynamic design analysis, especially where structural integrity is a primary requirement. These industries tend to demand highly efficient designs for their critical structures which, as a result, are increasingly operating in regimes where traditional linearity assumptions are no longer adequate. In particular, many modern structures are found to contain localized areas, often around joints or boundaries, where the actual mechanical behaviour is far from linear. Such structures need to have appropriate representation of these nonlinear features incorporated into the otherwise largely linear models that are used for design and operation. This paper proposes an approach to this task which is an extension of existing linear techniques, especially in the testing phase, involving only just as much nonlinear analysis as is necessary to construct a model which is good enough, or alid : i.e. capable of predicting the nonlinear response behaviour of the structure under all in-service operating and test conditions with a prescribed accuracy. A short-list of methods described in the recent literature categorized using our framework is given, which identifies those areas in which further development is most urgently required.
[84] Farrar C R, Worden K.2012.

Structural Health Monitoring: A Machine Learning Perspective

. New York: John Wiley & Sons.

[本文引用: 1]     

[85] Fox R L, Kapoor M P.1969.

Rates of change of eigenvalues and eigenvectors

. AIAA Journal, 6: 2426-2429.

[86] Friswell M I.1990.

Candidate Reduced Order Models for Structural Parameter Estimation

. Journal of Vibration and Acoustics, 112: 93-97.

DOI      URL      [本文引用: 1]     

[87] Friswell M, Mottershead J E.1995.

Finite Element Model Updating in Structural Dynamics

. New York: Springer Science & Business Media.

DOI      URL      [本文引用: 3]      摘要

Preface. 1. Introduction. 2. Finite Element Modelling. 3. Vibration Testing. 4. Comparing Numerical Data with Test Results. 5. Estimation Techniques. 6. Parameters for Model Updating. 7. Direct Methods Using Modal Data. 8. Iterative Methods Using Modal Data. 9. Methods Using Frequency Domain Data. 10. Case Study: an Automobile Body; M. Brughmans, J. Leuridan, K. Blauwkamp. 11. Discussion and Recommendations. Index.
[88] Friswell M I, Mottershead J E, Ahmadian H.2001.

Finite-element model updating using experimental test data: Parametrization and regularization

. Philosophical Transactions Mathematical Physical & Engineering Sciences, 359:169-186.

DOI      URL      [本文引用: 1]      摘要

Two critical issues in model updating are deciding how a finite-element model should be parametrized and estimating the unknown parameters from the resulting ill-conditioned equations. A lack of understanding of these issues will lead to updated models without physical meaning. This paper outlines the authors' approach to parametrization, using physical, geometric and generic element parameters. It also applies useful methods of regularization, namely parameter constraints, the singular-value decomposition, L-curves and cross-validation to model updating.
[89] Fritzen C P, Zhu S.1991.

Updating of finite element models by means of measured information

. Computers and Structures, 40: 475-486.

DOI      URL      [本文引用: 1]      摘要

The paper deals with the updating of computational models for elasto-mechanical vibrating structures. The updating procedure is based on system identification techniques using an incorrect model on the one hand and measured data of the real system on the other. Some approaches with different types of measurement data are discussed. For practical realization the finite element program ADINA is combined with a nonlinear optimization algorithm and measured transfer functions are used for the correction of the design parameters. The application of the procedure is demonstrated by means of three examples.
[90] Fritzen C P.1986.

Identification of mass, damping, and stiffness matrices of mechanical systems

. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 108: 9-16.

DOI      URL      [本文引用: 1]     

[91] Ghanem R G, Doostan A, Red-Horse J.2008.

A probabilistic construction of model validation

. Computer Methods in Applied Mechanics and Engineering, 197: 2585-2595.

DOI      URL      [本文引用: 1]      摘要

We describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty. The proposed approach is a functional analysis-based probabilistic approach for which we represent random quantities using polynomial chaos expansions (PCEs). The approach permits the formulation of the uncertainty assessment in validation, a significant component of the process, as a problem of approximation theory. It has two essential parts. First, a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model-based measurements. Such a calibration technique employs PCEs to represent the inherent uncertainty of the model parameters. Based on the asymptotic behavior of the statistical parameter estimator, the associated PCE coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information. Second, a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem. The above validation path is implemented for the case of dynamical system validation challenge exercise.
[92] Goge D.2003.

Automatic updating of large aircraft models using experimental data from ground vibration testing

. Aerospace Science and Technology, 7: 33-45.

DOI      URL      [本文引用: 1]      摘要

The aeroelastic stability certification of today's civil aircraft structures requires validated analytical models which have to meet high flutter calculation, fan-blade-off and windmilling certification calculation standards. The dynamic model of the aircraft must be validated in such a way that the dynamic behaviour of the aircraft is reproduced nearly exactly in order to reflect real scenarios when infuriate extreme or flight loads on the model. In addition, the dynamic model must be an accurate representation in order to predict the behaviour of the structure with regard to different boundary conditions. In view of shorter testing times or large-scale civil aircraft this topic will increase in importance in the future since correct free-free boundary conditions are very severe to realize during ground vibration testing (GVT). The above mentioned application fields illustrate the all-important role of the validated analytical model within the scope of civil aeronautics. The aim of this study was to find a new way of updating analytical models of large aircraft by using modal data obtained by GVT in order to save time during model validation. A strategy is presented in this article for validating the finite element (FE) model of a civil four-engine aircraft using a computational model updating (CMU) method.ZusammenfassungDie aeroelastische Stabilit01tszertifizierung der heutigen zivilen Luftfahrtstrukturen erfordert validierte mathematische Modelle, die hohe Anforderungen hinsichtlich Flatterrechnungen, Fan-Blade-Off und Windmilling Zertifizierungsberechnungen erfüllen müssen. Das dynamische Modell des Flugzeugs mu08 au08erdem in einer Art und Weise validiert sein, dass das dynamische Verhalten des Flugzeugs ann01hernd exakt wiedergegeben wird, um reale Szenarien zu reflektieren, wenn extreme Lasten oder wenn Fluglasten auf das Modell aufgebracht werden. Darüber hinaus mu08 das dynamische Modell die Realit01t widerspiegeln, um das Verhalten der Struktur hinsichtlich verschiedener Randbedingungen vorhersagen zu k02nnen. Die vorher erl01uterte Thematik wird in Zukunft immer mehr an Bedeutung gewinnen, insbesondere hinsichtlich kürzerer Testzeiten von sehr gro08en zivilen Flugzeugen, für die reale frei-frei Randbedingungen w01hrend des Standschwingungsversuchs nur sehr schwer zu realisieren sind. Die beschriebenen Anwendungsgebiete stellen die enorm wichtige Rolle des validierten analytischen Modells im Bereich der zivilen Luftfahrt dar. Das Ziel dieser Studie war es, eine neue Strategie zu entwickeln, um analytische Modelle von grossen Flugzeugen mit Hilfe modaler Daten von Standschwingungsversuchen zu korrigieren und die Zeit, die für eine Modellvalidierung ben02tigt wird zu reduzieren. In dieser Ver02ffentlichung wird diese Strategie vorgestellt. Das Finite Elemente (FE) Modell eines viermotorigen Langstreckengrossraumflugzeugs wird mit Hilfe eines Verfahrens zur computerunterstützten rechnerischen Anpassung von Modellparametern (engl.: Computational Model Updating) korrigiert.
[93] Gou B Y, Zhang W J, Lu Q H, et al.2016.

A successive selection method for finite element model updating

. Mechanical Systems and Signal Processing, 70: 320-333.

[本文引用: 1]      摘要

Finite Element (FE) model can be updated effectively and efficiently by using the Response Surface Method (RSM). However, it often involves performance trade-offs such as high computational cost for better accuracy or loss of efficiency for lots of design parameter updates. This paper proposes a Successive Selection Method (SSM), which is based on the linear Response Surface (RS) function and orthogonal design. SSM rewrites the linear RS function into a number of linear equations to adjust the Design of Experiment (DOE) after every FE calculation. SSM aims to interpret the implicit information provided by the FE analysis, to locate the Design of Experiment (DOE) points more quickly and accurately, and thereby to alleviate the computational burden. This paper introduces the SSM and its application, describes the solution steps of point selection for DOE in detail, and analyzes SSM壮s high efficiency and accuracy in the FE model updating. A numerical example of a simply supported beam and a practical example of a vehicle brake disc show that the SSM can provide higher speed and precision in FE model updating for engineering problems than traditional RSM.
[94] Grafe H.1999.

Model updating of large structural dynamics models using measured response functions. [PhD Thesis]

. London: University of London.

[本文引用: 1]     

[95] Guyan R J.1965.

Reduction of stiffness and mass matrices

. AIAA Journal, 3: 380-380.

DOI      URL      [本文引用: 1]      摘要

react-text: 307 It is known that a simple Bézout domain is the domain of elementary divisors if and only if it is 2-simple. The block-diagonal reduction of matrices over an n -simple Bézout domain (n ≥ 3) is realized. /react-text react-text: 308 /react-text
[96] Hanson K M, Hemez F M.2001.

A framework for assessing confidence in computational predictions

. Experimental Techniques, 25: 50-55.

DOI      URL      [本文引用: 1]      摘要

Abstract This article is the third in a series of papers concerning the importance of simulation code validation to the US Department of Energy Accelerated Strategic Computing Initiative (ASCI) program [1]. The series started with a review by John Garcia of the critical need for advanced validation techniques in the ASCI program, which was created to make up for the absence of nuclear testing through the use of simulation codes. Without testing, the simulation codes must be able to answer critical questions about the reliability of our aging stockpile of weapons. In the second paper, Bill Oberkampf gave an overview of validation concepts and described the requirements for a well-executed validation experiment. In this article we discuss the analysis of data obtained from validation experiments and motivate the use of uncertainties to quantify the accuracy of predictions made by simulation codes. This work represents merely a small fraction of the numerous verification and validation projects currently being conducted at the DOE National Laboratories and at several universities under the auspices of the ASCI program.
[97] Hasselman T, Lloyd G.2008.

A top-down approach to calibration, validation, uncertainty quantification and predictive accuracy assessment

. Computer Methods in Applied Mechanics and Engineering, 197: 2596-2606.

DOI      URL      [本文引用: 1]      摘要

This paper describes a “top-down” uncertainty quantification (UQ) approach for calibration, validation and predictive accuracy assessment of the SNL Validation Workshop Structural Dynamics Challenge Problem. The top-down UQ approach differs from the more conventional (“bottom-up”) approach in that correlated statistical analysis is performed directly with the modal characteristics (frequencies, mode shapes and damping ratios) rather than using the modal characteristics to derive the statistics of physical model parameters (springs, masses and viscous damping elements in the present application). In this application, a stochastic subsystem model is coupled with a deterministic subsystem model to analyze stochastic system response to stochastic forcing functions. The weak nonlinearity of the stochastic subsystem was characterized by testing it at three different input levels, low, medium and high. The calibrated subsystem models were validated with additional test data using published NASA and Air Force validation criteria. The validated subsystem models were first installed in the accreditation test bed where system response simulations involving stochastic shock-type force inputs were conducted. The validated stochastic subsystem model was then installed in the target application and simulations involving limited duration segments of stationary random vibration excitation were conducted.
[98] Hemez F M, Doebling S W.2000

a. Model Validation and Uncertainty Quantification//19th International Modal Analysis Conference, 2001, Kissimmee

, USA.

[本文引用: 1]     

[99] Hemez F M, Doebling S W.2000

b. Validation of structural dynamics models at Los Alamos National Laboratory

//41th Structures, Structural Dynamics, and Materials Conference and Exhibit, 2000, Atlanta, USA, 1437.

DOI      URL      [本文引用: 1]      摘要

No abstract prepared.
[100] Hemez F M, Doebling S W.2001.

Review and assessment of model updating for non-linear, transient dynamics

. Mechanical Systems and Signal Processing, 15: 45-74.

DOI      URL      [本文引用: 1]      摘要

The purpose of this publication is to motivate the need of validating numerical models based on time-domain data for non-linear, transient, structural dynamics and to discuss some of the challenges faced by this technology. Our approach is two-fold. First, several numerical and experimental testbeds are presented that span a wide variety of applications (from non-linear vibrations to shock response) and difficulty [from a single-degree-of-freedom (sdof) system with localised non-linearity to a three-dimensional (3-D), multiple-component assembly featuring non-linear material response and contact mechanics]. These testbeds have been developed at Los Alamos National Laboratory in support of the Advanced Strategic Computing Initiative and our code validation and verification program. Conventional, modal-based updating techniques are shown to produce erroneous models although the discrepancy between test and analysis modal responses can be bridged. This conclusion offers a clear justification that metrics based on modal parameters are not well suited to the resolution of inverse, non-linear problems. In the second part of this work, the state-of-the-art in the area of model updating for non-linear, transient dynamics is reviewed. The techniques identified as the most promising are assessed using data from our numerical or experimental testbeds. Several difficulties of formulating and solving inverse problems for non-linear structural dynamics are identified. Among them, we cite the formulation of adequate metrics based on time series and the need to propagate variability throughout the optimisation of the model's parameters. Another important issue is the necessity to satisfy continuity of the response when several finite element optimisations are successively carried out. An illustration of how this problem can be resolved based on the theory of optimal control is provided using numerical data from a non-linear Duffing oscillator. The publication concludes with a brief description of current research directions in inverse problem solving for structural dynamics.
[101] Hills R G, Pilch M, Dowding K J, et al.2008.

Validation challenge workshop

. Computer Methods and Applied Mechanics and Engineering, 197: 2375-2380.

DOI      URL      [本文引用: 1]     

[102] Horta L G, Kenny S P, Crespo L G, et al.2008.

NASA Langley's approach to the Sandia's structural dynamics challenge problem

. Computer Methods in Applied Mechanics and Engineering, 197: 2607-2620.

DOI      URL      [本文引用: 1]     

[103] Housner G, Bergman L A, Caughey T K, et al.1997.

Structural control: past, present, and future

. Journal of Engineering Mechanics, 123: 897-971.

DOI      URL      [本文引用: 1]     

[104] Hu S L J, Li H, Wang S.2007.

Cross-model cross-mode method for model updating

. Mechanical Systems and Signal Processing, 21: 1690-1703.

DOI      URL      [本文引用: 1]      摘要

Model updating has become a common method to improve the correlation between finite-element models and measured data. This paper develops a direct, physical property adjustment model updating method, named as cross-model cross-mode (CMCM) method. This new method is capable of updating the mass and stiffness matrices simultaneously based on very limited measured mode shapes and modal frequencies. Two structural models, a shear building model and a three-dimensional frame structural model, are demonstrated in the numerical examples. Numerical results indicate that an excellent updating is achievable by applying the CMCM method.
[105] Isasa I, Hot A, Cogan S, et al.2011.

Model updating of locally non-linear systems based on multi-harmonic extended constitutive relation error

. Mechanical Systems and Signal Processing, 25: 2413-2425.

DOI      URL      [本文引用: 1]      摘要

Improving the fidelity of numerical simulations using available test data is an important activity in the overall process of model verification and validation. While model updating or calibration of linear elastodynamic behaviors has been extensively studied for both academic and industrial applications over the past three decades, methodologies capable of treating non-linear dynamics remain relatively immature. The authors propose a novel strategy for updating an important subclass of non-linear models characterized by globally linear stiffness and damping behaviors in the presence of local non-linear effects. The approach combines two well-known methods for structural dynamic analysis. The first is the multi-harmonic balance (MHB) method for solving the non-linear equations of motion of a mechanical system under periodic excitation. This approach has the advantage of being much faster than time domain integration procedures while allowing a wide range of non-linear effects to be taken into account. The second method is the extended constitutive relation error (ECRE) that has been used in the past for error localization and updating of linear elastodynamic models. The proposed updating strategy will be illustrated using academic examples.
[106] Jafarkhani R, Masri S F.2011.

Finite Element Model Updating Using Evolutionary Strategy for Damage Detection

. Computer-Aided Civil Infrastructure Engineering, 26: 207-224.

DOI      URL      [本文引用: 1]      摘要

Abstract Abstract: Structural health monitoring through the use of finite element model updating techniques for dispersed civil infrastructures usually deals with minimizing a complex, nonlinear, nonconvex, high-dimensional cost function with several local minima. Hence, stochastic optimization algorithms with promising performance in solving global optimization problems have received considerable attention for finite element model updating purposes in recent years. In this study, the performance of an evolutionary strategy in the finite element model updating approach was investigated for damage detection in a quarter-scale two-span reinforced concrete bridge system which was tested experimentally at the University of Nevada, Reno. The damage sequence in the structure was induced by a range of progressively increasing excitations in the transverse direction of the specimen. Intermediate nondestructive white noise excitations and response measurements were used for system identification and damage detection purposes. It is shown that, when evaluated together with the strain gauge measurements and visual inspection results, the applied finite element model updating algorithm of this article could accurately detect, localize, and quantify the damage in the tested bridge columns throughout the different phases of the experiment.
[107] Jiang X M, Mahadevan S.2011.

Wavelet spectrum analysis approach to model validation of dynamic systems

. Mechanical Systems and Signal Processing, 25: 575-590.

DOI      URL      [本文引用: 1]      摘要

Feature-based validation techniques for dynamic system models could be unreliable for nonlinear, stochastic, and transient dynamic behavior, where the time series is usually non-stationary. This paper presents a wavelet spectral analysis approach to validate a computational model for a dynamic system. Continuous wavelet transform is performed on the time series data for both model prediction and experimental observation using a Morlet wavelet function. The wavelet cross-spectrum is calculated for the two sets of data to construct a time requency phase difference map. The Box-plot, an exploratory data analysis technique, is applied to interpret the phase difference for validation purposes. In addition, wavelet time requency coherence is calculated using the locally and globally smoothed wavelet power spectra of the two data sets. Significance tests are performed to quantitatively verify whether the wavelet time-varying coherence is significant at a specific time and frequency point, considering uncertainties in both predicted and observed time series data. The proposed wavelet spectrum analysis approach is illustrated with a dynamics validation challenge problem developed at the Sandia National Laboratories. A comparison study is conducted to demonstrate the advantages of the proposed methodologies over classical frequency-independent cross-correlation analysis and time-independent cross-coherence analysis for the validation of dynamic systems.
[108] Jin S S, Jung H J.2016.

Sequential surrogate modeling for efficient finite element model updating

. Computers & Structures, 168: 30-45.

DOI      URL      [本文引用: 1]      摘要

Despite the numerous studies concerning finite element model updating (FEMU), a challenging computational cost issue persists. Therefore, surrogate modeling has recently gained considerable attention in FEMU. Conventionally, surrogate models are constructed by identical samples for all outputs. It is very inefficient and subjective, if various response-surfaces exhibit even for identical parameters. Accordingly, we propose a sequential surrogate modeling for FEMU. It uses infill criteria to guide sampling for updating surrogate models automatically. The proposed method is successful to construct the different response-surfaces and apply FEMU. It is promising for constructing surrogate models with minimal user intervention and tremendous computational efficiency.
[109] Kabe A M.1985.

Stiffness matrix adjustment using mode data

. AIAA Journal, 23: 1431-1436.

DOI      URL      [本文引用: 1]     

[110] Kerschen G, Golinval J C.2004.

A model updating strategy of non-linear vibrating structures

. International Journal for Numerical Methods in Engineering, 60: 2147-2164.

DOI      URL      [本文引用: 1]      摘要

Abstract The objective of this paper is to present a model updating strategy of non-linear vibrating structures. Because modal analysis is no longer helpful in non-linear structural dynamics, a special attention is devoted to the features extracted from the proper orthogonal decomposition and one of its non-linear generalizations based on auto-associative neural networks. The efficiency of the proposed procedure is illustrated using simulated data from a three-dimensional portal frame. Copyright 2004 John Wiley & Sons, Ltd.
[111] Kerschen G, Golinval J C.2005.

Generation of accurate finite element models of nonlinear systems---Application to an aeroplane-like structure

. Nonlinear Dynamics, 39: 129-142.

DOI      URL      [本文引用: 1]      摘要

Model updating and validation is currently a central issue in the fields of computational structural mechanics and dynamics. The vast majority of applications however concerns linear structures. On the other hand, updating nonlinear models is something the structural dynamicist prefers to avoid mainly because tools such as modal analysis are no longer available. The objective of the present study is to propose a two-step methodology for dealing with nonlinear systems. Its most appealing feature is that it decouples the estimation of the linear and nonlinear parameters. A numerical application consisting of an aeroplane-like structure is used to assess the efficiency of the procedure.
[112] Kerschen G, Worden K, Vakakis A F, et al.2006.

Past, present and future of nonlinear system identification in structural dynamics

. Mechanical Systems and Signal Processing, 20: 505-592.

DOI      URL      [本文引用: 1]      摘要

This survey paper contains a review of the past and recent developments in system identification of nonlinear dynamical structures. The objective is to present some of the popular approaches that have been proposed in the technical literature, to illustrate them using numerical and experimental applications, to highlight their assets and limitations and to identify future directions in this research area. The fundamental differences between linear and nonlinear oscillations are also detailed in a tutorial.
[113] Kurt M, Eriten M, Mcfarland D M, et al.2015.

Methodology for model updating of mechanical components with local nonlinearities

. Journal of Sound and Vibration, 357: 331-348.

DOI      URL      [本文引用: 1]      摘要

In this work, we propose a new nonlinear model updating strategy based on global/local nonlinear system identification of the dynamics. The main objective of this study is to construct and update reduced-order models (ROM) of a dynamical system based solely on measured data. The approach relies on analyzing transient system responses (local dynamics) in the frequency–energy domain, and based on these, constructing damped frequency–energy plots – FEPs (global dynamics) under the assumption of weak damping. The system parameters are characterized and updated by matching the backbone branches of the FEPs with reduced-order model FEPs using experimental or computational data. The main advantage of this method is that the system model is updated solely based on simulation and/or experimental results. It follows that the approach is purely data-driven. By matching the frequency–energy dependences of the dynamics of the physical dynamical system and its reduced order model, we are able to identify, update and reconstruct not only the global features of the dynamics in the frequency and energy ranges of interest, but also the local dynamics, i.e., individual time series for specific initial or excitation conditions. Hence, this work paves the way toward a nonlinear model updating methodology with broad applicability. The main features of the proposed methodology are demonstrated with a system of nonlinearly coupled beams excited by a concentrated transient force.
[114] Kurt M, Moore K J, Eriten M, et al.2016.

Nonlinear Model Updating Methodology with Application to the IMAC XXXIII Round Robin Benchmark Problem//Nonlinear Dynamics

, Springer International Publishing, 1: 343-347.

DOI      URL      [本文引用: 1]      摘要

Abstract We develop a new nonlinear model updating strategy based on global/local nonlinear system identification of general mechanical systems. The approach relies on analyzing system time series in the frequency-energy domain by constructing Hamiltonian, and forced/damped frequency-energy plots (FEPs). The system parameters are then characterized and updated by matching the backbone branches of the FEPs with the frequency-energy wavelet transforms of experimental and/or computational time series. The main advantage of this method is that no nonlinearity model is assumed a priori, and the system model is updated solely based on simulation and/or experimental results. By matching the frequency-energy plots of the benchmark system and its reduced order model, we show that we are able to retrieve the global dynamics in the frequency and energy ranges of interest, identify bifurcations, characterize local nonlinearities, and accurately reconstruct time series.
[115] Kwon K S, Lin R M.2004.

Frequency selection method for FRF-based model updating

. Journal of Sound and Vibration, 278: 285-306.

DOI      URL      [本文引用: 1]      摘要

In this paper, a new frequency selection method for efficient FRF-based model updating is proposed. Using the proposed method, the frequency points used for updating can be selected in an automatic way such that the selected frequencies can carry as much information as possible with a limited number of frequencies. In order to demonstrate the effectiveness of the proposed method, a numerical example of truss structure is used. Different sets of frequency points for updating are compared in terms of numerical stability as well as updated results. Finally, the proposed method is extended to deal with rotor earing systems.
[116] Lenaerts V, Kerschen G, Golinval J C.2001.

Proper orthogonal decomposition for model updating of non-linear mechanical systems

. Mechanical Systems and Signal Processing, 15: 31-43.

DOI      URL      [本文引用: 1]      摘要

Proper orthogonal decomposition (POD), also known as Karhunen–Loeve (K–L) decomposition, is emerging as a useful experimental tool in dynamics and vibrations. The POD is a means of extracting spatial information from a set of time-series data available on a domain. The use of (K–L) transform is of great help in non-linear settings where traditional linear techniques such as modal-testing and power-spectrum analyses cannot be applied. These decomposition can be used as an orthogonal basis for efficient representation of the ensemble. The POM have been interpreted mainly as empirical system modes and the application of POD to measured displacements of a discrete structure with a known mass matrix leads to an estimation of the normal modes. We investigate the use of the proper orthogonal modes of displacements for the identification of parameters of non-linear dynamical structures with an optimisation procedure based on the difference between the experimental and simulated POM. A numerical example of a beam with a local non-linear component will illustrate the method.
[117] Levin R I, Lieven N.1998

a. Dynamic finite element model updating using simulated annealing and genetic algorithms

. Mechanical Systems and Signal Processing, 12: 91-120.

DOI      URL      [本文引用: 1]      摘要

Dynamic finite element (FE) model updating may be considered as an optimisation process. Over the past few years, two powerful new optimisation algorithms have been developed independently of each other; namely, the genetic algorithm (GA) and simulated annealing (SA). These algorithms are both probabilistic search algorithms capable of finding the global minimum amongst many local minima. This paper compares various implementations of the two algorithms for model updating purposes. A new variant of simulated annealing is suggested and is found to be the most effective of all the optimisation algorithms considered. This version of simulated annealing is then tested using several objective functions for simulated model updating in the frequency domain. In the second part of this paper, both SA and GAs are applied to a practical FE model updating problem using measured data. The new variation of the SA algorithm, termed the blended SA algorithm, performed better than the traditional GA algorithm. However, the results obtained show a significant dependence on the choice of updating parameters. It was concluded that model updating using these optimisation algorithms is a promising and viable approach, but the appropriate choice of updating parameters is of paramount importance.
[118] Levin R I, Lieven N.1998

b. Dynamic finite element model updating using neural networks

. Journal of Sound and Vibration, 210: 593-607.

DOI      URL      [本文引用: 1]      摘要

In this paper, a new method of finite element model updating using neural networks is presented. Many previous model updating techniques have exhibited inconsistent performance when subjected to noisy experimental data. From this background it is clear that a successful model updating method must be resistant to experimental noise. A well-known property of neural networks is robustness in the presence of noise, and it is hoped to exploit this property for model updating purposes. The proposed updating method is tested on a simple simulated model, both in the absence and presence of noise, with promising results. A further advantage of this updating method is the ability to work with a limited number of experimentally measured degrees of freedom and modes.
[119] Li Y, Astroza R, Conte J P, et al.2017.

Nonlinear FE model updating and reconstruction of the response of an instrumented seismic isolated bridge to the 2010 Maule Chile earthquake

. Earthquake Engineering & Structural Dynamics, 46: 2699-2716.

DOI      URL      [本文引用: 1]      摘要

Nonlinear finite element (FE) modeling has been widely used to investigate the effects of seismic isolation on the response of bridges to earthquakes. However, most FE models of seismic isolated bridges (SIB) have used seismic isolator models calibrated from component test data, while the prediction accuracy of nonlinear FE models of SIB is rarely addressed by using data recorded from instrumented bridges. In this paper, the accuracy of a state-of-the-art FE model is studied through nonlinear FE model updating (FEMU) of an existing instrumented SIB, the Marga-Marga Bridge located in Vina del Mar, Chile. The seismic isolator models are updated in 2 phases: component-wise and system-wise FEMU. The isolator model parameters obtained from 23 isolator component tests show large scatter, and poor goodness of fit of the FE-predicted bridge response to the 2010 Mw 8.8 Maule, Chile Earthquake is obtained when most of those parameter sets are used for the isolator elements of the bridge model. In contrast, good agreement is obtained between the FE-predicted and measured bridge response when the isolator model parameters are calibrated using the bridge response data recorded during the mega-earthquake. Nonlinear FEMU is conducted by solving single- and multiobjective optimization problems using high-throughput cloud computing. The updated FE model is then used to reconstruct response quantities not recorded during the earthquake, gaining more insight into the effects of seismic isolation on the response of the bridge during the strong earthquake.
[120] Lin R M, Ewins D.1994.

Analytical model improvement using frequency response functions

. Mechanical Systems and Signal Processing, 8: 437-458.

DOI      URL      [本文引用: 1]      摘要

In recent years, a number of different methods have been developed in order to update an analytical structural dynamics model using the limited data which can be obtained from vibration tests. The majority of these methods are based on correlation between the analytical and measured modal data and, in most cases, spatial completeness of the measured co-ordinates is critical. In this paper, a new method is presented which tackles the model updating problem by using an incomplete set of measured FRF data directly. It is shown that model updating methods based on modal data are, essentially, particular cases of the present generalised method in which FRF data at resonance frequencies only are employed. The advantages of using FRF data (over modal data) to update an analytical model are demonstrated. Special attention is given to the application of the method to the case where both measured modes and co-ordinates are incomplete. The practical applicability of the method is assessed based on the GARTEUR structure which has been used in a Europe-wide correlation exercise.
[121] Lin R M, Zhu J.2007.

Finite element model updating using vibration test data under base excitation

. Journal of Sound and Vibration, 303: 596-613.

DOI      URL      [本文引用: 1]      摘要

Most model updating methods use measured frequency response function (FRF) data to update analytical models whereas only response functions under base excitation can be obtained in practical vibration test due to difficulties and constraints which prevent conventional FRFs from being measured accurately. This paper presents a new model updating method, which can employ measured response function data under base excitation directly for updating. Mathematical formulations using measured response function data under base excitation to identify mass and stiffness modeling errors, have been established. Through simulated numerical case studies based on a cantilever beam as well as a practical GARTEUR structure, it has been proved that the proposed method is feasible and effective when applied to the identification of mass and stiffness modeling errors. It is also shown that this method has considerable noise-resisting ability in the case where the measured response function data are contaminated by certain level measurement noise.
[122] Link M, Friswell M.2003.

Working Group 1: Generation of validated structural dynamic models---results of a benchmark study utilising the garteur SM-AG19 test-bed

. Mechanical Systems and Signal Processing, 17: 9-20.

DOI      URL      [本文引用: 1]      摘要

Abstract In practice, the validation of analytical structural dynamic models is mainly based on comparing experimental modal analysis results with analytical predictions. Despite the high sophistication of analytical [finite element (FE)] modeling, practical applications often reveal considerable discrepancies between analytical and test results. In recent years, significant effort has therefore been expended on the development of mathematical procedures for updating analytical mass and stiffness matrices using dynamic test data. The success of these methods is governed not only by the skill of the analyst to assume an appropriate initial analysis model but also the source and the location of the erroneous parameters to be corrected. In practical applications, the source and location of the errors can be manifold, resulting in non-unique updated models with all of them fulfilling the mathematical criterion of minimizing the test/analysis discrepancies. The aim of the present benchmark study defined within the European COST Action F3 on tructural Dynamics was not only to compare the different computational model updating (CMU) procedures using a common test structure but also to see if the expected non-uniqueness of the results due to different computational methods, different structural idealizations and different parameter sets and, of course, different test data sets can be tolerated with regard to the intended utilization.
[123] Liu Y, Duan Z D.2012.

Fuzzy finite element model updating of bridges by considering the uncertainty of the measured modal parameters

. Science China Technological Sciences, 55: 3109-3117.

DOI      URL      [本文引用: 1]      摘要

AbstractIt is significant to consider the effect of uncertainty of the measured modal parameters on the updated finite element (FE) model, especially for updating the FE model of practical bridges, since the uncertainty of the measured modal parameters cannot be ignored owing to the application of output-only identification method and the existence of the measured noise. A reasonable method is to define the objective of the FE model updating as the statistical property of the measured modal parameters obtained by conducting couples of identical modal tests, however, it is usually impossible to implement repeated modal test due to the limit of practical situation and economic reason. In this study, a method based on fuzzy finite element (FFM) was proposed in order to consider the effect of the uncertainty of the measured modal parameters on the updated FE model by using the results of a single modal test. The updating parameters of bridges were deemed as fuzzy variables, and then the fuzz-ification of objective of the FE model updating was proposed to consider the uncertainty of the measured modal parameters. Finally, the effectiveness of the proposed method was verified by updating the FE model of a practical bridge with the measured modal parameters.
[124] Martin P.2000.

Guide for ASCI verification and validation planescontent V and format: version 2.0

. Sandia National Laborarories.

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[125] Marwala T.2010.

Finite Element Model Updating Using Computational Intelligence Techniques: Applications to Structural Dynamics

. New York: Springer Science &Business Media.

DOI      URL      [本文引用: 1]      摘要

FEM updating allows FEMs to be tuned better to reflect measured data. It can be conducted using two different statistical frameworks: the maximum likelihood approach and Bayesian approaches. This book applies both strategies to the field of structural mechanics, using vibration data. Computational intelligence techniques including: multi-layer perceptron neural networks; particle swarm and GA-based optimization methods; simulated annealing; response surface methods; and expectation maximization algorithms, are proposed to facilitate the updating process. Based on these methods, the most appropMarwala, Tshilidzi
[126] Mcfarland J, Mahadevan S.2008.

Error and variability characterization in structural dynamics modeling

. Computer Methods in Applied Mechanics and Engineering, 197: 2621-2631.

DOI      URL      [本文引用: 1]      摘要

The characterization of both modeling error and random variability has been the subject of recent interest within the modeling and simulation community. This paper addresses both of these topics, using as a testbed the structural dynamics validation challenge problem developed by Sandia National Laboratories. With a focus on the model intended use, two cases of model assessment are considered, illustrating what types of conclusions are appropriate when comparing model predictions and experimental observations. In addition, the random variability associated with the modal properties of a three degree-of-freedom subsystem is characterized using a non-parametric approach which incorporates kernel density estimation, principal component analysis, and Markov Chain Monte Carlo simulation.
[127] Meyer S, Link M.2003.

Modelling and updating of local non-linearities using frequency response residuals

. Mechanical Systems and Signal Processing, 17: 219-226.

DOI      URL      [本文引用: 1]      摘要

In this paper a method is presented to identify local non-linear stiffness and damping parameters from dynamic response data. The non-linearity is being taken into account by assembling non-linear two-degree-of-freedom elements into larger linear finite element models. Special care has been taken to keep the physical description of the structural model in the time domain. This model is linearized following the procedure of the harmonic balance method to get a suitable model description in the frequency domain. The response is calculated iteratively using dynamic condensation for the linear parts of the structure. Selected linear and non-linear parameters of an initial model are updated by minimising the deviations between analytical and measured displacement responses.
[128] Moaveni B, He X F, Conte J P, et al.2010.

Damage identification study of a seven-story full-scale building slice tested on the UCSD-NEES shake table

. Structural Safety, 32: 347-356.

DOI      URL      [本文引用: 1]      摘要

A full-scale seven-story reinforced concrete building section was tested on the UCSD-NEES shake table during the period October 2005 anuary 2006. The shake table tests were designed to damage the building progressively through four historical earthquake records. At various levels of damage, ambient vibration tests and low-amplitude white noise base excitations with root-mean-square accelerations of 0.03 g and 0.05 g were applied to the building, which responded as a quasi-linear system with parameters evolving as a function of structural damage. Modal parameters (natural frequencies, damping ratios and mode shapes) of the building were identified at different damage levels based on the response of the building to ambient as well as low-amplitude white noise base excitations, measured using DC coupled accelerometers. This paper focuses on damage identification of this building based on changes in identified modal parameters. A sensitivity-based finite element model updating strategy is used to detect, localize and quantify damage at each damage state considered. Three sets of damage identification results are obtained using modal parameters identified based on ambient, 0.03 g, and 0.05 g RMS white noise test data, respectively. The damage identification results obtained in all three cases do not exactly coincide, but they are consistent with the concentration of structural damage observed at the bottom two stories of the building. The difference in the identified damage results is mainly due to the significant difference in the identified modal parameters used in the three cases. The assumption of a quasi-linear dynamic system is progressively violated with increasing level of excitation. Therefore, application of nonlinear FE model updating strategies is recommended in future studies to resolve the errors caused by structural response nonlinearity.
[129] Moaveni B, Stavridis A, Lombaert G, et al.2013.

Finite-Element Model Updating for Assessment of Progressive Damage in a 3-Story Infilled RC Frame

. Journal of Structural Engineering, 139: 1665-1674.

DOI      URL      [本文引用: 1]      摘要

This paper presents a study on the identification of progressive damage, using an equivalent linear finite-element model updating strategy, in a masonry infilled RC frame that was tested on a shake table. A two-thirds-scale, 3-story, 2-bay, infilled RC frame was tested on the UCSD EES shake table to investigate the seismic performance of this type of construction. The shake table tests induced damage in the structure progressively through scaled historical earthquake records of increasing intensity. Between the earthquake tests and at various levels of damage, low-amplitude white-noise base excitations were applied to the infilled RC frame. In this study, the effective modal parameters of the damaged structure have been identified from the white-noise test data with the assumption that it responded in a quasi-linear manner. Modal identification has been performed using a deterministic-stochastic subspace identification method based on the measured input utput data. A sensitivity-based finite-element model updating strategy has been employed to detect, locate, and quantify damage (as a loss of effective local stiffness) based on the changes in the identified effective modal parameters. The results indicate that the method can reliably identify the location and severity of damage observed in the tests.
[130] Mottershead J E, Link M, Friswell M I.2011.

The sensitivity method in finite element model updating: A tutorial

. Mechanical Systems and Signal Processing, 25: 2275-2296.

DOI      URL      [本文引用: 2]      摘要

The sensitivity method is probably the most successful of the many approaches to the problem of updating finite element models of engineering structures based on vibration test data. It has been applied successfully to large-scale industrial problems and proprietary codes are available based on the techniques explained in simple terms in this article. A basic introduction to the most important procedures of computational model updating is provided, including tutorial examples to reinforce the reader’s understanding and a large scale model updating example of a helicopter airframe.
[131] Mottershead J, Friswell M.1993.

Model updating in structural dynamics: a survey

. Journal of Sound and Vibration, 167: 347-375.

DOI      URL      [本文引用: 1]      摘要

It is well known that finite element predictions are often called into question when they are in conflict with test results. The area known as model updating is concerned with the correction of finite element models by processing records of dynamic response from test structures. Model updating is a rapidly developing technology, and it is intended that this paper will provide an accurate review of the state of the art at the time of going to press. It is the authors' hope that this work will prove to be of value, especially to those who are getting acquainted with the research base and aim to participate in the application of model updating in industry, where a pressing need exists.
[132] Natke H.1988.

Updating computational models in the frequency domain based on measured data: a survey

. Probabilistic Engineering Mechanics, 3: 28-35.

DOI      URL      [本文引用: 2]      摘要

Investigations and qualifications of time-invariant linear elasto-mechanical systems often demand a computational model with high confidence. This can be achieved by updating computational models by measurements. The structure of the model with a finite number of degrees-of-freedom is assumed to be wellknown and the parameter values are given within general unknown confidence. In former times modal quantities were predominantly estimated and an adjustment of the computational model was performed by trial and error. Now procedures are available and have been investigated which use input and output measurements besides modal quantities. A brief survey of these procedures is given. Following the Bayes' approach some updating methods are discussed and compared. They are prepared in a way which is suitable for handling substructures. The paper shows the quality of the estimates. The signal-to-noise ratio influences mainly the precision of the estimates as well as the information contents of the signals. Statistically based weighting of the measured data is very important for accurate results.
[133] Nelson R B.1976.

Simplified calculation of eigenvector derivatives

. AIAA Journal, 14: 1201-1205.

DOI      URL      [本文引用: 1]      摘要

Not Available
[134] O'Callahan J C.1989.

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[135] O'Callahan J, Avitabile P, Riemer R.1989.

System equivalent reduction expansion process (SEREP)//7th International Modal Analysis Conference

, 1989, Schnectady, USA, Vol. 1, 29-37.

[136] Oberkampf W L, Roy C J.2010. Verification and Validation in Scientific Computing. Cambridge: Cambridge University Press.

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[137] Oden J T, Belytschko T, Babuska I, et al.2003.

Research directions in computational mechanics

. Computer Methods in Applied Mechanics and Engineering, 192: 913-922.

DOI      URL      [本文引用: 1]      摘要

This article is derived from a report prepared by the US National Committee on Theoretical and Applied Mechanics. It is part of that committee’s agenda to develop position papers on research directions in various areas of mechanics. This is the most recent work devoted to computational mechanics. The report was authored by a subcommittee consisting of Tinsley Oden (Chair), Ted Belytschko, Ivo Babuska and Thomas Hughes. It also incorporates suggestions made by the USNCTAM at large.
[138] Paez T L, Red-Horse J.2008.

Structural dynamics challenge problem: Summary

. Computer Methods in Applied Mechanics and Engineering, 197: 2660-2665.

DOI      URL      [本文引用: 1]      摘要

The six papers in this special issue that develop solutions to the Structural Dynamics Challenge Problem are summarized herein. The goal is to emphasize different tools and approaches applied to various parts of the structural dynamics problem. Specifically the following issues are considered: (1) Development of a mathematical framework for uncertainty quantification of a substructure. (2) Calibration of a mathematical model for the substructure. (3) Validation of the substructure mathematical model. (4) Use of the mathematical model of the substructure, in conjunction with another structure, to make a prediction of the probability that a regulatory limit is surpassed, and discussion of the uncertainty of the prediction. Different methodologies are presented and specific results vary, however, conclusions regarding satisfaction of the regulatory requirement match.
[139] Papadimitriou C, Papadioti D C.2013.

Component mode synthesis techniques for finite element model updating

. Computers and Structures, 126: 15-28.

DOI      URL      [本文引用: 1]      摘要

Deterministic and Bayesian finite element (FE) model updating techniques are computationally very demanding operations due to the large number of FE model re-analyses required. Component mode synthesis techniques are proposed to carry out the re-analyses efficiently in a substantially reduced space of generalized coordinates using exact component modes and characteristic interface modes computed only once from a reference FE model. The re-assembling of the reduced-order system matrices from components and interface modes is avoided. Theoretical and computational developments are demonstrated with model updating and damage identification applications for a highway bridge using a high fidelity model and simulated measurements.
[140] Paz M.1984.

Dynamic condensation

. AIAA Journal, 22: 724-727.

DOI      URL     

[141] Perera R, Fang S E, Ruiz A.2010.

Application of particle swarm optimization and genetic algorithms to multiobjective damage identification inverse problems with modelling errors

. Meccanica, 45: 723-734.

DOI      URL      [本文引用: 1]      摘要

Structural health monitoring has become an important research topic in conjunction with the damage assessment of structures. The use of system identification approaches for damage detection using inverse methods has become more widespread in recent years and their formulation in a multiobjective framework has become more usual. Inverse problems require the use of an initial baseline model of the undamaged structure. Modelling errors in the baseline model whose effects exceed the modal sensitivity to damage are critical and make an accurate estimation of damage impossible. Artificial intelligence techniques based on genetic algorithms are used increasingly as an alternative to more classical techniques to solve this kind of problem especially due to their feasibility for managing multiobjective problems. This paper outlines an understanding of how particle swarm optimization methods operate in damage identification problems based on multiobjective FE updating procedures and takes modelling errors into account. One experimental example is used to show their performance in comparison with genetic algorithms.
[142] Rebba R, Mahadevan S, Zhang R.2003.

Validation of uncertainty propagation models

//44th Structures, Structural Dynamics, and Materials Conference, Norfolk, USA, 1913.

DOI      URL      [本文引用: 1]      摘要

Differential cross sections for He single ionization in fast p and p04 impact are presented and compared to theory. To investigate possible correlation effects on the double ionization cross section in p04 collisions the in momentum space complete differential cross sections for fast p on He transfer ionization processes has been investigated and the influence of correlation effects, i.e. the influence of so called "off-shell" contributions in the He ground state wave function, has been measured.
[143] Rebba R, Mahadevan S.2006.

Model predictive capability assessment under uncertainty

. AIAA Journal, 44: 2376-2384.

DOI      URL      [本文引用: 1]      摘要

Not Available
[144] Red-Horse J, Paez T.2008.

Sandia national laboratories validation workshop: structural dynamics application

. Computer Methods in Applied Mechanics and Engineering, 197: 2578-2584.

DOI      URL      [本文引用: 1]      摘要

This article specifies a virtual structural dynamic subsystem and some systems, as well as mathematical models that approximately simulate them. The purpose is to define a setting for model validations conceived by participants in the Sandia National Laboratories Validation Workshop. Some broad guidelines for the model validations are set, as well as a regulatory requirement to be satisfied by the target system. Participants are directed to present techniques for subsystem model validation, to analytically predict whether or not the regulatory requirement will be met, and to estimate the likelihood of accuracy of the prediction.
[145] Ren W X, Chen H B.2010.

Finite element model updating in structural dynamics by using the response surface method

. Engineering Structures, 32: 2455-2465.

DOI      URL      摘要

Fast-running response surface models that approximate multivariate input/output relationships of time-consuming physical-based computer models enable effective finite element (FE) model updating analyses. In this paper, a response surface-based FE model updating procedure for civil engineering structures in structural dynamics is presented. The key issues to implement such a model updating are discussed such as sampling with design of experiments, selecting the significant updating parameters and constructing a quadratic polynomial response surface. The objective function is formed by the residuals between analytical and measured natural frequencies. Single-objective optimization with equal weights of natural frequency residual of each mode is used for optimization computation. The proposed procedure is illustrated by a simulated simply supported beam and a full-size precast continuous box girder bridge tested under operational vibration conditions. The results have been compared with those obtained from the traditional sensitivity-based FE model updating method. The real application to a full-size bridge has demonstrated that the FE model updating process is efficient and converges fast with the response surface to replace the original FE model. With the response surface at hand, an optimization problem is formulated explicitly. Hence, no FE calculation is required in each optimization iteration. The response surface-based FE model updating can be easily implemented in practice with available commercial FE analysis packages.
[146] Roy N A.1990.

A Survey of Finite Element Model Updating Methods

//Environmental Testing for Space Programmes Test: Facilities and Methods, 1990, Noordwijk, Netherlands, Vol. 304, 333.

URL      [本文引用: 1]      摘要

Not Available
[147] Rutherford B M.2008.

Computational modeling issues and methods for the ``regulatory problem'' in engineering--Solutions to the structural dynamics and static frame problems

. Computer Methods in Applied Mechanics and Engineering, 197: 2632-2644.

DOI      URL      [本文引用: 1]      摘要

http://linkinghub.elsevier.com/retrieve/pii/S0045782507005129
[148] Schedlinski C.2000.

Computational model updating of large scale finite element models

//18th International Modal Analysis Conference, 2000, San Antonio, USA, Vol. 4062, 476.

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Not Available
[149] Schmidt R.1994.

Updating nonlinear components

. Mechanical Systems and Signal Processing, 8: 679-690.

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[150] Schultze J F, Hemez F M, Doebling S W, et al.2001.

Application of non-linear system model updating using feature extraction and parameter effects analysis

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This research presents a new method to improve analytical model
[151] Schwer L E.2007.

An overview of the PTC 60/V & V 10: Guide for verification and validation in computational solid mechanics

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Guide for verification and validation in computational solid mechanics

. ASME Journal.

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[153] Shabbir F, Omenzetter P.2015.

Particle swarm optimization with sequential niche technique for dynamic finite element model updating

. Computer-Aided Civil Infrastructure Engineering, 30: 359-375.

DOI      URL      [本文引用: 1]      摘要

Abstract Due to uncertainties associated with material properties, structural geometry, boundary conditions, and connectivity of structural parts as well as inherent simplifying assumptions in the development of finite element (FE) models, actual behavior of structures often differs from model predictions. FE model updating comprises a multitude of techniques that systematically calibrate FE models in order to match experimental results. Updating of structural models can be posed as an optimization problem where model parameters that minimize the errors between the responses of the model and actual structure are sought. However, due to limited number of experimental responses and measurement errors, the optimization problem may have multiple admissible solutions in the search domain. Global optimization algorithms (GOAs) are useful and efficient tools in such situations as they try to find the globally optimal solution out of many possible local minima, but are not totally immune to missing the right minimum in complex problems such as those encountered in updating. A methodology based on particle swarm optimization (PSO), a GOA, with sequential niche technique (SNT) for FE model updating is proposed and explored in this article. The combination of PSO and SNT enables a systematic search for multiple minima and considerably increases the confidence in finding the global minimum. The method is applied to FE model updating of a pedestrian cable-stayed bridge using modal data from full-scale dynamic testing.
[154] Shahidi G, Pakzad S N.2013.

Response Surface Model Updating for Nonlinear Structures

. New York: Springer.

DOI      URL      [本文引用: 1]      摘要

Abstract This paper presents a procedure to update nonlinear finite element models in time. In the proposed method, accurate response surface models are constructed and evaluated to replace the finite element model at every time step of the analysis. Then, the optimization problem of model updating is formulated and solved iteratively leading to histograms of the updated model parameters. This methodology is beneficial in extracting more information from measured signals and compensate for the error present in the regressed response surface models. The proposed method was verified through a numerical case study of a steel frame with global nonlinearity. Appropriate design and model orders were successfully established and the optimization in time performed well in the simulated scenarios under the assumption of noise free and noisy measurement data.
[155] da Silva S, Cogan S, Foltête E, et al.2009.

Metrics for nonlinear model updating in structural dynamics

. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 31: 27-34.

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[156] Sipple J D, Sanayei M.2014.

Finite element model updating using frequency response functions and numerical sensitivities

. Structural Control and Health Monitoring, 21: 784-802.

DOI      URL      [本文引用: 1]      摘要

SUMMARYA new method is presented for finite element model (FEM) updating using frequency response functions and numerical sensitivities. The proposed method differs from methods currently in the literature by using numerical sensitivities to solve the inverse problem instead of analytical sensitivities. This method combines the usefulness of commercially available finite element modeling programs with advanced optimization algorithms to solve the inverse problem while requiring neither model reduction nor data expansion. The method is applied to several simulated test cases in which damage is detected and the usefulness of the method is shown. The method proved to be robust, stable, error tolerant, and successful in estimating unknown structural parameters using frequency response functions for FEM updating. Copyright 2013 John Wiley & Sons, Ltd.
[157] Smith M J, Hutton S G.1992.

Frequency modification using Newton's method and inverse iteration eigenvector updating

. AIAA Journal, 30: 1886-1891.

DOI      URL      摘要

http://arc.aiaa.org/doi/abs/10.2514/3.11151
[158] Song W, Dyke S, Harmon T.2012.

Application of nonlinear model updating for a reinforced concrete shear wall

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[159] Song W, Dyke S.2013a.

Development of a cyber-physical experimental platform for real-time dynamic model updating

. Mechanical Systems and Signal Processing, 37: 388-402.

DOI      URL      [本文引用: 1]      摘要

78 A cyber-physical real-time model updating experimental platform is developed. 78 The proposed platform enables updating nonlinear hysteretic systems in real-time. 78 Unscented Kalman filter (UKF) is selected for updating nonlinear hysteretic systems. 78 Implementation aspects related to “hard” real-time computing are discussed. 78 Results from two challenging experiments using the proposed platform are presented.
[160] Song W, Dyke S.2013

b. Real-time dynamic model updating of a hysteretic structural system

. Journal of Structural Engineering, 140: 04013082.

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[161] Sutter T R, Camarda C J, Walsh J L, et al.1989.

Comparison of several methods for calculating vibration mode shape derivatives

. AIAA Journal, 26: 1506-1511.

DOI      URL      [本文引用: 1]      摘要

Abstract Four methods for the calculation of derivatives of vibration mode shapes (eigenvectors) with respect to design parameters are described. These are finite-difference method, modal method, a modified modal method and R.B. Nelson's method. The methods are implemented in a general-purpose commercial finite-element program and applied to the following test problems: a cantilever beam and a stiffened cylinder with a cutout. Design variables are a beam tip mass, a beam root height, and specific dimensions of the cylinder model. The methods are compared on the basis of central processor (CP) seconds required to obtain the derivatives, and two of the methods are also evaluated for the rapidity of convergence. Results indicate an advantage in using Nelson's method because this method is exact and requires less CP time, especially when derivatives with respect to several design variables are computed.
[162] Venture C, Brincker R, Dascotte E, et al.2001.

FEM updating of the heritage court building structure

//19th International Modal Analysis Conference, 2001, Kissimmee, USA, Vol. 1, 324-330.

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[163] Wang H, LI A Q, Miao C Q.2005.

Finite element model updating and validating of Runyang Suspension Bridge based on SHMS

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[164] Wang X, Hill T L, Neild S A, et al.2018.

Model updating strategy for structures with localised nonlinearities using frequency response measurements

. Mechanical Systems and Signal Processing, 100: 940-961.

DOI      URL      [本文引用: 2]      摘要

This paper proposes a model updating strategy for localised nonlinear structures. It utilises an initial finite-element (FE) model of the structure and primary harmonic response data taken from low and high amplitude excitations. The underlying linear part of the FE model is first updated using low-amplitude test data with established techniques. Then, using this linear FE model, the nonlinear elements are localised, characterised, and quantified with primary harmonic response data measured under stepped-sine or swept-sine excitations. Finally, the resulting model is validated by comparing the analytical predictions with both the measured responses used in the updating and with additional test data. The proposed strategy is applied to a clamped beam with a nonlinear mechanism and good agreements between the analytical predictions and measured responses are achieved. Discussions on issues of damping estimation and dealing with data from amplitude-varying force input in the updating process are also provided.
[165] Wang Z C, Xin Y, Ren W X.2015.

Nonlinear structural model updating based on instantaneous frequencies and amplitudes of the decomposed dynamic responses

. Engineering Structures, 100: 189-200.

DOI      URL      [本文引用: 1]      摘要

This paper proposes a new nonlinear structural model updating method based on the instantaneous frequencies and amplitudes of the decomposed dynamic responses under forced vibration. The instantaneous frequencies and amplitudes of the decomposed mono-component are first extracted by analytical mode decomposition (AMD) and Hilbert transform. Then, an objective function based on the residuals of instantaneous frequencies and amplitudes between experimental structure and nonlinear model is created for calibration of the nonlinear model. In this paper, the structural nonlinear properties are simulated by using hysteresis material parameters of Bouc en model, and the optimal values of the hysteresis parameters are obtained by minimizing the objective function using the simulated annealing global optimization method. To validate the effectiveness of the proposed method, a three-story nonlinear shear type structure under earthquake and harmonic excitations is simulated as a numerical example. Then, the proposed method is verified by the shake table test of a real high voltage switch structure under forced vibration. The updated nonlinear structural model is further evaluated by the shake table test of the switch structure subjected to a new severe excitation. Both numerical and experimental results have shown that the proposed method can effectively update the nonlinear model and the updated model can be further used to predict the nonlinear responses due to new severe excitations.
[166] Wang Z C, Xin Y, Ren W X.2016.

Nonlinear structural joint model updating based on instantaneous characteristics of dynamic responses

. Mechanical Systems and Signal Processing, 76: 476-496.

DOI      URL      [本文引用: 1]      摘要

61This paper proposes a new nonlinear joint model updating method.61A novel objective function is created for calibration of the nonlinear joint model.61The optimal values of the nonlinear joint model are obtained by minimizing the objective function.61The accuracy of the proposed method is quantified both in numerical and experimental applications.
[167] Weng S, Xia Y, Xu Y L, et al.2011.

Substructure based approach to finite element model updating

. Computers and Structures, 89: 772-782.

DOI      URL      [本文引用: 1]      摘要

A substructure-based finite element model updating technique is proposed in this paper. A few eigenmodes of the independent substructures and their associated derivative matrices are assembled into a reduced eigenequation to recover the eigensolutions and eigensensitivities of the global structure. Consequently, only the concerned substructures and the reduced eigenequation are re-analyzed in the optimization process, thus reducing the computational load of the traditional model updating methods which perform on the global structure. Applications of the proposed substructure-based model updating to a frame structure and a practical bridge demonstrate that the present method is computationally effective and efficient.
[168] Worden K, Tomlinson G R.2000. Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. CRC Press.

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[169] Xu Y L, Zhang X H, Zhan S, et al.2011.

Testbed for structural health monitoring of long-span suspension bridges

. Journal of Bridge Engineering, 17: 896-906.

URL      [本文引用: 1]      摘要

Modern structural health monitoring systems have been developed to measure the loading environment and responses of long span suspension bridges. This book systematically introduces the fundamentals and outlines the advanced...
[170] Yuan P P, Wang Z C, Ren W X, et al.2016.

Nonlinear joint model updating using static responses

. Advances in Mechanical Engineering, 8: 1-15.

[171] Yuan Z X, Yu K P.2015.

Finite element model updating of damped structures using vibration test data under base excitation

. Journal of Sound and Vibration, 340: 303-316.

DOI      URL      [本文引用: 1]      摘要

A new method to update the finite element models of damped structures using vibration test data under base excitation is presented in this paper, which is used on the model updating of a simplified satellite structure. The main idea of the presented method is to update stiffness, mass and damping parameters of the finite element model using vibration test data under base excitation. Updating of damping parameters plays an important role in the calculation of system response. A simulated numerical case is used to prove the effectiveness of the proposed method, and the proper order of parameters to be updated is studied in the mean while. From the result of the comparative study, physical parameters like stiffness and mass should be updated prior to damping parameters. By the idea of “step by step” updating, the proposed method is used to update a simplified satellite model. In the process of updating, the sensitivity of parameters is studied before the updating, which leads to a better selection of updating parameter. After the updating, responses of key nodes predicted by the updated model matches the test result well in all measured direction.
[172] Zang C, Grafe H, Imregun M.2001.

Frequency--domain criteria for correlating and updating dynamic finite element models

. Mechanical Systems and Signal Processing, 15: 139-155.

DOI      URL      [本文引用: 1]      摘要

http://linkinghub.elsevier.com/retrieve/pii/S0888327000913578
[173] Zang C, Schwingshackl C W, Ewins D J.2008.

Model validation for structural dynamic analysis: An approach to the Sandia Structural Dynamics Challenge

. Computer Methods in Applied Mechanics and Engineering, 197: 2645-2659.

DOI      URL      [本文引用: 2]      摘要

The validation of mathematical models constructed for the dynamic analysis of critical structures is a very important, but complex, process. The essential requirement is to provide confirmation, using independent and more reliable data than that presented by the model in question, that the subject model is capable of describing the essential physics of the structure 檚 behaviour within the required accuracy. In this paper, the procedures of model validation using experimental data on a structure are summarised and applied to a structural dynamics validation problem developed by Sandia National Laboratories. One of the essential issues is to separate out any non-linear features of the system and to construct an appropriate linear model that is as accurate as possible to cope with variability of the subsystem structures. The linear model, which is constructed using simulated test data from an assembly of sample subsystems, is expressed as a mean model with a standard deviation. It is further used in the system response prediction for system accreditation and target application under specified excitation loads. The influence of the weak non-linearity features are neglected in the system response prediction because the experimental method used to derive the test data obscured the non-linear effects and precluded their identification. Further consideration of identification and modelling of the non-linear element for the Sandia 3DOF calibration system is discussed to evaluate its influence on the accuracy of the spatial model.
[174] Zhou S, Song W.2016. A Finite Element Model Updating Method Considering Environmental Impacts//Topics in Modal Analysis & Testing, Springer International Publishing, Vol. 10, 313-324.

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