为了保证结构的安全可靠，必须建立足够精确的有限元模型。由于各种简化和假定，以及不确定的边界约束， 利用实际工程结构的有限元模型得到的预测结果与实测响应之间不可避免地出现差异。模型修正的目的就是根据测量数据修正有限元模型的未知参数，使仿真结果与实际响应更吻合。考虑到结构模型和测量数据的随机性， 开展随机模型修正已成为必然。本文充分利用静力测量数据，基于随机同伦分析方法，提出了新的随机模型修正方法。 所提出的方法在测量误差或模型修正参数变异性较大时，能够有效地实现结构的模型修正。
本文主要研究工作如下：
1、 介绍了一种新的随机数值方法——随机同伦分析方法。 以待定参数的随机算子方程为基础，构造了关于待定参数的同伦形变方程，建立起算子方程的猜测解与真实解之间的同伦级数关系。求解同伦形变方程确定待定参数的同伦级数的各阶项系数。采用样本点法或概率残差最小化法来确定合适的辅助参数 h 值，以控制同伦级数的收敛域和收敛速度。 该方法突破了传统摄动方法的局限，不受随机参数小变异性的影响，能够解决较强非线性问题， 将关于单自变量的确定性同伦分析方法扩展到多随机变量的情况。
2、提出了基于同伦分析的静力随机有限元模型修正方法。基于同伦的概念，提出了一种新的静力随机有限元模型修正方法。 将结构参数和测量误差假定为有界分布的随机变量，用单元修正因子表示单元刚度的改变量， 建立了关于单元修正因子的随机有限元模型修正方程。构造了随机模型修正方程的同伦形变方程， 并求解得到了随机修正因子的同伦级数解。与低阶摄动方法不同，该方法在随机测量误差变异性较大时具有更好的修正精度。相比基于延缓拒绝自适应抽样的贝叶斯模型修正方法和快速贝叶斯模型修正方法，该方法在修正结果精度相当的情况下计算量更小。
3、提出了基于同伦分析的静力贝叶斯模型修正方法。该方法利用随机同伦分析方法建立了一种新的随机响应的代理模型，并将其与经典贝叶斯方法相结合，实现了结构的静力贝叶斯模型修正。 基于随机同伦分析方法构造了结构参数（物理特性、材料参数、几何尺寸等）与静力响应之间的同伦关系， 即随机代理模型。 以此为基础，建立了关于静力测量挠度的目标函数和似然函数。 选用高斯分布作为先验概率分布， 并利用经典延缓拒绝自适应抽样算法确定出模型修正参数的后验概率分布。 相比传统的贝叶斯模型修正方法，该方法不仅保证了模型修正的精度，而且提高了计算效率， 在随机测量误差的变异性增大时仍能保证较好的修正效果。
4、 提出了考虑静力测量误差相关性的同伦随机模型修正方法， 开展了静力测量误差相关性的研究。 利用 Karhunen-Loeve 展开将测量误差中相关的随机变量转化为独立随机变量的线性组合， 建立了静力随机模型修正方程。 基于随机同伦分析方法，求解了随机模型修正方程中的修正因子，研究了静力测量误差相关性对结构模型修正的影响。 不同于其他不确定性模型修正方法在处理测量误差的相关性时多采用的是区间模型和椭球凸模型，所提出的方法通过有限元模型考虑测量的相关性，能够在测量误差变异性较大时，保证修正结果与实测响应吻合得更好。 简支梁和连续梁的数值算例和试验结果说明，对于单跨简支梁，不同测点位移测量值之间的相关性对模型修正结果的影响很大，不容忽略； 两跨连续梁不同跨间的测量位移相关性对修正结果的影响较小，几乎可以忽略不计。因此，在实际工程结构的模型修正中，测量数据的相关性需得到重视。
 

[1] Mottershead J E, Friswell M I. Model Updating In Structural Dynamics: A Survey[J]. Journal of Sound & Vibration, 1993, 167(2):347-375.

[2] 张令弥. 动态有限元模型修正技术及其在航空航天结构中的应用[J]. 强度与环境, 1994,1 (2):10-17.

[3] 郭勤涛, 张令弥, 费庆国. 结构动力学有限元模型修正的发展-模型确认[J]. 力学进展, 2006, 36(1):38-44.

[4] Friswell M I. Finite Element Model Updating in Structural Dynamics[M]. Netherlands： Kluwer Academic Publishers, 1995.

[5] 朱宏平, 徐斌, 黄玉盈. 结构动力模型修正方法的比较研究及评估[J]. 力学进展, 2002，32(4):513-525

[6] Sehgal S, Kumar H. Structural Dynamic Model Updating Techniques: A State of the Art Review[J]. Archives of Computational Methods in Engineering, 2015, 23(3):1-19.

[7] Simoen E, Roeck G, Lombaert G. Dealing with uncertainty in model updating for damage assessment: A review [J]. Mechanical Systems & Signal Processing, 2015, 56-57(56):123-149.

[8] 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):543-575.

[9] 翁顺, 朱宏平. 基于有限元模型修正的土木结构损伤识别方法[J]. 工程力学, 2021, 38(3):1-16.

[10] 中华人民共和国交通运输部. JTG/TJ21-01-2015. 公路桥梁荷载试验规程[S]. 北京: 人民交通出版社, 2015.

[11] Sanayei M, Scampoli S F. Structural element stiffness identification from static test data[J].Journal of Engineering Mechanics, 1991, 117(5):1021-1036.

[12] Sanayei M, Onipede O. Damage assessment of structures using static test data[J]. AIAA Journal, 1991, 29(7):1174-1179.

[13] Sanayei M, Imbaro G R, McClain J A S, et al. Structural model updating using experimental static measurements[J]. Journal of structural engineering, 1997, 123(6):792-798.

[14] Sanayei M, Saletnik M J. Parameter Estimation of Structures from Static Strain Measurements.I: Formulation[J]. Journal of Structural Engineering, 1996, 122(5):555-562.

[15] Sanayei M, Saletnik M J. Parameter Estimation of Structures from Static Strain Measurements.II: Error Sensitivity Analysis[J]. Journal of Structural Engineering, 1996, 122(5):563-572.

[16] Banan M R, Hjelmstad K D. Parameter Estimation of Structures from Static Response. I.Computational Aspects[J]. Journal of Structural Engineering, 2015, 120(11):3243-3258.

[17] Banan M R, Hjelmstad K D. Parameter Estimation of Structures from Static Response. II: Numerical Simulation Studies[J]. Journal of Structural Engineering, 2015, 120(11):3259-3283.

[18] 李义强, 张彦兵, 王新敏. 基于参数识别的钢筋混凝土简支梁桥静力模型修正技术[J].石家庄铁道学院学报, 2006, 19(3):48-51.

[19] 钟颖. 基于静力测试数据的桥梁结构有限元模型修正[D]. 成都: 西南交通大学, 2009.

[20] 李旺东. 基于静力测试的拱桥模型修正及承载力评定[D]. 西安: 长安大学, 2010.

[21] 何志军. 桥梁结构刚度参数静力方法优化识别及有限元模型修正[D]. 长沙: 长沙理工大学, 2011.

[22] 周涵宇. 静力模型修正与统计分析在混凝土梁桥评定中的应用[D]. 哈尔滨: 哈尔滨工业大学, 2015.

[23] 毛建平. 基于神经网络的桥梁结构静力有限元模型修正[D]. 吉林: 吉林大学, 2011.

[24] Yuan P P, Wang Z C, Ren W X, et al. Nonlinear joint model updating using static responses[J].Advances in Mechanical Engineering, 2016, 8(12):1-15.

[25] 陈涛. 基于静荷载试验数据的桥梁结构参数识别[D]. 哈尔滨: 哈尔滨工业大学, 2013.

[26] 邓苗毅, 任伟新. 基于静力荷载试验的连续箱梁桥结构有限元模型修正[J]. 福州大学学报(自然科学版), 2009, 37(2):261-266.

[27] 向天宇, 赵人达, 蒲黔辉, 等. 基于静力测试数据的装配式混凝土简支梁有限元模型修正[J]. 公路交通科技, 2006, 23(10):79-82.

[28] 吴高杰. 加固后混凝土连续箱梁计算模型修正方法研究[D]. 南京: 东南大学, 2010.

[29] 郭力, 李兆霞, 高效伟. 基于复域灵敏度分析的静力模型修正方法研究[J]. 工程力学,2010, 27(8):100-106.

[30] 魏锦辉, 任伟新, 李广慧. 基于虚拟变形方法的结构有限元模型修正[J]. 工业建筑,2014, 1(11):85-90.

[31] 蒋赢达, 史莉娜. 某钢桁架桥基于静力数据的有限元模型修正[J]. 西部交通科技, 2010,1(9):70-73.

[32] 李书, 冯太华. 一种利用静力试验数据修正有限元模型的方法[J]. 应用力学学报, 1995,12(3):52-56.

[33] 邓苗毅,任伟新,王复明.基于静力响应面的结构有限元模型修正方法[J].试验力学, 2008,23(2):103-109.

[34] Ren W X, Fang S E, Deng M Y. Response Surface–Based Finite-Element-Model Updating Using Structural Static Responses[J]. Journal of Engineering Mechanics, 2011, 137(4):248-257.

[35] 代汉超, 石雪飞. 基于静力响应面的混凝土梁桥有限元模型修正[J]. 交通科学与工程,2013, 29(2):41-45.

[36] Liu Y, Ma J, Nie J, et al. Virtual Distortion Method-Based Finite Element Model Updating of Bridges by Using Static Deformation[J]. Journal of Engineering Mechanics, 2015, B4015003:1-9.

[37] Liu Y, Yang C, Tan Z. Hybrid element-based virtual distortion method for finite element model updating of bridges with wide-box girders[J]. Engineering Structures, 2017, 143(7):558-570.

[38] 杨秋伟, 刘济科, 李翠红. 基于局部静力测试的约束子结构修正法[J]. 振动与冲击,2012, 31(9):46-50.

[39] 郑惠强, 陈鹏程, 宓为建, 等. 大型桥吊结构动力有限元模型修正[J]. 同济大学学报(自然科学版), 2001, 29(12):1412-1415.

[40] 杜青, 蔡美峰, 张献民, 等. 钢筋混凝土桥梁结构动力有限元模型修正[J]. 公路交通科技, 2006, 23(1):60-62.

[41] 刘继承, 周传荣. 一个基于优化的有限元模型修正方法[J]. 振动与冲击, 2003, 22(2):33-34.

[42] Wu J R, Li Q S. Finite element model updating for a high-rise structure based on ambient vibration measurements[J]. Engineering Structures, 2004, 26(7):979-990.

[43] 费庆国, 张令弥, 李爱群, 等. 基于不同残差的动态有限元模型修正的比较研究[J]. 振动与冲击, 2005, 24(4):24-26.

[44] Sarmadi H, Karamodin A, Entezami A. A new iterative model updating technique based on least squares minimal residual method using measured modal data[J]. Applied Mathematical Modelling, 2016, 40(23-24):10323-10341.

[45] 袁永新, 戴华. 用不完全模态测量数据修正有限元分析模型[J]. 振动与冲击, 2006,25(6):154-156.

[46]袁永新, 蒋家尚. 一种运用不完全模态试验数据的无溢出模型修正方法[J]. 振动与冲击,2008, 27(7):1-3.

[47] Hu S, Li H, Wang S. Cross-model cross-mode method for model updating[J]. Mechanical Systems & Signal Processing, 2007, 21(4):1690-1703.

[48] 李剑. 有限元模型参数型修正方法的研究[D]. 上海: 上海交通大学, 2007.

[49] 侯吉林, 欧进萍. 基于局部模态的约束子结构模型修正法[J]. 力学学报, 2009,

41(5):748-756.

[50] 翁顺, 左越, 朱宏平, 等. 基于子结构的有限元模型修正方法[J]. 振动与冲击, 2017,36(4):99-104.

[51] Jaishi B, Ren W X. Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimisation technique[J]. Mechanical Systems and Signal Processing, 2007, 21(5):2295-2317.

[52] 魏锦辉, 任伟新, 万华平. 基于模态柔度的有限元模型修正方法[J]. 振动与冲击, 2013,32(13):106-111.

[53] Masoud P, Akbar E, Reza K M. Finite element model updating using strain‐ based power spectral density for damage detection[J]. Structural Control and Health Monitoring, 2016, 23(11):1314-1333.

[54] 徐张明, 高天明, 沈荣瀛,等. 一种改进的利用频响函数进行有限元模型修正的方法[J].振动与冲击, 2002, 21(3):43-45.

[55] 朱凼凼, 冯咬齐. 应用位移频响函数进行模型修正[J]. 宇航学报, 2006, 27(2):201-204.

[56] Lin R M, Zhu J. Finite element model updating using vibration test data under base excitation[J]. Journal of Sound & Vibration, 2007, 303(3-5):596-613.

[57] Sipple J D, Sanayei M. Finite element model updating using frequency response functions and numerical sensitivities[J]. Structural Control & Health Monitoring, 2014, 21(5):784-802.

[58] Garcia-Palencia A J, Santini-Bell E, Sipple J D, et al. Structural model updating of an in‐service bridge using dynamic data[J]. Structural Control and Health Monitoring, 2015, 22(10):1265-1281.

[59] Esfandiari A, Rahai A, Sanayei M, et al. Model Updating of a Concrete Beam with Extensive Distributed Damage Using Experimental Frequency Response Function[J]. Journal of Bridge Engineering, 2016, 21(4):04015081.

[60] Lin R M. Function-weighted frequency response function sensitivity method for analytical model updating[J]. Journal of Sound and Vibration, 2017, 403(1):59-74.

[61] 李伟明. 有限元模型修正方法及自由度匹配迭代技术研究[D]. 上海: 上海交通大学,2011.

[62] 张坤, 段忠东, 刘洋. 连续刚构桥动力特性参数识别与有限元模型修正[J]. 公路交通科技, 2008, 25(9):67-72.

[63] Cao Z, Fei Q, D Jiang, et al. Dynamic sensitivity-based finite element model updating for nonlinear structures using time-domain responses[J]. International Journal of Mechanical Sciences,2020, 15(184):105788.

[64] 姚昌荣, 李亚东. 基于静动力测试数据的斜拉桥模型修正[J]. 铁道学报, 2008, 30(3):65-70.

[65] 谢瑞杰. 基于静动力有限元模型修正的既有钢筋混凝土拱桥承载能力评估[D]. 长沙：中南大学, 2010.

[66] 韩万水, 王涛, 李永庆, 等. 大跨钢桁架悬索桥有限元模型实用修正方法[J]. 交通运输工程学报, 2011, 11 (5):18-27.

[67] 赵崇基. 基于模型修正的混凝土连续梁桥运营安全性能评估的试验研究[D]. 太原: 太原理工大学, 2016.

[68] Schommer S, Nguyen V H, Maas S, et al. Model updating for structural health monitoring using static and dynamic measurements[J]. Procedia engineering, 2017, 199:2146-2153.

[69]何涛, 张巍, 吴植安. 基于动静载试验数据的预应力混凝土梁模型修正方法试验研究[J].公路交通科技, 2015, 32(12):75-80.

[70] Jung D S, Kim C Y. Finite element model updating on small-scale bridge model using the hybrid genetic algorithm[J]. Structure & Infrastructure Engineering, 2013, 9(5):481-495.

[71] Shan D S, Li Q, Khan I, Zhou X H. A novel finite element model updating method based on substructure and response surface model[J]. Engineering Structures, 2015, 103(12):147-156.

[72] Hasancebi O, Dumlupinar T. Linear and nonlinear model updating of reinforced concrete Tbeam bridges using artificial neural networks[J]. Computers & Structures, 2013, 119(4):1-11.

[73] 魏锦辉, 任伟新. 基于响应面方法的桥梁静动力有限元模型修正[J]. 公路交通科技,2015, 32(2):68-73.

[74] 李刚. 基于静动力响应面的梁桥有限元模型修正方法研究[D]. 成都: 西南交通大学,2018.

[75] 张启伟, 范立础. 利用动静力测量数据的桥梁结构损伤识别[J]. 同济大学学报(自然科学版), 1998, 26(5):528-532.

[76] 田军. 有限元模型静力—模态协同修正技术[D]. 西安: 西北工业大学, 2004.

[77] 袁爱民. 基于灵敏度分析的有限元模型修正技术若干关键问题研究[D]. 南京: 东南大学, 2006.

[78] 夏樟华. 基于静动力的桥梁结构有限元模型修正[D]. 福州: 福州大学, 2006.

[79] Kim S, Kim N, Park Y S, Jin S S. A Sequential Framework for Improving Identifiability of FE Model Updating using Static and Dynamic Data [J]. Sensors, 2019, 19(23):5099.

[80] Cao W J, K Oh C, Smith I. Enhancing static-load-test identification of bridges using dynamic data [J]. Engineering Structures, 2019, 186(5):410-420.

[81] Steenackers G, Guillaume P. Finite element model updating taking into account the uncertainty on the modal parameters estimates[J]. Journal of Sound and Vibration, 2006, 296(4-5):919-934.

[82] Govers Y, Link M. Stochastic model updating—Covariance matrix adjustment from uncertain experimental modal data[J]. Mechanical Systems and Signal Processing, 2010, 24(3):696-706.

[83] Hua X G, Asce Y Q N M, Asce Z Q C M, et al. Monte Carlo Study of the Effect of Measurement Noise in Model Updating with Regularization[J]. Journal of Engineering Mechanics,2011, 138(1):71-81.

[84] Boulkaibet I, Mthembu L, Marwala T, et al. Finite Element Model Updating Using the Shadow Hybrid Monte Carlo Technique[M]. Springer International Publishing, 2014.

[85] Boulkaibet I, Mthembu L, Marwala T, et al. Finite element model updating using the shadow hybrid Monte Carlo technique[J]. Mechanical Systems and Signal Processing, 2015, 52:115-132.

[86] Savadkoohi A T, Molinari M, Bursi O S, et al. Finite element model updating of a semi‐ rigid moment resisting structure[J]. Structural Control and Health Monitoring, 2011, 18(2):149-168.

[87] Joubert D J, Marwala T. Monte Carlo Dynamically Weighted Importance Sampling For Finite Element Model Updating[M]. Springer International Publishing, 2016.

[88] Jasra A, Law K, Suciu C. Advanced Multilevel Monte Carlo Methods[J]. International Statistical Review, 2020, 88(3):548-579.

[89] Mares C, Mottershead J E, Friswell M I. Stochastic model updating: Part 1—theory and simulated example[J]. Mechanical Systems & Signal Processing, 2006, 20(7):1674-1695

[90] Mottershead J E, Mares C, James S, et al. Stochastic model updating: Part 2—application to a set of physical structures[J]. Mechanical Systems & Signal Processing, 2006, 20(8):2171-2185.

[91] Bi S, Deng Z, Chen Z. Stochastic validation of structural FE-models based on hierarchical cluster analysis and advanced Monte Carlo simulation[J]. Finite Elements in Analysis & Design,2013, 67(5):22-33.

[92] Deng Z, Bi S, Atamturktur S. Stochastic model updating using distance discrimination analysis[J]. Chinese Journal of Aeronautics, 2014, 27(5):1188-1198.

[93] Boulkaibet I, Mthembu L, Marwala T, et al. Finite Element Model Updating Using an Evolutionary Markov Chain Monte Carlo Algorithm[C]. Conference Proceedings of the Society for Experimental Mechanics Series. Dynamics of Civil Structures, 2015, 2 (1):245-253.

[94] Bao N, Wang C. A Monte Carlo simulation based inverse propagation method for stochastic model updating[J]. Mechanical Systems and Signal Processing, 2015, 60-61:928-944.

[95] Zhai X, Fei C W, Choy Y S, et al. A stochastic model updating strategy-based improved response surface model and advanced Monte Carlo simulation[J]. Mechanical Systems and Signal

Processing, 2017, 82:323-338.

[96] 费庆国, 张令弥, 李爱群, 等. 基于统计分析技术的有限元模型修正研究[J]. 振动与冲击, 2005, 24(3):23-26.

[97] 任伟新, 陈华斌. 基于响应面的桥梁有限元模型修正[J]. 土木工程学报, 2008,

41(12):73-78.

[98] 韩芳, 钟冬望, 龚相超. 基于信息融合技术的结构模型修正研究[J]. 固体力学学报,2011, 32(1):422-425.

[99] 万华平, 任伟新, 魏锦辉. 基于高斯过程响应面的结构有限元模型修正方法[J]. 振动与冲击, 2012, 31(24):82-87.

[100] 魏锦辉, 任伟新. 结构有限元模型修正的自适应响应面方法[J]. 振动与冲击, 2013,32(8):114-119.

[101] Deng L, Cai C S. Bridge Model Updating Using Response Surface Method and Genetic Algorithm[J]. Journal of Bridge Engineering, 2010, 15(5):553-564.

[102] 周林仁. 桥梁的几类 SHM Benchmark 及模型修正的子结构与响应面方法[D]. 大连:大连理工大学.2012.

[103] 郁胜. 悬索桥有限元模型修正的响应面方法[D]. 大连: 大连理工大学, 2015.

[104] Alpaslan E, Hac?efendio?lu K, Demir G, et al. Response surface‐ based finite‐ element model updating of a historic masonry minaret for operational modal analysis[J]. The Structural Design of Tall and Special Buildings, 2020, 29(9):e1733.

[105] Fang S E, Ren W X, R Perera. A stochastic model updating method for parameter variability quantification based on response surface models and Monte Carlo simulation[J]. Mechanical Systems & Signal Processing, 2012, 33(12):83-96.

[106] Fang S E, Zhang Q H, Ren W X. Parameter variability estimation using stochastic response surface model updating[J]. Mechanical Systems & Signal Processing, 2014, 49(1-2):249-263.

[107] Rui Q, Ouyang H J, Wang H Y. An efficient statistically equivalent reduced method on stochastic model updating[J]. Applied Mathematical Modelling, 2013, 37(8):6079-6096.

[108] 鲍诺, 王春洁, 赵军鹏,等. 基于响应面法的结构动力学模型修正[J]. 振动与冲击, 2013,32(16):54-58.

[109] Chakraborty S, Sen A. Adaptive response surface based efficient finite element model updating[J]. Finite Elements in Analysis and Design, 2014, 80:33-40.

[110] Fang S E, Zhang Q H, Ren W X. An interval model updating strategy using interval response surface models[J]. Mechanical Systems & Signal Processing, 2015, 60-61(8):909-927.

[111] 李中辉. 基于响应面模型和区间分析的斜拉桥有限元模型修正[D]. 成都: 西南交通大学, 2016.

[112] Beck J L, Katafygiotis L S. Updating models and their uncertainties. I: Bayesian statistical framework[J]. Journal of Engineering Mechanics, 1998, 124(4):455-461.

[113] Katafygiotis L S, Beck J L. Updating models and their uncertainties. II: Model identifiability[J]. Journal of Engineering Mechanics, 1998, 124(4):463-467.

[114] Kuok Y S C. Bayesian Methods for Updating Dynamic Models[J]. Applied Mechanics Reviews, 2011, 64(1):3-7.

[115] Boris A, Juan M, Wieger G, et al. Bayesian Finite Element Model Updating Using Static and Dynamic Data[C]. Conference Proceedings of the Society for Experimental Mechanics Series,Structural Dynamics, 2011, 3 (1): 395-402.

[116] 华宏星, 傅志方. 有限元模型修正中的 BAYES 方法的几点讨论[J]. 振动工程学报,1998, 11(1):110.

[117] Zhang E L, Feissel P, Antoni J. A comprehensive Bayesian approach for model updating and quantification of modeling errors[J]. Probabilistic engineering mechanics, 2011, 26(4):550-560.

[118] Lam H F, Yang J, Au S K. Bayesian model updating of a coupled-slab system using field test data utilizing an enhanced Markov chain Monte Carlo simulation algorithm[J]. Engineering Structures, 2015, 102(11):144-155.

[119] Lam H F, Hu J, Yang J H. Bayesian operational modal analysis and Markov chain Monte Carlo-based model updating of a factory building[J]. Engineering Structures, 2017, 132(2):314-336.

[120] Hu J, Lam H F, Yang J H. Operational modal identification and finite element model updating of a coupled building following Bayesian approach[J]. Structural Control & Health Monitoring,2018, 25(2):e2089.

[121] Lam H F, Yang J H, Au S K. Markov chain Monte Carlo‐ based Bayesian method for structural model updating and damage detection[J]. Structural Control & Health Monitoring, 2018,25(4):e2140.

[122] Beck J L, Au S K. Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation[J]. Journal of Engineering Mechanics, 2002, 128(4):380-391.

[123] Cheung S H, Beck J L. Bayesian model updating using hybrid Monte Carlo simulation with application to structural dynamic models with many uncertain parameters[J]. Journal of engineering mechanics, 2009, 135(4): 243-255.

[124] Bansal S. A new Gibbs sampling based Bayesian model updating approach using modal data from multiple setups[J]. International journal for uncertainty quantifications, 2015, 5(4):361-374.

[125] Sherri M, Boulkaibet I, Marwala T, et al. A Differential Evolution Markov Chain Monte Carlo Algorithm for Bayesian Model Updating[C]. Conference Proceedings of the Society for Experimental Mechanics Series, Special Topics in Structural Dynamics, 2018, 5:115-125.

[126] Gunawan D, Dang K D, Quiroz M, et al. Subsampling sequential Monte Carlo for static Bayesian models [J]. Statistics and Computing, 2020, 30(6): 1741-1758.

[127] Yuen K V, Beck J L, Katafygiotis L S. Efficient model updating and health monitoring methodology using incomplete modal data without mode matching[J]. Structural Control & Health Monitoring, 2010, 13(1):91-107.

[128] Yuen K V, Ortiz G A. Multiresolution Bayesian nonparametric general regression for structural model updating[J]. Structural Control & Health Monitoring, 2018, 25(2):e2077.1-14.

[129] Sun H, Büyük?ztürk O. Probabilistic updating of building models using incomplete modal data[J]. Mechanical Systems and Signal Processing, 2016, 75:27-40.

[130] Zhang F L, Ni Y C, Lam H F. Bayesian structural model updating using ambient vibration data collected by multiple setups[J]. Structural Control and Health Monitoring, 2017,24(12):e2023.1-18.

[131] Mustafa S, De bnath, N, Dutta A. Bayesian probabilistic approach for model updating and damage detection for a large truss bridge[J]. International Journal of Steel Structures, 2015,15(2):473-485.

[132] Mustafa S, Matsumoto Y. Bayesian Model Updating and Its Limitations for Detecting Local Damage of an Existing Truss Bridge[J]. Journal of Bridge Engineering, 2017, 22(7):04017019.

[133] 韩芳, 钟冬望, 汪君. 基于贝叶斯法的复杂有限元模型修正研究[J]. 振动与冲击, 2012,31(1):39-43.

[134] 尹涛, 王祥宇, 周越. 基于 Bayesian 证据推断与信息增益的参数化有限元修正模型选择[J]. 振动与冲击, 2018, 320(12):164-171.

[135] Jang J, Smyth A. Bayesian model updating of a full-scale finite element model with sensitivity-based clustering[J]. Structural Control and Health Monitoring, 2017, 24(11): e2004.

[136] VanDerHorn E, Mahadevan S. Bayesian model updating with summarized statistical and reliability data[J]. Reliability Engineering & System Safety, 2018, 172:12-24.

[137] Bartilson D T, Jang J, Smyth A W. Finite element model updating using objective-consistent sensitivity-based parameter clustering and Bayesian regularization[J]. Mechanical Systems and Signal Processing, 2019, 114(1):328-345.

[138] 张建新. 基于贝叶斯方法的有限元模型修正研究[D]. 重庆: 重庆大学, 2014.

[139] Jensen H A, Millas E, Kusanovic D, et al. Model-reduction techniques for Bayesian finite element model updating using dynamic response data[J]. Computer Methods in Applied Mechanics & Engineering, 2014, 279:301-324.

[140] Jensen H A, Esse C, Araya V, et al. Implementation of an adaptive meta-model for Bayesian finite element model updating in time domain[J]. Reliability Engineering & System Safety, 2016,160(4):174-190.

[141] Yan W J, Katafygiotis L S. A novel Bayesian approach for structural model updating utilizing statistical modal information from multiple setups[J]. Structural Safety, 2015, 52:260-271.

[142] Yan W J, Cao S Z, Ren W X, et al. Vectorization and distributed parallelization of Bayesian model updating based on a multivariate complex-valued probabilistic model of frequency response functions[J]. Mechanical Systems and Signal Processing, 2021, 156:107615.

[143] Wan H P, Ren W X. Stochastic model updating utilizing Bayesian approach and Gaussian process model[J]. Mechanical Systems & Signal Processing, 2016, 70–71(3):245-268.

[144] 万华平, 任伟新, 黄天立. 基于贝叶斯推理的随机模型修正方法[J]. 中国公路学报,2016, 29(4):67-76.

[145] 刘纲, 罗钧, 秦阳, 等. 基于改进 MCMC 方法的有限元模型修正研究[J]. 工程力学,2016, 33(6):138-145.

[146] Govers Y, Link M. Stochastic model updating—Covariance matrix adjustment from uncertain experimental modal data[J]. Mechanical Systems & Signal Processing, 2010, 24(3):696-706.

[147] Jacquelin E, Adhikari S, Friswell M I. A second-moment approach for direct probabilistic model updating in structural dynamics[J]. Mechanical systems and signal processing, 2012, 29(5):262-283.

[148] Khodaparast H H, Mottershead J E, Friswell M I. Perturbation methods for the estimation ofparameter variability in stochastic model updating[J]. Mechanical systems and signal processing,2008, 22(8):1751-1773.

[149] Husain N A, Khodaparast H H, Ouyang H J. Parameter selection and stochastic modelupdating using perturbation methods with parameter weighting matrix assignment[J]. Mechanical Systems & Signal Processing, 2012, 32:135-152.

[150] Hua X G, Ni Y Q, Chen Z Q, et al. An improved perturbation method for stochastic finite element model updating[J]. International Journal for Numerical Methods in Engineering, 2010,73(13):1845-1864.

[151] Xie D. A numerical method of structure-preserving model updating problem and its perturbation theory[J]. Applied Mathematics & Computation, 2011, 217(13):6364-6371.

[152] 张林林. 基于静力测量数据的梁式结构统计模型修正[D]. 武汉: 武汉理工大学, 2013.

[153] 姜东, 费庆国, 吴邵庆. 基于摄动法的不确定性有限元模型修正方法研究[J]. 计算力学学报, 2014, 31(4):431-437.

[154] 姜东. 不确定性结构动力学模型修正方法研究[D]. 南京: 东南大学, 2015.

[155] 陈喆, 何欢, 陈国平, 等. 考虑不确定性因素的有限元模型修正方法研究[J]. 振动工程学报, 2017, 30(6): 921-928.

[156] Chen H P, Maung T S. Regularised finite element model updating using measured incomplete modal data[J]. Journal of Sound & Vibration, 2014, 333(21):5566-5582.

[157] Wu Z F, Huang B, Li Y J, et al. A Statistical Model Updating Method of Beam Structures with Random Parameters under Static Load [J]. Applied Sciences, 2017, 7(6):601.

[158] Huang B, Chen H. A new approach for stochastic model updating using the hybrid perturbation-Garlekin method[J]. Mechanical Systems and Signal Processing, 2019, 129(8):1-19.

[159] Ma T, Zhang Y, Huang X. A novel approach for stochastic finite element model updating and parameter estimation[J]. Journal of Mechanical Engineering Science. 2014, 228(18):3329-3342.

[160] 梁锋. 基于摄动分析和子结构的有限元模型修正方法[D]. 武汉: 华中科技大学, 2016.

[161] 刘纲, 杨溥, 侍刚, 岳笛. 大跨度桥梁模型修正方法研究[J]. 桥梁建设, 2008, 4(1):19-22.

[162] Gautier G, Mevel L, Mencik J M, et al. Variance analysis for model updating with a finite element based subspace fitting approach[J]. Mechanical Systems & Signal Processing, 2017,91(7):142-156.

[163] Guo N, Yang Z, Wang L, et al. Dynamic model updating based on strain mode shape and natural frequency using hybrid pattern search technique[J]. Journal of Sound and Vibration, 2018,422:112-130.

[164] 闫桂荣, 段忠东, 欧进萍. 遗传算法在结构有限元模型修正中的应用[J]. 哈尔滨工业大学学报, 2007, 39(2):181-186.

[165] 陈凯. 基于粒子群优化算法的结构动力模型修改[D]. 南京: 南京理工大学, 2007.

[166] Qin S Q, Zhang Y Z, Zhou Y L, et al. Dynamic model updating for bridge structures using the kriging model and PSO algorithm ensemble with higher vibration modes[J]. Sensors, 2018,18(6):1879.

[167] 朱立伟, 李宏伟, 陈礼, 谢家兵. 基于人工鱼群算法的大跨度斜拉桥模型修正算法研究[J]. 公路交通技术, 2015, 2(1):34-38.

[168] Pouyan A, Shapour M, Rahim C. Updating Finite Element Model Using Stochastic Subspace Identification Method and Bees Optimization Algorithm[J]. Latin American Journal of Solids & Structures, 2018, 15(2):e12.

[169] 费庆国, 李爱群, 张令弥. 基于神经网络的非线性结构有限元模型修正研究[J]. 宇航学报, 2005, 26(3):267-269+281.

[170] 何浩祥, 闫维明, 王卓. 基于子结构和遗传神经网络的递推模型修正方法[J]. 工程力学, 2008, 25(4):99-105.

[171] 胡俊亮, 余晓琳, 郑恒斌, 等. 基于优化 BP 神经网络的梁结构有限元模型修正[J]. 华南理工大学学报(自然科学版), 2013,41(8):67-73.

[172] Teughels A, De Roeck G, Suykens J A K. Global optimization by coupled local minimizers and its application to FE model updating[J]. Computers & structures, 2003, 81(24-25):2337-2351.

[173] 刘洋. 高性能优化算法与结构模型修正的研究[D]. 哈尔滨: 哈尔滨工业大学, 2008.

[174] Astroza R, Nguyen L T, Nestorovi? T. Finite element model updating using simulated annealing hybridized with unscented Kalman filter[J]. Computers & Structures, 2016, 177:176-191.

[175] Liao S J. The proposed homotopy analysis technique for the solution of nonlinear problems[D]. Shanghai: Shanghai Jiao Tong University, 1992.

[176] Liao S J. On the homotopy analysis method for nonlinear problems[J]. Applied Mathematics and Computation, 2004, 147(2):499-513.

[177] Liao S J. An optimal homotopy-analysis approach for strongly nonlinear differential equations[J]. Communications in Nonlinear Science & Numerical Simulation, 2010, 15(8):2003-2016.

[178] 廖世俊, 刘曾. 同伦分析方法进展综述[J].力学进展,2019,49(1):237-273.

[179] 孙中奎, 徐伟, 杨晓丽, 等. 基于参数展开的同伦分析技术及其应用[J]. 力学学报,2005, 37(5):667-672.

[180] 徐伟, 孙中奎, 杨晓丽. 基于参数展开的同伦分析法在强非线性随机动力系统中的应用[J]. 物理学报, 2005, 54(11): 5069-5076

[181] Onyejekwe O N. Solution of some parabolic inverse problems by homotopy analysis method[J]. International Journal of Applied Mathematical Research, 2013, 3(1):81-87.

[182] Zhang H, Huang B. A new homotopy-based approach for structural stochastic analysis [J].Probabilistic Engineering Mechanics, 2019, 55:42-53.

[183] 张衡, 王鑫, 陈辉, 等. 基于同伦分析方法的随机结构静力响应求解[J]. 工程力学,2019, 36(11):36-42+70.

[184] Huang B, Zhang H, Phoon K-K. Homotopy approach for random eigenvalue problem [J].International Journal for Numerical Methods in Engineering, 2018, 113(3):450-478.

[185] Stefanou G. The stochastic finite element method: Past, present and future[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198(9-12):1031-1051.

[186] Gao W, Wu D, Song C M, et al. Hybrid probabilistic interval analysis of bar structures with uncertainty using a mixed perturbation Monte-Carlo method[J]. Finite Elements in Analysis and Design, 2011, 47(7):643-652.

[187] Apetre N, Ruzzene M. Spectral and perturbation analysis for ultrasonic guided waves[J].Sound and Vibration, 2012, 331(24):5358-5369.

[188] Sachdeva S K, Nair P B, Keane A J. Comparative study of projection schemes for stochastic finite element analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2006,195(19/22):2371-2392.

[189] Sachdeva S K, Nair P B, Keane A J. Hybridization of stochastic reduced basis methods with polynomial chaos expansions[J]. Probabilistic Engineering Mechanics, 2006, 21(2):182-192.

[190] 孙大风, 郭雪莽. 静力凝聚法及其在工程中的应用[J]. 华北水利水电学院学报, 1988,12(1):35-41.

[191] 张驰. 基于 Guyan 法的有限元模型缩聚技术研究[J]. 装备制造技术, 2013, 2(1):153-154.

[192] 李国庆, 罗帅, 张丽. 自由度缩聚的柔度损伤诊断法[J]. 力学季刊, 2020, 41(3):174-181.

[193] Ahmadian H, Mottershead J E, Friswell M I. Regularisation methods for finite element model updating[J]. Mechanical Systems and Signal Processing. 1998, 12(1):47-64.

[194] Michael I, Friswell J E M A. Finite element model updating using experimental test data:parametrization and regularization[J]. Philosophical Transactions Mathematical Physical & Engineering Sciences, 2001, 359(1778):169-186.

[195] 吴颉尔. 正则化方法及其在模型修正中的应用[D]. 南京： 南京航空航天大学, 2007.

[196] 邱飞力, 张立民, 张卫华. Tikhonov 方法在不适定模型修正中的应用[J]. 振动与冲击,2015, 34(12):126-131+150.

[197] 李英超, 王树青, 张敏. 正则化方法在结构模型修正中的应用研究[J]. 中国海洋大学学报:自然科学版, 2016, 46(9):107-115.

[198] Rubio P B, Louf F, Chamoin L. Fast model updating coupling Bayesian inference and PGD model reduction[J]. Computational Mechanics, 2018, 4(62):1485-1509.

[199] 中华人民共和国住房和城乡建设部. GB/T50152-201. 混凝土结构试验方法标准[S].北京：中国建筑工业出版社, 2012.

[200] Adhikari S, Friswell M I. Distributed parameter model updating using the Karhunen–Loèveexpansion[J]. Mechanical Systems and Signal Processing, 2010, 24(2):326-339.

[201] Machado M R, Adhikari S, Dos Santos J M C, et al. Estimation of beam material random field properties via sensitivity-based model updating using experimental frequency response functions [J]. Mechanical Systems and Signal Processing, 2018, 102:180-197.

[202] Reddy, S. K. Effects of Ignoring Correlated Measurement Error in Structural Equation Models [J]. Educational and Psychological Measurement, 1992, 52(3):549-570.

[203] Ouisse M, Foltête, Emmanuel. Model correlation and identification of experimental reduced models in vibroacoustical modal analysis [J]. Journal of Sound and Vibration, 2015, 342:200-217.

[204] Zimmerman D C, Simmermacher T. Model correlation using multiple static load and vibration tests[J]. AIAA Journal, 1995, 33(11):2182-2188.

[205] Koh B H, Dyke S J. Structural health monitoring for flexible bridge structures using correlation and sensitivity of modal data [J]. Computers and Structures, 2007, 85:117-130.

[206] An Y H, Ou J P. Experimental and numerical studies on model updating method of damage severity identification utilizing four cost functions[J]. Structural Control & Health Monitoring,2013, 20(1):107-120.

[207] Lim J H. A correlation study of satellite finite element model for coupled load analysis using transmissibility with modified correlation measures[J]. Aerospace Science and Technology. 2014,33(1):82-91.

[208] Guo N, Yang Z, Jia Y, et al. Model updating using correlation analysis of strain frequency response function[J]. Mechanical Systems and Signal Processing, 2016, 70-71:284-299.

[209] Modak S V. Model updating using uncorrelated modes[J]. Journal of Sound & Vibration,2014, 333(11):2297-2322.

[210] Kammer D C. Sensor placement for on-orbit modal identification and correlation of large space structures[J]. Journal of Guidance Control & Dynamics, 1991, 14(2):251-259.

[211] Papadimitriou C, Lombaert G. The effect of prediction error correlation on optimal sensor placement in structural dynamics[J].Mechanical Systems and Signal Processing, 2012, 28:105-127.

[212] Simoen E, Papadimitriou C, Lombaert G. On prediction error correlation in Bayesian modelupdating[J]. Journal of Sound and Vibration, 2013,32(18):4136-4152.

[213] Meggitt J. On the treatment of uncertainty in experimentally measured frequency response functions [J]. Metrologia, 2018, 55(6):806-818.

[214] Meggitt J, Moorhouse A T. A covariance based framework for the propagation of correlated uncertainty in frequency based dynamic sub-structuring [J]. Mechanical Systems and Signal Processing, 2020.136:106505.

[215] Liao B P, Zhao R, Yu K P, et al. A novel interval model updating framework based on correlation propagation and matrix-similarity method[J]. Mechanical Systems and Signal Processing, 2022, 162(2):108039.

[216] Ouyang H, Liu J, Han X, et al. Non-probabilistic uncertain inverse problem method considering correlations for structural parameter identification[J]. Structural and Multidisciplinary Optimization, 2021.64:1327-1342.

[217] Wu Z F, Huang B, Chen H, et al. A new homotopy approach for stochastic static model updating with large uncertain measurement errors [J]. Applied Mathematical Modelling, 2021, 98:758-782.