{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T12:52:47Z","timestamp":1777639967972,"version":"3.51.4"},"reference-count":49,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,6,4]],"date-time":"2020-06-04T00:00:00Z","timestamp":1591228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51708203"],"award-info":[{"award-number":["51708203"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["531118010047"],"award-info":[{"award-number":["531118010047"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Open Research Fund Program for innovation platforms of universities in the Hunan province from the Education Department of Hunan Province","award":["17K022"],"award-info":[{"award-number":["17K022"]}]},{"name":"Independent research project funds from the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body in Hunan University","award":["71860006"],"award-info":[{"award-number":["71860006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A novel probabilistic approach for model updating based on approximate Bayesian computation with subset simulation (ABC-SubSim) is proposed for damage assessment of structures using modal data. The ABC-SubSim is a likelihood-free Bayesian approach in which the explicit expression of likelihood function is avoided and the posterior samples of model parameters are obtained using the technique of subset simulation. The novel contributions of this paper are on three fronts: one is the introduction of some new stopping criteria to find an appropriate tolerance level for the metric used in the ABC-SubSim; the second one is the employment of a hybrid optimization scheme to find finer optimal values for the model parameters; and the last one is the adoption of an iterative approach to determine the optimal weighting factors related to the residuals of modal frequency and mode shape in the metric. The effectiveness of this approach is demonstrated using three illustrative examples.<\/jats:p>","DOI":"10.3390\/s20113197","type":"journal-article","created":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T03:32:21Z","timestamp":1591327941000},"page":"3197","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Probabilistic Updating of Structural Models for Damage Assessment Using Approximate Bayesian Computation"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8608-5767","authenticated-orcid":false,"given":"Zhouquan","family":"Feng","sequence":"first","affiliation":[{"name":"Key Laboratory of Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China"},{"name":"State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China"}]},{"given":"Yang","family":"Lin","sequence":"additional","affiliation":[{"name":"Key Laboratory of Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China"}]},{"given":"Wenzan","family":"Wang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6150-2563","authenticated-orcid":false,"given":"Xugang","family":"Hua","sequence":"additional","affiliation":[{"name":"Key Laboratory of Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China"},{"name":"State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China"}]},{"given":"Zhengqing","family":"Chen","sequence":"additional","affiliation":[{"name":"Key Laboratory of Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410082, China"},{"name":"State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1177\/1475921704045625","article-title":"Damage detection of reinforced concrete beams with novel distributed crack\/strain sensors","volume":"3","author":"Chen","year":"2004","journal-title":"Struct. Heal. Monit. An. Int. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1111\/j.1467-8667.2007.00467.x","article-title":"Parametric identification for a truss structure using axial strain","volume":"22","author":"Xu","year":"2007","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Fan, L., Bao, Y., and Chen, G. (2018). Feasibility of distributed fiber optic sensor for corrosion monitoring of steel bars in reinforced concrete. Sensors, 18.","DOI":"10.3390\/s18113722"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1098\/rsta.2000.0717","article-title":"Vibration-based structural damage identification","volume":"359","author":"Farrar","year":"2001","journal-title":"Philos. Trans. R. Soc. A Lond. Ser. Math. Phys. Eng. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.ymssp.2014.11.001","article-title":"Dealing with uncertainty in model updating for damage assessment: A review","volume":"56\u201357","author":"Simoen","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"e2146","DOI":"10.1002\/stc.2146","article-title":"Investigation of dynamic properties of long-span cable-stayed bridges based on one-year monitoring data under normal operating condition","volume":"25","author":"Mao","year":"2018","journal-title":"Struct. Control Health Monit."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1061\/(ASCE)0733-9399(1998)124:4(455)","article-title":"Updating models and their uncertainties. I: Bayesian statistical framework","volume":"124","author":"Beck","year":"1998","journal-title":"J. Eng. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1016\/S0168-874X(00)00071-8","article-title":"Civil structure condition assessment by FE model updating: Methodology and case studies","volume":"37","author":"Brownjohn","year":"2001","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1061\/(ASCE)0733-9445(2005)131:4(617)","article-title":"Structural finite element model updating using ambient vibration test results","volume":"131","author":"Jaishi","year":"2005","journal-title":"J. Struct. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1845","DOI":"10.1002\/nme.2151","article-title":"An improved perturbation method for stochastic finite element model updating","volume":"73","author":"Hua","year":"2008","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.strusafe.2014.06.004","article-title":"A novel Bayesian approach for structural model updating utilizing statistical modal information from multiple setups","volume":"52","author":"Yan","year":"2015","journal-title":"Struct. Saf."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Lee, Y.-J., and Cho, S. (2016). SHM-based probabilistic fatigue life prediction for bridges based on FE model updating. Sensors, 16.","DOI":"10.3390\/s16030317"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1016\/j.cma.2017.01.030","article-title":"Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment","volume":"318","author":"Huang","year":"2017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Asadollahi, P., Huang, Y., and Li, J. (2018). Bayesian finite element model updating and assessment of Cable-stayed bridges using wireless sensor data. Sensors, 18.","DOI":"10.3390\/s18093057"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Qin, S., Zhang, Y., Zhou, Y.-L., and Kang, J. (2018). Dynamic model updating for bridge structures using the Kriging model and PSO algorithm ensemble with higher vibration modes. Sensors, 18.","DOI":"10.3390\/s18061879"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.compstruc.2012.10.017","article-title":"Identification of structural models using a modified Artificial Bee Colony algorithm","volume":"116","author":"Sun","year":"2013","journal-title":"Comput. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1016\/j.ymssp.2007.10.004","article-title":"A multistage FE updating procedure for damage identification in large-scale structures based on multiobjective evolutionary optimization","volume":"22","author":"Perera","year":"2008","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1016\/j.jsv.2014.01.015","article-title":"A new multi-objective approach to finite element model updating","volume":"333","author":"Jin","year":"2014","journal-title":"J. Sound Vib."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1002\/stc.424","article-title":"Bayesian system identification based on probability logic","volume":"17","author":"Beck","year":"2010","journal-title":"Struct. Control Health Monit."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.ymssp.2015.12.024","article-title":"Probabilistic updating of building models using incomplete modal data","volume":"75","author":"Sun","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4136","DOI":"10.1016\/j.jsv.2013.03.019","article-title":"On prediction error correlation in Bayesian model updating","volume":"332","author":"Simoen","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1007\/s11222-011-9288-2","article-title":"Approximate Bayesian computational methods","volume":"22","author":"Marin","year":"2012","journal-title":"Stat. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jin, S.-S., and Jung, H.-J. (2016, January 21\u201324). Likelihood-free Bayesian computation for structural model calibration: A feasibility study. Proceedings of the SPIE 9803, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2016, Las Vegas, NV, USA.","DOI":"10.1117\/12.2222029"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.ymssp.2019.03.044","article-title":"Probabilistic damage identification incorporating approximate Bayesian computation with stochastic response surface","volume":"128","author":"Fang","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.ymssp.2017.06.017","article-title":"Model selection and parameter estimation in structural dynamics using approximate Bayesian computation","volume":"99","author":"Abdessalem","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.ymssp.2018.12.048","article-title":"Model selection and parameter estimation of dynamical systems using a novel variant of approximate Bayesian computation","volume":"122","author":"Abdessalem","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"A1339","DOI":"10.1137\/130932831","article-title":"Approximate Bayesian computation by Subset Simulation","volume":"36","author":"Chiachio","year":"2014","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/S0266-8920(01)00019-4","article-title":"Estimation of small failure probabilities in high dimensions by Subset Simulation","volume":"16","author":"Au","year":"2001","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.ymssp.2016.02.024","article-title":"Approximate Bayesian computation by Subset Simulation using hierarchical state-space models","volume":"84","author":"Vakilzadeh","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"B168","DOI":"10.1137\/16M1090466","article-title":"Using approximate Bayesian computation by Subset Simulation for efficient posterior assessment of dynamic state-space model classes","volume":"40","author":"Vakilzadeh","year":"2018","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Sunn\u00e5ker, M., Busetto, A.G., Numminen, E., Corander, J., Foll, M., and Dessimoz, C. (2013). Approximate Bayesian computation. PLoS Comput. Biol., 9.","DOI":"10.1371\/journal.pcbi.1002803"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Sisson, S.A., Fan, Y., and Beaumont, M. (2018). Handbook of Approximate Bayesian Computation, Chapman & Hall\/CRC. [1st ed.].","DOI":"10.1201\/9781315117195"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1098\/rsif.2008.0172","article-title":"Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems","volume":"6","author":"Toni","year":"2009","journal-title":"J. R. Soc. Interface"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.compstruc.2011.10.017","article-title":"Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions","volume":"92","author":"Zuev","year":"2012","journal-title":"Comput. Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1111\/j.1467-8667.2008.00589.x","article-title":"Uncertainty and sensitivity analysis of damage identification results obtained using finite element model updating","volume":"24","author":"Moaveni","year":"2009","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.strusafe.2010.03.006","article-title":"Damage identification study of a seven-story full-scale building slice tested on the UCSD-NEES shake table","volume":"32","author":"Moaveni","year":"2010","journal-title":"Struct. Saf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1861","DOI":"10.2514\/2.279","article-title":"Updating large finite element models in structural dynamics","volume":"36","author":"Moller","year":"1998","journal-title":"AIAA J."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Jia, X., and Papadimitriou, C. (2019, January 24\u201326). Data features-based likelihood-informed bayesian finite element model updating. Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Athens, Greece.","DOI":"10.7712\/120219.6328.18902"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"667","DOI":"10.12989\/sem.2009.32.5.667","article-title":"FE model updating based on hybrid genetic algorithm and its verification on numerical bridge model","volume":"32","author":"Jung","year":"2009","journal-title":"Struct. Eng. Mech."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1111\/mice.12142","article-title":"A hybrid optimization algorithm with Bayesian inference for probabilistic model updating","volume":"30","author":"Sun","year":"2015","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1093\/imamat\/6.1.76","article-title":"The convergence of a class of double-rank minimization algorithms 1. General considerations","volume":"6","author":"Broyden","year":"1970","journal-title":"IMA J. Appl. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1093\/comjnl\/13.3.317","article-title":"A new approach to variable metric algorithms","volume":"13","author":"Fletcher","year":"1970","journal-title":"Comput. J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1090\/S0025-5718-1970-0258249-6","article-title":"A family of variable-metric methods derived by variational means","volume":"24","author":"Goldfarb","year":"1970","journal-title":"Math. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1090\/S0025-5718-1970-0274029-X","article-title":"Conditioning of quasi-Newton methods for function minimization","volume":"24","author":"Shanno","year":"1970","journal-title":"Math. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"459","DOI":"10.12989\/sss.2012.10.4_5.459","article-title":"Finite element model updating of Canton Tower using regularization technique","volume":"10","author":"Truong","year":"2012","journal-title":"Smart Struct. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.ymssp.2006.05.011","article-title":"Structural identification based on optimally weighted modal residuals","volume":"21","author":"Christodoulou","year":"2007","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1177\/1475921704047499","article-title":"New Bayesian model updating algorithm applied to a structural health monitoring benchmark","volume":"3","author":"Ching","year":"2004","journal-title":"Struct. Health Monit."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Yi, T.-H., Li, H.-N., and Gu, M. (2011). Optimal sensor placement for health monitoring of high-rise structure based on genetic algorithm. Math. Probl. Eng., 2011.","DOI":"10.1155\/2011\/395101"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"612726","DOI":"10.1155\/2012\/612726","article-title":"Methodology developments in sensor placement for health monitoring of civil infrastructures","volume":"8","author":"Yi","year":"2012","journal-title":"Int. J. Distrib. Sens. Netw."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/11\/3197\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:35:45Z","timestamp":1760175345000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/11\/3197"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,4]]},"references-count":49,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["s20113197"],"URL":"https:\/\/doi.org\/10.3390\/s20113197","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,4]]}}}