{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T01:04:29Z","timestamp":1775783069933,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T00:00:00Z","timestamp":1614902400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T00:00:00Z","timestamp":1614902400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Grant No. 51975473"],"award-info":[{"award-number":["Grant No. 51975473"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Aviation Science Foundation for the Aviation Key Laboratory of Science and Technology on Life-support Technology","award":["Grant No.201929053001"],"award-info":[{"award-number":["Grant No.201929053001"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2022,8]]},"DOI":"10.1007\/s00366-021-01366-y","type":"journal-article","created":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T15:02:51Z","timestamp":1614956571000},"page":"3417-3437","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["An application of active learning Kriging for the failure probability and sensitivity functions of turbine disk with imprecise probability distributions"],"prefix":"10.1007","volume":"38","author":[{"given":"Pan","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zheng","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyu","family":"Huang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hanyuan","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,3,5]]},"reference":[{"issue":"1","key":"1366_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ress.2004.03.001","volume":"85","author":"JC Helton","year":"2004","unstructured":"Helton JC, Oberkampf WL (2004) Alternative representations of epistemic uncertainty. Reliab Eng Syst Saf 85(1):1\u201310. https:\/\/doi.org\/10.1016\/j.ress.2004.03.001","journal-title":"Reliab Eng Syst Saf"},{"key":"1366_CR2","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1016\/j.jhydrol.2011.12.027","volume":"420\u2013421","author":"S Sun","year":"2012","unstructured":"Sun S, Fu G, Djordjevi\u0107 S, Khu S-T (2012) Separating aleatory and epistemic uncertainties: probabilistic sewer flooding evaluation using probability box. J Hydrol 420\u2013421:360\u2013372. https:\/\/doi.org\/10.1016\/j.jhydrol.2011.12.027","journal-title":"J Hydrol"},{"issue":"7","key":"1366_CR3","doi-asserted-by":"publisher","first-page":"947","DOI":"10.1016\/j.ress.2006.07.002","volume":"92","author":"K Durga Rao","year":"2007","unstructured":"Durga Rao K, Kushwaha HS, Verma AK, Srividya A (2007) Quantification of epistemic and aleatory uncertainties in level-1 probabilistic safety assessment studies. Reliab Eng Syst Saf 92(7):947\u2013956. https:\/\/doi.org\/10.1016\/j.ress.2006.07.002","journal-title":"Reliab Eng Syst Saf"},{"key":"1366_CR4","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.ress.2013.02.014","volume":"116","author":"W Yao","year":"2013","unstructured":"Yao W, Chen X, Huang Y, van Tooren M (2013) An enhanced unified uncertainty analysis approach based on first order reliability method with single-level optimization. Reliab Eng Syst Saf 116:28\u201337. https:\/\/doi.org\/10.1016\/j.ress.2013.02.014","journal-title":"Reliab Eng Syst Saf"},{"issue":"4","key":"1366_CR5","doi-asserted-by":"publisher","first-page":"664","DOI":"10.1007\/BF03026971","volume":"21","author":"J Lee","year":"2007","unstructured":"Lee J, Yang I, Yang S, Kwak JS (2007) Uncertainty analysis and ANOVA for the measurement reliability estimation of altitude engine test. J Mech Sci Technol 21(4):664\u2013671. https:\/\/doi.org\/10.1007\/BF03026971","journal-title":"J Mech Sci Technol"},{"issue":"3","key":"1366_CR6","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1016\/S0951-8320(02)00056-X","volume":"77","author":"E Hofer","year":"2002","unstructured":"Hofer E, Kloos M, Krzykacz-Hausmann B, Peschke J, Woltereck M (2002) An approximate epistemic uncertainty analysis approach in the presence of epistemic and aleatory uncertainties. Reliab Eng Syst Saf 77(3):229\u2013238. https:\/\/doi.org\/10.1016\/S0951-8320(02)00056-X","journal-title":"Reliab Eng Syst Saf"},{"issue":"1","key":"1366_CR7","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1016\/j.ymssp.2013.01.024","volume":"37","author":"M Beer","year":"2013","unstructured":"Beer M, Ferson S, Kreinovich V (2013) Imprecise probabilities in engineering analyses. Mech Syst Signal Process 37(1):4\u201329. https:\/\/doi.org\/10.1016\/j.ymssp.2013.01.024","journal-title":"Mech Syst Signal Process"},{"key":"1366_CR8","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1016\/j.cma.2018.01.045","volume":"334","author":"J Zhang","year":"2018","unstructured":"Zhang J, Shields MD (2018) The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets. Comput Methods Appl Mech Eng 334:483\u2013506. https:\/\/doi.org\/10.1016\/j.cma.2018.01.045","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"14","key":"1366_CR9","doi-asserted-by":"publisher","first-page":"1048","DOI":"10.1016\/j.compstruc.2004.11.015","volume":"83","author":"SK Au","year":"2005","unstructured":"Au SK (2005) Reliability-based design sensitivity by efficient simulation. Comput Struct 83(14):1048\u20131061. https:\/\/doi.org\/10.1016\/j.compstruc.2004.11.015","journal-title":"Comput Struct"},{"issue":"1","key":"1366_CR10","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/j.probengmech.2006.05.002","volume":"22","author":"J Ching","year":"2007","unstructured":"Ching J, Hsieh Y-H (2007) Local estimation of failure probability function and its confidence interval with maximum entropy principle. Probab Eng Mech 22(1):39\u201349. https:\/\/doi.org\/10.1016\/j.probengmech.2006.05.002","journal-title":"Probab Eng Mech"},{"key":"1366_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.chaos.2012.12.001","volume":"48","author":"X Yuan","year":"2013","unstructured":"Yuan X, Xue Y, Liu M (2013) Analysis of an epidemic model with awareness programs by media on complex networks. Chaos Solitons Fract 48:1\u201311. https:\/\/doi.org\/10.1016\/j.chaos.2012.12.001","journal-title":"Chaos Solitons Fract"},{"key":"1366_CR12","doi-asserted-by":"publisher","first-page":"106975","DOI":"10.1016\/j.ress.2020.106975","volume":"201","author":"C Ling","year":"2020","unstructured":"Ling C, Lu Z, Zhang X (2020) An efficient method based on AK-MCS for estimating failure probability function. Reliab Eng Syst Saf 201:106975. https:\/\/doi.org\/10.1016\/j.ress.2020.106975","journal-title":"Reliab Eng Syst Saf"},{"issue":"10","key":"1366_CR13","doi-asserted-by":"publisher","first-page":"2244","DOI":"10.1016\/j.simpat.2011.08.003","volume":"19","author":"J Morio","year":"2011","unstructured":"Morio J (2011) Influence of input PDF parameters of a model on a failure probability estimation. Simul Model Pract Theory 19(10):2244\u20132255. https:\/\/doi.org\/10.1016\/j.simpat.2011.08.003","journal-title":"Simul Model Pract Theory"},{"key":"1366_CR14","doi-asserted-by":"publisher","first-page":"164","DOI":"10.1016\/j.ress.2018.06.008","volume":"178","author":"V Chabridon","year":"2018","unstructured":"Chabridon V, Balesdent M, Bourinet J-M, Morio J, Gayton N (2018) Reliability-based sensitivity estimators of rare event probability in the presence of distribution parameter uncertainty. Reliab Eng Syst Saf 178:164\u2013178. https:\/\/doi.org\/10.1016\/j.ress.2018.06.008","journal-title":"Reliab Eng Syst Saf"},{"issue":"10","key":"1366_CR15","doi-asserted-by":"publisher","first-page":"1443","DOI":"10.1016\/j.ress.2005.11.042","volume":"91","author":"JW Hall","year":"2006","unstructured":"Hall JW (2006) Uncertainty-based sensitivity indices for imprecise probability distributions. Reliab Eng Syst Saf 91(10):1443\u20131451. https:\/\/doi.org\/10.1016\/j.ress.2005.11.042","journal-title":"Reliab Eng Syst Saf"},{"issue":"1","key":"1366_CR16","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.probengmech.2004.05.005","volume":"20","author":"I Kaymaz","year":"2005","unstructured":"Kaymaz I, McMahon CA (2005) A response surface method based on weighted regression for structural reliability analysis. Probab Eng Mech 20(1):11\u201317. https:\/\/doi.org\/10.1016\/j.probengmech.2004.05.005","journal-title":"Probab Eng Mech"},{"key":"1366_CR17","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.compstruc.2017.09.002","volume":"194","author":"K Cheng","year":"2018","unstructured":"Cheng K, Lu Z (2018) Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression. Comput Struct 194:86\u201396. https:\/\/doi.org\/10.1016\/j.compstruc.2017.09.002","journal-title":"Comput Struct"},{"issue":"3","key":"1366_CR18","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1016\/j.strusafe.2003.05.002","volume":"26","author":"JE Hurtado","year":"2004","unstructured":"Hurtado JE (2004) An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory. Struct Saf 26(3):271\u2013293. https:\/\/doi.org\/10.1016\/j.strusafe.2003.05.002","journal-title":"Struct Saf"},{"issue":"32","key":"1366_CR19","doi-asserted-by":"publisher","first-page":"3491","DOI":"10.1016\/S0045-7825(02)00287-6","volume":"191","author":"M Papadrakakis","year":"2002","unstructured":"Papadrakakis M, Lagaros ND (2002) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191(32):3491\u20133507. https:\/\/doi.org\/10.1016\/S0045-7825(02)00287-6","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1366_CR20","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/j.ress.2019.03.002","volume":"188","author":"J Zhang","year":"2019","unstructured":"Zhang J, Xiao M, Gao L (2019) An active learning reliability method combining Kriging constructed with exploration and exploitation of failure region and subset simulation. Reliab Eng Syst Saf 188:90\u2013102. https:\/\/doi.org\/10.1016\/j.ress.2019.03.002","journal-title":"Reliab Eng Syst Saf"},{"issue":"2","key":"1366_CR21","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/j.strusafe.2004.09.001","volume":"27","author":"I Kaymaz","year":"2005","unstructured":"Kaymaz I (2005) Application of Kriging method to structural reliability problems. Struct Saf 27(2):133\u2013151. https:\/\/doi.org\/10.1016\/j.strusafe.2004.09.001","journal-title":"Struct Saf"},{"key":"1366_CR22","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.ress.2019.04.003","volume":"189","author":"W Chen","year":"2019","unstructured":"Chen W, Xu C, Shi Y, Ma J, Lu S (2019) A hybrid Kriging-based reliability method for small failure probabilities. Reliab Eng Syst Saf 189:31\u201341. https:\/\/doi.org\/10.1016\/j.ress.2019.04.003","journal-title":"Reliab Eng Syst Saf"},{"key":"1366_CR23","doi-asserted-by":"publisher","first-page":"113336","DOI":"10.1016\/j.cma.2020.113336","volume":"372","author":"N-C Xiao","year":"2020","unstructured":"Xiao N-C, Zhan H, Yuan K (2020) A new reliability method for small failure probability problems by combining the adaptive importance sampling and surrogate models. Comput Methods Appl Mech Eng 372:113336. https:\/\/doi.org\/10.1016\/j.cma.2020.113336","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"1366_CR24","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1556\/OH.2012.29250","volume":"153","author":"M Guarascio","year":"2012","unstructured":"Guarascio M, Huybrechts CJ, David M (2012) Advanced geostatistics in the mining industry: proceedings of the NATO Advanced Study Institute held at the Istituto di Geologia Applicata of the University of Rome, Italy, 13\u201325 October 1975. Orv Hetil 153(1):3\u201313","journal-title":"Orv Hetil"},{"issue":"10","key":"1366_CR25","doi-asserted-by":"publisher","first-page":"2459","DOI":"10.2514\/1.34321","volume":"46","author":"BJ Bichon","year":"2012","unstructured":"Bichon BJ, Eldred MS, Swiler LP, Mahadevan S, McFarland JM (2012) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46(10):2459\u20132468","journal-title":"AIAA J"},{"issue":"1","key":"1366_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.precisioneng.2012.07.007","volume":"37","author":"A Dumas","year":"2013","unstructured":"Dumas A, Echard B, Gayton N, Rochat O, Dantan J-Y, Van Der Veen S (2013) AK-ILS: an active learning method based on Kriging for the inspection of large surfaces. Precis Eng 37(1):1\u20139. https:\/\/doi.org\/10.1016\/j.precisioneng.2012.07.007","journal-title":"Precis Eng"},{"issue":"5","key":"1366_CR27","doi-asserted-by":"publisher","first-page":"1182","DOI":"10.1016\/j.camwa.2015.07.004","volume":"70","author":"Z Lv","year":"2015","unstructured":"Lv Z, Lu Z, Wang P (2015) A new learning function for Kriging and its applications to solve reliability problems in engineering. Comput Math Appl 70(5):1182\u20131197. https:\/\/doi.org\/10.1016\/j.camwa.2015.07.004","journal-title":"Comput Math Appl"},{"key":"1366_CR28","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1016\/j.ress.2016.09.003","volume":"157","author":"Z Sun","year":"2017","unstructured":"Sun Z, Wang J, Li R, Tong C (2017) LIF: a new Kriging based learning function and its application to structural reliability analysis. Reliab Eng Syst Saf 157:152\u2013165. https:\/\/doi.org\/10.1016\/j.ress.2016.09.003","journal-title":"Reliab Eng Syst Saf"},{"issue":"2","key":"1366_CR29","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/j.strusafe.2011.01.002","volume":"33","author":"B Echard","year":"2011","unstructured":"Echard B, Gayton N, Lemaire M (2011) AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation. Struct Saf 33(2):145\u2013154. https:\/\/doi.org\/10.1016\/j.strusafe.2011.01.002","journal-title":"Struct Saf"},{"issue":"1","key":"1366_CR30","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.probengmech.2008.01.005","volume":"24","author":"H Millwater","year":"2009","unstructured":"Millwater H (2009) Universal properties of kernel functions for probabilistic sensitivity analysis. Probab Eng Mech 24(1):89\u201399. https:\/\/doi.org\/10.1016\/j.probengmech.2008.01.005","journal-title":"Probab Eng Mech"},{"issue":"2","key":"1366_CR31","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1016\/j.compstruc.2003.09.002","volume":"82","author":"BD Youn","year":"2004","unstructured":"Youn BD, Choi KK (2004) A new response surface methodology for reliability-based design optimization. Comput Struct 82(2):241\u2013256. https:\/\/doi.org\/10.1016\/j.compstruc.2003.09.002","journal-title":"Comput Struct"},{"issue":"3","key":"1366_CR32","doi-asserted-by":"publisher","first-page":"439","DOI":"10.2307\/1425829","volume":"5","author":"G Matheron","year":"1973","unstructured":"Matheron G (1973) The intrinsic random functions and their applications. Adv Appl Probab 5(3):439\u2013468","journal-title":"Adv Appl Probab"},{"key":"1366_CR33","doi-asserted-by":"publisher","unstructured":"Koehler JR, Owen AB (1996) 9 Computer experiments. In: Handbook of Statistics, vol 13. Elsevier, pp 261\u2013308. doi:https:\/\/doi.org\/10.1016\/S0169-7161(96)13011-X","DOI":"10.1016\/S0169-7161(96)13011-X"},{"key":"1366_CR34","volume-title":"DACE\u2014a Matlab Kriging toolbox (version 2) informatics and mathematical modeling","author":"SN Lophaven","year":"2002","unstructured":"Lophaven SN, Nielsen HB, Sondergaard J (2002) DACE\u2014a Matlab Kriging toolbox (version 2) informatics and mathematical modeling. Technical University of Denmark, Copenhagen"},{"key":"1366_CR35","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1016\/j.ress.2012.10.008","volume":"111","author":"B Echard","year":"2013","unstructured":"Echard B, Gayton N, Lemaire M, Relun N (2013) A combined Importance Sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models. Reliab Eng Syst Saf 111:232\u2013240. https:\/\/doi.org\/10.1016\/j.ress.2012.10.008","journal-title":"Reliab Eng Syst Saf"},{"issue":"6","key":"1366_CR36","doi-asserted-by":"publisher","first-page":"2515","DOI":"10.1007\/s00158-017-1869-z","volume":"57","author":"Y Wang","year":"2018","unstructured":"Wang Y, Zhao W, Zhou G, Gao Q, Wang C (2018) Optimization of an auxetic jounce bumper based on Gaussian process metamodel and series hybrid GA-SQP algorithm. Struct Multidiscip Optim 57(6):2515\u20132525. https:\/\/doi.org\/10.1007\/s00158-017-1869-z","journal-title":"Struct Multidiscip Optim"},{"issue":"3","key":"1366_CR37","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.strusafe.2004.11.001","volume":"27","author":"L Schueremans","year":"2005","unstructured":"Schueremans L, Van Gemert D (2005) Benefit of splines and neural networks in simulation based structural reliability analysis. Struct Saf 27(3):246\u2013261. https:\/\/doi.org\/10.1016\/j.strusafe.2004.11.001","journal-title":"Struct Saf"},{"issue":"2","key":"1366_CR38","doi-asserted-by":"publisher","first-page":"1020","DOI":"10.1016\/j.matdes.2010.07.029","volume":"32","author":"NA Kadhim","year":"2011","unstructured":"Kadhim NA, Abdullah S, Ariffin AK (2011) Effect of the fatigue data editing technique associated with finite element analysis on the component fatigue design period. Mater Des 32(2):1020\u20131030. https:\/\/doi.org\/10.1016\/j.matdes.2010.07.029","journal-title":"Mater Des"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-021-01366-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-021-01366-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-021-01366-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,2]],"date-time":"2022-08-02T22:55:40Z","timestamp":1659480940000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-021-01366-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,5]]},"references-count":38,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["1366"],"URL":"https:\/\/doi.org\/10.1007\/s00366-021-01366-y","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,5]]},"assertion":[{"value":"22 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}