{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T23:33:36Z","timestamp":1770334416162,"version":"3.49.0"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,4,20]],"date-time":"2025-04-20T00:00:00Z","timestamp":1745107200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,20]],"date-time":"2025-04-20T00:00:00Z","timestamp":1745107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["No. NSFC 52375156"],"award-info":[{"award-number":["No. NSFC 52375156"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s00366-025-02135-x","type":"journal-article","created":{"date-parts":[[2025,4,20]],"date-time":"2025-04-20T11:10:46Z","timestamp":1745147446000},"page":"2953-2967","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A novel optimization-based adaptive Kriging method for structural reliability analysis"],"prefix":"10.1007","volume":"41","author":[{"given":"Yushan","family":"Liu","sequence":"first","affiliation":[]},{"given":"Luyi","family":"Li","sequence":"additional","affiliation":[]},{"given":"Zeming","family":"Chang","sequence":"additional","affiliation":[]},{"given":"Haohao","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,20]]},"reference":[{"key":"2135_CR1","volume-title":"Handbook of uncertainty quantification [M]","author":"LM Sekhohola-Dlamini","year":"2017","unstructured":"Sekhohola-Dlamini LM, Keshinro OM, Masudi WL, Cowan AK (2017) Handbook of uncertainty quantification [M]. Springer, New York"},{"issue":"3","key":"2135_CR2","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s00158-008-0234-7","volume":"37","author":"S Lee","year":"2009","unstructured":"Lee S, Chen W (2009) A comparative study of uncertainty propagation methods for black-box-type problems. Struct Multidisc Optim 37(3):239\u2013253","journal-title":"Struct Multidisc Optim"},{"key":"2135_CR3","doi-asserted-by":"crossref","DOI":"10.1016\/j.ress.2020.106908","volume":"199","author":"R Teixeira","year":"2020","unstructured":"Teixeira R, Nogal M, O\u2019Connor A et al (2020) Reliability assessment with density scanned adaptive Kriging. Reliab Eng Syst Safety 199:106908","journal-title":"Reliab Eng Syst Safety"},{"issue":"1","key":"2135_CR4","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1061\/JMCEA3.0001848","volume":"100","author":"AM Hasofer","year":"1974","unstructured":"Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div 100(1):111\u2013121","journal-title":"J Eng Mech Div"},{"issue":"5","key":"2135_CR5","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/0045-7949(78)90046-9","volume":"9","author":"R Rackwitz","year":"1978","unstructured":"Rackwitz R, Flessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489\u2013494","journal-title":"Comput Struct"},{"issue":"4","key":"2135_CR6","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1061\/JMCEA3.0002512","volume":"105","author":"B Fiessler","year":"1979","unstructured":"Fiessler B, Neumann HJ, Rackwitz R (1979) Quadratic limit states in structural reliability. J Eng Mech Div 105(4):661\u2013676","journal-title":"J Eng Mech Div"},{"key":"2135_CR7","first-page":"14","volume":"2005","author":"A Der Kiureghian","year":"2005","unstructured":"Der Kiureghian A (2005) First-and second-order reliability methods. Eng Des Reliab Handbook 2005:14","journal-title":"Eng Des Reliab Handbook"},{"issue":"1","key":"2135_CR8","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/S0167-4730(00)00027-8","volume":"23","author":"Y Zhao","year":"2001","unstructured":"Zhao Y, Ono T (2001) Moment methods for structural reliability. Struct Safety 23(1):47\u201375","journal-title":"Struct Safety"},{"key":"2135_CR9","volume":"400","author":"D Zhang","year":"2022","unstructured":"Zhang D, Shen S, Jiang C et al (2022) An advanced mixed-degree cubature formula for reliability analysis. Comput Meth Appl Mech Eng 400:115521","journal-title":"Comput Meth Appl Mech Eng"},{"key":"2135_CR10","volume-title":"Monte Carlo strategies in scientific computing [M]","author":"JS Liu","year":"2001","unstructured":"Liu JS, Liu JS (2001) Monte Carlo strategies in scientific computing [M]. Springer, New York"},{"issue":"1","key":"2135_CR11","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/0167-4730(89)90003-9","volume":"6","author":"RE Melchers","year":"1989","unstructured":"Melchers RE (1989) Importance sampling in structural systems. Struct Safety 6(1):3\u201310","journal-title":"Struct Safety"},{"issue":"4","key":"2135_CR12","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/S0266-8920(01)00019-4","volume":"16","author":"SK Au","year":"2001","unstructured":"Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Prob Eng Mech 16(4):263\u2013277","journal-title":"Prob Eng Mech"},{"issue":"4","key":"2135_CR13","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.probengmech.2004.05.001","volume":"19","author":"PS Koutsourelakis","year":"2004","unstructured":"Koutsourelakis PS, Pradlwarter HJ, Schu\u00ebller GI (2004) Reliability of structures in high dimensions, part I: algorithms and applications. Prob Eng Mech 19(4):409\u2013417","journal-title":"Prob Eng Mech"},{"key":"2135_CR14","volume":"228","author":"J Zhang","year":"2022","unstructured":"Zhang J, Gong W, Yue X et al (2022) Efficient reliability analysis using prediction-oriented active sparse polynomial chaos expansion. Reliab Eng Syst Safety 228:108749","journal-title":"Reliab Eng Syst Safety"},{"issue":"3","key":"2135_CR15","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s00158-010-0568-9","volume":"43","author":"C Hu","year":"2011","unstructured":"Hu C, Youn BD (2011) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidis Optim 43(3):419\u2013442","journal-title":"Struct Multidis Optim"},{"key":"2135_CR16","volume":"237","author":"C Luo","year":"2023","unstructured":"Luo C, Zhu SP, Keshtegar B et al (2023) An enhanced uniform simulation approach coupled with SVR for efficient structural reliability analysis. Reliab Eng Syst Safety 237:109377","journal-title":"Reliab Eng Syst Safety"},{"issue":"1","key":"2135_CR17","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s00366-023-01803-0","volume":"40","author":"JY Chen","year":"2024","unstructured":"Chen JY, Feng YW, Teng D et al (2024) Support vector machines-based pre-calculation error for structural reliability analysis. Eng Comput 40(1):477\u2013491","journal-title":"Eng Comput"},{"issue":"2","key":"2135_CR18","doi-asserted-by":"crossref","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 Safety 33(2):145\u2013154","journal-title":"Struct Safety"},{"issue":"2","key":"2135_CR19","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/s00366-023-01815-w","volume":"40","author":"J Wang","year":"2024","unstructured":"Wang J, Lu Z (2024) An efficient surrogate model method considering the candidate sample pool reduction by safety optimal hypersphere for random-interval mixed reliability analysis. Eng Comput 40(2):795\u2013811","journal-title":"Eng Comput"},{"key":"2135_CR20","volume":"432","author":"X Yang","year":"2024","unstructured":"Yang X, Zhang Y, Zhao J et al (2024) A novel active learning method based on matrix-operation RBF model for high-dimensional reliability analysis. Comput Methods Appl Mech Eng 432:117434","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2135_CR21","volume":"205","author":"HM Qian","year":"2021","unstructured":"Qian HM, Li YF, Huang HZ (2021) Time-variant system reliability analysis method for a small failure probability problem. Reliability Engineering & System Safety 205:107261","journal-title":"Reliability Engineering & System Safety"},{"key":"2135_CR22","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1016\/j.ress.2019.01.014","volume":"185","author":"X Zhang","year":"2019","unstructured":"Zhang X, Wang L, S\u00f8rensen JD (2019) REIF: a novel active-learning function toward adaptive Kriging surrogate models for structural reliability analysis. Reliab Eng Syst Safety 185:440\u2013454","journal-title":"Reliab Eng Syst Safety"},{"issue":"4","key":"2135_CR23","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1023\/A:1008306431147","volume":"13","author":"DR Jones","year":"1998","unstructured":"Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455\u2013492","journal-title":"J Glob Optim"},{"issue":"10","key":"2135_CR24","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.2514\/1.34321","volume":"46","author":"BJ Bichon","year":"2008","unstructured":"Bichon BJ, Eldred MS, Swiler LP et al (2008) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46(10):2459\u20132468","journal-title":"AIAA J"},{"issue":"2","key":"2135_CR25","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1007\/s00158-021-02892-5","volume":"64","author":"A Chaudhuri","year":"2021","unstructured":"Chaudhuri A, Marques AN, Willcox K (2021) mfEGRA: Multifidelity efficient global reliability analysis through active learning for failure boundary location. Struct Multidisc Optim 64(2):797\u2013811","journal-title":"Struct Multidisc Optim"},{"key":"2135_CR26","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.ress.2016.05.002","volume":"153","author":"Z Wen","year":"2016","unstructured":"Wen Z, Pei H, Liu H et al (2016) A Sequential Kriging reliability analysis method with characteristics of adaptive sampling regions and parallelizability. Reliab Eng Syst Safety 153:170\u2013179","journal-title":"Reliab Eng Syst Safety"},{"issue":"5","key":"2135_CR27","doi-asserted-by":"crossref","DOI":"10.1115\/1.4039339","volume":"140","author":"X Yang","year":"2018","unstructured":"Yang X, Liu Y, Mi C et al (2018) Active learning Kriging model combining with kernel-density-estimation-based importance sampling method for the estimation of low failure probability. J Mech Des 140(5):051402","journal-title":"J Mech Des"},{"key":"2135_CR28","doi-asserted-by":"crossref","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 et al (2013) A combined importance sampling and kriging reliability method for small failure probabilities with time-demanding numerical models. Reliab Eng Syst Safety 111:232\u2013240","journal-title":"Reliab Eng Syst Safety"},{"key":"2135_CR29","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.strusafe.2015.12.003","volume":"59","author":"X Huang","year":"2016","unstructured":"Huang X, Chen J, Zhu H (2016) Assessing small failure probabilities by AK\u2013SS: an active learning method combining Kriging and subset simulation. Struct Safety 59:86\u201395","journal-title":"Struct Safety"},{"key":"2135_CR30","volume":"228","author":"YZ Ma","year":"2022","unstructured":"Ma YZ, Zhu YC, Li HS et al (2022) Adaptive Kriging-based failure probability estimation for multiple responses. Reliab Eng Syst Safety 228:108771","journal-title":"Reliab Eng Syst Safety"},{"issue":"5","key":"2135_CR31","doi-asserted-by":"crossref","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","journal-title":"Comput Math Appl"},{"issue":"3","key":"2135_CR32","doi-asserted-by":"crossref","first-page":"2035","DOI":"10.1007\/s00366-021-01570-w","volume":"39","author":"H Liu","year":"2023","unstructured":"Liu H, He X, Wang P et al (2023) Time-dependent reliability analysis method based on ARBIS and Kriging surrogate model. Eng Comput 39(3):2035\u20132048","journal-title":"Eng Comput"},{"key":"2135_CR33","first-page":"1","volume":"107","author":"DW Jia","year":"2024","unstructured":"Jia DW, Wu ZY (2024) An improved adaptive Kriging model-based metamodel importance sampling reliability analysis method. Eng Comput 107:1\u201322","journal-title":"Eng Comput"},{"issue":"9","key":"2135_CR34","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1007\/s00158-024-03865-0","volume":"67","author":"X Fan","year":"2024","unstructured":"Fan X, Yang X, Liu Y (2024) A Kriging-assisted adaptive improved cross-entropy importance sampling method for random-interval hybrid reliability analysis. Struct Multidisc Optim 67(9):158","journal-title":"Struct Multidisc Optim"},{"key":"2135_CR35","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.ress.2013.10.010","volume":"123","author":"W Fauriat","year":"2014","unstructured":"Fauriat W, Gayton N (2014) AK-SYS: an adaptation of the AK-MCS method for system reliability. Reliab Eng Syst Safety 123:137\u2013144","journal-title":"Reliab Eng Syst Safety"},{"key":"2135_CR36","doi-asserted-by":"crossref","DOI":"10.1016\/j.ress.2022.108512","volume":"223","author":"AG Carlon","year":"2022","unstructured":"Carlon AG, Kroetz HM, Torii AJ et al (2022) Risk optimization using the Chernoff bound and stochastic gradient descent. Reliab Eng System Safety 223:108512","journal-title":"Reliab Eng System Safety"},{"issue":"6","key":"2135_CR37","first-page":"119","volume":"52","author":"DG Krige","year":"1951","unstructured":"Krige DG (1951) A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Southern African Institute of Mining and Metallurgy 52(6):119\u2013139","journal-title":"Journal of the Southern African Institute of Mining and Metallurgy"},{"issue":"8","key":"2135_CR38","doi-asserted-by":"crossref","first-page":"1246","DOI":"10.2113\/gsecongeo.58.8.1246","volume":"58","author":"G Matheron","year":"1963","unstructured":"Matheron G (1963) Principles of geostatistics. Econ Geol 58(8):1246\u20131266","journal-title":"Econ Geol"},{"key":"2135_CR39","volume-title":"DACE: a Matlab kriging toolbox","author":"SN Lophaven","year":"2002","unstructured":"Lophaven SN, Nielsen HB, S\u00f8ndergaard J (2002) DACE: a Matlab kriging toolbox. Informatics and Mathematical Modelling, The Technical University of Denmark, IMM"},{"key":"2135_CR40","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.ress.2017.12.014","volume":"172","author":"B Keshtegar","year":"2018","unstructured":"Keshtegar B, Chakraborty S (2018) An efficient-robust structural reliability method by adaptive finite-step length based on Armijo line search. Reliab Eng Syst Saf 172:195\u2013206","journal-title":"Reliab Eng Syst Safety"},{"key":"2135_CR41","doi-asserted-by":"crossref","DOI":"10.1016\/j.ress.2020.107286","volume":"206","author":"K Cheng","year":"2021","unstructured":"Cheng K, Lu Z (2021) Adaptive Bayesian support vector regression model for structural reliability analysis. Reliab Eng Syst Saf 206:107286","journal-title":"Reliab Eng Syst Safety"},{"issue":"2","key":"2135_CR42","first-page":"D40160002","volume":"3","author":"R Sch\u00f6bi","year":"2016","unstructured":"Sch\u00f6bi R, Sudret B, Marelli S (2016) Rare event estimation using polynomial-chaos kriging. ASCE-ASME J Risk Uncert Eng Syst Civil Eng 3(2):D40160002","journal-title":"ASCE-ASME J Risk Uncert Eng Syst Civil Eng"},{"issue":"3","key":"2135_CR43","doi-asserted-by":"crossref","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 Safety 27(3):246\u2013261","journal-title":"Struct Safety"},{"key":"2135_CR44","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.ress.2012.09.005","volume":"110","author":"P Wei","year":"2013","unstructured":"Wei P, Lu Z, Yuan X (2013) Monte Carlo simulation for moment-independent sensitivity analysis. Reliab Eng Syst Safety 110:60\u201367","journal-title":"Reliab Eng Syst Safety"},{"issue":"4","key":"2135_CR45","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1007\/s00158-018-1975-6","volume":"58","author":"W Yun","year":"2018","unstructured":"Yun W, Lu Z, Jiang X (2018) An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability. Struct Multidisc Optim 58(4):1383\u20131393","journal-title":"Struct Multidisc Optim"},{"key":"2135_CR46","doi-asserted-by":"crossref","first-page":"1144","DOI":"10.1016\/j.jcp.2016.06.005","volume":"321","author":"K Konakli","year":"2016","unstructured":"Konakli K, Sudret B (2016) Polynomial meta-models with canonical low-rank approximations. J Comput Phys 321:1144\u20131169","journal-title":"J Comput Phys"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-025-02135-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-025-02135-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-025-02135-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T21:15:30Z","timestamp":1764105330000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-025-02135-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,20]]},"references-count":46,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["2135"],"URL":"https:\/\/doi.org\/10.1007\/s00366-025-02135-x","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,20]]},"assertion":[{"value":"15 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 2025","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 declared that they have no conflicts of interest in this work, we declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}