{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T14:54:46Z","timestamp":1769525686561,"version":"3.49.0"},"reference-count":45,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T00:00:00Z","timestamp":1627689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In practical engineering, due to the lack of information, it is impossible to accurately determine the distribution of all variables. Therefore, time-variant reliability problems with both random and interval variables may be encountered. However, this kind of problem usually involves a complex multilevel nested optimization problem, which leads to a substantial computational burden, and it is difficult to meet the requirements of complex engineering problem analysis. This study proposes a decoupling strategy to efficiently analyze the time-variant reliability based on the mixed uncertainty model. The interval variables are treated with independent random variables that are uniformly distributed in their respective intervals. Then the time-variant reliability-equivalent model, containing only random variables, is established, to avoid multi-layer nesting optimization. The stochastic process is first discretized to obtain several static limit state functions at different times. The time-variant reliability problem is changed into the conventional time-invariant system reliability problem. First order reliability analysis method (FORM) is used to analyze the reliability of each time. Thus, an efficient and robust convergence hybrid time-variant reliability calculation algorithm is proposed based on the equivalent model. Finally, numerical examples shows the effectiveness of the proposed method.<\/jats:p>","DOI":"10.3390\/a14080229","type":"journal-article","created":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T21:51:07Z","timestamp":1627854667000},"page":"229","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0591-1834","authenticated-orcid":false,"given":"Fangyi","family":"Li","sequence":"first","affiliation":[{"name":"Center for Research on Leading Technology of Special Equipment, School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou 510006, China"},{"name":"Key Laboratory for Safety Control of Bridge Engineering, Ministry of Education and Hunan Province, Changsha University of Science and Technology, Changsha 410114, China"}]},{"given":"Yufei","family":"Yan","sequence":"additional","affiliation":[{"name":"Key Laboratory for Safety Control of Bridge Engineering, Ministry of Education and Hunan Province, Changsha University of Science and Technology, Changsha 410114, China"}]},{"given":"Jianhua","family":"Rong","sequence":"additional","affiliation":[{"name":"Key Laboratory for Safety Control of Bridge Engineering, Ministry of Education and Hunan Province, Changsha University of Science and Technology, Changsha 410114, China"}]},{"given":"Houyao","family":"Zhu","sequence":"additional","affiliation":[{"name":"Center for Research on Leading Technology of Special Equipment, School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou 510006, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,31]]},"reference":[{"key":"ref_1","unstructured":"Zhang, D., Zhang, N., Ye, N., Fang, J., and Han, X. (2020). Hybrid learning algorithm of radial basis function networks for reliability analysis. IEEE Trans. Reliab., 1\u201314."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"107106","DOI":"10.1016\/j.ymssp.2020.107106","article-title":"On reliability analysis method through rotational sparse grid nodes","volume":"147","author":"Wu","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s00158-015-1334-9","article-title":"Layout optimization of continuum structures considering the probabilistic and fuzzy directional uncertainty of applied loads based on the cloud model","volume":"53","author":"Liu","year":"2016","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1007\/s00158-020-02536-0","article-title":"An efficient evolutionary structural optimization method for multi-resolution designs","volume":"62","author":"Wang","year":"2020","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"91838","DOI":"10.1109\/ACCESS.2019.2924931","article-title":"Robust Optimization for micromachine design problems involving multimodal distributions","volume":"7","author":"Huang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1061\/JMCEA3.0001848","article-title":"Exact and invariant second-moment code format","volume":"100","author":"Hasofer","year":"1974","journal-title":"J. Eng. Mech. Div."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/0045-7949(78)90046-9","article-title":"Structural reliability under combined random load sequences","volume":"9","author":"Rackwitz","year":"1978","journal-title":"Comput. Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/0047-259X(89)90089-4","article-title":"Asymptotic approximations for multivariate integrals with an application to multinormal probabilities","volume":"30","author":"Breitung","year":"1989","journal-title":"J. Multivar. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0167-4730(90)90061-S","article-title":"Structural analysis methods for system reliability","volume":"7","author":"Bennett","year":"1990","journal-title":"Struct. Saf."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Jiang, C., Huang, X.P., Han, X., and Zhang, D.Q. (2014). A Time-variant reliability analysis method based on stochastic process discretization. J. Mech. Des., 136.","DOI":"10.1115\/1.4027865"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.apm.2018.12.019","article-title":"Structural design optimization based on hybrid time-variant reliability measure under non-probabilistic convex uncertainties","volume":"69","author":"Wang","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"081010","DOI":"10.1115\/1.4027636","article-title":"Time-dependent mechanism reliability analysis with envelope functions and first-order approximation","volume":"136","author":"Du","year":"2014","journal-title":"J. Mech. Des."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"110588","DOI":"10.1016\/j.engstruct.2020.110588","article-title":"A time-variant reliability analysis method for non-linear limit-state functions with the mixture of random and interval variables","volume":"213","author":"Li","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1002\/j.1538-7305.1944.tb00874.x","article-title":"Mathematical analysis of random noise","volume":"23","author":"Rice","year":"1944","journal-title":"Bell Syst. Tech. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.ress.2003.10.005","article-title":"The PHI2 method: A way to compute time-variant reliability","volume":"84","author":"Sudret","year":"2004","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1080\/15732470701270058","article-title":"Analytical derivation of the outcrossing rate in time-variant reliability problems","volume":"4","author":"Sudret","year":"2008","journal-title":"Struct. Infrastruct. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/0167-4730(93)90018-V","article-title":"Time-dependent system reliability analysis by adaptive importance sampling","volume":"12","author":"Mori","year":"1993","journal-title":"Struct. Saf."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Singh, A., Mourelatos, Z.P., and Nikolaidis, E. (2011). An importance sampling approach for time-dependent reliability. Proc. Pap., 1077\u20131088.","DOI":"10.1115\/DETC2011-47200"},{"key":"ref_19","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_20","doi-asserted-by":"crossref","unstructured":"Wang, Z., Mourelatos, Z.P., Li, J., Baseski, I., and Singh, A. (2014). Time-dependent reliability of dynamic systems using subset simulation with splitting over a series of correlated time intervals. J. Mech. Des., 136.","DOI":"10.1115\/1.4027162"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.ress.2018.05.007","article-title":"Experimental estimation of time variant system reliability of vibrating structures based on subset simulation with Markov chain splitting","volume":"178","author":"Sonal","year":"2018","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.strusafe.2006.02.002","article-title":"The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters","volume":"29","author":"Chen","year":"2007","journal-title":"Struct. Saf."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Hu, Z., and Du, X. (2013). A sampling approach to extreme value distribution for time-dependent reliability analysis. J. Mech. Des., 135.","DOI":"10.1115\/1.4023925"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.cma.2012.02.013","article-title":"Accelerated subset simulation with neural networks for reliability analysis","volume":"223\u2013224","author":"Papadopoulos","year":"2012","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Wang, Z., and Wang, P. (2012). A Nested extreme response surface approach for time-dependent reliability-based design optimization. J. Mech. Des., 134.","DOI":"10.1115\/1.4007931"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.ress.2013.02.017","article-title":"A new approach for reliability analysis with time-variant performance characteristics","volume":"115","author":"Wang","year":"2013","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1023\/A:1008306431147","article-title":"Efficient Global optimization of expensive black-box functions","volume":"13","author":"Jones","year":"1998","journal-title":"J. Glob. Optim."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Zhang, D., Han, X., Jiang, C., Liu, J., and Li, Q. (2017). Time-dependent reliability analysis through response surface method. J. Mech. Des., 139.","DOI":"10.1115\/1.4035860"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1061\/(ASCE)0733-9399(1993)119:6(1136)","article-title":"Optimal discretization of random fields","volume":"119","author":"Li","year":"1993","journal-title":"J. Eng. Mech. Asce"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1002\/qre.750","article-title":"Time-variant reliability of nonlinear structures: Application to a representative part of a plate floor","volume":"22","author":"Cazuguel","year":"2006","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"101864","DOI":"10.1016\/j.strusafe.2019.05.001","article-title":"An efficient time-dependent reliability method","volume":"81","author":"Gong","year":"2019","journal-title":"Struct. Saf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s10999-015-9324-z","article-title":"A time-variant reliability analysis method for structural systems based on stochastic process discretization","volume":"13","author":"Jiang","year":"2017","journal-title":"Int. J. Mech. Mater. Des."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1935","DOI":"10.1007\/s00158-018-2002-7","article-title":"An improved TRPD method for time-variant reliability analysis","volume":"58","author":"Jiang","year":"2018","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0167-4730(94)90013-2","article-title":"A non-probabilistic concept of reliability","volume":"14","year":"1994","journal-title":"Struct. Saf."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Du, X., and Sudjianto, A. (2021, January 17\u201319). Reliability-Based Design with the Mixture of Random and Interval Variables. Proceedings of the ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Online, Virtual.","DOI":"10.1115\/DETC2003\/DAC-48709"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s10999-017-9401-6","article-title":"Robustness and reliability of composite structures: Effects of different sources of uncertainty","volume":"15","year":"2019","journal-title":"Int. J. Mech. Mater. Des."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1109\/TR.2017.2703593","article-title":"A single-loop approach for time-variant reliability-based design optimization","volume":"66","author":"Huang","year":"2017","journal-title":"IEEE Trans. Reliab."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"041402","DOI":"10.1115\/1.4035792","article-title":"An outcrossing rate model and its efficient calculation for time-dependent system reliability analysis","volume":"139","author":"Jiang","year":"2017","journal-title":"J. Mech. Des."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/0266-8920(89)90024-6","article-title":"Asymptotic approximations for probability integrals","volume":"4","author":"Breitung","year":"1989","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_40","unstructured":"Madsen, H.O., Krenk, S., and Lind, N.C. (2006). Method of Structural Safety, Courier Corporation."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/0167-4730(85)90027-X","article-title":"First order vs. second order reliability analysis of series structures","volume":"2","author":"Madsen","year":"1985","journal-title":"Struct. Saf."},{"key":"ref_42","unstructured":"Nocedal, J., and Wright, S.J. (2006). Numerical Optimization, Springer. [2nd ed.]."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/s00158-010-0582-y","article-title":"A robust iterative algorithm for structural reliability analysis","volume":"43","author":"Gong","year":"2011","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1728","DOI":"10.1016\/j.cpc.2012.03.014","article-title":"Efficient sampling methods for global reliability sensitivity analysis","volume":"183","author":"Wei","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1007\/s00158-014-1189-5","article-title":"An active learning kriging model for hybrid reliability analysis with both random and interval variables","volume":"51","author":"Yang","year":"2015","journal-title":"Struct. Multidiscip. Optim."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/8\/229\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:38:32Z","timestamp":1760164712000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/8\/229"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,31]]},"references-count":45,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["a14080229"],"URL":"https:\/\/doi.org\/10.3390\/a14080229","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,31]]}}}