{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:28:10Z","timestamp":1760146090482,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,24]],"date-time":"2024-09-24T00:00:00Z","timestamp":1727136000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LGF22E080021","LY24E080010"],"award-info":[{"award-number":["LGF22E080021","LY24E080010"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The stochastic finite element method is an important tool for structural reliability analysis. In order to improve the calculation efficiency, a stochastic finite element method based on the Krylov subspace is proposed for the static reliability analysis of structures. The first step of the proposed method is to preprocess the static response equation considering randomness to reduce the condition number of the coefficient matrix. The second step of the proposed method is to construct a Krylov subspace based on the preprocessed static response equation. Then, the static displacement of random sampling is expressed as a linear combination of subspace basis vectors to achieve the purpose of a fast solution. Finally, statistics and failure probability are calculated according to the static response obtained from thousands of random samples. Three numerical examples are given to compare the proposed method with the stochastic finite element method based on the Neumann series. The results show that the stochastic finite element method based on the Krylov subspace is more accurate and efficient than the stochastic finite element method based on the Neumann series.<\/jats:p>","DOI":"10.3390\/a17100424","type":"journal-article","created":{"date-parts":[[2024,9,24]],"date-time":"2024-09-24T08:54:17Z","timestamp":1727168057000},"page":"424","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace"],"prefix":"10.3390","volume":"17","author":[{"given":"Jianyun","family":"Huang","sequence":"first","affiliation":[{"name":"Ningbo Communications Engineering Construction Group Co., Ltd., Ningbo 315000, China"},{"name":"Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315211, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiuwei","family":"Yang","sequence":"additional","affiliation":[{"name":"Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315211, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongfei","family":"Cao","sequence":"additional","affiliation":[{"name":"Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315211, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiwei","family":"Ma","sequence":"additional","affiliation":[{"name":"Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315211, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"102019","DOI":"10.1016\/j.strusafe.2020.102019","article-title":"Adaptive approaches in metamodel-based reliability analysis: A review","volume":"89","author":"Teixeira","year":"2021","journal-title":"Struct. Saf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108223","DOI":"10.1016\/j.ress.2021.108223","article-title":"Machine learning-based methods in structural reliability analysis: A review","volume":"219","author":"Afshari","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"108731","DOI":"10.1016\/j.ress.2022.108731","article-title":"Bayesian post-processing of Monte Carlo simulation in reliability analysis","volume":"227","author":"Betz","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"114218","DOI":"10.1016\/j.cma.2021.114218","article-title":"Hybrid enhanced Monte Carlo simulation coupled with advanced machine learning approach for accurate and efficient structural reliability analysis","volume":"388","author":"Luo","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/S0167-8191(99)00022-8","article-title":"Krylov subspace methods for structural finite element analysis","volume":"25","author":"Lenhardt","year":"1999","journal-title":"Parallel Comput."},{"key":"ref_6","first-page":"305","article-title":"Overview of Reliability Analysis Methods Based on Random Finite Element Method","volume":"70","author":"Shi","year":"2023","journal-title":"China Rubber Ind."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"108817","DOI":"10.1016\/j.soildyn.2024.108817","article-title":"Seismic reliability analysis of high earth-rockfill dams subjected to mainshock-aftershock sequences using a novel noninvasive stochastic finite element method","volume":"183","author":"Pang","year":"2024","journal-title":"Soil Dyn. Earthq. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.dt.2023.07.015","article-title":"Random vibration analysis of FGM plates subjected to moving load using a refined stochastic finite element method","volume":"34","author":"Do","year":"2024","journal-title":"Def. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1002\/nag.3684","article-title":"Stochastic finite element modeling of heterogeneities in massive concrete and reinforced concrete structures","volume":"48","author":"Ghannoum","year":"2024","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kami\u0144ski, M., Guminiak, M., Lenartowicz, A., \u0141asecka-Plura, M., Przychodzki, M., and Sumelka, W. (2023). Eigenvibrations of Kirchhoff Rectangular Random Plates on Time-Fractional Viscoelastic Supports via the Stochastic Finite Element Method. Materials, 16.","DOI":"10.3390\/ma16247527"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1007\/s11803-023-2217-5","article-title":"Resonance analysis of a high-speed railway bridge using a stochastic finite element method","volume":"22","author":"Xiang","year":"2023","journal-title":"Earthq. Eng. Eng. Vib."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"116815","DOI":"10.1016\/j.compstruct.2023.116815","article-title":"Non-destructive strength prediction of composite laminates utilizing deep learning and the stochastic finite element methods","volume":"311","author":"Nastos","year":"2023","journal-title":"Compos. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"115860","DOI":"10.1016\/j.cma.2022.115860","article-title":"A stochastic finite element scheme for solving partial differential equations defined on random domains","volume":"405","author":"Zheng","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"103414","DOI":"10.1016\/j.probengmech.2023.103414","article-title":"Efficient structural reliability analysis via a weak-intrusive stochastic finite element method","volume":"71","author":"Zheng","year":"2023","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"106099","DOI":"10.1016\/j.compag.2021.106099","article-title":"Modal properties of macaw palm fruit-rachilla system: An approach by the stochastic finite element method (SFEM)","volume":"184","author":"Santos","year":"2021","journal-title":"Comput. Electron. Agric."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1007\/s00419-020-01819-8","article-title":"Stochastic finite element method based on point estimate and Karhunen\u2013Lo\u00e9ve expansion","volume":"91","author":"Liu","year":"2021","journal-title":"Arch. Appl. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"106326","DOI":"10.1016\/j.compstruc.2020.106326","article-title":"Extended stochastic finite element method enhanced by local mesh refinement for random voids analysis","volume":"239","author":"Han","year":"2020","journal-title":"Comput. Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1080\/17499518.2019.1690151","article-title":"Reliability-based assessment of foundations under HM combined loading using random finite element method","volume":"14","author":"Kormi","year":"2020","journal-title":"Georisk Assess. Manag. Risk Eng. Syst. Geohazards"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ahmadi Moghaddam, H., and Mertiny, P. (2020). Stochastic finite element analysis framework for modelling electrical properties of particle-modified polymer composites. Nanomaterials, 10.","DOI":"10.3390\/nano10091754"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1002\/nag.3137","article-title":"Dynamic stochastic finite element method using time-dependent generalized polynomial chaos","volume":"45","author":"Lacour","year":"2021","journal-title":"Int. J. Numer. Anal. Methods Geomech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"110800","DOI":"10.1016\/j.nucengdes.2020.110800","article-title":"Stochastic finite elements analysis of large concrete structures\u2019 serviceability under thermo-hydro-mechanical loads\u2013Case of nuclear containment buildings","volume":"370","author":"Bouhjiti","year":"2020","journal-title":"Nucl. Eng. Des."},{"key":"ref_22","first-page":"1431","article-title":"An interval finite element method based on the neumann series expansion","volume":"52","author":"Wu","year":"2020","journal-title":"Chin. J. Theor. Appl. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Shinozuka, M., and Yamazaki, F. (2020). Stochastic finite element analysis: An introduction. Stochastic Structural Dynamics, Chapman and Hall\/CRC.","DOI":"10.1201\/9781003076582-14"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"115812","DOI":"10.1016\/j.cma.2022.115812","article-title":"Strain energy density as a Gaussian process and its utilization in stochastic finite element analysis: Application to planar soft tissues","volume":"404","author":"Aggarwal","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105578","DOI":"10.1016\/j.jmps.2024.105578","article-title":"Mechanistic map of random fields for stochastic finite element simulations of quasibrittle fracture","volume":"186","author":"Vievering","year":"2024","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"108723","DOI":"10.1016\/j.soildyn.2024.108723","article-title":"Seismic analysis of gravity dam-foundation systems using stochastic spectral finite element method","volume":"182","author":"Zeng","year":"2024","journal-title":"Soil Dyn. Earthq. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4431","DOI":"10.1016\/j.apm.2009.02.012","article-title":"Model reduction by Neumann series expansion","volume":"33","author":"Yang","year":"2009","journal-title":"Appl. Math. Model."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1002\/nme.7129","article-title":"highly efficient method for structural model reduction","volume":"124","author":"Yang","year":"2023","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_29","first-page":"325025","article-title":"Generalized Neumann expansion and its application in stochastic finite element methods","volume":"2013","author":"Wang","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"967","DOI":"10.2514\/1.C033883","article-title":"Improved Neumann expansion method for stochastic finite element analysis","volume":"54","author":"Bae","year":"2017","journal-title":"J. Aircr."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1299\/mel.20-00228","article-title":"Influence of higher orders of Neumann expansion on accuracy of stochastic linear elastic finite element method with random physical parameters","volume":"6","author":"Degeneve","year":"2020","journal-title":"Mech. Eng. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1007\/s00419-023-02369-5","article-title":"The Neumann\u2013Monte Carlo methodology applied to the quantification of uncertainty in the problem stochastic bending of the Levinson\u2013Bickford beam","volume":"93","author":"Squarcio","year":"2023","journal-title":"Arch. Appl. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Golub, G.H., and Van Loan, C.F. (2013). Matrix Computations, JHU Press.","DOI":"10.56021\/9781421407944"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/10\/424\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:01:01Z","timestamp":1760112061000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/10\/424"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,24]]},"references-count":33,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["a17100424"],"URL":"https:\/\/doi.org\/10.3390\/a17100424","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2024,9,24]]}}}