{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T08:43:45Z","timestamp":1777884225439,"version":"3.51.4"},"reference-count":41,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100009103","name":"Education Department of Shaanxi Provincial government","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100009103","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017596","name":"Natural Science Basic Research Program of Shaanxi Province","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100017596","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Advances in Engineering Software"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1016\/j.advengsoft.2026.104146","type":"journal-article","created":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T01:59:13Z","timestamp":1773107953000},"page":"104146","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Structural stiffness topology optimization considering nonlinear continuous cumulative damage"],"prefix":"10.1016","volume":"217","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9031-2195","authenticated-orcid":false,"given":"Yongsheng","family":"Han","sequence":"first","affiliation":[]},{"given":"Linfeng","family":"Han","sequence":"additional","affiliation":[]},{"given":"Runze","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.advengsoft.2026.104146_bib0001","doi-asserted-by":"crossref","unstructured":"M.P. Bends\u00f8e, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput Methods Appl Mech Eng 71(2) (1988) 197-224. DOI: 10.1016\/0045-7825(88)90086-2.","DOI":"10.1016\/0045-7825(88)90086-2"},{"key":"#cr-split#-10.1016\/j.advengsoft.2026.104146_bib0002.1","doi-asserted-by":"crossref","unstructured":"J.D. Deaton, R.V. Grandhi, A survey of structural and multidisciplinary continuum topology optimization: post 2000, Struct Multidiscip Optim 49","DOI":"10.1007\/s00158-013-0956-z"},{"key":"#cr-split#-10.1016\/j.advengsoft.2026.104146_bib0002.2","doi-asserted-by":"crossref","unstructured":"(1) (2014) 1-38. DOI: 10.1007\/s00158-013-0956-z.","DOI":"10.1007\/s00158-013-0956-z"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0003","doi-asserted-by":"crossref","unstructured":"J.H. Zhu, W.H. Zhang, L. Xia, Topology optimization in aircraft and aerospace structures design, Arch Comput Method Eng 23(4) (2016) 595-622. DOI: 10.1007\/s11831-015-9151-2.","DOI":"10.1007\/s11831-015-9151-2"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0004","doi-asserted-by":"crossref","unstructured":"K. Suzuki, N. Kikuchi, A homogenization method for shape and topology optimization, Comput Methods Appl Mech Eng 93(3) (1991) 291-318. DOI: 10.1016\/0045-7825(91)90245-2.","DOI":"10.1016\/0045-7825(91)90245-2"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0005","doi-asserted-by":"crossref","unstructured":"G.I.N. Rozvany, Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics, Struct Multidiscip Optim 21(2) (2001) 90-108. DOI: 10.1007\/s001580050174.","DOI":"10.1007\/s001580050174"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0006","doi-asserted-by":"crossref","unstructured":"J.A. Sethian, A. Wiegmann, Structural boundary design via level set and immersed interface methods, J Comput Phys 163(2) (2000) 489-528. DOI: 10.1006\/jcph.2000.6581.","DOI":"10.1006\/jcph.2000.6581"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0007","doi-asserted-by":"crossref","unstructured":"M.Y. Wang, X.M. Wang, D.M. Guo, A level set method for structural topology optimization, Comput Methods Appl Mech Eng 192(1-2) (2003) 227-246. DOI: 10.1016\/S0045-7825(02)00559-5.","DOI":"10.1016\/S0045-7825(02)00559-5"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0008","doi-asserted-by":"crossref","unstructured":"X. Guo, W.S. Zhang, W.L. Zhong, Doing topology optimization explicitly and geometrically-A new moving morphable components based framework, J Appl Mech Trans ASME. 81(8) (2014) 081009. DOI: 10.1115\/1.4027609.","DOI":"10.1115\/1.4027609"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0009","doi-asserted-by":"crossref","unstructured":"X. Guo, W.S. Zhang, J. Zhang, J. Yuan, Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons, Comput Methods Appl Mech Eng 310 (2016) 711-748. DOI: 10.1016\/j.cma.2016.07.018.","DOI":"10.1016\/j.cma.2016.07.018"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0010","doi-asserted-by":"crossref","unstructured":"X.P. Qian, Topology optimization in B-spline space, Comput Methods Appl Mech Eng 265 (2013) 15-35. DOI: 10.1016\/j.cma.2013.06.001.","DOI":"10.1016\/j.cma.2013.06.001"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0011","doi-asserted-by":"crossref","unstructured":"J.H. Rong, Y.M. Xie, X.Y. Yang, An improved method for evolutionary structural optimization against buckling, Comput Struct 79(3) (2001) 253-263. DOI: 10.1016\/S0045-7949(00)00145-0.","DOI":"10.1016\/S0045-7949(00)00145-0"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0012","doi-asserted-by":"crossref","unstructured":"X.D. Huang, Y.M. Xie, Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials, Comput Mech 43(3) (2009) 393-401. DOI: 10.1007\/s00466-008-0312-0.","DOI":"10.1007\/s00466-008-0312-0"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0013","doi-asserted-by":"crossref","unstructured":"O. Sigmund, K. Maute, Topology optimization approaches-a comparative review, Struct Multidiscip Optim 48(6) (2013) 1031-1055. DOI: 10.1007\/s00158-013-0978-6.","DOI":"10.1007\/s00158-013-0978-6"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0014","doi-asserted-by":"crossref","unstructured":"M. Kranz, J.K. L\u00fcdeker, B. Kriegesmann, An empirical study on stress-based fail-safe topology optimization and multiple load path design, Struct Multidiscip Optim 64(4) (2021) 2113-2134. DOI: 10.1007\/s00158-021-02969-1.","DOI":"10.1007\/s00158-021-02969-1"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0015","doi-asserted-by":"crossref","unstructured":"J. Mart\u00ednez-Frutos, R. Ortigosa, Risk-averse approach for topology optimization of fail-safe structures using the level-set method, Comput Mech 68(5) (2021) 1039-1061. DOI: 10.1007\/s00466-021-02058-6.","DOI":"10.1007\/s00466-021-02058-6"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0016","doi-asserted-by":"crossref","unstructured":"P.F. Li, Y. Wu, J. Yvonnet, A SIMP-Phase field topology optimization framework to maximize quasi-brittle fracture resistance of 2D and 3D composites, Theor Appl Fract Mech 114 (2021) 102919. DOI: 10.1016\/j.tafmec.2021.102919.","DOI":"10.1016\/j.tafmec.2021.102919"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0017","doi-asserted-by":"crossref","unstructured":"C. Wu, J.G. Fang, S.W. Zhou, Z.P. Zhang, G.Y. Sun, G.P. Steven, Q. Li, A path-dependent level set topology optimization with fracture criterion, Comput Struct 249 (2021) 106515. DOI: 10.1016\/j.compstruc.2021.106515.","DOI":"10.1016\/j.compstruc.2021.106515"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0018","doi-asserted-by":"crossref","unstructured":"M.P. Bends\u00f8e, A.R. D\u00edaz, A method for treating damage related criteria in optimal topology design of continuum structures, Struct Optim 16(2-3) (1998) 108-115. DOI: 10.1007\/BF01202821.","DOI":"10.1007\/BF01202821"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0019","doi-asserted-by":"crossref","unstructured":"O. Amir, O. Sigmund, Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization, Struct Multidiscip Optim 47(2) (2013) 157-174. DOI: 10.1007\/s00158-012-0817-1.","DOI":"10.1007\/s00158-012-0817-1"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0020","doi-asserted-by":"crossref","unstructured":"O. Amir, A topology optimization procedure for reinforced concrete structures, Comput Struct 114 (2013) 46-58. DOI: 10.1016\/j.compstruc.2012.10.011.","DOI":"10.1016\/j.compstruc.2012.10.011"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0021","doi-asserted-by":"crossref","unstructured":"K.A. James, H. Waisman, Failure mitigation in optimal topology design using a coupled nonlinear continuum damage model, Comput Methods Appl Mech Eng 268 (2014) 614-631. DOI: 10.1016\/j.cma.2013.10.022.","DOI":"10.1016\/j.cma.2013.10.022"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0022","doi-asserted-by":"crossref","unstructured":"K.A. James, H. Waisman, Topology optimization of structures under variable loading using a damage superposition approach, Int J Numer Methods Eng 101(5) (2015) 375-406. DOI: 10.1002\/nme.4810.","DOI":"10.1002\/nme.4810"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0023","doi-asserted-by":"crossref","unstructured":"L. Li, G.D. Zhang, K. Khandelwal, Topology optimization of energy absorbing structures with maximum damage constraint, Int J Numer Methods Eng 112(7) (2017) 737-775. DOI: 10.1002\/nme.5531.","DOI":"10.1002\/nme.5531"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0024","doi-asserted-by":"crossref","unstructured":"L. No\u00ebl, P. Duysinx, K. Maute, Level set topology optimization considering damage, Struct Multidiscip Optim 56(4) (2017) 737-753. DOI: 10.1007\/s00158-017-1724-2.","DOI":"10.1007\/s00158-017-1724-2"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0025","doi-asserted-by":"crossref","unstructured":"J.Z. Du, Y.H. Guo, Z.M. Chen, Y.K. Sui, Topology optimization of continuum structures considering damage based on independent continuous mapping method, Acta Mech Sin PRC 35(2) (2019) 433-444. DOI: 10.1007\/s10409-018-0807-7.","DOI":"10.1007\/s10409-018-0807-7"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0026","doi-asserted-by":"crossref","unstructured":"L. Li, G.D. Zhang, K. Khandelwal, Failure resistant topology optimization of structures using nonlocal elastoplastic damage model, Struct Multidiscip Optim 58(4) (2018)1589-1618. DOI: 10.1007\/s00158-018-1984-5.","DOI":"10.1007\/s00158-018-1984-5"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0027","doi-asserted-by":"crossref","unstructured":"R. Alberdi, K. Khandelwal, Topology optimization of pressure dependent elastoplastic energy absorbing structures with material damage constraints, Finite Elem Anal Des 133 (2017) 42-61. DOI: 10.1016\/j.finel.2017.05.004.","DOI":"10.1016\/j.finel.2017.05.004"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0028","doi-asserted-by":"crossref","unstructured":"D.C. Da, J. Yvonnet, L. Xia, G.Y. Li, Topology optimization of particle-matrix composites for optimal fracture resistance taking into account interfacial damage, Int J Numer Methods Eng 115(5) (2018) 604-626. DOI: 10.1002\/nme.5818.","DOI":"10.1002\/nme.5818"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0029","doi-asserted-by":"crossref","unstructured":"M. Jansen, G. Lombaert, M. Schevenels, O. Sigmund, Topology optimization of fail-safe structures using a simplified local damage model, Struct Multidiscip Optim 49(4) (2014) 657-666. DOI: 10.1007\/s00158-013-1001-y.","DOI":"10.1007\/s00158-013-1001-y"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0030","doi-asserted-by":"crossref","unstructured":"Z. Chen, K. Long, P. Wen, S. Nouman, Fatigue-resistance topology optimization of continuum structure by penalizing the cumulative fatigue damage, Adv Eng Softw 150 (2020) 102924. DOI: 10.1016\/j.advengsoft.2020.102924.","DOI":"10.1016\/j.advengsoft.2020.102924"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0031","doi-asserted-by":"crossref","unstructured":"K. Long, X. Wang, Y.X. Du, Robust topology optimization formulation including local failure and load uncertainty using sequential quadratic programming, Int J Mech Mater Des 15 (2019) 317-332. DOI: 10.1007\/s10999-018-9411-z.","DOI":"10.1007\/s10999-018-9411-z"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0032","doi-asserted-by":"crossref","unstructured":"Z.S. Li, L. Wang, Y. Chai, L. Zhang, Y.R. Liu, Topology design of multi-scale thermo-elastic structures considering performance reduction and disturbance based on interval models, Appl Math Model 146 (2025) 116167. DOI: 10.1016\/j.apm.2025.116167.","DOI":"10.1016\/j.apm.2025.116167"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0033","doi-asserted-by":"crossref","unstructured":"Z.S. Li, L. Wang, F. Li, Model-and-data-driven robust concurrent optimization for structural topology and device layout integrating load location and non-design domain effects, Aerosp Sci Technol 170 (2026) 111566. DOI: 10.1016\/j.ast.2025.111566.","DOI":"10.1016\/j.ast.2025.111566"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0034","doi-asserted-by":"crossref","unstructured":"Z.S. Li, L. Wang, K.X. Gu, Y.R. Liu, Interval-based scale-span topology optimization considering reliable manufacturing limits via the parameterized level set method and image recognition techniques, Thin Walled Struct 214 (2025) 113353. DOI: 10.1016\/j.tws.2025.113353.","DOI":"10.1016\/j.tws.2025.113353"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0035","doi-asserted-by":"crossref","unstructured":"X.Y. Zhao, L. Wang, Double-scale time-dependent reliable topology optimization based on the first-passage failure and interval process theories, Comput Methods Appl Mech Eng 443 (2025) 118088. DOI: 10.1016\/j.cma.2025.118088.","DOI":"10.1016\/j.cma.2025.118088"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0036","doi-asserted-by":"crossref","unstructured":"L. Xia, L. Zhang, Q. Xia, T.L. Shi, Stress-based topology optimization using bi-directional evolutionary structural optimization method, Comput Methods Appl Mech Eng 333 (2018) 356-370. DOI: 10.1016\/j.cma.2018.01.035.","DOI":"10.1016\/j.cma.2018.01.035"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0037","doi-asserted-by":"crossref","unstructured":"F. Zhao, L. Xia, W.X. Lai, Q. Xia, T.L. Shi, Evolutionary topology optimization of continuum structures with stress constraints, Struct Multidiscip Optim 59(2) (2019) 647-658. DOI: 10.1007\/s00158-018-2090-4.","DOI":"10.1007\/s00158-018-2090-4"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0038","doi-asserted-by":"crossref","unstructured":"B. Xu, Y.S. Han, L. Zhao, Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints, Appl Math Model 80 (2020) 771-791. DOI: 10.1016\/j.apm.2019.12.009.","DOI":"10.1016\/j.apm.2019.12.009"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0039","doi-asserted-by":"crossref","unstructured":"B. Xu, Y.S. Han, L. Zhao, Bi-directional evolutionary stress-based topology optimization of material nonlinear structures, Struct Multidiscip Optim 63(3) (2020) 1287-1305. DOI: 10.1007\/s00158-020-02757-3.","DOI":"10.1007\/s00158-020-02757-3"},{"key":"10.1016\/j.advengsoft.2026.104146_bib0040","doi-asserted-by":"crossref","unstructured":"O. Sigmund, A 99 line topology optimization code written in Matlab, Struct Multidiscip Optim 21(2) (2001) 120-127. DOI: 10.1007\/s001580050176.","DOI":"10.1007\/s001580050176"}],"container-title":["Advances in Engineering Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0965997826000529?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0965997826000529?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T02:17:28Z","timestamp":1777601848000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0965997826000529"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6]]},"references-count":41,"alternative-id":["S0965997826000529"],"URL":"https:\/\/doi.org\/10.1016\/j.advengsoft.2026.104146","relation":{},"ISSN":["0965-9978"],"issn-type":[{"value":"0965-9978","type":"print"}],"subject":[],"published":{"date-parts":[[2026,6]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Structural stiffness topology optimization considering nonlinear continuous cumulative damage","name":"articletitle","label":"Article Title"},{"value":"Advances in Engineering Software","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.advengsoft.2026.104146","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"104146"}}