{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T09:52:43Z","timestamp":1777024363994,"version":"3.51.4"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T00:00:00Z","timestamp":1769990400000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100014013","name":"UK Research and Innovation","doi-asserted-by":"publisher","award":["EP\/Y037103\/1"],"award-info":[{"award-number":["EP\/Y037103\/1"]}],"id":[{"id":"10.13039\/100014013","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s00366-025-02237-6","type":"journal-article","created":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T04:01:16Z","timestamp":1770004876000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A differential equation-driven update strategy for density-based topology optimization: implementation with MATLAB codes"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8907-4388","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6380-9259","authenticated-orcid":false,"given":"Wei","family":"Tan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,2]]},"reference":[{"issue":"1","key":"2237_CR1","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.jcp.2003.09.032","volume":"194","author":"G Allaire","year":"2004","unstructured":"Allaire G, Jouve F (2004) Toader A-M Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363\u2013393","journal-title":"J Comput Phys"},{"issue":"2","key":"2237_CR2","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.jcp.2005.12.015","volume":"216","author":"S Amstutz","year":"2006","unstructured":"Amstutz S (2006) Andr\u00e4 H A new algorithm for topology optimization using a level-set method. J Comput Phys 216(2):573\u2013588","journal-title":"J Comput Phys"},{"issue":"2","key":"2237_CR3","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1007\/s00158-020-02527-1","volume":"62","author":"CS Andreasen","year":"2020","unstructured":"Andreasen CS, Elingaard MO (2020) Aage N Level set topology and shape optimization by density methods using cut elements with length scale control. Struct Multidiscip Optim 62(2):685\u2013707","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00158-010-0594-7","volume":"43","author":"E Andreassen","year":"2011","unstructured":"Andreassen E, Clausen A, Schevenels M, Lazarov BS (2011) Sigmund O Efficient topology optimization in matlab using 88 lines of code. Struct Multidiscip Optim 43:1\u201316","journal-title":"Struct Multidiscip Optim"},{"issue":"1","key":"2237_CR5","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s00158-019-02353-0","volume":"61","author":"J Bai","year":"2020","unstructured":"Bai J (2020) Zuo W Hollow structural design in topology optimization via moving morphable component method. Struct Multidiscip Optim 61(1):187\u2013205","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR6","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03115-5","volume-title":"Optimization of structural topology, shape, and material,","author":"MP Bends\u00f8e","year":"1995","unstructured":"Bends\u00f8e MP (1995) Optimization of structural topology, shape, and material, vol 414. Springer"},{"key":"2237_CR7","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s004190050248","volume":"69","author":"MP Bends\u00f8e","year":"1999","unstructured":"Bends\u00f8e MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635\u2013654","journal-title":"Arch Appl Mech"},{"issue":"2","key":"2237_CR8","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0045-7825(88)90086-2","volume":"71","author":"MP Bends\u00f8e","year":"1988","unstructured":"Bends\u00f8e MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197\u2013224","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2237_CR9","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/s00158-009-0430-0","volume":"41","author":"VJ Challis","year":"2010","unstructured":"Challis VJ (2010) A discrete level-set topology optimization code written in matlab. Struct Multidiscip Optim 41:453\u2013464","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR10","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1007\/s11831-019-09362-8","volume":"27","author":"S Coniglio","year":"2020","unstructured":"Coniglio S, Morlier J, Gogu C, Amargier R (2020) Generalized geometry projection: a unified approach for geometric feature based topology optimization. Archives of Computational Methods in Engineering 27:1573\u20131610","journal-title":"Archives of Computational Methods in Engineering"},{"key":"2237_CR11","doi-asserted-by":"crossref","first-page":"2211","DOI":"10.1007\/s00158-020-02629-w","volume":"62","author":"F Ferrari","year":"2020","unstructured":"Ferrari F, Sigmund O (2020) A new generation 99 line matlab code for compliance topology optimization and its extension to 3d. Struct Multidiscip Optim 62:2211\u20132228","journal-title":"Struct Multidiscip Optim"},{"issue":"4","key":"2237_CR12","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1007\/s00158-020-02760-8","volume":"63","author":"O Giraldo-Londo\u00f1o","year":"2021","unstructured":"Giraldo-Londo\u00f1o O, Paulino GH (2021) Polystress: a matlab implementation for local stress-constrained topology optimization using the augmented lagrangian method. Struct Multidiscip Optim 63(4):2065\u20132097","journal-title":"Struct Multidiscip Optim"},{"issue":"8","key":"2237_CR13","volume":"81","author":"X Guo","year":"2014","unstructured":"Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically-a new moving morphable components based framework. J Appl Mech 81(8):081009","journal-title":"J Appl Mech"},{"key":"2237_CR14","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s00466-008-0312-0","volume":"43","author":"X Huang","year":"2009","unstructured":"Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43:393\u2013401","journal-title":"Comput Mech"},{"key":"2237_CR15","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1007\/s00158-010-0487-9","volume":"41","author":"X Huang","year":"2010","unstructured":"Huang X, Xie Y-M (2010) A further review of eso type methods for topology optimization. Struct Multidiscip Optim 41:671\u2013683","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR16","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s00158-019-02396-3","volume":"61","author":"Y Liang","year":"2020","unstructured":"Liang Y, Cheng G (2020) Further elaborations on topology optimization via sequential integer programming and canonical relaxation algorithm and 128-line matlab code. Struct Multidiscip Optim 61:411\u2013431","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR17","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1007\/s00158-014-1107-x","volume":"50","author":"K Liu","year":"2014","unstructured":"Liu K, Tovar A (2014) An efficient 3d topology optimization code written in matlab. Struct Multidiscip Optim 50:1175\u20131196","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR18","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1016\/j.cma.2018.04.034","volume":"340","author":"Y Liu","year":"2014","unstructured":"Liu Y, Li Z, Wei P, Wang W (2014) Parameterized level-set based topology optimization method considering symmetry and pattern repetition constraints. Comput Methods Appl Mech Eng 340:1079\u20131101","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2237_CR19","doi-asserted-by":"publisher","unstructured":"Liu Y, Xia B, Li Z, Ren H, Du J, Tan W (2026) Co-design of structures via generalized evolution. Comput Meth Appl Mech Eng 450:118584. https:\/\/doi.org\/10.1016\/j.cma.2025.118584","DOI":"10.1016\/j.cma.2025.118584"},{"key":"2237_CR20","volume":"387","author":"Y Liu","year":"2021","unstructured":"Liu Y, Yang C, Wei P, Zhou P, Du J (2021) An ode-driven level-set density method for topology optimization. Comput Methods Appl Mech Eng 387:114159","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"2237_CR21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/nme.2092","volume":"76","author":"Z Luo","year":"2008","unstructured":"Luo Z, Wang MY, Wang S, Wei P (2008) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Meth Eng 76(1):1\u201326","journal-title":"Int J Numer Meth Eng"},{"key":"2237_CR22","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.cma.2015.05.005","volume":"293","author":"JA Norato","year":"2015","unstructured":"Norato JA, Bell BK, Tortorelli DA (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comput Methods Appl Mech Eng 293:306\u2013327","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"2237_CR23","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","volume":"79","author":"S Osher","year":"1988","unstructured":"Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulations. J Comput Phys 79(1):12\u201349","journal-title":"J Comput Phys"},{"key":"2237_CR24","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1007\/s00158-014-1190-z","volume":"51","author":"M Otomori","year":"2015","unstructured":"Otomori M, Yamada T, Izui K, Nishiwaki S (2015) Matlab code for a level set-based topology optimization method using a reaction diffusion equation. Struct Multidiscip Optim 51:1159\u20131172","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR25","doi-asserted-by":"crossref","first-page":"2727","DOI":"10.1007\/s00158-018-2094-0","volume":"58","author":"ED Sanders","year":"2018","unstructured":"Sanders ED, Pereira A, Aguil\u00f3 MA, Paulino GH (2018) Polymat: an efficient matlab code for multi-material topology optimization. Struct Multidiscip Optim 58:2727\u20132759","journal-title":"Struct Multidiscip Optim"},{"issue":"2","key":"2237_CR26","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1006\/jcph.2000.6581","volume":"163","author":"JA Sethian","year":"2000","unstructured":"Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489\u2013528","journal-title":"J Comput Phys"},{"key":"2237_CR27","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1007\/s001580050176","volume":"21","author":"O Sigmund","year":"2001","unstructured":"Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21:120\u2013127","journal-title":"Struct Multidiscip Optim"},{"issue":"6","key":"2237_CR28","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1007\/s00158-013-0978-6","volume":"48","author":"O Sigmund","year":"2013","unstructured":"Sigmund O, Maute K (2013) Topology optimization approaches: A comparative review. Struct Multidiscip Optim 48(6):1031\u20131055","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR29","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/s00158-016-1420-7","volume":"54","author":"O Sigmund","year":"2016","unstructured":"Sigmund O, Aage N, Andreassen E (2016) On the (non-) optimality of michell structures. Struct Multidiscip Optim 54:361\u2013373","journal-title":"Struct Multidiscip Optim"},{"issue":"3","key":"2237_CR30","doi-asserted-by":"crossref","first-page":"1579","DOI":"10.1007\/s00158-020-02552-0","volume":"62","author":"H Smith","year":"2020","unstructured":"Smith H, Norato JA (2020) A matlab code for topology optimization using the geometry projection method. Struct Multidiscip Optim 62(3):1579\u20131594","journal-title":"Struct Multidiscip Optim"},{"issue":"7","key":"2237_CR31","doi-asserted-by":"crossref","first-page":"2697","DOI":"10.1016\/j.jcp.2009.12.017","volume":"229","author":"A Takezawa","year":"2010","unstructured":"Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697\u20132718","journal-title":"J Comput Phys"},{"issue":"3","key":"2237_CR32","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s00158-011-0706-z","volume":"45","author":"C Talischi","year":"2012","unstructured":"Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct Multidiscip Optim 45(3):309\u2013328","journal-title":"Struct Multidiscip Optim"},{"key":"2237_CR33","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s00158-011-0696-x","volume":"45","author":"C Talischi","year":"2012","unstructured":"Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) Polytop: a matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45:329\u2013357","journal-title":"Struct Multidiscip Optim"},{"issue":"5","key":"2237_CR34","doi-asserted-by":"crossref","first-page":"2827","DOI":"10.1007\/s00158-021-03050-7","volume":"64","author":"C Wang","year":"2021","unstructured":"Wang C, Zhao Z, Zhou M, Sigmund O, Zhang XS (2021) A comprehensive review of educational articles on structural and multidisciplinary optimization. Struct Multidiscip Optim 64(5):2827\u20132880","journal-title":"Struct Multidiscip Optim"},{"issue":"1\u20132","key":"2237_CR35","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/S0045-7825(02)00559-5","volume":"192","author":"MY Wang","year":"2003","unstructured":"Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1\u20132):227\u2013246","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"2237_CR36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.32604\/cmes.2023.027603","volume":"137","author":"Y Wang","year":"2023","unstructured":"Wang Y, Li X, Long K, Wei P (2023) Open-source codes of topology optimization: A summary for beginners to start their research. CMES-Computer Modeling in Engineering & Sciences 137(1):1\u201334","journal-title":"CMES-Computer Modeling in Engineering & Sciences"},{"key":"2237_CR37","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1007\/s00158-018-1904-8","volume":"58","author":"P Wei","year":"2018","unstructured":"Wei P, Li Z, Li X, Wang MY (2018) An 88-line matlab code for the parameterized level set method based topology optimization using radial basis functions. Struct Multidiscip Optim 58:831\u2013849","journal-title":"Struct Multidiscip Optim"},{"issue":"6","key":"2237_CR38","doi-asserted-by":"crossref","first-page":"1229","DOI":"10.1007\/s00158-015-1294-0","volume":"52","author":"L Xia","year":"2015","unstructured":"Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in matlab. Struct Multidiscip Optim 52(6):1229\u20131241","journal-title":"Struct Multidiscip Optim"},{"issue":"5","key":"2237_CR39","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1016\/0045-7949(93)90035-C","volume":"49","author":"YM Xie","year":"1993","unstructured":"Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Computers & structures 49(5):885\u2013896","journal-title":"Computers & structures"},{"key":"2237_CR40","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1007\/s00158-020-02540-4","volume":"62","author":"M Yaghmaei","year":"2020","unstructured":"Yaghmaei M, Ghoddosian A, Khatibi MM (2020) A filter-based level set topology optimization method using a 62-line matlab code. Struct Multidiscip Optim 62:1001\u20131018","journal-title":"Struct Multidiscip Optim"},{"issue":"45\u201348","key":"2237_CR41","doi-asserted-by":"crossref","first-page":"2876","DOI":"10.1016\/j.cma.2010.05.013","volume":"199","author":"T Yamada","year":"2010","unstructured":"Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199(45\u201348):2876\u20132891","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2237_CR42","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1007\/s00158-015-1372-3","volume":"53","author":"W Zhang","year":"2016","unstructured":"Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model. Struct Multidiscip Optim 53:1243\u20131260","journal-title":"Struct Multidiscip Optim"},{"issue":"1","key":"2237_CR43","doi-asserted-by":"crossref","DOI":"10.1115\/1.4034972","volume":"84","author":"W Zhang","year":"2017","unstructured":"Zhang W, Yang W, Zhou J, Li D, Guo X (2017) Structural topology optimization through explicit boundary evolution. J Appl Mech 84(1):011011","journal-title":"J Appl Mech"},{"key":"2237_CR44","doi-asserted-by":"crossref","DOI":"10.1016\/j.advengsoft.2020.102955","volume":"152","author":"Y Zhao","year":"2021","unstructured":"Zhao Y, Hoang V-N, Jang G-W, Zuo W (2021) Hollow structural topology optimization to improve manufacturability using three-dimensional moving morphable bars. Adv Eng Softw 152:102955","journal-title":"Adv Eng Softw"},{"issue":"1\u20133","key":"2237_CR45","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0045-7825(91)90046-9","volume":"89","author":"M Zhou","year":"1991","unstructured":"Zhou M, Rozvany GIN (1991) The coc algorithm, part ii: Topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1\u20133):309\u2013336","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2237_CR46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cma.2016.06.027","volume":"310","author":"Y Zhou","year":"2016","unstructured":"Zhou Y, Zhang W, Zhu J, Xu Z (2016) Feature-driven topology optimization method with signed distance function. Comput Methods Appl Mech Eng 310:1\u201332","journal-title":"Comput Methods Appl Mech Eng"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-025-02237-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-025-02237-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-025-02237-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T09:01:20Z","timestamp":1777021280000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-025-02237-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2]]},"references-count":46,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["2237"],"URL":"https:\/\/doi.org\/10.1007\/s00366-025-02237-6","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2]]},"assertion":[{"value":"30 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 February 2026","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 no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"To replicate the results presented in this work, readers can refer to the MATLAB code provided in the Appendix. Additionally, the source MATLAB code is available for download on GitHub.:","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Replication of results"}}],"article-number":"34"}}