{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T06:13:07Z","timestamp":1773468787249,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:00:00Z","timestamp":1773446400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:00:00Z","timestamp":1773446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100011392","name":"State Key Laboratory of Supramolecular Structure and Materials","doi-asserted-by":"publisher","award":["GZ231004"],"award-info":[{"award-number":["GZ231004"]}],"id":[{"id":"10.13039\/501100011392","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Hunan Science and Technology Innovative Research Team in Higher Educational Institutions of Key Technology and Application of Green Concrete Structure"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s10957-025-02870-7","type":"journal-article","created":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T05:30:05Z","timestamp":1773466205000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reconstruction of Structural Topology Optimization Mapping: A Novel Two-Stage Mapping Modeling and Solution Approach"],"prefix":"10.1007","volume":"209","author":[{"given":"Xi-Rong","family":"Peng","sequence":"first","affiliation":[]},{"given":"Yun-Kang","family":"Sui","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,14]]},"reference":[{"issue":"1","key":"2870_CR1","doi-asserted-by":"publisher","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, B.S., Sigmund, O.: Efficient topology optimization in MATLAB using 88 lines of code. Struct. Multidiscip. Optim. 43(1), 1\u201316 (2011)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"1","key":"2870_CR2","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/BF01197709","volume":"17","author":"M Beckers","year":"1999","unstructured":"Beckers, M.: Topology optimization using a dual method with discrete variables. Struct. Optim. 17(1), 14\u201324 (1999)","journal-title":"Struct. Optim."},{"issue":"4","key":"2870_CR3","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/BF01650949","volume":"1","author":"MP Bends\u00f8e","year":"1989","unstructured":"Bends\u00f8e, M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1(4), 193\u2013202 (1989)","journal-title":"Struct. Optim."},{"issue":"2","key":"2870_CR4","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/0045-7825(88)90086-2","volume":"71","author":"MP Bends\u00f8e","year":"1988","unstructured":"Bends\u00f8e, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197\u2013224 (1988)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"9","key":"2870_CR5","doi-asserted-by":"publisher","first-page":"2143","DOI":"10.1002\/nme.116","volume":"50","author":"B Bourdin","year":"2001","unstructured":"Bourdin, B.: Filters in topology optimization. Numer. Meth Engineering. 50(9), 2143\u20132158 (2001)","journal-title":"Numer. Meth Engineering."},{"issue":"5","key":"2870_CR6","doi-asserted-by":"publisher","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.: Generalized geometry projection: a unified approach for geometric feature based topology optimization. Arch. Computat. Methods Eng. 27(5), 1573\u20131610 (2020)","journal-title":"Arch. Computat. Methods Eng."},{"issue":"1","key":"2870_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00158-013-0956-z","volume":"49","author":"JD Deaton","year":"2014","unstructured":"Deaton, J.D., Grandhi, R.V.: A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49(1), 1\u201338 (2014)","journal-title":"Struct. Multidiscip. Optim."},{"key":"2870_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.advengsoft.2020.102921","volume":"150","author":"Y-F Fu","year":"2020","unstructured":"Fu, Y.-F., Rolfe, B., Chiu, L.N.S., Wang, Y., Huang, X., Ghabraie, K.: SEMDOT: smooth-edged material distribution for optimizing topology algorithm. Adv. Eng. Softw. 150, 102921 (2020)","journal-title":"Adv. Eng. Softw."},{"issue":"2","key":"2870_CR9","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1007\/s10957-023-02222-3","volume":"198","author":"Y-F Fu","year":"2023","unstructured":"Fu, Y.-F., Long, K., Rolfe, B.: On non-penalization SEMDOT using discrete variable sensitivities. J. Optim. Theory Appl. 198(2), 644\u2013677 (2023)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"2870_CR10","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1186\/s10033-020-00503-w","volume":"33","author":"J Gao","year":"2020","unstructured":"Gao, J., Xiao, M., Zhang, Y., Gao, L.: A comprehensive review of isogeometric topology optimization: methods, applications and prospects. Chin. J. Mech. Eng. 33(1), 87 (2020)","journal-title":"Chin. J. Mech. Eng."},{"issue":"2","key":"2870_CR11","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1002\/nme.1064","volume":"61","author":"JK Guest","year":"2004","unstructured":"Guest, J.K., Pr\u00e9vost, J.H., Belytschko, T.: Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Numer. Meth Eng.. 61(2), 238\u2013254 (2004)","journal-title":"Numer. Meth Eng.."},{"issue":"8","key":"2870_CR12","doi-asserted-by":"publisher","DOI":"10.1115\/1.4027609","volume":"81","author":"X Guo","year":"2014","unstructured":"Guo, X., Zhang, W., Zhong, W.: Doing topology optimization explicitly and geometrically\u2014a new moving morphable components based framework. J. Appl. Mech. 81(8), 081009 (2014)","journal-title":"J. Appl. Mech."},{"key":"2870_CR13","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/j.cma.2016.12.004","volume":"317","author":"VN Hoang","year":"2017","unstructured":"Hoang, V.N., Jang, G.W.: Topology optimization using moving morphable bars for versatile thickness control. Comput. Methods Appl. Mech. Eng. 317, 153\u2013173 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2870_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.engstruct.2020.110330","volume":"208","author":"X Huang","year":"2020","unstructured":"Huang, X.: Smooth topological design of structures using the floating projection. Eng. Struct. 208, 110330 (2020)","journal-title":"Eng. Struct."},{"key":"2870_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.advengsoft.2020.102942","volume":"151","author":"X Huang","year":"2021","unstructured":"Huang, X.: On smooth or 0\/1 designs of the fixed-mesh element-based topology optimization. Adv. Eng. Softw. 151, 102942 (2021)","journal-title":"Adv. Eng. Softw."},{"key":"2870_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115444","volume":"399","author":"X Huang","year":"2022","unstructured":"Huang, X., Li, W.: Three-field floating projection topology optimization of continuum structures. Comput. Methods Appl. Mech. Eng. 399, 115444 (2022)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2870_CR17","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1016\/j.cma.2018.10.050","volume":"348","author":"Y Liang","year":"2019","unstructured":"Liang, Y., Cheng, G.: Topology optimization via sequential integer programming and canonical relaxation algorithm. Comput. Methods Appl. Mech. Eng. 348, 64\u201396 (2019)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"2870_CR18","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/s00158-019-02396-3","volume":"61","author":"Y Liang","year":"2020","unstructured":"Liang, Y., Cheng, G.: Further elaborations on topology optimization via sequential integer programming and canonical relaxation algorithm and 128-line MATLAB code. Struct. Multidiscip. Optim. 61(1), 411\u2013431 (2020)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"6","key":"2870_CR19","doi-asserted-by":"publisher","first-page":"2851","DOI":"10.1007\/s00158-020-02693-2","volume":"62","author":"Y Liang","year":"2020","unstructured":"Liang, Y., Sun, K., Cheng, G.: Discrete variable topology optimization for compliant mechanism design via sequential approximate integer programming with trust region (SAIP-TR). Struct. Multidiscip. Optim. 62(6), 2851\u20132879 (2020)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"6","key":"2870_CR20","doi-asserted-by":"publisher","first-page":"2457","DOI":"10.1007\/s00158-018-1994-3","volume":"57","author":"J Liu","year":"2018","unstructured":"Liu, J., Gaynor, A.T., Chen, S., Kang, Z., Suresh, K., Takezawa, A., Li, L., Kato, J., Tang, J., Wang, C.C.L., Cheng, L., Liang, X., To, A.C.: Current and future trends in topology optimization for additive manufacturing. Struct. Multidiscip. Optim. 57(6), 2457\u20132483 (2018)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"1","key":"2870_CR21","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","volume":"79","author":"S Osher","year":"1988","unstructured":"Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12\u201349 (1988)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"2870_CR22","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/s00158-007-0217-0","volume":"37","author":"GIN Rozvany","year":"2009","unstructured":"Rozvany, G.I.N.: A critical review of established methods of structural topology optimization. Struct. Multidiscip. Optim. 37(3), 217\u2013237 (2009)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"4","key":"2870_CR23","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1080\/08905459708945415","volume":"25","author":"O Sigmund","year":"1997","unstructured":"Sigmund, O.: On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach. 25(4), 493\u2013524 (1997)","journal-title":"Mech. Struct. Mach."},{"issue":"4\u20135","key":"2870_CR24","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/s00158-006-0087-x","volume":"33","author":"O Sigmund","year":"2007","unstructured":"Sigmund, O.: Morphology-based black and white filters for topology optimization. Struct. Multidisc. Optim. 33(4\u20135), 401\u2013424 (2007)","journal-title":"Struct. Multidisc. Optim."},{"issue":"6","key":"2870_CR25","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1007\/s00158-013-0978-6","volume":"48","author":"O Sigmund","year":"2013","unstructured":"Sigmund, O., Maute, K.: Topology optimization approaches: a comparative review. Struct. Multidiscip. Optim. 48(6), 1031\u20131055 (2013)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"1","key":"2870_CR26","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1007\/BF01214002","volume":"16","author":"O Sigmund","year":"1998","unstructured":"Sigmund, O., Petersson, J.: Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Optim. 16(1), 68\u201375 (1998)","journal-title":"Struct. Optim."},{"key":"2870_CR27","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.finel.2017.10.006","volume":"139","author":"R Sivapuram","year":"2018","unstructured":"Sivapuram, R., Picelli, R.: Topology optimization of binary structures using integer linear programming. Finite Elem. Anal. Des. 139, 49\u201361 (2018)","journal-title":"Finite Elem. Anal. Des."},{"issue":"3","key":"2870_CR28","doi-asserted-by":"publisher","first-page":"1579","DOI":"10.1007\/s00158-020-02552-0","volume":"62","author":"H Smith","year":"2020","unstructured":"Smith, H., Norato, J.A.: A MATLAB code for topology optimization using the geometry projection method. Struct. Multidisc. Optim. 62(3), 1579\u20131594 (2020)","journal-title":"Struct. Multidisc. Optim."},{"issue":"5","key":"2870_CR29","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1002\/nme.700","volume":"57","author":"M Stolpe","year":"2003","unstructured":"Stolpe, M., Svanberg, K.: Modelling topology optimization problems as linear mixed 0\u20131 programs. Numer. Meth Eng.. 57(5), 723\u2013739 (2003)","journal-title":"Numer. Meth Eng.."},{"key":"2870_CR30","volume-title":"Solving and application for topology optimization of continuum structures: ICM method based on step function","author":"YK Sui","year":"2018","unstructured":"Sui, Y.K., Peng, X.R.: Solving and application for topology optimization of continuum structures: ICM method based on step function. Elsevier Inc., Cambridge (2018)"},{"issue":"4","key":"2870_CR31","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s00158-007-0118-2","volume":"34","author":"K Svanberg","year":"2007","unstructured":"Svanberg, K., Werme, M.: Sequential integer programming methods for stress constrained topology optimization. Struct. Multidisc. Optim. 34(4), 277\u2013299 (2007)","journal-title":"Struct. Multidisc. Optim."},{"issue":"5","key":"2870_CR32","doi-asserted-by":"publisher","first-page":"885","DOI":"10.1016\/0045-7949(93)90035-C","volume":"49","author":"YM Xie","year":"1993","unstructured":"Xie, Y.M., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49(5), 885\u2013896 (1993)","journal-title":"Comput. Struct."},{"issue":"11","key":"2870_CR33","doi-asserted-by":"publisher","first-page":"1483","DOI":"10.2514\/2.626","volume":"37","author":"XY Yang","year":"1999","unstructured":"Yang, X.Y., Xie, Y.M.: Bidirectional evolutionary method for stiffness optimization. AIAA J. 37(11), 1483\u20131488 (1999)","journal-title":"AIAA J."},{"issue":"5","key":"2870_CR34","doi-asserted-by":"publisher","first-page":"1173","DOI":"10.1007\/s00158-016-1466-6","volume":"54","author":"S Zhang","year":"2016","unstructured":"Zhang, S., Norato, J.A., Gain, A.L., Lyu, N.: A geometry projection method for the topology optimization of plate structures. Struct. Multidisc. Optim. 54(5), 1173\u20131190 (2016)","journal-title":"Struct. Multidisc. Optim."},{"key":"2870_CR35","doi-asserted-by":"publisher","first-page":"590","DOI":"10.1016\/j.cma.2017.05.002","volume":"322","author":"W Zhang","year":"2017","unstructured":"Zhang, W., Chen, J., Zhu, X., Zhou, J., Xue, D., Lei, X., Guo, X.: Explicit three dimensional topology optimization via moving morphable void (MMV) approach. Comput. Methods Appl. Mech. Eng. 322, 590\u2013614 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2870_CR36","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/0045-7825(91)90046-9","volume":"89","author":"M Zhou","year":"1991","unstructured":"Zhou, M., Rozvany, G.I.N.: The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89, 309\u2013336 (1991)","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02870-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02870-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02870-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T05:30:06Z","timestamp":1773466206000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02870-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,14]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["2870"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02870-7","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,14]]},"assertion":[{"value":"27 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 October 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 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 author declared that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"4"}}