{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T08:21:44Z","timestamp":1782462104001,"version":"3.54.5"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2017,3,15]],"date-time":"2017-03-15T00:00:00Z","timestamp":1489536000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"publisher","award":["201506965015"],"award-info":[{"award-number":["201506965015"]}],"id":[{"id":"10.13039\/501100004543","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":[[2017,10]]},"DOI":"10.1007\/s00366-017-0503-4","type":"journal-article","created":{"date-parts":[[2017,3,15]],"date-time":"2017-03-15T05:07:58Z","timestamp":1489554478000},"page":"871-884","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["A meshless method for multi-material topology optimization based on the alternating active-phase algorithm"],"prefix":"10.1007","volume":"33","author":[{"given":"Mingtao","family":"Cui","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongfang","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jingling","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fanglin","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2017,3,15]]},"reference":[{"key":"503_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00158-013-0956-z","volume":"49","author":"JD Deaton","year":"2013","unstructured":"Deaton JD, Grandhi RV (2013) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49:1\u201338","journal-title":"Struct Multidiscip Optim"},{"key":"503_CR2","volume-title":"Topology optimization: theory, methods and applications","author":"MP Bends\u00f8e","year":"2003","unstructured":"Bends\u00f8e MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin"},{"key":"503_CR3","first-page":"1","volume":"43","author":"E Andreassen","year":"2010","unstructured":"Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2010) Efficient topology optimization in MATLAB using 88 lines of code. Struct. Multidiscip. Optim 43:1\u201316","journal-title":"Optim"},{"key":"503_CR4","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 YM (2010) A further review of ESO type methods for topology optimization. Struct Multidiscip Optim 41:671\u2013683","journal-title":"Struct Multidiscip Optim"},{"key":"503_CR5","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s00466-008-0312-0","volume":"43","author":"X Huang","year":"2008","unstructured":"Huang X, Xie YM (2008) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43:393\u2013401","journal-title":"Comput Mech"},{"key":"503_CR6","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 192:227\u2013246","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"503_CR7","first-page":"59","volume":"34","author":"G Allaire","year":"2005","unstructured":"Allaire G, Gournay FD, Jouve F, Toader AM (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybernetics 34(1):59\u201380","journal-title":"Control Cybernetics"},{"key":"503_CR8","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/s00158-013-0912-y","volume":"48","author":"NP Dijk","year":"2013","unstructured":"Dijk NP, Maute K, Langelaar M, Keulen FV (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48:437\u2013472","journal-title":"Struct Multidiscip Optim"},{"key":"503_CR9","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/j.matcom.2008.01.003","volume":"79","author":"VP Nguyen","year":"2008","unstructured":"Nguyen VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79:763\u2013813","journal-title":"Math Comput Simul"},{"key":"503_CR10","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1086\/112164","volume":"82","author":"LB Lucy","year":"1977","unstructured":"Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013\u20131024","journal-title":"Astron J"},{"key":"503_CR11","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1093\/mnras\/181.3.375","volume":"181","author":"RA Gingold","year":"1977","unstructured":"Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices R Astron Soc 181:375\u2013389","journal-title":"Monthly Notices R Astron Soc"},{"key":"503_CR12","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1002\/nme.1620370205","volume":"37","author":"T Belytschko","year":"1994","unstructured":"Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229\u2013256","journal-title":"Int J Numer Methods Eng"},{"key":"503_CR13","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1002\/fld.1650200824","volume":"20","author":"WK Liu","year":"1995","unstructured":"Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Eng 20:1081\u20131106","journal-title":"Int J Numer Methods Eng"},{"key":"503_CR14","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s004660050346","volume":"22","author":"SN Atluri","year":"1998","unstructured":"Atluri SN, Zhu T (1998) A new meshless local Petrov\u2013Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117\u2013127","journal-title":"Comput Mech"},{"key":"503_CR15","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s004660050467","volume":"25","author":"SN Atluri","year":"2000","unstructured":"Atluri SN, Zhu T (2000) The meshless local Petrov\u2013Galerkin (MLPG) approach for solving problems in elasto-statics. Comput Mech 25:169\u2013179","journal-title":"Comput Mech"},{"key":"503_CR16","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1016\/j.enganabound.2010.03.001","volume":"34","author":"J Zheng","year":"2010","unstructured":"Zheng J, Long SY, Li GY (2010) The topology optimization design for continuum structures based on the element free Galerkin method. Eng Anal Bound Elem 34:666\u2013672","journal-title":"Eng Anal Bound Elem"},{"key":"503_CR17","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1002\/nme.3325","volume":"90","author":"Z Luo","year":"2012","unstructured":"Luo Z, Zhang N, Gao W, Ma H (2012) Structural shape and topology optimization using a meshless Galerkin level set method. Int J Numer Meth Eng 90:369\u2013389","journal-title":"Int J Numer Meth Eng"},{"key":"503_CR18","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1002\/nme.4394","volume":"93","author":"Z Luo","year":"2013","unstructured":"Luo Z, Zhang N, Wang Y, Gao W (2013) Topology optimization of structures using meshless density variable approximants. Int J Numer Meth Eng 93:443\u2013464","journal-title":"Int J Numer Meth Eng"},{"key":"503_CR19","doi-asserted-by":"crossref","first-page":"1625","DOI":"10.1016\/j.enganabound.2013.09.010","volume":"37","author":"F Zhao","year":"2013","unstructured":"Zhao F (2013) A meshless Pareto-optimal method for topology optimization. Eng Anal Bound Elem 37:1625\u20131631","journal-title":"Eng Anal Bound Elem"},{"key":"503_CR20","doi-asserted-by":"crossref","first-page":"1851","DOI":"10.1177\/1045389X10389345","volume":"21","author":"Z Luo","year":"2010","unstructured":"Luo Z, Gao W, Song C (2010) Design of multi-phase piezoelectric actuators. J Intell Mater Syst Struct 21:1851\u20131865","journal-title":"J Intell Mater Syst Struct"},{"key":"503_CR21","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"},{"key":"503_CR22","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s00158-011-0625-z","volume":"43","author":"CF Hvejsel","year":"2011","unstructured":"Hvejsel CF, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43:811\u2013825","journal-title":"Struct Multidiscip Optim"},{"key":"503_CR23","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s00158-006-0035-9","volume":"33","author":"S Zhou","year":"2006","unstructured":"Zhou S, Wang MY (2006) Multimaterial structural topology optimization with a generalized Cahn\u2013Hilliard model of multiphase transition. Struct Multidiscip Optim 33:89\u2013111","journal-title":"Struct Multidiscip Optim"},{"key":"503_CR24","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/j.cma.2014.04.005","volume":"276","author":"R Tavakoli","year":"2014","unstructured":"Tavakoli R (2014) Multimaterial topology optimization by volume constrained Allen\u2013Cahn system and regularized projected steepest descent method. Comput Methods Appl Mech Eng 276:534\u2013565","journal-title":"Comput Methods Appl Mech Eng"},{"key":"503_CR25","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1016\/j.cma.2003.10.008","volume":"193","author":"MY Wang","year":"2004","unstructured":"Wang MY, Wang X (2004) \u201cColor\u201d level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Methods Appl Mech Eng 193:469\u2013496","journal-title":"Comput Methods Appl Mech Eng"},{"key":"503_CR26","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1016\/j.cma.2013.10.003","volume":"268","author":"X Guo","year":"2014","unstructured":"Guo X, Zhang W, Zhong W (2014) Stress-related topology optimization of continuum structures involving multi-phase materials. Comput Methods Appl Mech Eng 268:632\u2013655","journal-title":"Comput Methods Appl Mech Eng"},{"key":"503_CR27","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/j.cad.2004.03.007","volume":"37","author":"MY Wang","year":"2005","unstructured":"Wang MY, Wang X (2005) A level-set based variational method for design and optimization of heterogeneous objects. Comput-Aided Des 37:321\u2013337","journal-title":"Comput-Aided Des"},{"key":"503_CR28","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1051\/cocv\/2013076","volume":"20","author":"G Allaire","year":"2014","unstructured":"Allaire G, Dapogny C, Delgado G, Michailidis G (2014) Multi-phase structural optimization via a level set method. ESAIM Control Optim Calc Var 20:576\u2013611","journal-title":"ESAIM Control Optim Calc Var"},{"key":"503_CR29","first-page":"623","volume":"50","author":"N Vermaak","year":"2014","unstructured":"Vermaak N, Michailidis G, Parry G, Estevez R, Allaire G, Br\u00e9chet Y (2014) Material interface effects on the topology optimizationof multi-phase structures using a level set method. Struct. Multidiscip. Optim 50:623\u2013644","journal-title":"Optim"},{"key":"503_CR30","doi-asserted-by":"crossref","first-page":"1570","DOI":"10.1016\/j.cma.2014.11.002","volume":"283","author":"YQ Wang","year":"2015","unstructured":"Wang YQ, Luo Z, Zhang K, Zhang N (2015) A multi-material level set-based topology and shape optimization method. Comput Methods Appl Mech Eng 283:1570\u20131586","journal-title":"Comput Methods Appl Mech Eng"},{"key":"503_CR31","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1007\/s00158-013-0999-1","volume":"49","author":"R Tavakoli","year":"2014","unstructured":"Tavakoli R, Mohseni SM (2014) Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidiscip Optim 49:621\u2013642","journal-title":"Struct Multidiscip Optim"},{"key":"503_CR32","unstructured":"Shepard D (1968) A two-dimensional function for irregularly spaced points. Proceedings of the 23rd ACM National Conference, pp 517\u2013524"},{"issue":"4","key":"503_CR33","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1111\/j.1467-8659.2005.00903.x","volume":"24","author":"KW Brodlie","year":"2005","unstructured":"Brodlie KW, Asim MR, Unsworth K (2005) Constrained visualization using the Shepard interpolation family. Comput Graph Forum 24(4):809\u2013820","journal-title":"Comput Graph Forum"},{"key":"503_CR34","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/S0045-7825(96)01079-1","volume":"139","author":"S Beisse","year":"1996","unstructured":"Beisse S, Belytschkob T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Engrg 139:49\u201374","journal-title":"Comput Methods Appl Mech Engrg"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00366-017-0503-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-017-0503-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-017-0503-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,9,16]],"date-time":"2017-09-16T02:17:06Z","timestamp":1505528226000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00366-017-0503-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,15]]},"references-count":34,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,10]]}},"alternative-id":["503"],"URL":"https:\/\/doi.org\/10.1007\/s00366-017-0503-4","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,15]]}}}