{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T15:38:02Z","timestamp":1770910682110,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T00:00:00Z","timestamp":1625184000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"New Energy and Industrial Technology Development Organization","award":["Project to support the discovery of young researchers by the public and private sectors."],"award-info":[{"award-number":["Project to support the discovery of young researchers by the public and private sectors."]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Additive manufacturing (AM) is employed for fabricating industrial products with complex geometries. As topological optimization is suitable for designing complex geometries, studies have combined AM and topological optimization, evaluating the density optimization of lattice structures as a variant of topological optimization. The lattice structures of components fabricated via AM comprise voids. Models designed using topological optimization should be modified to ensure structures suitable for AM. As the lattice unit can be easily fabricated using AM with fewer design modifications, this study uses lattice density optimization for an industrial AM product. We propose a method of optimizing the lattice distribution for controlling the surface temperature uniformity of industrial products, such as molds. The effective thermal conductivity of the lattice is calculated using the homogenization and finite element methods. The effective thermal conductivity changes depending on the internal pore sizes. The proposed methodology is validated using a 3D example; the minimization problem of surface temperature variations in the target domain is considered. The variable density of the embedded lattice in the target domain is optimized, and we experimentally validated the performance of the lattice unit cell and optimal 3D structure using metal powder bed fusion AM.<\/jats:p>","DOI":"10.3390\/sym13071194","type":"journal-article","created":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T10:06:34Z","timestamp":1625220394000},"page":"1194","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Temperature Distribution Design Based on Variable Lattice Density Optimization and Metal Additive Manufacturing"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0171-5836","authenticated-orcid":false,"given":"Akira","family":"Ueno","sequence":"first","affiliation":[{"name":"AGC Inc., 1-1 Suehiro-cho, Tsurumi-ku, Yokohama-shi, Kanagawa 230-0045, Japan"},{"name":"Department of Transportation and Environmental Engineering, Graduate School of Advanced Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashihiroshima-shi, Hiroshima 739-8527, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Honghu","family":"Guo","sequence":"additional","affiliation":[{"name":"Department of Applied Mechanics and Aerospace Engineering, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8197-4306","authenticated-orcid":false,"given":"Akihiro","family":"Takezawa","sequence":"additional","affiliation":[{"name":"Department of Applied Mechanics and Aerospace Engineering, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ryota","family":"Moritoyo","sequence":"additional","affiliation":[{"name":"Department of Transportation and Environmental Engineering, Graduate School of Advanced Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashihiroshima-shi, Hiroshima 739-8527, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4766-5889","authenticated-orcid":false,"given":"Mitsuru","family":"Kitamura","sequence":"additional","affiliation":[{"name":"Department of Transportation and Environmental Engineering, Graduate School of Advanced Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashihiroshima-shi, Hiroshima 739-8527, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1116\/1.3056171","article-title":"Efficient mold manufacturing for precision glass molding","volume":"27","author":"Dambon","year":"2009","journal-title":"J. Vac. Sci. Tech. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1016\/j.icheatmasstransfer.2010.07.012","article-title":"Rapid mold temperature control in injection molding by using steam heating","volume":"30","author":"Jeng","year":"2010","journal-title":"Int. Commun. Heat Mass"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1016\/j.matdes.2009.06.010","article-title":"Research of thermal response simulation and mold structure optimization for rapid heat cycle molding processes, respectively, with steam heating and electric heating","volume":"31","author":"Wang","year":"2010","journal-title":"Mater. Des."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gibson, I., Rosen, D., and Stucker, B. (2010). Additive Manufacturing Technologies, Springer.","DOI":"10.1007\/978-1-4419-1120-9"},{"key":"ref_5","first-page":"472","article-title":"Additive manufacturing-driven mold design for castings","volume":"22","author":"Kang","year":"2018","journal-title":"Addit. Manuf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0045-7825(88)90086-2","article-title":"Generating optimal topologies in structural design using a homogenization method","volume":"71","author":"Kikuchi","year":"1988","journal-title":"Comput. Comput. Method. Appl. M"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bends\u00f8e, M.P., and Sigmund, O. (2003). Topology Optimization: Theory, Methods, and Applications, Springer.","DOI":"10.1007\/978-3-662-05086-6"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2452","DOI":"10.1299\/kikaic.74.2452","article-title":"Topology optimization for heat convection problems including design-dependent effects","volume":"74","author":"Iga","year":"2008","journal-title":"Trans. Jpn. Soc. Mech. Eng."},{"key":"ref_9","first-page":"841","article-title":"A review about the engineering design of optimal heat transfer systems using topology optimization","volume":"112","author":"Dbouk","year":"2017","journal-title":"Appl. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s10999-017-9396-z","article-title":"A 2D topology optimisation algorithm in NURBS framework with geometric constraints","volume":"14","author":"Costa","year":"2018","journal-title":"Int. J. Mech. Mater. Des."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1080\/15376494.2019.1582826","article-title":"NURBS hypersurfaces for 3D topology optimisation problems","volume":"28","author":"Costa","year":"2021","journal-title":"Mech. Adv. Mater. Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1016\/j.cma.2019.05.026","article-title":"Minimum length scale control in a NURBS-based SIMP method","volume":"354","author":"Costa","year":"2019","journal-title":"Comput. Method Appl. Mech. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.cirp.2019.04.048","article-title":"Maximum length scale requirement in a topology optimisation method based on NURBS hyper-surfaces","volume":"68","author":"Costa","year":"2019","journal-title":"CIRP Ann. Manuf. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Montemurro, M., and Refai, K. (2021). Topology optimization method based on non-uniform rational basis spline hyper-surfaces for heat conduction problems. Symmetry, 13.","DOI":"10.3390\/sym13050888"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s11831-019-09331-1","article-title":"From topology optimization design to additive manufacturing: Today\u2019s success and tomorrow\u2019s roadmap","volume":"27","author":"Meng","year":"2020","journal-title":"Arch. Comput. Method Eng."},{"key":"ref_16","first-page":"305","article-title":"Cellular lattices of biomedical Co-Cr-Mo-alloy fabricated by electron beam melting with the aid of shape optimization","volume":"12","author":"Koizumi","year":"2016","journal-title":"Addit. Manuf."},{"key":"ref_17","first-page":"194","article-title":"High-stiffness and strength porous maraging steel via topology optimization and selective laser melting","volume":"18","author":"Takezawa","year":"2017","journal-title":"Addit. Manuf."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1108\/RPJ-04-2016-0069","article-title":"Efficient design optimization of variable-density cellular structures for additive manufacturing: Theory and experimental validation","volume":"23","author":"Cheng","year":"2017","journal-title":"Rapid Prototyp. J."},{"key":"ref_19","first-page":"189","article-title":"Natural frequency optimization of 3D printed variable-density honeycomb structure via a homogenization-based approach","volume":"20","author":"Wang","year":"2018","journal-title":"Addit. Manuf."},{"key":"ref_20","first-page":"285","article-title":"Method to optimize an additively-manufactured functionally-graded lattice structure for effective liquid cooling","volume":"28","author":"Takezawa","year":"2019","journal-title":"Addit. Manuf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"118564","DOI":"10.1016\/j.ijheatmasstransfer.2019.118564","article-title":"Optimization of an additively manufactured functionally graded lattice structure with liquid cooling considering structural performances","volume":"143","author":"Takezawa","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1115\/1.3167081","article-title":"Analysis of composite materials: A survey","volume":"50","author":"Hansin","year":"1983","journal-title":"J. Appl. Mech."},{"key":"ref_23","first-page":"131","article-title":"On constitutive macro-variables for heterogeneous solids at finite strain","volume":"326","author":"Hill","year":"1972","journal-title":"Proc. Roy. Soc. A Math. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.cma.2014.04.003","article-title":"Structural topology optimization with strength and heat conduction constraints","volume":"276","author":"Takezawa","year":"2014","journal-title":"Comput. Method Appl. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1002\/nme.1620240207","article-title":"The method of moving asymptotes: A new method for structural optimization","volume":"24","author":"Svanberg","year":"1987","journal-title":"Int. J. Numer. Methods Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1194\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:25:09Z","timestamp":1760163909000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1194"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,2]]},"references-count":25,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071194"],"URL":"https:\/\/doi.org\/10.3390\/sym13071194","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,2]]}}}