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To express the highly constrained triply periodic minimal surfaces (TPMS) in this manner, we present the first fully automated version of the conjugate surface construction method, which allows novices to create complex TPMS from intuitive input. We demonstrate our representation’s expressiveness, accuracy, and compactness by constructing a wide range of established structures and hundreds of novel structures with diverse architectures and material properties. We also conduct a user study to verify our representation’s ease-of-use and ability to expand engineers’ capacity for exploration."}],"doi":"10.1145/3605389","external_id":{"isi":["001086833300007"]},"project":[{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","grant_number":"101045083","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088"}],"volume":42,"acknowledgement":"The authors thank Mina Konaković Luković and Michael Foshey for their early contributions to this project, David Palmer and Paul Zhang for their insightful discussions about minimal surfaces and the CSCM, Julian Panetta for providing the Elastic Textures code, and Hannes Hergeth for his feedback and support. We also thank our user study participants and anonymous reviewers.\r\nThis material is based upon work supported by the National Science Foundation\r\n(NSF) Graduate Research Fellowship under Grant No. 2141064; the MIT Morningside\r\nAcademy for Design Fellowship; the Defense Advanced Research Projects Agency\r\n(DARPA) Grant No. FA8750-20-C-0075; the ERC Consolidator Grant No. 101045083,\r\n“CoDiNA: Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena”; and the NewSat project, which is co-funded by the Operational Program for Competitiveness and Internationalisation (COMPETE2020), Portugal 2020, the European Regional Development Fund (ERDF), and the Portuguese Foundation for Science and Technology (FTC) under the MIT Portugal program.","file_date_updated":"2023-12-04T08:04:14Z","isi":1,"publisher":"Association for Computing Machinery","publication":"ACM Transactions on Graphics","type":"journal_article","day":"01","date_updated":"2025-09-09T13:33:58Z","status":"public","date_created":"2023-11-29T15:02:03Z","article_type":"original","year":"2023","oa_version":"Published Version","publication_status":"published","date_published":"2023-10-01T00:00:00Z","month":"10","issue":"5","citation":{"chicago":"Makatura, Liane, Bohan Wang, Yi-Lu Chen, Bolei Deng, Chris Wojtan, Bernd Bickel, and Wojciech Matusik. “Procedural Metamaterials: A Unified Procedural Graph for Metamaterial Design.” <i>ACM Transactions on Graphics</i>. 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Procedural metamaterials: A unified procedural graph for metamaterial design. <i>ACM Transactions on Graphics</i>. 2023;42(5). doi:<a href=\"https://doi.org/10.1145/3605389\">10.1145/3605389</a>"},"author":[{"first_name":"Liane","last_name":"Makatura","full_name":"Makatura, Liane"},{"full_name":"Wang, Bohan","last_name":"Wang","first_name":"Bohan"},{"full_name":"Chen, Yi-Lu","first_name":"Yi-Lu","last_name":"Chen","id":"0b467602-dbcd-11ea-9d1d-ed480aa46b70"},{"full_name":"Deng, Bolei","last_name":"Deng","first_name":"Bolei"},{"orcid":"0000-0001-6646-5546","id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","full_name":"Wojtan, Christopher J","first_name":"Christopher J","last_name":"Wojtan"},{"first_name":"Bernd","last_name":"Bickel","full_name":"Bickel, Bernd","id":"49876194-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6511-9385"},{"first_name":"Wojciech","last_name":"Matusik","full_name":"Matusik, Wojciech"}],"keyword":["Computer Graphics and Computer-Aided Design"],"department":[{"_id":"GradSch"},{"_id":"ChWo"},{"_id":"BeBi"}],"has_accepted_license":"1","ddc":["531","006"],"oa":1,"publication_identifier":{"issn":["0730-0301"],"eissn":["1557-7368"]},"file":[{"checksum":"0192f597d7a2ceaf89baddfd6190d4c8","file_name":"tog-22-0089-File004.zip","relation":"main_file","content_type":"application/zip","date_updated":"2023-11-29T15:16:01Z","access_level":"open_access","date_created":"2023-11-29T15:16:01Z","file_size":95467870,"file_id":"14630","success":1,"creator":"yichen"},{"relation":"main_file","content_type":"application/zip","file_name":"tog-22-0089-File005.zip","checksum":"7fb024963be81933494f38de191e4710","success":1,"file_id":"14631","file_size":103731880,"date_created":"2023-11-29T15:16:01Z","creator":"yichen","date_updated":"2023-11-29T15:16:01Z","access_level":"open_access"},{"date_updated":"2023-12-04T08:04:14Z","access_level":"open_access","date_created":"2023-12-04T08:04:14Z","file_size":57067476,"success":1,"file_id":"14638","creator":"dernst","checksum":"b7d6829ce396e21cac9fae0ec7130a6b","file_name":"2023_ACMToG_Makatura.pdf","relation":"main_file","content_type":"application/pdf"}],"title":"Procedural metamaterials: A unified procedural graph for metamaterial design","_id":"14628","quality_controlled":"1","intvolume":"        42"},{"language":[{"iso":"eng"}],"article_number":"171","scopus_import":"1","article_processing_charge":"No","acknowledged_ssus":[{"_id":"M-Shop"}],"abstract":[{"text":"The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions, and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length.\r\nWe also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods, and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.","lang":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"name":"MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and Modeling","grant_number":"715767","call_identifier":"H2020","_id":"24F9549A-B435-11E9-9278-68D0E5697425"}],"doi":"10.1145/3606033","external_id":{"isi":["001086833300010"]},"volume":42,"acknowledgement":"We thank the anonymous reviewers for their generous feedback, and Julian Fischer for his help in proving Proposition 1. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715767).","publisher":"Association for Computing Machinery","ec_funded":1,"publication":"ACM Transactions on Graphics","isi":1,"file_date_updated":"2023-07-04T08:11:28Z","date_updated":"2026-04-03T22:30:20Z","type":"journal_article","day":"20","status":"public","date_created":"2023-07-04T07:41:30Z","corr_author":"1","article_type":"original","date_published":"2023-09-20T00:00:00Z","year":"2023","oa_version":"Submitted Version","publication_status":"published","month":"09","issue":"5","citation":{"chicago":"Hafner, Christian, and Bernd Bickel. “The Design Space of Kirchhoff Rods.” <i>ACM Transactions on Graphics</i>. Association for Computing Machinery, 2023. <a href=\"https://doi.org/10.1145/3606033\">https://doi.org/10.1145/3606033</a>.","short":"C. Hafner, B. 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Wiley, 2022. <a href=\"https://doi.org/10.1111/cgf.14581\">https://doi.org/10.1111/cgf.14581</a>.","short":"T. Alderighi, L. Malomo, T. Auzinger, B. Bickel, P. Cignoni, N. Pietroni, Computer Graphics Forum 41 (2022) 435–452.","ista":"Alderighi T, Malomo L, Auzinger T, Bickel B, Cignoni P, Pietroni N. 2022. State of the art in computational mould design. Computer Graphics Forum. 41(6), 435–452.","ieee":"T. Alderighi, L. Malomo, T. Auzinger, B. Bickel, P. Cignoni, and N. Pietroni, “State of the art in computational mould design,” <i>Computer Graphics Forum</i>, vol. 41, no. 6. Wiley, pp. 435–452, 2022.","apa":"Alderighi, T., Malomo, L., Auzinger, T., Bickel, B., Cignoni, P., &#38; Pietroni, N. (2022). State of the art in computational mould design. <i>Computer Graphics Forum</i>. Wiley. <a href=\"https://doi.org/10.1111/cgf.14581\">https://doi.org/10.1111/cgf.14581</a>","mla":"Alderighi, Thomas, et al. “State of the Art in Computational Mould Design.” <i>Computer Graphics Forum</i>, vol. 41, no. 6, Wiley, 2022, pp. 435–52, doi:<a href=\"https://doi.org/10.1111/cgf.14581\">10.1111/cgf.14581</a>.","ama":"Alderighi T, Malomo L, Auzinger T, Bickel B, Cignoni P, Pietroni N. State of the art in computational mould design. <i>Computer Graphics Forum</i>. 2022;41(6):435-452. doi:<a href=\"https://doi.org/10.1111/cgf.14581\">10.1111/cgf.14581</a>"},"has_accepted_license":"1","ddc":["000"],"oa":1,"department":[{"_id":"BeBi"}],"quality_controlled":"1","intvolume":"        41","publication_identifier":{"issn":["0167-7055"],"eissn":["1467-8659"]},"file":[{"file_name":"star_molding_preprint.pdf","checksum":"c40cc8ceb7b7f0512172b883d712198e","content_type":"application/pdf","relation":"main_file","access_level":"open_access","date_updated":"2022-08-28T18:18:08Z","title":"pre-peer reviewed version","creator":"bbickel","description":"This is the pre-peer reviewed version of the following article: Alderighi, T., Malomo, L., Auzinger, T., Bickel, B., Cignoni, P. and Pietroni, N. (2022), State of the Art in Computational Mould Design. Computer Graphics Forum, which has been published in final form at https://doi.org/10.1111/cgf.14581. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.","file_id":"11994","file_size":32480850,"date_created":"2022-08-28T18:18:08Z"}],"title":"State of the art in computational mould design","_id":"11993","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","abstract":[{"lang":"eng","text":"Moulding refers to a set of manufacturing techniques in which a mould, usually a cavity or a solid frame, is used to shape a liquid or pliable material into an object of the desired shape. The popularity of moulding comes from its effectiveness, scalability and versatility in terms of employed materials. Its relevance as a fabrication process is demonstrated by the extensive literature covering different aspects related to mould design, from material flow simulation to the automation of mould geometry design. In this state-of-the-art report, we provide an extensive review of the automatic methods for the design of moulds, focusing on contributions from a geometric perspective. We classify existing mould design methods based on their computational approach and the nature of their target moulding process. We summarize the relationships between computational approaches and moulding techniques, highlighting their strengths and limitations. 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Also includes pre-built binaries of the HyENA library, but not sources - please contact the HyENA authors to obtain these sources if required (https://mech.tugraz.at/hyena)","lang":"eng"}],"datarep_id":"73","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-12T12:31:35Z","status":"public","date_updated":"2025-04-14T09:25:58Z","day":"16","type":"research_data","publisher":"Institute of Science and Technology Austria","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","name":"Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0)","short":"CC BY-SA (4.0)","image":"/images/cc_by_sa.png"},"ec_funded":1,"file_date_updated":"2020-07-14T12:47:04Z","keyword":["Boundary elements","brittle fracture","computer graphics","fracture simulation"],"author":[{"id":"357A6A66-F248-11E8-B48F-1D18A9856A87","first_name":"David","last_name":"Hahn","full_name":"Hahn, David"}],"citation":{"short":"D. 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