[{"quality_controlled":"1","language":[{"iso":"eng"}],"publication_identifier":{"issn":["0730-0301"],"eissn":["1557-7368"]},"_id":"17203","day":"01","has_accepted_license":"1","conference":{"start_date":"2024-07-28","end_date":"2024-08-01","location":"Denver, Colorado"},"file":[{"date_updated":"2024-07-05T12:05:17Z","access_level":"open_access","file_name":"sif-final.pdf","relation":"main_file","creator":"chafner","success":1,"checksum":"0dc9f5a6422b8a49a79026900f349ee5","file_id":"17204","file_size":7225150,"date_created":"2024-07-05T12:05:17Z","content_type":"application/pdf"},{"checksum":"cde433c6a40688d5f1187fb5721f6f94","content_type":"application/pdf","date_created":"2024-07-05T12:06:03Z","file_size":397262,"file_id":"17205","date_updated":"2024-07-05T12:06:03Z","creator":"chafner","relation":"supplementary_material","file_name":"sif-supp-final.pdf","access_level":"open_access"},{"checksum":"c0457a09c2ab9a1c2935c995dcc84907","file_size":170001305,"file_id":"17276","content_type":"video/mp4","date_created":"2024-07-17T09:29:13Z","title":"Submission Video","date_updated":"2024-07-17T09:29:13Z","access_level":"open_access","relation":"supplementary_material","creator":"chafner","file_name":"sif-video-final.mp4"}],"keyword":["Topology Optimization","Mass Moments","Computational Geometry"],"external_id":{"isi":["001289270900045"]},"fulldoi":"https://doi.org/10.1145/3658194","volume":43,"citation":{"short":"C. Hafner, M. Ly, C. Wojtan, Transactions on Graphics 43 (2024).","ieee":"C. Hafner, M. Ly, and C. Wojtan, “Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments,” <i>Transactions on Graphics</i>, vol. 43, no. 4. Association for Computing Machinery, 2024.","ista":"Hafner C, Ly M, Wojtan C. 2024. Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. Transactions on Graphics. 43(4), 78.","chicago":"Hafner, Christian, Mickaël Ly, and Chris Wojtan. “Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only on Mass Moments.” <i>Transactions on Graphics</i>. Association for Computing Machinery, 2024. <a href=\"https://doi.org/10.1145/3658194\">https://doi.org/10.1145/3658194</a>.","apa":"Hafner, C., Ly, M., &#38; Wojtan, C. (2024). Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. <i>Transactions on Graphics</i>. Denver, Colorado: Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3658194\">https://doi.org/10.1145/3658194</a>","ama":"Hafner C, Ly M, Wojtan C. Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. <i>Transactions on Graphics</i>. 2024;43(4). doi:<a href=\"https://doi.org/10.1145/3658194\">10.1145/3658194</a>","mla":"Hafner, Christian, et al. “Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only on Mass Moments.” <i>Transactions on Graphics</i>, vol. 43, no. 4, 78, Association for Computing Machinery, 2024, doi:<a href=\"https://doi.org/10.1145/3658194\">10.1145/3658194</a>."},"date_created":"2024-07-05T12:08:57Z","corr_author":"1","acknowledgement":"We thank Gianmarco Cherchi for his help in tailoring the Mesh Booleans code for this project, Stefan Jeschke for his help with the photographs, Malina Strugaru and Aleksei Kalinov for their help with the samples, and the anonymous reviewers as well as the members of the ISTA Visual Computing Group for their feedback. This project was funded in part by the European Research Council (ERC Consolidator Grant 101045083 CoDiNA).","department":[{"_id":"ChWo"}],"article_processing_charge":"Yes (via OA deal)","type":"journal_article","abstract":[{"lang":"eng","text":"The behavior of a rigid body primarily depends on its mass moments, which consist of the mass, center of mass, and moments of inertia. It is possible to manipulate these quantities without altering the geometric appearance of an object by introducing cavities in its interior. Algorithms that find cavities of suitable shapes and sizes have enabled the computational design of spinning tops, yo-yos, wheels, buoys, and statically balanced objects. Previous work is based, for example, on topology optimization on voxel grids, which introduces a large number of optimization variables and box constraints, or offset surface computation, which cannot guarantee that solutions to a feasible problem will always be found.\r\n\r\nIn this work, we provide a mathematical analysis of constrained topology optimization problems that depend only on mass moments. This class of problems covers, among others, all applications mentioned above. Our main result is to show that no matter the outer shape of the rigid body to be optimized or the optimization objective and constraints considered, the optimal solution always features a quadric-shaped interface between material and cavities. This proves that optimal interfaces are always ellipsoids, hyperboloids, paraboloids, or one of a few degenerate cases, such as planes.\r\n\r\nThis insight lets us replace a difficult topology optimization problem with a provably equivalent non-linear equation system in a small number (<10) of variables, which represent the coefficients of the quadric. This system can be solved in a few seconds for most examples, provides insights into the geometric structure of many specific applications, and lets us describe their solution properties. Finally, our method integrates seamlessly into modern fabrication workflows because our solutions are analytical surfaces that are native to the CAD domain."}],"oa_version":"Published Version","article_type":"original","ddc":["516"],"publication":"Transactions on Graphics","year":"2024","isi":1,"date_updated":"2025-09-08T08:29:09Z","oa":1,"status":"public","month":"07","scopus_import":"1","title":"Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments","issue":"4","author":[{"full_name":"Hafner, Christian","last_name":"Hafner","first_name":"Christian","id":"400429CC-F248-11E8-B48F-1D18A9856A87"},{"id":"6340d7f0-b48d-11eb-b10d-b7487e71d9f1","first_name":"Mickaël","full_name":"Ly, Mickaël","last_name":"Ly"},{"id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","first_name":"Christopher J","orcid":"0000-0001-6646-5546","last_name":"Wojtan","full_name":"Wojtan, Christopher J"}],"file_date_updated":"2024-07-17T09:29:13Z","intvolume":"        43","publication_status":"published","date_published":"2024-07-01T00:00:00Z","doi":"10.1145/3658194","article_number":"78","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","project":[{"name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","grant_number":"101045083","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088"}],"publisher":"Association for Computing Machinery"},{"extern":"1","date_created":"2026-03-30T12:22:47Z","citation":{"ista":"Roques-Carmes C, Lin Z, Christiansen RE, Salamin Y, Kooi SE, Joannopoulos JD, Johnson SG, Soljačić M. 2022. Toward 3D-printed inverse-designed metaoptics. ACS Photonics. 9(1), 43–51.","short":"C. Roques-Carmes, Z. Lin, R.E. Christiansen, Y. Salamin, S.E. Kooi, J.D. Joannopoulos, S.G. Johnson, M. Soljačić, ACS Photonics 9 (2022) 43–51.","ieee":"C. Roques-Carmes <i>et al.</i>, “Toward 3D-printed inverse-designed metaoptics,” <i>ACS Photonics</i>, vol. 9, no. 1. American Chemical Society, pp. 43–51, 2022.","chicago":"Roques-Carmes, Charles, Zin Lin, Rasmus E. Christiansen, Yannick Salamin, Steven E. Kooi, John D. Joannopoulos, Steven G. Johnson, and Marin Soljačić. “Toward 3D-Printed Inverse-Designed Metaoptics.” <i>ACS Photonics</i>. American Chemical Society, 2022. <a href=\"https://doi.org/10.1021/acsphotonics.1c01442\">https://doi.org/10.1021/acsphotonics.1c01442</a>.","ama":"Roques-Carmes C, Lin Z, Christiansen RE, et al. Toward 3D-printed inverse-designed metaoptics. <i>ACS Photonics</i>. 2022;9(1):43-51. doi:<a href=\"https://doi.org/10.1021/acsphotonics.1c01442\">10.1021/acsphotonics.1c01442</a>","apa":"Roques-Carmes, C., Lin, Z., Christiansen, R. E., Salamin, Y., Kooi, S. E., Joannopoulos, J. D., … Soljačić, M. (2022). Toward 3D-printed inverse-designed metaoptics. <i>ACS Photonics</i>. American Chemical Society. <a href=\"https://doi.org/10.1021/acsphotonics.1c01442\">https://doi.org/10.1021/acsphotonics.1c01442</a>","mla":"Roques-Carmes, Charles, et al. “Toward 3D-Printed Inverse-Designed Metaoptics.” <i>ACS Photonics</i>, vol. 9, no. 1, American Chemical Society, 2022, pp. 43–51, doi:<a href=\"https://doi.org/10.1021/acsphotonics.1c01442\">10.1021/acsphotonics.1c01442</a>."},"page":"43-51","volume":9,"OA_place":"repository","article_processing_charge":"No","type":"journal_article","OA_type":"green","day":"07","_id":"21527","publication_identifier":{"eissn":["2330-4022"]},"language":[{"iso":"eng"}],"quality_controlled":"1","fulldoi":"https://doi.org/10.1021/acsphotonics.1c01442","external_id":{"arxiv":["2105.11326"]},"keyword":["metasurfaces","inverse design","multilayered metaoptics","3D printing","topology optimization"],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2105.11326","open_access":"1"}],"date_published":"2022-01-07T00:00:00Z","publication_status":"published","intvolume":"         9","publisher":"American Chemical Society","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","arxiv":1,"doi":"10.1021/acsphotonics.1c01442","year":"2022","publication":"ACS Photonics","ddc":["530"],"article_type":"letter_note","oa_version":"Preprint","abstract":[{"text":"Optical metasurfaces have been heralded as the platform to integrate multiple functionalities in a compact form-factor, with the potential to replace bulky optical components. A central stepping stone toward realizing this promise is the demonstration of multifunctionality under several constraints (e.g., at multiple incident wavelengths and/or angles) in a single device, an achievement being hampered by design limitations inherent to single-layer planar geometries. Here, we propose a framework for the inverse design of multilayer metaoptics via topology optimization, showing that even few-wavelength thick devices can achieve high-efficiency multifunctionality, such as multiangle light concentration and plan-achromaticity. We embody our framework in multiple closely spaced patterned layers of a low-index polymer, with fabrication constraints specific to this platform enforced in the optimization process. We experimentally demonstrate our approach with an inverse-designed 3D-printed light concentrator working at five different nonparaxial angles of incidence. Our framework paves the way toward realizing multifunctional ultracompact 3D nanophotonic devices.","lang":"eng"}],"issue":"1","author":[{"id":"e2e68fc9-6505-11ef-a541-eb4e72cc3e82","first_name":"Charles","last_name":"Roques-Carmes","full_name":"Roques-Carmes, Charles"},{"last_name":"Lin","full_name":"Lin, Zin","first_name":"Zin"},{"last_name":"Christiansen","full_name":"Christiansen, Rasmus E.","first_name":"Rasmus E."},{"full_name":"Salamin, Yannick","last_name":"Salamin","first_name":"Yannick"},{"last_name":"Kooi","full_name":"Kooi, Steven E.","first_name":"Steven E."},{"full_name":"Joannopoulos, John D.","last_name":"Joannopoulos","first_name":"John D."},{"last_name":"Johnson","full_name":"Johnson, Steven G.","first_name":"Steven G."},{"first_name":"Marin","full_name":"Soljačić, Marin","last_name":"Soljačić"}],"title":"Toward 3D-printed inverse-designed metaoptics","scopus_import":"1","month":"01","status":"public","oa":1,"date_updated":"2026-04-27T09:14:46Z"}]