article published yes Liane Makatura author Bohan Wang author Yi-Lu Chen author 0b467602-dbcd-11ea-9d1d-ed480aa46b70 Bolei Deng author Christopher J Wojtan author 3C61F1D2-F248-11E8-B48F-1D18A9856A870000-0001-6646-5546 Bernd Bickel author 49876194-F248-11E8-B48F-1D18A9856A870000-0001-6511-9385 Wojciech Matusik author GradSch department ChWo department BeBi department Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena project We introduce a compact, intuitive procedural graph representation for cellular metamaterials, which are small-scale, tileable structures that can be architected to exhibit many useful material properties. Because the structures’ “architectures” vary widely—with elements such as beams, thin shells, and solid bulks—it is difficult to explore them using existing representations. Generic approaches like voxel grids are versatile, but it is cumbersome to represent and edit individual structures; architecture-specific approaches address these issues, but are incompatible with one another. By contrast, our procedural graph succinctly represents the construction process for any structure using a simple skeleton annotated with spatially varying thickness. 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. https://research-explorer.ista.ac.at/download/14628/14630/tog-22-0089-File004.zip application/zipno https://research-explorer.ista.ac.at/download/14628/14631/tog-22-0089-File005.zip application/zipno https://research-explorer.ista.ac.at/download/14628/14638/2023_ACMToG_Makatura.pdf application/pdfno Association for Computing Machinery2023 eng Computer Graphics and Computer-Aided Design 0730-0301 1557-7368 00108683330000710.1145/3605389 425 Makatura, L., Wang, B., Chen, Y.-L., Deng, B., Wojtan, C., Bickel, B., &#38; Matusik, W. (2023). Procedural metamaterials: A unified procedural graph for metamaterial design. <i>ACM Transactions on Graphics</i>. Association for Computing Machinery. <a href="https://doi.org/10.1145/3605389">https://doi.org/10.1145/3605389</a> L. Makatura, B. Wang, Y.-L. Chen, B. Deng, C. Wojtan, B. Bickel, W. Matusik, ACM Transactions on Graphics 42 (2023). Makatura L, Wang B, Chen Y-L, et al. 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> L. Makatura <i>et al.</i>, “Procedural metamaterials: A unified procedural graph for metamaterial design,” <i>ACM Transactions on Graphics</i>, vol. 42, no. 5. Association for Computing Machinery, 2023. Makatura, Liane, et al. “Procedural Metamaterials: A Unified Procedural Graph for Metamaterial Design.” <i>ACM Transactions on Graphics</i>, vol. 42, no. 5, 168, Association for Computing Machinery, 2023, doi:<a href="https://doi.org/10.1145/3605389">10.1145/3605389</a>. 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>. Association for Computing Machinery, 2023. <a href="https://doi.org/10.1145/3605389">https://doi.org/10.1145/3605389</a>. Makatura L, Wang B, Chen Y-L, Deng B, Wojtan C, Bickel B, Matusik W. 2023. Procedural metamaterials: A unified procedural graph for metamaterial design. ACM Transactions on Graphics. 42(5), 168. 146282023-11-29T15:02:03Z2025-09-09T13:33:58Z