---
_id: '14628'
abstract:
- lang: eng
  text: 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.
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."
article_number: '168'
article_processing_charge: Yes (in subscription journal)
article_type: original
author:
- first_name: Liane
  full_name: Makatura, Liane
  last_name: Makatura
- first_name: Bohan
  full_name: Wang, Bohan
  last_name: Wang
- first_name: Yi-Lu
  full_name: Chen, Yi-Lu
  id: 0b467602-dbcd-11ea-9d1d-ed480aa46b70
  last_name: Chen
- first_name: Bolei
  full_name: Deng, Bolei
  last_name: Deng
- first_name: Christopher J
  full_name: Wojtan, Christopher J
  id: 3C61F1D2-F248-11E8-B48F-1D18A9856A87
  last_name: Wojtan
  orcid: 0000-0001-6646-5546
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
- first_name: Wojciech
  full_name: Matusik, Wojciech
  last_name: Matusik
citation:
  ama: '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>'
  apa: '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>'
  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>. Association
    for Computing Machinery, 2023. <a href="https://doi.org/10.1145/3605389">https://doi.org/10.1145/3605389</a>.'
  ieee: '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.'
  ista: '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.'
  mla: '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>.'
  short: L. Makatura, B. Wang, Y.-L. Chen, B. Deng, C. Wojtan, B. Bickel, W. Matusik,
    ACM Transactions on Graphics 42 (2023).
date_created: 2023-11-29T15:02:03Z
date_published: 2023-10-01T00:00:00Z
date_updated: 2023-12-04T08:09:05Z
day: '01'
ddc:
- '531'
- '006'
department:
- _id: GradSch
- _id: ChWo
- _id: BeBi
doi: 10.1145/3605389
file:
- access_level: open_access
  checksum: 0192f597d7a2ceaf89baddfd6190d4c8
  content_type: application/zip
  creator: yichen
  date_created: 2023-11-29T15:16:01Z
  date_updated: 2023-11-29T15:16:01Z
  file_id: '14630'
  file_name: tog-22-0089-File004.zip
  file_size: 95467870
  relation: main_file
  success: 1
- access_level: open_access
  checksum: 7fb024963be81933494f38de191e4710
  content_type: application/zip
  creator: yichen
  date_created: 2023-11-29T15:16:01Z
  date_updated: 2023-11-29T15:16:01Z
  file_id: '14631'
  file_name: tog-22-0089-File005.zip
  file_size: 103731880
  relation: main_file
  success: 1
- access_level: open_access
  checksum: b7d6829ce396e21cac9fae0ec7130a6b
  content_type: application/pdf
  creator: dernst
  date_created: 2023-12-04T08:04:14Z
  date_updated: 2023-12-04T08:04:14Z
  file_id: '14638'
  file_name: 2023_ACMToG_Makatura.pdf
  file_size: 57067476
  relation: main_file
  success: 1
file_date_updated: 2023-12-04T08:04:14Z
has_accepted_license: '1'
intvolume: '        42'
issue: '5'
keyword:
- Computer Graphics and Computer-Aided Design
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: ACM Transactions on Graphics
publication_identifier:
  issn:
  - 0730-0301
  - 1557-7368
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
status: public
title: 'Procedural metamaterials: A unified procedural graph for metamaterial design'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 42
year: '2023'
...