---
_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: 2025-09-09T13:33:58Z
day: '01'
ddc:
- '531'
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department:
- _id: GradSch
- _id: ChWo
- _id: BeBi
doi: 10.1145/3605389
external_id:
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keyword:
- Computer Graphics and Computer-Aided Design
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
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  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Procedural metamaterials: A unified procedural graph for metamaterial design'
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 42
year: '2023'
...
---
_id: '13188'
abstract:
- lang: eng
  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."
acknowledged_ssus:
- _id: M-Shop
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).
article_number: '171'
article_processing_charge: No
article_type: original
author:
- first_name: Christian
  full_name: Hafner, Christian
  id: 400429CC-F248-11E8-B48F-1D18A9856A87
  last_name: Hafner
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
citation:
  ama: Hafner C, Bickel B. The design space of Kirchhoff rods. <i>ACM Transactions
    on Graphics</i>. 2023;42(5). doi:<a href="https://doi.org/10.1145/3606033">10.1145/3606033</a>
  apa: Hafner, C., &#38; Bickel, B. (2023). The design space of Kirchhoff rods. <i>ACM
    Transactions on Graphics</i>. Association for Computing Machinery. <a href="https://doi.org/10.1145/3606033">https://doi.org/10.1145/3606033</a>
  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>.
  ieee: C. Hafner and B. Bickel, “The design space of Kirchhoff rods,” <i>ACM Transactions
    on Graphics</i>, vol. 42, no. 5. Association for Computing Machinery, 2023.
  ista: Hafner C, Bickel B. 2023. The design space of Kirchhoff rods. ACM Transactions
    on Graphics. 42(5), 171.
  mla: Hafner, Christian, and Bernd Bickel. “The Design Space of Kirchhoff Rods.”
    <i>ACM Transactions on Graphics</i>, vol. 42, no. 5, 171, Association for Computing
    Machinery, 2023, doi:<a href="https://doi.org/10.1145/3606033">10.1145/3606033</a>.
  short: C. Hafner, B. Bickel, ACM Transactions on Graphics 42 (2023).
corr_author: '1'
date_created: 2023-07-04T07:41:30Z
date_published: 2023-09-20T00:00:00Z
date_updated: 2026-04-03T22:30:20Z
day: '20'
ddc:
- '516'
department:
- _id: BeBi
doi: 10.1145/3606033
ec_funded: 1
external_id:
  isi:
  - '001086833300010'
file:
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  creator: chafner
  date_created: 2023-07-04T08:11:28Z
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  date_updated: 2023-07-04T07:46:28Z
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  content_type: application/pdf
  creator: chafner
  date_created: 2023-07-04T07:46:30Z
  date_updated: 2023-07-04T07:46:30Z
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  file_size: 430086
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  checksum: c0fd9a57d012046de90c185ffa904b76
  content_type: video/mp4
  creator: chafner
  date_created: 2023-07-04T07:46:39Z
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  creator: chafner
  date_created: 2023-07-04T07:47:10Z
  date_updated: 2023-07-04T07:47:10Z
  file_id: '13193'
  file_name: matlab-submission.zip
  file_size: 25790
  relation: supplementary_material
  title: Matlab Source Code with Example
file_date_updated: 2023-07-04T08:11:28Z
has_accepted_license: '1'
intvolume: '        42'
isi: 1
issue: '5'
keyword:
- Computer Graphics
- Computational Design
- Computational Geometry
- Shape Modeling
language:
- iso: eng
month: '09'
oa: 1
oa_version: Submitted Version
project:
- _id: 24F9549A-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '715767'
  name: 'MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and
    Modeling'
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
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scopus_import: '1'
status: public
title: The design space of Kirchhoff rods
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 42
year: '2023'
...
---
_id: '11993'
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. Finally, we discuss potential future
    research directions.
article_processing_charge: No
article_type: original
author:
- first_name: Thomas
  full_name: Alderighi, Thomas
  last_name: Alderighi
- first_name: Luigi
  full_name: Malomo, Luigi
  last_name: Malomo
- first_name: Thomas
  full_name: Auzinger, Thomas
  id: 4718F954-F248-11E8-B48F-1D18A9856A87
  last_name: Auzinger
  orcid: 0000-0002-1546-3265
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
- first_name: Paulo
  full_name: Cignoni, Paulo
  last_name: Cignoni
- first_name: Nico
  full_name: Pietroni, Nico
  last_name: Pietroni
citation:
  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>
  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>
  chicago: Alderighi, Thomas, Luigi Malomo, Thomas Auzinger, Bernd Bickel, Paulo Cignoni,
    and Nico Pietroni. “State of the Art in Computational Mould Design.” <i>Computer
    Graphics Forum</i>. Wiley, 2022. <a href="https://doi.org/10.1111/cgf.14581">https://doi.org/10.1111/cgf.14581</a>.
  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.
  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.
  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>.
  short: T. Alderighi, L. Malomo, T. Auzinger, B. Bickel, P. Cignoni, N. Pietroni,
    Computer Graphics Forum 41 (2022) 435–452.
corr_author: '1'
date_created: 2022-08-28T18:17:01Z
date_published: 2022-09-01T00:00:00Z
date_updated: 2024-10-09T21:03:21Z
day: '01'
ddc:
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doi: 10.1111/cgf.14581
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  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.'
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file_date_updated: 2022-08-28T18:18:08Z
has_accepted_license: '1'
intvolume: '        41'
isi: 1
issue: '6'
keyword:
- Computer Graphics and Computer-Aided Design
language:
- iso: eng
month: '09'
oa: 1
oa_version: Submitted Version
page: 435-452
publication: Computer Graphics Forum
publication_identifier:
  eissn:
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  issn:
  - 0167-7055
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: State of the art in computational mould design
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 41
year: '2022'
...
---
_id: '5568'
abstract:
- lang: eng
  text: Includes source codes, test cases, and example data used in the thesis Brittle
    Fracture Simulation with Boundary Elements for Computer Graphics. 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)
article_processing_charge: No
author:
- first_name: David
  full_name: Hahn, David
  id: 357A6A66-F248-11E8-B48F-1D18A9856A87
  last_name: Hahn
citation:
  ama: 'Hahn D. Source codes: Brittle fracture simulation with boundary elements for
    computer graphics. 2017. doi:<a href="https://doi.org/10.15479/AT:ISTA:73">10.15479/AT:ISTA:73</a>'
  apa: 'Hahn, D. (2017). Source codes: Brittle fracture simulation with boundary elements
    for computer graphics. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT:ISTA:73">https://doi.org/10.15479/AT:ISTA:73</a>'
  chicago: 'Hahn, David. “Source Codes: Brittle Fracture Simulation with Boundary
    Elements for Computer Graphics.” Institute of Science and Technology Austria,
    2017. <a href="https://doi.org/10.15479/AT:ISTA:73">https://doi.org/10.15479/AT:ISTA:73</a>.'
  ieee: 'D. Hahn, “Source codes: Brittle fracture simulation with boundary elements
    for computer graphics.” Institute of Science and Technology Austria, 2017.'
  ista: 'Hahn D. 2017. Source codes: Brittle fracture simulation with boundary elements
    for computer graphics, Institute of Science and Technology Austria, <a href="https://doi.org/10.15479/AT:ISTA:73">10.15479/AT:ISTA:73</a>.'
  mla: 'Hahn, David. <i>Source Codes: Brittle Fracture Simulation with Boundary Elements
    for Computer Graphics</i>. Institute of Science and Technology Austria, 2017,
    doi:<a href="https://doi.org/10.15479/AT:ISTA:73">10.15479/AT:ISTA:73</a>.'
  short: D. Hahn, (2017).
datarep_id: '73'
date_created: 2018-12-12T12:31:35Z
date_published: 2017-08-16T00:00:00Z
date_updated: 2025-04-14T09:25:58Z
day: '16'
ddc:
- '004'
department:
- _id: ChWo
doi: 10.15479/AT:ISTA:73
ec_funded: 1
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  file_name: IST-2017-73-v1+1_FractureRB_v1.1_2017_07_20_final_public.zip
  file_size: 199353471
  relation: main_file
file_date_updated: 2020-07-14T12:47:04Z
has_accepted_license: '1'
keyword:
- Boundary elements
- brittle fracture
- computer graphics
- fracture simulation
license: https://creativecommons.org/licenses/by-sa/4.0/
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: 2533E772-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '638176'
  name: 'Big Splash: Efficient Simulation of Natural Phenomena at Extremely Large
    Scales'
publisher: Institute of Science and Technology Austria
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status: public
title: 'Source codes: Brittle fracture simulation with boundary elements for computer
  graphics'
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  short: CC BY-SA (4.0)
type: research_data
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...