“Generalized deployable elastic geodesic grids” by Pillwein and Musialski
Conference:
Type(s):
Title:
- Generalized deployable elastic geodesic grids
Session/Category Title: Fabrication
Presenter(s)/Author(s):
Abstract:
Given a designer created free-form surface in 3d space, our method computes a grid composed of elastic elements which are completely planar and straight. Only by fixing the ends of the planar elements to appropriate locations, the 2d grid bends and approximates the given 3d surface. Our method is based purely on the notions from differential geometry of curves and surfaces and avoids any physical simulations. In particular, we introduce a well-defined elastic grid energy functional that allows identifying networks of curves that minimize the bending energy and at the same time nestle to the provided input surface well. Further, we generalize the concept of such grids to cases where the surface boundary does not need to be convex, which allows for the creation of sophisticated and visually pleasing shapes. The algorithm finally ensures that the 2d grid is perfectly planar, making the resulting gridshells inexpensive, easy to fabricate, transport, assemble, and finally also to deploy. Additionally, since the whole structure is pre-strained, it also comes with load-bearing capabilities. We evaluate our method using physical simulation and we also provide a full fabrication pipeline for desktop-size models and present multiple examples of surfaces with elliptic and hyperbolic curvature regions. Our method is meant as a tool for quick prototyping for designers, architects, and engineers since it is very fast and results can be obtained in a matter of seconds.
References:
1. Changyeob Baek and Pedro M. Reis. 2019. Rigidity of hemispherical elastic gridshells under point load indentation. Journal of the Mechanics and Physics of Solids 124 (Mar 2019), 411–426.
2. Changyeob Baek, Andrew O. Sageman-Furnas, Mohammad K. Jawed, and Pedro M. Reis. 2018. Form finding in elastic gridshells. Proceedings of the National Academy of Sciences of the United States of America 115, 1 (Jan 2018), 75–80.
3. Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun. 2008. Discrete elastic rods. ACM Trans. Graph. 27, 3 (Aug 2008), 1.
4. Tian Chen, Julian Panetta, Max Schnaubelt, and Mark Pauly. 2021. Bistable Auxetic Surface Structures. ACM Trans. Graph. 40, 4, Article 39 (July 2021), 9 pages.
5. Keenan Crane, Marco Livesu, Enrico Puppo, and Yipeng Qin. 2020. A Survey of Algorithms for Geodesic Paths and Distances. arXiv (Jul 2020). arXiv:2007.10430 http://arxiv.org/abs/2007.10430
6. Levi H. Dudte, Etienne Vouga, Tomohiro Tachi, and L. Mahadevan. 2016. Programming curvature using origami tessellations. Nature Materials 15, 5 (May 2016), 583–588.
7. Akash Garg, Andrew O. Sageman-Furnas, Bailin Deng, Yonghao Yue, Eitan Grinspun, Mark Pauly, and Max Wardetzky. 2014. Wire mesh design. ACM Trans. Graph. 33, 4 (Jul 2014), 1–12.
8. Christoph Gengnagel, Julian Lienhard, Holger Alpermann, Christoph Gengnagel, and Jan Knippers. 2013. Active bending, a review on structures where bending is used as a self-formation process. International Journal of Space Structures 28, 3-4 (2013), 187–196.
9. David E. Goldberg. 1989. Genetic Algorithms in Search, Optimization and Machine Learning (1st ed.). Addison-Wesley Longman Publishing Co., Inc., USA.
10. Ruslan Guseinov, Connor McMahan, Jesús Pérez, Chiara Daraio, and Bernd Bickel. 2020. Programming temporal morphing of self-actuated shells. Nature Communications 11, 1 (Dec 2020), 1–7.
11. Ruslan Guseinov, Eder Miguel, and Bernd Bickel. 2017. CurveUps. ACM Trans. Graph. 36, 4 (Jul 2017), 1–12.
12. Christian Hafner and Bernd Bickel. 2021. The Design Space of Plane Elastic Curves. ACM Trans. Graph. 40, 4, Article 126 (July 2021), 20 pages.
13. Edmund Happold and Ian Liddell. 1975. Timber Lattice Roof for the Mannheim Bundesgartenschau. The Structural Engineer 53, 3 (1975).
14. Alexandra Ion, Michael Rabinovich, Philipp Herholz, and Olga Sorkine-Hornung. 2020. Shape approximation by developable wrapping. ACM Trans. Graph. 39, 6 (Nov 2020), 1–12.
15. Florin Isvoranu, Julian Panetta, Tian Chen, Etienne Bouleau, and Mark Pauly. 2019. X-Shell Pavilion: A Deployable Elastic Rod Structure. In Proceedings of IASS Annual Symposia, Vol. 2019. International Association for Shell and Spatial Structures, 1–8.
16. Caigui Jiang, Florian Rist, Helmut Pottmann, and Johannes Wallner. 2020. Freeform Quad-Based Kirigami. ACM Trans. Graph. 39, 6, Article 209 (Nov. 2020), 11 pages.
17. Martin Kilian, Simon Flöry, Zhonggui Chen, Niloy J. Mitra, Alla Sheffer, and Helmut Pottmann. 2008. Curved folding. ACM Trans. Graph. 27, 3 (Aug 2008), 1.
18. Martin Kilian, Aron Monszpart, and Niloy J. Mitra. 2017. String Actuated Curved Folded Surfaces. ACM Trans. Graph. 36, 3 (May 2017), 1–13.
19. Mina Konaković, Keenan Crane, Bailin Deng, Sofien Bouaziz, Daniel Piker, and Mark Pauly. 2016. Beyond developable. ACM Trans. Graph. 35, 4 (Jul 2016), 1–11.
20. Mina Konaković-Luković, Julian Panetta, Keenan Crane, and Mark Pauly. 2018. Rapid deployment of curved surfaces via programmable auxetics. ACM Trans. Graph. 37, 4 (Jul 2018), 1–13.
21. Francesco Laccone, Luigi Malomo, Jesùs Pérez, Nico Pietroni, Federico Ponchio, Bernd Bickel, and Paolo Cignoni. 2019. FlexMaps Pavilion: a twisted arc made of mesostructured flat flexible panels. In Proceedings of IASS Annual Symposia, Vol. 2019. International Association for Shell and Spatial Structures (IASS), 1–7.
22. Max Lagally. 1910. Über die Verbiegung geodätischer Netze. Sitzungsbericht der Bayerischen Akademie der Wissenschaften, Vol. 1910,10. Verl.d.K.B.Akad.d.Wiss., München. http://publikationen.badw.de/de/003396114
23. Marc S. Lavine. 2015. Popping materials and devices from 2D into 3D. Science 347, 6218 (Jan 2015), 141–143.
24. Bruno Lévy, Sylvain Petitjean, Nicolas Ray, and Jérome Maillot. 2002. Least Squares Conformal Maps for Automatic Texture Atlas Generation. ACM Trans. Graph. 21, 3 (July 2002), 362–371.
25. Julian Lienhard, Holger Alpermann, Christoph Gengnagel, and Jan Knippers. 2013. Active Bending, a Review on Structures where Bending is Used as a Self-Formation Process. International Journal of Space Structures 28, 3-4 (Sep 2013), 187–196.
26. Julian Lienhard and Christoph Gengnagel. 2018. Recent developments in bending-active structures. In Creativity in Structural Design, annual Symposium of the IASS – International Association for Shell and Spatial Structures. Boston.
27. Mingchao Liu, Lucie Domino, and Dominic Vella. 2020. Tapered elasticæ as a route for axisymmetric morphing structures. Soft Matter 16, 33 (Sep 2020), 7739–7750.
28. Luigi Malomo, Jesús Pérez, Emmanuel Iarussi, Nico Pietroni, Eder Miguel, Paolo Cignoni, and Bernd Bickel. 2018. FlexMaps. ACM Trans. Graph. 37, 6 (Dec 2018), 1–14.
29. Fady Massarwi, Craig Gotsman, and Gershon Elber. 2007. Papercraft Models using Generalized Cylinders. In 15th Pacific Conference on Computer Graphics and Applications (PG’07). IEEE, 148–157.
30. Eleanna Panagoulia and Simon Schleicher. 2016. Bending-active Structures: A Case study for an Office Chaise Lounge. In eCAADe – Computing for a better tomorrow, Anetta Kȩpczyńska-Walczak and Sebastian Białkowski (Eds.). eCAADe (Education and Research in Computer Aided Architectural Design in Europe), Lodz, 621–630. http://ecaade.org/downloads/eCAADe-2018-Volume1.pdf
31. Julian Panetta, Florin Isvoranu, Tian Chen, Emmanuel Siéfert, Benoît Roman, and Mark Pauly. 2021. Computational Inverse Design of Surface-Based Inflatables. ACM Trans. Graph. 40, 4, Article 40 (July 2021), 14 pages.
32. Julian Panetta, Mina Konaković-Luković, Florin Isvoranu, Etienne Bouleau, and Mark Pauly. 2019. X-Shells: a new class of deployable beam structures. ACM Trans. Graph. 38, 4 (Jul 2019), 1–15.
33. Jesús Pérez, Miguel A. Otaduy, and Bernhard Thomaszewski. 2017. Computational design and automated fabrication of kirchhoff-plateau surfaces. ACM Trans. Graph. 36, 4 (Jul 2017), 1–12.
34. Jesús Pérez, Bernhard Thomaszewski, Stelian Coros, Bernd Bickel, José A. Canabal, Robert Sumner, and Miguel A. Otaduy. 2015. Design and fabrication of flexible rod meshes. ACM Trans. Graph. 34, 4 (Jul 2015), 138:1–138:12.
35. Nico Pietroni, Marco Tarini, Amir Vaxman, Daniele Panozzo, and Paolo Cignoni. 2017. Position-based tensegrity design. ACM Trans. Graph. 36, 6 (Nov 2017), 1–14.
36. Stefan Pillwein, Johanna Kübert, Florian Rist, and Przemyslaw Musialski. 2020a. Design and Fabrication of Elastic Geodesic Grid Structures. In Symposium on Computational Fabrication. ACM, New York, NY, USA, 1–11. arXiv:2010.08062
37. Stefan Pillwein, Johanna Kübert, Florian Rist, and Przemyslaw Musialski. 2021. Design and fabrication of multi-patch elastic geodesic grid structures. Computers & Graphics 98 (2021), 218–230.
38. Stefan Pillwein, Kurt Leimer, Michael Birsak, and Przemyslaw Musialski. 2020b. On Elastic Geodesic Grids and Their Planar to Spatial Deployment. ACM Trans. Graph. 39, 4 (Jun 2020), 125:1–125:12. arXiv:2007.00201
39. Helmut Pottmann, Qixing Huang, Bailin Deng, Alexander Schiftner, Martin Kilian, Leonidas Guibas, and Johannes Wallner. 2010. Geodesic patterns. ACM Trans. Graph. 29, 4 (Jul 2010), 1–10.
40. Yipeng Qin, Xiaoguang Han, Hongchuan Yu, Yizhou Yu, and Jianjun Zhang. 2016. Fast and Exact Discrete Geodesic Computation Based on Triangle-Oriented Wavefront Propagation. ACM Trans. Graph. 35, 4 (July 2016), 1–13.
41. Gregory Quinn and Christoph Gengnagel. 2018. Full Scale Prototype for the Pneumatic Erection of Elastic Gridshells. In Proceedings of IASS Annual Symposia, Vol. 2018. International Association for Shell and Spatial Structures (IASS), 1–8.
42. Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. 2018. Discrete Geodesic Nets for Modeling Developable Surfaces. ACM Trans. Graph. 37, 2 (feb 2018), 1–17.
43. Eike Schling, Martin Kilian, Hui Wang, Jonas Schikore, and Helmut Pottmann. 2018. Design and construction of curved support structures with repetitive parameters. In Advances in Architectural Geometry (AAG) 2018.
44. Vladimir Shukhov. 1896. Rotunda of the Panrussian Exposition (Nizhny Novgorod, 1896) | Structurae. https://structurae.net/en/structures/rotunda-of-the-panrussian-exposition
45. Enrique Soriano. 2017. Low-Tech Geodesic Gridshell: Almond Pavilion. archidoct 4 (2017), 29.
46. Enrico Soriano, Ramon Sastre, and Dionis Boixader. 2019. G-shells: Flat collapsible geodesic mechanisms for gridshells. In IASS Annual Symposium 2019 – Structural Membranes. Barcelona.
47. Enrique Soriano, Pep Tornabell, Dragos Naicu, and Günther H Filz. 2015. Topologically-based curvature in thin elastic shell networks.
48. Oded Stein, Eitan Grinspun, and Keenan Crane. 2018. Developability of triangle meshes. ACM Trans. Graph. 37, 4 (Aug 2018), 1–14.
49. Vitaly Surazhsky, Tatiana Surazhsky, Danil Kirsanov, Steven J. Gortler, and Hugues Hoppe. 2005. Fast Exact and Approximate Geodesics on Meshes. ACM Trans. Graph. 24, 3 (July 2005), 553–560.
50. Demetri Terzopoulos, John Platt, Alan Barr, and Kurt Fleischer. 1987. Elastically deformable models. ACM SIGGRAPH Computer Graphics 21, 4 (Aug 1987), 205–214.
51. Josh Vekhter, Jiacheng Zhuo, Luisa F Gil Fandino, Qixing Huang, and Etienne Vouga. 2019. Weaving geodesic foliations. ACM Trans. Graph. 38, 4 (Jul 2019), 1–22.
52. Aurel Voss. 1907. Über diejenigen Flächen, welche durch zwei Scharen von Kurven konstanter geodätischer Krümmung in infinitesimale Rhomben zerlegt werden. Sitzungsbericht der Bayerischen Akademie der Wissenschaften, Vol. 36,7. Verl.d.K.B.Akad.d.Wiss., München. http://publikationen.badw.de/de/003388868
53. Johannes Wallner, Alexander Schiftner, Martin Kilian, Simon Flöry, Mathias Höbinger, Bailin Deng, Qixing Huang, and Helmut Pottmann. 2010. Tiling Freeform Shapes With Straight Panels: Algorithmic Methods. In Advances in Architectural Geometry 2010. Springer Vienna, Vienna, 73–86.
54. Hui Wang, Davide Pellis, Florian Rist, Helmut Pottmann, and Christian Müller. 2019. Discrete geodesic parallel coordinates. ACM Trans. Graph. 38, 6 (Nov 2019), 1–13.
55. William Welch and Andrew Witkin. 1992. Variational surface modeling. ACM SIGGRAPH Computer Graphics 26, 2 (Jul 1992), 157–166.
56. Hongyi Xu, Espen Knoop, Stelian Coros, and Moritz Bächer. 2018. Bend-It: Design and Fabrication of Kinetic Wire Characters. ACM Trans. Graph. 37, 6, Article 239 (Dec. 2018), 15 pages.
