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The design space of plane elastic curves

Published: 19 July 2021 Publication History

Abstract

Elastic bending of initially flat slender elements allows the realization and economic fabrication of intriguing curved shapes. In this work, we derive an intuitive but rigorous geometric characterization of the design space of plane elastic rods with variable stiffness. It enables designers to determine which shapes are physically viable with active bending by visual inspection alone. Building on these insights, we propose a method for efficiently designing the geometry of a flat elastic rod that realizes a target equilibrium curve, which only requires solving a linear program.
We implement this method in an interactive computational design tool that gives feedback about the feasibility of a design, and computes the geometry of the structural elements necessary to realize it within an instant. The tool also offers an iterative optimization routine that improves the fabricability of a model while modifying it as little as possible. In addition, we use our geometric characterization to derive an algorithm for analyzing and recovering the stability of elastic curves that would otherwise snap out of their unstable equilibrium shapes by buckling. We show the efficacy of our approach by designing and manufacturing several physical models that are assembled from flat elements.

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References

[1]
Marco Attene, Marco Livesu, Sylvain Lefebvre, Thomas Funkhouser, Szymon Rusinkiewicz, Stefano Ellero, Jonás Martínez, and Amit Haim Bermano. 2018. Design, Representations, and Processing for Additive Manufacturing. Synthesis Lectures on Visual Computing 10, 2 (2018), 1--146.
[2]
B. Audoly and Y. Pomeau. 2010. Elasticity and Geometry: From Hair Curls to the Nonlinear Response of Shells. Oxford University Press. https://books.google.at/books?id=FMQRDAAAQBAJ
[3]
Milan Batista. 2015. On stability of elastic rod planar equilibrium configurations. International Journal of Solids and Structures 72 (2015), 144 -- 152.
[4]
Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun. 2008. Discrete elastic rods. In ACM Transactions on Graphics. ACM Press.
[5]
Amit H. Bermano, Thomas Funkhouser, and Szymon Rusinkiewicz. 2017. State of the Art in Methods and Representations for Fabrication-Aware Design. Computer Graphics Forum 36, 2 (2017), 509--535.
[6]
Florence Bertails, Basile Audoly, Marie-Paule Cani, Bernard Querleux, Frédéric Leroy, and Jean-Luc Lévêque. 2006. Super-helices for predicting the dynamics of natural hair. ACM Transactions on Graphics 25, 3 (2006), 1180--1187.
[7]
Florence Bertails-Descoubes, Alexandre Jourdan, Victor Romero, and Arnaud Lazarus. 2018. Inverse design of an isotropic suspended Kirchhoff rod: theoretical and numerical results on the uniqueness of the natural shape. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, 2212 (2018), 20170837.
[8]
Gaurav Bharaj, David I. W. Levin, James Tompkin, Yun Fei, Hanspeter Pfister, Wojciech Matusik, and Changxi Zheng. 2015. Computational Design of Metallophone Contact Sounds. ACM Transactions on Graphics 34, 6, Article 223 (2015), 13 pages.
[9]
Bernd Bickel, Paolo Cignoni, Luigi Malomo, and Nico Pietroni. 2018. State of the Art on Stylized Fabrication. Computer Graphics Forum 37, 6 (2018), 325--342.
[10]
Max Born. 1906. Untersuchungen über die Stabilität der elastische Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen. Göttingen, Dieterich'sche Univ.-Buchdruckerei.
[11]
S.B. Coşkun. 2010. Analysis of Tilt-Buckling of Euler Columns with Varying Flexural Stiffness Using Homotopy Perturbation Method. Mathematical Modelling and Analysis 15, 3 (2010), 275--286.
[12]
Pierre Cuvilliers, Paul Mayencourt, and Caitlin Mueller. 2018. The Arc Lamp workshop at AAG 2018: active bending and digital fabrication. http://digitalstructures.mit.edu/page/blog#the-arc-lamp-workshop-at-aag-2018-active-bending-and-digital-fabrication Accessed: 2021-01-27.
[13]
Alexandre Derouet-Jourdan, Florence Bertails-Descoubes, and Joëlle Thollot. 2010. Stable Inverse Dynamic Curves. ACM Transactions on Graphics 29 (2010).
[14]
Levi H Dudte, Etienne Vouga, Tomohiro Tachi, and Lakshminarayanan Mahadevan. 2016. Programming curvature using origami tessellations. Nature materials 15, 5 (2016), 583--588.
[15]
Moritz Fleischmann and Achim Menges. 2011. ICD/ITKE Research Pavilion: A Case Study of Multi-disciplinary Collaborative Computational Design. In Computational Design Modelling. Springer Berlin Heidelberg, 239--248.
[16]
Akash Garg, Andrew O. Sageman-Furnas, Bailin Deng, Yonghao Yue, Eitan Grinspun, Mark Pauly, and Max Wardetzky. 2014. Wire mesh design. ACM Transactions on Graphics 33, 4 (2014), 1--12.
[17]
Konstantinos Gavriil, Ruslan Guseinov, Jesús Pérez, Davide Pellis, Paul Henderson, Florian Rist, Helmut Pottmann, and Bernd Bickel. 2020. Computational Design of Cold Bent Glass Façades. ACM Transactions on Graphics 39, 6, Article 208 (2020), 16 pages.
[18]
I. M. Gelfand and S. V. Fomin. 1963. Calculus of Variations. Prentice Hall.
[19]
Ruslan Guseinov, Connor McMahan, Jesús Pérez, Chiara Daraio, and Bernd Bickel. 2020. Programming temporal morphing of self-actuated shells. Nature communications 11, 1 (2020), 1--7.
[20]
Ruslan Guseinov, Eder Miguel, and Bernd Bickel. 2017. CurveUps. ACM Transactions on Graphics 36, 4 (2017), 1--12.
[21]
Liang He, Huaishu Peng, Michelle Lin, Ravikanth Konjeti, François Guimbretière, and Jon E. Froehlich. 2019. Ondulé: Designing and Controlling 3D Printable Springs. In Proceedings of the 32nd Annual ACM Symposium on User Interface Software and Technology. ACM.
[22]
Alexandra Ion, Michael Rabinovich, Philipp Herholz, and Olga Sorkine-Hornung. 2020. Shape approximation by developable wrapping. ACM Transactions on Graphics 39, 6 (2020), 1--12.
[23]
David Jourdan, Melina Skouras, Etienne Vouga, and Adrien Bousseau. 2020. Printing-on-Fabric Meta-Material for Self-Shaping Architectural Models. In Advances in Architectural Geometry. http://www-sop.inria.fr/reves/Basilic/2020/JSVB20
[24]
Jan Knippers, Florian Scheible, Matthias Oppe, and Hauke Jungjohann. 2012. Bioinspired Kinetic GFRP-façade for the Thematic Pavilion of the EXPO 2012 in Yeosu. In Proceedings of the IASS-APCS-Symposium, IASS (Ed.). Seoul, Korea.
[25]
Mina Konaković-Luković, Julian Panetta, Keenan Crane, and Mark Pauly. 2018. Rapid deployment of curved surfaces via programmable auxetics. ACM Transactions on Graphics 37, 4 (2018), 1--13.
[26]
Joon Kyu Lee and Byoung Koo Lee. 2018. Elastica and buckling loads of nonlinear elastic tapered cantilever columns. Engineering Solid Mechanics 6 (2018), 39--50.
[27]
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.
[28]
Mingchao Liu, Lucie Domino, and Dominic Vella. 2020. Tapered elasticæ as a route for axisymmetric morphing structures. Soft Matter 16 (2020), 7739--7750. Issue 33.
[29]
J. H. Maddocks. 1981. Analysis of nonlinear differential equations governing the equilibria of an elastic rod and their stability. Ph.D. Dissertation. University of Oxford.
[30]
Luigi Malomo, Jesús Pérez, Emmanuel Iarussi, Nico Pietroni, Eder Miguel, Paolo Cignoni, and Bernd Bickel. 2019. FlexMaps. ACM Transactions on Graphics 37, 6 (2019), 1--14.
[31]
Robert S. Manning, Kathleen A. Rogers, and John H. Maddocks. 1998. Isoperimetric Conjugate Points with Application to the Stability of DNA Minicircles. Proceedings: Mathematical, Physical and Engineering Sciences 454, 1980 (1998), 3047--3074. http://www.jstor.org/stable/53424
[32]
Eder Miguel, Mathias Lepoutre, and Bernd Bickel. 2016. Computational design of stable planar-rod structures. ACM Transactions on Graphics 35, 4 (2016), 1--11.
[33]
Dinesh K. Pai. 2002. STRANDS: Interactive Simulation of Thin Solids using Cosserat Models. Computer Graphics Forum 21, 3 (2002), 347--352.
[34]
J. Panetta, M. Konaković-Luković, F. Isvoranu, E. Bouleau, and M. Pauly. 2019. X-Shells: A New Class of Deployable Beam Structures. ACM Transactions on Graphics 38, 4 (2019), 1--15.
[35]
Jesús Pérez, Miguel A. Otaduy, and Bernhard Thomaszewski. 2017. Computational design and automated fabrication of kirchhoff-plateau surfaces. ACM Transactions on Graphics 36, 4 (2017), 1--12.
[36]
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 Transactions on Graphics 34, 4 (2015), 1--12.
[37]
Stefan Pillwein, Kurt Leimer, Michael Birsak, and Przemyslaw Musialski. 2020. On elastic geodesic grids and their planar to spatial deployment. ACM Transactions on Graphics 39, 4 (2020).
[38]
Yuri Sachkov and Stanislav Levyakov. 2010. Stability of inflectional elasticae centered at vertices or inflection points. Proceedings of the Steklov Institute of Mathematics 271 (2010), 177--192.
[39]
Oded Stein, Eitan Grinspun, and Keenan Crane. 2018. Developability of triangle meshes. ACM Transactions on Graphics 37, 4 (2018), 1--14.
[40]
Ioanna Symeonidou. 2015. Analogue and digital form-finding of bending rod structures. In Proceedings of IASS Annual Symposia, Vol. 2015. International Association for Shell and Spatial Structures (IASS), 1--12.
[41]
Nico Van der Aa, H.G. Morsche, and R.R.M. Mattheij. 2007. Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem. The Electronic Journal of Linear Algebra 16 (2007).
[42]
Guanyun Wang, Ye Tao, Ozguc Bertug Capunaman, Humphrey Yang, and Lining Yao. 2019. A-line: 4D Printing Morphing Linear Composite Structures. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems. ACM.
[43]
Katja Wolff, Roi Poranne, Oliver Glauser, and Olga Sorkine-Hornung. 2018. Packable Springs. Computer Graphics Forum 37, 2 (2018), 251--262.
[44]
Hongyi Xu, Espen Knoop, Stelian Coros, and Moritz Bächer. 2019. Bend-it: Design and Fabrication of Kinetic Wire Characters. ACM Transactions on Graphics 37, 6 (2019), 1--15.
[45]
Xu Zheng, Zhichao Fan, Haoran Fu, Yuan Liu, Yanyang Zi, Yonggang Huang, and Yihui Zhang. 2019. Optimization-Based Approach for the Inverse Design of Ribbon-Shaped Three-Dimensional Structures Assembled Through Compressive Buckling. Physical Review Applied 11 (2019).

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cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 40, Issue 4
August 2021
2170 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/3450626
Issue’s Table of Contents
This work is licensed under a Creative Commons Attribution International 4.0 License.

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Association for Computing Machinery

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Publication History

Published: 19 July 2021
Published in TOG Volume 40, Issue 4

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Author Tags

  1. active bending
  2. computational design
  3. computational fabrication
  4. elastic rods
  5. interactive design

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