“MERCI: Mixed curvature-based elements for computing equilibria of thin elastic ribbons”
Conference:
Type(s):
Title:
- MERCI: Mixed curvature-based elements for computing equilibria of thin elastic ribbons
Presenter(s)/Author(s):
Abstract:
Ribbons are thin elastic structures lying in between rods and plates. Relying on the 1D Wunderlich ribbon model, we propose mixed position-curvature elements for computing ribbon equilibria subject to arbitrary boundary constraints. Our simulator is carefully validated on subtle scenarios such as M?bius bands or confined and lateral transverse buckling.
References:
[1]
D. Arnold and F. Brezzi. 1997. Locking free finite element for shells. Math. Comput. 66, 217 (01 1997), 1?14. DOI:
[2]
Basile Audoly and S?bastien Neukirch. 2021. A one-dimensional model for elastic ribbons: A little stretching makes a big difference. Journal of the Mechanics and Physics of Solids 153 (2021), 104457. DOI:
[3]
Basile Audoly and Yves Pomeau. 2010. Elasticity and Geometry: From Hair Curls to the Nonlinear Response of Shells. Oxford University Press, Oxford.
[4]
Basile Audoly and G.H.M. van der Heijden. 2022. Analysis of cone-like singularities in twisted elastic ribbons. Journal of the Mechanics and Physics of Solids 171 (2022), 105131. DOI:
[5]
D. Baraff and A. Witkin. 1998. Large steps in cloth simulation. In Computer Graphics Proceedings (Proc. ACM SIGGRAPH?98). ACM, Orlando, FL, 43?54. Retrieved from http://www.cs.cmu.edu/baraff/papers/sig98.pdf
[6]
S?ren Bartels and Peter Hornung. 2015. Bending paper and the M?bius strip. Journal of Elasticity 119, 1 (2015), 113?136.
[7]
M. Bergou, B. Audoly, E. Vouga, M. Wardetzky, and E. Grinspun. 2010. Discrete viscous threads. ACM Transactions on Graphics 29, 4, Article 116 (2010), 10 pages. DOI:
[8]
M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, and E. Grinspun. 2008. Discrete elastic rods. ACM Transactions on Graphics (Proc. ACM SIGGRAPH?08) 27, 3 (2008), 1?12. DOI:
[9]
F. Bertails, B. Audoly, M.-P. Cani, B. Querleux, F. Leroy, and J.-L. L?v?que. 2006. Super-helices for predicting the dynamics of natural hair. ACM Transactions on Graphics (Proc. ACM SIGGRAPH?06) 25, 3 (2006), 1180?1187. DOI:
[10]
Davide Bigoni. 2012. Nonlinear Solids Mechanics. Cambridge University Press, Cambridge, UK.
[11]
Pengbo Bo and Wenping Wang. 2007. Geodesic-controlled developable surfaces for modeling paper bending. Computer Graphics Forum 26, 3 (2007), 365?374. DOI:
[12]
B. Boehm. 1981. Software Engineering Economics. Prentice-Hall, Englewood Cliffs, N.J.
[13]
Friedrich B?s, Max Wardetzky, Etienne Vouga, and Omer Gottesman. 2017. On the incompressibility of cylindrical origami patterns. Journal of Mechanical Design 139, 2 (2017), 021404. DOI:
[14]
Richard Bradshaw, David Campbell, Mousa Gargari, Amir Mirmiran, and Patrick Tripeny. 2002. Special structures: Past, present, and future. Journal of Structural Engineering 128, 6 (2002), 691?709.
[15]
R. Bridson, S. Marino, and R. Fedkiw. 2003. Simulation of clothing with folds and wrinkles. In ACM SIGGRAPH – EG Symposium on Computer Animation (SCA?03). The Eurographics Association, Goslar, 28?36. Retrieved from http://www.cs.ubc.ca/rbridson/docs/cloth2003.pdf
[16]
Michel Carignan, Ying Yang, Nadia Magnenat Thalmann, and Daniel Thalmann. 1992. Dressing animated synthetic actors with complex deformable clothes. SIGGRAPH Comput. Graph. 26, 2 (jul 1992), 99?104. DOI:
[17]
R. Casati. 2015. Quelques contributions ? la mod?lisation num?rique de structures ?lanc?es pour l?informatique graphique. phdthesis. Universit? Grenoble Alpes. Retrieved from http://www.theses.fr/2015GREAM053/document
[18]
R. Casati and F. Bertails-Descoubes. 2013. Super space clothoids. ACM Transactions on Graphics 32, 4 (2013), 48. DOI:
[19]
Herzl Chai. 2006. On the crush worthiness of a laterally confined bar under axial compression. Transactions of the ASME 73, 5 (2006), 834?841. DOI:
[20]
Rapha?l Charrondi?re, Florence Bertails-Descoubes, S?bastien Neukirch, and Victor Romero. 2020. Numerical modeling of inextensible elastic ribbons with curvature-based elements. Computer Methods in Applied Mechanics and Engineering 364 (2020), 112922. DOI:
[21]
H.-Y. Chen, A. Sastry, W. van Rees, and E. Vouga. 2018. Physical simulation of environmentally induced thin shell deformation. ACM Trans. Graph. 37, 4, Article 146 (July 2018), 13 pages. DOI:
[22]
K.-J. Choi and H.-S. Ko. 2002. Stable but responsive cloth. ACM Trans. Graph. 21, 3 (July 2002), 604?611. DOI:
[23]
Julien Chopin, Vincent D?mery, and Benny Davidovitch. 2015. Roadmap to the morphological instabilities of a stretched twisted ribbon. Journal of Elasticity 119 (2015), 137?189. DOI:
[24]
St?phanie Deboeuf, Suzie Proti?re, and Eytan Katzav. 2024. Yin-yang spiraling transition of a confined buckled elastic sheet. Physical Review Research 6, 1 (2024), 013100. DOI:
[25]
Philippe Decaudin, Dan Julius, Jamie Wither, Laurence Boissieux, Alla Sheffer, and Marie-Paule Cani. 2006. Virtual garments: A fully geometric approach for clothing design. Computer Graphics Forum 25, 3 (2006), 625?634. DOI:
[26]
Marcelo A. Dias and Basile Audoly. 2014. A non-linear rod model for folded elastic strips. Journal of the Mechanics and Physics of Solids 62 (2014), 57?80. DOI:
[27]
Marcelo A. Dias and Basile Audoly. 2015. ?Wunderlich, meet kirchhoff?: A general and unified description of elastic ribbons and thin rods. Journal of Elasticity 119, 1 (2015), 49?66. DOI:
[28]
E. Dill. 1992. Kirchhoff?s theory of rods. Archive for History of Exact Sciences 44, 1 (1992), 1?23.
[29]
Manfredo P Do Carmo. 2016. Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. Courier Dover Publications, Mineola, New York.
[30]
Eusebius Doedel, Herbert B. Keller, and Jean Pierre Kernevez. 1991. Numerical analysis and control of bifurcation problems (II): Bifurcation in infinite dimensions. International Journal of Bifurcation and Chaos 01, 4 (1991), 745?772.
[31]
G. Domokos, P. Holmes, and B. Royce. 1997. Constrained euler buckling. Journal of Nonlinear Science 7, 3 (1997), 281?314.
[32]
Levi H. Dudte, Etienne Vouga, Tomohiro Tachi, and L. Mahadevan. 2016. Programming curvature using origami tessellations. Nature Materials 15, 5 (2016), 583?588. DOI:
[33]
Herv? Elettro, Paul Grandgeorge, and S?bastien Neukirch. 2017. Elastocapillary coiling of an elastic rod inside a drop. Journal of Elasticity 127, 2 (2017), 235?247.
[34]
E. English and R. Bridson. 2008. Animating developable surfaces using nonconforming elements. ACM Transactions on Graphics (Proc. of SIGGRAPH?08) 27, 3 (2008), 1?5. DOI:
[35]
Roger Fosdick and Eliot Fried (Eds.). 2015. The Mechanics of Ribbons and Moebius Bands. Springer.
[36]
Lorenzo Freddi, Peter Hornung, Maria Giovanna Mora, and Roberto Paroni. 2016. A corrected sadowsky functional for inextensible elastic ribbons. Journal of Elasticity 123, 2 (2016), 125?136.
[37]
R. Goldenthal, D. Harmon, R. Fattal, M. Bercovier, and E. Grinspun. 2007. Efficient simulation of inextensible cloth. ACM Transactions on Graphics (Proc. ACM SIGGRAPH?07) 26, 3, Article 49 (2007), 7 pages. DOI:
[38]
E. Grinspun, A. Hirani, M. Desbrun, and P. Schr?der. 2003. Discrete shells. In ACM SIGGRAPH – EG Symposium on Computer Animation (SCA?03). The Eurographics Association, Goslar, Germany, 62?67. Retrieved from http://www.multires.caltech.edu/pubs/ds.pdf
[39]
M. Habera, J. S. Hale, A. Logg, C. Richardson, J. Ring, M. E. Rognes, N. Sime, and G. N. Wells. 2018. The Fenics Project. (2018). Retrieved from https://fenicsproject.org
[40]
S. Hadap. 2006. Oriented strands – Dynamics of stiff multi-body system. In ACM SIGGRAPH – EG Symposium on Computer Animation (SCA?06). Eurographics Association, Vienna, Austria, 91?100.
[41]
Christian Hafner and Bernd Bickel. 2021. The design space of plane elastic curves. ACM Transactions on Graphics (TOG) 40, 4 (2021), 1?20. DOI:
[42]
Jack S. Hale, Matteo Brunetti, St?phane P. A. Bordas, and Corrado Maurini. 2018. Simple and extensible plate and shell finite element models through automatic code generation tools. Computers & Structures 209 (2018), 163?181. DOI:
[43]
M. Hauth, O. Etzmuss, and W. Strasser. 2003. Analysis of numerical methods for the simulation of deformable models. The Visual Computer 19 (2003), 581?600. DOI:
[44]
Denis F. Hinz and Eliot Fried. 2015. Translation of michael sadowsky?s paper ?the differential equations of the M?bius band?. Journal of Elasticity 119, 1 (2015), 19?22.
[45]
Weicheng Huang, Chao Ma, Qiang Chen, and Longhui Qin. 2022. A discrete differential geometry-based numerical framework for extensible ribbons. International Journal of Solids and Structures 248 (2022), 111619. DOI:
[46]
Weicheng Huang, Yunbo Wang, Xuanhe Li, and Mohammad K. Jawed. 2020. Shear induced supercritical pitchfork bifurcation of pre-buckled bands, from narrow strips to wide plates. Journal of the Mechanics and Physics of Solids 145 (2020), 104168. DOI:
[47]
Victor Ceballos Inza, Florian Rist, Johannes Wallner, and Helmut Pottmann. 2023. Developable quad meshes and contact element nets. Transactions on Graphics (TOG) 42, 6 (2023), 183. DOI: https://doi.org/10.1145/
[48]
David Jourdan, M?lina Skouras, Etienne Vouga, and Adrien Bousseau. 2022. Computational design of self-actuated surfaces by printing plastic ribbons on stretched fabric. Computer Graphics Forum 41, 2 (April 2022), 1?14. Retrieved from https://hal.inria.fr/hal-03588434
[49]
Martin Kilian, Simon Fl?ry, Zhonggui Chen, Niloy J. Mitra, Alla Sheffer, and Helmut Pottmann. 2008. Curved folding. ACM Trans. Graph. 27, 3 (aug 2008), 1?9. DOI:
[50]
Kevin Korner, Basile Audoly, and Kaushik Bhattacharya. 2021. Simple deformation measures for discrete elastic rods and ribbons. Proceedings of the Royal Society A 477, 2256 (2021), 20210561.
[51]
Arun Kumar, Poornakanta Handral, Darshan Bhandari, and Ramsharan Rangarajan. 2021. More views of a one-sided surface: Mechanical models and stereo vision techniques for M?bius strips. Proceedings of the Royal Society A 477, 2250 (2021), 20210076. DOI:
[52]
A. Kumar, P. Handral, C. S. Darshan Bhandari, A. Karmakarn, and R. Rangarajan. 2020. An investigation of models for elastic ribbons: Simulations & experiments. Journal of the Mechanics and Physics of Solids 143 (2020), 104070.
[53]
Claire Lestringant and Dennis M. Kochmann. 2020. Modeling of flexible beam networks and morphing structures by geometrically exact discrete beams. Journal of Applied Mechanics 87, 8 (05 2020), 081006. DOI:
[54]
Yang Liu, Helmut Pottmann, Johannes Wallner, Yong-Liang Yang, and Wenping Wang. 2006. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph. 25, 3 (jul 2006), 681?689. DOI:
[55]
A. E. H. Love. 1944. A Treatise on the Mathematical Theory of Elasticity (4th ed.). Dover Publications, New York.
[56]
David G. Luenberger. 1973. Introduction to Linear and Nonlinear Programming (3rd ed.). Addison-Wesley, New York.
[57]
A. I. Lurie. 2005. Theory of Elasticity. Springer, Berlin.
[58]
L. Mahadevan and J. B. Keller. 1993. The shape of a M?bius band. Proc. R. Soc. Lond. A 440, 1908 (1993), 149?162. DOI:
[59]
A. G. M. Michell. 1899. Elastic stability of long beams under transverse forces. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 48, 292 (1899), 298?309.
[60]
J. T. Miller, T. Su, J. Pabon, N. Wicks, K. Bertoldi, and P.M. Reis. 2015. Buckling of a thin elastic rod inside a horizontal cylindrical constraint. Extreme Mechanics Letters 3 (2015), 36?44. DOI:
[61]
Alexander Moore and Timothy Healey. 2019. Computation of elastic equilibria of complete M?bius bands and their stability. Mathematics and Mechanics of Solids 24, 4 (2019), 939?967. DOI:
[62]
P. M. Naghdi. 1963. Foundations of elastic shell theory. In Progress in Solid Mechanics, I. N. Sneddon and R. Hill (Eds.). Vol. 4. North-Holland, Amsterdam, 1?90.
[63]
R. Narain, A. Samii, and J. O?Brien. 2012. Adaptive anisotropic remeshing for cloth simulation. ACM Trans. Graph. 31, 6, Article 152 (Nov. 2012), 10 pages. DOI:
[64]
S?bastien Neukirch and Basile Audoly. 2021. A convenient formulation of Sadowsky?s model for elastic ribbons. Proc. R. Soc. A 477, 2255 (2021), 20210548. DOI:
[65]
Jorge Nocedal, Andreas W?chter, and Richard A. Waltz. 2009. Adaptive barrier update strategies for nonlinear interior methods. SIAM Journal on Optimization 19, 4 (2009), 1674?1693. DOI:
[66]
D. Pai. 2002. Strands: Interactive simulation of thin solids using cosserat models. Computer Graphics Forum (Proc. Eurographics?02) 21, 3 (2002), 347?352.
[67]
Zherong Pan, Jin Huang, and Hujun Bao. 2016. Modelling developable ribbons using ruling bending coordinates. Computing Research Repository abs/1603.04060 (2016), 1?9. arXiv:1603.04060. Retrieved from https://arxiv.org/abs/1603.04060
[68]
H. Pham. 2006. System Software Reliability. Springer, London.
[69]
Nico Pietroni, Corentin Dumery, Raphael Falque, Mark Liu, Teresa Vidal-Calleja, and Olga Sorkine-Hornung. 2022. Computational pattern making from 3D garment models. ACM Transactions on Graphics 41, 4 (2022), 157:1?14.
[70]
A. Pocheau and B. Roman. 2004. Uniqueness of solutions for constrained Elastica. Physica D: Nonlinear Phenomena 192, 3-4 (2004), 161?186.
[71]
Helmut Pottmann and Gerald Farin. 1995. Developable rational B?zier and B-spline surfaces. Computer Aided Geometric Design 12, 5 (1995), 513?531. DOI:
[72]
Xavier Provot. 1995. Deformation constraints in a mass-spring model to describe rigid cloth behaviour. In Proceedings of Graphics Interface?95 (GI?95). Canadian Human-Computer Communications Society, Toronto, Ontario, Canada, 147?154. Retrieved from http://graphicsinterface.org/wp-content/uploads/gi1995-17.pdf
[73]
Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. 2018a. Discrete geodesic nets for modeling developable surfaces. ACM Trans. Graph. 37, 2, Article 16 (feb 2018), 17 pages. DOI:
[74]
Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. 2018b. The shape space of discrete orthogonal geodesic nets. ACM Transactions on Graphics (proceedings of ACM SIGGRAPH ASIA) 37, 6, Article 228 (2018), 17 pages. DOI:
[75]
E. Reissner. 1989. Lateral buckling of beams. Computers & Structures 33, 5 (1989), 1289?1306.
[76]
Yingying Ren, Julian Panetta, Tian Chen, Florin Isvoranu, Samuel Poincloux, Christopher Brandt, Alison Martin, and Mark Pauly. 2021. 3D weaving with curved ribbons. ACM Trans. Graph. 40, 4, Article 127 (July 2021), 15 pages. DOI:
[77]
B. Roman and A. Pocheau. 1999. Buckling cascade of thin plates: Forms, constraints and similarity. Europhysics Letters (EPL) 46, 5 (1999), 602?608. DOI:
[78]
B. Roman and A. Pocheau. 2002. Postbuckling of bilaterally constrained rectangular thin plates. Journal of the Mechanics and Physics of Solids 50 (2002), 2379?2401. DOI:
[79]
Victor Romero, Micka?l Ly, Abdullah-Haroon Rasheed, Rapha?l Charrondi?re, Arnaud Lazarus, S?bastien Neukirch, and Florence Bertails-Descoubes. 2021. Physical validation of simulators in Computer Graphics: A new framework dedicated to slender elastic structures and frictional contact. ACM Transactions on Graphics 40, 4 (2021), 1?18. Retrieved from https://hal.inria.fr/hal-03217459
[80]
Michael Sadowsky. 1929. Die differentialgleichungen des M?BIUSschen bandes. Jahresbericht der Deutschen Mathematiker-Vereinigung 39 (1929), 49?51.
[81]
Tomohiko G. Sano and Hirofumi Wada. 2018. Snap-buckling in asymmetrically constrained elastic strips. Phys. Rev. E 97, 1 (Jan. 2018), 013002. DOI:
[82]
Tomohiko G. Sano and Hirofumi Wada. 2019. Twist-induced snapping in a bent elastic rod and ribbon. Phys. Rev. Lett. 122, 11 (2019), 114301. DOI:
[83]
Camille Schreck, Damien Rohmer, Stefanie Hahmann, Marie-Paule Cani, Shuo Jin, Charlie C. L. Wang, and Jean-Francis Bloch. 2015. Non-smooth developable geometry for interactively animating paper crumpling. ACM Transactions on Graphics 35, 1 (Dec. 2015), 18. DOI:
[84]
G. E. Schwarz. 1990. The dark side of the moebius strip. The American Mathematical Monthly 97, 10 (1990), 890?897.
[85]
Zhongwei Shen, Jin Huang, Wei Chen, and Hujun Bao. 2015. Geometrically exact simulation of inextensible ribbon. Computer Graphics Forum 34, 7 (2015), 145?154. DOI:
[86]
R. T. Shield. 1992. Bending of a beam or wide strip. Quarterly Journal of Mechanics and Applied Mathematics 45, 4 (1992), 567?573.
[87]
Madlaina Signer, Alexandra Ion, and Olga Sorkine-Hornung. 2021. Developable metamaterials: Mass-fabricable metamaterials by laser-cutting elastic structures. In CHI?21: Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems. Association for Computing Machinery, New York, NY, 1?13. DOI:
[88]
Justin Solomon, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2012. Flexible developable surfaces. Comput. Graph. Forum 31, 5 (aug 2012), 1567?1576. DOI:
[89]
Valentin Sonneville and Olivier Br?ls. 2014. A formulation on the special euclidean group for dynamic analysis of multibody systems. Journal of Computational and Nonlinear Dynamics 9, 4 (2014), 041002. DOI:
[90]
J. Spillmann and M. Teschner. 2007. CoRdE: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects. In ACM SIGGRAPH – EG Symposium on Computer Animation (SCA?07). ACM-EG SCA, San Diego, CA, 63?72.
[91]
E. L. Starostin and G. H. M. van der Heijden. 2007. The shape of a M?bius strip. Nature Materials 6, 8 (2007), 563. DOI:
[92]
E. L. Starostin and G. H. M. van der Heijden. 2015. Equilibrium shapes with stress localisation for inextensible elastic M?bius and other strips. Journal of Elasticity 119, 1 (2015), 67?112.
[93]
Chengcheng Tang, Pengbo Bo, Johannes Wallner, and Helmut Pottmann. 2016. Interactive design of developable surfaces. ACM Trans. Graph. 35, 2, Article 12 (jan 2016), 12 pages. DOI:
[94]
Demetri Terzopoulos, John Platt, Alan Barr, and Kurt Fleischer. 1987. Elastically deformable models. ACM Transactions on Graphics 21, 4 (1987), 205?214. DOI:
[95]
Bernhard Thomaszewski, Simon Pabst, and Wolfgang Stra?er. 2009. Continuum-based strain limiting. Computer Graphics Forum 28, 2 (2009), 569?576. DOI:
[96]
J. M. T. Thompson, M. Silveira, G. H. M. van der Heijden, and M. Wiercigroch. 2012. Helical post-buckling of a rod in a cylinder: With applications to drill-strings. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 468, 2142 (2012), 1591?1614.
[97]
Russell E. Todres. 2015. Translation of W. Wunderlich?s ?On a developable M?bius band?. Journal of Elasticity 119, 1 (2015), 23?34.
[98]
Petia Svetomirova Tzokova. 2020. Confined wrinkling of thin elastic rods, sheets and cones. Ph.D. Dissertation. University of Cambridge. Retrieved from www.repository.cam.ac.uk/handle/1810/310886
[99]
J. Vekhter, J. Zhuo, L. Fandino, Q. Huang, and E. Vouga. 2019. Weaving geodesic foliations. ACM Trans. Graph. 38, 4, Article 34 (July 2019), 22 pages. DOI:
[100]
Andreas W?chter and Lorenz T. Biegler. 2006. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming 106, 1 (2006), 25?57. DOI:
[101]
Thomas Wolf, Victor Cornill?re, and Olga Sorkine-Hornung. 2021. Physically-based book simulation with freeform developable surfaces. Computer Graphics Forum (proceedings of EUROGRAPHICS 2021) 40, 2 (2021), 449?460. DOI:
[102]
W. Wunderlich. 1962. ?ber ein abwickelbares M?biusband. Monatshefte f?r Mathematik 66, 3 (1962), 276?289. DOI:
[103]
Tian Yu, Ignacio Andrade-Silva, Marcelo A. Dias, and J.A. Hanna. 2021. Cutting holes in bistable folds. Mechanics Research Communications 124 (2021), 103700. DOI:
