×

Numerical modeling of inextensible elastic ribbons with curvature-based elements. (English) Zbl 1442.74098

Summary: We propose a robust and efficient numerical model to compute stable equilibrium configurations of clamped elastic ribbons featuring arbitrarily curved natural shapes. Our spatial discretization scheme relies on elements characterized by a linear normal curvature and a quadratic geodesic torsion with respect to arc length. Such a high-order discretization allows for a great diversity of kinematic representations, while guaranteeing the surface of the ribbon to remain perfectly inextensible. Stable equilibria are calculated by minimizing the sum of the gravitational and elastic energies of the ribbon, under a developability constraint. Our algorithm compares favorably to standard shooting and collocation methods, as well as to experiments. It furthermore shows significant differences in behavior compared to numerical models for thin elastic rods, while yielding a substantial speed-up compared to a more general thin elastic shell simulator. These results confirm the benefit of designing a special numerical model dedicated to ribbons.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34A26 Geometric methods in ordinary differential equations
49M15 Newton-type methods
49M25 Discrete approximations in optimal control
53Z30 Applications of differential geometry to engineering
65D17 Computer-aided design (modeling of curves and surfaces)
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Sadowsky, M., Die Differentialgleichungen des MÖBIUSschen bandes, Jahresber. Dtsch. Math. -Ver., 49-51 (1929), translated in [2] · JFM 56.0614.21
[2] (Fosdick, R.; Fried, E., The Mechanics of Ribbons and Moebius Bands (2015), Springer), Previously published in Journal of Elasticity Volume 119, 2015 · Zbl 1320.74006
[3] Audoly, B.; Pomeau, Y., Elasticity and Geometry: from Hair Curls to the Nonlinear Response of Shells (2010), Oxford University Press · Zbl 1223.74001
[4] Chopin, J.; Kudrolli, A., Helicoids, wrinkles, and loops in twisted ribbons, Phys. Rev. Lett., 111, Article 174302 pp. (2013)
[5] Moebius, F. A., Über die Bestimmung des Inhaltes eines Polyeders (On the determination of the volume of a polyhedron), Ber. Verh. Sächs. Ges. Wiss., 17, 31-68 (1865), see also Gesammelte Werke, Band II (Collected Works, vol. II), p. 484. Hirzel, Leipzig (1886)
[6] Hinz, D. F.; Fried, E., Translation of Michael Sadowsky’s Paper ’An Elementary Proof for the Existence of a Developable Möbius Band and the Attribution of the Geometric Problem to a Variational Problem’, J. Elasticity, 119, 1, 3-6 (2015) · Zbl 1414.74017
[7] Wunderlich, W., Über ein abwickelbares Möbiusband, Monatsh. Math., 66, 3, 276-289 (1962) · Zbl 0105.14802
[8] Starostin, E. L.; van der Heijden, G. H.M., The shape of a Möbius strip, Nature Mater., 6, 563 (2007)
[9] Moore, A.; Healey, T., Computation of elastic equilibria of complete Möbius bands and their stability, Math. Mech. Solids, 24, 4, 939-967 (2019) · Zbl 1446.74154
[10] Doedel, E. J.; Oldeman, B. E., AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential EquationsTech. Rep (2012), Concordia University
[11] Dias, M. A.; Audoly, B., “Wunderlich, Meet Kirchhoff”: A general and unified description of elastic ribbons and thin rods, J. Elasticity, 119, 1, 49-66 (2015) · Zbl 1455.74058
[12] Audoly, B.; Seffen, K. A., Buckling of naturally curved elastic strips: The ribbon model makes a difference, J. Elasticity, 119, 1, 293-320 (2015) · Zbl 1423.74455
[13] Moulton, D. E.; Grandgeorge, P.; Neukirch, S., Stable elastic knots with no self-contact, J. Mech. Phys. Solids, 116, 33-53 (2018)
[14] Sano, T. G.; Wada, H., Twist-induced snapping in a bent elastic rod and ribbon, Phys. Rev. Lett., 122, Article 114301 pp. (2019)
[15] Simo, J. C., A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Comput. Methods Appl. Mech. Engrg., 49, 1, 55-70 (1985) · Zbl 0583.73037
[16] Borri, M.; Bottasso, C., An intrinsic beam model based on a helicoidal approximation-Part I: Formulation, Internat. J. Numer. Methods Engrg., 37, 13, 2267-2289 (1994), URL http://dx.doi.org/10.1002/nme.1620371308 · Zbl 0806.73028
[17] Crisfield, M. A.; Jelenić, G., Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation, Proc. R. Soc. A, 455, 1983, 1125-1147 (1998) · Zbl 0926.74062
[18] Pai, D., Strands: Interactive simulation of thin solids using Cosserat models, Comput. Graph. Forum (Proc. Eurographics’02), 21, 3, 347-352 (2002)
[19] Bertails, F.; Audoly, B.; Cani, M.-P.; Querleux, B.; Leroy, F.; Lévêque, J.-L., Super-helices for predicting the dynamics of natural hair, ACM Trans. Graph., 25, 1180-1187 (2006)
[20] Spillmann, J.; Becker, M.; Teschner, M., Non-iterative computation of contact forces for deformable objects, J. WSCG, 15, 33-40 (2007)
[21] Goyal, S.; Perkins, N.; Lee, C., Non-linear dynamic intertwining of rods with self-contact, Int. J. Non-Linear Mech., 43, 1, 65-73 (2008), URL http://www.sciencedirect.com/science/article/pii/S0020746207001989 · Zbl 1203.74081
[22] Bergou, M.; Wardetzky, M.; Robinson, S.; Audoly, B.; Grinspun, E., Discrete elastic rods, ACM Trans. Graph. (Proc. ACM SIGGRAPH’08), 27, 3, 1-12 (2008), URL http://www.cs.columbia.edu/cg/rods/
[23] Lang, H.; Linn, J.; Arnold, M., Multi-body dynamics simulation of geometrically exact Cosserat rods, Multibody Syst. Dyn., 25, 3, 285-312 (2011), URL https://doi.org/10.1007/s11044-010-9223-x · Zbl 1271.74264
[24] Sonneville, V.; Cardona, A.; Brüls, O., Geometrically exact beam finite element formulated on the special Euclidean group SE(3), Comput. Methods Appl. Mech. Engrg., 268, 451-474 (2014), URL http://www.sciencedirect.com/science/article/pii/S0045782513002600 · Zbl 1295.74050
[25] Renda, F.; Boyer, F.; Dias, J.; Seneviratne, L., Discrete Cosserat approach for multisection soft manipulator dynamics, IEEE Trans. Robot., 34, 6, 1518-1533 (2018)
[26] Shen, Z.; Huang, J.; Chen, W.; Bao, H., Geometrically exact simulation of inextensible ribbon, Comput. Graph. Forum, 34, 7, 145-154 (2015)
[27] Pan, Z.; Huang, J.; Bao, H., Modelling developable ribbons using ruling bending coordinates, Comput. Res. Repos. (2016), URL https://arxiv.org/abs/1603.04060
[28] Casati, R.; Bertails-Descoubes, F., Super space clothoids, ACM Trans. Graph., 32, 4, 48 (2013) · Zbl 1305.68215
[29] Wächter, A.; Biegler, L. T., On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106, 1, 25-57 (2006) · Zbl 1134.90542
[30] Hinz, D. F.; Fried, E., Translation of Michael Sadowsky’s Paper ‘The Differential Equations of the Möbius Band’, J. Elasticity, 119, 1, 19-22 (2015) · Zbl 1414.74019
[31] Casati, R., Quelques contributions à la modélisation numérique de structures élancées pour l’informatique graphique (2015), Université Grenoble Alpes, http://www.theses.fr/2015GREAM053/document
[32] Demailly, J.-P., Analyse numérique et équations différentielles (2006), EDP Sciences
[33] Adrianova, L. Y., Introduction to Linear Systems of Diferential Equations, Translations of Mathematical Monographs (1995), American Mathematical Society · Zbl 0844.34001
[34] Bertails, F., Linear time super-helices, Comput. Graph. Forum (Proc. Eurographics’09), 28, 2 (2009)
[35] Neher, M., An enclosure method for the solution of linear ODEs with polynomial coefficients, Numer. Funct. Anal. Optim., 20, 779-803 (1999) · Zbl 0936.65084
[36] Goldberg, D., What every computer scientist should know about floating-point arithmetic, ACM Comp. Surv., 23, 5-48 (1991)
[37] Nocedal, J.; Wright, S., Numerical Optimization (2006), Springer: Springer New York, NY, USA · Zbl 1104.65059
[38] Broyden, C. G., The convergence of a class of double-rank minimization algorithms 1. General considerations, IMA J. Appl. Math., 6, 1, 76-90 (1970) · Zbl 0223.65023
[39] Antman, S. S., Nonlinear Problems of Elasticity (2004), Springer-Verlag: Springer-Verlag New York · Zbl 0820.73002
[40] Lurie, A. I., (Theory of Elasticity. Theory of Elasticity, Foundations of Engineering Mechanics (2005), Springer)
[41] Antman, S. S., Kirchhoff’s problem for nonlinearly elastic rods, Quart. Appl. Math., 32, 3, 221-240 (1974) · Zbl 0302.73031
[42] Naghdi, P. M., Foundations of elastic shell theory, (Sneddon, I. N.; Hill, R., Progress in Solid Mechanics, Vol. 4 (1963), North-Holland), 1-90 · Zbl 0116.15902
[43] Simo, J.; Fox, D., On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization, Comput. Methods Appl. Mech. Engrg., 72, 3, 267-304 (1989) · Zbl 0692.73062
[44] Arnold, D. N.; Brezzi, F., Locking-free finite element methods for shells, Math. Comp., 66, 1-14 (1997) · Zbl 0854.65095
[45] Hale, J. S.; Brunetti, M.; Bordas, S. P.; Maurini, C., Simple and extensible plate and shell finite element models through automatic code generation tools, Comput. Struct., 209, 163-181 (2018)
[46] Wolfram Research, J. S., Mathematica (2019), Version 11, Champaign, IL, USA
[47] Doedel, E.; Keller, H. B.; Kernevez, J. P., Numerical analysis and control of bifurcation problems (I) bifurcation in finite dimensions, Int. J. Bifurcation Chaos, 1, 3, 493-520 (1991) · Zbl 0876.65032
[48] Doedel, E.; Keller, H. B.; Kernevez, J. P., Numerical analysis and control of bifurcation problems (II): Bifurcation in infinite dimensions, Int. J. Bifurcation Chaos, 01, 04, 745-772 (1991) · Zbl 0876.65060
[49] Duclaux, V., Pulmonary Occlusions, Eyelid Entropion and Aneurysm: A Physical Insight in Physiology (2006), Université de Provence - Aix-Marseille I, URL https://tel.archives-ouvertes.fr/tel-00130610
[50] Arriagada, O. A.; Massiera, G.; Abkarian, M., Curling and rolling dynamics of naturally curved ribbons, Soft Matter, 10, 3055-3065 (2014)
[51] Zhang, Z., A flexible new technique for Camera Calibration, IEEE Trans. Pattern Anal. Mach. Intell., 22, 11, 1330-1334 (2000)
[52] Reissner, E., On one-dimensional finite-strain beam theory: The plane problem, Z. Angew. Math. Phys., 23, 795-804 (1972) · Zbl 0248.73022
[53] Cowper, G. R., The shear coefficient in Timoshenko’s beam theory, J. Appl. Mech., 33, 2, 335-340 (1966) · Zbl 0151.37901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.