ABSTRACT
We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description---we represent the material frame by its angular deviation from the natural Bishop frame---as well as in the dynamical treatment---we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.
Supplemental Material
- Audoly, B., and Pomeau, Y. 2008. Elasticity and geometry: from hair curls to the nonlinear response of shells. Oxford University Press. To appear.Google Scholar
- Audoly, B., Clauvelin, N., and Neukirch, S. 2007. Elastic knots. Physical Review Letters 99, 16, 164301.Google ScholarCross Ref
- Audoly, B., Clauvelin, N., and Neukirch, S. 2008. Instabilities of elastic knots under twist. In preparation.Google Scholar
- Bertails, F., Audoly, B., Cani, M.-P., Querleux, B., Leroy, F., and Lévêque, J.-L. 2006. Super-helices for predicting the dynamics of natural hair. In ACM TOG, 1180--1187. Google ScholarDigital Library
- Bobenko, A. I., and Suris, Y. B. 1999. Discrete time lagrangian mechanics on lie groups, with an application to the lagrange top. Comm. in Mathematical Physics 204, 1, 147--188.Google ScholarCross Ref
- Brown, J., Latombe, J.-C., and Montgomery, K. 2004. Real-time knot-tying simulation. Vis. Comp. V20, 2, 165--179. Google ScholarDigital Library
- Chang, J., Shepherd, D. X., and Zhang, J. J. 2007. Cosserat-beam-based dynamic response modelling. Computer Animation and Virtual Worlds 18, 4--5, 429--436. Google ScholarCross Ref
- Choe, B., Choi, M. G., and Ko, H.-S. 2005. Simulating complex hair with robust collision handling. In SCA '05, 153--160. Google ScholarDigital Library
- de Vries, R. 2005. Evaluating changes of writhe in computer simulations of supercoiled DNA. J. Chem. Phys. 122, 064905.Google ScholarCross Ref
- Falk, R. S., and Xu, J.-M. 1995. Convergence of a second-order scheme for the nonlinear dynamical equations of elastic rods. SJNA 32, 4, 1185--1209. Google ScholarDigital Library
- Fuller, F. B. 1971. The writhing number of a space curve. PNAS 68, 4, 815--819.Google ScholarCross Ref
- Fuller, F. B. 1978. Decomposition of the linking number of a closed ribbon: a problem from molecular biology. PNAS 75, 8, 3557--3561.Google ScholarCross Ref
- Goldenthal, R., Harmon, D., Fattal, R., Bercovier, M., and Grinspun, E. 2007. Efficient simulation of inextensible cloth. In ACM TOG, 49. Google ScholarDigital Library
- Goldstein, R. E., and Langer, S. A. 1995. Nonlinear dynamics of stiff polymers. Phys. Rev. Lett. 75, 6 (Aug), 1094--1097.Google ScholarCross Ref
- Goriely, A., and Tabor, M. 1998. Spontaneous helix hand reversal and tendril perversion in climbing plants. Physical Review Letters 80, 7 (Feb), 1564--1567.Google ScholarCross Ref
- Goriely, A. 2006. Twisted elastic rings and the rediscoveries of Michell's instability. Journal of Elasticity 84, 281--299.Google ScholarCross Ref
- Goyal, S., Perkins, N. C., and Lee, C. L. 2007. Non-linear dynamic intertwining of rods with self-contact. International Journal of Non-Linear Mechanics In Press, Corrected Proof.Google Scholar
- Grégoire, M., and Schömer, E. 2007. Interactive simulation of one-dimensional flexible parts. CAD 39, 8, 694--707. Google ScholarDigital Library
- Grinspun, E., Ed. 2006. Discrete differential geometry: an applied introduction. ACM SIGGRAPH 2006 Courses.Google ScholarDigital Library
- Hadap, S., Cani, M.-P., Lin, M., Kim, T.-Y., Bertails, F., Marschner, S., Ward, K., and Kačić-Alesić, Z. 2007. Strands and hair: modeling, animation, and rendering. ACM SIGGRAPH 2007 Courses, 1--150. Google ScholarDigital Library
- Hadap, S. 2006. Oriented strands: dynamics of stiff multi-body system. In SCA '06, 91--100. Google ScholarDigital Library
- Hairer, E., Lubich, C., and Wanner, G. 2006. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Series in Comp. Math. Springer.Google Scholar
- Hanson, A. J. 2006. Visualizing Quaternions. MK/Elsevier.Google Scholar
- Klapper, I. 1996. Biological applications of the dynamics of twisted elastic rods. J. Comp. Phys. 125, 2, 325--337. Google ScholarDigital Library
- Lanczos, C. 1986. The Variational Principles of Mechanics, 4th ed. Dover.Google Scholar
- Langer, J., and Singer, D. A. 1996. Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review 38, 4, 605--618. Google ScholarDigital Library
- Lenoir, J., Cotin, S., Duriez, C., and Neumann, P. 2006. Interactive physically-based simulation of catheter and guidewire. Computer & Graphics 30, 3, 417--423. Google ScholarDigital Library
- Lin, C.-C., and Schwetlick, H. R. 2004. On the geometric flow of Kirchhoff elastic rods. SJAM 65, 2, 720--736.Google Scholar
- Maddocks, J. H., and Dichmann, D. J. 1994. Conservation laws in the dynamics of rods. Journal of elasticity 34, 83--96.Google ScholarCross Ref
- Maddocks, J. H. 1984. Stability of nonlinearly elastic rods. Archive for Rational Mechanics and Analysis 85, 4, 311--354.Google ScholarCross Ref
- Pai, D. K. 2002. STRANDS: interactive simulation of thin solids using Cosserat models. CGF 21, 3.Google ScholarCross Ref
- Phillips, J., Ladd, A., and Kavraki, L. 2002. Simulated knot tying. In Proceedings of ICRA 2002, vol. 1, 841--846.Google Scholar
- Press, W. H., Vetterling, W. T., Teukolsky, S. A., and Flannery, B. P. 2002. Numerical Recipes in C++: the art of scientific computing. Cambridge University Press. Google ScholarDigital Library
- Rubin, M. B. 2000. Cosserat Theories: Shells, Rods and Points. Kluwer Academic Publishers.Google ScholarCross Ref
- Shabana, A. 2001. Computational Dynamics, 2nd ed. John Wiley & Sons, New York.Google Scholar
- Smith, R., 2008. Open dynamics engine. http://www.ode.org/.Google Scholar
- Sommerfeld, A. 1964. Mechanics of deformable bodies, vol. 2 of Lectures on theoretical physics. Academic Press.Google Scholar
- Spillmann, J., and Teschner, M. 2007. Corde: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects. In SCA '07, 63--72. Google ScholarDigital Library
- Spillmann, J., and Teschner, M. 2008. An adaptative contact model for the robust simulation of knots. CGF 27.Google Scholar
- Terzopoulos, D., Platt, J., Barr, A., and Fleischer, K. 1987. Elastically deformable models. In Proceedings of SIGGRAPH '87, 205--214. Google ScholarDigital Library
- Theetten, A., Grisoni, L., Andriot, C., and Barsky, B. 2006. Geometrically exact dynamic splines. Tech. rep., INRIA.Google Scholar
- van der Heijden, G. H. M., and Thompson, J. M. T. 2000. Helical and localised buckling in twisted rods: A unified analysis of the symmetric case. Nonlinear Dynamics 21, 1, 71--99.Google ScholarCross Ref
- van der Heijden, G. H. M., Neukirch, S., Goss, V. G. A., and Thompson, J. M. T. 2003. Instability and self-contact phenomena in the writhing of clamped rods. Int. J. Mech. Sci. 45, 161--196.Google ScholarCross Ref
- Ward, K., Bertails, F., Kim, T.-Y., Marschner, S. R., Cani, M.-P., and Lin, M. C. 2007. A survey on hair modeling: Styling, simulation, and rendering. IEEE TVCG 13, 2 (Mar./Apr.), 213--234. Google ScholarDigital Library
- Yang, Y., Tobias, I., and Olson, W. K. 1993. Finite element analysis of DNA supercoiling. J. Chem. Phys. 98, 2, 1673--1686.Google ScholarCross Ref
Index Terms
- Discrete elastic rods
Recommendations
Discrete viscous threads
We present a continuum-based discrete model for thin threads of viscous fluid by drawing upon the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes space-time ...
Discrete elastic rods
We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach ...
Super-helices for predicting the dynamics of natural hair
Simulating human hair is recognized as one of the most difficult tasks in computer animation. In this paper, we show that the Kirchhoff equations for dynamic, inextensible elastic rods can be used for accurately predicting hair motion. These equations ...
Comments