“Discrete elastic rods” by Bergou, Wardetzky, Robinson, Audoly and Grinspun

  • ©Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun

Conference:


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Title:

    Discrete elastic rods

Presenter(s)/Author(s):



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.

References:


    1. 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
    2. Audoly, B., Clauvelin, N., and Neukirch, S. 2007. Elastic knots. Physical Review Letters 99, 16, 164301.Google ScholarCross Ref
    3. Audoly, B., Clauvelin, N., and Neukirch, S. 2008. Instabilities of elastic knots under twist. In preparation.Google Scholar
    4. 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
    5. 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
    6. Brown, J., Latombe, J.-C., and Montgomery, K. 2004. Real-time knot-tying simulation. Vis. Comp. V20, 2, 165–179. Google ScholarDigital Library
    7. 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
    8. 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
    9. de Vries, R. 2005. Evaluating changes of writhe in computer simulations of supercoiled DNA. J. Chem. Phys. 122, 064905.Google ScholarCross Ref
    10. 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
    11. Fuller, F. B. 1971. The writhing number of a space curve. PNAS 68, 4, 815–819.Google ScholarCross Ref
    12. 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
    13. Goldenthal, R., Harmon, D., Fattal, R., Bercovier, M., and Grinspun, E. 2007. Efficient simulation of inextensible cloth. In ACM TOG, 49. Google ScholarDigital Library
    14. Goldstein, R. E., and Langer, S. A. 1995. Nonlinear dynamics of stiff polymers. Phys. Rev. Lett. 75, 6 (Aug), 1094–1097.Google ScholarCross Ref
    15. 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
    16. Goriely, A. 2006. Twisted elastic rings and the rediscoveries of Michell’s instability. Journal of Elasticity 84, 281–299.Google ScholarCross Ref
    17. 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
    18. Grégoire, M., and Schömer, E. 2007. Interactive simulation of one-dimensional flexible parts. CAD 39, 8, 694–707. Google ScholarDigital Library
    19. Grinspun, E., Ed. 2006. Discrete differential geometry: an applied introduction. ACM SIGGRAPH 2006 Courses.Google ScholarDigital Library
    20. 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
    21. Hadap, S. 2006. Oriented strands: dynamics of stiff multi-body system. In SCA ’06, 91–100. Google ScholarDigital Library
    22. 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
    23. Hanson, A. J. 2006. Visualizing Quaternions. MK/Elsevier.Google Scholar
    24. Klapper, I. 1996. Biological applications of the dynamics of twisted elastic rods. J. Comp. Phys. 125, 2, 325–337. Google ScholarDigital Library
    25. Lanczos, C. 1986. The Variational Principles of Mechanics, 4th ed. Dover.Google Scholar
    26. Langer, J., and Singer, D. A. 1996. Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review 38, 4, 605–618. Google ScholarDigital Library
    27. 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
    28. Lin, C.-C., and Schwetlick, H. R. 2004. On the geometric flow of Kirchhoff elastic rods. SJAM 65, 2, 720–736.Google Scholar
    29. Maddocks, J. H., and Dichmann, D. J. 1994. Conservation laws in the dynamics of rods. Journal of elasticity 34, 83–96.Google ScholarCross Ref
    30. Maddocks, J. H. 1984. Stability of nonlinearly elastic rods. Archive for Rational Mechanics and Analysis 85, 4, 311–354.Google ScholarCross Ref
    31. Pai, D. K. 2002. STRANDS: interactive simulation of thin solids using Cosserat models. CGF 21, 3.Google ScholarCross Ref
    32. Phillips, J., Ladd, A., and Kavraki, L. 2002. Simulated knot tying. In Proceedings of ICRA 2002, vol. 1, 841–846.Google Scholar
    33. 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
    34. Rubin, M. B. 2000. Cosserat Theories: Shells, Rods and Points. Kluwer Academic Publishers.Google ScholarCross Ref
    35. Shabana, A. 2001. Computational Dynamics, 2nd ed. John Wiley & Sons, New York.Google Scholar
    36. Smith, R., 2008. Open dynamics engine. http://www.ode.org/.Google Scholar
    37. Sommerfeld, A. 1964. Mechanics of deformable bodies, vol. 2 of Lectures on theoretical physics. Academic Press.Google Scholar
    38. 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
    39. Spillmann, J., and Teschner, M. 2008. An adaptative contact model for the robust simulation of knots. CGF 27.Google Scholar
    40. Terzopoulos, D., Platt, J., Barr, A., and Fleischer, K. 1987. Elastically deformable models. In Proceedings of SIGGRAPH ’87, 205–214. Google ScholarDigital Library
    41. Theetten, A., Grisoni, L., Andriot, C., and Barsky, B. 2006. Geometrically exact dynamic splines. Tech. rep., INRIA.Google Scholar
    42. 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
    43. 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
    44. 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
    45. 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


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