“Discrete viscous threads” by Bergou, Audoly, Vouga, Wardetzky and Grinspun

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 symmetry, which sheds light on the role of time-parallel transport in eliminating—without approximation—all but an O(n) band of entries of the physical system’s energy Hessian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluid-mechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics.

1. Andreassen, E., Gundersen, E., Hinrichsen, E. L., and Langtangen, H. P. 1997. Numerical Methods and Software Tools in Industrial Mathematics. Birkhäueser, Boston, ch. A mathematical model for the melt spinning of polymer fibers, 195–212.Google Scholar
2. Bargteil, A. W., Wojtan, C., Hodgins, J. K., and Turk, G. 2007. A Finite Element Method for Animating Large Viscoplastic Flow. ACM TOG 26, 3 (Jul), 16:1–16:8. Google ScholarDigital Library
3. Batty, C., and Bridson, R. 2008. Accurate Viscous Free Surfaces for Buckling, Coiling, and Rotating Liquids. In SCA ’08, 219–226. Google ScholarDigital Library
4. Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., and Grinspun, E. 2008. Discrete Elastic Rods. ACM TOG 27, 3 (Aug), 63:1–63:12. Google ScholarDigital Library
5. 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. ACM TOG 25, 3 (Jul), 1180–1187. Google ScholarDigital Library
6. Bonito, A., Picasso, M., and Laso, M. 2006. Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phys. 215, 2, 691–716. Google ScholarDigital Library
7. Boyer, F., and Primault, D. 2004. Finite element of slender beams in finite transformations: a geometrically exact approach. Int. J. Numer. Methods Eng. 59, 5, 669–702.Google ScholarCross Ref
8. Bridson, R., and Müller-Fischer, M. 2007. Fluid Simulation. SIGGRAPH 2007 Course Notes. Google ScholarDigital Library
9. Chang, Y., Bao, K., Liu, Y., Zhu, J., and Wu, E. 2009. A particle-based method for viscoelastic fluids animation. VRST ’09 (Nov), 111–117. Google ScholarDigital Library
10. Chentanez, N., Alterovitz, R., Ritchie, D., Cho, L., Hauser, K. K., Goldberg, K., Shewchuk, J. R., and O’Brien, J. F. 2009. Interactive Simulation of Surgical Needle Insertion and Steering. ACM TOG 28, 3 (Jul), 88:1–88:10. Google ScholarDigital Library
11. Chiu-Webster, S., and Lister, J. R. 2006. The fall of a viscous thread onto a moving surface: a ‘fluid-mechanical sewing machine’. J. Fluid Mech. 569, 89–111.Google ScholarCross Ref
12. Clavet, S., Beaudoin, P., and Poulin, P. 2005. Particle-based Viscoelastic Fluid Simulation. In SCA ’05. Google ScholarDigital Library
13. Desbrun, M., and Gascuel, M. 1996. Smoothed particles: A new paradigm for animating highly deformable bodies. Computer Animation and Simulation (Jan), 61–76. Google ScholarDigital Library
14. Dewynne, J. N., Ockendon, J. R., and Wilmott, P. 1992. A systematic derivation of the leading-order equations for extensional flows in slender geometries. J. Fluid Mech. 24, 323–338.Google ScholarCross Ref
15. DiVerdi, S., Krishnaswamy, A., and Hadap, S. 2010. Industrial-Strength Painting with a Bristle Brush Simulation. submitted.Google Scholar
16. Eggers, J., and Dupont, T. F. 1994. Drop formation in a one-dimensional approximation of the Navier-Stokes equation. J. Fluid Mech. 262, 205–221.Google ScholarCross Ref
17. Entov, V. M., and Yarin, A. L. 1984. The dynamics of thin liquid jets in air. J. Fluid Mech. 140, 91–111.Google ScholarCross Ref
18. Fedkiw, R., Stam, J., and Jensen, H. W. 2001. Visual Simulation of Smoke. In SIGGRAPH 2001, 15–22. Google ScholarDigital Library
19. Foster, N., and Fedkiw, R. 2001. Practical Animation of Liquids. In SIGGRAPH 2001, 23–30. Google ScholarDigital Library
20. Foster, N., and Metaxas, D. 1997. Modeling the Motion of a Hot, Turbulent Gas. In SIGGRAPH 97, 181–188. Google ScholarDigital Library
21. Gerszewski, D., Bhattacharya, H., and Bargteil, A. W. 2009. A Point-based Method for Animating Elastoplastic Solids. In SCA ’09. Google ScholarDigital Library
22. Goktekin, T. G., Bargteil, A. W., and O’Brien, J. F. 2004. A method for animating viscoelastic fluids. ACM TOG 23, 3 (Aug), 463–468. Google ScholarDigital Library
23. Grégoire, M., and Schömer, E. 2007. Interactive simulation of one-dimensional flexible parts. Comput.-Aided Des. 39, 8, 694–707. Google ScholarDigital Library
24. Hauth, M., Etzmuss, O., and Strasser, W. 2003. Analysis of numerical methods for the simulation of deformable models. Vis. Comp. 19, 7–8, 581–600.Google ScholarDigital Library
25. Hong, J.-M., and Kim, C.-H. 2005. Discontinuous fluids. ACM TOG 24, 3 (Jul), 915–920. Google ScholarDigital Library
26. Irving, G. 2007. Methods for the physically based simulation of solids and fluids. PhD thesis, Stanford University.Google Scholar
27. Kaldor, J. M., James, D. L., and Marschner, S. 2010. Efficient Yarn-based Cloth with Adaptive Contact Linearization. ACM TOG 29, 4 (Jul). Google ScholarDigital Library
28. Kharevych, L., Yang, W., Tong, Y., Kanso, E., Marsden, J. E., Schröder, P., and Desbrun, M. 2006. Geometric, Variational Integrators for Computer Animation. In SCA ’06, 43–51. Google ScholarDigital Library
29. Kim, D., Song, O.-Y., and Ko, H.-S. 2009. Stretching and Wiggling Liquids. ACM TOG 28, 5 (Dec), 120:1–120:7. Google ScholarDigital Library
30. Kirchhoff, G. 1859. Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. Journal für die reine und angewandte Mathematik 56, 285–313.Google ScholarCross Ref
31. Landau, L. D., and Lifshitz, E. M. 1981. Theory of Elasticity (Course of Theoretical Physics), 2nd ed. Pergamon Press.Google Scholar
32. Langer, J., and Singer, D. 1996. Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review, 605–618. Google ScholarDigital Library
33. Le Merrer, M., Seiwert, J., Quéré, D., and Clanet, C. 2008. Shapes of hanging viscous filaments. EPL 84, 56004.Google ScholarCross Ref
34. Lee, S., Olsen, S., and Gooch, B. 2006. Interactive 3D fluid jet painting. NPAR ’06 (Jun), 97–104. Google ScholarDigital Library
35. Losasso, F., Gibou, F., and Fedkiw, R. 2004. Simulating water and smoke with an octree data structure. ACM TOG 23, 3 (Aug), 457–462. Google ScholarDigital Library
36. Miller, G., and Pearce, A. 1989. Globular dynamics: A connected particle system for animating viscous fluids. COMP. GRAPH. (Jan), 305–309.Google Scholar
37. Morris, S. W., Dawes, J. H. P., Ribe, N. M., and Lister, J. R. 2008. Meandering instability of a viscous thread. Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 77, 6, 066218.Google Scholar
38. Müller, M., Charypar, D., and Gross, M. 2003. Particle-based fluid simulation for interactive applications. In SCA ’03, 154–159. Google ScholarDigital Library
39. Müller, M., Keiser, R., Nealen, A., Pauly, M., Gross, M., and Alexa, M. 2004. Point based animation of elastic, plastic and melting objects. In SCA ’04, 141–151. Google ScholarDigital Library
40. Nealen, A., Müller, M., Keiser, R., Boxerman, E., and Carlson, M. 2006. Physically Based Deformable Models in Computer Graphics. CGF 25, 4, 809–836.Google ScholarCross Ref
41. O’Brien, J. F., Bargteil, A. W., and Hodgins, J. K. 2002. Graphical Modeling and Animation of Ductile Fracture. ACM TOG 21, 3 (Jul), 291–294. Google ScholarDigital Library
42. Oishi, C. M., Tomé, M. F., Cuminato, J. A., and McKee, S. 2008. An implicit technique for solving 3D low Reynolds number moving free surface flows. J. Comput. Phys. 227, 16, 7446–7468. Google ScholarDigital Library
43. Pai, D. K. 2002. STRANDS: Interactive Simulation of Thin Solids using Cosserat Models. CGF 21, 3, 347–352.Google ScholarCross Ref
44. Panda, S., Marheineke, N., and Wegener, R. 2008. Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers. Math. Meth. Appl. Sci. 31, 10, 1153–1173.Google ScholarCross Ref
45. Radovitzky, R., and Ortiz, M. 1999. Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Comput. Methods Appl. Mech. Eng 172, 1–4, 203–240.Google ScholarCross Ref
46. Rafiee, A., Manzari, M. T., and Hosseini, M. 2007. An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows. Int. J. Non Linear Mech. 42, 10, 1210–1223.Google ScholarCross Ref
47. Ribe, N. M., Huppert, H. E., Hallworth, M. A., Habibi, M., and Bonn, D. 2006. Multiple coexisting states of liquid rope coiling. J. Fluid Mech. 555, 1, 275–297.Google ScholarCross Ref
48. Ribe, N. M. 2004. Coiling of viscous jets. Proc. Math., Phys. and Eng. Sci., 3223–3239.Google ScholarCross Ref
49. Skorobogatiy, M., and Mahadevan, L. 2000. Folding of viscous sheets and filaments. EPL 52, 5, 532–538.Google ScholarCross Ref
50. 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
51. Spillmann, J., and Teschner, M. 2008. An Adaptive Contact Model for the Robust Simulation of Knots. CGF 27, 2, 497–506.Google ScholarCross Ref
52. Stam, J. 1999. Stable Fluids. In SIGGRAPH 99, 121–128. Google ScholarDigital Library
53. Steele, K., Cline, D., Egbert, P., and Dinerstein, J. 2004. Modeling and rendering viscous liquids. CAVW (Jan), 183–192. Google ScholarDigital Library
54. Stora, D., Agliati, P.-O., Cani, M.-P., Neyret, F., and Gascuel, J.-D. 1999. Animating lava flows. GI ’99 (Jan), 203–210. Google ScholarDigital Library
55. Strutt, J. W. 1945. Theory of Sound, vol. 2. Dover Publications.Google Scholar
56. Taylor, G. I. 1968. Instability of jets, threads, and sheets of viscous fluid. In Proc. 12th Intl Congr. Appl. Mech., Stanford, Springer, Ed., 382.Google Scholar
57. Terzopoulos, D., and Fleischer, K. 1988. Modeling Inelastic Deformation: Viscoelasticity, Plasticity, Fracture. In SIGGRAPH 88, 269–278. Google ScholarDigital Library
58. Terzopoulos, D., Platt, J., and Fleischer, K. 1991. Heating and melting deformable models. J. Visual. Comp. Animat. 2, 2, 68–73.Google ScholarCross Ref
59. Theetten, A., Grisoni, L., Duriez, C., and Merlhiot, X. 2007. Quasi-dynamic splines. In SPM ’07, ACM, New York, 409–414. Google ScholarDigital Library
60. Theetten, A., Grisoni, L., Andriot, C., and Barsky, B. 2008. Geometrically exact dynamic splines. Comput.-Aided Des. 40, 1, 35–48. Google ScholarDigital Library
61. Trouton, F. R. S. 1906. On the coefficient of viscous traction and its relation to that of viscosity. Proc. Royal Soc. London, A 77, 426–440.Google ScholarCross Ref
62. Witkin, A., and Baraff, D. 2001. Physically Based Modeling: Principles and Practice. SIGGRAPH 2001 Course Notes.Google Scholar
63. Wojtan, C., and Turk, G. 2008. Fast Viscoelastic Behavior with Thin Features. ACM TOG 27, 3 (Aug), 47:1–47:8. Google ScholarDigital Library