“Schrödinger’s smoke”

  • ©Albert Chern, Felix Knöppel, Ulrich Pinkall, Peter Schröder, and Steffen Weißmann




    Schrödinger's smoke

Session/Category Title: FLUIDS SIMULATION




    We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, the fluid state is represented by a C2-valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type. The latter ensures that dynamics due to thin vortical structures, all important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity. We demonstrate its use in a variety of scenarios, compare it with experiments, and evaluate it against benchmark tests. A full implementation is included in the ancillary materials.


    1. Abraham, R., Marsden, J. E., and Ratiu, T. 2001. Manifolds, Tensor Analysis and Applications. No. 75 in Appl. Math. Sci. Springer.Google Scholar
    2. Al-Mohy, A. H., and Higham, N. J. 2011. Computing the Action of the Matrix Exponential with an Application to Exponential Integrators. SIAM J. Sci. Comp. 33, 2, 488–511. Google ScholarDigital Library
    3. Angelidis, A., and Neyret, F. 2005. Simulation of Smoke based on Vortex Filament Primitives. In Proc. Symp. Comp. Anim., ACM SIGGRAPH/Eurographics, 87–96. Google ScholarDigital Library
    4. Angot, P., Bruneau, C.-H., and Fabrie, P. 1999. A Penalization Method to Take into account Obstacles in Incompressible Viscous Flows. Num. Math. 81, 4, 497–520.Google ScholarCross Ref
    5. Arquis, E., and Caltagirone, J.-P. 1984. Sur les conditions hydrodynamiques au voisinage d’une interface milieu fluide-mileu poreux: application á convection naturelle. CR Acad. Sci. Paris, Série II 299, 1, 1–4.Google Scholar
    6. Bridson, R., Houriham, J., and Nordenstam, M. 2007. Curlnoise for Procedural Fluid Flow. ACM Trans. Graph. 26, 3, 46:1–3. Google ScholarDigital Library
    7. Brochu, T., Keeler, T., and Bridson, R. 2012. Linear-Time Smoke Animation with Vortex Sheet Meshes. In Proc. Symp. Comp. Anim., ACM SIGGRAPH/Eurographics, 87–95. Google ScholarDigital Library
    8. Carbou, G., and Fabrie, P. 2003. Boundary Layer for a Penalization Method for Viscous Incompressible Flow. Adv. Diff. Eq. 8, 12, 1453–1480.Google Scholar
    9. Cayley, A. 1845. On Certain Results Relating to Quaternions. Phil. Mag. 26, 141–145.Google Scholar
    10. Cheney, W., and Goldstein, A. A. 1959. Proximity Maps for Convex Sets. Proc. AMS 10, 3, 448–450.Google ScholarCross Ref
    11. Clebsch, A. 1859. Ueber die Integration der hydrodynamischen Gleichungen. J. Reine Angew. Math. 56, 1–10.Google ScholarCross Ref
    12. Cooper, N. R. 1999. Propagating Magnetic Vortex Rings in Ferromagnets. Phys. R. Lett. 82, 7, 1554–1557.Google ScholarCross Ref
    13. Cottet, G.-H., and Koumoutsakos, P. D. 2000. Vortex Methods:Theory and Practice. Cam. U. Press.Google Scholar
    14. Crane, K., de Goes, F., Desbrun, M., and Schröder, P. 2013. Digital Geometry Processing with Discrete Exterior Calculus. In Courses, ACM SIGGRAPH. Google ScholarDigital Library
    15. Deng, J., Hou, T. Y., and Yu, X. 2005. A Level Set Formulation for the 3D Incompressible Euler Equations. Methods Appl. Anal. 12, 4, 427–440.Google ScholarCross Ref
    16. Desbrun, M., Kanso, E., and Tong, Y. 2008. Discrete Differential Forms for Computational Modeling. In Discrete Differential Geometry, A. I. Bobenko, P. Schröder, J. M. Sullivan, and G. M. Ziegler, Eds., Vol. 38 of Oberwolfach Seminars. Birkhäuser Verlag.Google Scholar
    17. Elcott, S., Tong, Y., Kanso, E., Schröder, P., and Desbrun, M. 2007. Stable, Circulation-Preserving, Simplicial Fluids. ACM Trans. Graph. 26, 1, 4:1–12. Google ScholarDigital Library
    18. Fedkiw, R., Stam, J., and Jensen, H. W. 2001. Visual Simulation of Smoke. In Proc. ACM/SIGGRAPH Conf, ACM, 15–22. Google ScholarDigital Library
    19. Feynman, R. P. 1955. Application of Quantum Mechanics to Liquid Helium, Vol. 1 of Progress in Low Temperature Physics. North-Holland, Ch. II, 17–53.Google Scholar
    20. Frisch, T., Pomeau, Y., and Rica, S. 1992. Transition to Dissipation in a Model of Superflow. Phys. R. Lett. 69, 11, 1644–1647.Google ScholarCross Ref
    21. Ginsburg, V. L., and Pitaevskii, L. P. 1958. On the Theory of Superfluidity. J. Exp. Theor. Phys. 7, 5, 858–861.Google Scholar
    22. Gross, E. P. 1961. Structure of a Quantized Vortex in Boson Systems. Il Nuovo Cimento 20, 3, 454–477.Google ScholarCross Ref
    23. Hall, H. E., and Vinen, W. F. 1956. The Rotation of Liquid Helium II. II. The Theory of Mutual Friction in Uniformly Rotating Helium II. Proc. R. Soc. L. A, Math. Phys. S. 238, 1213, 215–234.Google Scholar
    24. Hanson, A. J. 2005. Visualizing Quaternions. Morgan Kaufmann. Google ScholarDigital Library
    25. Hasimoto, H. 1972. A Soliton on a Vortex Filament. J. Fl. Mech. 51, 3, 477–485.Google ScholarCross Ref
    26. Hopf, H. 1931. Über die Abbildungen der Dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 1, 637–665.Google ScholarCross Ref
    27. Jause-Labert, C., Godeferd, F. S., and Favier, B. 2012. Numerical Validation of the Volume Penalization Method in Three-Dimensional Pseudo-Spectral Simulations. Comp. & Fl. 67, 41–56.Google Scholar
    28. Jerrard, R. L., and Spirn, D. 2015. Hydrodynamic Limit of the Gross-Pitaevskii Equation. Comm. Par. Diff. Eq. 40, 2, 135–190.Google ScholarCross Ref
    29. Kim, T., Thürey, N., James, D., and Gross, M. 2008. Wavelet Turbulence for Fluid Simulation. ACM Trans. Graph. 27, 3, 50:1–6. Google ScholarDigital Library
    30. Kim, D., young Song, O., and Ko, H.-S. 2009. Stretching and Wiggling Liquids. ACM Trans. Graph. 28, 5, 120:1–7. Google ScholarDigital Library
    31. Kleckner, D., and Irvine, W. T. M. 2013. Creation and Dynamics of Knotted Vortices. Nature Physics 9, 253–258. Video at http://www.nature.com/nphys/journal/v9/n4/extref/nphys2560-s7.mov.Google ScholarCross Ref
    32. Knöppel, F., Crane, K., Pinkall, U., and Schröder, P. 2015. Stripe Patterns on Surfaces. ACM Trans. Graph. 34, 4, 39:1–11. Google ScholarDigital Library
    33. Koumoutsakos, P., Cottet, G.-H., and Rossinelli, D. 2008. Flow Simulations Using Particles: Bridging Computer Graphics and CFD. In ACM/SIGGRAPH Conf. Classes, ACM/SIGGRAPH, 25:1–25:73. Google ScholarDigital Library
    34. Kuznetsov, E. A., and Mikhailov, A. V. 1980. On the Topological meaning of Canonical Clebsch Variables. Phys. Lett. A 77, 1, 37–38.Google ScholarCross Ref
    35. Landau, L., and Lifshits, E. 1935. On the Theory of the Dispersion of Magnetic Permeability in Ferromagnetic Bodies. Phys. Zeitsch. Sow. 8, 153–169.Google Scholar
    36. Leonard, A. 1980. Vortex Methods for Flow Simulation. J. Comput. Phys. 37, 3, 289–335.Google ScholarCross Ref
    37. Lim, T. T. 1989. An Experimental Study of a Vortex Ring Interacting with an Inclined Wall. Exp. in Fl. 7, 7, 453–463. Videos at http://serve.me.nus.edu.sg/limtt/video/Oblique_collison_front.mpg and http://serve.me.nus.edu.sg/limtt/video/Oblique_collison_top.mpg.Google ScholarCross Ref
    38. Lim, T. T. 1997. A Note on the Leapfrogging between two Coaxial Vortex Rings at low Reynolds Number. Phys. Fluids 9, 1, 239–241. Video at http://serve.me.nus.edu.sg/limtt/video/leapfrog.mpeg.Google ScholarCross Ref
    39. Lin, F.-H., and Xin, J. X. 1999. On the Incompressible Fluid Limit and the Vortex Motion Law of the Nonlinear Schrödinger Equation. Comm. Math. Phys. 200, 2, 249–274.Google ScholarCross Ref
    40. Lyons, D. W. 2003. An Elementary Introduction to the Hopf Fibration. Math. Mag. 76, 2, 87–98.Google ScholarCross Ref
    41. Madelung, E. 1926. Eine anschauliche Deutung der Gleichung von Schrödinger. Nat. Wiss. 14, 45, 1004–1004.Google ScholarCross Ref
    42. Madelung, E. 1927. Quantentheorie in hydrodynamischer Form. Z. Phys. 40, 3-4, 322–326.Google ScholarCross Ref
    43. McAdams, A., Sifakis, E., and Teran, J. 2010. A Parallel Multigrid Poisson Solver for Fluids Simulation on large Grids. In Proc. Symp. Comp. Anim., ACM SIGGRAPH/Eurographics, 65–74. Google ScholarDigital Library
    44. Niemi, A. J., and Sutcliffe, P. 2014. Leapfrogging Vortex Rings in the Landau-Lifshitz Equation. Nonlinearity 27, 9.Google ScholarCross Ref
    45. Onsager, L. 1949. Statistical Hydrodynamics. Il Nuovo Cimento 6, 2, 279–287. Footnote (1).Google ScholarCross Ref
    46. Osher, S., and Fedkiw, R. 2003. Level Set Methods and Dynamic Implicit Surfaces, Vol. 153 of Appl. Math. Sci. Springer.Google Scholar
    47. Packard, R. E., and Sanders, T. M. 1969. Detection of Single Quantized Vortex Lines in Rotating HE II. Phys. R. Lett. 22, 16, 823–826.Google ScholarCross Ref
    48. Park, S. I., and Kim, M. J. 2005. Vortex Fluids for Gaseous Phenomena. In Proc. Symp. Comp. Anim., ACM SIGGRAPH/Eurographics, 261–270. Google ScholarDigital Library
    49. Pfaff, T., Thürey, N., and Gross, M. 2012. Lagrangian Vortex Sheets for Animating Fluids. ACM Trans. Graph. 31, 4, 112:1–8. Google ScholarDigital Library
    50. Pitaevskii, L. P. 1961. Vortex Lines in an Imperfect Bose Gas. J. Exp. Theor. Phys. 13, 2, 451–454.Google Scholar
    51. Reeves, M. T., Billam, T. P., Anderson, B. P., and Bradley, A. S. 2015. Identifying a Superfluid Reynolds Number via Dynamical Similarity. Phys. R. Lett. 114, 15, 155302 (5pp).Google Scholar
    52. Rios, L. S. D. 1906. Sul moto d’un liquido indefinito con un filetto vorticoso di forma qualunque. Rend. Cir. Mat. Pal. 22, 1, 177–135.Google Scholar
    53. Rosenhead, L. 1931. The Formation of Vortices from a Surface of Discontinuity. Proc. R. Soc. Lond. A 134, 823, 170–192.Google ScholarCross Ref
    54. Saffman, P. G. 1992. Vortex Dynamics. Cam. U. Press.Google Scholar
    55. Sasaki, K., Suzuki, N., and Saito, H. 2010. Bénard — von Kármán Vortex Street in a Bose-Einstein Condensate. Phys. R. Lett. 104, 15–16, 150404 (4pp).Google Scholar
    56. Schönberg, M. 1954. On the Hydrodynamical Model of the Quantum Mechanics. Il Nuovo Cimento 12, 1, 103–133.Google ScholarCross Ref
    57. Schwarz, K. W. 1985. Three-Dimensional Vortex Dyanmics in Superfluid 4He: Line-Line and Line-Boundary Interactions. Phys. R. B 31, 9, 5782–5804.Google ScholarCross Ref
    58. Schwarz, G. 1995. Hodge Decomposition – A Method for Solving Boundary Value Problems. Springer.Google Scholar
    59. Seifert, H. 1935. Über das Geschlecht von Knoten. Math. Ann. 110, 1, 571–592.Google ScholarCross Ref
    60. Selle, A., Rasmussen, N., and Fedkiw, R. 2005. A Vortex Particle Method for Smoke, Water and Explosions. ACM Trans. Graph. 24, 3, 910–914. Google ScholarDigital Library
    61. Selle, A., Fedkiw, R., Kim, B., and Rossignac, J. 2008. An Unconditionally Stable MacCormack Method. J. Sci. Comp. 35, 2-3, 350–371. Google ScholarDigital Library
    62. Sorokin, A. L. 2001. Madelung Transformation for Vortex Flows of a Perfect Liquid. Doklady Physics 46, 8, 576–578.Google ScholarCross Ref
    63. Stagg, G. W., Parker, N. G., and Barenghi, C. F. 2014. Quantum Analogues of Classical Wakes in Bose-Einstein Condensates. J. Phys. B: At. Mol. Opt. Phys. 47, 9, 095304 (8pp).Google ScholarCross Ref
    64. Stam, J. 1999. Stable Fluids. In Proc. ACM/SIGGRAPH Conf, ACM, 121–128. Google ScholarDigital Library
    65. Steinhoff, J., and Underhill, D. 1994. Modification of the Euler Equations for “Vortciticy Confinement:” Application to the Computation of Interacting Vortex Rings. Phys. Fluids 6, 8, 2738–2744.Google ScholarCross Ref
    66. Stock, M. J., Dahm, W. J. A., and Tryggvason, G. 2008. Impact of a Vortex Ring on a Density Interface using a Regularized Inviscid Vortex Sheet Method. J. Comput. Phys. 227, 21, 9021–9043. Google ScholarDigital Library
    67. Strouhal, V. 1878. Ueber eine besondere Art der Tonerregung. Ann. Ph. Ch., Series III 5, 10, 216–250.Google Scholar
    68. Sutcliffe, P. 2007. Vortex Rings in Ferromagnets: Numerical Simulations of the Time-Dependent Three-Dimensional Landau-Lifshitz Equation. Phys. R. B 76, 18, 184439 (6pp).Google Scholar
    69. Volovik, G. E. 2003. Classical and Quantum Regimes of Superfluid Turbulence. JETP Letters 78, 9, 533–537.Google ScholarCross Ref
    70. Weissmann, S., and Pinkall, U. 2010. Filament-Based Smoke with Vortex Shedding and Variational Reconnection. ACM Trans. Graph. 29, 4, 115:1–12. Google ScholarDigital Library
    71. Weissmann, S., Pinkall, U., and Schröder, P. 2014. Smoke Rings from Smoke. ACM Trans. Graph. 33, 4, 140:1–8. Google ScholarDigital Library
    72. Zhang, X., Bridson, R., and Greif, C. 2015. Restoring the Missing Vorticity in Advection-Projection Fluid Solvers. ACM Trans. Graph. 34, 4, 52:1–8. Google ScholarDigital Library

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