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Stable, circulation-preserving, simplicial fluids

Published: 01 January 2007 Publication History

Abstract

Visual quality, low computational cost, and numerical stability are foremost goals in computer animation. An important ingredient in achieving these goals is the conservation of fundamental motion invariants. For example, rigid and deformable body simulation benefits greatly from the conservation of linear and angular momenta. In the case of fluids, however, none of the current techniques focuses on conserving invariants, and consequently, often introduce a visually disturbing numerical diffusion of vorticity. Just as important visually is the resolution of complex simulation domains. Doing so with regular (even if adaptive) grid techniques can be computationally delicate. In this article, we propose a novel technique for the simulation of fluid flows. It is designed to respect the defining differential properties, that is, the conservation of circulation along arbitrary loops as they are transported by the flow. Consequently, our method offers several new and desirable properties: Arbitrary simplicial meshes (triangles in 2D, tetrahedra in 3D) can be used to define the fluid domain; the computations involved in the update procedure are efficient due to discrete operators with small support; and it preserves discrete circulation, avoiding numerical diffusion of vorticity.

References

[1]
Abraham, R., Marsden, J., and Ratiu, T., Eds. 1988. Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, vol. 75. Springer Verlag.
[2]
Alliez, P., Cohen-Steiner, D., Yvinec, M., and Desbrun, M. 2005. Variational tetrahedral meshing. ACM Trans. Graph. 24, 3, 617--625.
[3]
Angelidis, A., Neyret, F., Schpok, J., Dwyer, W. T., and Ebert, D. S. 2005. Modeling and animating gases with simulation features. In Proceedings of the ACM/Eurographics Symposium on Computer Animation. 97--106.
[4]
Angelidis, A., Neyret, F., Singh, K., and Nowrouzezahrai, D. 2006. A controllable, fast and stable basis for vortex based smoke simulation. In Proceedings of the ACM/Eurographics Symposium on Computer Animation. 25--32.
[5]
Bochev, P. B. and Hyman, J. M. 2006. Principles of mimetic discretizations of differential operators. Compatible Spatial Discretization Series/IMA vol. 142 in Mathematics and Applications, 89--120.
[6]
Bossavit, A. 1998. Computational Electromagnetism. Academic Press, Boston.
[7]
Bossavit, A. and Kettunen, L. 1999. Yee-Like schemes on a tetrahedral mesh. Int. J. Num. Modelling: Electr. Net. Dev. Fields 12, 129--142.
[8]
Chang, W., Giraldo, F., and Perot, B. 2002. Analysis of an exact fractional step method. J. Comput. Phys. 180, 3 (Nov.), 183--199.
[9]
Chorin, A. and Marsden, J. 1979. A Mathematical Introduction to Fluid Mechanics, 3rd Ed. Springer Verlag.
[10]
Desbrun, M., Kanso, E., and Tong, Y. 2006. Discrete differential forms for computational sciences. ACM SIGGRAPH 2006 Courses, 39--54.
[11]
E, W. and Liu, J.-G. 1996a. Finite difference schemes for imcompressible flows in vorticity formulations. 1, 181--195.
[12]
E, W. and Liu, J.-G. 1996b. Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys. 124, 2, 368--382.
[13]
Elcott, S. and Schröder, P. 2006. Building your own dec at home. In Discrete Differential Geometry, E. Grinspun, et al., Eds. Course Notes. ACM SIGGRAPH Comput. Graph.
[14]
Fedkiw, R., Stam, J., and Jensen, H. W. 2001. Visual simulation of smoke. In Proceedings of the ACM SIGGRAPH Conference. Computer Graphics Proceedings, Annual Conference Series. 15--22.
[15]
Feldman, B. E., O'Brien, J. F., and Klingner, B. M. 2005. Animating gases with hybrid meshes. ACM Trans. Graph. 24, 3, 904--909.
[16]
Foster, N. and Fedkiw, R. 2001. Practical animation of liquids. In Proceedings of the ACM SIGGRAPH Conference. Computer Graphics Proceedings, Annual Conference Series. 23--30.
[17]
Foster, N. and Metaxas, D. 1997. Modeling the motion of a hot, turbulent gas. In Proceedings of the ACM SIGGRAPH Conference. Computer Graphics Proceedings, Annual Conference Series. 181--188.
[18]
Gamito, M. N., Lopes, P. F., and Gomes, M. R. 1995. Two-Dimensional simulation of gaseous phenomena using vortex particles. In Proceedings of the 6th Eurographics Workshop on Computer Animation and Simulation. 3--15.
[19]
Goktekin, T. G., Bargteil, A. W., and O'Brien, J. F. 2004. A method for animating viscoelastic fluids. ACM Trans. Graph. 23, 3 (Aug.), 463--468.
[20]
Guendelman, E., Selle, A., Losasso, F., and Fedkiw, R. 2005. Coupling water and smoke to thin deformable and rigid shell. ACM Trans. Graph. 24, 3 (Aug.), 973--981.
[21]
Hirani, A. 2003. Discrete exterior calculus. Ph.D. thesis, California Institute of Technology.
[22]
Klingner, B. M., Feldman, B. E., Chentanez, N., and O'Brien, J. F. 2006. Fluid animation with dynamic meshes. ACM Trans. Graph. 25, 3 (Aug.), 820--825.
[23]
Langtangen, H.-P., Mardal, K.-A., and Winter, R. 2002. Numerical methods for incompressible viscous flow. Adv. Water Resources 25, 8--12 (Aug--Dec), 1125--1146.
[24]
Losasso, F., Gibou, F., and Fedkiw, R. 2004. Simulating water and smoke with an octree data structure. ACM Trans. Graph. 23, 3 (Aug.), 457--462.
[25]
Marsden, J. E. and Wenstein, A. 1983. Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Physica D 7, 305--323.
[26]
Marsden, J. E. and West, M. 2001. Discrete mechanics and variational integrators. Acta Numerica 10, 357--515.
[27]
McNamara, A., Treuille, A., Popovic, Z., and Stam, J. 2004. Fluid control using the adjoint method. ACM Trans. Graph. 23, 3 (Aug.), 449--456.
[28]
Munkres, J. R. 1984. Elements of Algebraic Topology. Addison-Wesley.
[29]
Nicolaides, R. A. and Wu, X. 1997. Covolume solutions of three-dimensional div-curl equations. SIAM J. Numer. Anal. 34, 2195--2203.
[30]
Park, S. I. and Kim, M. J. 2005. Vortex fluid for gaseous phenomena. In Proceedings of the ACM/Eurographics Symposium on Computer Animation. 261--270.
[31]
Pighin, F., Cohen, J. M., and Shah, M. 2004. Modeling and editing flows using advected radial basis functions. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation. 223--232.
[32]
Selle, A., Rasmussen, N., and Fedkiw, R. 2005. A vortex particle method for smoke, water and explosions. ACM Trans. Graph. 24, 3 (Aug.), 910--914.
[33]
Shi, L. and Yu, Y. 2004. Inviscid and incompressible fluid simulation on triangle meshes. J. Comput. Animation Virtual Worlds 15, 3--4 (June), 173--181.
[34]
Shi, L. and Yu, Y. 2005. Controllable smoke animation with guiding objects. ACM Trans. Graph. 24, 1 (Jan.), 140--164.
[35]
Shiels, D. and Leonard, A. 2001. Investigation of a drag reduction on a circular cylinder in rotary oscillation. J. Fluid Mech. 431, 297--322.
[36]
Stam, J. 1999. Stable fluids. In Proceedings of the ACM SIGGRAPH Conference. Computer Graphics Proceedings, Annual Conference Series. 121--128.
[37]
Stam, J. 2001. A simple fluid solver based on the FFT. J. Graph. Tools 6, 2, 43--52.
[38]
Stam, J. 2003. Flows on surfaces of arbitrary topology. ACM Trans. Graph. 22, 3 (July), 724--731.
[39]
Steinhoff, J. and Underhill, D. 1994. Modification of the Euler equations for vorticity confinement: Applications to the computation of interacting vortex rings. Phys. Fluids 6, 8 (Aug.), 2738--2744.
[40]
Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. In Proceedings of the ACM/Eurographics Symposium on Geometry Processing. 201--210.
[41]
Tong, Y., Lombeyda, S., Hirani, A. N., and Desbrun, M. 2003. Discrete multiscale vector field decomposition. ACM Trans. Graph. 22, 3, 445--452.
[42]
Treuille, A., McNamara, A., Popović, Z., and Stam, J. 2003. Keyframe control of smoke simulations. ACM Trans. Graph. 22, 3 (July), 716--723.
[43]
Warren, J., Schaefer, S., Hirani, A., and Desbrun, M. 2007. Barycentric coordinates for convex sets. Adv. Comput. Math. to appear.
[44]
Weißmann, S. 2006. Real time simulation of fluid flow. M.S. thesis, Technische Universität Berlin.
[45]
Wendt, J., Baxter, W., Oguz, I., and Lin, M. 2005. Finite volume flow simulations on arbitrary domains. Tech. Rep. TR05-019, UNC-CH Computer Science. Sept.
[46]
Yaeger, L., Upson, C., and Myers, R. 1986. Combining physical and visual simulation---Creation of the planet Jupiter for the film 2010. In Proceedings of the ACM SIGGRAPH Conference on Computer Graphics 20, 4, 85--93.

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  1. Stable, circulation-preserving, simplicial fluids

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    cover image ACM Transactions on Graphics
    ACM Transactions on Graphics  Volume 26, Issue 1
    January 2007
    96 pages
    ISSN:0730-0301
    EISSN:1557-7368
    DOI:10.1145/1189762
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    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 01 January 2007
    Published in TOG Volume 26, Issue 1

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    Author Tags

    1. Fluid animation
    2. Lie advection
    3. stable fluids
    4. vorticity preservation

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