Energy-preserving integrators for fluid animation

ABSTRACT

Numerical viscosity has long been a problem in fluid animation. Existing methods suffer from intrinsic artificial dissipation and often apply complicated computational mechanisms to combat such effects. Consequently, dissipative behavior cannot be controlled or modeled explicitly in a manner independent of time step size, complicating the use of coarse previews and adaptive-time stepping methods. This paper proposes simple, unconditionally stable, fully Eulerian integration schemes with no numerical viscosity that are capable of maintaining the liveliness of fluid motion without recourse to corrective devices. Pressure and fluxes are solved efficiently and simultaneously in a time-reversible manner on simplicial grids, and the energy is preserved exactly over long time scales in the case of inviscid fluids. These integrators can be viewed as an extension of the classical energy-preserving Harlow-Welch / Crank-Nicolson scheme to simplicial grids.

REFERENCES

• 1: Benzi, M., Golub, G. H., and Liesen, J. 2005. Numerical solution of saddle point problems. Acta Numerica 14, 1--137.
• 2: Miklós Bergou , Saurabh Mathur , Max Wardetzky , Eitan Grinspun, TRACKS: toward directable thin shells, ACM Transactions on Graphics (TOG), v.26 n.3, July 2007
• 3: J U Brackbill , H M Ruppel, FLIP: A method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions, Journal of Computational Physics, v.65 n.2, p.314-343, Aug. 1986 [doi>10.1016/0021-9991(86)90211-1]
• 4: Nuttapong Chentanez , Bryan E. Feldman , François Labelle , James F. O'Brien , Jonathan R. Shewchuk, Liquid simulation on lattice-based tetrahedral meshes, Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on Computer animation, August 02-04, 2007, San Diego, California
• 5: Chorin, A., and Marsden, J. 1979. A Mathematical Introduction to Fluid Mechanics, 3rd edition ed. Springer-Verlag.
• 6: M. Duponcheel , P. Orlandi , G. Winckelmans, Time-reversibility of the Euler equations as a benchmark for energy conserving schemes, Journal of Computational Physics, v.227 n.19, p.8736-8752, October, 2008 [doi>10.1016/j.jcp.2008.06.020]
• 7: Sharif Elcott , Yiying Tong , Eva Kanso , Peter Schröder , Mathieu Desbrun, Stable, circulation-preserving, simplicial fluids, ACM Transactions on Graphics (TOG), v.26 n.1, p.4-es, January 2007 [doi>10.1145/1189762.1189766]
• 8: Ronald Fedkiw , Jos Stam , Henrik Wann Jensen, Visual simulation of smoke, Proceedings of the 28th annual conference on Computer graphics and interactive techniques, p.15-22, August 2001 [doi>10.1145/383259.383260]
• 9: Bryan E. Feldman , James F. O'Brien , Okan Arikan, Animating suspended particle explosions, ACM Transactions on Graphics (TOG), v.22 n.3, July 2003
• 10: Bryan E. Feldman , James F. O'Brien , Bryan M. Klingner, Animating gases with hybrid meshes, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 11: Nick Foster , Dimitris Metaxas, Modeling the motion of a hot, turbulent gas, Proceedings of the 24th annual conference on Computer graphics and interactive techniques, p.181-188, August 1997 [doi>10.1145/258734.258838]
• 12: Gresho, P. M., and Sani, R. L. 2000. Incompressible Flow and the Finite Element Method. J. Wiley & Sons.
• 13: Harlow, F. H., and Welch, J. E. 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids 8, 12 (Dec.), 2182--2189.
• 14: ByungMoon Kim , Yingjie Liu , Ignacio Llamas , Jarek Rossignac, Advections with Significantly Reduced Dissipation and Diffusion, IEEE Transactions on Visualization and Computer Graphics, v.13 n.1, p.135-144, January 2007 [doi>10.1109/TVCG.2007.3]
• 15: Frank Losasso , Frédéric Gibou , Ron Fedkiw, Simulating water and smoke with an octree data structure, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 16: K. Mahesh , G. Constantinescu , P. Moin, A numerical method for large-eddy simulation in complex geometries, Journal of Computational Physics, v.197 n.1, p.215-240, 10 June 2004 [doi>10.1016/j.jcp.2003.11.031]
• 17: Molemaker, J., Cohen, J. M., Patel, S., and Yong Noh, J. 2008. Low viscosity flow simulations for animation. In Symposium on Computer Animation, 9--18.
• 18: Pavlov, D. 2009. Structure-preserving Discretizations of Incompressible Fluids. PhD dissertation in Mathematics, California Institute of Technology.
• 19: Blair Perot, Conservation properties of unstructured staggered mesh schemes, Journal of Computational Physics, v.159 n.1, p.58-89, Mar.20.2000 [doi>10.1006/jcph.2000.6424]
• 20: Schechter, H., and Bridson, R. 2008. Evolving sub-grid turbulence for smoke animation. In Symposium on Computer Animation, 1--8.
• 21: Andrew Selle , Nick Rasmussen , Ronald Fedkiw, A vortex particle method for smoke, water and explosions, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 22: Andrew Selle , Ronald Fedkiw , Byungmoon Kim , Yingjie Liu , Jarek Rossignac, An Unconditionally Stable MacCormack Method, Journal of Scientific Computing, v.35 n.2-3, p.350-371, June 2008 [doi>10.1007/s10915-007-9166-4]
• 23: Shi, L., and Yu, Y. 2002. Visual smoke simulation with adaptive octree refinement. Computer Graphics and Imaging.
• 24: Simo, J., and Armero, F. 1994. Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations. Computer Methods in Applied Mechanics and Engineering 111, 1--2, 111--154.
• 25: Jos Stam, Stable fluids, Proceedings of the 26th annual conference on Computer graphics and interactive techniques, p.121-128, July 1999 [doi>10.1145/311535.311548]
• 26: Steinhoff, J., and Underhill, D. 1994. Modification of the euler equations for Vorticity Confinement. Physics of Fluids 6, 8 (Aug.), 2738--2744.
• 27: Ari Stern , Mathieu Desbrun, Discrete geometric mechanics for variational time integrators, ACM SIGGRAPH 2006 Courses, July 30-August 03, 2006, Boston, Massachusetts [doi>10.1145/1185657.1185669]
• 28: Xing Zhang , David Schmidt , Blair Perot, Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics, Journal of Computational Physics, v.175 n.2, p.764-791, January 20, 2002 [doi>10.1006/jcph.2001.6973]
• 29: Yongning Zhu , Robert Bridson, Animating sand as a fluid, ACM SIGGRAPH 2005 Papers, July 31-August 04, 2005, Los Angeles, California

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.7 Three-Dimensional Graphics and Realism
          Subjects: Animation

Keywords:
Eulerian fluid animation, energy preservation, time integration