Model reduction for real-time fluids

ABSTRACT

We present a new model reduction approach to fluid simulation, enabling large, real-time, detailed flows with continuous user interaction. Our reduced model can also handle moving obstacles immersed in the flow. We create separate models for the velocity field and for each moving boundary, and show that the coupling forces may be reduced as well. Our results indicate that surprisingly few basis functions are needed to resolve small but visually important features such as spinning vortices.

REFERENCES

• 1: Ausseur, J., Pinier, J., Glauser, M., and Higuchi, H. 2004. Predicting the Dynamics of the Flow over a NACA 4412 using POD. APS Meeting Abstracts (Nov.), D8.
• 2: Jernej Barbič , Doug L. James, Real-Time subspace integration for St. Venant-Kirchhoff deformable models, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 3: Jeff Bolz , Ian Farmer , Eitan Grinspun , Peter Schröoder, Sparse matrix solvers on the GPU: conjugate gradients and multigrid, ACM Transactions on Graphics (TOG), v.22 n.3, July 2003
• 4: M. Couplet , C. Basdevant , P. Sagaut, Calibrated reduced-order POD-Galerkin system for fluid flow modelling, Journal of Computational Physics, v.207 n.1, p.192-220, 20 July 2005 [doi>10.1016/j.jcp.2005.01.008]
• 5: Elcott, S., Tong, Y., Kanso, E., Schröder, P., and Desbrun, M. 2005. Stable, circulation-preserving, simplicial fluids. In Discrete Differential Geometry, Chapter 9 of Course Notes. ACM SIGGRAPH.
• 6: 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
• 7: Fogleman, M., Lumley, J., Rempfer, D., and Haworth, D. 2004. Application of the proper orthogonal decomposition to datasets of internal combustion engine flows. Journal of Turbulence 5, 23 (June).
• 8: Nick Foster , Dimitri Metaxas, Realistic animation of liquids, Graphical Models and Image Processing, v.58 n.5, p.471-483, Sept. 1996 [doi>10.1006/gmip.1996.0039]
• 9: Nolan Goodnight , Cliff Woolley , Gregory Lewin , David Luebke , Greg Humphreys, A multigrid solver for boundary value problems using programmable graphics hardware, Proceedings of the ACM SIGGRAPH/EUROGRAPHICS conference on Graphics hardware, July 26-27, 2003, San Diego, California
• 10: Mark J. Harris , Greg Coombe , Thorsten Scheuermann , Anselmo Lastra, Physically-based visual simulation on graphics hardware, Proceedings of the ACM SIGGRAPH/EUROGRAPHICS conference on Graphics hardware, September 01-02, 2002, Saarbrucken, Germany
• 11: Holmes, P., Lumley, J. L., and Berkooz, G. 1996. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge, MA.
• 12: Doug L. James , Kayvon Fatahalian, Precomputing interactive dynamic deformable scenes, ACM Transactions on Graphics (TOG), v.22 n.3, July 2003
• 13: Jens Krüger , Rüdiger Westermann, Linear algebra operators for GPU implementation of numerical algorithms, ACM Transactions on Graphics (TOG), v.22 n.3, July 2003
• 14: Li, W., Wei, X., and Kaufman, A. 2003. Implementing lattice Boltzmann computation on graphics hardware. The Visual Computer 19, 7--8, 444--456.
• 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: Lumley, J. L. 1970. Stochastic Tools in Turbulence, vol. 12 of Applied Mathematics and Mechanics. Academic Press, New York.
• 17: M. Marion , R. Temam, Nonlinear Galerkin methods, SIAM Journal on Numerical Analysis, v.26 n.5, p.1139-1157, Oct. 1989 [doi>10.1137/0726063]
• 18: Maurel, S., Boree, J., and Lumley, J. 2001. Extended proper orthogonal decomposition: Application to jet/vortex interaction. Flow, Turbulence and Combustion 67, 2 (June), 125--36.
• 19: Sang Il Park , Myoung Jun Kim, Vortex fluid for gaseous phenomena, Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation, July 29-31, 2005, Los Angeles, California [doi>10.1145/1073368.1073406]
• 20: Clarence W. Rowley , Jerrold E. Marsden, Reconstruction equations and the Karhunen—Loéve expansion for systems with symmetry, Physica D, v.142 n.1-2, p.1-19, Aug. 2000 [doi>10.1016/S0167-2789(00)00042-7]
• 21: Rowley, C. W., Kevrekidis, I. G., Marsden, J. E., and Lust, K. 2003. Reduction and reconstruction for self-similar dynamical systems. Nonlinearity 16 (July), 1257--1275.
• 22: Rowley, C., Williams, D., Colonius, T., Murray, R., and MacMartin, D. 2006. Linear models for control of cavity flow oscillations. J. Fluid Mech. (Jan.).
• 23: Schmit, R., and Glasuer, M. 2002. Low dimensional tools for flow-structure interaction problems: Application to micro air vehicles. APS Meeting Abstracts (Nov.), D1+.
• 24: 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
• 25: S. Sirisup , G. E. Karniadakis, A spectral viscosity method for correcting the long-term behavior of POD models, Journal of Computational Physics, v.194 n.1, p.92-116, 10 February 2004 [doi>10.1016/j.jcp.2003.08.021]
• 26: Sirovich, L. 1987. Turbulence and the dynamics of coherent structures. I - Coherent structures. II - Symmetries and transformations. III - Dynamics and scaling. Quarterly of Applied Mathematics 45 (Oct.), 561--571.
• 27: 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]
• 28: Jos Stam, A simple fluid solver based on the FFT, Journal of Graphics Tools, v.6 n.2, p.43-52, 2001
• 29: Hongcheng Wang , Qing Wu , Lin Shi , Yizhou Yu , Narendra Ahuja, Out-of-core tensor approximation of multi-dimensional matrices of visual data, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
• 30: Enhua Wu , Youquan Liu , Xuehui Liu, An improved study of real-time fluid simulation on GPU: Research Articles, Computer Animation and Virtual Worlds, v.15 n.3-4, p.139-146, July 2004 [doi>10.1002/cav.v15:3/4]

INDEX TERMS

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

Keywords:
flow with boundaries, model reduction, proper orthogonal decomposition