Modular bases for fluid dynamics

Abstract

We present a new approach to fluid simulation that balances the speed of model reduction with the flexibility of grid-based methods. We construct a set of composable reduced models, or tiles, which capture spatially localized fluid behavior. We then precompute coupling terms so that these models can be rearranged at runtime. To enforce consistency between tiles, we introduce constraint reduction. This technique modifies a reduced model so that a given set of linear constraints can be fulfilled. Because dynamics and constraints can be solved entirely in the reduced space, our method is extremely fast and scales to large domains.

References

1. Adams, B., Pauly, M., Keiser, R., and Guibas, L. J. 2007. Adaptively sampled particle fluids. In Proc. SIGGRAPH '07.
2. Angelidis, A., and Neyret, F. 2005. Simulation of smoke based on vortex filament primitives. In Proc. SCA '05.
3. Angelidis, A., Neyret, F., Singh, K., and Nowrouzezahrai, D. 2006. A controllable, fast and stable basis for vortex based smoke simulation. In Proc. SCA '06.
4. 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, D8.
5. Babuška, I. 1973. The finite element method with Lagrangian multipliers. Numer. Math. 20, 3.
6. Barbič, J., and James, D. 2005. Real-time subspace integration for St. Venant-Kirchhoff deformable models. In Proc. SIGGRAPH '05.
7. Barbič, J., and Popović, J. 2008. Real-time control of physically based simulations using gentle forces. ACM Transactions on Graphics 27, 5.
8. Bolz, J., Farmer, I., Grinspun, E., and Schröder, P. 2003. Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid. In Proc. SIGGRAPH '03.
9. Borggaard, J., Gugercin, S., and Iliescu, T. 2006. A domain decomposition approach to POD. IEEE Conference on Decision and Control.
10. Chenney, S. 2004. Flow tiles. In Proc. SCA '04.
11. Cohen, M. F., Shade, J., Hiller, S., and Deussen, O. 2003. Wang tiles for image and texture generation. In Proc. SIGGRAPH '03.
12. Couplet, M., Basdevant, C., and Sagaut, P. 2005. Calibrated reduced-order POD-Galerkin system for fluid flow modelling. J. Comput. Phys. 207, 1.
13. 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.
14. Farhat, C., Tezaur, R., and Toivanen, J. 2000. A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers. Int. J. Numer. Meth. Engng.
15. Farhat, C., Harari, I., and Franca, L. P. 2001. The discontinuous enrichment method. Comput. Methods Appl. Mech. Engrg. 190.
16. Farhat, C., Harari, I., and Hetmaniuk, U. 2003. A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Applied Mechanics and Engineering 192, 1389--1419.
17. Feldman, B. E., O'Brien, J. F., and Klingner, B. M. 2005. Animating gases with hybrid meshes. In Proc. SIGGRAPH '05.
18. Foster, N., and Metaxas, D. 1996. Realistic animation of liquids. Graphical Models and Image Processing 58, 5.
19. Goodnight, N., Woolley, C., Luebke, D., and Humphreys, G. A. 2003. Multigrid solver for boundary value problems using programmable graphics hardware. In Proceeding of Graphics Hardware.
20. Harris, M. J., Coombe, G., Scheuermann, T., and Lastra, A. 2002. Physically-based visual simulation on graphics hardware. In Graphics Hardware 2002, 109--118.
21. Holmes, P., Lumley, J. L., and Berkooz, G. 1996. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.
22. James, D. L., and Fatahalian, K. 2003. Precomputing interactive dynamic deformable scenes. In Proc. SIGGRAPH '03.
23. Keiser, R., Adams, B., Gasser, D., Bazzi, P., Dutre, P., and Gross, M. 2005. A unified lagrangian approach to solidfluid animation. In Proceedings Symposium Point-Based Graphics.
24. Krüger, J., and Westermann, R. 2003. Linear algebra operators for GPU implementation of numerical algorithms. In Proc. SIGGRAPH '03.
25. LeGresley, P. A., and Alonso, J. J. 2003. Dynamic domain decomposition and error correction for reduced order models. 41st AIAA Aerospace Sciences Meeting and Exhibit.
26. Li, W., Wei, X., and Kaufman, A. 2003. Implementing lattice Boltzmann computation on graphics hardware. The Visual Computer 19, 7--8.
27. Losasso, F., Gibou, F., and Fedkiw, R. 2004. Simulating water and smoke with an octree data structure. In Proc. SIGGRAPH '04.
28. Lucia, D. J., and King, P. I. 2002. Domain decomposition for reduced-order modeling of a flow with moving shocks. AIAA Journal 40, 11, 2360--2362.
29. Lumley, J. L. 1970. Stochastic Tools in Turbulence, vol. 12 of Applied Mathematics and Mechanics. Academic Press.
30. Marion, M., and Temam, R. 1989. Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26, 5, 1139--1157.
31. Müller, M., Charypar, D., and Gross, M. 2003. Particle-Based Fluid Simulation for Interactive Applications. In Proc. SCA '03.
32. Park, S. I., and Kim, M. J. 2005. Vortex fluid for gaseous phenomena. In Proc. SCA '05.
33. Rowley, C., Williams, D., Colonius, T., Murray, R., and MacMartin, D. 2006. Linear models for control of cavity flow oscillations. J. Fluid Mech. 547, 317--330.
34. Selle, A., Rasmussen, N., and Fedkiw, R. 2005. A vortex particle method for smoke, water and explosions. In Proc. SIGGRAPH '05.
35. Shewchuk, J. R. 1994. An introduction to the conjugate gradient method without the agonizing pain. Tech. Rep. CS-94-125, Carnegie Mellon University, Pittsburgh, PA, USA.
36. Sirisup, S., and Karniadakis, G. E. 2004. A spectral viscosity method for correcting the long-term behavior of POD models. J. Comput. Phys. 194, 1, 92--116.
37. 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.
38. Sloan, P.-P., Kautz, J., and Snyder, J. 2002. Precomputed radiance transfer for real-time rendering in dynamic, lowfrequency lighting environments. In Proc. SIGGRAPH '02.
39. Stam, J. 1999. Stable Fluids. In Computer Graphics (SIGGRAPH 99).
40. Tezaur, R., and Farhat, C. 2006. Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Meth. Engng 66.
41. Tezaur, R., Zhang, L., and Farhat, C. 2008. A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems. Comput. Methods Appl. Mech. Engrg. 197.
42. Toselli, A., and Widlund, O. 2005. Domain Decomposition Methods - Algorithms and Theory. Springer.
43. Treuille, A., Lewis, A., and Popović, Z. 2006. Model reduction for real-time fluids. In Proc. SIGGRAPH '06.
44. Wu, E., Liu, Y., and Liu, X. 2005. An improved study of realtime fluid simulation on GPU. Computer Animation and Virtual Worlds 15, 3--4.
45. Zhang, L., Tezaur, R., and Farhat, C. 2006. The discontinuous enrichment method for elastic wave propagation in the medium-frequency regime. Internat. J. Numer. Methods Engrg. 66.
46. Zhu, Y., and Bridson, R. 2005. Animating sand as a fluid. In Proc. SIGGRAPH '05.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

  • Modeling and simulation

    • Simulation types and techniques

      • Simulation by animation