Simulating water and smoke with an octree data structure

Abstract

We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric positive definite enabling the use of fast solution methods such as preconditioned conjugate gradients, whereas the standard approximation to the Poisson equation on an octree grid results in a non-symmetric linear system which is more computationally challenging to invert. The semi-Lagrangian characteristic tracing technique is used to advect the velocity, smoke density, and even the level set making implementation on an octree straightforward. In the case of smoke, we have multiple refinement criteria including object boundaries, optical depth, and vorticity concentration. In the case of water, we refine near the interface as determined by the zero isocontour of the level set function.

References

1. ALMGREN, A., BELL, J., COLELLA, P., HOWELL, L., AND WELCOME, M. 1998. A conservative adaptive projection method for the variable density incompressible navier-stokes equations. J. Comput. Phys. 142, 1--46.
2. BERGER, M., AND COLELLA, P. 1989. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64--84.
3. BERGER, M., AND OLIGER, J. 1984. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484--512.
4. CARLSON, M., MUCHA, P., VAN HORN III, R., AND TURK, G. 2002. Melting and flowing. In ACM SIGGRAPH Symposium on Computer Animation, 167--174.
5. CHEN, J., AND LOBO, N. 1994. Toward interactive-rate simulation of fluids with moving obstacles using the navier-stokes equations. Computer Graphics and Image Processing 57, 107--116.
6. CHEN, S., MERRIMAN, B., OSHER, S., AND SMEREKA, P. 1997. A simple level set method for solving stefan problems. 8--29.
7. DAY, M., COLELLA, P., LIJEWSKI, M., RENDLEMAN, C., AND MARCUS, D. 1998. Embedded boundary algorithms for solving the poisson equation on complex domains. Tech. rep., Lawrence Berkeley National Laboratory (LBNL-41811).
8. ENRIGHT, D., FEDKIW, R., FERZIGER, J., AND MITCHELL, I. 2002. A hybrid particle level set method for improved interface capturing. J. Comp. Phys. 183, 83--116.
9. ENRIGHT, D., MARSCHNER, S., AND FEDKIW, R. 2002. Animation and rendering of complex water surfaces. ACM Trans. Graph. (SIGGRAPH Proc.) 21, 3, 736--744.
10. ENRIGHT, D., LOSASSO, F., AND FEDKIW, R. 2004. A fast and accurate semi-Lagrangian particle level set method. Computers and Structures, (in press).
11. FEDKIW, R., STAM, J., AND JENSEN, H. 2001. Visual simulation of smoke. In Proc. of ACM SIGGRAPH 2001, 15--22.
12. FELDMAN, B. E., O'BRIEN, J. F., AND ARIKAN, O. 2003. Animating suspended particle explosions. ACM Trans. Graph. (SIGGRAPH Proc.) 22, 3, 708--715.
13. FOSTER, N., AND FEDKIW, R. 2001. Practical animation of liquids. In Proc. of ACM SIGGRAPH 2001, 23--30.
14. FOSTER, N., AND METAXAS, D. 1996. Realistic animation of liquids. Graph. Models and Image Processing 58, 471--483.
15. FOSTER, N., AND METAXAS, D. 1997. Controlling fluid animation. In Computer Graphics International 1997, 178--188.
16. FOSTER, N., AND METAXAS, D. 1997. Modeling the motion of a hot, turbulent gas. In Proc. of SIGGRAPH 97, 181--188.
17. FRISKEN, S., PERRY, R., ROCKWOOD, A., AND JONES, T. 2000. Adaptively sampled distance fields: a general representation of shape for computer graphics. In Proc. SIGGRAPH 2000, 249--254.
18. GIBOU, F., FEDKIW, R., CHENG, L.-T., AND KANG, M. 2002. A second--order--accurate symmetric discretization of the poisson equation on irregular domains. 205--227.
19. HAM, F., LIEN, F., AND STRONG, A. 2002. A fully conservative secondorder finite difference scheme for incompressible flow on nonuniform grids. J. Comput. Phys. 117, 117--133.
20. HARLOW, F., AND WELCH, J. 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182--2189.
21. HONG, J.-M., AND KIM, C.-H. 2003. Animation of bubbles in liquid. Comp. Graph. Forum (Eurographics Proc.) 22, 3, 253--262.
22. JU, T., LOSASSO, F., SCHAEFER, S., AND WARREN, J. 2002. Dual contouring of Hermite data. ACM Trans. Graph. (SIGGRAPH Proc.) 21, 3, 339--346.
23. KASS, M., AND MILLER, G. 1990. Rapid, stable fluid dynamics for computer graphics. In Computer Graphics (Proc. of SIGGRAPH 90), vol. 24, 49--57.
24. LAMORLETTE, A., AND FOSTER, N. 2002. Structural modeling of natural flames. ACM Trans. Graph. (SIGGRAPH Proc.) 21, 3, 729--735.
25. LEE, H., DESBRUN, M., AND SCHRODER, P. 2003. Progressive encoding of complex isosurfaces. ACM Trans. Graph. (SIGGRAPH Proc.) 22, 3, 471--476.
26. MIYAZAKI, R., DOBASHI, Y., AND NISHITA, T. 2002. Simulation of cumuliform clouds based on computational fluid dynamics. Proc. Eurographics 2002 Short Presentation, 405--410.
27. MÜLLER, M., CHARYPAR, D., AND GROSS, M. 2003. Particle-based fluid simulation for interactive applications. In Proc. of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 154--159.
28. NGUYEN, D., FEDKIW, R., AND JENSEN, H. 2002. Physically based modeling and animation of fire. In ACM Trans. Graph. (SIGGRAPH Proc.), vol. 29, 721--728.
29. OHTAKE, Y., BELYAEV, A., ALEXA, M., TURK, G., AND SEIDEL, H. 2003. Multi-Level Partition of Unity Implicits. ACM Trans. Graph. (SIGGRAPH Proc.) 22, 3, 463--470.
30. PERRY, R., AND FRISKEN, S. 2001. Kizamu: a system for sculpting digital characters. In Proc. SIGGRAPH 2001, vol. 20, 47--56.
31. POPINET, S. 2003. Gerris: A tree-based adaptive solver for the incompressible euler equations in complex geometries. J. Comp. Phys. 190, 572--600.
32. PREMOZE, S., TASDIZEN, T., BIGLER, J., LEFOHN, A., AND WHITAKER, R. 2003. Particle-based simulation of fluids. In Comp. Graph. Forum (Eurographics Proc.), vol. 22, 401--410.
33. RASMUSSEN, N., NGUYEN, D., GEIGER, W., AND FEDKIW, R. 2003. Smoke simulation for large scale phenomena. ACM Trans. Graph. (SIGGRAPH Proc.) 22, 703--707.
34. ROUSSEL, O., SCHNEIDER, K., TSIGULIN, A., AND BOCKHORN, H. 2003. A conservative fully adaptive multiresolution algorithm for parabolic pdes. J. Comput. Phys. 188, 493--523.
35. SAAD, Y. 1996. Iterative methods for sparse linear systems. PWS Publishing. New York, NY.
36. SAMET, H. 1989. The Design and Analysis of Spatial Data Structures. Addison-Wesley, New York.
37. SETHIAN, J. 1996. A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591--1595.
38. SOCHNIKOV, V., AND EFRIMA, S. 2003. Level set calculations of the evolution of boundaries on a dynamically adaptive grid. Int. J. Num. Methods in Eng. 56, 1913--1929.
39. STAM, J. 1999. Stable fluids. In Proc. of SIGGRAPH 99, 121--128.
40. STAM, J. 2003. Flows on surfaces of arbitrary topology. ACM Trans. Graph. (SIGGRAPH Proc.) 22, 724--731.
41. STRAIN, J. 1999. Fast tree-based redistancing for level set computations. J. Comput. Phys. 152, 664--686.
42. STRAIN, J. 1999. Tree methods for moving interfaces. J. Comput. Phys. 151, 616--648.
43. STRAIN, J. 2000. A fast modular semi-lagrangian method for moving interfaces. J. Comput. Phys. 161, 512--536.
44. SUSSMAN, M., ALMGREN, A., BELL, J., COLELLA, P., HOWELL, L., AND WELCOME, M. 1999. An adaptive level set approach for incompressible two-phase flows. J. Comput. Phys. 148, 81--124.
45. SUSSMAN, M. 2003. A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles. J. Comp. Phys. 187, 110--136.
46. TAKAHASHI, T., FUJII, H., KUNIMATSU, A., HIWADA, K., SAITO, T., TANAKA, K., AND UEKI, H. 2003. Realistic animation of fluid with splash and foam. Comp. Graph. Forum (Eurographics Proc.) 22, 3, 391--400.
47. TREUILLE, A., MCNAMARA, A., POPOVIĆ, Z., AND STAM, J. 2003. Keyframe control of smoke simulations. ACM Trans. Graph. (SIGGRAPH Proc.) 22, 3, 716--723.
48. TSITSIKLIS, J. 1995. Efficient algorithms for globally optimal trajectories. IEEE Trans. on Automatic Control 40, 1528--1538.
49. WESTERMANN, R., KOBBELT, L., AND ERTL, T. 1999. Real-time exploration of regular volume data by adaptive reconstruction of isosurfaces. The Vis. Comput. 15, 2, 100--111.
50. YNGVE, G., O'BRIEN, J., AND HODGINS, J. 2000. Animating explosions. In Proc. SIGGRAPH 2000, vol. 19, 29--36.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation