Advections with Significantly Reduced Dissipation and Diffusion
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Abstract—Back and Forth Error Compensation and Correction (BFECC) was recently developed for interface computation using a level set method. We show that BFECC can be applied to reduce dissipation and diffusion encountered in a variety of advection steps, such as velocity, smoke density, and image advections on uniform and adaptive grids and on a triangulated surface. BFECC can be implemented trivially as a small modification of the first-order upwind or semi-Lagrangian integration of advection equations. It provides second-order accuracy in both space and time. When applied to level set evolution, BFECC reduces volume loss significantly. We demonstrate the benefits of this approach on image advection and on the simulation of smoke, bubbles in water, and the highly dynamic interaction between water, a solid, and air. We also apply BFECC to dye advection to visualize vector fields.
[1] 135 S. Osher and J.A. Sethian, “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations,” J. Computational Physics, vol. 79, pp. 12-49, 1988.
[2] S.J. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag, 2002.
[3] J.A. Sethian, Level Set Methods and Fast Marching Methods. Cambridge Univ. Press, 1999.
[4] J. Stam, “Stable Fluids,” Proc. ACM SIGGRAPH, pp. 121-128, 1999.
[5] B. Jobard, G. Erlebacher, and M.Y. Hussaini, “Lagrangian-Eulerian Advection of Noise and Dye Textures for Unsteady Flow Visualization,” IEEE Trans. Visualization and Computer Graphics, vol. 8, no. 3, July-Sept. 2002.
[6] D. Enright, F. Losasso, and R. Fedkiw, “A Fast and Accurate Semi-Lagrangian Particle Level Set Method,” Computers and Structures, vol. 83, pp. 479-490, 2005.
[7] R. Fedkiw, J. Stam, and H. Jensen, “Visual Simulation of Smoke,” Proc. ACM SIGGRAPH, pp. 23-30, 2001.
[8] A. Selle, N. Rasmussen, and R. Fedkiw, “A Vortex Particle Method for Smoke, Water and Explosions,” Proc. ACM SIGGRAPH, 2005.
[9] S.I. Park and M.J. Kim, “Vortex Fluid for Gaseous Phenomena,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, 2005.
[10] Y. Zhu and R. Bridson, “Animating Sand as a Fluid,” Proc. ACM SIGGRAPH, 2005.
[11] X.D. Liu, S. Osher, and T. Chan, “Weighted Essentially Non-Oscillatory Schemes,” J. Computational Physics, vol. 115, no. 1, pp.200-212, 1994.
[12] G. Jiang and C.-W. Shu, “Efficient Implementation of Weighted ENO Schemes,” J. Computational Physics, vol. 126, pp. 202-228, 1996.
[13] T. Takahashi, H. Fujii, A. Kunimatsu, K. Hiwada, T. Saito, K. Tanaka, and H. Ueki, “Realistic Animation of Fluid with Splash and Foam,” Proc. EUROGRAPHICS, vol. 22, 2003.
[14] O. Song, H. Shin, and H. Ko, “Stable but Nondissipative Water,” ACM Trans. Graphics, vol. 24, no. 1, pp. 81-97, 2005.
[15] D. Enright, S. Marschner, and R. Fedkiw, “Animation and Rendering of Complex Water Surfaces,” Proc. ACM SIGGRAPH, 2002.
[16] T.F. Dupont and Y. Liu, “Back and Forth Error Compensation and Correction Methods for Removing Errors Induced by Uneven Gradients of the Level Set Function,” J. Computational Physics, vol. 190, no. 1, pp. 311-324, 2003.
[17] T.F. Dupont and Y. Liu, “Back and Forth Error Compensation and Correction Methods for Semi-Lagrangian Schemes with Application to Level Set Interface Computations,” Math. of Computation, 2006.
[18] B. Kim, Y. Liu, I. Llamas, and J. Rossignac, “Flowfixer: Using BFECC for Fluid Simulation,” Proc. Eurographics Workshop Natural Phenomena, 2005.
[19] N. Foster and D. Metaxas, “Realistic Animation of Liquids,” Graphical Models and Image Processing, vol. 58, no. 5, pp. 471-483, 1996.
[20] N. Foster and R. Fedkiw, “Practical Animation of Liquids,” Proc. ACM SIGGRAPH, pp. 15-22, 2001.
[21] M. Carlson, P.J. Mucha, and G. Turk, “Rigid Fluid: Animating the Interplay between Rigid Bodies and Fluid,” Proc. ACM SIGGRAPH, 2004.
[22] T.G. Goktekin, A.W. Bargteil, and J.F. O'Brien, “A Method for Animating Viscoelastic Fluids,” Proc. ACM SIGGRAPH, pp. 463-468, 2004.
[23] J.-M. Hong and C.-H. Kim, “Discontinuous Fluids,” Proc. ACM SIGGRAPH, 2005.
[24] Z. Fan, Y. Zhao, A. Kaufman, and Y. He, “Adapted Unstructured lBM for Flow Simulation on Curved Surfaces,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, 2005.
[25] J. Stam, “Flows on Surfaces of Arbitrary Topology,” Proc. ACM SIGGRAPH, pp. 724-731, 2003.
[26] L. Shi and Y. Yu, “Inviscid and Incompressible Fluid Simulation on Triangle Meshes,” Computer Animation and Virtual Worlds, vol. 15, nos. 3-4, pp. 173-181, 2004.
[27] M. Sussman, P. Smereka, and S. Osher, “A Levelset Approach for Computing Solutions to Incompressible Two-Phase Flow,” J.Computational Physics, vol. 114, no. 1, pp. 146-159, 1994.
[28] M. Oevermann, R. Klein, M. Berger, and J. Goodman, “A Projection Method for Two-Phase Incompressible Flow with Surface Tension and Sharp Interface Resolution” Technical Report ZIB-Report 00-17, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 2000.
[29] H. Haario, Z. Korotkaya, P. Luukka, and A. Smolianski, “Computational Modelling of Complex Phenomena in Bubble Dynamics: Vortex Shedding and Bubble Swarms,” Proc. European Congress Computational Methods in Applied Sciences and Eng. (ECCOMAS), 2004.
[30] J.-M. Hong and C.-H. Kim, “Animation of Bubbles in Liquid,” Proc. EUROGRAPHICS, vol. 22, 2003.
[31] D. Weiskopf, “Dye Advection without the Blur: A Level-Set Approach for Texture-Based Visualization of Unsteady Flow,” Proc. EUROGRAPHICS, vol. 23, 2004.
[32] M.P. do Carmo, Riemannian Geometry. Birkhauser, 1994.
[33] F. Losasso, F. Gibou, and R. Fedkiw, “Simulating Water and Smoke with an Octree Data Structure,” Proc. ACM SIGGRAPH, pp.457-462, 2004.
[34] A.S. Almgren, J.B. Bell, and W.G. Szymczak, “A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection,” SIAM J. Scientific Computing, vol. 17, no. 2, Mar. 1996.
[35] M. Sussman, A. Almgren, J. Bell, P. Colella, L. Howell, and M. Welcome, “An Adaptive Level Set Approach for Incompressible Two-Phase Flow,” J. Computational Physics, vol. 148, pp. 81-124, 1999.
[36] E. Guendelman, A. Selle, F. Losasso, and R. Fedkiw, “Coupling Water and Smoke to Thin Deformable and Rigid Shells,” Proc. ACM SIGGRAPH, 2005.
Index Terms:
Advection, diffusion, dissipation, fluid, smoke.
Citation:
ByungMoon Kim, Yingjie Liu, Ignacio Llamas, Jarek Rossignac, "Advections with Significantly Reduced Dissipation and Diffusion," IEEE Transactions on Visualization and Computer Graphics, vol. 13, no. 1, pp. 135-144, Jan./Feb. 2007, doi:10.1109/TVCG.2007.3
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