“Fluid Cohomology” by Yin, Nabizadeh, Wu, Wang and Chern
Conference:
Type:
Title:
- Fluid Cohomology
Session/Category Title: Going With The Flow
Presenter(s)/Author(s):
Moderator(s):
Abstract:
The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.
References:
1. Ralph Abraham, Jerrold E Marsden, and Tudor Ratiu. 2012. Manifolds, tensor analysis, and applications. Vol. 75. Springer Science & Business Media.
2. Christopher R Anderson. 1989. Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows. J. Comput. Phys. 80, 1 (1989), 72–97.
3. Ryoichi Ando, Nils Thuerey, and Chris Wojtan. 2015a. A Stream Function Solver for Liquid Simulations. ACM Transactions on Graphics (TOG) 34, 4 (2015), 53:1–52:9.
4. Ryoichi Ando, Nils Thuerey, and Chris Wojtan. 2015b. A Hindsight on the Stream Function Solver for Liquid Simulation (Errata for [Ando et al. 2015a]). https://ryichando.graphics/download/hindsight_streamfunc.pdf
5. Alexis Angelidis and Fabrice Neyret. 2005. Simulation of smoke based on vortex filament primitives. In Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation. 87–96.
6. Vladimir I. Arnold. 1966. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. In Annales de l’institut Fourier, Vol. 16. 319–361.
7. Vladimir I. Arnold. 1969. On one-dimensional cohomology of the Lie algebra of divergence-free vector fields and on rotation numbers of dynamic systems. In Vladimir I. Arnold-Collected Works. Springer, 179–182.
8. Vladimir I. Arnold and Boris A. Khesin. 1998. Topological Methods in Hydrodynamics. Springer.
9. Omri Azencot, Steffen Weißmann, Maks Ovsjanikov, Max Wardetzky, and Mirela Ben-Chen. 2014. Functional fluids on surfaces. In Computer Graphics Forum, Vol. 33. Wiley Online Library, 237–246.
10. Augustin Banyaga. 1978. Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. Commentarii Mathematici Helvetici 53, 1 (1978), 174–227.
11. Ayan Biswas, Richard Strelitz, Jonathan Woodring, Chun-Ming Chen, and Han-Wei Shen. 2016. A scalable streamline generation algorithm via flux-based isocontour extraction. In Proceedings of the 16th Eurographics Symposium on Parallel Graphics and Visualization. 69–78.
12. Stefanella Boatto and Jair Koiller. 2015. Vortices on closed surfaces. In Geometry, mechanics, and dynamics. Springer, 185–237.
13. Alain Bossavit. 1998. Computational Electromagnetism. Academic Press. https://www.sciencedirect.com/science/article/pii/B9780121187101500015
14. Robert Bridson. 2015. Fluid simulation for computer graphics (2 ed.). CRC press.
15. Robert Bridson, Jim Houriham, and Marcus Nordenstam. 2007. Curl-noise for procedural fluid flow. ACM Transactions on Graphics (TOG) 26, 3 (2007), 46–es.
16. Tyson Brochu, Todd Keeler, and Robert Bridson. 2012. Linear-time smoke animation with vortex sheet meshes. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Citeseer, 87–95.
17. Jumyung Chang, Vinicius C. Azevedo, and Christopher Batty. 2019. Divergence-Free and Boundary-Respecting Velocity Interpolation Using Stream Functions. In Proceedings of the 18th Annual ACM SIGGRAPH/Eurographics Symposium on Computer Animation (Los Angeles, California) (SCA ’19). Association for Computing Machinery, New York, NY, USA, 7:1–7:2.
18. Jumyung Chang, Ruben Partono, Vinicius C Azevedo, and Christopher Batty. 2022. Curl-Flow: Boundary-Respecting Pointwise Incompressible Velocity Interpolation for Grid-Based Fluids. ACM Transactions on Graphics (TOG) 41, 6 (2022), 243:1–243:21.
19. Albert Chern, Felix Knöppel, Ulrich Pinkall, Peter Schröder, and Steffen Weißmann. 2016. Schrödinger’s smoke. ACM Transactions on Graphics (TOG) 35, 4 (2016), 77:1–77:13.
20. Alexandre Joel Chorin and Jerrold E Marsden. 1990. A mathematical introduction to fluid mechanics. Vol. 168. Springer.
21. Qiaodong Cui, Timothy Langlois, Pradeep Sen, and Theodore Kim. 2021. Spiral-spectral fluid simulation. ACM Transactions on Graphics (TOG) 40, 6 (2021), 202:1–202:16.
22. Tyler De Witt, Christian Lessig, and Eugene Fiume. 2012. Fluid simulation using laplacian eigenfunctions. ACM Transactions on Graphics (TOG) 31, 1 (2012), 10:1–10:11.
23. Paweł Dłotko. 2012. A Fast Algorithm to Compute Cohomology Group Generators of Orientable 2-Manifolds. Pattern Recogn. Lett. 33, 11 (2012), 1468–1476.
24. Weinan E and Jian-Guo Liu. 1996. Vorticity boundary condition and related issues for finite difference schemes. Journal of computational physics 124, 2 (1996), 368–382.
25. Sharif Elcott, Yiying Tong, Eva Kanso, Peter Schröder, and Mathieu Desbrun. 2007. Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics (TOG) 26, 1 (2007), 4–es.
26. David Eppstein. 2003. Dynamic Generators of Topologically Embedded Graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, Maryland) (SODA ’03). Society for Industrial and Applied Mathematics, USA, 599–608.
27. Jeff Erickson and Kim Whittlesey. 2005. Greedy Optimal Homotopy and Homology Generators. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Vancouver, British Columbia) (SODA ’05). Society for Industrial and Applied Mathematics, USA, 1038–1046.
28. Manuel Noronha Gamito, Pedro Faria Lopes, and Mário Rui Gomes. 1995. Two-dimensional simulation of gaseous phenomena using vortex particles. In Computer Animation and Simulation’95. Springer, 3–15.
29. JS Godfrey. 1989. A Sverdrup model of the depth-integrated flow for the world ocean allowing for island circulations. Geophysical & Astrophysical Fluid Dynamics 45, 1–2 (1989), 89–112.
30. Abhinav Golas, Rahul Narain, Jason Sewall, Pavel Krajcevski, Pradeep Dubey, and Ming Lin. 2012. Large-scale fluid simulation using velocity-vorticity domain decomposition. ACM Transactions on Graphics (TOG) 31, 6 (2012), 148:1–148:9.
31. Francis H Harlow and J Eddie Welch. 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. The physics of fluids 8, 12 (1965), 2182–2189.
32. A. Hatcher. 2002. Algebraic Topology. Cambridge University Press.
33. Anil Nirmal Hirani. 2003. Discrete exterior calculus. California Institute of Technology.
34. Weizhen Huang, Julian Iseringhausen, Tom Kneiphof, Ziyin Qu, Chenfanfu Jiang, and Matthias B Hullin. 2020. Chemomechanical simulation of soap film flow on spherical bubbles. ACM Transactions on Graphics (TOG) 39, 4 (2020), 41:1–41:13.
35. Sadashige Ishida, Peter Synak, Fumiya Narita, Toshiya Hachisuka, and Chris Wojtan. 2020. A model for soap film dynamics with evolving thickness. ACM Transactions on Graphics (TOG) 39, 4 (2020), 31:1–31:11.
36. Sadashige Ishida, Chris Wojtan, and Albert Chern. 2022. Hidden Degrees of Freedom in Implicit Vortex Filaments. ACM Transactions on Graphics (TOG) 41, 6 (2022), 24:1–24:12.
37. BA Khesin and Yu V Chekanov. 1989. Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity in D dimensions. Physica D: Nonlinear Phenomena 40, 1 (1989), 119–131.
38. Boris Khesin, Daniel Peralta-Salas, and Cheng Yang. 2022. The helicity uniqueness conjecture in 3D hydrodynamics. Trans. Amer. Math. Soc. 375, 02 (2022), 909–924.
39. ByungMoon Kim, Yingjie Liu, Ignacio Llamas, and Jaroslaw R Rossignac. 2005. Flowfixer: Using BFECC for fluid simulation. Technical Report. Georgia Institute of Technology.
40. John M Lee. 2013. Smooth manifolds. In Introduction to smooth manifolds. Springer, 1–31.
41. CC Lin. 1941. On the motion of vortices in two dimensions: I. Existence of the Kirchhoff-Routh function. Proceedings of the National Academy of Sciences 27, 12 (1941), 570–575.
42. Beibei Liu, Gemma Mason, Julian Hodgson, Yiying Tong, and Mathieu Desbrun. 2015. Model-reduced variational fluid simulation. ACM Transactions on Graphics (TOG) 34, 6 (2015), 244:1–244: 12.
43. Jerrold Marsden and Alan Weinstein. 1983. Coadjoint Orbits, Vortices, and Clebsch Variables for Incompressible Fluids. Physica D: Nonlinear Phenomena 7, 1 (1983), 305–323.
44. Akira Mizukami. 1983. A stream function-vorticity finite element formulation for Navier-Stokes equations in multi-connected domain. International journal for numerical methods in engineering 19, 9 (1983), 1403–1409.
45. Philip J Morrison. 1998. Hamiltonian description of the ideal fluid. Reviews of modern physics 70, 2 (1998), 467.
46. Mohammad Sina Nabizadeh, Stephanie Wang, Ravi Ramamoorthi, and Albert Chern. 2022. Covector fluids. ACM Transactions on Graphics (TOG) 41, 4 (2022), 113:1–113:16.
47. Steven A Orszag and Moshe Israeli. 1974. Numerical simulation of viscous incompressible flows. Annual Review of Fluid Mechanics 6, 1 (1974), 281–318.
48. VI Oseledets. 1989. On a new way of writing the Navier-Stokes equation. The Hamiltonian formalism. Russ. Math. Surveys 44 (1989), 210–211.
49. Marcel Padilla. 2018. Point vortex dynamics on closed surfaces. Master thesis, Technische Universität Berlin (2018).
50. David Palmer, Dmitriy Smirnov, Stephanie Wang, Albert Chern, and Justin Solomon. 2022. DeepCurrents: Learning Implicit Representations of Shapes with Boundaries. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2022).
51. Sang Il Park and Myoung Jun Kim. 2005. Vortex fluid for gaseous phenomena. In Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation. 261–270.
52. Dmitry Pavlov, Patrick Mullen, Yiying Tong, Eva Kanso, Jerrold E Marsden, and Mathieu Desbrun. 2011. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena 240, 6 (2011), 443–458.
53. Joseph Pedlosky, Lawrence J Pratt, Michael A Spall, and Karl R Helfrich. 1997. Circulation around islands and ridges. Journal of Marine Research 55, 6 (1997), 1199–1251.
54. Konstantin Poelke and Konrad Polthier. 2016. Boundary-aware Hodge decompositions for piecewise constant vector fields. Computer-Aided Design 78 (2016), 126–136.
55. Luigi Quartapelle. 1993. Numerical solution of the incompressible Navier-Stokes equations. Vol. 113. Springer Science & Business Media.
56. L Quartapelle and F Valz-Gris. 1981. Projection conditions on the vorticity in viscous incompressible flows. International Journal for Numerical Methods in Fluids 1, 2 (1981), 129–144.
57. M Rank and A Voigt. 2021. Active flows on curved surfaces. Physics of Fluids 33, 7 (2021), 072110.
58. Dietmar Rempfer. 2006. On boundary conditions for incompressible Navier-Stokes problems. Applied Mechanics Reviews 59 (2006). Issue 3.
59. Damien Rioux-Lavoie, Ryusuke Sugimoto, Tümay Özdemir, Naoharu H. Shimada, Christopher Batty, Derek Nowrouzezahrai, and Toshiya Hachisuka. 2022. A Monte Carlo Method for Fluid Simulation. ACM Transactions on Graphics (TOG) 41, 6 (2022), 240:1–240:16.
60. Rick Salmon. 1988. Hamiltonian fluid mechanics. Annual review of fluid mechanics 20, 1 (1988), 225–256.
61. Syuhei Sato, Yoshinori Dobashi, Kei Iwasaki, Tsuyoshi Yamamoto, and Tomoyuki Nishita. 2014. Deformation of 2D Flow Fields Using Stream Functions. In SIGGRAPH Asia 2014 Technical Briefs (Shenzhen, China) (SA ’14). Association for Computing Machinery, New York, NY, USA, 4:1–4:4.
62. Syuhei Sato, Yoshinori Dobashi, and Theodore Kim. 2021. Stream-Guided Smoke Simulations. ACM Transactions on Graphics (TOG) 40, 4 (2021), 161:1–161:7.
63. Günter Schwarz. 2006. Hodge Decomposition-A method for solving boundary value problems. Springer.
64. Andrew Selle, Ronald Fedkiw, Byungmoon Kim, Yingjie Liu, and Jarek Rossignac. 2008. An unconditionally stable MacCormack method. Journal of Scientific Computing 35, 2 (2008), 350–371.
65. Lin Shi and Yizhou Yu. 2004. Inviscid and incompressible fluid simulation on triangle meshes. Computer Animation and Virtual Worlds 15, 3–4 (2004), 173–181.
66. Clayton Shonkwiler. 2009. Poincaré duality angles on Riemannian manifolds with boundary. Ph. D. Dissertation. University of Pennsylvania.
67. Yousuf Soliman, Albert Chern, Olga Diamanti, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2021. Constrained Willmore surfaces. ACM Transactions on Graphics (TOG) 40, 4 (2021), 112:1–112:17.
68. Jos Stam. 2003. Flows on surfaces of arbitrary topology. ACM Transactions On Graphics (TOG) 22, 3 (2003), 724–731.
69. Cedric Taylor and Paul Hood. 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Computers & Fluids 1, 1 (1973), 73–100.
70. Tayfun E Tezduyar, Roland Glowinski, and J Liou. 1988. Petrov-Galerkin methods on multiply connected domains for the vorticity-stream function formulation of the incompressible Navier-Stokes equations. International journal for numerical methods in fluids 8, 10 (1988), 1269–1290.
71. Alexander Thom. 1933. The flow past circular cylinders at low speeds. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 141, 845 (1933), 651–669.
72. William Thomson. 1868. On Vortex Motion. Earth and Environmental Science Transactions of the Royal Society of Edinburgh 25, 1 (1868), 217–260.
73. Joris Vankerschaver, Eva Kanso, and Jerrold E Marsden. 2009. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics 1, 2 (2009), 223–266.
74. Mauricio Vines, Ben Houston, Jochen Lang, and Won-Sook Lee. 2013. Vortical inviscid flows with two-way solid-fluid coupling. IEEE Transactions on Visualization and Computer Graphics 20, 2 (2013), 303–315.
75. Stephanie Wang and Albert Chern. 2021. Computing Minimal Surfaces with Differential Forms. ACM Transactions on Graphics (TOG) 40, 4 (2021), 113:1–113:14.
76. Alan Weinstein. 1998. Poisson geometry. Differential Geometry and its applications 9, 1–2 (1998), 213–238.
77. Steffen Weißmann. 2014. Hamiltonian dynamics of several rigid bodies interacting with point vortices. Journal of Nonlinear Science 24 (2014), 359–382.
78. Steffen Weißmann and Ulrich Pinkall. 2010. Filament-Based Smoke with Vortex Shedding and Variational Reconnection. ACM Transactions on Graphics (TOG) 29, 4 (2010), 115:1–115:12.
79. Steffen Weißmann and Ulrich Pinkall. 2012. Underwater rigid body dynamics. ACM Transactions on Graphics (TOG) 31, 4 (2012), 104:1–104:7.
80. Shiying Xiong, Rui Tao, Yaorui Zhang, Fan Feng, and Bo Zhu. 2021. Incompressible Flow Simulation on Vortex Segment Clouds. ACM Transactions on Graphics (TOG) 40, 4 (2021), 98:1–98:11.
81. Larry Yaeger, Craig Upson, and Robert Myers. 1986. Combining Physical and Visual Simulation—Creation of the Planet Jupiter for the Film “2010” (SIGGRAPH ’86). Association for Computing Machinery, New York, NY, USA, 85–93.
82. Shuqi Yang, Shiying Xiong, Yaorui Zhang, Fan Feng, Jinyuan Liu, and Bo Zhu. 2021. Clebsch gauge fluid. ACM Transactions on Graphics (TOG) 40, 4 (2021), 99:1–99:11.
83. Xinxin Zhang and Robert Bridson. 2014. A PPPM fast summation method for fluids and beyond. ACM Transactions on Graphics (TOG) 33, 6 (2014), 206:1–206:11.
84. Xinxin Zhang, Robert Bridson, and Chen Greif. 2015. Restoring the missing vorticity in advection-projection fluid solvers. ACM Transactions on Graphics (TOG) 34, 4 (2015), 52:1–52:8.
85. Rundong Zhao, Mathieu Desbrun, Guo-Wei Wei, and Yiying Tong. 2019. 3D Hodge decompositions of edge-and face-based vector fields. ACM Transactions on Graphics (TOG) 38, 6 (2019), 181:1–181:13.
