“Hidden Degrees of Freedom in Implicit Vortex Filaments” by Ishida, Wojtan and Chern
Conference:
Type(s):
Title:
- Hidden Degrees of Freedom in Implicit Vortex Filaments
Session/Category Title: Fluid Simulation
Presenter(s)/Author(s):
Abstract:
This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.
References:
1. Luigi Ambrosio and Halil Mete Soner. 1996. Level set approach to mean curvature flow in arbitrary codimension. Journal of differential geometry 43, 4 (1996), 693–737.
2. Alexis Angelidis. 2017. Multi-scale vorticle fluids. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1–12.
3. 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.
4. Vladimir I. Arnold and Boris A. Khesin. 1998. Topological Methods in Hydrodynamics. Springer.
5. James Arvo. 1995. Applications of irradiance tensors to the simulation of non-lambertian phenomena. Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, 335–342.
6. Miklós Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, and Eitan Grinspun. 2008. Discrete Elastic Rods., Article 63 (2008), 12 pages.
7. Fabrice Bethuel, Haïm Brezis, Frédéric Hélein, et al. 1994. Ginzburg-landau vortices. Vol. 13. Springer.
8. Michael G. Bevis and Greg Cambareri. 1987. Computing the area of a spherical polygon of arbitrary shape. Mathematical Geology 19 (1987), 335–346.
9. Jack Binysh and Gareth P Alexander. 2018. Maxwell’s theory of solid angle and the construction of knotted fields. Journal of Physics A: Mathematical and Theoretical 51, 38 (aug 2018), 385202.
10. 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.
11. Paul Burchard, Li-Tien Cheng, Barry Merriman, and Stanley Osher. 2001. Motion of curves in three spatial dimensions using a level set approach. J. Comput. Phys. 170, 2 (2001), 720–741.
12. Albert Chern. 2017. Fluid dynamics with incompressible Schrödinger flow. Ph.D. Dissertation. California Institute of Technology.
13. Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2017. Inside fluids: Clebsch maps for visualization and processing. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1–11.
14. Albert Chern, Felix Knöppel, Ulrich Pinkall, Peter Schröder, and Steffen Weißmann. 2016. Schrödinger’s smoke. ACM Transactions on Graphics (TOG) 35, 4 (2016), 1–13.
15. Alexandre Joel Chorin. 1990. Hairpin removal in vortex interactions. J. Comput. Phys. 91, 1 (1990), 1–21.
16. A. Clebsch. 1859. Ueber die Integration der hydrodynamischen Gleichungen. Journal für die reine und angewandte Mathematik 56 (1859), 1–10. English translation by D. H. Delphenich, http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/clebsch_-_clebsch_variables.pdf.
17. Georges-Henri Cottet, Petros D Koumoutsakos, et al. 2000. Vortex methods: theory and practice. Vol. 8. Cambridge university press Cambridge.
18. Fang Da, Christopher Batty, Chris Wojtan, and Eitan Grinspun. 2015. Double bubbles sans toil and trouble: Discrete circulation-preserving vortex sheets for soap films and foams. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1–9.
19. 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.
20. 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.
21. Evgenii A Kuznetsov and Aleksandr V Mikhailov. 1980. On the topological meaning of canonical Clebsch variables. Physics Letters A 77, 1 (1980), 37–38.
22. Horace Lamb. 1895. Hydrodynamics. Cambridge University Press.
23. William E. Lorensen and Harvey E. Cline. 1987. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. SIGGRAPH Comput. Graph. 21, 4 (aug 1987), 163–169.
24. Jerrold Marsden and Alan Weinstein. 1983. Coadjoint Orbits, Vortices, and Clebsch Variables for Incompressible Fluids. Physica D: Nonlinear Phenomena 7, 1 (1983), 305–323.
25. Chohong Min. 2004. Local level set method in high dimension and codimension. Journal of computational physics 200, 1 (2004), 368–382.
26. Philip J Morrison. 1998. Hamiltonian description of the ideal fluid. Reviews of modern physics 70, 2 (1998), 467.
27. Kunio Murasugi. 2008 – 1996. Knot theory & its applications. Birkhäuser, Boston.
28. Ken Museth, Nick Avramoussis, and Dan Bailey. 2019. OpenVDB. In ACM SIGGRAPH 2019 Courses. 1–56.
29. Mohammad Sina Nabizadeh, Albert Chern, and Ravi Ramamoorthi. 2021. Kelvin Transformations for Simulations on Infinite Domains. ACM Transactions on Graphics (TOG) 40, 4 (2021), 97:1–97:15.
30. Stanley Osher and James A Sethian. 1988. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of computational physics 79, 1 (1988), 12–49.
31. Marcel Padilla, Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2019. On bubble rings and ink chandeliers. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–14.
32. David Palmer, David Bommes, and Justin Solomon. 2020. Algebraic representations for volumetric frame fields. ACM Transactions on Graphics (TOG) 39, 2 (2020), 1–17.
33. 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.
34. Tobias Pfaff, Nils Thuerey, and Markus Gross. 2012. Lagrangian vortex sheets for animating fluids. ACM Transactions on Graphics (TOG) 31, 4 (2012), 1–8.
35. Len M Pismen, Len M Pismen, et al. 1999. Vortices in nonlinear fields: from liquid crystals to superfluids, from non-equilibrium patterns to cosmic strings. Vol. 100. Oxford University Press.
36. Ziyin Qu, Xinxin Zhang, Ming Gao, Chenfanfu Jiang, and Baoquan Chen. 2019. Efficient and conservative fluids using bidirectional mapping. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–12.
37. Steven J Ruuth, Barry Merriman, Jack Xin, and Stanley Osher. 2001. Diffusion-generated motion by mean curvature for filaments. Journal of Nonlinear Science 11, 6 (2001), 473–493.
38. Philip Geoffrey Saffman. 1992. Vortex Dynamics.
39. H. Seifert. 1935. Über das Geschlecht von Knoten. Math. Ann. 110, 1 (1935), 571–592.
40. 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.
41. Andrew Selle, Nick Rasmussen, and Ronald Fedkiw. 2005. A vortex particle method for smoke, water and explosions. In ACM SIGGRAPH 2005 Papers. 910–914.
42. Justin Solomon, Amir Vaxman, and David Bommes. 2017. Boundary element octahedral fields in volumes. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1.
43. Steffen Weißmann and Ulrich Pinkall. 2009. Real-time Interactive Simulation of Smoke Using Discrete Integrable Vortex Filaments. In Workshop in Virtual Reality Interactions and Physical Simulation “VRIPHYS” (2009), Hartmut Prautzsch, Alfred Schmitt, Jan Bender, and Matthias Teschner (Eds.). The Eurographics Association.
44. Steffen Weißmann and Ulrich Pinkall. 2010. Filament-Based Smoke with Vortex Shedding and Variational Reconnection. ACM Trans. Graph. 29, 4, Article 115 (jul 2010), 12 pages.
45. Steffen Weißmann and Ulrich Pinkall. 2012. Underwater Rigid Body Dynamics. ACM Trans. Graph. 31, 4, Article 104 (jul 2012), 7 pages.
46. Steffen Weißmann, Ulrich Pinkall, and Peter Schröder. 2014. Smoke Rings from Smoke. ACM Trans. Graph. 33, 4, Article 140 (jul 2014), 8 pages.
47. 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.
48. 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), 1–11.
49. Xinxin Zhang and Robert Bridson. 2014. A PPPM fast summation method for fluids and beyond. ACM Transactions on Graphics (TOG) 33, 6 (2014), 1–11.
50. 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), 1–8.
