An adaptive variational finite difference framework for efficient symmetric octree viscosity

Abstract

While pressure forces are often the bottleneck in (near-)inviscid fluid simulations, viscosity can impose orders of magnitude greater computational costs at lower Reynolds numbers. We propose an implicit octree finite difference discretization that significantly accelerates the solution of the free surface viscosity equations using adaptive staggered grids, while supporting viscous buckling and rotation effects, variable viscosity, and interaction with scripted moving solids. In experimental comparisons against regular grids, our method reduced the number of active velocity degrees of freedom by as much as a factor of 7.7 and reduced linear system solution times by factors between 3.8 and 9.4. We achieve this by developing a novel adaptive variational finite difference methodology for octrees and applying it to the optimization form of the viscosity problem. This yields a linear system that is symmetric positive definite by construction, unlike naive finite difference/volume methods, and much sparser than a hypothetical finite element alternative. Grid refinement studies show spatial convergence at first order in L_∞ and second order in L₁, while the significantly smaller size of the octree linear systems allows for the solution of viscous forces at higher effective resolutions than with regular grids. We demonstrate the practical benefits of our adaptive scheme by replacing the regular grid viscosity step of a commercial liquid simulator (Houdini) to yield large speed-ups, and by incorporating it into an existing inviscid octree simulator to add support for viscous flows. Animations of viscous liquids pouring, bending, stirring, buckling, and melting illustrate that our octree method offers significant computational gains and excellent visual consistency with its regular grid counterpart.

References

1. Mridul Aanjaneya, Ming Gao, Haixiang Liu, Christopher Batty, and Eftychios Sifakis. 2017. Power diagrams and sparse paged grids for high resolution adaptive liquids. ACM Trans. Graph. (SIGGRAPH) 36, 4 (2017), 140.
2. Ann S. Almgren, John B. Bell, and Phillip Colella. 1998. A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. J. Comp. Phys. 142, 1 (1998), 1--46.
3. Ryoichi Ando, Nils Thuerey, and Chris Wojtan. 2013. Highly adaptive liquid simulations on tetrahedral meshes. ACM Trans. Graph. (SIGGRAPH) 32, 4 (2013), 103.
4. Ryoichi Ando, Nils Thuerey, and Chris Wojtan. 2015a. A dimension-reduced pressure solver for liquid simulations. Computer Graphics Forum (Eurographics) 34, 2 (2015), 473--480.
5. Ryoichi Ando, Nils Thuerey, and Chris Wojtan. 2015b. A stream function solver for liquid simulations. ACM Trans. Graph. (SIGGRAPH) 34, 4 (2015), 53.
6. Luiz Fernando de Souza Andrade, Marcos Sandim, Fabiano Petronetto, Paulo Pagliosa, and Afonso Paiva. 2015. Particle-based fluids for viscous jet buckling. Computers and Graphics 52 (2015), 106--115.
7. Adam W. Bargteil, Chris Wojtan, Jessica K. Hodgins, and Greg Turk. 2007. A finite element method for animating large viscoplastic flow. ACM Trans. Graph. (SIGGRAPH) 26, 3 (2007), 16.
8. Christopher Batty. 2017. A cell-centred finite volume method for the Poisson problem on non-graded quadtrees with second order accurate gradients. J. Comp. Phys. 331 (2017), 49--72.
9. Christopher Batty, Florence Bertails, and Robert Bridson. 2007. A fast variational framework for accurate solid-fluid coupling. ACM Trans. Graph. (SIGGRAPH) 26, 3 (2007), 100.
10. Christopher Batty and Robert Bridson. 2008. Accurate viscous free surfaces for buckling, coiling, and rotating liquids. In Symposium on Computer Animation. 219--228.
11. Christopher Batty and Ben Houston. 2011. A simple finite volume method for adaptive viscous liquids. In Symposium on Computer Animation. 111--118.
12. Christopher Batty, Andres Uribe, Basile Audoly, and Eitan Grinspun. 2012. Discrete viscous sheets. ACM Trans. Graph. (SIGGRAPH) 31, 4 (2012), 113.
13. Christopher Batty, Stefan Xenos, and Ben Houston. 2010. Tetrahedral embedded boundary methods for accurate and flexible adaptive fluids. Computer Graphics Forum (Eurographics) 29, 2 (2010), 695--704.
14. Miklos Bergou, Basile Audoly, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2010. Discrete viscous threads. ACM Trans. Graph. (SIGGRAPH) 29, 4 (2010), 116.
15. Andrea Bonito, Marco Picasso, and Manuel Laso. 2006. Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comp. Phys. 215, 2 (2006), 691--716.
16. Robert Bridson. 2015. Fluid simulation for computer graphics, 2nd edition. AK Peters / CRC Press.
17. Tyson Brochu, Christopher Batty, and Robert Bridson. 2010. Matching fluid simulation elements to surface geometry and topology. ACM Trans. Graph. (SIGGRAPH) 29, 4 (2010), 47.
18. Steve Capell, Seth Green, Brian Curless, Tom Duchamp, and Zoran Popović. 2002. A multiresolution framework for dynamic deformations. In Symposium on Computer Animation. 41--47.
19. Mark Carlson, Peter J. Mucha, R. Van Horn, and Greg Turk. 2002. Melting and flowing. In Symposium on Computer Animation. 167--174.
20. Nuttapong Chentanez, Bryan E. Feldman, François Labelle, James F. O'Brien, and Jonathan Richard Shewchuk. 2007. Liquid simulation on lattice-based tetrahedral meshes. In Symposium on Computer Animation. 219--228.
21. Pascal Clausen, Martin Wicke, Jonathan Richard Shewchuk, and James F. O'Brien. 2013. Simulating liquids and solid-liquid interactions with Lagrangian meshes. ACM Trans. Graph. 32, 2 (2013), 17.
22. Sharif Elcott, Yiying Tong, Eva Kanso, Peter Schröder, and Mathieu Desbrun. 2007. Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 1 (2007), 4.
23. Henrik Fält and Doug Roble. 2003. Fluids with extreme viscosity. In SIGGRAPH Sketches. 1.
24. Florian Ferstl, Rudiger Westermann, and Christian Dick. 2014. Large-scale liquid simulation on adaptive hexahedral grids. IEEE TVCG 20, 10 (2014), 1405--1417.
25. Nick Foster and Dimitris Metaxas. 1996. Realistic animation of liquids. Graphical Models and Image Processing 58, 5 (1996), 471--483.
26. Ming Gao, Andre Pradhana Tampubulon, Chenfanfu Jiang, and Eftychios Sifakis. 2017. An adaptive generalized interpolation material point method for simulating elastoplastic materials. ACM Trans. Graph. (SIGGRAPH Asia) 36, 6 (2017), 223.
27. T.V. Gerya, D. A. May, and T. Duretz. 2013. An adaptive staggered grid finite difference method for modeling geodynamic Stokes flows with strongly variable viscosity. Geochemistry, Geophysics, Geosystems 14, 4 (2013), 1200--1225.
28. Tolga G. Goktekin, Adam W. Bargteil, and James F. O'Brien. 2004. A method for animating viscoelastic fluids. ACM Trans. Graph. (SIGGRAPH) 23, 3 (2004), 463--468.
29. Eitan Grinspun, Peter Krysl, and Peter Schröder. 2002. CHARMS: A simple framework for adaptive simulation. ACM Trans. Graph. (SIGGRAPH) 21, 3 (2002), 281--290.
30. Arthur Guittet, Maxime Theillard, and Frédéric Gibou. 2015. A stable projection method for the incompressible Navier-Stokes equations on arbitrary geometries and adaptive Quad/Octrees. J. Comp. Phys. 292 (2015), 215--238.
31. Jeong-Mo Hong and Chang-Hun Kim. 2005. Discontinuous fluids. ACM Trans. Graph. (SIGGRAPH) 24, 3 (2005), 915--920.
32. Lior Horesh and Eldad Haber. 2011. A second order discretization of Maxwell's equations in the quasi-static regime on octree grids. SIAM J. Sci. Comput. 33, 5 (2011), 2805--2822.
33. Bryan M. Klingner, Bryan E. Feldman, Nuttapong Chentanez, and James F. O'Brien. 2006. Fluid animation with dynamic meshes. ACM Trans. Graph. (SIGGRAPH) 25, 3 (2006), 820--825.
34. Egor Larionov, Christopher Batty, and Robert Bridson. 2017. Variational stokes: a unified pressure-viscosity solver for accurate viscous liquids. ACM Trans. Graph. (SIGGRAPH) 36, 4 (2017), 101.
35. Frank Losasso, Ronald Fedkiw, and Stanley Osher. 2006. Spatially adaptive techniques for level set methods and incompressible flow. Computers & Fluids 35, 10 (2006), 995--1010.
36. Frank Losasso, Frédéric Gibou, and Ron Fedkiw. 2004. Simulating water and smoke with an octree data structure. ACM Trans. Graph. (SIGGRAPH) 23, 3 (2004), 457--462.
37. Chohong Min and Frédéric Gibou. 2006. A second order accurate projection method for the incompressible Navier-Stokes equations on non-graded adaptive grids. J. Comp. Phys. 219, 2 (2006), 912--929.
38. Marek Misztal, Robert Bridson, Kenny Erleben, Andreas Baerentzen, and Francois Anton. 2010. Optimization-based fluid simulation on unstructured meshes. In VRIPHYS. 11--20.
39. Marek Misztal, Kenny Erleben, Adam W. Bargteil, B. Bunch Christensen, Andreas Baerentzen, and Robert Bridson. 2014. Multiphase flow of immiscible fluids on unstructured moving meshes. IEEE TVCG 20, 1 (2014), 4--16.
40. Patrick Mullen, Keenan Crane, Dmitry Pavlov, Yiying Tong, and Mathieu Desbrun. 2009. Energy-preserving integrators for fluid animation. ACM Trans. Graph. (SIGGRAPH) 28, 3 (2009), 38.
41. Rahul Narain, Abhinav Golas, and Ming C. Lin. 2010. Free-flowing granular materials with two-way solid coupling. ACM Trans. Graph. (SIGGRAPH Asia) 29, 6 (2010), 173.
42. Michael B. Nielsen and Robert Bridson. 2016. Spatially adaptive FLIP fluid simulations in Bifrost. In ACM SIGGRAPH talks. 41.
43. K. D. Nikitin, M. A. Olshanskii, K. M. Terekhov, and Y. V. Vassilevski. 2011. A numerical method for the simulation of free surface flows of viscoplastic fluid in 3D. Journal of Computational Mathematics (2011), 605--622.
44. K. D. Nikitin and Yuri V. Vassilevski. 2008. Free surface flow modelling on dynamically refined hexahedral meshes. Russian Journal of Numerical Analysis and Mathematical Modelling 23, 5 (2008), 469--485.
45. Maxim A. Olshanskii, Kirill M. Terekhov, and Yuri V. Vassilevski. 2013. An octree-based solver for the incompressible Navier-Stokes equations with enhanced stability and low dissipation. Computers & Fluids 84 (2013), 231--246.
46. Andreas Peer, Markus Ihmsen, Jens Cornelis, and Matthias Teschner. 2015. An implicit viscosity formulation for SPH fluids. ACM Trans. Graph. (SIGGRAPH) 34, 4 (2015), 114.
47. Daniel Ram, Theodore Gast, Chenfanfu Jiang, Craig Schroeder, Alexey Stomakhin, Joseph Teran, and Pirouz Kavehpour. 2015. A material point method for viscoelastic fluids, foams and sponges. In Symposium on Computer Animation. 157--163.
48. Nick Rasmussen, Doug Enright, Duc Nguyen, Sebastian Marino, N. Sumner, Willi Geiger, Samir Hoon, and Ron Fedkiw. 2004. Directable photorealistic liquids. In Symposium on Computer Animation. 193--202.
49. Rajsekhar Setaluri, Mridul Aanjaneya, Sean Bauer, and Eftychios Sifakis. 2014. SPGrid: A sparse paged grid structure applied to adaptive smoke simulation. ACM Trans. Graph. (SIGGRAPH Asia) 33, 6 (2014), 205.
50. Lin Shi and Yizhou Yu. 2004. Visual smoke simulation with adaptive octree refinement. Computer Graphics and Imaging (2004), 13--19.
51. Side Effects Software. 2018. Houdini.
52. Jos Stam. 1999. Stable fluids. In SIGGRAPH. 121--128.
53. Alexey Stomakhin, Craig Schroeder, Chenfanfu Jiang, Lawrence Chai, Joseph Teran, and Andrew Selle. 2014. Augmented MPM for phase-change and varied materials. ACM Trans. Graph. (SIGGRAPH) 33, 4 (2014), 138.
54. Tetsuya Takahashi, Yoshinori Dobashi, Issei Fujishiro, Tomoyuki Nishita, and Ming C. Lin. 2015. Implicit formulation for SPH-based viscous fluids. Computer Graphics Forum (Eurographics) 34, 2 (2015), 493--502.
55. Jeremy D. Wendt, William Baxter, Ipek Oguz, and Ming C. Lin. 2007. Finite volume flow simulations on arbitrary domains. Graphical Models 69, 1 (2007), 19--32.
56. Chris Wojtan and Greg Turk. 2008. Fast viscoelastic behavior with thin features. ACM Trans. Graph. (SIGGRAPH) 27, 3 (2008), 47.
57. Yonghao Yue, Breannan Smith, Christopher Batty, Changxi Zheng, and Eitan Grinspun. 2015. Continuum foam: A material point method for shear-dependent flows. ACM Trans. Graph. 34, 5 (2015), 160.
58. Yonghao Yue, Breannan Smith, Peter Yichen Chen, Maytee Chantharayukhonthorn, Ken Kamrin, and Eitan Grinspun. 2018. Hybrid Grains: Adaptive Coupling of Discrete and Continuum Simulations of Granular Media. ACM Trans. Graph. (SIGGRAPH Asia) 37, 6 (2018), 283.
59. Bo Zhu, Minjae Lee, Ed Quigley, and Ronald Fedkiw. 2015. Codimensional non-Newtonian fluids. ACM Trans. Graph. (SIGGRAPH) 34, 4 (2015), 115.
60. Yongning Zhu, Eftychios Sifakis, Joseph Teran, and Achi Brandt. 2010. An efficient multigrid method for the simulation of high-resolution elastic solids. ACM Trans. Graph. 29, 2 (2010), 16.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

      • Physical simulation