Functional Fluids on Surfaces
Fluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triang...
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| Published in | Computer graphics forum Vol. 33; no. 5; pp. 237 - 246 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Blackwell Publishing Ltd
01.08.2014
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| Subjects | |
| Online Access | Get full text |
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| Abstract | Fluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triangle mesh. Unlike the commonly used Eulerian description of the fluid using its time‐varying velocity field, we propose to model fluids using their vorticity, i.e., by a (time varying) scalar function on the surface. During each time step, we advance scalar vorticity along two consecutive, stationary velocity fields. This approach leads to a variational integrator in the space continuous setting. In addition, using this approach, the update rule amounts to manipulating functions on the surface using linear operators, which can be discretized efficiently using the recently introduced functional approach to vector fields. Combining these time and space discretizations leads to a conceptually and algorithmically simple approach, which is efficient, time‐reversible and conserves vorticity by construction. We further demonstrate that our method exhibits no numerical dissipation and is able to reproduce intricate phenomena such as vortex shedding from boundaries. |
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| AbstractList | Fluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triangle mesh. Unlike the commonly used Eulerian description of the fluid using its time‐varying velocity field, we propose to model fluids using their vorticity, i.e., by a (time varying) scalar function on the surface. During each time step, we advance scalar vorticity along two consecutive, stationary velocity fields. This approach leads to a variational integrator in the space continuous setting. In addition, using this approach, the update rule amounts to manipulating functions on the surface using linear operators, which can be discretized efficiently using the recently introduced functional approach to vector fields. Combining these time and space discretizations leads to a conceptually and algorithmically simple approach, which is efficient, time‐reversible and conserves vorticity by construction. We further demonstrate that our method exhibits no numerical dissipation and is able to reproduce intricate phenomena such as vortex shedding from boundaries. Fluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triangle mesh. Unlike the commonly used Eulerian description of the fluid using its time-varying velocity field, we propose to model fluids using their vorticity, i.e., by a (time varying) scalar function on the surface. During each time step, we advance scalar vorticity along two consecutive, stationary velocity fields. This approach leads to a variational integrator in the space continuous setting. In addition, using this approach, the update rule amounts to manipulating functions on the surface using linear operators, which can be discretized efficiently using the recently introduced functional approach to vector fields. Combining these time and space discretizations leads to a conceptually and algorithmically simple approach, which is efficient, time-reversible and conserves vorticity by construction. We further demonstrate that our method exhibits no numerical dissipation and is able to reproduce intricate phenomena such as vortex shedding from boundaries. [PUBLICATION ABSTRACT] |
| Author | Ovsjanikov, Maks Wardetzky, Max Weißmann, Steffen Ben-Chen, Mirela Azencot, Omri |
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| Copyright | 2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd. |
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| References_xml | – reference: Ovsjanikov M., Ben-Chen M., Solomon J., Butscher A., Guibas L.: Functional maps: a flexible representation of maps between shapes. ACM Trans. Graph. 31, 4 (July 2012), 30:1-30:11. 2, 3, 4 – reference: Zhu J., Liu Y., Chang Y., Wu E.: Animating turbulent water by vortex shedding in pic/flip. Science China Information Sciences 56, 3 (2013), 1-11. 3 – reference: Pavlov D., Mullen P., Tong Y., Kanso E., Marsden J.E., Desbrun M.: Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena 240, 6 (2011), 443-458. 3, 5, 6, 7 – reference: Moser J., Veselov A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Communications in Mathematical Physics 139, 2 (1991), 217-243. 2, 5 – reference: Crowdy D., Marshall J.: Analytical solutions for rotating vortex arrays involving multiple vortex patches. Journal of Fluid Mechanics 523 (2005), 307-337. 1 – reference: McKenzie A.: HOLA: a High-Order Lie Advection of discrete differential forms, with applications in fluid dynamics. PhD thesis, California Institute of Technology, 2007. 7, 8 – reference: Weissmann S., Pinkall U.: Filament-based smoke with vortex shedding and variational reconnection. ACM Trans. Graph. 29, 4 (2010), 115:1-115:12. 3 – reference: Auer S., Macdonald C.B., Treib M., Schneider J., Westermann R.: Real-time fluid effects on surfaces using the Closest Point Method. Computer Graphics Forum 31, 6 (2012), 1909-1923. 2 – reference: Polthier K., Preuss E.: Identifying vector field singularities using a discrete hodge decomposition. Visualization and Mathematics 3 (2003), 113-134. 5 – reference: Marsden J.E., Pekarsky S., Shkoller S.: Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity 12, 6 (1999), 1647-1662. 2, 5 – reference: Pfaff T., Thuerey N., Selle A., Gross M.: Synthetic turbulence using artificial boundary layers. ACM Trans. Graph. 28, 5 (2009), 121:1-121:10. 3 – reference: Anderson Jr J.D.: Ludwig Prandtl's boundary layer. Physics Today 58, 12 (December 2005). 3 – reference: Azencot O., Ben-Chen M., Chazal F., Ovsjanikov M.: An operator approach to tangent vector field processing. In Comp. Graph. Forum (Proc. SGP) (2013), vol. 32, pp. 73-82. 2, 3, 4, 5, 6, 9 – reference: Arnold V.I., Khesin B.A.: Topological Methods in Hydrodynamics. Springer, 1998. 3 – reference: Chorin A.J.: Vorticity and Turbulence. Springer, 1994. 7 – reference: Miller J., Weichman P.B., Cross M.C.: Statistical mechanics, Euler's equation, and Jupiter's Red Spot. Phys. Rev. A 45 (Feb 1992), 2328-2359. 1 – reference: Al-Mohy A.H., Higham N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM journal on scientific computing 33, 2 (2011), 488-511. 6, 7 – reference: Mullen P., Crane K., Pavlov D., Tong Y., Desbrun M.: Energy-preserving integrators for fluid animation. ACM Trans. Graph. 28, 3 (2009), 38:1-38:8. 3, 5, 6, 7, 8 – reference: De Witt T., Lessig C., Fiume E.: Fluid simulation using Laplacian eigenfunctions. ACM Transactions on Graphics (TOG) 31, 1 (2012), 10. 2 – reference: Crane K., Weischedel C., Wardetzky M.: Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow. ACM Trans. Graph. 32 (2013). 8 – reference: Bobenko A.I., Suris Y.B.: Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top. Comm. Math. Phys. 204, 1 (1999), 147-188. 2, 5 – reference: Bridson R.: Fluid Simulation for Computer Graphics. A K Peters, 2008. 1 – reference: Elcott S., Tong Y., Kanso E., Schröder P., Desbrun M.: Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics (TOG) 26, 1 (2007), 4. 2 – reference: Davidson P.A.: Turbulence: an introduction for scientists and engineers. Oxford University Press, Oxford, UK New York, 2004. 3 – reference: Hegeman K., Ashikhmin M., Wang H., Qin H., Gu X.: GPU-based conformal flow on surfaces. Commun. Inf. Syst. 9, 2 (2009), 197-212. 2 – reference: Baek S.K.: Vortex interaction on curved surfaces. Phys. Rev. E 86 (Nov 2012), 056603. 1 – reference: Saffman P.G.: Vortex Dynamics. Cambridge Univ Pr, 1992. 2, 3 – reference: Palacios J., Zhang E.: Interactive visualization of rotational symmetry fields on surfaces. Visualization and Computer Graphics, IEEE Transactions on 17, 7 (2011), 947-955. 7 – reference: Shi L., Yu Y.: Inviscid and incompressible fluid simulation on triangle meshes. Computer Animation and Virtual Worlds 15, 3-4 (2004), 173-181. 2 – reference: Chorin A.J., Marsden J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, 2000. 3 – reference: San O., Staples A.E.: A coarse-grid projection method for accelerating incompressible flow computations. Journal of Computational Physics 233 (2013), 480-508. 8 – volume: 9 start-page: 197 issue: 2 year: 2009 end-page: 212 article-title: GPU‐based conformal flow on surfaces publication-title: Commun. Inf. Syst – start-page: 30:1 issue: 4 year: 2012 end-page: 30:11 article-title: Functional maps: a flexible representation of maps between shapes publication-title: ACM Trans. Graph – volume: 56 start-page: 1 issue: 3 year: 2013 end-page: 11 article-title: Animating turbulent water by vortex shedding in pic/flip publication-title: Science China Information Sciences – volume: 139 start-page: 217 issue: 2 year: 1991 end-page: 243 article-title: Discrete versions of some classical integrable systems and factorization of matrix polynomials publication-title: Communications in Mathematical Physics – volume: 3 start-page: 113 year: 2003 end-page: 134 article-title: Identifying vector field singularities using a discrete hodge decomposition publication-title: Visualization and Mathematics – year: 2007 – year: 2000 – volume: 28 start-page: 121:1 issue: 5 year: 2009 end-page: 121:10 article-title: Synthetic turbulence using artificial boundary layers publication-title: ACM Trans. Graph – start-page: 1 year: 2013 end-page: 37 – volume: 22 start-page: 724 year: 2003 end-page: 731 – start-page: 121 year: 1999 end-page: 128 – volume: 204 start-page: 147 issue: 1 year: 1999 end-page: 188 article-title: Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top publication-title: Comm. Math. Phys – year: 1992 – year: 1994 – year: 1998 – start-page: 307 year: 2005 end-page: 319 – volume: 33 start-page: 488 issue: 2 year: 2011 end-page: 511 article-title: Computing the action of the matrix exponential, with an application to exponential integrators publication-title: SIAM journal on scientific computing – volume: 523 start-page: 307 year: 2005 end-page: 337 article-title: Analytical solutions for rotating vortex arrays involving multiple vortex patches publication-title: Journal of Fluid Mechanics – volume: 45 start-page: 2328 year: 1992 end-page: 2359 article-title: Statistical mechanics, Euler's equation, and Jupiter's Red Spot publication-title: Phys. Rev. A – volume: 29 start-page: 115:1 issue: 4 year: 2010 end-page: 115:12 article-title: Filament‐based smoke with vortex shedding and variational reconnection publication-title: ACM Trans. Graph – volume: 26 start-page: 4 issue: 1 year: 2007 article-title: Stable, circulation‐preserving, simplicial fluids publication-title: ACM Transactions on Graphics (TOG) – volume: 15 start-page: 173 issue: 3 year: 2004 end-page: 4 181 article-title: Inviscid and incompressible fluid simulation on triangle meshes publication-title: Computer Animation and Virtual Worlds – volume: 12 start-page: 1647 issue: 6 year: 1999 end-page: 1662 article-title: Discrete Euler‐Poincaré and Lie‐Poisson equations publication-title: Nonlinearity – volume: 17 start-page: 947 issue: 7 year: 2011 end-page: 955 article-title: Interactive visualization of rotational symmetry fields on surfaces publication-title: Visualization and Computer Graphics, IEEE Transactions on – start-page: 261 year: 2005 end-page: 270 – volume: 31 start-page: 10 issue: 1 year: 2012 article-title: Fluid simulation using Laplacian eigenfunctions publication-title: ACM Transactions on Graphics (TOG) – volume: 58 issue: 12 year: 2005 article-title: Ludwig Prandtl's boundary layer publication-title: Physics Today – year: 2008 – volume: 28 start-page: 38:1 issue: 3 year: 2009 end-page: 38:8 article-title: Energy‐preserving integrators for fluid animation publication-title: ACM Trans. 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| Snippet | Fluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean... |
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| SubjectTerms | Algorithms Analysis Categories and Subject Descriptors (according to ACM CCS) Computational fluid dynamics Computer graphics Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-Physically based modeling Computer Graphics [I.3.7]: Three-Dimensional Graphics and Realism-Animation Computer simulation Fluid flow Fluids Mathematical analysis Mathematical models Scalars Simulation Studies Vorticity |
| Title | Functional Fluids on Surfaces |
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