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Constraint Fluids
March 2012 (vol. 18 no. 3)
pp. 516-526
K. Bodin, Umed Univ., Umea, Sweden
C. Lacoursiere, Umed Univ., Umea, Sweden
M. Servin, Dept. of Phys., Umed Univ., Umea, Sweden
We present a fluid simulation method based on Smoothed Particle Hydrodynamics (SPH) in which incompressibility and boundary conditions are enforced using holonomic kinematic constraints on the density. This formulation enables systematic multiphysics integration in which interactions are modeled via similar constraints between the fluid pseudoparticles and impenetrable surfaces of other bodies. These conditions embody Archimede's principle for solids and thus buoyancy results as a direct consequence. We use a variational time stepping scheme suitable for general constrained multibody systems we call SPOOK. Each step requires the solution of only one Mixed Linear Complementarity Problem (MLCP) with very few inequalities, corresponding to solid boundary conditions. We solve this MLCP with a fast iterative method. Overall stability is vastly improved in comparison to the unconstrained version of SPH, and this allows much larger time steps, and an increase in overall performance by two orders of magnitude. Proof of concept is given for computer graphics applications and interactive simulations.

[1] M. Desbrun and M.-P. Cani, “Smoothed Particles: A New Paradigm for Animating Highly Deformable Bodies,” Proc. Eurographics Workshop Computer Animation and Simulation, pp. 61-76, 1996.
[2] D. Stora, P.-O. Agliati, M.-P. Cani, F. Neyret, and J.-D. Gascuel, “Animating Lava Flows,” Proc. Conf. Graphics Interface (GI '99), pp. 203-210, , June 1999.
[3] M. Müller, D. Charypar, and M. Gross, “Particle-Based Fluid Simulation for Interactive Applications,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation (SCA '03), pp. 154-159, 2003.
[4] M. Müller, S. Schirm, and M. Teschner, “Interactive Blood Simulation for Virtual Surgery Based on Smoothed Particle Hydrodynamics,” J. Technology and Health Care, vol. 12, no. 1, pp. 25-31, 2004.
[5] W. Liu, C. Sewell, N. Blevins, K. Salisbury, K. Bodin, and N. Hjelte, “Representing Fluid with Smoothed Particle Hydrodynamics in a Cranial Base Simulator,” Proc. Medicine Meets Virtual Reality (MMVR) Conf., pp. 257-259, 2008.
[6] M. Müller, B. Solenthaler, R. Keiser, and M. Gross, “Particle-Based Fluid-Fluid Interaction,” Proc. Eurographics/ACM SIGGRAPH Symp. Computer Animation '05, pp. 1-7, 2005.
[7] T. Lenaerts and P. Dutré, “Mixing Fluids and Granular Materials,” Proc. 30th Ann. Conf. European Assoc. for Computer Graphics (Eurograpics '09), P. Dutré and M. Stamminger, eds., 2009.
[8] M. Müller, R. Keiser, A. Nealen, M. Pauly, M. Gross, and M. Alexa, “Point Based Animation of Elastic, Plastic and Melting Objects,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, Aug. 2004.
[9] L.B. Lucy, “A Numerical Approach to the Testing of the Fission Hypothesis,” The Astronomical J., vol. 82, pp. 1013-1024, 1977.
[10] R.A. Gingold and J.J. Monaghan, “Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars,” Monthly Notices of the Royal Astronomical Soc., vol. 181, pp. 375-398, 1977.
[11] S. Li and W.K. Liu, “Meshfree and Particle Methods and Their Applications,” Applied Mechanics Rev., vol. 55, pp. 1-34, Jan. 2002.
[12] G.R.G.-R Liu and M.B. Liu, Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, 2003.
[13] J.J. Monaghan, “Smoothed Particle Hydrodynamics,” Reports on Progress in Physics, vol. 68, no. 8, pp. 1703-1759,, 2005.
[14] S. Rusinkiewicz and M. Levoy, “QSplat: A Multiresolution Point Rendering System for Large Meshes,” Proc. ACM SIGGRAPH, pp. 343-352, 2000.
[15] M. Zwicker, M. Pauly, O. Knoll, and M. Gross, “Pointshop 3D: An Interactive System for Point-Based Surface Editing,” Proc. ACM SIGGRAPH, pp. 322-329, 2002.
[16] M. Levoy, “Display of Surfaces from Volume Data,” IEEE Computer Graphics and Applications, vol. 8, no. 3, pp. 29-37, May 1988.
[17] W. McNeely, K. Puterbaugh, and J. Troy, “Six Degree-of-Freedom Haptic Rendering Using Voxel Sampling,” Proc. ACM SIGGRAPH, pp. 401-408, Aug. 1999.
[18] K.L. Palmerius, M. Cooper, and A. Ynnerman, “Haptic Rendering of Dynamic Volumetric Data,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 2, pp. 263-276, Mar. 2008.
[19] W.T. Reeves, “Particle System—A Technique for Modeling a Class of Fuzzy Objects,” Proc. ACM SIGGRAPH, vol. 17, pp. 359-376, July 1983.
[20] M. Levoy and T. Whitted, “The Use of Points as a Display Primitive,” Technical Report 85-022, Computer Science Dept., Jan. 1985.
[21] H. Pfister, M. Zwicker, J. van Baar, and M. Gross, “Surfels: Surface Elements as Rendering Primitives,” Proc. ACM SIGGRAPH, 2000.
[22] M. Gross and H. Pfister, Point-Based Graphics. Morgan-Kauffman, July 2007.
[23] R. Courant, K. Friedrichs, and H. Lewy, “On the Partial Difference Equations of Mathematical Physics,” IBM J. Research and Development, vol. 11, pp. 215-234, 1967.
[24] R. Bridson, Fluid Simulation for Computer Graphics. A.K. Peters, 2008.
[25] S.J. Cummins and M. Rudman, “An SPH Projection Method,” J. Computational Physics, vol. 152, pp. 584-607, 1999.
[26] S. Koshizuka and Y. Oka, “Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid,” Nuclear Science Eng., vol. 123, pp. 421-434, July 1996.
[27] S. Premoze, T. Tasdizen, J. Bigler, A. Lefohn, and R.T. Whitaker, “Particle-Based Simulation of Fluids,” Proc. Ann. Conf. European Assoc. for Computer Graphics (Eurographics '03), vol. 22, no. 3, pp. 401-410, 2003.
[28] E.-S. Lee, C. Moulinec, R. Xuc, D. Violeau, D. Laurence, and P. Stansby, “Comparisons of Weakly Compressible and Truly Incompressible Algorithms for the SPH Mesh Free Particle Method,” J. Computational Physics, vol. 227, pp. 8417-8436, 2008.
[29] M. Ellero, M. Serrano, and P. Español, “Incompressible Smoothed Particle Hydrodynamics,” J. Computational Physics, vol. 226, no. 2, pp. 1731-1752, B6WHY-4P2YWYC-5/2611488181b994c812ccd286bcef63457 , 2007.
[30] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, vol. 31, Springer-Verlag, 2001.
[31] B. Solenthaler and R. Pajarola, “Predictive-Corrective Incompressible SPH,” Proc. ACM SIGGRAPH '09 papers, pp. 1-6, 2009.
[32] M. Müller, B. Heidelberger, M. Hennix, and J. Ratcliff, “Position Based Dynamics,” J. Visual Comm. and Image Representation, vol. 18, no. 2, pp. 109-118, 2007.
[33] G. Batchelor, An Introduction to Fluid Dynamics. Cambridge Univ. Press, 1967.
[34] M. Becker and M. Teschner, “Weakly Compressible SPH for Free Surface Flows,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, pp. 63-72, 2007.
[35] C. Lacoursière, “Ghosts and Machines: Regularized Variational Methods for Interactive Simulations of Multibodies with Dry Frictional Contacts,” PhD Dissertation, Dept. of Computing Science, Umeå Univ., http://urn.kb.seresolve?urn=urn:nbn:se: umu:diva-1143 , June 2007.
[36] U. Ascher, H. Chin, L. Petzold, and S. Reich, “Stabilization of Constrained Mechanical Systems with DAEs and Invariant Manifolds,” J. Mechanics of Structures and Machines, vol. 23, pp. 135-158, 1995.
[37] F.A. Bornemann, Homogenization in Time of Singularly Perturbed Mechanical Systems. Springer, 1998.
[38] H. Rubin and P. Ungar, “Motion under a Strong Constraining Force,” Comm. on Pure and Applied Math., vol. 10, pp. 65-87, 1957.
[39] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, second ed., vol. 14, Springer-Verlag, 1996.
[40] J. Baumgarte, “Stabilization of Constraints and Integrals of Motion in Dynamical Systems,” Computer Methods in Applied Mechanics and Eng., vol. 1, no. 1, pp. 1-16, 1972.
[41] C. Lanczos, The Variational Principles of Mechanics, fourth ed., Dover Publications, 1986.
[42] V.I. Arnold, Mathematical Methods of Classical Mechanics, second ed., vol. 60, translated from the Russian by K. Vogtmann and A. Weinstein, Springer-Verlag, 1989.
[43] A.J. Kurdila and F.J. Narcowich, “Sufficient Conditions for Penalty Formulation Methods in Analytical Dynamics,” Computational Mechanics, vol. 12, pp. 81-96, 1993.
[44] M. Servin and C. Lacoursière, “Rigid Body Cable for Virtual Environments,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 4, pp. 783-796, July/Aug. 2008.
[45] M. Servin, C. Lacoursière, and N. Melin, “Interactive Simulation of Elastic Deformable Materials,” Proc. SIGRAD Conf., pp. 22-32, 2006.
[46] C. Lacoursière, “Regularized, Stabilized, Variational Methods for Multibodies,” Proc. 48th Scandinavian Conf. Simulation and Modeling (SIMS '07), D.F. Peter Bunus and C. Führer, eds., pp. 40-48, Oct. 2007.
[47] C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, vol. 16, SIAM, 1995.
[48] M. Yildiz, R.A. Rook, and A. Suleman, “SPH with the Multiple Boundary Tangent Method,” Int'l J. Numerical Methods in Eng., vol. 77, pp. 1416-1438,, 2009.
[49] B.V. Mirtich, “Impulse-Based Dynamic Simulation of Rigid Body Systems,” PhD dissertation, Univ. of California at Berkeley, 1996.
[50] AgX, “AgX Multiphysics SDK,” http://www.algoryx.seagx, 2009.
[51] POV-Ray, “Persistence of Vision Raytracer v.3.6,” http:/, 2008.
[52] Algodoo, “Algodoo 2D Physics Sandbox v5.42,” http://www. algoryx.sealgodoo, 2009.
[53] S. Kitsionas and A.P. Whitworth, “Smoothed Particle Hydrodynamics with Particle Splitting, Applied to Self-gravitating Collapse,” Monthly Notices of the Royal Astronomical Soc., vol. 330, pp. 129-136, 2002.
[54] B. Adams, M. Pauly, R. Keiser, and L.J. Guibas, “Adaptively Sampled Particle Fluids,” ACM Trans. Graphics, vol. 26, no. 3, 2007.
[55] M. Müller, B. Heidelberger, M. Hennix, and J. Ratcliff, “Position Based Dynamics,” Proc. Third Workshop Virtual Reality Interactions and Physical Simulation, 2006.
[56] L. Kharevych, W. Yang, Y. Tong, E. Kanso, J.E. Marsden, P. Schröder, and M. Desbrun, “Geometric Variational Integrators for Computer Animation,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation (SCA '06), pp. 43-51, 2006.
[57] H.C. Elman, D.J. Silvester, and A.J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford Univ. Press, 2005.
[58] S. Koshizuka, Y. Oka, and H. Tamako, “A Particle Method for Calculating Splashing of Incompressible Viscous Fluid,” Proc. Int'l Conf. Math. and Computations, Reactor Physics and Environmental Analysis, vol. 2, pp. 1514-1521, 1995.
[59] F. Losasso, T. Shinar, A. Selle, and R. Fedkiw, “Multiple Interacting Liquids,” Proc. ACM SIGGRAPH, vol. 25, pp. 812-819, 2006.
[60] M. Carlson, P.J. Mucha, R.B. Van HornIII, and G. Turk, “Melting and Flowing,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation (SCA '02), pp. 167-174, 2002.
[61] Y. Zhu and R. Bridson, “Animating Sand as a Fluid,” Proc. ACM SIGGRAPH, 2005.
[62] F. Colin, R. Egli, and F. Lin, “Computing a Null Divergence Velocity Field Using Smoothed Particle Hydrodynamics,” J. Computational Physics, vol. 217, pp. 680-692, 2006.
[63] W.G. Hoover, Smooth Particle Applied Mechanics: The State of the Art. World Scientific, 2006.
[64] M. Kass and G. Miller, “Rapid, Stable Fluid Dynamics for Computer Graphics,” Proc. ACM SIGGRAPH, vol. 24, pp. 49-57, 1990.
[65] J. Stam, “Stable Fluids,” Proc. ACM SIGGRAPH, pp. 121-128, 1999.
[66] J. Chen and N. Lobo, “Toward Interactive-Rate Simulation of Fluids with Moving Obstacle Using the Navier-Stokes Equations,” Computer Graphics and Image Processing, vol. 57, pp. 107-116, 1994.
[67] N. Foster and D. Metaxas, “Realistic Animation of Liquids,” Graphical Models and Image Processing, vol. 58, no. 5, pp. 471-483, 1996.
[68] N. Foster and R. Fedkiw, “Practical Animation of Liquids,” Proc. ACM SIGGRAPH, pp. 23-30, 2001.
[69] D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell, “Animation and Rendering of Complex Water Surfaces,” ACM Trans. Graphics, vol. 21, no. 3, pp. 736-744, 2002.
[70] M.R. Hestenes and E. Stiefel, “Methods of Conjugate Gradients for Solving Linear Systems,” J. Research of the Nat'l Bureau of Standards, vol. 49, pp. 409-436, 1952.
[71] F. Losasso, J.O. Talton, N. Kwatra, and R. Fedkiw, “English Two-Way Coupled SPH and Particle Level Set Fluid Simulation,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 4, pp. 797-804, July/Aug. 2008.
[72] G.R.G.-R Liu and M.B. Liu, Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, 2003.
[73] K. Lundin, M. Sillen, M. Cooper, and A. Ynnerman, “Haptic Visualization of Computational Fluid Dynamics Data Using Reactive Forces,” Proc. Conf. Visualization and Data Analysis, R.F. Erbacher, J.C. Roberts, M.T. Grohn, and K. Borner, eds., pp. 31-41, 2005.
[74] M. Anitescu, “Optimization-Based Simulation of Nonsmooth Rigid Multibody Dynamics,” Math. Programming, vol. 105, no. 1, pp. 113-143, 2006.
[75] R. Goldenthal, D. Harmon, R. Fattal, M. Bercovier, and E. Grinspun, “Efficient Simulation of Inextensible Cloth,” Proc. ACM SIGGRAPH, 2007.

Index Terms:
iterative methods,computational fluid dynamics,computer graphics,digital simulation,hydrodynamics,interactive simulations,constraint fluids,fluid simulation method,smoothed particle hydrodynamics,incompressibility conditions,boundary conditions,holonomic kinematic constraints,systematic multiphysics integration,fluid pseudoparticles,Archimedes principle,buoyancy,SPOOK,mixed linear complementarity problem,fast iterative method,computer graphics applications,Force,Equations,Mathematical model,Computer graphics,Approximation methods,Computational modeling,Stability analysis,variational integrator.,SPH,incompressible,constraints,fluid simulation
K. Bodin, C. Lacoursiere, M. Servin, "Constraint Fluids," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 3, pp. 516-526, March 2012, doi:10.1109/TVCG.2011.29
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