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A Prediction-Correction Approach for Stable SPH Fluid Simulation from Liquid to RigidFrançois DagenaisJonathan GagnonEric Paquette
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Melting and solidification•Animation of transition between
▫Liquid phase▫Rigid phase
•Non-elastic materials• Lagrangian simulation
▫Almost rigid longer computational times
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Goals• Improved lagrangian simulation of melting objects
▫Improved stability▫Shorter computational times▫Easier control
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Overview•Previous work•Proposed Approach
▫Melting and solidification▫Constraints propagation▫Stability improvements
•Results• Limitations and conclusion
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Previous work•Melting and solidification
▫Solved for eulerian approaches[Stam 1999] [Carlson et al. 2002][Fält and Roble 2003] [Rasmussen et al. 2004][Batty and Bridson 2008]
▫Still a challenge for lagrangianapproaches
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Carlson et al. 2002
Batty and Bridson 2008
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Previous work• Lagrangian
Variable viscosity[Muller et al. 2003]
Elastic [Solenthaler et al. 2007] [Chang et al. 2009]
Plastic[Paiva et al. 2006]
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[Paiva et al. 2006]
[Solenthaler et al. 2007]
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Overview•Previous work•Proposed Approach
▫Melting and solidification▫Constraints propagation▫Stability improvements
•Results• Limitations and conclusion
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Melting and solidification• Integrated in a SPH fluid solver
•Minimisation problem
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Deformation error•Difference between
▫Current deformation▫Target deformation
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Target Deformation•Based on relative position of neighbors
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Rigidity forces correction11
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Rigidity forces correction12
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Rigidity forces correction13
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Integration14
Compute density and pressure
Compute forces (SPH)
Update velocity and position
t > tend ?no
END
yes
Compute rigidity forces
Initialize rigidity forces
Predict particles position
Adjust rigidity forces
Stopping criterion
met?
no
yes
Compute particles deformation error
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Integration15
Initialise rigidity forces
Predict particles position
Adjust rigidity forces
Stopping criterion
met?
no
yes
Compute particles deformation error
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Overview•Previous work•Proposed Approach
▫Melting and solidification▫Constraints propagation▫Stability improvements
•Results• Limitations and conclusion
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Why?•Particles only affect neighbors
▫Slow convergence•Early termination
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Almost no variation of !
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Constraints propagation18
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Constraints propagation19
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Constraints propagation20
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Constraints propagation21
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Overview•Previous work•Proposed Approach
▫Melting and solidification▫Constraints propagation▫Stability improvements
•Results• Limitations and conclusion
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Stability•Other sources of instability
▫Pressure forces▫Heat diffusion
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Adaptative time step•Advantages
▫Stable simulation▫Shorter computational times
•« Courant–Friedrichs–Lewy » condition
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Adaptative time step•Maximum velocity estimation
▫Previous maximal velocity▫Maximal acceleration
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Heat diffusion• Increases simulation realism•A temperature Ti is assigned to each particle
▫Specified by the user▫Updated using heat diffusion equation▫Temperature affects rigidity
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Heat diffusion•Unstable when
▫Large time step▫Large heat diffusion coefficient
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Heat diffusion•Proposed approach
▫Implicit formulation▫Handle individually each pair of neighbor particles
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Heat diffusion – Implicit formulation
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Heat diffusion - video30
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Overview•Previous work•Proposed Approach
▫Melting and solidification▫Constraints propagation▫Stability improvements
•Results• Limitations and conclusion
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Video32
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Example timeper
frame
timeper
iteration
avg.Δt
Ratiotrigide/ttotal
Blocs si = 0.00 17.0s 1.0s 0.00257s
0.33
Blocs si = 0.25 88.1s 9.0s 0.00429s
0.88
Blocs si = 0.50 90.2s 9.9s 0.00463s
0.89
Blocs si = 0.75 56.8s 7.4s 0.00548s
0.91
Blocs si = 0.90 94.5s 14.5s 0.00651s
0.92
Blocs si = 0.99 65.5s 17.1s 0.01096s
0.94
Blocs si = 1.00 23.5s 21.4s 0.03787s
0.97
Stanford’s bunny 480.1s 50.3s 0.00438s
0.97
Stanford’s Armadillo
165.2s 14.1s 0.00359s
0.92
« h » 619.7s 49.3s 0.00333s
0.97
« h » 2 848.7s 53.1s 0.00262s
0.98
Rigid forces computation takes most of the computational timesTime per iteration increases as the fluid become more rigidTimestep independent of rigidityVariable rigidity = longer computational time, because of the propagation conditions
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Comparison with traditionnal viscosity34
μi = 1 000 μ
i = 10 000 μ
i = 100 000
si = 0.75 s
i = 0.92 s
i = 0.98
Traditionnal viscosity Our approachμi Δt Total time si
avg. Δt Total time1 000 6.1x10-4
s47.80 min 0.75 4.05x10-3
s85.03 min
10 000 6.1x10-5 s
484.81 min 0.92 4.80x10-3 s
103.70 min
100 000
5.9x10-6
s4474.26
min0.98 6.36x10-3
s161.65
min
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Overview•Previous work•Proposed Approach
▫Melting and solidification▫Constraints propagation▫Stability improvements
•Results• Limitations and conclusion
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Limitations•Model does not support rotationnal mouvements•Too slow for small si
•Not physically exact, but visually plausible
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Conclusion• Improved lagrangian simulation of melting and
solidification▫Smaller computational times▫Improved stability and control
•Futur works▫Handle rotational behaviors▫Further improve computational times
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Thank you!
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Heat diffusion•Proposed approach
▫Implicit formulation▫Handle individually each pair of neighbor particles
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1
2
3 4
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Heat diffusion•Neighbors traversal order affects results•Solutions
▫Randomize traversal order▫Average of normal and reverse order
Used in our examples
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Adaptive time step41