A Prediction-Correction Approach for Stable SPH Fluid Simulation from Liquid to Rigid François Dagenais Jonathan Gagnon Eric Paquette

Melting and solidification Animation of transition between ▫ Liquid phase ▫ Rigid phase Non-elastic materials Lagrangian simulation ▫ Almost rigid  longer computational times 2

Goals Improved lagrangian simulation of melting objects ▫ Improved stability ▫ Shorter computational times ▫ Easier control 3

Overview Previous work Proposed Approach ▫ Melting and solidification ▫ Constraints propagation ▫ Stability improvements Results Limitations and conclusion 4

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 lagrangian approaches 5 Carlson et al Batty and Bridson 2008

Previous work Lagrangian  Variable viscosity [Muller et al. 2003]  Elastic [Solenthaler et al. 2007] [Chang et al. 2009]  Plastic [Paiva et al. 2006] 6 [Paiva et al. 2006] [Solenthaler et al. 2007]

Overview Previous work Proposed Approach ▫ Melting and solidification ▫ Constraints propagation ▫ Stability improvements Results Limitations and conclusion 7

Melting and solidification Integrated in a SPH fluid solver Minimisation problem 8

Deformation error Difference between ▫ Current deformation ▫ Target deformation 9

Target Deformation Based on relative position of neighbors 10

Rigidity forces correction 11

Rigidity forces correction 12

Rigidity forces correction 13

Integration 14 Compute density and pressure Compute forces (SPH) Update velocity and position t > t end ? no END yes Compute rigidity forces Initialize rigidity forces Predict particles position Adjust rigidity forces Stopping criterion met? no yes Compute particles deformation error

Integration 15 Initialise rigidity forces Predict particles position Adjust rigidity forces Stopping criterion met? no yes Compute particles deformation error

Overview Previous work Proposed Approach ▫ Melting and solidification ▫ Constraints propagation ▫ Stability improvements Results Limitations and conclusion 16

Why? Particles only affect neighbors ▫ Slow convergence Early termination 17 Almost no variation of !

Constraints propagation 18

Constraints propagation 19

Constraints propagation 20

Constraints propagation 21

Overview Previous work Proposed Approach ▫ Melting and solidification ▫ Constraints propagation ▫ Stability improvements Results Limitations and conclusion 22

Stability Other sources of instability ▫ Pressure forces ▫ Heat diffusion 23

Adaptative time step Advantages ▫ Stable simulation ▫ Shorter computational times « Courant–Friedrichs–Lewy » condition 24

Adaptative time step Maximum velocity estimation ▫ Previous maximal velocity ▫ Maximal acceleration 25

Heat diffusion Increases simulation realism A temperature T i is assigned to each particle ▫ Specified by the user ▫ Updated using heat diffusion equation ▫ Temperature affects rigidity 26

Heat diffusion Unstable when ▫ Large time step ▫ Large heat diffusion coefficient 27

Heat diffusion Proposed approach ▫ Implicit formulation ▫ Handle individually each pair of neighbor particles 28

Heat diffusion – Implicit formulation 29

Heat diffusion - video 30

Overview Previous work Proposed Approach ▫ Melting and solidification ▫ Constraints propagation ▫ Stability improvements Results Limitations and conclusion 31

Video 32

33 Exampletime per frame time per iteration avg. Δ t Ratio t rigide /t total Blocs s i = s1.0s s0.33 Blocs s i = s9.0s s0.88 Blocs s i = s9.9s s0.89 Blocs s i = s7.4s s0.91 Blocs s i = s14.5s s0.92 Blocs s i = s17.1s s0.94 Blocs s i = s21.4s s0.97 Stanford’s bunny480.1s50.3s s0.97 Stanford’s Armadillo165.2s14.1s s0.92 « h »619.7s49.3s s0.97 « h » s53.1s s0.98 Rigid forces computation takes most of the computational times Time per iteration increases as the fluid become more rigid Timestep independent of rigidityVariable rigidity = longer computational time, because of the propagation conditions

Comparison with traditionnal viscosity 34 Traditionnal viscosityOur approach μiμi ΔtΔt Total timesisi avg. Δt Total time x10 -4 s47.80 min x10 -3 s85.03 min x10 -5 s min x10 -3 s min x10 -6 s min x10 -3 s min

Overview Previous work Proposed Approach ▫ Melting and solidification ▫ Constraints propagation ▫ Stability improvements Results Limitations and conclusion 35

Limitations Model does not support rotationnal mouvements Too slow for small s i Not physically exact, but visually plausible 36

Conclusion Improved lagrangian simulation of melting and solidification ▫ Smaller computational times ▫ Improved stability and control Futur works ▫ Handle rotational behaviors ▫ Further improve computational times 37

Thank you! 38

Heat diffusion Proposed approach ▫ Implicit formulation ▫ Handle individually each pair of neighbor particles

Heat diffusion Neighbors traversal order affects results Solutions ▫ Randomize traversal order ▫ Average of normal and reverse order  Used in our examples 40

Adaptive time step 41