1. A Unified Lagrangian Approach to Solid-Fluid Animation Richard Keiser, Bart Adams, Dominique Gasser, Paolo Bazzi, Philip Dutré, Markus Gross

2. Motivation Increasing importance of realistic animation of physics phenomena –Deformable solids and fluids –Phase transitions, melting and freezing User interaction –Animations in interactive time

3. Motivation Solving the continuum mechanics equations using –Eulerian methods –Lagrangian methods Meshfree particle methods have become popular Implicit handling of topological changes Simple advection Boundary conditions Incompressibility Müller et al., SCA 2005

4. Motivation Challenge: Surface reconstruction –Represent fine detail for solids –Smooth surface for fluids –Handle topological changes Explicit/implicit surface? Explicit: Detail representation Implicit: Topological changes

5. Related Work Carlson et al. [02] –Model different materials by varying the viscosity Müller et al. [04] –Mesh-free continuum- mechanics-based model for animating elasto- plastic objects Goktekin et al. [04] –Viscoelastic fluids by adding an elastic term to the Navier- Stokes equations

6. Overview Governing Equations Lagrangian Approach for Solid-Fluid Simulations Melting & Freezing Hybrid Explicit-Implicit Surface Implicit Surface Deformation Results Conclusions

7. Navier-Stokes Equations Momentum equation Continuity equation

8. Navier-Stokes Equations Conservation of momentum

9. Navier-Stokes Equations Conservation of momentum Material Derivative in Eulerian setting:

10. Navier-Stokes Equations Conservation of momentum Material Derivative in Eulerian setting: Material Derivative in Lagrangian setting:

11. Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, …

12. Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress

13. Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress Viscosity stress

14. Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress Viscosity stress

15. Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress Viscosity stress

16. Deformable Solids Conservation of momentum Deformed configurationReference configuration u(x)u(x) xx+u(x)

17. Lagrangian Approach Deformable Solids Fluids Conservation of mass

18. Lagrangian Approach Conservation of mass Deformable Solids Fluids

19. Lagrangian Approach Conservation of mass Deformable Solids Fluids

20. Lagrangian Approach Conservation of mass Deformable Solids Fluids

21. Lagrangian Approach Conservation of mass Deformable Solids Fluids

22. Lagrangian Approach Conservation of mass mass moves with particles Deformable Solids Fluids

23. Lagrangian Approach Merged Equation Elastic, pressure and viscous stress Body force f –Gravity, surface tension, …

24. Forces Viscous, pressure and surface tension forces are derived using Smoothed Particle Hydrodynamics (SPH) Derive elastic body forces via strain energy Explicit integration using leap-frog

25. Material Properties Animation control: –Stiffness (Young’s Modulus E ) –Compressibility (Poisson’s ratio) –Plasticity –Viscosity (µ) –Cohesion / surface tension Elasto-plastic behavior Fluid behavior

26. Viscoelastic Materials Fluid: No elastic forces ( E = 0 ) Solid: No viscosity (μ = 0 ) and surface tension Viscoelastic materials: couple parameters to scalar a elastic solid fluid

27. Demo

28. Melting and Freezing Define properties per particle Change properties depending on a scalar T (called temperature) Heat transfer between particles –Solve heat equation using SPH:

29. Surface Solid surface –Highly detailed Fluid surface –Smooth surface due to surface tension –Inherent topological changes Local changes from solid to fluid surfaces for melting and freezing

30. Hybrid Surface Point-sampled surface –wrapped around the particles Hybrid implicit-explicit –Explicit representation for solids Exploit displacement field –Implicit representation for fluids defined as iso-value from particle density field –Blend locally between implicit / explicit surfaces for melting and freezing Depending on temperature T

31. Implicit Surface Problems of implicit surface defined by particles: –“blobby” surface –Surface with large offset to particles Control surface by defining energy potentials

32. Potentials Implicit potential

33. Potentials Implicit potential Smoothing potential

34. Potentials Implicit potential Smoothing potential Attracting potential

35. Potentials Implicit potential Smoothing potential Attracting potential Repulsion potential

36. Forces Potential energy of a surfel is the weighted sum of the potentials Derive forces which minimize potential energy: –Apply implicit, attraction and smoothing force in new normal direction –Apply repulsion force in tangential direction

37. Melting # particles: 3.9k, avg. # surfels: 58k Timings per frame: physics: 3.1 s, surface: 21 s

38. Freezing # particles: 2.4k, avg. # surfels: 3.4k Timings per frame: physics: 0.4 s, surface: 1.2 s

39. Conclusion Lagrangian approach for physics –Wide range of materials from stiff solids, elasto-plastic and visco-elastic objects, to fluids –Stable and efficient –Simple to program Lagrangian approach for surface –Hybrid implicit-explicit approach allows both detailed and smooth surfaces undergoing rapid topological changes –Potentials for better surface control

40. Discussion

42. Fluid Forces Viscous, pressure and surface tension forces are derived using Smoothed Particle Hydrodynamics (SPH):

43. Elastic Force Derive elastic body forces via strain energy Green-Saint-Venant strain tensor Hookean Material

44. Integration Elastic, pressure, viscosity, surface tension and external forces Explicit integration using Leap-frog Animation control: –Stiffness (Young’s Modulus E ) –Compressibility (Poisson’s ratio) –Plasticity –Viscosity (µ) –Cohesion / surface tension Elasto-plastic behavior Fluid behavior

45. Constraints Restrict position and movement of surface Implicit constraint –Restrict surfel to be within a minimal iso- level –Enforces automatic splitting External constraint –For adapting to a contact surface –Potentials prevent discontinuities

46. Contributions Framework for animation of both solids and fluids, and phase transitions Lagrangian approach for both physics and surface Hybrid implicit-explicit surface generation Surface control by defining potentials and geometric constraints