A Unified Lagrangian Approach to Solid-Fluid Animation

1. A Unified Lagrangian Approach to Solid-Fluid Animation 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 Reference configuration Deformed configuration x x+u(x) u(x)

17. Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

18. Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

19. Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

20. Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

21. Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

22. Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass mass moveswith particles

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 fluid elastic solid

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