1. Particle-Based non-Newtonian Fluid Animation for Melting Objects Afonso Paiva Fabiano P. do Carmo Thomas Lewiner Geovan Tavares Matmidia - Departament of Mathematics – PUC-Rio

2. Fluid for Animation

3. Melt and Flow Visually Realistic Computer Animations for Melting Objects

4. Overview Related Works Governing Equations Viscoplastic Model Solid-Liquid Transition SPH Basics Implementation Tracking Free Surface Results

5. Related Works Particle-based Keiser et al., 2005 Grid-based Carlson et al., 2002

6. Smoothed Particle Hydrodynamics - SPH Popular in astrophysics because: –Resolution automatically adapts to density –Easy to combine with N- Body algorithms –Modeling compressible fluids –Can be extend to incompressible fluids Gingold & Monaghan (1977) and Lucy (1977)

7. Lagrangian Formulation of Navier-Stokes Equations PDE → ODEs Particle Methods to perform CFD –SPH, PIC, MAC, MPS, … Continuity Equation Momentum Equation

8. Viscoplastic Model P. R. S. Mendes, E. S. S. Dutra, J. R. R. Siffert, and M. F. Naccache. “Gas displacement of viscoplastic liquids in cappilary tubes” - Journal of Non-Newtonian Fluid Mechanics, 2005. Based on Generalized Newtonian Liquid model Stress Tensor:

9. Viscoplastic Model Viscosity Function n – power-law index J – jump number: –yield stress –low shear rate viscosity –consistency index

10. Solid-Liquid Transition Heat Equation Viscosity Function Temperature > Fusion PointLiquidNS Equations TemperatureViscosityJump Number linearly

11. Solid-Liquid Transition

12. Overview Related Works Governing Equations Viscoplastic Model Solid-Liquid Transition SPH Basics Implementation Tracking Free Surface Results

13. SPH Basics Meshless Method –Particles are moving interpolation centers for fluid quantities –Easy to track fluid free surface SPH Average Operator Quintic Spline

14. SPH Gradient Scalar Vector Divergent

15. SPH Density SPH Approximation of Density SPH Approximation of Continuity Equation

16. SPH Density SPH Approximation of Continuity Equation

17. SPH Momentum Equation Pressure Equation of state –we can approximate the incompressible fluid by a quasi- compressible fluid (Batchelor, 1974)

18. SPH Momentum Equation Stress Tensor

19. SPH Laplacian Traditionally in CG: Inspired in SPH Projection Method (Cummins and Rudman, 1999)

20. Implementation Leap-Frog Scheme Numerical Stability –Artificial Viscosity –XSPH –CFL Condition: Tree Search Method –Complexity:

21. Rendering the Free Surface SPH Characteristic Function Topological Marching Cubes –T. Lewiner, H. Lopes, A. W. Vieira e G. Tavares. “Efficient implementation of marching cubes with topological guarantees” - JGT, 2003

22. Rendering the Free Surface

24. Work in Progress Phase Transition → Multiphase Flow Rendering the Free Surface → Free Surface Tracking Octree Particle Search → Pair-List

25. That’s all Folks http://www.mat.puc-rio.br/~apneto