1. Point Based Animation of Elastic, Plastic and Melting Objects Matthias Müller Richard Keiser Markus Gross Mark Pauly Andrew Nealen Marc Alexa ETH Zürich TU Darmstadt Stanford

2. Motivation

3. ► Point-based approach for physically- based simulation Point based volume (physics) “Phyxels” = physical elements Point based surface (appearance) “Surfels” = surface elements → No connectivity needed → Inspired by Mesh Free Methods

4. Mesh Free Physics ► Advantages No volumetric mesh needed Natural adaptation to topological changes ► Disadvantages Difficulty of getting sharp fracture lines Neighboring Phyxels are not explicitly given ─Throughout this work we use Spatial Hashing [Teschner et al. 03] for fast neighbor search (when needed)

5. Related Work ► Desbrun & Cani [95,96,99] Physics: Smoothed Particle Hydrodynamics (SPH) Surface: Implicit with suppressed distance blending ► Tonnesen [98] Physics: Lennard-Jones based forces Surface: Particles with orientation

6. Main Contributions ► Point based elasticity model derived from continuum mechanics SPH designed for the simulation of stars/galaxies Lennard-Jones designed for molecular interactions ► Dynamically adapting, point sampled surface animation Densely sampled surface Coupled with coarse physical model (phyxels) High quality visuals at no simulation cost

7. Let‘s start with some Physics The volumetric, physical model phyxels in yellow

8. Body forces + new external forces = next integration step We start with pure elasticity, and add plasticity/flow thereafter Simulation Loop ► Start with undeformed object and apply external forces (per phyxel) Add external forces Time integrationGradient of displacement fieldStrain (Greens strain)Stress (Hookean material)Body forces (from elastic energy)

9. Continuum Elasticity Deformed configuration Reference configuration Displacement (vector) field: u(x) = (u(x,y,z), v(x,y,z), w(x,y,z)) u(x)u(x) xx+u(x) x’x’ u(x’) x’+u(x’)

10. Elastic Strain → Strain depends on the spatial derivatives of u(x) no strain strain u(x) Next: Compute spatial derivatives of the x component u

11. Estimation of Derivatives u(xi)u(xi) xixi xjxj xx u(xj)u(xj) ► Computation of the unknown u x, u y and u z at x i by Linear approximation ► Minimize → MLS approximation of derivatives

12. Elastic Energy Density ► Strain from  u ► Stress via material law (Hooke) ► Energy density (scalar)

13. Elastic Forces ► Estimate volume v i represented by phyxel i via SPH ► Elastic energy of phyxel i ► Depends on u i and u j of all neighbors j ► Phyxel i and all neighbors j receive a force

14. Plasticity ► Change of reference frame (flow) After each time step: Copy deformed frame to reference frame: ► Strain State Plasticity [O’Brien et al. 02] Reference frame stays fixed

15. Benefits and Limitations ► Pros Approach based on continuum elasticity Strain- and stress tensors can be used for plasticity and fracture ► Cons Sparse phyxel sampling → MLS problems Currently only works well for volumetric objects Close phyxels always interact → Extension needed for fracture simulation

16. So what about the Surface? The volumetric, physical model phyxels (in yellow) This works, as previously discussed The detailed surface model surfels (in blue) Two methods...

17. Displacement Approach ► Displace surfels along with phyxels Approximation of surfel displacement based on displacement field u Must be invariant under linear transformations ─Reuse first order MLS approximation of  u

18. Displacement Approach Deforming Reference Frame (30 FPS, P4 3GHz)

19. Displacement Approach + Fast, easy to implement (in Pointshop3D) + Handles very detailed surfaces + Explicit surface model - Doesn’t support topological changes - Needs self-collision detection / response

20. ► Generate implicit surface L I from phyxels This guarantees a consistent surface when splitting and merging ► Blend between implicit and detailed surface model L D Displace detailed surface using the previously described displacement approach Project displaced surfels onto the implicit surface Blend between these two surfel positions and normals Resample (for resampling details, see paper) Multirepresentation Approach

21. ► Blending factor based on estimate of local change of topology Fracture: inside phyxels become boundary phyxels Merging: boundary phyxels become inside phyxels ► Use eigenanalysis of the local covariance to estimate topological change Multirepresentation Approach

22. Offline simulation at ca. 5 sec./frame

23. Multirepresentation Approach Offline simulation at ca. 8 sec./frame

24. Multirepresentation Approach + Can handle arbitrary topological changes + Very detailed models + Interpolation of attributes (Zombie surfels, see paper) - Not interactive

25. Future Work ► Adaptive phyxel resampling ► Collision detection and response ► Physical modeling of fracture ► Surface modeling of fracture ► Physical modeling of adhesion/cohesion ► Physical modeling of surfaces and strands

26. Thank you! ► Contact Information Project Website: http://www.pointbasedanimation.org Andrew Nealen Marc Alexa {nealen,alexa}@informatik.tu-darmstadt.de Matthias Müller Richard Keiser Markus Gross {muellerm,keiser,grossm}@inf.ethz.ch Mark Pauly mapauly@stanford.edu