Smoothed Particle: a new paradigm for animating highly deformable bodies

1. Smoothed Particle:a new paradigm for animating highly deformable bodies 1996 Eurographics Workshop Mathieu Desbrun, Marie-Paule Gascuel

2. Abstract • Smoothed particle • Sample points • Approximation of the value • Derivatives of local physical quantities • Goal • Animation of inelastic bodies with a wide range of stiffness and viscosity • Coherent definition of surface • Efficient integration scheme

3. 1 Introduction • Mesh deformation • Finite-defference or finite-element methods • Doesn’t fit to large inelastic deformations • Particle system • Interactions are not dependant to connections but on distance • Good for large changes in shape and in topology

4. 1.1 Previous approaches • Particle system • Moving point • Widely used for simulation inelastic deformation and even fluids • Most methods use same attraction-repulsion force interaction • Derives from the Lennard-Jones potential • O(n2) calculation • Interaction forces are clamped to zero at a cutoff radious

5. Variety problems of particle system • Lennard-Jones interaction forces are not easy to manipulate • Finding values that will result in a desired global behavior is quite difficult • Time integration • No stability criterion is provided • Lack of definition of the surface • For collision and contact

6. 1.2 Overview • Extend the Smoothed Particle Hydrodynamics (SPH) for fluid simulation • Particles can be considered as matter elements, for sample points • Smoothed particles are used to approximate the values and derivatives of continuous physical quantities • Smoothed particles ensure valid and stable simulation of physical behavior

7. 2 Smoothed Particle Hydrodynamics • Simulating a fluid consists in computing the variation of continuous functions • Mass density, speed, pressure, or temperature over space and time • Eulerian approach • Dividing space into a fixed grid of voxels • Division of huge empty volumes • Not intuitive • Lagrangian approach • Evolution of selected fluid elements over space and time

8. 2.1 Discrete formulation of continuous fields • Denotation • mj : mass, rj: position, vj: velocity, ρj: density • As a sample point, it can also carry physical fields values • Ex: pressure or temperature • Similar to Monte-Carlo techniques • Fields and derivatives can be approximated by a discrete sum • Smoothed Particle • Smeared out according to a smoothing kernel Wh • h: distribution smoothing length

9. Basis equations of the SPH formalism • mj : mass, rj: position, vj: velocity, ρj: density • f: a continuous field, fj: f(rj) – value of f at particle j • Mass density

10. Verification of equation (2)

11. 2.2 Pressure forces • Symmetric expression of the pressure force on particle i • If the Pi is known at each particle i • ∇iWhij : Wh(ri – rj) • P is computed from PV = k

12. Verification of equation (4)

13. 2.3 Viscosity • Express by adding a damping force term • C : • sound of speed • Fastest velocity • Speed of deformation will be transmitted to the whole material • Πij : • 1st - shear and bulk viscosity • 2nd - prevents particle interpenetration at high speed

14. 3 Simulating highly deformable bodies with smoothed particles • The SPH approach provides a robust and reliable tool for fluid simulation • But SPH does not directly apply to Computer Graphics • Several additions and modifications

15. 3.1 Interaction Force Design • Pressure and cohesion forces • We would like to animate materials with constant density at rest • Needs some internal cohesion • Resulting in attraction-repulsion forces like LJ • (P+P0)V = k, V = 1/ρ, P0 = kρ0

16. Advantage & Force equation • Advantage : • If same mass, evenly distributed • Good for sample point approximating • If constant density, constant volume • Force equation

17. Interpretation • First term • Density gradient descent • Minimize the difference between current and desired densities • Second term • Symmetry term • Ensures the action-reaction principle • K determines the strength of the density recovery • Large : stiff material, small : soft material

18. 3.2 Choice of a smoothing kernel • Smoothing kernel Wh • Very important • Sample point • Approximate values and derivatives of various functions • Small matter elements • Extent of a particle in space • h: radius of influence of interaction forces • Kernel’s support is related to the computational complexity of the simulation

19. Spline Gaussian kernel • Most researches used • Finite radius of influence • Simpler computation • Difficult to evaluate interaction forces • Getting closer, repulsive forces are attenuated • Because of ∇Wh • Results clustering

20. New kernel • Designed to handle nearby particles • Attraction/repulsion force looks very similar to Lennard-Jones attraction/repulsion force

21. 3.3 Results • Density values are displayed in shades of gray • 80 smoothed particles • Parameters : k = 10, c = 2, h is constrained by ρ0 • c represents viscosity, k represents stiffness

22. Discussion • Parallels and differences between smoothed and standard particle system • Cohesion/pressure forces • similar to Lennard Jones forces • Different to microscopic observation, derived from a global equation • Easy to generalize to other materials • Viscosity • Very close to previous ad-hoc models • Computed from relative speeds and proximities

23. Discussion (cont’) • Symmetric pairwise forces • Smoothed particles ensure both stability and accuracy • Because of Monte Carlo approaches • Naturally defines a surface around a deformable body • Gives stability criteria that help efficiency

24. 4 Associating a surface to smoothed particles • Computer Graphics needs continuous representation for discretized model • Particle systems have often been coated with implicit functions • For tight and constant volume, coherent definition are required • SPH has natural way of defining a surface

25. 4.1 Level Set of Mass Density • Density ρ • Continuous function • Indicates where and how mass is distributed in space • Isovalues of density define implicit surfaces • The choice of adequate isovalue should lead to volume preservation at no extra cost

26. 4.2 Coherent choice of Iso-Density • Iso-contour value • Distance of 2h apart has no interaction • Surface should be located at a distance h • Display using Iso-value of density

27. Volume variation • variations of maximum ten percent • Preserving its surface area • Resulting in smooth and realistic shapes

28. 5 Implementation issues • O(n2) • Large number of particles • Very short time step • To avoid divergences or oscillations • Smoothed particles linear time simulation • Time step & adaptive integration

29. 5.1 Neighbor search Acceleration • Bottleneck • Force evaluation • Nearest neighbor search must be performed • Grid of voxels of size 2h • Evaluation of forces on particles : O(n) • Creating the grid of voxels and finding particles lying in each voxel : O(n)

30. 5.2 Locally adaptive integration • Time step • Avoids divergence and ensures efficiency • Local stability criteria • Greatly reduce the computation • Use adapted integration time steps • Reduce computation • Automatically avoid divergence

31. Time Stepping • Courant condition • vδt/δx ≤ 1 • δt : the time step used for integration • v : velocity • δx : grid size • Some grid point do not leaped

32. Translate into smoothed particle • Each particle i must not be passed by • δti ≤ h/c • h : smoothing length • c : sound speed • Using viscosity • α : Courant number, (approx. 0.3) • Our implementation

33. Adaptive Time Integration • Global adapted time step : δt = mini δti • Only a few particles needs a precise integration • Use individual particle time steps • Δt : • User-defined simulation rate • Power of two subdivisions • Position are advanced at every smallest time step • Force evaluations are performed at each individual time step

34. Integration scheme • Leapfrog integrator • Position correction • Time step is totally managed by physical and numerical stability criterion

35. 6 Conclusion • Smoothed particles as samples of mass smeared out in space • Each particles is integrated at individual time steps • Coherent implicit representation from the spatial density • Efficient complexity • Intuitive parameters for viscous material