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

2. Abstract Smoothed particle Goal 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 Particle system
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 : Πij :
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
If same mass, evenly distributed
Good for sample point approximating
If constant density, constant volume
Force equation

17. Interpretation First term Second 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
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
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