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Smoothed Particle: a new paradigm for animating highly deformable bodies. 1996 Eurographics Workshop Mathieu Desbrun, Marie-Paule Gascuel. Abstract. Smoothed particle Sample points Approximation of the value Derivatives of local physical quantities Goal

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smoothed particle a new paradigm for animating highly deformable bodies

Smoothed Particle:a new paradigm for animating highly deformable bodies

1996 Eurographics Workshop

Mathieu Desbrun, Marie-Paule Gascuel

abstract
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
1 introduction
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
1 1 previous approaches
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
variety problems of particle system
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
1 2 overview
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
2 smoothed particle hydrodynamics
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
2 1 discrete formulation of continuous fields
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
basis equations of the sph formalism
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
2 2 pressure forces
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
2 3 viscosity
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
3 simulating highly deformable bodies with smoothed particles
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
3 1 interaction force design
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
advantage force equation
Advantage & Force equation
  • Advantage :
    • If same mass, evenly distributed
      • Good for sample point approximating
    • If constant density, constant volume
  • Force equation
interpretation
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
3 2 choice of a smoothing kernel
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
spline gaussian kernel
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
new kernel
New kernel
  • Designed to handle nearby particles
  • Attraction/repulsion force looks very similar to Lennard-Jones attraction/repulsion force
3 3 results
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
discussion
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
discussion cont
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
4 associating a surface to smoothed particles
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
4 1 level set of mass density
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
4 2 coherent choice of iso density
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
volume variation
Volume variation
  • variations of maximum ten percent
  • Preserving its surface area
  • Resulting in smooth and realistic shapes
5 implementation issues
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
5 1 neighbor search acceleration
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)
5 2 locally adaptive integration
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
time stepping
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
translate into smoothed particle
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
adaptive time integration
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
integration scheme
Integration scheme
  • Leapfrog integrator
  • Position correction
  • Time step is totally managed by physical and numerical stability criterion
6 conclusion
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