Andrew Butts 1

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ImplementingParticle Systems

• CS 468 Spring 2004
• Andrew Butts

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Whats going on?

• Just integrate the particle equations of motion.
• P V
• V A F/m
• All the fun is in computing F
• And in rendering!

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Simulation Loop

• PA 4 Requires this loop
• Initialize/Emit particles
• Run integrator (evaluate derivatives)
• Update particle states
• Render
• Repeat!

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System states

• Every particle has a state s
• s (position, velocity, mass, age, color, )
• p and v are the only ones that must vary with
  time
• The entire system state is S
• S (p1, v1, p2, v2, p3, v3, )
• Each p and v is a 3-vector
• Can think of S as just a vector in 6n dimensions
• P, V, A, and F are 3n-vectors

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System states

• 2.5 ways to implement
• Particle object array
• Intuitive. Store all particle attributes in one
  place
• Only need to deal with Particle objects
• Clean implementation if we pick just one
  integration scheme
• System state array
• Store 6n vector S as an array of doubles
• Simplifies the integrator interface
• Or transform back and forth

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Integration

• How do we implement an integrator?
• Write a black-box that works on any G function
• Takes an initial value G at time t, a function
  G(value, time) and timestep h. Returns G(th).
• The integrator can be completely separate from
  the particle representations.
• If your system has complex forces, repeated G
  evaluations become the bottleneck

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Integration

• Euler Method
• S(th) S(t) hS( S(t), t )
• Whats S ?
• S (P, V) (V, A) (V, F/m)
• Simple to implement
• Requires only one evaluation of S
• Simple enough to be coded directly into the
  simulation loop
• Error is O(h)

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Integration

• Midpoint Method
• S(th) S(t) hS( Sm, th/2)
• Sm S(t) 0.5h S( S(t), t )
• A little less simple
• Must compute S twice
• Must keep temporary system state arrays
• Error is O(h2)

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Integration

• Numerical Explosion
• Can combat error with smaller timesteps at
  additional computational cost
• Plagues stiff systems with forces prone to
  oscillation
• Whys it called explosion?
• Demo! (Andrews Euler cloth explooodes!)
• Implicit integration schemes deal with stiff
  systems
• Are somewhat painful to implement

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The Derivative Function

• How to implement the S function
• Want V and A
• Know V is just the particles current velocity
• A F/m. Evaluate forces here.

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Forces

• A F/m
• Particle masses wont change
• But need to evaluate F at every time step.
• The force on one particle may depend on the
  positions of all the others

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Forces

• Typically, have multiple independent forces.
• For each force, add its contribution to each
  particle.
• Need a force accumulator variable per particle
• Or accumulate force in the acceleration variable,
  and divide by m after all forces are accumulated

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Forces

• Example forces
• Earth gravity, air resistance
• Springs, mutual gravitation
• Force fields
• Wind
• Attractors/Repulsors
• Vortices

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Forces

• Earth Gravity
• f -9.81(particle mass in Kg)Y
• Drag
• f -kv
• Uniform Wind
• f k

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Forces

• Simple Random Wind
• After each timestep, add a random offset to the
  direction
• Noisy Random Wind
• Acts within a bounding box
• Define a grid of random directions in the box
• Trilinear interpolation to get f
• After each timestep, add a random offset to each
  direction and renormalize

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Forces

• Attractors/Repulsors
• Special force object at position x
• Only affects particles within a certain distance
• Within the radius, distance-squared falloff
• if x-p lt d
• v (x-p)/x-p
• f k/x2 x
• else
• f 0
• Use the regular grid optimization from lecture

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Emitters

• What is it?!
• Object with position, orientation
• Regulates particle birth and death
• Usually 1 per particle system
• More than 1 can make controlling particle death
  inconvenient

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Emitters

• Regulating particles
• At birth, reset the particles parameters
• Free to set them arbitrarily!
• For death, a few possibilities
• If a particle is past a certain age, reset it.
• Keep an index into the particle array, and reset
  a group of K particles at each timestep.
• Should allocate new particles only once!
• Recycle their objects or array positions.

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Emitters

• Fountain
• Given the emitter position and direction, we have
  a few possibilities
• Choose particle velocity by jittering the
  direction vector
• Choose random spherical coordinates for the
  direction vector
• Demo
• http//www.delphi3d.net/download/vp_sprite.zip

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Rendering

• Spheres are easy but boring.
• Combine points, lines, and alpha blending for
  moderately interesting effects.
• Render oriented particle meshes
• Store rotation info per-particle
• Keep meshes facing forward along their paths
• Can arbitrarily pick up vector

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Rendering

• Render billboards
• Want to represent particles by textures
• Should always face the viewer
• Should get smaller with distance
• Want to avoid OpenGLs 2d functions

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Rendering

• Render billboards (one method)
• Draws an image-plane aligned, diamond-shaped quad
• Given a particle at p, and the eyes basis
  (u,v,w), draw a quad with vertices
• q0 eye.u
• q1 eye.v
• q2 -eye.u
• q3 -eye.v
• Translate it to p
• Will probably want alpha blending enabled for
  smoke, fire, pixie dust, etc. See the Red Book
  for more info.

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Simulation Loop Recap

• A recap of the loop
• Initialize/Emit particles
• Run integrator (evaluate derivatives)
• Update particle states
• Render
• Repeat!
• Particle Illusion Demo

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Resources

• The Big One
• www.google.com
• Numerical Recipes - more on integration
• www.library.cornell.edu/nr/
• ParticleIllusion Demos
• www.wondertouch.com
• Physically Based Modeling Notes
• http//www.pixar.com/companyinfo/research/pbm2001/
  index.html

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Adaptive Stepsize

• Goal improve stability without destroying
  performance
• Define a measure for step success
• Springs Has any spring stretched more than 10
  during this step?
• If the step was unsuccessful,
• Roll back and try with half the stepsize
• Else if the last 2 steps were successful,
• Try doubling the stepsize

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Particle-Object Collision Detection

• With very simple objects, this is easy
• Plane Test the sign of (x-p) n
• Box Check six planes!
• Jon Moons Head Cast ray from p to some outside
  point. If it intersects Jon Moon an odd number
  of times, p is inside.
• Relies on having a CLOSED object!
• Should accelerate intersection with a grid or
  octree

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Collision Response

• If the step carried a particle into an object,
• A few possibilities
• Accurate Solve for t at impact and roll the
  simulation back.
• Must process collisions in chronological order
• Might iterate forever!
• Hack Dont even bother rolling back! Teleport
  the particle to the surface of the object
• Add a response force
• Bounce, slide