Particle system dynamics
Jiří Chmelík, Marek Trtík
PA199

►Motivation
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►Motion of a single particle: Equations of motion
►Use of an ODE solver
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►Motion of many particles
►
►Forces
►Gravity, drag, spring, local interaction
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►Collision: particle vs. plane
►Detection, response, simple friction
Outline

Motivation
3
https://experiments.withgoogle.com/fluid-particles
https://en.wikipedia.org/wiki/Particle_system

Particle definition
4


Particle equations of motion
5


Solving equations of motion
6


Solving equations of motion
7


void getState(Particle const& p, std::vector<float>& y0) {
y0.push_back(p.position.x);
y0.push_back(p.position.y);
y0.push_back(p.position.z);
y0.push_back(p.velocity.x);
y0.push_back(p.velocity.y);
y0.push_back(p.velocity.z);
}
Building initial state for ODE
8

Building derivatives for ODE
9


void doSimulationStep(Particle& p, float& t, float const dt) {
UpdateForce(p,t,dt); // Applies external forces and impulses.
std::vector<float> y0, y;
std::vector<F_y_t> Fyt;
getState(p, y0);
getDerivative(p, Fyt);
ODE(y0, Fyt, t, dt, y); // Computes y and updates t (t += dt).
setState(p, y.begin());
}
Simulation step for single particle
10

void setState(Particle& p, std::vector<float>::const_iterator& it) {
p.position.x = *it; ++it;
p.position.y = *it; ++it;
p.position.z = *it; ++it;
p.velocity.x = *it; ++it;
p.velocity.y = *it; ++it;
p.velocity.z = *it; ++it;
}
Saving ODE results
11

Data flow in simulation step
12
p.position
p.velocity
p.position
p.velocity
p.force / p.mass
p.velocity

Particle system
13


void getState(ParticleSystem const& ps, std::vector<float>& y0) {
for (Particle const& p : ps) getState(p,y0);
}
void getDerivative(ParticleSystem const& ps, std::vector<F_y_t>& Fyt) {
for (Particle const& p : ps) getDerivatives(p, Fyt);
}
void setState(ParticleSystem& ps, std::vector<float>::const_iterator& it) {
for (Particle& p : ps) setState(p, it);
}
ODE helper functions
14

void doSimulationStep(ParticleSystem& ps, float& t, float const dt) {
UpdateForce(ps,t,dt); // Applies external forces and impulses.
std::vector<float> y0, y;
std::vector<F_y_t> Fyt;
getState(ps, y0);
getDerivative(ps, Fyt);
ODE(y0, Fyt, t, dt, y); // Computes y and updates t (t += dt).
setState(ps, y.begin());
}
Simulation step for whole system
15

Data flow in simulation step
16
ps[0]
ps[1]
ps[2]
…

void UpdateForce(ParticleSystem& ps, float const t, float const dt) {
clearForce(ps);
applyForce(ps,t,dt);  // Add all forces and impulses to all particles.
}
void clearForce(ParticleSystem& ps) {
for (Particle& p : ps) p.force = Vector3(0,0,0);
}
►Next we discuss what forces we can add to particles inside the function applyForce().
Forces
17

Gravity
18


Viscous Drag
19


Spring
20


Local interaction
21
https://experiments.withgoogle.com/fluid-particles

►We often want particles to collide with the ground or a wall. These boundaries can be approximated
by planes.
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►The process consists of two parts:
►Detection of a collision.
►Response to the collision.
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Collision: particle vs. plane
22
https://github.com/LakshithaMadushan/Unity-Particle-System

Collision detection
23


Collision response
24


Simple friction
25


►We defined particle and particle system.
►
►We learned Newton’s equations of motion for a particle, i.e., a system of 1st order ODEs.
►
►We learned how to use ODE solver for the simulation.
►
►We learned several kinds of forces which we can apply to particles.
►
►We know how to compute and respond to collision of a particle with a plane, including application
of a friction force.
Summary
26

►[1] Andrew Witkin; Physically Based Modeling: Principles and Practice Particle System Dynamics;
Robotics Institute, Carnegie Mellon University, 1997.
References
27