particle dynamics - college of computing

Particle dynamics

• Particle overview

• Particle system

• Forces

• Constraints

• Second order motion analysis

Particle system

• Particles are objects that have mass, position, and velocity, but without spatial extent

• Particles are the easiest objects to simulate but they can be made to exhibit a wide range of objects

• Each particle has a position, mass, and velocity

• maybe color, age, temperature

• Seeded randomly at start

• maybe some created each frame

• Move each frame according to physics

• Eventually die when some condition met

Particle animation

Sparks from a campfire

• Add 2-3 particles at each frame

• initialize position and temperature randomly

• Move in specified turbulent smoke flow and decrease temperature as evolving

• Render as a glowing dot

• Kill when too cold to glow visibly

Rendering

• Simplest rendering: color dots

• Animated sprites

• Deformable blobs

• Transparent spheres

• Shadows

A Newtonian particle• First order motion is sufficient, if

• a particle state only contains position

• no inertia

• particles are extremely light

• Most likely particles have inertia and are affected by gravity and other forces

• This puts us in the realm of second order motion

Second-order ODE

f = ma

What is the differential equation that describes the behavior of a mass point?

What does f depend on?

x(t) =f(x(t), x(t))

m

Second-order ODE

Add a new variable, v(t), to get a pair of coupled first order equations

This is not a first oder ODE because it has second derivatives

{x = v

v = f/m

x(t) =f(x(t), x(t))

m= f(x, x)

Phase space

[

x

v

]

=

x1

x2

x3

v1

v2

v3

Concatenate position and velocity to form a 6-vector: position in phase space

First order differential equation: velocity in the phase space

[

x

v

]

=

[

v

f

m

]


• Particle system

• Forces

• Constraints


Particle structure

x

v

f

m

Particle

position

velocity

force accumulator

mass

a point in the phase space

Solver interface

system solversolver

interface

x

v

f

m

particle GetDim 6

Get/Set State

x

v

Deriv Evalv

f

m

system

Particle system structure

n time

...

x1

v1

f1

m1

x2

v2

f2

m2

xn

vn

fn

mn

particles

system

Particle system structure

solversolver

interface

GetDim 6n

Get/Set State

x1

v1

x2

v2

xn

vn

. . .

Deriv Evalvn

fnmn

v1

f1m1

v2

f2m2

. . .

particles

timen

Deriv Eval

Clear forces: loop over particles, zero force accumulatorCalculate forces: sum all forces into accumulator

Gather: loop over particles, copy v and f/m into destination array


• Particle system

• Forces

• Constraints


Forces

• Constant• gravity

• Position/time dependent• force fields, springs

• Velocity dependent• drag

system

particles

n time

Particle systems with forces

...

x1

v1

f1

m1

x2

v2

f2

m2

xn

vn

fn

mn

F1 F2 Fm...forces

Force structure

• Unlike particles, forces are heterogeneous (type-dependent)

• Each force object “knows”

• which particles it influences

• how much contribution it adds to the force accumulator

system

particles

n time

Particle systems with forces

...

x1

v1

f1

m1

x2

v2

f2

m2

xn

vn

fn

mn

forces F1 F2 Fm...

F

Gravity

p

sys apply_fun

Gparticle system

. . .

x1

v1

f1

m1

x2

v2

f2

m2

xn

vn

fn

mn

p->f += p->m*F->G

Unary force: f = mG

Exerting a constant force on each particle

F

Viscous drag

k

sys

particle system

p . . .

x1

v1

f1

m1

x2

v2

f2

m2

xn

vn

fn

mn

apply_fun

p->f += p->v*F->k

At very low speeds for small particles, air resistance is approximately:

fdrag = −kdragv

AttractionAct on any or all pairs of particles, depending on their positions

xp

xq

l

fq = −fp

fp = −kmpmq

|l|2l

|l|

l = xp − xq

Attraction

p

sys apply_fun

kparticle system

F

xp

vp

fp

mp

xq

vq

fq

mqfp = −kmpmq

|l|2l

|l|

Damped spring

fp = −

[

ks(|l|− r) + kdl · l

|l|

]

l

|l|

r| l |

xp

xql = xp − xq

fq = −fp

Damped spring

p

sys apply_fun

ksparticle system

F

xp

vp

fp

mp

xq

vq

fq

mq

kd

r

fp = −

[

ks(|l|− r) + kdl · l

|l|

]

l

|l|

Deriv Eval

...

x1

v1

f1

m1

x2

v2

f2

m2

xn

vn

fn

mn

1. Clear force accumulators

F1 F2 Fm...

2. Invoke apply_force functions

3. Return derivatives to solver[

x

v

]

=

[

v

f

m

]

ODE solver

Euler’s method: x(t0 + h) = x(t0) + hf(x, t)

xt+1 = xt + hxt

vt+1 = vt + hvt

2.4.Get/Set State

Euler step

system

GetDim

Deriv Eval

solversolver

interfacext+1 = xt + hxt

vt+1 = vt + hvt

3.

time5. Advance time

Deriv Eval

1.. . .

vn

fnmn

v1

f1m1

v2

f2m2

x1

v1

x2

v2

xn

vn

. . .

particles


• Particle system

• Forces

• Constraints


Particle Interaction

• We will revisit collision when we talk about rigid body simulation

• For now, just simple point-plane collisions

Collision detection

vT

vNv

x

N

Normal and tangential components

vN = (N · v)N

vT = v − vN

Collision detection

vT

vNv

p

x

N

Particle is on the legal side if

Particle is heading in if

Particle is within of the wall ifε

(x − p) · N ≥ 0

(x − p) · N < ε

v · N < 0

Collision response

v

vT

vN

Before collision

v′

vT

−krvN

After collision

coefficient of restitution: 0 ≤ kr < 1

v′= vT − krvN

Contact

p

N

Conditions for resting contact:

f fN

fT

If a particle is pushed into the contact plane a contact force fc is exerted to cancel the normal component of f

x

1. particle is on the collision surface

v

2. zero normal velocity


• Particle system

• Forces

• Constraints


Linear analysis

• Linearly approximate acceleration

• Split up analysis into different cases

• constant acceleration

• linear acceleration

a(x,v) ≈ a0 − Kx − Dv

Constant acceleration

• Solution is

• v(t) only needs 1st order accuracy, but x(t) demands 2nd order accuracy

v(t) = v0 + a0t

x(t) = x0 + v0t +1

2a0t

2

Linear acceleration

• When K (or D) dominates ODE, what type of motion does it correspond to?

• Need to compute the eigenvalues of A

a(x,v) = −Kx − Dv

d

dt

[

x

v

]

=

[

0 I

−K −D

] [

x

v

]

= A

[

x

v

]

Linear acceleration

Assume is an eigenvalue of A, is the corresponding eigenvector

α

[

u1

u2

]

[

0 I

−K −D

] [

u1

u2

]

= α

[

u1

u2

]

The eigenvector of A has the form[

u

αu

]

Often, D is linear combination of K and I (Rayleigh damping)That means K and D have the same eigenvectors

Linear acceleration

Now assume u is an eigenvector for both K and D

For any u, if is an eigenvector of A, following must be true

[

u

αu

]

[

0 I

−K −D

] [

u

αu

]

= α

[

u

αu

]

−λku − αλdu = α2u

α = −1

2λd ±

√

(1

2λd)2 − λk

Eigenvalue approximation

• If D dominates

• exponential decay

• If K dominates

• oscillation

α ≈ −λd, 0

α ≈ ±√−1

√

λk

Analysis• Constant acceleration (e.g. gravity)

• demands 2nd order accuracy for position

• Position dependence (e.g. spring force)

• demands stability but low or zero damping

• looks at imaginary axis

• Velocity dependence (e.g. damping)

• demands stability, exponential decay

• looks at negative real axis

Explicit methods• First-order explicit Euler method

• constant acceleration:• position dependence:• velocity dependence:

• RK3 and RK4• constant acceleration:• position dependence:• velocity dependence:

bad (1st order)

very bad (unstable)

ok (conditionally stable)

great (high order)



Implicit methods• Implicit Euler method

• constant acceleration:• position dependence:• velocity dependence:

• Trapezoidal rule• constant acceleration:• position dependence:• velocity dependence:

bad (1st order)

ok (stable but damped)

great (monotone)

great (2nd order)

great (stable and no damp)

good (stable, not monotone)

What’s next?

• How do we enforce constraints on the particles?

• Read (optional): Particle animation and rendering using data parallel computation, SIG90, Karl Sims

particle dynamics - college of computing

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