Particle Systems Two - Computer Animation - Lecture Slides, Slides of Computer Graphics and Animation

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Forces

Uniform Gravity

 A very simple, useful force is the uniform gravity field:  It assumes that we are near the surface of a planet with a huge enough mass that we can treat it as infinite  As we don’t apply any equal and opposite forces to anything, it appears that we are breaking Newton’s third law, however we can assume that we are exchanging forces with the infinite mass, but having no relevant affect on it

[ ]

(^02) 0

s

m

m

gravity

g

f g

Gravity

 The law describes an equal and opposite force

exchanged between two bodies, where the force

is proportional to the product of the two masses

and inversely proportional to their distance

squared. The force acts in a direction e along a

line from one particle to the other (in an

attractive direction)

f e

2 1 2

d

Gm m

gravity

1 2 1 2

r r

r r

e

Gravity

 The equation describes the gravitational force between two particles  To compute the forces in a large system of particles, every pair must be considered  This gives us an N 2 loop over the particles  Actually, there are some tricks to speed this up, but we won’t look at those

Aerodynamic Drag

 Like gravity, the aerodynamic drag force

appears to violate Newton’s Third Law, as we

are applying a force to a particle but no equal

and opposite force to anything else

 We can justify this by saying that the particle is

actually applying a force onto the surrounding air,

but we will assume that the resulting motion is

just damped out by the viscosity of the air

Springs

 A simple spring force can be described as:  Where k is a ‘spring constant’ describing the stiffness of the spring and x is a vector describing the displacement

f x

spring s

= − k

Springs

 As springs apply equal and opposite forces to two particles, they should obey conservation of momentum  As it happens, the springs will also conserve energy, as the kinetic energy of motion can be stored in the deformation energy of the spring and later restored  In practice, our simple implementation of the particle system will guarantee conservation of momentum, due to the way we formulated it  It will not, however guarantee the conservation of energy, and in practice, we might see a gradual increase or decrease in system energy over time  A gradual decrease of energy implies that the system damps out and might eventually come to rest. A gradual increase, however, it not so nice… (more on this later)

Dampers

 We can also use damping forces between particles:  Dampers will oppose any difference in velocity between particles  The damping forces are equal and opposite, so they conserve momentum, but they will remove energy from the system  In real dampers, kinetic energy of motion is converted into complex fluid motion within the damper and then diffused into random molecular motion causing an increase in temperature. The kinetic energy is effectively lost.

f v

damp d

= − k

Dampers

 To compute the damping force, we need to

know the closing velocity of the two particles, or

the speed at which they are approaching each

other

 This gives us the instantaneous closing velocity

of the two particles

v e v e r r r r e = ⋅ − ⋅ − − = 1 2 1 2 1 2 v

Dampers

 Another way we could compute the closing

velocity is to compare the distance between the

two particles to their distance from last frame

 The difference is that this is a numerical

computation of the approximate derivative, while

the first approach was an exact analytical

computation

t d v Δ − − = 1 2 0 r r

Force Fields

 We can also define any arbitrary force field that we want. For example, we might choose a force field where the force is some function of the position within the field  We can also do things like defining the velocity of the air by some similar field equation and then using the aerodynamic drag force to compute a final force  Using this approach, one can define useful turbulence fields, vortices, and other flow patterns f ∝ f^ ( r^ ) field

Collisions & Impulse

 A collision is assumed to be instantaneous

 However, for a force to change an object’s

momentum, it must operate over some time

interval

 Therefore, we can’t use actual forces to do

collisions

 Instead, we introduce the concept of an impulse,

which can be though of as a large force acting

over a small time

Impulse

 The addition of impulses makes a slight modification to our particle simulation: ( ) t t m i i ′ = + ′ Δ ′ = + Δ + = = ∑ ∑ r r v v v f j j j f f 0 0 1 // Integrate to get new velocity& position // Computeallforcesand impulses

Collisions

 Today, we will just consider the simple case of a particle colliding with a static object  The particle has a velocity of v before the collision and collides with the surface with a unit normal n  We want to find the collision impulse j applied to the particle during the collision