SC1 P HYSICALLY B ASED M ODELING Overview Overview One Lousy - - PowerPoint PPT Presentation

SC1 SIGGRAPH 2001 COURSE NOTES PHYSICALLY BASED MODELING

Overview Overview

• One Lousy Particle
• Particle Systems
• Forces: gravity, springs …
• Implementation and Interaction
• Simple collisions

SC2 SIGGRAPH 2001 COURSE NOTES PHYSICALLY BASED MODELING

A Newtonian Particle A Newtonian Particle

• Differential equation: f = ma
• Forces can depend on:

– Position, Velocity, Time

( , , ) t m = f x x x ɺ ɺɺ

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Second Order Equations Second Order Equations

Not in our standard form because it has 2nd derivatives Add a new variable, v, to get a pair of coupled 1st order equations.

( , , ) t m = f x x x ɺ ɺɺ m =   =  x v v f ɺ ɺ

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Phase Space Phase Space

Concatenate x and v to make a 6-vector: Position in Phase Space. Velocity in Phase Space: another 6-vector. A vanilla 1st-order differential equation.

      x v       x v ɺ ɺ m     =         x v v f ɺ ɺ

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Particle Structure Particle Structure

x v f

m

Position Velocity Force Accumulator mass Position in Phase Space

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Solver Interface Solver Interface

x v f

m

Get/Set State Deriv Eval

6

Dim(state)

v

f m

x v

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Particle Systems Particle Systems

x v f

m

x v f

m

x v f

m

x v f

m

x v f

m

particles n time

…

x v f

m

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Solver Interface Solver Interface

6n x1 v1 x2 v2 xn vn v1 f1 m1 v2 f2 m2 vn fn mn

particles n time

Deriv Eval Deriv Eval Get/Set State Get/Set State Dim(State) Dim(State)

Particle System Diffeq Solver

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Deriv Eval Loop Deriv Eval Loop

• Clear forces

– Loop over particles, zero force accumulators.

• Calculate forces

– Sum all forces into accumulators.

• Gather

– Loop over particles, copying v and f/m into destination array.

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Forces Forces

• Constant

gravity

• Position/time dependent

force fields

• Velocity-Dependent

drag

• n-ary

springs

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Force Structures Force Structures

• Unlike particles, forces are

heterogeneous.

• Force Objects:

– black boxes – point to the particles they influence – add in their own forces (type dependent)

• Global force calculation:

– loop, invoking force objects

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Particle Systems, with forces Particle Systems, with forces

x v f

m

x v f

m …

x v f

m

particles n time forces nforces

… F

F

F F F

A list of force

• bjects to invoke

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Gravity Gravity

x v f

m

G apply_fun p sys

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

F Force Law:

Particle system

grav

m = f G

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Viscous Drag Viscous Drag

x v f

m

k apply_fun p sys

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

F Force Law:

Particle system

drag drag

k = − f v

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Damped Spring Damped Spring

x v f

m

F

x v f

m

ks p2 p1 kd apply_fun sys Particle system

Force Law:

( )

1 2 1 s d

k r k     ∆ ⋅∆ ∆ = − ∆ − +     ∆ ∆       = − v x x f x x x f f

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Deriv Eval Loop Deriv Eval Loop

… F

F

F F F

Clear Force Accumulators Invoke apply_force functions

x v f

m

x v f

m …

x v f

m

x v f

m

x v f

m …

x v f

m

Return [v, f/m,…] to solver.

1 2 3

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Solver Interface Solver Interface

Get/Set State

System

Dim(state)

Solver

Deriv Eval

You are Here

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Bouncing off the Walls Bouncing off the Walls

• Later: rigid body

collision and contact.

• For now, just simple

point-plane collisions.

• Add-ons for a particle

simulator.

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Normal and Tangential Components Normal and Tangential Components

VN = (N⋅V)N VT = V - VN

VT VN V N

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Collision Detection Collision Detection

N X V P

N⋅V < 0

( )

X - P ⋅N < ε

• Within ε of the wall.
• Heading in.
• Within ε of the wall.
• Heading in.

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Collision Response Collision Response

VN VT

• krVN

V V′ = VT - krVN V′ VT

Before After

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Conditions for Contact Conditions for Contact

N P V F X

• On the wall
• Moving along the wall
• Pushing against the wall
• On the wall
• Moving along the wall
• Pushing against the wall

N⋅V < ε

( )

X - P ⋅N < ε

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Contact Force Contact Force

The wall pushes back, cancelling the normal component of F. (An example of a constraint force.) F F′ N

F′ = FT

• FN

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Basic 2-D Interaction Basic 2-D Interaction

X X X

Operations: – Create – Attach – Drag – Nail

Cursor Cursor Mouse Spring Mouse Spring Nail Nail

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Try this at home! Try this at home!

The notes give you everything you need to build a basic interactive mass/spring simulator—try it.

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