Bending, Breaking and Bending, Breaking and Squishing StuffSquishing Stuff

Marq SingerMarq SingerRed Storm EntertainmentRed Storm Entertainment

marqs@redstorm.commarqs@redstorm.com

Synopsis

This is the last lecture of the day, so I’ll try to be nice

Stuff that’s cool, but not essential Soft body dynamics Breaking and bending stuff Generating sounds

Squishing Stuff

Soft Body Dynamics

The Basics

Use constraints to limit behavior For our purposes, we will treat each

discreet entity as one particle in a system

Particles can be doors on hinges, bones in a skeleton, points on a piece of cloth, etc.

Spring Constraints

Seems like a reasonable choice for soft body dynamics (cloth)

In practice, not very useful Unstable, quickly explodes

Stiff Constraints

A special spring case does work Ball and Stick/Tinkertoy Particles stay a fixed distance apart Basically an infinitely stiff spring Simple Not as prone to explode

Cloth Simulation

Use stiff springs Solving constraints by relaxation Solve with a linear system

Cloth Simulation0P

1P

5P

1,0C 2,1C5,0C

Cloth Simulation

Forces on our cloth

m

tFtta

tatvttv

tvtpttp

ii

iii

iii

)()(

)()(

)()(

Cloth Simulation

Relaxation is simple Infinitely rigid springs are stable

1. Predetermine Ci distance between particles

2. Apply forces (once per timestep)3. Calculate for two particles4. If move each particle half the distance5. If n = 2, you’re done!

Relaxation Methods

0P 1P

1,0C

2)()(,

2)()(:0 1100

ptPttP

ptPttPp

Relaxation Methods

0P 1P

1,0C

0P 1P 2)()(,

2)()(:0 1100

ptPttP

ptPttPp

Relaxation Methods

0P 1P

1,0C

0P 1P 2)()(,

2)()(:0 1100

ptPttP

ptPttPp

Relaxation Methods

1,0C

0P 1P 2)()(,

2)()(:0 1100

ptPttP

ptPttPp

0P 1P

Relaxation Methods

1,0C

0P 1P 2)()(,

2)()(:0 1100

ptPttP

ptPttPp

0P 1P

Relaxation Methods

1,0C

0P 1P 2)()(,

2)()(:0 1100

ptPttP

ptPttPp

0P 1P

Cloth Simulation

When n > 2, each particle’s movement influenced by multiple particles

Satisfying one constraint can invalidate another

Multiple iterations stabilize system converging to approximate constraints

Forces applied (once) before iterations Fixed timestep (critical)

More Cloth Simulation

Use less rigid constraints Vary the constraints in each

direction (i.e. horizontal stronger than vertical)

Warp and weft constraints

Still More Cloth Simulation Sheer Springs

Still More Cloth Simulation Flex Springs

Using a Linear System

Can sum up forces and constraints Represent as system of linear

equations Solve using matrix methods

Basic Stuff

Systems of linear equations

Where:A = matrix of coefficientsx = column vector of variablesb = column vector of solutions

bAx

Basic Stuff

Populating matricies is a bit tricky, see [Boxerman] for a good example

Isolating the ith equation:

i

n

j

jij bxa 1

Jacobi Iteration

Solve for xi (assume other entries in x unchanged):

ii

ij

kjiji

ki a

xabx

)1(

)(

(Which is basically what we did a few slides back)

Jacobi Iteration

In matrix form:

bDxULDx kk 1)1(1)( )(

D, -L, -U are subparts of AD = diagonal-L = strictly lower triangular-U = strictly upper triangular

Jacobi IterationDefinition (diagonal, strictly lower, strictly upper):

A = D - L - U

DLLL

UDLL

UUDL

UUUD

Lots More Math(not covered here) I highly recommend [Shewchuk 1994] Gauss-Seidel Successive Over Relaxation (SOR) Steepest Descent Conjugate Gradient Newton’s Method (in some cases) Hessian Newton variants (Discreet, Quasi,

Truncated)