Bending Breaking and Squishing Stuff Marq Singer Red

Bending, Breaking and Squishing Stuff Marq Singer > Red Storm Entertainment > marqs@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 Simulation

Cloth Simulation > Forces on our cloth

Cloth Simulation > > Relaxation is simple Infinitely rigid springs are stable 1. 2. 3. 4. 5. Predetermine Ci distance between particles Apply forces (once per timestep) Calculate D for two particles If D != 0, move each particle half the distance If n = 2, you’re done!

Relaxation Methods

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 coefficients x = column vector of variables b = column vector of solutions

Basic Stuff > Populating matricies is a bit tricky, see [Boxerman] for a good example Isolating the ith equation:

Jacobi Iteration Solve for xi (assume other entries in x unchanged): (Which is basically what we did a few slides back)

Jacobi Iteration In matrix form: D, -L, -U are subparts of A D = diagonal -L = strictly lower triangular -U = strictly upper triangular

Jacobi Iteration Definition (diagonal, strictly lower, strictly upper): A=D-L-U

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)