Bending, Breaking and Squishing Stuff

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Bending, Breaking and Squishing Stuff

• Marq Singer
• Red Storm Entertainment
• marqs_at_redstorm.com

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Synopsis

• This is the last lecture of the day, so Ill try
  to be nice
• Stuff thats cool, but not essential
• Soft body dynamics
• Breaking and bending stuff
• Generating sounds

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Squishing Stuff

• Soft Body Dynamics

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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.

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Spring Constraints

• Seems like a reasonable choice for soft body
  dynamics (cloth)
• In practice, not very useful
• Unstable, quickly explodes

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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

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Cloth Simulation

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

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Cloth Simulation
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Cloth Simulation

• Forces on our cloth

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Cloth Simulation

• Relaxation is simple
• Infinitely rigid springs are stable
• 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, youre done!

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Relaxation Methods
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Relaxation Methods
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Relaxation Methods
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Relaxation Methods
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Relaxation Methods
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Relaxation Methods
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Cloth Simulation

• When n gt 2, each particles 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)

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More Cloth Simulation

• Use less rigid constraints
• Vary the constraints in each direction (i.e.
  horizontal stronger than vertical)
• Warp and weft constraints

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Still More Cloth Simulation

• Sheer Springs

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Still More Cloth Simulation

• Flex Springs

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Using a Linear System

• Can sum up forces and constraints
• Represent as system of linear equations
• Solve using matrix methods

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Basic Stuff

• Systems of linear equations
• Where
• A matrix of coefficients
• x column vector of variables
• b column vector of solutions

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Basic Stuff

• Populating matricies is a bit tricky, see
  Boxerman for a good example
• Isolating the ith equation

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Jacobi Iteration

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

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

• In matrix form

D, -L, -U are subparts of A D diagonal -L
strictly lower triangular -U strictly upper
triangular
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Jacobi Iteration

• Definition (diagonal, strictly lower, strictly
  upper)
• A D - L - U

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Lots More Math(not covered here)

• I highly recommend Shewchuk 1994
• Gauss-Seidel
• Successive Over Relaxation (SOR)
• Steepest Descent
• Conjugate Gradient
• Newtons Method (in some cases)
• Hessian
• Newton variants (Discreet, Quasi, Truncated)