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Bending, Breaking and Squishing Stuff Marq Singer Red Storm Entertainment marqs@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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Forces on our cloth

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

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Relaxation Methods

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Cloth Simulation When n > 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 x i (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)

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Lecture 7 - Systems of Equations CVEN 302 June 17, 2002.

Lecture 7 - Systems of Equations CVEN 302 June 17, 2002.

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