Bending, Breaking and Squishing Stuff

1. Bending, Breaking and Squishing Stuff Marq Singer Red Storm Entertainment marqs@redstorm.com

2. 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

3. Squishing Stuff • Soft Body Dynamics

4. 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.

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

6. 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

7. Cloth Simulation • Use stiff springs • Solving constraints by relaxation • Solve with a linear system

8. Cloth Simulation

9. Cloth Simulation • Forces on our cloth

10. 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, you’re done!

11. Relaxation Methods

12. Relaxation Methods

13. Relaxation Methods

14. Relaxation Methods

15. Relaxation Methods

16. Relaxation Methods

17. 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)

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

19. Still More Cloth Simulation • Sheer Springs

20. Still More Cloth Simulation • Flex Springs

21. Using a Linear System • Can sum up forces and constraints • Represent as system of linear equations • Solve using matrix methods

22. Basic Stuff Systems of linear equations Where: A = matrix of coefficients x = column vector of variables b = column vector of solutions

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

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

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

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

27. 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)