Animation Example: Cloth time evolution of a mesh subject to - PDF document

Animation Example: Cloth  time evolution of a mesh subject to Simulation with Meshes  internal forces  stretch and bending  stretch and bending  external forces From Volino & Magnenat-Thalmann  gravity, drag, user applied forces  setting it up CS 176 Winter 2011 CS 176 Winter 2011 1 2 Setup Discretization Vertices as functions of time Time stepping  position, velocity, acceleration  forward Euler  time dependence (first order) d d ( d )  backward Euler b k d l  implicit equation! BUT: stable! CS 176 Winter 2011 CS 176 Winter 2011 3 4 Implicit Solution Time Stepping Simple approach Final equation  linearize f (Taylor series to 1 st order)  now just need f…  classic approach: potential energy  force follows as negative gradient CS 176 Winter 2011 CS 176 Winter 2011 5 6

Energies and Forces Baraff & Witkin Approaches Constraints as key element  continuum models  forces given directly  discretization through finite  in a moment… elements/volumes/differences elements/volumes/differences  equilibrium as vanishing condition ilib i i hi diti  discrete models  simplest example: mass/spring constraint forces  ancillary energy systems numerical device (not physics!) CS 176 Winter 2011 CS 176 Winter 2011 7 8 Forces Continued C(x) Functions Need further derivatives What do we want?  stiffness matrix  resist stretch and shear  measure with the deformation tensor tensor original configuration i i l fi i d f deformed configuration d fi i w b v a CS 176 Winter 2011 CS 176 Winter 2011 9 10 Deformation Gradient Damping Forces w Necessary for simulation b  in direction of force gradient in v direction of a velocity  proportional to velocity constant… direction magnitude compare to  Hessian CS 176 Winter 2011 CS 176 Winter 2011 11 12

Damping Forces Oy Ve… Hessian What else?  get rid of non-symmetric term  actual constraints…  point constraints easy  (let’s just leave it at that for now) (l t’ j t l it t th t f ) Bending  depends on velocity  much smaller component but can be important CS 176 Winter 2011 CS 176 Winter 2011 13 14 Bending Guess what: dihedral angle  next time CS 176 Winter 2011 15