1. Comp259 / Spring 2002 Samir Naik 2/6/02
Cloth Simulation
Comp259 / Spring 2002
Samir Naik
2/6/02

2. Overview Applications of cloth Cloth Modeling
Cloth Dynamics and Simulation
Demos

3. Applications of cloth Movies, games, VR.
Try to find a room without cloth!
We can usually just texture map static objects
Try to find a person not wearing clothes!
Texture mapping clothes doesn’t look good on animated characters
Need to simulate movement of the cloth

4. Motivation
Texture mapped clothes vs. simulated

5. Cloth Modeling
D.E. Breen, D.H. House and M.J. Wozny, “Predicting the Drape of Woven Cloth Using Interacting Particles,” SIGGRAPH '94 Conference Proceedings, (Orlando, FL, July 1994) pp
Getting the final configuration (drape) of cloth to look right.
Static, not dynamic modeling.
What would this shirt look like made from cotton rather than silk?
Would this dress have a more pleasing drape if made from silk rather than light wool?

6. Cloth Modeling
Example of draping cloth over objects

7. Particle-based model Cloth fibers and crossings
Approximated by mass-spring particle system.

8. Particle-based model

9. Particle-based model Categorizing thread motion Energy equation
Thread collision
Thread stretching
Out-of-plane bending
Trellising (shearing)
Energy equation
Ui = Urepel i + Ustretch i + Ubend i + Utrellis i + Ugrav i

10. Energy equation Energy equation
Ui = Urepel i + Ustretch i + Ubend i + Utrellis i + Ugrav I
Urepel. Artifical repulsion energy, keeps particles a minimum distance apart. Simple approach to collision detection.
Ustretch. Energy of tensile strain between each particle and it’s 4 neighbors
Ubend. Energy due to out of plane bending.
Utrellis. Energy due to bending around a thread crossing.
Ugrav. Potential energy due to gravity.

11. Repel and stretch

12. { { Repel and Stretch Repel forces Stretch forces rij rij> rij{
rij>
rij {
rij>

13. Repel and stretch
Repel and stretch energy function are stiff since most cloth doesn’t deform too much.

14. Energy equations
Repel and stretch equations are about the same for all types of cloth
Can derive equations from trial and error
Bend/shear equations are different for different types of cloth (wool, cotton, etc). Changes.
Derive equations from empirical mechanical data.

15. Kawabata system Physically measuring cloth properties.
Approximate actual cloth energy functions with simpler functions.
Use piecewise linear and quadratic polynomials that interpolate inflection points of real curve

16. Bending equation

17. Bending For each particle i, Mi is the set containing 6 ij angles i1

18. Bending
Bend energy equation comes from measurement

19. Bending
Bend energy in one direction

20. Trellising (shearing)

21. Trellising(shearing)
Trellising energy equation also comes from measurment

22. Trellising (shearing)
Shear in one direction

23. Results

24. Results

25. Cloth Dynamics Breen wanted cloth to look good in static poses
Time step could be very small because simulation time was done off line
VR/games want cloth to look good when a user is interacting with it.
Need simulation to be real time, so we need larger time steps (dt >.03 sec)

26. Integration Purpose of integration
Given the known position x(to) and velocity v(to) at time to
Find a new position x(t0+h) and velocity v(to+h) at time t0+h

27. Explicit integration Forward Euler’s method
Based solely on conditions at to
Takes no notice of wildly changing derivatives and proceeds forward blindly

28. Explicit integration
BAD! Step size must be really small or simulation will blow up.
Could use RK4
Better smoothing function
Force is sampled more times during time step.
A little more stable, but still has same problems
Collisions introduce discontinuties, and RK4 doesn’t like discontinuities.

29. Explicit integration
DEMO - when things blow up
Clothy by Jeff Lander

30. Implicit integration
Backwards Euler method

31. Backwards Euler
Start from output state (x0+dx, v0+dv) and use a forwards Euler step to run the system backward in time.
i.e. taking the step -h(v(t0+h), f(x(t0+h),v(t0+h))) brings you back to (x0,v0)
Forces an output state whose derivative at least points back to where you came from.
Sanity checking!

32. Backwards Euler
We still consider forces to be constant within time step
But now output state will have consistent forces that will not give rise to instabilities

33. Backwards Euler We must compute f (x0+dx, v0+dv)
To avoid iteration do a first order approximation
Substituting this in we get

34. Backwards Euler
Grouping all v terms together
Solve for v and x

35. Baraff/Witkin
David Baraff, Andrew Witkin, “Large Steps in Cloth Simulation,” SIGGRAPH '98 Conference Proceedings, (Orlando, FL, July 1998) pp

36. Cloth Energy
Breen describes cloth behavior as energy U, which was to be minimized
Baraff describes behavior in terms of a grid of conditions
can be simple: C(x) = |x - |
or more complicated: Cij(x) =
we want C(x) == 0

37. Forces
Energy function
k is a stiffness constant
Force function

38. Force derivatives Need force derivatives for backwards Euler
With respect to x
K is a sparse matrix
With respect to y
Since C(x) doesn’t depend on v, the matrix f/v is zero

39. Stretch/compresson Particles have fixed plane coordinate (ui,vi)
w(u,v) is a linear function over each triangle
maps from plane to world space
Stretch can be measured by examining derivatives
wu= w/u
wv= w/v

40. Stretch/compression What is this function “w”?
Approximate as a linear function over triangle
x1
x2
World coords
(u1, v1)
Plane coords
(u2, v2)

41. Stretch/compression Condition for stretch energy
a is area of triangle in uv (plane) coords
bu and bv are usually 1, but can be changed to lengthen and widen certain parts of cloth
i.e. the cuff on a shirt sleeve

42. Shear and bend forces
Inner product should be minimized wv wu
= 0
!= 0

43. Damping Damping should only occur in direction of systems velocity
This looks similar to our force equation

44. Derivative of damping force
With respect to x
drop term for symmetry’s sake
With respect to v

45. Mass Modification Tricky way to constrain points
Make mass infinity. It won’t move!
We actually have a 3x3 inverse mass matrix
so mass is actually a vector
can constrain motion in each axis separately
e.g. no acceleration in z axis

46. Results

47. Verlet Integration Implicit Velocity-less Update step
store current position x and prev position x*
Update step
x = x-x* + ah2
= x-x* + h2(f0/M)

48. Verlet integration Update step x-x* is approximate velocity
x = x-x* + h2(f0/M)
x-x* is approximate velocity
Since velocity is implicitly given, it is hard for velocity and position to come out of sync.
Damping
x = .99(x-x*) + h2(f0/M)

49. Verlet cloth Thomas Jakobsen, “Advanced Character Physics”, GDC 2001
Simplify everything by constraining particles by rods not springs
At each iteration, rod goes back to it’s rest length
Solve by local iteration
Each constraint tries to satisfy itself, let the other constraints worry about themselves

50. Verlet cloth Pros Cons fast stable not physically based
loss of energy in system
can’t minimize bending/shearing easily

51. Verlet cloth
Demo