Large steps in cloth simulation

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Large steps in cloth simulation
David Baraff - Andrew Witkin SIGGraph 1998
Presented by Ohad Barzilay Computer Graphics Lab,
Hebrew University
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Why Cloth Simulation?

• Cloth simulation adds realism
• Cloth simulation is an Open Problem
• Cloth simulation is COOL

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Applications

• Entertainment
• Movies (Special F/X, Animation)
• Video Games (Real-time, FMV)
• Also
• Fashion (Designing, Virtual shows)

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Stuart Little Cloths

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Cloth Problem

• Cloth is a deformable object with complex
  behavior. Simulating realistic cloth is hard.
• Fooling people is near impossible. People see
  cloth in various/extreme conditions all their
  lives.

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Cloth Problem

• Simulating realistic looking cloth
• A triangle mesh of particles
• Material properties (mass, flexibility, etc.)
• Reaction to external forces (wind, gravity)
• Collisions (cloth-cloth, cloth-object)

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Cloth Problem

• Mass-Spring System
• Velocities and positions change over time
• Masses triangulated to form mesh

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Solving Method

• Apply external forces
• Gravity
• Viscous Drag
• Wind, etc.
• Apply internal forces
• Stretching
• Shearing
• Bending

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Stretching

• Problem Not enough to hold a shape

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Shearing

• Problem dropping on floor wads up in big mess of
  spring spaghetti

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Bending

• Fibers in actual cloth run the length of the
  fabric and resist folding and bending
• Well stretch a spring across 2 cells (as shown
  in next slide)

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Internal Forces Overview

• Cloth grid

?Stretch Springs
?Shear Springs
?Bend Springs
(All springs are damped, linear springs)
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System Overview

• Particles and forces are in R3
• Handles contact geometric constrains in direct
  fashion

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Solution Techniques

• Common Differential Equation
• Where
• x (vector) and M (diagonal Matrix) represents
  the geometric state mass distrobution of the
  cloth.
• E (scalar of x) is the Internal Energy, F
  (function of x and ) are the External Forces.

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Solution Techniques

• Given an initial condition x(t0) x0
• Find an approximation for x(t0h) for some time
  step, h
• We must have a way of evaluating the derivative
  function ?x(t)/ ?t f(x,t).

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Solution Techniques
Implicit Integration

• Implicit Integration
• Explicit Integration stretch correction
• Semi-implicit Integration stretch correction

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Implicit Integration

• Given the system of ODEs
• The forward Euler method defines
• A backwards Euler method defines

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Implicit Integration

• Let ?xx(t0h)-x(t0) and ? v v(t0h)-v(t0)
• Using a Taylor series expansion for f(x0? x,
  v0? v)
• With some substitutions

Eq. 1
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Condition Functions

• We need to determine forces and derivatives for
  Eq. 1
• Establish condition functions for
• Stretching
• Shearing
• Bending
• Compute forces

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Condition Functions

• Given a condition function C(p)???
• The force on particle, i, is defined as
• Its derivative with respect to particle, j

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Condition Functions

• Damping forces are defined
• Where

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Sparse Matrices

• Dimension is 3n X 3n for n particles
• Most elements are zero
• Connected particles produce non-zero elements
• Non-zero data is stored compactly
• Array of column indices for each row
• Array of columns
• Array of data corresponding to columns

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Sparse Matrices

• Simple Example

Rows 0, 1, 3, 5, 7 Columns 0, 1,
3, 2, 3, 1, 2 Data 8, 4, 3, 5, 7,
3, 7
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Sparse Matrices

• For symmetric matrices, dont store anything
  below the main diagonal
• Arithmetic routines need to be aware of this

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Solving the System

• Form
• Use the conjugate-gradient iterative method to
  solve A?vb for ?v
• Constraints enforced using a filter function

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Implementation 2

• Derive forces from condition functions
• Use implicit integration
• Solve the linear system iteratively
• Enforce constraints on particles through filter

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Implicit Integration

• More complex mathematically
• Need force derivatives
• Form and solve large linear systems
• Iterative methods used
• Larger time steps possible

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Challenges

• Debugging is VERY difficult
• Huge mistakes can still produce close results

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Challenges

• Collision Detection
• Friction
• Seaming (for clothing)

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Conclusion

• Cloth simulation research is ongoing and
  advancing
• Real-time simulation of simple cloth is
  possible (we give away accuracy for speed)

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Checking up close

Movie Speed
Slow Motion
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Statistics Table Map

• Cloth (vertices / triangles)
• 2,602 / 4,944
• Solid (vertices / triangles)
• 322 / 640
• Time/frame
• 2.23
• Time Step (ms) (min / max)
• 16.5 / 33
• Total frames/steps
• 75 / 80

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T H E E N D