Writing a Hair Dynamics Solver

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Writing a Hair Dynamics Solver

• Tae-Yong Kim
• Rhythm Hues Studios
• tae_at_rhythm.com

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Hair Simulation Overview

• Understanding the Problem
• State of the Art
• Mass-Spring for Hair
• Length Preservation
• Angle Preservation
• Numerical Integration
• Advanced Issues

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Uniqueness of Hair Dynamics

• Hair does not stretch at all
• But, (long) hair bends quite a lot
• Upon extreme bending, hair becomes stiff

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Uniqueness of Hair Dynamics

• Hair does not stretch at all
• Stiffness Issue in length
• But, (long) hair bends quite a lot
• Preservation of Angular Momentum
• Upon extreme bending, hair becomes stiff
• Stiffness issue in bending

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State of the art

• Rigid joint approach
• Mass-Spring (Clumped Particle Model)
• Continuum approach

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Rigid Joints

• Simulate hair as connected rigid joints
• Developed in robotics field (e.g. Featherstones
  algorithm)
• Reduced Coordinate Formulation (Hadap2001)
• Generalized Serial Rigid Multi-body chain
  (Chang2002)

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Rigid joints state of the art

• Pros
• Natural Model for Hair
• Stiffness in stretch is avoided by formulation
• Cons
• Explicit integration only (Runge Kutta, etc.)
• Another stiffness problem in angles
• Collision response difficult (IK required)
• Hair tip touching shoulder problem
• Fails on stiff hair, fast collision

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Mass Spring

• Simulate hair as connected particle mass
• Early work on hair simulation
• Mass-spring-hinge model
• Explicit integrators

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Mass Spring state of the art

• Pros
• Well understood in other field (e.g. cloth)
• No restriction in collision handling
• More stable integrator available (implicit)
• Cons
• Stiffness issue in explicit integrator
• Angles difficult to deal with in implicit one

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Hair is a non-linear problem

• Rigid Joint (linear in angles)
• Non-linear constraints
• Mass Spring (linear in position)
• Non-linear in angles

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Our choice - Mass Spring

• Friendly for production environment
• Robust Collision Response
• Efficient
• Challenges
• Standard mass spring integrator fails for a
  non-linear problem
• Implicit integrator helps, but not fully.

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Basic Mass Spring System
xi
xj
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Basic Mass Spring System
xi
xj
L
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Basic Mass Spring System
xi
xj
Fj
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Basic Mass Spring System
xi
xj
Fj
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Basic Mass Spring System
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How large is k ?
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k
?

• Hair does not stretch at all
• k is close to infinity

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k
?
n

• Explicit integrator diverges on
• Large dt (smaller dt? slow!)
• Large k (smaller k? looks bad!)

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k
?

• Implicit integrator adds stability
• Large dt (OK)
• Large k (OK)

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k
?
? implicit integration?
n1
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Implicit integration a closer look
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Implicit integration a closer look
Damping
Filter

• Implicit integrator adds stability thru
• Wider filtering of velocity
• Artificial Damping

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Implicit integration as a filter
Damping
Filter


Velocity Field
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As k becomes larger, Implicit filter widens
Smaller k


larger k
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As K??, implicit filter makes things move together
Smaller k
k??

• Stability gain good for cloth, not quite for hair
• Too large k results in excessive smoothing
• ?Loss of angular momentum
• Too small k results in stretching of hair

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Implicit integration as a damping force

• Artificial damping is automatically added
• Damping term depends on k and dt
• ?Increase in stability, but loss of accuracy
• ?too much damping as k approaches ?

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k
?
? implicit integration?
n1

• Implicit integrator adds stability
• Loss of angular momentum
• Good Jacobian (filter) very important

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k
Is infinity!

• Well, how do we preserve length then?
• ? use non-linear correction

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non-linear post correction
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non-linear post correction
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non-linear post correction
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non-linear post correction

• Post solve correction
• Successive relaxation until convergence
• Guaranteed length preservation
• Cheap simulation of k?infinity

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non-linear post correction

• How to implement?
• Cloth simulation literatures
• Provot 1995 (position only)
• Bridson 2002 (impulse)
• Hair-specific relaxation possible

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non-linear post correction

• Provot1995
• Simulates biphasic spring
• Apply correction until convergence (or time
  limited)
• Non-linear, Non-dynamic inverse kinematic
  procedure

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non-linear post correction

• Bridson2002
• Momentum preserving impulse
• Non-linear Jacobi vs Gauss-Seidel

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k
?
Recap

• Large k not good
• Explicit integrator blows up
• Loss of angular momentum in implicit solvers

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k
?
Recap

• Use post-step relaxation
• Very small k ok
• Explicit integrators ok

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Angle preservation

• Length preservation isnt enough
• Hair tends to go back to original shape

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Angle preservation
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Angle preservation

• Flexion spring?
• Linear spring between distant nodes.

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Flexion spring (x)

• Linear Spring ? Loss of angular momentum
• Ambiguity in direction for angles gt 180?
• Unwanted wrinkles in hair shape

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Angle preservation

• Derivation on angles
• Two angles suffice (no torsion)
• Energy derived from changes in angles

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Angle preservation
?0
F
?
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Angle preservation

• Derivation on angles
• Additional anchor point needed at the root

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Angle preservation

• Derivation on angles
• Full angle required (360 ?)
• Axis generation an issue

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Angle preservation

• Implicit integration used on angles
• Non-linear problem again
• Jacobian treatment needed

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Predictor-corrector scheme

• Implicit integration as a predictor for angles
• Non-linear corrector for length
• Two-pass implicit filtering with non-linear
  corrector in the middle of integration
• Linearized Implicit integrator augmented with a
  non-linear optimizer

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Predictor-corrector scheme

• Implicit Filter (Predictor)

• Sharpener (Corrector)

• Implicit Filter (Predictor)

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1.First pass-implicit integration

• First implicit solve to get new velocity

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2.First pass-implicit integration

• Advance position with the predicted mid-step
  velocity

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3.Non-linear Correction

• Apply non-linear corrector to get position
  (length) right

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4.Impulse

• Change velocity due to length preservation
• Velocity may be out of sync after impulse

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5.Second implicit integration

• Filters out velocity field
• Velocity field in sync again

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Recap - Simulating Hair with Mass Spring System

• Hair dynamics is a non-linear numerical
  integration problem
• Standard mass spring integrator fails
• Implicit integrator helps, but not enough.
• Predictor-corrector scheme

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Numerical integration

• Hair system is a banded system
• Block-tridiagonal (3 bands, length only)
• Block-diagonal (5 bands including angles)
• non-iterative, fast solver exists
• blockwise cholesky, LU decomposition
• 10-20x faster than Conjugate Gradient

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Advanced Issues

• Collision handling
• Standard particle-based collision handler works
  seamlessly
• Flexibility in collision response
• Projection Invariant in implicit integration
  (Baraff-Witkin Style)
• Direct position alternation (corrector term)

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Advanced Issues

• Hair-hair interaction
• Many ideas from existing simulation literature
  such as
• Proximity-based spring forces
• Continuous collision
• Rigid body like interaction
• Fluid-like interaction
• .

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Movies and QA

• Hair without bending force
• Hair with bending force
• Non-straight rest shape
• Collision of stiff hair
• Hair hair interaction type I
• Hair hair interaction type II