Implicit Methods

1
Implicit Methods

• Patrick Quirk
• February 10, 2005

2
Introduction, Motivation

• Explicit methods have a difficult time with some
  types of ODEs
• They guess where the function is heading
• Stiff ODEs
• One component big for a short time
• Does not contribute later
• Must make time step small for explicit to handle
  it

3
Solution Implicit Methods

• Backward Euler Method
• Evaluates ƒ at the point were aiming at
• Less guessing, more accuracy
• Consequently, the step size can be larger

4
Implicit Methods

• Last example was simplisitc
• Typically cannot directly solve for yn1
• Unless ƒ is linear
• Soapproximate ƒ with its Taylor Expansion

5
Implicit Methods

• More equation manipulations are done
• Allows us to approximate ?x
• Albeit with an inverse of a matrix!
• In general this extra computation isnt bad
• Larger time step offsets this

6
An Aside - Explicit Method

• Verlet Algorithm
• Does away with the velocity term
• Stores only current, previous position
• Approximates velocity by xn-xn-1

7
An Aside - Explicit Method

• But its explicit!
• Its a second-order method, higher accuracy
• Has the computational cost of a first-order
• No numerical drift like in some other explicit
  and implicit methods
• Easy to code!
• Widely used in games and physics
• Often chosen over implicit methods for these
  reasons

8
Big Picture - Implicit Methods

• Explicit Method code is easy
• Implicit Method code is harder
• Try to make problem un-stiff if possible
• Then use an explicit method and small ?t
• Otherwise write an implicit solver
• Use the larger time step to your advantage

9
Second Order ODEs

• Dynamics problems are often second order
• Need to convert it to a first order problem

10
Second Order ODEs

• Introduce new variables to get first order
• But this gives a 2nx2n linear system
• Bad news for BEM, has to solve that system
• Can we reduce the size?

11
Second Order ODEs

• Can reduce it to nxn
• Introduce some more new variables
• Taylor Expand ƒ (in two dimensions)
• Put that back into the equation

12
Second Order ODEs

• Substitute the first row into the second
• Regroup terms and solve for ?v
• If there is a variance in time, an extra term

13
Sources

• SIGGRAPH Course Notes
• http//www.pixar.com/companyinfo/research/pbm2001/
  notesd.pdf
• Mathworld References
• http//mathworld.wolfram.com
• A Practical Dynamics System. Kacic-Alesic,
  Nordenstam, Bullock. Eurographics/SIGGRAPH 2003.
• http//portal.acm.org/citation.cfm?id846276.84627
  8
• Advanced Character Physics. Thomas Jakobsen.
  Gamasutra. 2003
• http//www.gamasutra.com/resource_guide/20030121/j
  acobson_01.shtml

14
Interactive Animation of Structured Deformable
Objects

• Mathieu Desbrun, Peter Schröder, Alan Barr
• Caltech

15
Introduction, Motivation

• Interactive deformation is tough
• Simulation methods are too slow
• Usually used in VR, user has control
• Limited greatly by time step
• Too big means instability in simulation
• Too small means alteration in reality
• Implicit Integration
• Offers low computational time
• Nearly arbitrary time step

16
Implicit Integration 1D case

• Replace all forces at time t by t1
• Now positions are not blindly reached, but are
  logically inferred.

n1

• In other words
• Explicit steps into the unknown with only
  initial conditions.
• Implicit tries to hit the next position
  correctly.

17
Implicit Integration 1D case

• Must compute Fn1 at tdt (spring forces)
• Use a first order approximation (since we dont
  know what exactly Fn1(tdt) is)

• Here ?n1x is called the backward difference
  operator

18
Implicit Integration 1D case

• Add artificial viscosity
• Increases stability
• Force filtering
• Uses matrix multiplication (HdF/dx)
• Global force effects
• Smoothes large force differences
• Pull it all together

19
Implicit Integration 1D case

• Final Form for 1D
• Look closely, is it really implicit?

Damping Force
Force Filter Matrix
20
Implicit Integration 1D case
Good
Bad

• Loss in accuracy
• Force smoothing
• Artificial viscosity
• Force filtering matrix changes at each time step
  (2D/3D only)
• Forces you to solve a linear system

• Gain in speed
• Gain in speed
• Gain in speed

21
Implicit Integration 1D case

• Some of the Bads arent all that bad
• Artificial viscosity
• It is frequently added to simulations.
• If it is already taken care of, so much the
  better.
• Changing filter matrix
• Even if this does take time, the computational
  advantages are already so significant it doesnt
  matter too much.
• Approximating it makes it a constant
• Compute it once, use it forever

22
Implicit Integration 2D/3D case

• Similar to 1D case, though the smoothing matrix
  changes with each step.
• This paper removes the need to solve a linear
  system
• Splits the force into a linear and non-linear
  part
• Linear is easily solved, non-linear approximated
  to zero
• Magnitude does not change (internal forces), just
  direction
• Preserve linear/angular momenta
• Includes inverse dynamics to remove stretching

23
Implicit Integration 2D/3D case

• In 2D/3D, implicit integration is done by
  computing the Hessian matrix H
• It depends on position of particle nodes,
  changing
• Equation for one element of H is too long to show
  here
• Each node is a 3x3 matrix, total of 3nx3n
  elements
• Too large to solve efficiently
• Approximate it

24
Implicit Integration 2D/3D case

• Matrix split into 2 parts
• Linear
• Non-Linear
• Simple, assume its zero
• Its constant in magnitude, but will rotate

25
Implicit Integration 2D/3D case

• Check if linear, angular momenta are conserved
• Linear is
• Sum of artificial viscosity forces is zero
• Force filtering has no effect
• Angular is not!
• Because of that assumption about the non-linear
  hessian
• Desbrun corrects this with some inverse dynamics

26
Results
27
Sources

• Interactive Animation of Structured Deformable
  Objects. Desbrun, Schröder, Barr. Graphics
  Interface. 1999
• http//www.multires.caltech.edu/pubs/GI99.pdf