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Implicit Methods

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Explicit methods have a difficult time with some types of ODE's ... Kacic-Alesic, Nordenstam, Bullock. Eurographics/SIGGRAPH 2003. ... – PowerPoint PPT presentation

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Title: 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
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