Variational Time Integrators Ari Stern Mathieu Desbrun Geometric Variational Integrators for Computer Animation L Kharevych Weiwei Y Tong E Kanso J E Marsden P Schr ö ID: 320020
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Slide1
Discrete Geometric Mechanics for Variational Time Integrators
Ari SternMathieu Desbrun
Geometric, Variational Integrators for Computer Animation
L.
Kharevych
Weiwei
Y. Tong
E.
Kanso
J. E. Marsden
P.
Schr
ö
der
M.
DesbrunSlide2
Time Integration
Interested in Dynamic SystemsAnalytical solutions usually difficult or impossible
Need numerical methods to compute time progressionSlide3
Local vs. Global Accuracy
Local accuracy (in scientific applications)In CG, we care more for qualitative behaviorGlobal behavior > Local behavior for our purposesA
geometric approach can guarantee bothSlide4
Simple Example: Swinging Pendulum
Equation of motion:
Rewrite as first-order equations:
Slide5
Discretizing the Problem
Break time into equal steps of length
:
Replace continuous functions
and
with
discrete functions
and
Approximate the differential equation
by finding values
for
Various methods to compute
Slide6
Taylor Approximation
First order approximation using tangent to curve:
v
As
, approximations approach continuous values
Slide7
Explicit Euler Method
Direct first order approximations:
Pros:
Fast
Cons:
Energy “blows up”
Numerically unstable
Bad global accuracy
Slide8
Implicit Euler Method
Evaluate RHS using next time step:
Pros:
Numerically stable
Cons:
Energy dissipation
Needs non-linear solver
Bad global accuracy
Slide9
Symplectic Euler Method
Evaluate
explicitly, then
:
Energy is conserved!
Numerically stable
Fast
Good global accuracy
Slide10
Symplecticity
Sympletic motions preserve thetwo-form:
For a trajectory of points in
phase space:
A
rea of 2D-phase-space region is
preserved
in time
Liouville’s
Theorem
Slide11
Geometric View: Lagrangian Mechanics
Lagrangian:
Action Functional:
Least Action Principle:
Action Functional
“Measure of Curvature”
Least Action
“Curvature” is
extremized
Slide12
Euler-Lagrange Equation
=
= 0
Slide13
Lagrangian Example: Falling Mass
Slide14
The Discrete Lagrangian
Derive discrete equations of motion from a Discrete Lagrangian
to recover symplecticity:
RHS can be approximated using one-point quadrature:
Slide15
The Discrete Action Functional
Continuous version:
Discrete version:
Slide16
Discrete Euler-Lagrange Equation
Slide17
Discrete Lagrangian Example: Falling Mass
Slide18
More General: Hamilton-Pontryagin Principle
Equations of motion given by critical points of Hamilton-Pontryagin
action
3 variations now:
is a
Lagrange Multiplier
to equate
and
Analog to Euler-Lagrange equation:
Slide19
Discrete Hamilton-Pontryagin Principle
Slide20
Faster Update via Minimization
Minimization > Root-FindingVariational Integrability Assumption:
Above satisfied by most current models in computer animation
Slide21
Minimization: The Lilyan
Slide22
Results
http://tinyurl.com/n5sn3xq