discrete geometric mechanics for variational time integrators
DESCRIPTION
Geometric, Variational Integrators for Computer Animation. Discrete Geometric Mechanics for Variational Time Integrators. L. Kharevych Weiwei Y. Tong E. Kanso J. E. Marsden P. Schr ö der M. Desbrun. Ari Stern Mathieu Desbrun. Time Integration. Interested in D ynamic Systems - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/1.jpg)
Discrete Geometric Mechanics for
Variational Time Integrators
Ari Stern
Mathieu Desbrun
Geometric, Variational
Integrators for Computer Animation
L. KharevychWeiweiY. Tong
E. KansoJ. E. MarsdenP. SchröderM. Desbrun
![Page 2: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/2.jpg)
Time Integration
• Interested in Dynamic Systems
• Analytical solutions usually difficult or impossible
• Need numerical methods to compute time progression
![Page 3: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/3.jpg)
Local vs. Global Accuracy
• Local accuracy (in scientific applications)
• In CG, we care more for qualitative behavior
• Global behavior > Local behavior for our purposes
• A geometric approach can guarantee both
![Page 4: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/4.jpg)
Simple Example: Swinging Pendulum
• Equation of motion:
• Rewrite as first-order equations:
𝑞 (𝑡)
𝑙
![Page 5: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/5.jpg)
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
![Page 6: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/6.jpg)
Taylor Approximation
• First order approximation using tangent to curve:
v
• As , approximations approach continuous values
(𝑞𝑘 ,𝑣𝑘)
(𝑞𝑘+1 ,𝑣𝑘+1)
![Page 7: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/7.jpg)
Explicit Euler Method
• Direct first order approximations:
• Pros:• Fast
• Cons:• Energy “blows up”• Numerically unstable• Bad global accuracy
![Page 8: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/8.jpg)
Implicit Euler Method
• Evaluate RHS using next time step:
• Pros:• Numerically stable
• Cons:• Energy dissipation• Needs non-linear solver• Bad global accuracy
![Page 9: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/9.jpg)
Symplectic Euler Method
• Evaluate explicitly, then :
• Energy is conserved!• Numerically stable• Fast• Good global accuracy
![Page 10: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/10.jpg)
Symplecticity
• Sympletic motions preserve thetwo-form:
• For a trajectory of points inphase space:
• Area of 2D-phase-space region is preserved in time
• Liouville’s Theorem
![Page 11: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/11.jpg)
Geometric View: Lagrangian Mechanics
• Lagrangian: • Action Functional:• Least Action Principle:
• Action Functional “Measure of Curvature”• Least Action “Curvature” is extremized
𝑡 0
𝑇
![Page 12: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/12.jpg)
Euler-Lagrange Equation
=
= 0
![Page 13: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/13.jpg)
Lagrangian Example: Falling Mass
![Page 14: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/14.jpg)
The Discrete Lagrangian
• Derive discrete equations of motion from a Discrete Lagrangian to recover symplecticity:
• RHS can be approximated using one-point quadrature:
![Page 15: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/15.jpg)
The Discrete Action Functional
• Continuous version:
• Discrete version:
![Page 16: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/16.jpg)
Discrete Euler-Lagrange Equation
![Page 17: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/17.jpg)
Discrete Lagrangian Example: Falling Mass
![Page 18: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/18.jpg)
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:
![Page 19: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/19.jpg)
Discrete Hamilton-Pontryagin Principle
![Page 20: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/20.jpg)
Faster Update via Minimization
• Minimization > Root-Finding
• Variational Integrability Assumption:
• Above satisfied by most current models in computer animation
![Page 21: Discrete Geometric Mechanics for Variational Time Integrators](https://reader036.vdocuments.mx/reader036/viewer/2022062408/56814020550346895dab7a9e/html5/thumbnails/21.jpg)
Minimization: The Lilyan