UNC Chapel Hill M. C. Lin Disclaimer The following slides reuse materials from SIGGRAPH 2001 Course...
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![Page 1: UNC Chapel Hill M. C. Lin Disclaimer The following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright 2001.](https://reader030.fdocuments.net/reader030/viewer/2022032800/56649d445503460f94a2176c/html5/thumbnails/1.jpg)
UNC Chapel Hill M. C. Lin
Disclaimer
The following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright 2001 by David Baraff at Pixar).
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UNC Chapel Hill M. C. Lin
Determining Step Size
Explicit Integration– Too big, unstable!– Too small, too slow– Adaptive, maybe– Ultimately the constants decide!
Implicit Methods– Taking large steps when possible
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UNC Chapel Hill M. C. Lin
An Example
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UNC Chapel Hill M. C. Lin
Speed Limitation of Euler’s Method
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UNC Chapel Hill M. C. Lin
Stiff Equations
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UNC Chapel Hill M. C. Lin
A Stiff Energy Landscape
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UNC Chapel Hill M. C. Lin
Example: Particle-on-line
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UNC Chapel Hill M. C. Lin
Example: Particle-on-line
![Page 9: UNC Chapel Hill M. C. Lin Disclaimer The following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright 2001.](https://reader030.fdocuments.net/reader030/viewer/2022032800/56649d445503460f94a2176c/html5/thumbnails/9.jpg)
UNC Chapel Hill M. C. Lin
Example: Particle-on-line
![Page 10: UNC Chapel Hill M. C. Lin Disclaimer The following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright 2001.](https://reader030.fdocuments.net/reader030/viewer/2022032800/56649d445503460f94a2176c/html5/thumbnails/10.jpg)
UNC Chapel Hill M. C. Lin
Example: Particle-on-line
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UNC Chapel Hill M. C. Lin
Explicit Integration
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UNC Chapel Hill M. C. Lin
Problems
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UNC Chapel Hill M. C. Lin
Explicit vs. Implicit Euler Method
vs.
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UNC Chapel Hill M. C. Lin
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UNC Chapel Hill M. C. Lin
One Step: Implicit vs. Explicit
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UNC Chapel Hill M. C. Lin
Large Systems
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UNC Chapel Hill M. C. Lin
Implicit Integration
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UNC Chapel Hill M. C. Lin
Implicit Integration
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UNC Chapel Hill M. C. Lin
Implicit Integration
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UNC Chapel Hill M. C. Lin
Linearized Implicit Integration
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UNC Chapel Hill M. C. Lin
Single-Step Implicit Euler Method
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UNC Chapel Hill M. C. Lin
Solving Large Systems
Matrix structure reflects force-coupling:(i , j)th entry exists iff fi depends on Xj
Conjugate gradient a good first choice