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).

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

UNC Chapel Hill M. C. Lin An Example

UNC Chapel Hill M. C. Lin Speed Limitation of Euler’s Method

UNC Chapel Hill M. C. Lin Stiff Equations

UNC Chapel Hill M. C. Lin A Stiff Energy Landscape

UNC Chapel Hill M. C. Lin Example: Particle-on-line

UNC Chapel Hill M. C. Lin Example: Particle-on-line

UNC Chapel Hill M. C. Lin Example: Particle-on-line

UNC Chapel Hill M. C. Lin Example: Particle-on-line

UNC Chapel Hill M. C. Lin Explicit Integration

UNC Chapel Hill M. C. Lin Problems

UNC Chapel Hill M. C. Lin Explicit vs. Implicit Euler Method vs.

UNC Chapel Hill M. C. Lin

UNC Chapel Hill M. C. Lin One Step: Implicit vs. Explicit

UNC Chapel Hill M. C. Lin Large Systems

UNC Chapel Hill M. C. Lin Implicit Integration

UNC Chapel Hill M. C. Lin Implicit Integration

UNC Chapel Hill M. C. Lin Implicit Integration

UNC Chapel Hill M. C. Lin Linearized Implicit Integration

UNC Chapel Hill M. C. Lin Single-Step Implicit Euler Method

UNC Chapel Hill M. C. Lin Solving Large Systems Matrix structure reflects force-coupling: (i, j)th entry exists iff f i depends on X j Conjugate gradient a good first choice