DisclaimerDetermining Step SizeAn ExampleSpeed Limitation of Euler’s MethodStiff EquationsA Stiff Energy LandscapeExample: Particle-on-lineSlide 8Slide 9Slide 10Explicit IntegrationProblemsExplicit vs. Implicit Euler MethodSlide 14One Step: Implicit vs. ExplicitLarge SystemsImplicit IntegrationSlide 18Slide 19Linearized Implicit IntegrationSingle-Step Implicit Euler MethodSolving Large SystemsUNC Chapel HillM. C. LinDisclaimerThe following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright 2001 by David Baraff at Pixar).UNC Chapel HillM. C. LinDetermining Step SizeExplicit Integration–Too big, unstable!–Too small, too slow–Adaptive, maybe–Ultimately the constants decide!Implicit Methods–Taking large steps when possibleUNC Chapel HillM. C. LinAn ExampleUNC Chapel HillM. C. LinSpeed Limitation of Euler’s MethodUNC Chapel HillM. C. LinStiff EquationsUNC Chapel HillM. C. LinA Stiff Energy LandscapeUNC Chapel HillM. C. LinExample: Particle-on-lineUNC Chapel HillM. C. LinExample: Particle-on-lineUNC Chapel HillM. C. LinExample: Particle-on-lineUNC Chapel HillM. C. LinExample: Particle-on-lineUNC Chapel HillM. C. LinExplicit IntegrationUNC Chapel HillM. C. LinProblemsUNC Chapel HillM. C. LinExplicit vs. Implicit Euler Methodvs.UNC Chapel HillM. C. LinUNC Chapel HillM. C. LinOne Step: Implicit vs. ExplicitUNC Chapel HillM. C. LinLarge SystemsUNC Chapel HillM. C. LinImplicit IntegrationUNC Chapel HillM. C. LinImplicit IntegrationUNC Chapel HillM. C. LinImplicit IntegrationUNC Chapel HillM. C. LinLinearized Implicit IntegrationUNC Chapel HillM. C. LinSingle-Step Implicit Euler MethodUNC Chapel HillM. C. LinSolving Large SystemsMatrix structure reflects force-coupling:(i , j)th entry exists iff fi depends on XjConjugate gradient a good first
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