Advanced Computer Graphics (Fall 2010) CS 283, Lecture 23: Physical Simulation 2 Ravi Ramamoorthi Most slides courtesy James O’Brien from CS Fall 2009

Outline  Rigid bodies: basic physics  Simple springs  Stress and Strain  Numerical Discretization  Transition from continuous physics model to something we can actually do on a computer  Important topic in numerical analysis and graphics  Modal Analysis  Simplified linear system (reduced dimensional)

Finite Element Method  Disjoint elements tile material domain  Derivatives from shape functions  Nodes shared by adjacent elements

Finite Element Method  Disjoint elements tile material domain  Derivatives from shape functions  Nodes shared by adjacent elements

FEM Discretization

Tetrahedral Discretization

Element Shape Functions

Material Derivatives

Recall Elasticity

Node Forces (Green Strain)

Corotational Method  Factor out rotation using polar decomposition  Cauchy strain without errors due to rotations

Node Forces and Jacobian

Deformable Object

Outline  Rigid bodies: basic physics  Simple springs  Stress and Strain  Numerical Discretization  Transition from continuous physics model to something we can actually do on a computer  Important topic in numerical analysis and graphics  Modal Analysis  Simplified linear system (reduced dimensional)

Modal Analysis

Deformation

Example

Sound

Sound Example

Modal Decomposition

Linearization

Normalize for Mass

Diagonalize

Diagonalize

Modal Decomposition

Individual Modes

Fast Computation

PS 2 Example

Sound Examples