1. Advanced Computer Graphics (Fall 2010) CS 283, Lecture 23: Physical Simulation 2 Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283/fa10 Most slides courtesy James O’Brien from CS294-13 Fall 2009

2. 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)

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

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

5. FEM Discretization

6. Tetrahedral Discretization

7. Element Shape Functions

8. Material Derivatives

9. Recall Elasticity

10. Node Forces (Green Strain)

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

12. Node Forces and Jacobian

13. Deformable Object

14. 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)

15. Modal Analysis

16. Deformation

17. Example

18. Sound

19. Sound Example

20. Modal Decomposition

21. Linearization

22. Normalize for Mass

23. Diagonalize

24. Diagonalize

25. Modal Decomposition

26. Individual Modes

27. Fast Computation

28. PS 2 Example

29. Sound Examples