1. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Lecture VIII Deformable Bodies CS274: Computer Animation and Simulation

2. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Overview Deformable Bodies Many objects are not rigid  jello  mud  gases/liquids  etc. Two main techniques:  Geometric deformations  Physically-based methods

3. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Geometric Deformations Deform the object’s geometry directly Two main techniques:  control point / vertex manipulation  space warping

4. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Control Point / Vertex Manipulation Edit the surface vertices or control points directly

5. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Space Warping Deform the object by deforming the space it is in Two main techniques:  Nonlinear Deformation  Free Form Deformation (FFD) Independent of object representation

6. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Nonlinear Global Deformation Objects are defined in a local object space Deform this space using a combination of:  Non-uniform Scaling  Tapering  Twisting  Bending

7. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Nonlinear Global Deformation

8. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Nonlinear Global Deformation Good for modeling [Barr 87] Animation is harder

9. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Free Form Deformation (FFD) Deform space by deforming a lattice around an object The deformation is defined by moving the control points Imagine it as if the object were encased in rubber

10. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Free Form Deformation (FFD) The lattice defines a Bezier volume Compute lattice coordinates Alter the control points Compute the deformed points

11. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer FFD Example

12. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer FFD Example

13. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer FFD Animation Animate a reference and a deformed lattice referencedeformedmorphed

14. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer FFD Animation Animate the object through the lattice referencedeformedmorphed

15. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Extended Free Form Deformations Extended FFDs:  noncubical lattice  arbitrary parameterization Dirichlet FFDs:  use Delaunay triangulation of the control points as the lattice  use Sibson coordinates as the lattice coordinates

16. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Factor Curves Modify the transformation applied to the object based on where and when it is applied

17. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Factor Curves Scripted animation can lead to complex motions (depending on animator skill) Deformations can be nested

18. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Physically-Based Deformations Deform the object according to physical laws Two main techniques:  mass-spring systems  finite element methods

19. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Connect the particles with springs:  structural springs  shear springs  bending springs  etc. Mass-Spring Systems Treat the object as a collection of particles Simulate using standard particle dynamics

20. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Finite Element Methods Mass-Spring systems are not very realistic We need:  more accurate physical laws  error control Finite Element Methods (FEM) offer a way to solve the physical equations we wish to simulate.

21. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Elastic Models Deformations and forces are related by: deformation energy density and dampening

22. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Elastic Models Deformation energy approximated by: Simulate using finite elements/finite differences 1 st Fundamental Form2 nd Fundamental Form measures curvaturemeasures length

23. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Green Deformation Model Relates the stress and strain: Simulate using finite elements/finite differences

24. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Green Deformation Model Stress Tensor: Strain Tensor: represents the force distribution within an object

25. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Finite Element Methods We need a way to solve the equations Finite Element Method:  discretize the object into elements  represent the solution as a sum of basis functions  compute the solution such that the residual is orthogonal to a set of test functions Example:

26. CS274 Spring 01 Lecture 8 Copyright © Mark Meyer Weak Form Sometimes we need to solve: But our basis functions may not have second derivatives?!? Integration by parts can move derivatives to the test functions!! This is known as the weak form.