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Lecture VIII

CS274: Computer Animation and Simulation. Lecture VIII. Deformable Bodies. Overview. Deformable Bodies. Many objects are not rigid jello mud gases/liquids etc. Two main techniques: Geometric deformations Physically-based methods. Geometric Deformations.

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Lecture VIII

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  1. CS274: Computer Animation and Simulation Lecture VIII Deformable Bodies

  2. Overview Deformable Bodies • Many objects are not rigid • jello • mud • gases/liquids • etc. • Two main techniques: • Geometric deformations • Physically-based methods

  3. Geometric Deformations Deform the object’s geometry directly • Two main techniques: • control point / vertex manipulation • space warping

  4. Control Point /Vertex Manipulation Edit the surface vertices or control points directly

  5. 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. 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. Nonlinear Global Deformation

  8. Nonlinear Global Deformation Good for modeling [Barr 87] Animation is harder

  9. 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. Free Form Deformation (FFD) The lattice defines a Bezier volume Compute lattice coordinates Alter the control points Compute the deformed points

  11. FFD Example

  12. FFD Example

  13. FFD Animation Animate a reference and a deformed lattice reference deformed morphed

  14. FFD Animation Animate the object through the lattice reference deformed morphed

  15. 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. Factor Curves Modify the transformation applied to the object based on where and when it is applied

  17. Factor Curves Scripted animation can lead to complex motions (depending on animator skill) Deformations can be nested

  18. Physically-Based Deformations Deform the object according to physical laws • Two main techniques: • mass-spring systems • finite element methods

  19. Mass-Spring Systems Treat the object as a collection of particles • Connect the particles with springs: • structural springs • shear springs • bending springs • etc. Simulate using standard particle dynamics

  20. 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. deformation energy density and dampening Elastic Models Deformations and forces are related by:

  22. Elastic Models Deformation energy approximated by: 1st Fundamental Form 2nd Fundamental Form measures length measures curvature Simulate using finite elements/finite differences

  23. Green Deformation Model Relates the stress and strain: Simulate using finite elements/finite differences

  24. Green Deformation Model Stress Tensor: Strain Tensor: represents the force distribution within an object

  25. Finite Element Methods We need a way to solve the equations Example: • 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

  26. 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.

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