Overview Class

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OverviewClass 6 (Tues, Feb 4)

• Begin deformable models!!
• Background on elasticity
• Elastostatics generalized 3D springs
• Boundary integral formulation of linear
  elasticity (from ARTDEFO (SIGGRAPH 99))

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Equations of Elasticity

• Full equations of nonlinear elastodynamics
• Nonlinearities due to
• geometry (large deformation rotation of local
  coord frame)
• material (nonlinear stress-strain curve volume
  preservation)
• Simplification for small-strain (linear
  geometry)
• Dynamic and quasistatic cases useful in different
  contexts
• Very stiff almost rigid objects
• Haptics
• Animation style

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Deformation and Material Coordinates

• w undeformed world/body material coordinate
• xx(w) deformed material coordinate
• ux-w displacement vector of material point

Body Frame
w
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Green Cauchy Strain Tensors

• 3x3 matrix describing stretch (diagonal) and
  shear (off-diagonal)

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Green Cauchy Strain Tensors

• 3x3 matrix describing stretch (diagonal) and
  shear (off-diagonal)

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Stress Tensor

• Describes forces acting inside an object

n
w
dA (tiny area)
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Stress Tensor

• Describes forces acting inside an object

n
w
dA (tiny area)
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Body Forces

• Body forces follow by Greens theorem, i.e.,
  related to divergence of stress tensor

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Body Forces

• Body forces follow by Greens theorem, i.e.,
  related to divergence of stress tensor

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Newtons 2nd Law of Motion

• Simple (finite volume) discretization

w
dV
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Newtons 2nd Law of Motion

• Simple (finite volume) discretization

w
dV
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Stress-strain Relationship

• Still need to know this to compute anything
• An inherent material property

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Stress-strain Relationship

• Still need to know this to compute anything
• An inherent material property

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Strain Rate Tensor Damping
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Strain Rate Tensor Damping
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Naviers Eqn of Linear Elastostatics

• Linear Cauchy strain approx.
• Linear isotropic stress-strain approx.
• Time-independent equilibrium case

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Naviers Eqn of Linear Elastostatics

• Linear Cauchy strain approx.
• Linear isotropic stress-strain approx.
• Time-independent equilibrium case

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Material properties G,n provide easy way
to specify physical behavior
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Solution Techniques

• Many ways to approximation solutions to Naviers
  (and full nonlinear) equations
• Will return to this later.
• Detour ArtDefo paper
• ArtDefo - Accurate Real Time Deformable
  ObjectsDoug L. James, Dinesh K. Pai.Proceedings
  of SIGGRAPH 99. pp. 65-72. 1999.

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Boundary Conditions
Specify interaction with environment

• Types
• Displacements u on Gu(aka Dirichlet)
• Tractions (forces) p on Gp (aka Neumann)
• ? Boundary Value Problem (BVP)

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Boundary Integral Equation Form
Directly relates u and p on the boundary!
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Boundary Element Method (BEM)

• Define ui, pi at nodes

H u G p
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Solving the BVP
H u G p H,G large dense

• ? A v z, A large, dense

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BIE, BEM and Graphics

• No interior meshing
• Smaller (but dense) system matrices
• Sharp edges easy with constant elements
• Easy tractions (for haptics)
• Easy to handle mixed and changing BC
  (interaction)
• More difficult to handle complex inhomogeneity,
  non-linearity

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ArtDefo Movie Preview