overview class #6 (tues, feb 4)
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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))
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
Deformation and Material Coordinates
• w: undeformed world/body material coordinate• x=x(w): deformed material coordinate• u=x-w: displacement vector of material point
Body Framewx
u
Green & Cauchy Strain Tensors• 3x3 matrix describing stretch (diagonal) and shear (off-diagonal)
dA (tiny area)
Stress Tensor
• Describes forces acting inside an object
dA) surface material orientedon (Force ˆ
)(symmetric ][)(
333231
232221
131211
dA
w ij
nf
n
w
Body Forces
• Body forces follow by Green’s theorem, i.e., related to divergence of stress tensor
Newton’s 2nd Law of Motion
• Simple (finite volume) discretization…
w dV
Stress-strain Relationship
• Still need to know this to compute anything
• An inherent material property
Strain Rate Tensor & Damping
Navier’s Eqn of Linear Elastostatics
• Linear Cauchy strain approx.
• Linear isotropic stress-strain approx.
• Time-independent equilibrium case:
Material properties G, provide easy way tospecify physical behavior
Solution Techniques
• Many ways to approximation solutions to Navier’s (and full nonlinear) equations
• Will return to this later.
• Detour: ArtDefo paper– ArtDefo - Accurate Real Time Deformable Objects
Doug L. James, Dinesh K. Pai.Proceedings of SIGGRAPH 99. pp. 65-72. 1999.
Boundary Conditions
•Types:– Displacements u on u
(aka Dirichlet)
– Tractions (forces) p on p
(aka Neumann)
Boundary Value Problem (BVP)
Specify interaction with environmentSpecify interaction with environment
Boundary Integral Equation Form
dd p *u u *p uc
0
d *u u N
0u N
Integration by partsChoose u*, p* as “fundamental solutions”
Weaken
Directly relates u and p on the boundary!
Boundary Element Method (BEM)
•Define ui, pi at nodes
dd p*u u*puc
n
jjij
n
jjijii
11
ˆ pg uhuc
n
jjj xx
1
u u
H u = G pH u = G p
Constant Elements
PointLoadat j
i
gij
Solving the BVP
A v = z, A large, dense
Red: BV specifiedRed: BV specifiedYellow: BV unknownYellow: BV unknown
H u = G p H,G large & dense
Specify boundary conditions
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
ArtDefo Movie Preview
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