Overview Class #6 (Tues, Feb 4) Begin deformable models!! Background on elasticity Elastostatics: generalized 3D springs Boundary integral formulation.

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