Overview Class #6 (Tues, Feb 4) Begin deformable models!! Background on elasticity Elastostatics: generalized 3D springs Boundary integral formulation of linear elasticity (from A RT D EFO (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 Frame w x u

Green & Cauchy Strain Tensors 3x3 matrix describing stretch (diagonal) and shear (off-diagonal)

Green & Cauchy Strain Tensors 3x3 matrix describing stretch (diagonal) and shear (off-diagonal)

dA (tiny area) Stress Tensor Describes forces acting inside an object n w

dA (tiny area) Stress Tensor Describes forces acting inside an object n w

Body Forces Body forces follow by Green’s theorem, i.e., related to divergence of stress tensor

Body Forces Body forces follow by Green’s theorem, i.e., related to divergence of stress tensor

Newton’s 2 nd Law of Motion Simple (finite volume) discretization… w dV

Newton’s 2 nd Law of Motion Simple (finite volume) discretization… w dV

Stress-strain Relationship Still need to know this to compute anything An inherent material property

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:

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 to specify 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

Boundary Conditions Types: –Displacements u on  u (aka Dirichlet) –Tractions (forces) p on  p (aka Neumann)  Boundary Value Problem (BVP) Specify interaction with environment

Boundary Integral Equation Form Integration by parts Choose u*, p* as “fundamental solutions” Weaken Directly relates u and p on the boundary!

Boundary Element Method (BEM) Define u i, p i at nodes H u = G p Constant Elements Point Load at j i g ij

Solving the BVP  A v = z, A large, dense Red: BV specified Yellow: 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

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