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Overview Class #6 (Tues, Feb 4)

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

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Overview Class #6 (Tues, Feb 4)

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  1. 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))

  2. 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

  3. u x 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

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

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

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

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

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

  9. Strain Rate Tensor & Damping

  10. Navier’s Eqn of Linear Elastostatics • Linear Cauchy strain approx. • Linear isotropic stress-strain approx. • Time-independent equilibrium case:

  11. Material properties G,n provide easy way to specify physical behavior

  12. 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 ObjectsDoug L. James, Dinesh K. Pai.Proceedings of SIGGRAPH 99. pp. 65-72. 1999.

  13. Boundary Conditions • Types: • Displacements u onGu(aka Dirichlet) • Tractions (forces) p on Gp(aka Neumann) •  Boundary Value Problem (BVP) Specify interaction with environment

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

  15. Constant Elements i Point Load at j gij Boundary Element Method (BEM) • Define ui, pi at nodes H u = G p

  16. Specify boundary conditions Red: BV specified Yellow: BV unknown Solving the BVP •  A v = z, A large, dense H u = G p H,G large & dense

  17. 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

  18. ArtDefo Movie Preview

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