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Introduction to Deformable Models and Elasticity in Real-Time Animation

This overview discusses the principles of deformable models, focusing on elasticity and elastostatics. Key topics include the boundary integral formulation of linear elasticity, full equations of nonlinear elastodynamics, and the effects of geometry and material properties on deformation. It covers essential concepts such as the Green and Cauchy strain tensors, stress tensors, and boundary value problems. Techniques for solving Navier's equations and applications in haptics and animation are highlighted, drawing on insights from the ArtDefo paper presented at SIGGRAPH '99.

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Introduction to Deformable Models and Elasticity in Real-Time Animation

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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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