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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))
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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
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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
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Green & Cauchy Strain Tensors 3x3 matrix describing stretch (diagonal) and shear (off-diagonal)
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Green & Cauchy Strain Tensors 3x3 matrix describing stretch (diagonal) and shear (off-diagonal)
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dA (tiny area) Stress Tensor Describes forces acting inside an object n w
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dA (tiny area) Stress Tensor Describes forces acting inside an object n w
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Body Forces Body forces follow by Green’s theorem, i.e., related to divergence of stress tensor
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Body Forces Body forces follow by Green’s theorem, i.e., related to divergence of stress tensor
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Newton’s 2 nd Law of Motion Simple (finite volume) discretization… w dV
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Newton’s 2 nd Law of Motion Simple (finite volume) discretization… w dV
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Stress-strain Relationship Still need to know this to compute anything An inherent material property
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Stress-strain Relationship Still need to know this to compute anything An inherent material property
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Strain Rate Tensor & Damping
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Navier’s Eqn of Linear Elastostatics Linear Cauchy strain approx. Linear isotropic stress-strain approx. Time-independent equilibrium case:
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Navier’s Eqn of Linear Elastostatics Linear Cauchy strain approx. Linear isotropic stress-strain approx. Time-independent equilibrium case:
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Material properties G, provide easy way to specify physical behavior
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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.
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Boundary Conditions Types: –Displacements u on u (aka Dirichlet) –Tractions (forces) p on p (aka Neumann) Boundary Value Problem (BVP) Specify interaction with environment
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Boundary Integral Equation Form Integration by parts Choose u*, p* as “fundamental solutions” Weaken Directly relates u and p on the boundary!
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Boundary Element Method (BEM) Define u i, p i at nodes H u = G p Constant Elements Point Load at j i g ij
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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
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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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ArtDefo Movie Preview
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