Model Reduction

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Motivation 72
3D Continuum Mechanics + FEM: • u = deformation vector Equations of Motion • Supports large deformations • Nonlinear . . . [Euler, Lagrange] 73
Page 1 and 2:
FEM Simulation of 3D Deformable Sol
Page 3 and 4:
• Background: BSc Mathematics PhD
Page 5 and 6:
Computer graphics and animation App
Page 7 and 8:
Deformable object simulations Task:
Page 9 and 10:
Real-time deformable objects • 30
Page 11 and 12:
Outline • Vega FEM • Introducti
Page 13 and 14:
Vega 13 • Free and open source (B
Page 15 and 16:
Deformable Models in Vega • Linea
Page 17 and 18:
Integrators in Vega • Implicit Ne
Page 19 and 20:
Materials in Vega • Linear materi
Page 21 and 22: Sparse Linear Solvers: • Jacobi-p
Page 23 and 24: Real-time Interaction 23
Page 25 and 26: Fun Shing Sin, Daniel Schroeder, Je
Page 27 and 28: Why PCG times depend on stiffness
Page 29 and 30: Vega Live Demo 29 29
Page 31 and 32: Online Course Notes: http://www.fem
Page 33 and 34: Model Reduction + Faster computatio
Page 35 and 36: Other Names for Projection-Based Mo
Page 37 and 38: Model Reduction Outside of Computer
Page 39 and 40: Outline • Vega FEM • Introducti
Page 41 and 42: Notation: Deformation Vector (3D me
Page 43 and 44: Linear Equations of Motion of 3D So
Page 45 and 46: Columns of U are deformation basis
Page 47 and 48: Linear Modes Linear Modes k = 4 sho
Page 49 and 50: Linear Modes 49
Page 51 and 52: Linear Modes are Shapes with the Le
Page 53 and 54: Linear Modes in Computer Graphics [
Page 55 and 56: Computing Linear Modes • Remove r
Page 57 and 58: • Works very well ARPACK • Writ
Page 59 and 60: Free-fly Modes 59
Page 61 and 62: Independent modal oscillators • I
Page 63 and 64: [James and Pai 2002] Linear modal s
Page 65 and 66: BD-Tree Collision Detection for Red
Page 67 and 68: Software for model reduction (by Je
Page 69 and 70: Live Demo: Building a Reduced Nonli
Page 71: Model Reduction of Nonlinear Deform
Page 75 and 76: Reduced equations of motion Substit
Page 77 and 78: Reduced internal forces: slow to ev
Page 79 and 80: How to select the basis ? Basis mus
Page 81 and 82: Motion basis from modal derivatives
Page 83 and 84: Basis Runtime simulation: Modal Der
Page 85 and 86: Comparison: Full simulation vs redu
Page 87 and 88: Modal Derivatives What load causes
Page 89 and 90: Simplified Heart (modal derivatives
Page 91 and 92: Nonlinear materials + reduction •
Page 93 and 94: Combining full simulation with mode
Page 95 and 96: Haptics (Greek; pertaining to the s
Page 97 and 98: Virtual assembly (aircraft geometry
Page 99 and 100: [Barbic, da Silva and Popovic 2009]
Page 101 and 102: [Barbic and Popovic 2008] Real-time
Page 103 and 104: Multibody dynamics with self-collis
Page 105 and 106: Model Reduction + FEM + Domain Deco
Page 107 and 108: Space Station dynamics: 75 fps, 250
Page 109: Acknowledgments • National Scienc
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Model Reduction

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