1. Preface
3. Simulation with Optimization
4. 1. Discrete Space and Time
5. 1. 1.1. Representations of a Solid Geometry
   2. 1.2. Newton's Second Law
   3. 1.3. Time Integration
   4. 1.4. Explicit Time Integration
   5. 1.5. Implicit Time integration
   6. 1.6. Summary
6. 2. Optimization Framework
7. 1. 2.1. Optimization Time Integrator
   2. 2.2. Dirichlet Boundary Conditions
   3. 2.3. Contact
   4. 2.4. Friction
   5. 2.5. Summary
8. 3. Projected Newton
9. 1. 3.1. Convergence of Newton's Method
   2. 3.2. Line Search
   3. 3.3. Gradient-Based Optimization
   4. 3.4. Summary
10. 4. Case Study: 2D Mass-Spring*
11. 1. 4.1. Spatial and Temporal Discretizations
    2. 4.2. Inertia Term
    3. 4.3. Mass-Spring Potential Energy
    4. 4.4. Optimization Time Integrator
    5. 4.5. Simulation with Visualization
    6. 4.6. GPU-Accelerated Simulation
    7. 4.7. Summary
13. Boundary Treatments
14. 5. Dirichlet Boundary Conditions*
15. 1. 5.1. Equality Constraint Formulation
    2. 5.2. DOF Elimination Method
    3. 5.3. Case Study: Hanging Sqaure*
    4. 5.4. Summary
16. 6. Slip Dirichlet Boundary Conditions
17. 1. 6.1. Axis-Aligned Slip DBC
    2. 6.2. Change of Variables
    3. 6.3. General Slip DBC
    4. 6.4. Summary
18. 7. Distance Barrier for Nonpenetration
19. 1. 7.1. Signed Distances
    2. 7.2. Distance Barrier
    3. 7.3. Solution Accuracy
    4. 7.4. Summary
20. 8. Filter Line Search*
21. 1. 8.1. Tunneling Issue
    2. 8.2. Penetration-free Trajectory
    3. 8.3. Case Study: Square Drop*
    4. 8.4. Summary
22. 9. Frictional Contact
23. 1. 9.1. Smooth Dynamic-Static Transition
    2. 9.2. Semi-Implicit Discretization
    3. 9.3. Fixed-Point Iteration
    4. 9.4. Summary
24. 10. Case Study: Square On Slope*
25. 1. 10.1. From Ground To Slope
    2. 10.2. Slope Friction
    3. 10.3. Summary
26. 11. Moving Boundary Conditions*
27. 1. 11.1. Penalty Method
    2. 11.2. Case Study: Compressing Square*
    3. 11.3. Summary
29. Hyperelasticity
30. 12. Kinematics Theory
31. 1. 12.1. Continuum Motion
    2. 12.2. Deformation
    3. 12.3. Summary
32. 13. Strain Energy
33. 1. 13.1. Rigid Null Space and Rotation Invariance
    2. 13.2. Polar Singular Value Decomposition
    3. 13.3. Simplified Models and Invertibility
    4. 13.4. Summary
34. 14. Stress and Its Derivatives
35. 1. 14.1. Stress
    2. 14.2. Computing Stress
    3. 14.3. Computing Stress Derivatives
    4. 14.4. Summary
36. 15. Case Study: Inversion-free Elasticity*
37. 1. 15.1. Linear Triangle Elements
    2. 15.2. Computing Energy, Gradient, and Hessian
    3. 15.3. Filter Line Search for Non-Inversion
    4. 15.4. Summary
39. Governing Equations
40. 16. Strong and Weak Forms
41. 1. 16.1. Conservation of Mass
    2. 16.2. Conservation of Momentum
    3. 16.3. Weak Form
    4. 16.4. Summary
42. 17. Discretization of Weak Forms
43. 1. 17.1. Discrete Space
    2. 17.2. Discrete Time
    3. 17.3. Summary
44. 18. Boundary Conditions and Frictional Contact
45. 1. 18.1. Incorporating Boundary Conditions
    2. 18.2. Normal Contact for Nonpenetration
    3. 18.3. Barrier Potential
    4. 18.4. Friction Force
    5. 18.5. Summary
47. Finite Element Method
48. 19. Linear Finite Elements
49. 1. 19.1. Piecewise Linear Displacement Field
    2. 19.2. Mass Matrix and Lumping
    3. 19.3. Elasticity Term
    4. 19.4. Summary
50. 20. Piecewise Linear Boundaries
51. 1. 20.1. Boundary Conditions
    2. 20.2. Solid-Obstacle Contact
    3. 20.3. Self-Contact
    4. 20.4. Summary
52. 21. Case Study: 2D Self-Contact*
53. 1. 21.1. Scene Setup and Boundary Element Collection
    2. 21.2. Point-Edge Distance
    3. 21.3. Barrier Energy and Its Derivatives
    4. 21.4. Continuous Collision Detection
    5. 21.5. Summary
54. 22. 2D Frictional Self-Contact*
55. 1. 22.1. Discretization and Approximation
    2. 22.2. Precomputing Normal and Tangent Information
    3. 22.3. Friction Energy and Its Derivatives
    4. 22.4. Summary
56. 23. 3D Elastodynamics
57. 1. 23.1. Kinematics
    2. 23.2. Mass Matrix
    3. 23.3. Elasticity
    4. 23.4. Summary
58. 24. 3D Frictional Self-Contact
59. 1. 24.1. Barrier and Distances
    2. 24.2. Collision Detection
    3. 24.3. Friction
    4. 24.4. Summary
61. Spatial Reductions
62. 25. Rigid Body Simulation*
63. 1. 25.1. Rigid Body Dynamics
    2. 25.2. Subspace Simulation
    3. 25.3. Affine Body Dynamics
    4. 25.4. Case Study: ABD Square Drop*
    5. 25.5. Summary
64. 26. Modal Reductions*
65. 1. 26.1. Linear Modal Analysis
    2. 26.2. Linear Demo*
    3. 26.3. Non-linear Reductions
    4. 26.4. Reduction Basis
    5. 26.5. Summary
67. Material Point Method
68. 27. Spatial and Temporal Discretization
69. 1. 27.1. Material Particles
    2. 27.2. Interpolating Functions
    3. 27.3. Particle-Grid Transfers
    4. 27.4. Deformation Gradient and Particle State Update
    5. 27.5. Summary
70. 28. Beyond Elasticity: Plasticity and Viscosity
71. 1. 28.1. Discretization of Plastic Flow
    2. 28.2. Yield Condition and Return Mapping
    3. 28.3. Summary
72. 29. Boundary Treatments
73. 1. 29.1. Boundary Conditions
    2. 29.2. Frictional Contact on Material Particles
    3. 29.3. Summary
74. 30. Case Study: Two Colliding Elastic Blocks in 2D*
75. 1. 30.1. Simulation Setup
    2. 30.2. Particle-In-Cell Transfer
    3. 30.3. Summary
76. 31. Case Study: Plastic Material Models*
77. 1. 31.1. Drucker-Prager Elastoplasticity
    2. 31.2. SDF-based Sphere Collider
    3. 31.3. Affine Particle-In-Cell Transfer
    4. 31.4. 2D Viscoplastic Bars
    5. 31.5. Summary
79. Position Based Simulations
80. 32. Position Based Dynamics (PBD)
81. 1. 32.1. Particle Simulation Preliminaries
    2. 32.2. Core Algorithm / Position Based Framework
    3. 32.3. Constraint Formulation
    4. 32.4. Constraint Solver
    5. 32.5. Extended PBD (XPBD): Stiffness and Damping
    6. 32.6. Summary
82. 33. Case Study: Cloth Simulation*
83. 1. 33.1. Stretching Constraints
    2. 33.2. Bending Constraints
    3. 33.3. Complete Cloth Simulation
84. 34. Case Study: Mesh Simulation*
85. 1. 34.1. Volume Constraints
    2. 34.2. Edge Constraints
    3. 34.3. Complete Mesh Simulation
86. 35. Case Study: Position Based Fluids*
87. 1. 35.1. Density Constraints
    2. 35.2. Tensile Instability Correction
    3. 35.3. Advanced Fluid Effects
    4. 35.4. Complete PBF Simulation
88. 36. Other Common Constraints
89. 1. 36.1. Collision Constraints
    2. 36.2. Continuum-based Constraints
    3. 36.3. Summary
91. Special Topics
92. 37. Linear System
93. 1. 37.1. Direct Solver
    2. 37.2. Basic Iterative Methods
    3. 37.3. Conjugate Gradient
    4. 37.4. Summary
94. Bibliography