14.1 Physically Based Simulation II Mass-Spring Systems Hao Li - PowerPoint PPT Presentation

Fall 2014 CSCI 420: Computer Graphics 14.1 Physically Based Simulation II � Mass-Spring Systems Hao Li http://cs420.hao-li.com 1

Mass-Spring Systems • The 101 of Physics Simulation � � • What do we want to simulate? Deformable Objects � • Design a model. Mass points + springs. • Write differential equations. Newton’s 2nd Law (Hooke) • Discretize equations. Integration methods for ODEs • Add interaction. Collision detection + response • Simulate! 2

Mass-Spring Systems • Simulation of cloth based on deformable surfaces (Polygonal mesh) • Realistic simulation of cloth with different fabrics such as wool, cotton, or silk for garment design 3

Facial Animation • Simulation of facial expressions based on deformable surfaces/volumes/muscles • Animation of face models from speech and mimic parameters 4

Medical Simulation • Simulation of deformable soft tissue • Surgical planning • Medical training 5

Overview • Model and Physics • Implementation Hints • Time-Discretization • Collision Response • (Simulation Loop) 6

Mass-Point System • Discretization of an object into mass points (gas, fluid, elastic object, inelastic object) • System with multiple mass centers (Planetary System) • Interaction between points i and j ≠ i based on internal forces • All other forces at point i are external forces • Overall force 7

Mass-Point System • Discretization of an object into mass points • Representation of forces between masses by springs • Computation of dynamics

Mass-Points

Physically-based Equations

Elastic Forces: Springs

Elastic Energies

Forces at a Mass Point

Dissipative Forces

System Equations

System Equations

Overview • Model and Physics • Implementation Hints • Time-Discretization • Collision Response • (Simulation Loop) 17

Elastic Spring 18

Mass Point 19

Force Computation 20

Overview • Model and Physics • Implementation Hints • Time-Discretization • Collision Response • (Simulation Loop) 21

System Equations 22

Solution 23

Problem 24

Numerical Integration Methods 25

Numerical Integration Methods 26

Explicit Integration 27

Explicit Integration 28

Explicit Integration 29

Error Accumulation 30

Problems 31

Improving Accuracy - Leap Frog 32

Improving Accuracy - Runge Kutta 2nd order (midpoint method) 33

Midpoint vs Euler 34

Implementation 35

Avoiding Instability 36

Topology and Stability 37

Volumetric Models - Tet Meshes 38

Topology Ambiguity Problem 39

Cloth Forces 40

Cloth Springs 41

Cloth Springs 42

Cloth Springs 43

http://cs420.hao-li.com Thanks! 44