Mass Spring System(弹簧系统)

• Motion follows from
Newton’s 2nd Law
Fi mi ai
Research
Equations of Motion
Newton’s 2nd Law
Fi mi ai
Equations of motion for one mass point (3 equations) Equations of motion for system of mass points (3n equations)
F k l L
xj
Force in 3D
F
Fi k xi x j L

x x
i
xi x j
j
Initial spring length L Current spring length l Spring stiffness k
Research
Elastic Energy
Research
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Dynamics
Force is known for every particle. How do we determine motion xi t ?
• Kinematic relations
Velocity
v i (t ) d x i (t ) dt
Acceleration 2 d v i (t ) d x i (t ) a i (t ) dt dt 2
l
xj
F
L
Elastic spring energy gradient of Force = – energy
1 E W k (l L) 2 2
E Fi xi
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Forces at Mass Point
Internal forces F int
External forces F ext
Each point holds properties
mi • Mass • Position x i (t ) • Velocity v i (t )
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11
Outline
Mass-Spring Systems Steps towards simulation
1. 2. 3. 4. Spatial discretization: sample object with mass points Forces: define internal (springs!) and external forces Dynamics: set up equations of motion Temporal discretization: solve equations of motion
Numerical Solution
Research
6
Mass-Spring Systems
Mass-spring systems are particle systems with special interaction forces Steps towards simulation
1. 2. 3. 4. Spatial discretization: sample object with mass points Forces: define internal (springs!) and external forces Dynamics: set up equations of motion Temporal discretization: solve equations of motion
Research
7
Applications
Cloth Simulation
Bridson et al., 2002
Choi & Ko, 2002
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8
Applications
Hair animation Facial animation
Selle et al., 2008
Lee et al., 1995
L
l
F
Initial spring length L Current spring length l Spring stiffness k
F k l L Hooke’s Law
Research
Internal Forces: Elastic Springs
xi
L
l
F
Force in 1D
– – – – Elastic: deformations are reversible Viscous: amplitude of oscillations is reduced Plastic: irreversible deformations Any combination thereof
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Acceleration d 2 x i (t ) a i (t ) dt 2
d 2 xi t int ext mi Fi t Fi t 2 dt
d 2 xt int ext t M F t F 2 dt
Mass-Spring Systems
A Basic Tool for Modeling Deformable Objects
Part 1
Research
Physical Simulation: How to… ?
Physical System Particle System (Discrete Model)
• Controllable dissipation useful for physics simulations Do we want things to move indefinitely?
• Dissipation for mass-spring systems Point damping F pd t v (t )
Deformable Objects Particle System (Discrete Model)
PDE (Continuous Model)
Modeling
Finite Differences
Finite Elements
Ordinary Differential Equations
Numerical Solution
Research
10
Spatial Discretization
Sample object with mass points
• Total mass of object: M • Number of mass points: n • Mass of each point: m=M/n (uniform distribution)
is damping coefficient
+ Simple and efficient – Damps all motion (translations and rotations)
Research
Outline
Mass-Spring Systems Steps towards simulation
1. 2. 3. 4. Spatial discretization: sample object with mass points Forces: define internal (springs!) and external forces Dynamics: set up equations of motion Temporal discretization: solve equations of motion
4
Physical Simulation: How to… ?
Deformable Objects Particle System (Discrete Model)
PDE (Continuous Model)
Modeling
Finite Differences
Finite Elements
Ordinary Differential Equations
Numerical Solution
Research
5
Physical Simulation: How to… ?
Deformable Objects Mass-Spring Systems (Discrete Model)
Modeling
Ordinary Differential Equations
Research
3
Deformable Objects
• Deformable objects
– change size and shape due to applied forces – can be deformed but resist deformation
• Common material properties
PDE (Continuous Model)
Modeling
Finite Differences
Finite Elements
Ordinary Differential Equations
Numerical Solution
Research
2
Physical Simulation: How to… ?
Total force Fi Fiint Fiext Note: forces are 3D, Fi R 3
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Internal Forces: Elastic Springs
F
Elasticity: Ability of a spring to return to its initial form when the deforming force is removed. Spring Force: • Ceiiinosssttuv. (Hooke, 1676) • Ut tensio, sic vis. (Hooke, 1678) Force is linear w.r.t. extension!