Mass Spring System(弹簧系统)

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
1. 2. 3. 4. Spatial discretization: samplewk.baidu.comobject with mass points Forces: define internal (springs!) and external forces Dynamics: set up equations of motion Temporal discretization: solve equations of motion
• 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 )
Mass-Spring Systems
A Basic Tool for Modeling Deformable Objects
Part 1
Research
Physical Simulation: How to… ?
Physical System Particle System (Discrete Model)
4
Physical Simulation: How to… ?
Deformable Objects Particle System (Discrete Model)
PDE (Continuous Model)
Modeling
Finite Differences
Finite Elements
Ordinary Differential Equations
• 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)
– – – – Elastic: deformations are reversible Viscous: amplitude of oscillations is reduced Plastic: irreversible deformations Any combination thereof
Research
• Gravity • Contact forces • All forces that are not caused by springs
x3
L3 , k3
x1
L1 , k1
x0
L2 , k 2
x2
Total spring force xi x0 ki li Li li i| i 1,2,3 Resulting force at point i
PDE (Continuous Model)
Modeling
Finite Differences
Finite Elements
Ordinary Differential Equations
Numerical Solution
Research
2
Physical Simulation: How to… ?
Research
3
Deformable Objects
• Deformable objects
– change size and shape due to applied forces – can be deformed but resist deformation
• Common material properties
Research
7
Applications
Cloth Simulation
Bridson et al., 2002
Choi & Ko, 2002
Research
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Applications
Hair animation Facial animation
Selle et al., 2008
Lee et al., 1995
Medical simulation
Kuehnapfel et al., 1993
Research
9
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
Deformable Objects Particle System (Discrete Model)
PDE (Continuous Model)
Modeling
Finite Differences
Finite Elements
Ordinary Differential Equations
Numerical Solution
Research
20
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
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
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
is damping coefficient
+ Simple and efficient – Damps all motion (translations and rotations)
Research
Outline
Mass-Spring Systems Steps towards simulation
Research
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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)
F
int 0
Fi Fiint Fiext
Research
Dissipative Forces
• Real-world mechanical systems dissipate energy over time Internal friction Thermal energy (irreversible process)
Total force Fi Fiint Fiext Note: forces are 3D, Fi R 3
Research
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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!
xi
L
l
F
For purely elastic springs (materials) • Force depends only on position • No energy lost during deformation
Work done by forces W k x L dx
Each point holds properties
mi • Mass • Position x i (t ) • Velocity v i (t )
Research
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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
Research
12
Forces
What are the forces that act on particle i?
External forces
– Gravity
g
Internal forces
0 m Fi mi 0 s2 9.81
– Elastic spring forces – Viscous damping forces
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
Numerical Solution
Research
5
Physical Simulation: How to… ?
Deformable Objects Mass-Spring Systems (Discrete Model)
Modeling
Ordinary Differential Equations
l
xj
F
L
Elastic spring energy gradient of Force = – energy
1 E W k (l L) 2 2
E Fi xi
Research
Forces at Mass Point
Internal forces F int
External forces F ext