1 Static Equilibrium From Static Eq. to Dynamic Eq. System of mass - - PDF document

SLIDE 1

1

I ntroduction to

Static/ Dynamic Deformation with Mass-Spring Systems

Min Gyu Choi Graphics & Media Lab.@SNU

Static/ Dynamic Deformation

• Static deformation
• Dynamic deformation

undeformed shape deformed shape

finternal fexternal + = finertia = 0

Mass-Spring Systems

Using spring forces to connect mass points

Mass-Spring Systems

undeformed deformed

Using spring forces to connect mass points

Elastic Springs

Spring characteristics

• Stiffness constant: k
• Initial spring length: l
• Current spring length: L

Hooke’s law: f = k(L - l)

f l L

Forces at the Mass Points

Internal force

• External force
• Gravity, etc.

Resulting force

( )

∑

=

− − − − =

3 1 int i i i i i i

l k x x x x x x f

x

1

x

2

x

3

x

1 1,k

l

2 2,k

l

3 3,k

l int i

f

model problem

ext i

f

ext int i i i

f f f + =

SLIDE 2

2

Static Equilibrium

System of mass points Static equilibrium

) (

ext =

+ = f x K f

x

1

x

2

x

3

x

1 1,k

l

2 2,k

l

3 3,k

l

model problem

) (

int

x K f =

From Static Eq. to Dynamic Eq.

Static equilibrium Equation of motion

) ( ) (

2 2

t dt t d f x M = ) (

ext =

+ = f x K f

( )

) ( ) (

ext =

+ = f x K f t t

From Static Eq. to Dynamic Eq.

Static equilibrium Equation of motion with damping

) ( ) ( ) (

2 2

t dt t d dt t d f x C x M = + ) (

ext =

+ = f x K f

( )

) ( ) (

ext =

+ = f x K f t t

Mass-Spring Dynamics

Equation of motion for mass point i at time t

• 2nd order differential equation
• f is used for acceleration and damping
• Without damping term, simply f = ma

mass position damping coefficient spring + external damping acceleration

) ( ) ( ) (

2 2

t dt t d c dt t d m

i i i i i

f x x = +

Numerical I ntegration

2nd order ODE ⇒ two coupled 1st order ODEs

) ( ) ( ) (

2 2

t dt t d c dt t d m

i i i i i

f x x = + ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ − = =

i i i i i i i

m t c t dt t d t dt t d ) ( ) ( ) ( ) ( ) ( v f v v x

velocity acceleration

Damping Revisited

Point damping

• Damping force in opposite direction to the velocity

Damped spring

• Damping force proportional to the velocity of the spring

i i i i i

m t c t dt t d ) ( ) ( ) ( v f v − =

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Topology and Stability

Stability with respect to deformation

not stable stable but not general much more resistant in direction than in Requires well-chosen stable topologies!

Finite Element Method with Lumped Mass Formulation

Min Gyu Choi Graphics & Media Lab.@SNU

Mass-Spring Systems vs. FEM

• Mass-spring dynamics
• Explicit integration
• Elasto-dynamic equations
• Explicit integration

f Ku u C u M = + + & & &

( ) ( )

Ku u C f u M u u u − − = ∆ ∆ + = ∆ & & & & h h ) ( ) ( ) (

2 2

t dt t d c dt t d m

i i i i i

f x x = + matrix inversion small displacements stable topology appropriate constants

( )

i i i i i i i i

c m h h x f x x x x & & & & − = ∆ ∆ + = ∆ ) (

Mass-Tensor Systems

• Mass-spring dynamics
• Explicit integration
• Elasto-dynamic equations
• Explicit integration

f Ku u C u M = + + & & &

( ) ( )

Ku u C f u M u u u − − = ∆ ∆ + = ∆ & & & & h h ) ( ) ( ) (

2 2

t dt t d c dt t d m

i i i i i

f x x = +

( )

i i i i i i i i

c m h h x f x x x x & & & & − = ∆ ∆ + = ∆ ) ( Providing force from strain energy rather than spring energy!

Lumped Mass Formulation

• Mass-spring dynamics
• Explicit integration
• Elasto-dynamic equations
• Explicit integration

f Ku u C u M = + + & & &

( ) ( )

Ku u C f u M u u u − − = ∆ ∆ + = ∆ & & & & h h ) ( ) ( ) (

2 2

t dt t d c dt t d m

i i i i i

f x x = +

( )

i i i i i i i i

c m h h x f x x x x & & & & − = ∆ ∆ + = ∆ ) ( mass lumping

≅

3 3×

Lumped Mass Formulation

• Mass-spring dynamics
• Explicit integration
• Elasto-dynamic equations
• Explicit integration

f Ku u C u M = + + & & &

( ) ( )

Ku u C f u M u u u − − = ∆ ∆ + = ∆ & & & & h h ) ( ) ( ) (

2 2

t dt t d c dt t d m

i i i i i

f x x = +

( )

i i i i i i i i

c m h h x f x x x x & & & & − = ∆ ∆ + = ∆ ) (

( ) ( )

Ku u C f u D u u u − − = ∆ ∆ + = ∆ & & & & h h

m

( ) ( )

Ku u C f D u u u u − − = ∆ ∆ + = ∆

−

& & & &

1 m

h h Mass Lum ping

SLIDE 4

4

Mass-Spring vs. Lumped Mass FEM

• Mass-spring dynamics

Large displacements

are supported inherently.

Damped spring

• Elasto-dynamic equations

Large displacements Strain rate tensor ) ( ) ( ) (

2 2

t dt t d c dt t d m

i i i i i

f x x = +

j i j i ij

u u u u ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ = x x x x & & ν

ij j i ij

u u δ ε − ∂ ∂ ⋅ ∂ ∂ = x x Adopt nonlinear visco-elastic formulation! f Ku u C u M = + + & & &

Visco-Elastic Formulation

Green’s strain tensor

ij j i ij

u u δ ε − ∂ ∂ ⋅ ∂ ∂ = x x I J J ε − =

T

) (u x u

[ ]

3 3×

∂ ∂ =

j i

u x J

Visco-Elastic Formulation

Green’s strain tensor Strain rate tensor

ij j i ij

u u δ ε − ∂ ∂ ⋅ ∂ ∂ = x x I J J ε − =

T j i j i ij ij

u u u u dt d ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ = = x x x x & & ε ν J J J J ν & &

T T +

=

[ ]

3 3×

∂ ∂ =

j i

u x & & J

[ ]

3 3×

∂ ∂ =

j i

u x J

Visco-Elastic Formulation

Stress tensor Strain energy ε I ε σ µ λ

ε

2 ) tr( + = ν I ν σ ψ φ

ν

2 ) tr( + =

elastic stress tensor viscous stress tensor

∫

Ω

Ω = d U

ij ijε

σ ε

ε

2 1

∫

Ω

Ω = d U

ij ijν

σ ν

ν

2 1

elastic potential energy damping potential energy

Finite Element Discretization

Refer to the paper

James F. O’Brien and Jessica K. Hogdins Graphical Modeling and Animation of Brittle Fracture Computer Graphics (Proc. SIGGRAPH ’99), pp. 137-146