Material Point Method (MPM)

Wednesday, 10/2/2002

Variational principleParticle discretizationGrid interpolation

Lennard-Jones Potential

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φ(rij )=4εσrij

⎛

⎝ ⎜

⎞

⎠ ⎟ 12

−σrij

⎛

⎝ ⎜

⎞

⎠ ⎟

6⎡

⎣ ⎢

⎤

⎦ ⎥

potential force

Handout: Molecular dynamics of simple systemsError, page 7

Motion equations

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σij, j +ρbi =ρai

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∂σ xx

∂x+

∂σxy

∂y+ρbx =ρax

∂σ xy

∂x+

∂σyy

∂y+ρby =ρay

Tensor expression:

Engineering expression:

Discretization in Material Point Method (MPM)

Solid line is an outline of the body analyzed.Black dots are the material points.Dashed lines show a regular, background grid for calculation.

Motion equations (Variational form)

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∂σ xx

∂x+

∂σxy

∂y+ρbx −ρax

⎛

⎝ ⎜ ⎞

⎠ Ω∫ ψdΩ =0

∂σ xy

∂x+

∂σyy

∂y+ρby −ρay

⎛ ⎝ ⎜ ⎞

⎠ Ω∫ ψdΩ =0

: arbitrary spatial function

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∂σ xx

∂x+

∂σxy

∂y+ρbx −ρax =0

∂σ xy

∂x+

∂σyy

∂y+ρby −ρay =0

Elastic Bar Dropping

€

dσdx

+ρb=ρa

€

b=g

Acceleration due to gravity

Particle Discretization?How to solve?

€

dσdx

+ρb=ρa

Direct solution

€

dσdx

+ρb=ρa

€

x← x+vdt

v← v+adt a=1ρ

dσdx

+b

σ ← σ +Edvdx

dt

Solution:

Variational Form

€

dσdx

+ρb−ρa⎛ ⎝

⎞ ⎠ a

b∫ ψdx=0

€

dσdx

+ρb=ρa

: arbitrary spatial function

Particle Discretization?

€

dσdx

+ρg−ρa⎛ ⎝

⎞ ⎠ a

b∫ ψdx=0

Variational Equation

€

dσdx

+bρ−aρ⎛ ⎝

⎞ ⎠ a

b

∫ ψdx=0

€

d(σψ)dx

−σdψdx

+ bρ−aρ( )ψ⎡ ⎣

⎤ ⎦ dx

a

b∫ =0

€

(σψ)b −(σψ) a + bρψdxa

b∫ − σdψdx

dxa

b∫ = aρψdxa

b∫

Considering free ends at a and b

€

bρψdxa

b

∫ − σdψdx

dxa

b

∫ = aρψdxa

b

∫

Particle Discretization

€

m(p)b(p)ψ (p)

p

∑ − V(p)σ (p) dψ (p)

dxp

∑ = m(p)a(p)ψ (p)

p

∑

€

bρψdxa

b

∫ − σdψdx

dxa

b

∫ = aρψdxa

b

∫

Spatial grid

Shape function

€

φ(x)=(x2 −x)x2 −x1

φ (1) +(−x1 +x)x2 −x1

φ (2)

€

φ(x)= φ (n)N(n)(x)n

∑

€

N(1)(x) =(x2 −x)x2 −x1

N(2)(x)=(−x1 +x)x2 −x1

Grid Interpolation

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m(p)b(p)ψ (p)

p

∑ − V(p)σ (p) dψ (p)

dxp

∑ = m(p)a(p)ψ (p)

p

∑

€

ψ ( p) = ψ (n)N(n,p)

n

∑

€

a(p) = a (n)N (n,p)

n

∑

€

m( p)b( p) Nn

∑ (n,p)ψ (n)

p

∑ − V( p)σ( p) dN(n,p)

dxψ (n)

n

∑p

∑

= m( p) N (n',p)

n'

∑ a(n') N(n,p)

n

∑ ψ (n)

p

∑

Grid Equation

€

m( p)b( p) Nn

∑ (n,p)ψ (n)

p

∑ − V( p)σ( p) dN(n,p)

dxψ (n)

n

∑p

∑

= m( p) N (n',p)

n'

∑ a(n') N(n,p)

n

∑ ψ (n)

p

∑

€

m(p)b(p)N (n,p)

p

∑ − V(p)σ (p) dN(n,p)

dxp

∑ = N (n',p)N (n,p)

p

∑ m(p)

n'

∑ a(n')

Since is arbitrary:

€

ψ (n)

Grid Equation (F=ma)

€

m(p)b(p)N (n,p)

p

∑ − V(p)σ (p) dN(n,p)

dxp

∑ = N (n',p)N (n,p)

p

∑ m(p)

n'

∑ a(n')

€

F (n) = m(p)b(p)N (n,p)

p

∑

€

f (n) =− V(p)σ (p) dN(n,p)

dxp

∑

€

m (n,n') = N (n',p)N (n,p)

p

∑ m(p)

External force

Internal force

Mass matrix

€

F (n) +f (n) = m (n,n')

n'

∑ a(n')

Derivative of Shape Function

€

N(1)(x) =(x2 −x)x2 −x1

N(2)(x)=(−x1 +x)x2 −x1

€

ddx

N (1)(x)=−1

x2 −x1

ddx

N (2)(x) =1

x2 −x1

Summary

Variational principleParticle discretizationGrid interpolation

€

F (n) +f (n) = m (n,n')

n'

∑ a(n')

Next class: MPM procedure