Material Point MethodGrid Equations

Monday, 10/7/2002

Mass matrixLumped mass matrix

Elastic Bar Dropping

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dσdx

+ρb=ρa

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b=g

Acceleration due to gravity

Variational Form

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dσdx

+ρb−ρa⎛ ⎝

⎞ ⎠ a

b∫ ψdx=0

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dσdx

+ρb=ρa

: arbitrary spatial function

Particle Discretization?

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dσdx

+ρg−ρa⎛ ⎝

⎞ ⎠ a

b∫ ψdx=0

Grid Equations of Motion

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

p

∑

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f (n) =− V(p)σ (p) dN(n,p)

dxp

∑

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

p

∑ m(p)

External force

Internal force

Mass matrix

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

n'

∑ a (n')

Shape function between node n and particle p:

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N(n,p) =N (n)(x( p))

Shape Functions (1D)

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N(1)(x) =(x2 −x)x2 −x1

N(2)(x)=(−x1 +x)x2 −x1

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ddx

N (1)(x)=−1

x2 −x1

ddx

N (2)(x) =1

x2 −x1

Elastic Bar Dropping (coordinates)

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x (1) =1m

x (2) =2m

x (3) =3m

Nodes:

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x(1) =1.25m

x(2) =1.75m

x(3) =2.25m

x(4) =2.75m

Particles:

Elastic Bar Dropping (particle mass)

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m(1) =m(2) =m(3) =m(4) =1000kg

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ρ=2000kg m3Mass density:

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A=1m2Cross section area:

Elastic Bar Dropping (mass matrix)

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

p

∑ m(p)Mass matrix

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m =

? ? 0

? ? ?

0 ? ?

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Elastic Bar Dropping (grid mass)

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

p

∑ m(p)Mass matrix

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N(1)(x) =(x2 −x)x2 −x1

N(2)(x)=(−x1 +x)x2 −x1

Grid mass (1,1)

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N(1,1) =34

, N (1,2) =14

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m (1,1) =34

⋅34

⋅1000kg+14

⋅14

⋅1000kg

=625kg

Grid mass (1,2)

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N(1,1) =N (2,2) =34

, N (2,1) =N (2,1) =14

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m (1,2) = N(1,p)N(1,p)

p

∑ m( p)

=34

⋅14

⋅1000kg+14

⋅34

⋅1000kg=375kg

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m (1,3) =0

Grid mass (2,1)

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m (2,1) = N(1,p)N(2,p)

p

∑ m( p) =m (1,2)

Grid mass (2,2)

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m (2,2) = N(2,p)( )

2

p

∑ m( p)

= N (2,1)( )

2+ N (2,2)

( )2+ N (2,3)

( )2+ N (2,4)

( )2

[ ]⋅1000kg

=14

⎛ ⎝

⎞ ⎠

2

+34

⎛ ⎝

⎞ ⎠

2

+34

⎛ ⎝

⎞ ⎠

2

+14

⎛ ⎝

⎞ ⎠

2⎡

⎣ ⎢ ⎤

⎦ ⎥ ⋅1000kg

=1250kg

Grid equations

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m =

625 375 0

375 1250 375

0 375 625

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ kg

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

p

∑

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f (n) =− V(p)σ (p) dN(n,p)

dxp

∑

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

n'

∑ a (n')

Mass matrix

Compact mass matrix:Only matrix elements close to diagonal are not zero.

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

n'

∑ a (n')

Lumped Mass Matrix

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M (n) ≈ m (n,n')

n'

∑ = N (n',p)N (n,p)

p

∑ m(p)

n'

∑Lumped grid mass :

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

n'

∑ a (n') ≈M (n)a (n)

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Q N(n',p)

n'

∑ =1

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M (n) ≈ m( p)N(n,p) N(n',p)

n'

∑⎛

⎝ ⎜ ⎞

⎠ ⎟

p

∑ = m( p)N (n,p)

p

∑

Mass matrix vs. lumped mass matrix

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m =

625 375 0

375 1250 375

0 375 625

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ kg

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M =

1000 0 0

0 2000 0

0 0 1000

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ kg

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

n'

∑ a (n') ≈M (n)a (n)