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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)