Wing Kam Liu* and Wei Chen, Northwestern University ... - SAMSI
Wing Kam Liu* and Wei Chen, Northwestern University ... - SAMSI
Wing Kam Liu* and Wei Chen, Northwestern University ... - SAMSI
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Stochastic Formulation for Multiresolution Continua<br />
Solid Mechanics<br />
( v) ⎡ + �( ) � � ⎤(<br />
v)<br />
⎡T + S + S ⎤ = −T<br />
N f + f g h<br />
⎣ Z⎦ ⎣ Z ⎦<br />
Intrinsic stress<br />
( )<br />
σ · ∇−Σ: ∇⊗∇ + f = � γ<br />
ρ ρ ρ<br />
S · ∇−s + B = Γ � , ρ = 1, 2, …,<br />
N<br />
Σ · n = 0<br />
σ · n−Σ: ∇⊗n<br />
S<br />
ρ<br />
Macro double stress<br />
Spatial operator<br />
Boundary conditions<br />
( )<br />
body force<br />
= t<br />
· n = T, ρ = 1,2, …,<br />
N<br />
acceleration<br />
Temporal operator<br />
( ( t ) ( t ) ( t ) )<br />
n n n n<br />
σβ , , β = f , f , f sym ∇v x, , Z , L x, , Z −L, ∇L<br />
x, , Z<br />
σ<br />
Z(<br />
s)<br />
x1<br />
β β<br />
R<strong>and</strong>om dimension<br />
Z<br />
x3<br />
Z()<br />
i<br />
t1<br />
x1<br />
VELOCITY<br />
x2<br />
x3<br />
(1)<br />
t1<br />
x2<br />
x1<br />
t2<br />
x3<br />
RANDOM SOLUTIONS<br />
t1<br />
x2<br />
t2<br />
MICRO-VELOCITY GRADIENT<br />
t2<br />
In stochastic collocation, Galerkin<br />
projection, r<strong>and</strong>om dimension<br />
discretized like space <strong>and</strong> time<br />
Constitutive law an implicit function of a denumerable set of r<strong>and</strong>om<br />
variables
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