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University of Houston Cullen College of Engineering neering
Introduction to SCM (cont.)
To use such a rule, one needs to know the simulation output
at each of the quadrature points
- - for this purpose, one can build a polynomial approximation
( ) f ξ r
{ }
1 Q q q
ξ
=
r
( ( )) G f ξ r % ( ( )) G f ξ r trical & Computer Engin and then evaluate that approximation at the quadrature points
- - the simplest means of doing this is to use the set of Lagrange
interpolation polynomials corresponding to the sample points
( ) { }
1
sample
N s s
L ξ
=
r
( ) ( )
( )
( ) ( )
sample
N s s
G f G f L ξ ξ ξ = ∑ r r %
{ }
1
sample
N s s
ξ
=
r
1
( ( )) ( ) ( ) (( ( )))
Q q q q q
G f d G f ξ ρ ξ ξ ω ρ ξ ξ
Γ =
=
∫ ∑
r r r r r
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29
- Then we have the approximation for the mean:
( ) ( )
( )
1
( ) ( )
s s s
G f G f L ξ ξ ξ
=
∑
[ ]
( )
1 1
( ) ( ) ( ) ( )
sample
N Q s q s q q s q
E f f d f L ξ ρ ξ ξ ξ ω ρ ξ ξ
Γ = =
= = ∑
∑ ∫
r r r r r r ( )
1 1 1
( ( )) ( )
sample
q N Q s q s q q s q
G f L ξ ω ρ ξ ξ
= = =
= ∑
∑
r r r
University of Houston Cullen College of Engineering neering
i
' ε
_
= 1.0 m,
1, = =
r host host x y z
μ
ε σ = =
l l l
Numerical example
trical & Computer Engin
_
[ ]:15% [ 8, [
] ] 1
r inclusion
inclusion
E VF E E ε σ =
=
Mean values:
Goal: Evaluating the global sensitivity of effective permittivity
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30
g g y p y due to variation in certain mixing parameters
Varying parameters: inclusion relative permittivity , inclusion conductivity
and volume fraction; all of which are assumed to be Gaussian variables.
