Environmental Engineering Reference
In-Depth Information
b. For a one-dimensional vertical random field generated in the interval [−
a
y
,
a
y
],
the eigenvalues and eigenfunctions are calculated using the same equations after
replacing the horizontal coordinate (
x
), the horizontal half-width (
a
x
) of the domain
and the horizontal autocorrelation length (
l
x
), respectively, by the vertical coordi-
nate (
y
), the vertical half-depth (
a
y
) of the domain, and the vertical autocorrela-
tion length (
l
y
).
c. In the case of a 2D random field, the eigenvalues are calculated as the product of all
possible combinations of eigenvalues of the 1D random fields of each direction as
follows:
y
2D
=
x
λ
λ λ
(15.8)
i
j
k
where
λ
i
2D
are the eigenvalues of the 2D random field, λ
j
x
{
j
= 1
, ...,
M
}
are the eigenvalues
of the horizontal direction, and λ
y
{
}
…
k
=
1,
,
M
are the eigenvalues of the vertical direc-
tion. Similarly
φ
2D
(, )
xy
=
φ
( )
x
φ
(
y
(15.9)
i
j
k
in which φ
i
2D
(, ) are the eigenfunctions of the 2D random field, φ
j
xy
(){
xj
= 1
, ...,
M
}
are the
= 1
…
eigenfunctions of the horizontal direction, and φ
k
(){
yk
,
,
M
}
are the eigenfunctions of
the vertical direction.
Notice that the eigenvalues (and the corresponding eigenfunctions) of the 2D random
field retained in the analysis are the highest
M
ones in the list of values obtained after
arranging these eigenvalues in a decreasing order. It should be emphasized that the choice
of the number
M
of terms retained in the K-L expansion (cf.
Equation 15.2
)
depends on
the desired accuracy of the problem being treated. In the case of a Gaussian random field,
the error estimate of the K-L expansion with
M
terms can be calculated as follows (Sudret
and Berveiller 2008):
M
∑
ε
rr xy
(, )
=
11
−
(
/
σ
)
λφ
2
( ,)
xy
(15.10)
ii
i
=
1
in which σ is the standard deviation of the Gaussian random field. In the case of a log-nor-
deviation of the underlying normal random field. It should be mentioned that in this case,
the K-L expansion given in
Equation 15.2
becomes (Cho and Park 2010)
M
∑
1
Rx,y,
(
θ
) ≈
exp
µ
+
λ
φ
(, )()
xy
ξ θ
(15.11)
ln
i
i
i
i
=
where μ
ln
is the mean value of the underlying normal random field. Notice that σ
ln
and μ
ln
can be computed using the following equations:
(15.12)
σ
=
ln
(
1
+
(
σ µ
/
))
2
ln
µ
=
ln
µ
− 05
.
σ
2
(15.13)
ln
ln
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