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-
mal random field, one should use σ ln instead of σ in Equation 15.10 , where σ ln is the standard
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
 
Search WWH ::




Custom Search