Environmental Engineering Reference
In-Depth Information
the vector {ξ
if
}
if
=1
, …,
M
. This ensures that the SS technique does not affect the correlation
structure of the random field.
The basic idea of the link is that for a given random field realization obtained by K-L
expansion, the vector {ξ
if
}
if
=1
, …,
M
represents a sample of the SS method for which the system
response is calculated in two steps. The first step is to substitute the vector {ξ
if
}
if
=1
, …,
M
in
the K-L expansion to calculate the values of the random field at the centers of the different
elements of the deterministic mesh according to their coordinates. The second step is to use
the deterministic model to calculate the corresponding system response.
The algorithm of the SS approach in the case of a spatially varying soil property can be
described as follows:
1. Choose the number
M
of terms of K-L expansion. This number must be sufficient to
accurately represent the target random field.
2. Generate a vector of (
M
) standard normal random variables {ξ
1
, …, ξ
if
, …, ξ
M
} by
direct MCS.
3. Substitute the vector {ξ
1
, …, ξ
if
, …, ξ
M
} in the K-L expansion to obtain the first real-
ization of the random field. Then, use the deterministic model to calculate the corre-
sponding system response.
4. Repeat steps 2 and 3 until obtaining a prescribed number
N
s
of realizations of the
random field and their corresponding values of the system response. Then, evaluate
the corresponding values of the performance function to obtain the vector
G
0
, where
GG GG
……
. Notice that the values of the performance function of the
different realizations are arranged in an increasing order in the vector
G
0
. Notice also
that the subscripts “0” refer to the first level (level 0) of the SS approach.
1
k
Ns
= {,,
,
,
}
0
0
0
5. Prescribe a constant intermediate conditional failure probability
p
0
for the failure
regions
F
j
(
j
= 1, 2, …,
m
− 1) and evaluate the first failure threshold
C
1
, which cor-
responds to the failure region
F
1
, where
C
1
is equal to the [(
N
s
×
p
0
) + 1]th value in the
increasing list of elements of the vector
G
0
. This ensures that the value of
P
(
F
1
) will be
equal to the prescribed
p
0
value.
6. Among the
N
s
realizations, there are [
N
s
×
p
0
] ones whose values of the performance
function are less than
C
1
(i.e., they are located in the failure region
F
1
). The corre-
sponding vectors of standard normal random variables are used as “mother vectors”
to generate
N
s
new vectors of standard normal random variables using Markov chain
method based on the modified M-H algorithm by Santoso et al. (2011). These new
vectors are substituted in the K-L expansion to obtain the random field realizations
of level 1.
7. The values of the performance function corresponding to the realizations of level 1
are gathered in an increasing order within a vector of performance function values
GG GG
1
……
k
Ns
= {,,
,
,
}
1
1
1
8. Evaluate the second failure threshold
C
2
as the [
(
Np
s
×+
)
1
]th
value in the increasing
0
list of elements of the vector
G
1
.
9. Repeat steps 6-8 to evaluate the failure thresholds
C
3
,
C
4
, …,
C
m
corresponding to
the failure regions
F
3
,
F
4
, …,
F
m
. Notice that contrary to all other thresholds, the last
failure threshold
C
m
is negative. Thus,
C
m
is set to zero and the conditional failure
probability of the last level, that is,
P
(
F
m
|
F
m
−1) is calculated as
Ns
1
=
∑
P(
F
m
F
)
Is
(
)
(15.16)
m
−
1
Fk
N
m
s
k
=
1
where
I
F
m
= 1, if the performance function
G
(
s
k
) is negative and
I
F
m
= 0 otherwise.
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