Geology Reference
In-Depth Information
y
k
is tested if it is ac-
ceptable in terms of a conservative
criterion, i.e. if
f
STOCHASTIC ANALYSIS
3. The new point
{
*
}
*
k
*
k
The characterization of the sub-optimization
problems
P
({
y
})
≥
f
({
y
})
and
cq
=
0 1 2
requires the estima-
tion of failure probabilities and their sensitivities.
For that purpose an advanced simulation technique
is adopted and implemented in the present for-
mulation.
}
,
k
, ,
, ...
if
h
* *
≥ =
1
If
these conditions are satisfied (conserva-
tive step) the point
{
({
y
k
})
h
({
y
k
}),
j
, ...
n
.
y
k
{
cqj
j
c
y
k
is used as the
current design for the next cycle, that
is,
{
*
}
y
k
does not represent a conservative step
an inner loop is initiated to effect con-
servatism. For functions which are not
conservative at
{
k
+
1
*
k
y
}
=
{
y
}.
If the design
{
}
Reliability Estimation
Subset simulation is adopted in this formulation
for the purpose of estimating the corresponding
failure probabilities during the design process (Au
and Beck, 2001). In the approach, the failure
probabilities are expressed as a product of con-
ditional probabilities of some chosen intermediate
failure events, the evaluation of which only re-
quires simulation of more frequent events. There-
fore, a rare event simulation problem is con-
verted into a sequence of more frequent event
simulation problems. For example, the failure
probability
P
y
k
the corresponding
coefficients of the second-order diagonal
terms are increased by multiplying the
scalars
χ
i
by a constant greater than one.
The modified approximations are used
to construct a new sub-optimization
problem to obtain a new point.
4. The overall design process is continued until
some convergence criterion is satisfied.
*
}
It is noted that the requirement of a conserva-
tive step in the algorithm can be relaxed and
demand that a feasible descent step is made instead,
i . e . i f
f
F
j
({ })
can be expressed as the
y
product
*
k
*(
k
−
1
)
({
y
})
<
f
({
y
})
a n d i f
m
−
1
*
≤
0 1
In this case, the con-
servatism is only enforced when a feasible descent
step could not be made. This approach which is
called, relaxed conservatism, inherits the global
convergence properties of the algorithm that
enforces conservatism at each design cycle. In
general the above optimization scheme is very
effective when the curvatures of the functions
involved in the optimization problem are not too
large and relatively uniform throughout the design
space. For other cases, methods based on local
response surfaces, trust regions and line search
methodologies may be more appropriate (Alex-
androv et al., 1998; Bucher and Bourgund, 1990;
Jensen et al., 2009).
h
({
y
k
})
,
j
, ...
n
.
F
j
∏
P
({ })
y
P F
(
({ }))
y
P F
(
({ }) /
y
F
({ }))
y
=
j
c
F
j
,
1
j k
,
+
1
j k
,
j
k
=
1
(32)
where
F
({ })
y
=
F y
({ })
is the target failure
j m
,
j
F
j
event and:
F
({ })
y
⊂
F
({ })
y
...
⊂
F
({ })
y
j m
,
j m
,
−
1
j
1
,
F
j
F
j
is a nested sequence of failure events. Equation
(32) expresses the failure probability
P
F
j
({ })
as
y
a product of
P F
(
({ }))
y
and the conditional
j
,1
probabilities:
P F
(
({ }) /
y
F
({ })),
y
k
=
1
, ...,
m
−
1
.
j k
,
+
1
j k
,
F
j
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