Geology Reference
In-Depth Information
Box 5.
0 2
∑
f
∂
({ })
y
y
0
∂
f
({ }) (
y
y
0
y
y
)
∂
f
({ })
y
y
0
y
y
−
∑
∑
y
−
i
2
χ
f
y
(
i
−
2
)
(30)
i
i
i
0
∂
∂
∂
+
−
−
(
i
)
i
(
i
)
i
i
(
i
)
i
i
Box 6.
∑
h
t
y
y
0
h
t
y
y
0
h
t
y
0
∂
({ })
∂
({ })
(
0 2
∂
({ })
y
y
)
x
y
−
t
(
h
(
≤
j
j
j
y
−
i
2
χ
y
(
i
−
2
)
j
i
i
i
0
∂
∂
∂
y
(31)
+
−
−
(
i
)
i
i
)
i
i
i
)
i
i
j
j
j
−
h
t
({ })
y
0
+
ln[
P
*
] ,
j
=
1
, ...,
n
j
F
c
j
Solution Scheme
t
0
∂
h
({ })
y
y
∑
t
0
t
0
j
0
h
({ })
y
=
h
({ })
y
−
y
j
j
i
∂
+
(
i
j
)
i
The solution scheme of the optimization process
proceeds as follows:
t
0
∑
h
∂
({ })
y
y
t
h
j
0
2
−
y
(
χ
−
1
)
j
∂
i
i
−
(
i
)
i
(28)
j
1. Start from a feasible design. At the beginning
of the
k
th
design cycle (
k
=
0 1 2
, ,
, ...
) the
∑
and
i
j
−
∑
mean summation over the
where
objective function
f
({ })
and constraint
y
(
)
(
)
i
j
+
functions
h
=
1
are approxi-
mated by using the approach introduced
previously. The approximations require
function evaluations (
f
({ }),
y
j
, ...,
n
variables belonging to group
(
i
j
−
re-
spectively. Then, the approximate reliability
constraints can be written as
i
j
+
and
(
),
j
c
)
y
k
({ })
,
k
h
({ }),
y
j
=
1
, ...,
n
) and sensitivity analy-
t
*
h
({ })
y
=
h
({ })
y
−
ln[
P
]
≤
0
,
j
=
1
, ...,
n
j
c
cqj
cqj
F
c
k
k
({ }), ({ }), , ...,1
). In
this step, all design variables are assumed
to be continuous.
2. Using this information an explicit sub-
optimization problem is constructed. The
explicit sub-optimization problem can be
solved either by standard methods that treat
the problem directly in the primal design
variable space (branch and bound techniques,
combinatorial methods, evolution-based
optimization techniques, etc.) or by dual
methods (Fleury and Braibant, 1986). In the
present implementation, a genetic algorithm
is applied to solve the sub-optimization
problem.
ses (
∇
f
y
∇
h
y
j
=
n
j
(29)
j
c
Approximate Optimization Problem
Using the above approximations for the objective
and constraint functions, the approximate sub-
optimization problem defined at the point
{ }
y
0
,
0
takes the form in Box 5 and 6.
P
{ }
0
: Minimize (see Box 5) subject to (see
Box 6)
j
P
{
}
,
=
1, ...,
, where all terms have been
previously defined.
n
c
Search WWH ::
Custom Search