Geology Reference
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stochastic dynamic responses and their sensitivi-
ties for evaluating the stochastic constraint of the
related optimization problem have been avoided
by applying the
MLSM
based adaptive
RSM.
The
proposed
RDO
approach is elucidated through the
optimization of a three-storied concrete frame. The
numerical results obtained by the conventional
and the proposed
RDO
approaches are presented
to demonstrate the effectiveness of the proposed
RDO
approach.
for a performance function, f (u), its nominal
value is obtained as:
f
=
(u
)
, where
u
is a vec-
tor comprising of nominal values of u. Generally,
the RDO is performed by improving the robust-
ness of the performance function by minimizing
a gradient index obtained through first-order
Taylor series expansion (Lee, & Perk, 2001) as
defined below:
N
∂
∂
f
u
∑
1
∆
f
∆
u
(2)
=
i
i
=
i
The Conventional RDO Approach
Most of the developments in the field of
RDO
as
discussed in the previous section primarily use
the weighted sum method (
WSM
) for examples as
adopted by Du &Chen (2000), Lee & Perk (2001).
This approach is termed in the present study as
'the conventional
RDO
approach' in which the
robustness of a design performance is expressed
in terms of the dispersion of performance function
from its nominal value.
Let, u = (x, z) is a vector composed of n-di-
mensional DVs,
x
=
[
where, N is the total number of DVs and DPs and
∆
u
i
quantifies their uncertainty amplitudes. The
objective of an RDO is to achieve optimum per-
formance of the design as well as its less sensitiv-
ity with respect to the variations of DVs and DPs
due to uncertainty. This leads to a dual criteria
performance function. This dual criteria perfor-
mance function is transformed to an equivalent
single objective function as following:
x x
,
, ..
x
n
]
and l-dimen-
1
2
f
f
∆
∆
f
f
1 2
. The lower and
upper bounds of the ith UBB type DV or DP,
u
i
I
are denoted by
u
l
and
u
i
u
, respectively. In interval
mathematics
u
i
I
is expressed in Box 1 (Moore,
1979).
In the above,
u
i
is the nominal value of
u
i
, and
∆
u
i
denotes the maximum variation of
u
i
from
its' nominal value, termed as dispersion. If a
practical estimate of the nominal value is avail-
able, it can be directly assigned to
u
i
. In absence
of that,
u
i
is usually taken as,
u
sional DPs,
z
=
[
z z
,
, ..,
z
l
]
minimize:
(
1
−
α
)
+
α
,
0
≤ ≤
α
1
*
*
(3)
where,
α
is a weighting factor i
n
the above bi-
objective optimization problem,
f
*
and
∆
f
*
are
the optimal solutions at two ideal situations ob-
tained for
α
=1.0 and 0.0, respectively. The
maximum robustness will be achieved for α =0.0,
and α = 1.0 indicates optimization without any
consideration for robustness.
(
)
/ 2
. Then,
l
u
+
u
i
i
Box 1.
u
I
=
u u
l
u
=
u
−
∆
u u
+
∆
u
=
u u
+ −
∆ ∆
u
u
(1)
[
,
]
[
,
]
[
,
]
[
,
]
i
i
i
i
i
i
i
i
i
i
i
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