Geology Reference
In-Depth Information
The constraint functions are exactly satisfied
in a DDO. However, these are expected to vary
due to the presence of uncertainty in the DVs and
DPs. As a consequence, the final design obtained
by the DDO approach may become infeasible in
presence of uncertainty and the constraint func-
tions are required to be revised to include the
effect of their variations due to uncertainty. The
robustness of a constraint is the feasibility of the
constraint that needs to be guaranteed for the
considered uncertainty ranges of the DVs and
DPs. Further details of this can be found in the
works of Lee, & Park (2001); Wang, Peng, Hu,
& Cao (2009). When the DVs and DPs are char-
acterized by random variables with known prob-
ability density function (pdf), the probabilistic
feasibility of the constraint can be approximated
ensuring a target reliability level for the considered
constraint (Zang, Friswell, & Mottershead, 2005).
A general probabilistic feasibility formulation for
jth constraint can be expressed as:
P g
to uncertainty in the DVs and DPs can be esti-
mated through first-order Taylor series expansion
of the constraint function (Du, & Chen, 2000) as:
g
u
N
1
j
g
u
(4)
=
j
i
i
=
i
To further enhance the quality of robustness
of the constraint, it is multiplied by a penalty
factor, k j and the constraint function can be ex-
pressed as (Lee, & Park, 2001):
g
( u = +
g
k
g
0
(5)
j
j
j
j
It may be noted that a direct relationship to
ensure a target reliability level can not be achieved
in case of UBB type DVs and DPs and the value
of k j is introduced in equivalence to the probabi-
listic feasibility formulation, thereby indirectly
ensuring safety requirement of a design. The
selection of k j is somewhat ad hoc. Obviously,
the larger value of k j means one is more conser-
vative to enhance the feasibility of the associated
constraint.
Finally, the conventional RDO formulation
is expressed by combining Eqs. (3) and (5) as:
( ) ≤ ≥ 0 1 , where, P oj is
the probability one desires to satisfy for the jth
constraint feasibility. Assuming g j ( ) to be nor-
mally distributed, this probabilistic feasibility of
the constraint can be approximated as
g
[
]
P j
,
, ....,
m
j
oj
( u + σ 0 , where, g j ( u and σ g j are the
mean and standard deviation of g j ( ) . The pen-
alty factor, kj is used to enhance the feasibility of
g j ( ) and can be obtained from, kj= ¦ -1 (
k
j
j
g j
f
f
f
f
minimize:
(
1
α
)
+
α
,
0
≤ ≤
α
1
*
*
P oj ,
where ¦ -1 (.) is the inverse of the cumulative
density function of a standard normal distribution.
For example, to ensure a reliability level of P oj
=99.87%, one should take kj=3.0.
However, when the DVs and DPs are of UBB
type, above probabilistic feasibility formulation
for the constraints cannot be adopted. In order to
enhance the feasibility of the jth constraint, an
additional quantity ( g j ) is introduced to con-
sider the effect of uncertainty in the DVs and DPs.
The maximum variation ( g j ) of the jth constraint
with respect to its nominal value, g
)
such that:
g
+
k
g
0
,
j
=
1
,
2
, ......,
m
.
j
j
j
(6)
The conventional RDO approach as presented
above and also by the proposed RDO formulation
to be presented in the later part of the chapter are
based on linear perturbation based approximation
of response functions about the mean values of the
UBB parameters. The accuracy and efficiency of
such linear perturbation based approach are well
documented in stochastic finite element literatures
(Vanmarcke et al., 1986; Ghanem, & Spanos, 1990;
Kleiber, & Hien, 1992). The study of accuracy
= (u ) due
g
j
 
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