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complex sensitivity analysis of the stochastic
dynamic system (Chaudhuri, & Chakraborty,
2004) as the RDO problem requires the gradients
of the objective function and associated con-
straints. For systems of practical interest, re-
peated evaluations of dynamic responses and their
sensitivities will be extremely time-consuming.
Thus, the use of direct optimization procedure is
not suitable to perform RDO under stochastic
excitation. In the present study, an alternative to
the direct optimization methods is proposed. An
RSM based approximation is adopted judicious-
ly to approximate the dynamic response required
to obtain the stochastic constraint of the related
SSO problem. The MLSM, a local approximation
approach is observed to be elegant in this regard
(Kim, Wang, & Choi, 2005) is adopted in the
present study. The essential concept of the MLSM
is briefly described here in order to outline the
procedure. The further details about this may be
found elsewhere (Kim, Wang, & Choi, 2005;
Bhattacharjya, & Chakraborty, 2009; Kang, Koh,
& Choo, 2010).
The MLSM based RSM is a weighted LSM
that has varying weight functions with respect to
the position of approximation. The weight associ-
ated with a particular sampling point xi decays
as the prediction point x moves away from xi.
The weight function is defined around the predic-
tion point x and its magnitude changes with x. If
yi is the ith response (i=1,2,….,q) with respect to
the variable xij, which denotes the ith observation
of the jth variable xj obtained by a suitable design
of experiments (DOE), following matrix form can
express the relationship between the responses (
y
) and the variables (
x
),
y
=
xa
+
µ
(21)
y
In the above,
x
,
y
,
a
and
µ
y
are the design
matrices containing the input data obtained by
the DOE, the response vector, the unknown coef-
ficient vector and the error vector, respectively.
The least-squares function
L x
y
( )
can be defined
as the sum of the weighted errors,
q
xa
∑
=
2
T
T
L x
( )=
w
ε
µ
W
( )
x
µ
= −
(
y
xa W y
)
( )(
x
−
)
y
(if
(if
(if
=
1
(22)
where,
W
(
x
is the diagonal matrix of the weight
function. It can be obtained by utilizing the weight-
ing function such as constant, linear, quadratic,
higher order polynomials, exponential functions,
etc. (Kim, Wang, & Choi, 2005), as described in
Box 3.
R
I
is the approximate radius of the sphere of
influence, chosen as twice the distance between
the centre point and extreme most experimental
point. The value of
R
I
is chosen to secure sufficient
number of neighbouring experimental points to
Box 3.
w x x
( -
) = ( )
w d
(if
Constant:
1
.0
Linear:
1-
d R
d R
/
/
I
2
(if
d R
/
≤
1.0),
Quadratic:
1- (
)
I
I
=
th
2
3
4
4 order polynomi
al: 1- 6(
d R
/
) + 8(
d R
/
) -3(
d R
/
)
I
I
I
Exponential:
exp
(-
d
/
R
)
I
(if
d R
/
≥
1
.0),
0.0
I
(23)
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