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mulae for updating design variables. Moharrami
(2006) improved the convergence of this type of
OC algorithm by establishing and solving a QP
sub-problem for finding the Lagrange Multipliers.
This improvement completes the efficiency and
robustness of the algorithm.
of FD method depends on the degree of nonlinear-
ity of the function and the value of
δ
d
i
. Habibi
and Moharrami (2010) showed that FD method
not only does not give accurate sensitivity values
but also sometimes results in false evaluation.
Analytical Sensitivity Analysis
SENSITIVITY ANALYSIS
Another way of evaluating the sensitivity of a
structural behavior is to differentiate the basic
equation from which that behavior is evaluated.
As an example, the sensitivity of the displace-
ment vector when a structure behaves linearly can
be derived as follows:
The basic equation that the displacement vector
is calculated from is:
The change in a behaviour
B
j
of structure due to
change in a design variable
d
i
that is expressed as
B
d
∂
∂
j
i
is called the sensitivity of
B
j
with respect to
d
i
.
The sensitivity calculation is performed in several
ways. Here, we point to some of them.
K
∆ =
P
(18)
Differentiating this equation with respect to
any generic design variable,
d
i
results in:
Finite Difference Method
As was mentioned before, the behaviour B is usu-
ally implicit function of design variable. Therefore,
it is not possible to find the sensitivity via a direct
differentiation of a function. As a remedy for this
problem, the Finite Difference (FD) method has
been widely used. In this method, the difference
between the values of a function at two adjacent
points are used to find the approximate slope of the
curve (sensitivity) at the desired point as follows:
∂
∂
K
d
∂
∂
∆
=
∂
∂
P
d
∆
+
K
(19)
d
i
i
i
Rearranging the above equation for
∂ ∆
d
i
results in the sensitivity equation as follows:
P
d
K
d
∂
∂
∆
∂
∂
−
∂
∂
K
−
1
=
∆
(20)
d
i
i
i
1
2
∂
∂
B
d
B d
(
)
−
−
B d
(
)
j
j
j
Note that all terms in the right hand side of
the above equation is known and although Δ is
not known explicitly in terms of design variables,
its derivatives can be calculated accurately. To
calculate the sensitivity
∂ ∆
d
i
from the above
equation efficiently, the paper by Arora and
Haug,(1979) is recommended for study.
As another example, consider the evaluation
of sensitivity of a number of vibrating frequencies
of a structure. The basic equation for calculating
≈
(17)
1
2
(
d
d
)
i
i
i
The numerator of the right side of Eq.(17) is
the difference between the values of
B
j
at two
adjacent design points
d
1
and
d
2
where all com-
ponents of vector of design variables remain
unchanged except
d
i
that can assume any value
of
d
1
±
δ
i.e., the current point and a point before,
or current point and the point after, or the points
before and after the current point. The accuracy
d
i
i
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