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max
β
eigenvalues are differentiable (in the Fréchet
sense). Note that in this approach the symmetry
reduction is not applied in the analysis and the
analysis is based on the full structure. Using
symmetry reduced structure for analysis cannot
be validated in eigenfrequency optimization since
the eigenmodes are not necessarily symmetric
even for a symmetric structure.
Another approach to avoid repeated eigenfre-
quencies is to add an extra condition to the optimi-
zation problem ensuring that the eigenfrequencies
are distant from each other. For example one may
reformulate problem (1.15) as follows (Bendsøe
and Sigmund 2003).
x x
,
,
,
x
1
2
N
j
such that
α λ
β
λ
,
j
=
1 2
1 2
,
,
,
N
j
d
K
ϕ
=
M
ϕ
,
j
=
,
,
,
N
j
j
j
d
N
x v
v
e e
e
=
1
0
≤ ≤
x
1
,
e
=
1 2
,
,
,
N
e
(1.32)
with α < 1, for example α = 0.95. This formulation
is known as bound formulation . Note that with α =
1, Problem (1.32) is equivalent to Problem (1.15).
It is also possible to turn around the multiple
eigenvalue problem by including adjacent eigen-
frequencies in the objective function. For example
Yang et al. (1999b) used the arithmetic mean of
the eigenvalues as the objective function when
Figure 6. Solving example 2, using mean eigenfrequency as objective function: final solution (a) and
evolution of the objective function and the first three eigenfrequencies (b)
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