WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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respect to the higher bound C1 we calculate the eigenfrequencies and the sensitivities on the basis
of the interpolation function (34) with L40. The cost of using this method is that we have to
calculate the eigenfrequencies twice.
3.4. Results for a square design domain
In the first examples we use material parameters corresponding to
mA ¼ 2:25; mB ¼ 2.
First, we maximize the gap between eigenfrequencies for a square domain. A square design
inherently has double eigenfrequencies even with only one material. If we want to maximize the
gap between the first and second eigenfrequencies it is not possible to start from an initial design
where all of the design values have been assigned a uniform value (e.g. te ¼ 0:5). To overcome this
problem it is chosen to start from a design in which all design variables are assigned a finite value,
e.g. te ¼ 0:5, except for one element in which the value te ¼ 0 is used. With this small variation,
there are no initial double eigenfrequency and the optimization works. We apply a continuation
scheme in which the optimization is started with L ¼ 0 and the value of jLj is then increased
during the optimization to ensure a final 0–1 design. This scheme also reduces the possibility of
ending in a local minimum. Additionally, we always perform the optimization with different
initial designs in order to ensure that we converge to the same minimum.
In Fig. 10, the results of optimizing the gap between eigenfrequencies are shown for n 2½1 : 12Š.
The design domain is discretized in 100 100 elements. As it appears from the figure, the new
penalization scheme has enabled us to obtain optimized structures with a well-defined distribution
of the two materials. Only very few gray elements appear such as those near the corners for n ¼ 7
and n ¼ 12. For some modes it appears that the pattern observed in 1D is valid here as well, i.e.
the optimal design is periodic like with the periodicity increasing with increasing mode order. For
other modes (n ¼ 5, n ¼ 10, n ¼ 11 and n ¼ 12) the optimized design show a different topological
distribution of the two materials.
3.5. Results for a rectangular design domain
ARTICLE IN PRESS
J.S. Jensen, N.L. Pedersen / Journal of Sound and Vibration 289 (2006) 967–986
In the next examples the same method of optimization has been used but in this case the design
domain size is changed so that the ratio of the side length is 2/1. The results of the optimizations
are shown in Fig. 11. The design domain is here discretized in 100 50 elements.
The results depicted in Fig. 11 are similar to the results for the square domain with some
topologies being periodic like and others being qualitatively different. We now try to examine if
the general results from 1D can be transferred to the 2D case.
3.6. Maximizing the ratio of adjacent eigenfrequencies
As in the 1D case we now change the objective function to that in Eq. (13) so that the ratio
between adjacent eigenfrequencies is maximized, but we still use the double-bound formulation
introduced in Eq. (30). We repeat the optimization for the rectangular domain for the case where
n ¼ 7, i.e. corresponding to the design in Fig. 11(g).