WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

904 S. Halkjær, O. Sigmund and J. S. Jensen
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ture. Here, the periodicity of the structure has been chosen
to be 8 cm which corresponds to dx 1.546 m. It is noted
that the results compare well with the results obtained in
the previous section for the 2D crystals. Studying the optimized
topology in Fig. 13 reveals that the Aluminum to
base cell size ratio is approximately 0.42 – exactly like the
result obtained for an infinite periodic medium in Fig. 7.
One may also compare the dispersion diagram in Fig. 7
and the transmission spectrum in Fig. 13 and find nearly
perfect agreement as should be expected.
Case of dx
long.
shear
0 5 10 15 20 25 30
Frequency [kHz]
Fig. 13. Top: optimized design for a plane shear wave propagating
from left to right. Design domain dimensions: dx ¼ 10 2pcs=w and
dy ¼ dx=10. Bottom: corresponding transmission spectrum for a shear
and a longitudinal wave (periodicity of the design chosen to be 8 cm).
a b
Fig. 14. Optimized designs for the case of dx ¼ 2 2pcs=w and
dy ¼ dx. (a): longitudinal wave, (b): shear wave.
wavelength
The slab dimension is now reduced so that
dx ¼ 2 2pcs=w, i.e. the axial dimension is just two wavelengths
of a shear wave. The slab section height is
doubled so that the design domain now is quadratic
(dy ¼ dx).
Instead of considering only a single plane wave, we
now optimize the design for five different waves simultaneously.
Each wave propagates from x ¼ 0 but only from
1/5th of the section height. In this way we can ensure that
the resulting design reflects incoming waves from multiple
directions. Additionally, we treat waves propagating both
from left to right and vice versa, so that a total of 10 load
cases are treated.
Figure 14 shows the optimized designs obtained for a
longitudinal wave and for a shear wave, separately. It is
apparent that the designs are no longer periodic-like but
are strongly influenced by the limited axial dimension.
The wavelength of the longitudinal wave is longer than the
shear wavelength (cc ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ð2 2nÞ=ð1 2nÞ cs ¼ ffiffi p
6 cs
for material 1), which implies that dx is less that a single
a b
Fig. 15. Optimized design for the case of dx ¼ 2 2pcs=w and
dy ¼ dx. (a): longitudinal and shear wave propagating in both directions,
(b): longitudinal and shear wave propagating from left to right.
wavelength. This is reflected in the designs as well as in
the performance. Due to the larger wavelength the longitudinal
wave cannot be reflected very efficiently, and with
an incoming plane wave we compute F 0:27, which is
only a 1.5 dB improvement compared to the case where
the whole design domain is filled with material 2. For
the shorter shear wave the corresponding optimized design
performs better, and with a plane wave we find
F 0:00071 which is a 23 dB improvement compared to
a completely scatter-filled design domain.
Finally, we try to design the slab so that it minimizes
the transmission of longitudinal and shear waves simultaneously.
Figure 15 shows the optimized designs. The design
to the left was obtained for 10 load cases just as the
previous examples, whereas the design to the right was
obtained by considering only waves propagating from left
to right. For the symmetric design we obtain F ¼ 0:60 for
an incoming plane longitudinal wave and F ¼ 0:00078
for a shear wave, whereas we compute F ¼ 0:58 and
F ¼ 0:00081 for the asymmetrical design.
In order to study the influence of the load cases, the
optimization has been performed also with 10 wave load
cases from each side instead of five, and using the previously
obtained designs as a starting guess. This caused
only very small changes of the optimized designs, which
indicates that the designs are quite robust with respect to
the angle of incidence of the incoming waves.
Conclusion and outlook
The topology optimization method has been used to design
infinite periodic beams and plates as well as finite
structures with maximum band gaps from two a priori
chosen materials; one heavy and stiff (Aluminum), the
other light and soft (PMMA).
For the beam case we have designed two beams, which
prevent either longitudinal or bending waves with frequencies
in certain intervals from propagating in them, while a
third beam has been designed which prevents both wave
types from propagating.
Producing 2d gratings from the optimized beam topologies
is not feasible since in-plane shear modes and out-ofplane
torsional modes ‘‘pollute” the band gaps. Instead,
we have performed simple parameter studies as well as
free material distribution by topology optimization in order
to find the optimal cell geometries of infinite sonic
crystals.