WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

898 S. Halkjær, O. Sigmund and J. S. Jensen
Alu
PMMA
f [kHz]
100
80
60
40
20
1 2 3 4 5 6 7 8
[cm]
0
0 0.5 1 1.5
k
B
2 2.5 3
Fig. 1. Final design corresponding to maximum relative band gap between
first and second band (thick lines) for longitudinal waves. Full
lines: Longitudinal waves. Dashed lines: Bending waves.
Table 2. Extreme frequency band values for longitudinal waves in
Figure 1. Unit is kHz.
k B ¼ 0 k B ¼ p
f1 0 10.367
f2 40.798 30.452
f3 40.844 51.191
f4 81.603 71.283
design. It should be noted, that if another base cell length
was chosen, no qualitative change would be observed and
the ratio of the two material lengths would be the same.
With this said, PMMA occupies approximately a fraction
of 0.29 of the base cell length. That is, the larger part of
the base cell is made up of the heavy and stiff material
while a smaller part is occupied by the light and soft material.
Figure 1 (bottom) also shows the lowest frequency
bands for both longitudinal and bending waves, the latter
just to indicate their position. The extreme edge values of
the bands are listed in Table 2. The first three relative
band gap sizes are listed in Table 6 (second column). The
symbols in the first column denote band gap number and
wave type considered, e.g. F L 12
denotes the band gap be-
tween the first and second frequency band and L indicates,
that longitudinal waves are considered.
As a second example the relative band gap between the
third and fourth band for bending waves has been maximized.
The resulting design and frequency bands are
shown in Fig. 2. In this case, increasing or decreasing the
base cell length results in an increasing relative band gap
size mainly due to vertical displacements of the frequency
bands. This effect has been excluded by choosing the
same constant base cell length as in the previous case
d ¼ 8:70 cm. This design problem appears to have many
local minima, some of which result in designs consisting
of mixtures of the two materials (i.e. z e 6¼ 0; 1), depending
on the starting point. The design shown in Fig. 2 has the
largest relative band gap of several trials and furthermore
Alu
PMMA
f [kHz]
80
60
40
20
1 2 3 4 [cm] 5 6 7 8
0
0 0.5 1 1.5
k
B
2 2.5 3
Fig. 2. Final design corresponding to maximum relative band gap between
third and fourth band (thick lines) for bending waves. Full
lines: Longitudinal waves. Dashed lines: Bending waves.
Table 3. Extreme frequency band values for bending waves in Fig. 2.
Unit is kHz.
k B ¼ 0 k B ¼ p
f1 0 1.1508
f2 5.2781 1.5692
f3 5.9316 10.363
f4 21.302 15.194
f5 21.415 27.832
f6 39.457 34.075
has a pure 0/1 design. The majority of the cell is again
occupied by the heavy and stiff material while a smaller
part (around a fraction of 0.33) is occupied by the light
and soft material. However, this time the light material is
distributed in two different parts. The extreme frequency
band values are listed in Table 3. The first three relative
band gaps are listed in Table 6 (third column).
As the last optimization example, the design resulting
from maximizing the overlap between the first and second
longitudinal band and the third and fourth bending band is
shown in Fig. 3 together with the frequency bands. These
band gaps are chosen because they have overlaps initially.
An optimized base cell length of d ¼ 8:70 cm is found by
the algorithm is this case (which is the reason for choosing
this value in the previous two cases). The unique solution
is a result of the frequency bands for the two wave
types moving in the same direction but with different
speeds as a function of base cell length. The design is
quite similar to the design in the bending case. A larger
part of the light material occupies the same amount of the
cell as in the bending case, while the narrow part has decreased
compared to the bending case. The extreme values
for the different frequency bands are tabulated in Table 4.
The relative band gaps for the separate wave types as well
as for the combined problem are shown in Table 6 (fourth
column) with F denoting the overlapping band gap for
the combined problem. As expected, the first longitudinal