WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Phononic band-gap optimization
(a) (b)
(c)
2 W00 X g 0
II . ... I
~,o=O, 20 I I ( =0, I=0
0 | a=O, 3= 50000 -----1-
S4-20 )-v yJ1A g target frequency
-40
4 ) .....A.AN
---60-
-80 - ..'
. . . .. ,
20 40 60 80 1 00
frequ.ency, f (k-Iz)
Figure 12. Optimization of structure for minimum response at Q = 55 kHz. (a) Structural
domain and periodic boundary loading in the optimization procedure; (b) optimized topology;
(c) average response at the right boundary when subjected to loading at the left boundary.
Dashed line a = 0, P = 50 x 103; solid line, a = =- 0. High-contrast case.
figure 7b). As a result, the optimized topology obtained for f = 52 kHz is no longer
periodic (see figure lib). The response shown in figure 1lc only has a small reduction
in the response when the damping is removed. Since we tried several different starting
guesses that always led to the same conclusion, this example strongly indicates that
design of a band gap for the low-contrast case and in-plane modelling is not possible.
Figure 12 shows the result of an optimization problem resembling the one in figure
8. Here the structure is subjected to loading at only two opposite boundaries
(modelled as two separate loading cases). The optimized topology (figure 12b) is
now seen to resemble a one-dimensional Bragg grating with some modifications near
the boundaries. The response obtained with damping included (figure 12c) shows a
reduction in response for a large frequency range (up to more than 80 kHz). However,
the response with the damping removed is dominated by resonance peaks due
to reflections at the boundaries.
Phil. Trans. R. Soc. Lond. A (2003)
1017