WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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(a) free boundary
free boundary
(d)
Phononic band-gap optimization
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(b)
Figure 8. Damping of wave propagation in a quadratic plate for f = 41 kHz. (a) Design domain
and boundary conditions; (b) optimized structure for high-contrast case; (c) optimized structure
for low-contrast case; (d) frequency response for the high-contrast case. Dashed line, a = 0,
3 = 25 x 103; solid line, c = 3 = 0.
In the following we show a number of examples with different combinations of
object, boundary condition and out-of-plane/in-plane modelling. The examples are
selected for illustrative purposes, and any other combinations of the above can be
solved by simple changes in an input file.
(i) Results for out-of-plane waves
Figure 8 shows an example where the suggested optimization procedure is used
to minimize wave propagation through a square plate. The left edge is subjected to
forced vibrations with frequency f = 41 kHz (corresponding to the centre frequency
of the gap of figure 3d); the left and right edges have absorbing boundary conditions,
and the top and bottom edges are free. The objective is to minimize the average
amplitude at the right edge. The resulting topologies are, not unexpectedly, a grid
of alternating phase 1 and phase 2 materials corresponding to Bragg gratings. This
structure is known to reflect one-dimensional (horizontally propagating) waves. The
frequency response for the high-contrast case is shown in figure 8d. It is seen that
there is a large band gap around the excitation frequency f = 41 kHz. Compared
with the response of the square-inclusion structure from figure 5, where the input
Phil. Trans. R. Soc. Lond. A (2003)
111111
(c)
1013
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