WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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22 Chapter 3 Bandgap structures as optimal designs
Reflectance
a) b)
Reflectance
c) d)
1.0
0.8
0.6
0.4
0.2
0.0
0 400 800 1200 1600
1.0
0.8
0.6
0.4
0.2
Frequency (Hz)
0.0
0 400 800 1200 1600
Frequency (Hz)
Figure 3.4 Optimized distribution of scattering (black) and host material (white) for
maximum reflectance in a frequency interval around 788Hz and reflectance curves for the
optimized designs for pressure (solid line) and shear waves (dashed line). a,b) ±10%
frequency interval, c,d) ±25% frequency interval. From paper [6].
are given for a pressure wave (a), a shear wave (b) and for both a shear and a
pressure wave (c). The appearance of the structures reflects the multi-angle wave
incidence that provokes the two-dimensional nature of the material distribution.
The restricted spatial dimension also gives the structures appearances that are not
regularly periodic.
The results show that adding a third material to the design problem has no
beneficial effect on reducing the transmission of waves. In all reported cases the best
performance is obtained for a structure that is composed of the two materials having
the largest contrast between their material properties. For a further discussion of
this observation see the comments in the last paragraph of this chapter.
3.3 Eigenfrequency separation
The optimized structures strongly depend on the specific location of the input force
and/or the direction of the incoming wave. Hence, a unidirectional wave usually
results in a one-dimensional structure (Bragg grating) optimized for exactly this
form of excitation. Other excitation types as in the form of multi-directional waves
result in different structures. If the structure is to be optimized for all relevant
loading conditions, the problem of how to specify these loading conditions arises.
An alternative to this approach is to consider the structural eigenfrequencies. If
frequency ranges with no eigenfrequencies can be created, then no loading condition