WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

(a)
(b)
Phononic band-gap optimization
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Figure 6. Material optimization based on out-of-plane modelling. Optimized topologies for maxi-
mization of relative size of the band gap between the first and the second bands: (a) high-contrast
case; (b) low-contrast case. In both cases the base cell was discretized by 30 x 30 bilinear finite
elements.
inclusions with small 'ears' almost independent of the phase contrast. The relative
band-gap sizes are 0.21 for the high-contrast case and 0.001 (i.e. hardly a band gap)
for the low-contrast case. For comparison, the relative band-gap sizes for perfect
square inclusions with relative sizes 0.17 are 0.19 and -0.001, respectively. In this
case, it therefore cannot be concluded that the perfectly square inclusion is the
optimal solution.
(b) Structural optimization
The material design problem in the previous subsection assumed infinite period-
icity of the material. This means that neither the influence of boundaries nor the
defects in the periodic structure could be modelled. In order to model finite domains,
we use the wave equation (2.11) and the objective function here may be to minimize
the magnitude of the wave at the boundaries (hinder wave propagation) or to max-
imize the wave magnitude at certain points in the structure (waveguiding).
An optimization problem solving the problem of minimizing the wave magnitude at
a point, a line, or an area of a structure subjected to periodic loading with frequency
Q can be written as
min ulTLlul s.t.: (K + iC- 22M)u f, < Xe 1, e = 1,..., N, (3.6)
x
where L is a zero matrix with ones at the diagonal elements corresponding to the
degrees of freedom of the nodes, lines or areas to be damped. Due to the complex
Phil. Trans. R. Soc. Lond. A (2003)
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