WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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O. Sigmund and J. S. Jensen
we get
(K + i2C - 22M)u = f, (2.11)
where we have added boundary (harmonic) forces f and a damping matrix C stemming
from absorbing boundaries (2.7) or structural damping. In this work we include
absorbing boundaries only in the out-of-plane case. Structural damping is included
in the form of standard Rayleigh damping,
C =aK + 3M, (2.12)
with a the stiffness-proportional and / the mass-proportional damping constants.
Figure 4 shows three different structures and the corresponding steady-state
responses when subjected to harmonic loading at the left boundary. In-plane vibra-
tions are considered for the high-contrast case and the response is calculated as the
average amplitude along the right boundary. The structure in figure 4c is made from
a periodic material consisting of a matrix of material 1 (grey) with 36 square inclu-
sions of material 2 (black) (the inclusions cover 36% of the base cells, corresponding
to the periodic material in figure 3c). From figure 3c it was found that the infinitely
periodic material has a band gap between the third and the fourth propagation
modes around f = 60 kHz. Figure 4d shows the corresponding response of the finite
structure when subjected to the external loading. The band gap is identified from the
drop in response around the gap frequency, but the drop is not as large as might have
been expected and there are two peaks in the middle of the interval. This can partly
be ascribed to the small number of base cells (6 x 6) included in the structure, but
more importantly to resonance effects due to the reflections from the non-absorbing
boundaries.
Figure 4d also shows the response calculated with mass-proportional damping
included (a = 0, 3 = 50 x 103). The damping is seen to remove the resonance peaks
in the response, but the reduction in response near the band gap is still seen and the
peaks inside the band-gap range have been removed. This effect of smoothing of the
response by including damping will be used later in the optimization procedure. The
responses for the structures made of pure material 1 and material 2, respectively, are
shown in figure 4d for comparison.
A response diagram for the structure in figure 4c but modelling the out-of-plane
case and including absorbing boundary conditions on the left and right edges is shown
in figure 5. Compared with the response curve for the in-plane and non-absorbing
boundary case from figure 4, the response is seen to be much cleaner and free from
local peaks. Also, the band gap is seen to correspond very well to the partial band
gaps from figure 3d.
From the results shown in figures 3 and 4, two optimization problems naturally
emerge. For the material problem the size of the band gaps should be maximized to
increase the frequency range for which waves cannot propagate through the material.
For the structural problem, the response of the structure may be minimized for a
given frequency, or alternatively maximized in certain structural areas to create
wave-damping or wave-guiding structures, respectively.
3. Topology optimization
We follow the standard approach to topology optimization (cf. Bends0e 1995;
Bends0e & Sigmund 2003). The topology-optimization problem consists of distribut-
Phil. Trans. R. Soc. Lond. A (2003)