WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

902 S. Halkjær, O. Sigmund and J. S. Jensen
f [kHz]
35
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5
0
GΓ X
Fig. 8. Top: 5 by 4 unit cells of a topology optimized crystal structure
consisting of Aluminum (black) in a PMMA material. Bottom:
Dispersion diagram for in-plane polarized waves in the topology optimized
crystal. The relative band gap is 0.525.
To compare the results with the beam case, we initially
perform an analysis of bending waves propagating along
the x-axis through a grated plate consisting of alternating
PMMA and Aluminum slabs with the slab to cell width
ratios of 0.29 and and 0.71, respectively, as given by the
result from Fig. 1. The base cell size is again 8 by 8 cm
and the thickness of the cell is 1cm. The resulting dispersion
diagram for the first 15 eigenvalues for horizontally
propagating waves is shown in Fig. 10. Note that all the
shear wave bands are double modes. The thick dashed
lines indicate the bending modes and the thin lines indicate
torsional modes. The thick dashed lines for the bending
waves do not fully match the full lines for the beam
bending waves (dashed lines in Fig. 1) because of slightly
different base cell size, Poisson’s ratio effect and because
the beam has a circular cross section. Compared to the
beam case, the dispersion diagram for the lowest modes is
similar, however, the higher bending modes are ‘‘polluted”
with the torsional waves (thin lines). A parameter study
where the Aluminum slab to cell width ratio is optimized
reveals that the largest relative size of the first band gap is
0.53 and is obtained for Aluminum slab to cell width ratio
of 0.2 and that the largest relative size of the second band
gap is 0.42 and is obtained for Aluminum slab to cell
width ratio of 0.44. In contrast to the in-plane case above,
we did not find any non-symmetric cell topologies that
could improve the size of the relative band gaps of the
grating like structures for the first and second gaps. This
can be explained by the fact that the first and second gaps
only are controlled by the bending modes whereas for the
in-plane case the upper boundary of the gaps where limited
by a combination of longitudinal and shear wave
modes.
f [kHz]
40
35
30
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0
GΓ M K GΓ
Fig. 9. Dispersion diagram and optimized topology for in-plane polarized
waves for a rhombic base cell with waves propagating in all
directions. The complete relative band gap size is 0.505.
To create a band gap for all wave directions simultaneously,
we optimize the band gap for all wave-vectors on
the edges of the triangular areas indicated in Fig. 5. We did
not manage to open up a gap for the square cell case (for the
considered materials and geometry) but a topology with a
resulting relative band gap size of 0.02 (very small) for the
rhombic base cell is shown in Fig. 11. It is seen that the optimized
topology again is a hexagonal array of nearly circular
Aluminum inclusions in a PMMA matrix, however, this time
with a smaller radius than for the in-plane polarized case.
In future work we plan to study the case of combined
bending and in-plane plane waves modelled by the plane
stress assumption, i.e. the shell case.
Design of finite structures: 2D wavereflectors
In the previous two examples it was demonstrated how
infinite phononic crystals can be optimized with respect to
maximum band gap size. This can lead to a significant
improvement of the performance of potential phononic
crystal-based devices, such as e.g. mechanical filters and
vibration insulators. However, the analysis of the crystals
is based on the assumption of infinite media. Thus, if the
dimensions of the device are small, approaching that of a
single crystal, edge effects become important and must be
considered.
In this example we consider wave propagation through
a finite slab of elastic material and aim to design the slab
for minimum transmission of longitudinal and shear
waves. Figure 12 shows the model. We consider a wave
(longitudinal or shear) propagating in a matrix material
denoted 1 (material properties: E1; q 1; n1). The goal is
now to optimize the distribution of the matrix material and
a scattering material (E2; q 2; n2) in the slab (design domain)
such that the wave energy transmitted through the