WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Inverse design of phononic crystals by tobology optimization 901
f [kHz]
80
70
60
50
40
30
20
10
0
8cm 5.7cm
Γ
X
Fig. 6. Top: 5 by 4 unit cells of a simple slab-like crystal consisting
of 71% Aluminum inclusions (black) in a PMMA material. Bottom:
Dispersion diagram for in-plane polarized waves in a PMMA/Aluminum
grating with Aluminum slab to cell width ratio of 0.71. The first
25 modes for propagation in the horizontal direction are shown and
bold lines indicate longitudinal modes. The grey areas indicate band
gaps.
Numerical examples
First we consider in-plane polarized elastic waves in a 2D
elastic crystal with a square base cell size measuring 8 cm
by 8 cm. To compare the results with the beam case, we
initially perform an analysis of 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 resulting dispersion diagram for
the first 25 eigenvalues for horizontally propagating waves
(i.e. wave vectors along the line G X in Fig. 5) is shown
in Fig. 6. The thick lines indicate the longitudinal modes
and the thin lines indicate in-plane elastic shear waves.
The thick lines for longitudinal waves do not fully match
the full lines for the beam longitudinal waves in Fig. 1
because of slightly different base cell sizes, and the Poisson’s
ratio effect. As opposed to the beam case, however,
the dispersion diagram is “polluted” with in-plane elastic
shear waves (thin lines) and there are only four narrow
band gaps as indicated by the grey boxes. It is not obvious
between which bands the gap will be largest for
waves propagating along the x-axis. In order to optimize
the structure, we first perform a simple parameter study
where we vary the relative width of the Aluminum slabs
and find the maximum band gap for the grated structure.
The result of this study is that the largest gap is found
between the 3rd and the 4th band for an Aluminum slab
to cell width ratio of 0.42. The dispersion diagram for this
structure is shown in Fig. 7 and the size of the relative
band gap is 0.452. The dispersion diagram in Fig. 7 can
be compared with the transmission diagram for the optimized
finite periodic structure in Fig. 13.
f [kHz]
30
20
10
35
25
15
5
0
0 0.5 1 1.5 2 2.5 3
Γ
8cm 3.4cm
Fig. 7. Top: 5 by 4 unit cells of a simple optimized slab-like crystal
consisting of 42% Aluminum inclusions (black) in a PMMA material.
Bottom: Dispersion diagram for in-plane polarized waves propagating
in the x-direction. The bold lines indicate longitudinal modes. The
grey areas indicate band gaps and the relative size of the first band
gap is 0.45.
Now one may ask the question whether it is possible to
obtain a larger band gap for waves propagating in the
x-direction if one allows the topology of the grating to
vary? Indeed, this is possible. The result of a topology
optimization process where no symmetry was imposed on
the square base cell and the gap between the 3rd and the
4th bands was optimized is seen in Fig. 8. The optimization
problem corresponded to Eq. (11), except that the design
variables for this case were the 3364 nodal density
variables in the FE mesh used to discretize the base cell.
The obtained relative band gap size is 0.525, i.e. approximately
15% better than the simple grating from Fig. 7.
The topology in Fig. 8 is an interesting variation of the
simple grating that apparently raises the shear mode bands
that define the upper edge of the gap.
If we want to create a band gap for all wave directions
simultaneously, we have to optimize the band gap for all
wave-vectors on the edges of the triangular areas indicated
in Fig. 5. The results of this study for the square base cell
is a square Aluminum inclusion with slightly rounded corners
which has a relative band gap of 0.428. For the rhombic
base cell the optimized topology is shown in Fig. 9.
The bases cells were both discretized by 6728 triangular
first order elements. The optimized relative band gap size
is 0.505 for the rhombic base cell. Both values are, as
expected, lower than for the directionally optimized topology
in Fig. 7. It is seen that the topology with the largest
complete band gap is a hexagonal array of PMMA matrix
material with circular inclusions of Aluminum (Fig. 9). As
discussed in Ref. [11], the optimal topology depends on
the material properties and must therefore always be optimized
for the particular materials considered.
Now we repeat the same study for bending waves.
X