WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Inverse design of phononic crystals by tobology optimization 903
f [kHz]
20
10
25
15
5
0
0 0.5 1 1.5 2 2.5 3
Γ
slab is minimized. In this example, the material properties
again correspond to PMMA and Aluminum, respectively.
As in Eqs. (13)–(14) we again consider in-plane polarized
waves with the plane strain assumption for the following
examples.
In order to avoid artificial reflections of waves propagating
in the negative x direction at x ¼ 0 and in the positive
x direction at x ¼ L, a Perfectly Matched Layer
(PML) is added at x < 0 and x > L, respectively. Details of
PML modelling for elastic waves can be found e.g. in [18].
The slab is modelled as being infinite in the y direction
by using periodic boundary conditions:
uðx; dyÞ ¼uðx; 0Þ ; ð19Þ
vðx; dyÞ ¼vðx; 0Þ : ð20Þ
The cost function F for the optimization problem is the
squared amplitude of the transmitted wave evaluated at
x ¼ L:
F ¼ 1
dy
ðdy
0
8cm 5.7cm
Fig. 10. Dispersion diagram for bending waves along the x-axis in a
PMMA/Aluminum grating with Aluminum slab to cell width ratio of
0.71. The first 15 modes are shown and bold dashed lines indicate
pure bending modes. The grey areas indicate band gaps.
ðuðL; yÞ 2 þ vðL; yÞ 2 Þ dy ; ð21Þ
thus, giving a measure of the average transmitted wave
energy.
As in the previous examples we use a FE discretization
of the governing equation:
ðK w 2 MÞ u ¼ f ; ð22Þ
where u ¼fu1; v1; ...; uN; vNg are the discretized nodal
values of the displacement field and where the stiffness
matrix K and the mass matrix M are complex due to the
added PML damping layers. The load vector f specifies an
incoming longitudinal or shear wave. Quadratic first order
elements have been used in the FE analysis.
X
f [kHz]
20
15
10
5
0
GΓ M Γ K Γ
Fig. 11. Dispersion diagram and optimized topology for the bending
case. The relative band gap size is 0.02 between the 3rd and the 4th
bands.
dy
y
ρ1
E1
ν1
In the following examples PMMA is used as a matrix
material and aluminum as the scattering material, with the
material properties as in the previous examples.
Case of dx
design domain
dx
wavelength
We first consider the case where the slab dimension dx is
large compared to the wavelength in the matrix material,
the wavelength being 2pc=w where c is the wave speed.
Specifically, we consider a propagating p shear wave and let
ρ1
E1
ν1
Fig. 12. Computational model. The inverse design problem is to find
the optimal distribution of matrix and scattering material in the design
domain in order to have the minimum wave transmission
through the domain.
dx ¼ 10 2pcs=w, where cs ¼ ffiffiffiffiffiffiffiffiffiffiffi
m1 =q1 is the shear wave
velocity. Thus, the axial dimension of the design domain
is 10 wavelengths in the matrix material. Additionally, we
consider a thin slab (dy ¼ 1 =10dx) in order to avoid any
design variation in the y direction.
Figure 13 shows the optimized design obtained for a
plane shear wave propagating from left to right. As appears,
the design is periodic-like consisting of almost identical
base cells. The bottom picture depicts the transmission
spectrum, defined as the logarithm to the ratio between the
squared incident and transmitted amplitudes, for a longitudinal
and a shear wave propagation in the optimized struc-
L
x