WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Topological material layout in plates for vibration suppression and wave propagation control
Fig. 3 Displacement of the plate shown in Fig. 2 bottom for the
frequency ω = 9000 rad
s . Top: The optimized plate. The largest
displacements are located around the plate center. Bottom: The
original plate
on static stiffness in order to obtain results that can be
used in practice.
7 Examples: energy transport
In this section we look at the optimization of the energy
transport in different plates. Although we present academic
problems they are relevant for guiding mechanical
vibrations in certain direction with the purpose of
absorbing them or alternatively harvesting the energy.
First consider the problem sketched in Fig. 4a. The
figure shows a plate subjected to a harmonically varying
out-of-plane force on the left edge and damped in all
degrees of freedom along the remaining edges. The
objective is to maximize the energy transport in the
horizontal direction through the black part to the right
which consists of 4 by 8 elements. The black areas
also indicate that the densities are fixed at one, i.e.
they represent steel. The frequencies of excitation are
ω = 11900, 12000, 12100 rad
. We use three frequencies
s
in order to optimize for a wider frequency span and also
it seems to provide more black and white designs. The
plate is quite heavily damped with damping coefficients
α = 1 · 10−4 and β = 1 · 10−5 and damping applied to
a b
Fig. 4 Sketch of optimization problem. a The objective is to
maximize the energy in the horizontal direction through the black
area to the right which consists of 4x8 elements. b The objective
is to maximize the energy in the vertical direction towards the
edges through the black areas to the right. Each area is made by
4x3 elements
the three edges as well. The plate is discretized into
100 by 100 elements which combined with a wavelength
of 0.064m in the pure polycarbonate means that there
are approximately 13 elements per wavelength. The
wavelenght in steel is longer giving more elements per
wavelength in steel regions. We will allow at most 50%
steel to be used in the design.
In Jensen and Sigmund (2005) artificial damping
(pamping) was proposed to penalize intermediate densities
and obtain more black and white structures. By
adding the term
C e P = ɛPϱ e (1 − ϱ e )ωM e
(49)
to the element damping matrix intermediate density
elements will act as dampers. In (49) ɛP is a parameter
controlling the magnitude of damping and M e is the element
mass matrix. When maximizing energy transport
this will favor 0-1 designs.
The results are shown in Fig. 5 as a plot of the
topology and vectors showing the energy transport.
We see a structure with a cone-like shape that guides
the energy from the excitation towards the optimization
Fig. 5 Topology and energy transport for the problem sketched
in Fig. 4a. Only energy vectors larger than 10% of the maximum
energy vector are shown for clarity