WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

domain. To the very left the passive elements are seen
and immediately to the right of these a thin strip of
white (polycarbonate) before a large domain of steel.
Further to the right the structure consists of steel inclusions
that deflect the energy towards the optimization
area and keeps it away from the damped edges. Only
energy vectors larger than 10% of the largest vector are
shown in order to make the figure more clear. From
that we can see that very little energy is transported
outside the cone-shaped part. This is partly due to
damping and partly due to the optimized design that
guides the energy. Right at the optimization area an
interesting effect is seen, as the energy after passing
these elements follows a circular path back in order to
increase the objective value.
A closer look at the results shows that during the
optimization the objective energy is increased by a
factor of 63 while the energy supplied to the structure
is increased by a factor of 5.7. This means that a part of
the increase in objective energy is due to a larger energy
input to the structure and a part is due to an improved
design that is able to guide the energy more efficiently.
As we are maximizing the energy through some part of
the structure an increase in input energy is an efficient
way to obtain a larger objective value.
We now look at a slightly different example as shown
in Fig. 4b, consisting of the same plate as before but
only loaded along the central part of the left edge. The
objective is now to maximize the energy transport in
the vertical direction through two symmetrically placed
areas near the top and bottom edge. The structure is
thus supposed to split the energy and lead it towards
the edges. The topology and energy transport is shown
in Fig. 6.
We see that the structure is designed in such a way
that the energy is kept from going to the edges. Instead
it is being led to the right where it is bend towards the
optimization areas. Looking closely we recognize the
same behaviour as before where the energy turns after
Fig. 6 Topology and energy transport for the problem sketched
in Fig. 4b. Only energy vectors larger than 10% of the maximum
energy vector are shown for clarity
A.A. Larsen et al.
leaving the optimization area in that way increasing the
objective energy.
As in the previous example both the input and
objective energies are increased. The input energy is
increased by a factor of 5.4 while the objective energy
is increased by a factor of 306. This is a very
large improvement compared to the input energy and
compared to the last example. The large improvement
is possible because the objective is to maximize the
energy in the vertical direction and for the original plate
the energy propagates from the applied force in a way
such that the vertical component of the energy through
the optimization areas is very small. In this case the
increase is therefore mostly due to an improvement in
the design of the plate.
In both examples, shown here, it is seen that the
material boundaries, i.e. where steel and polycarbonate
meet, deflect the energy such that it is guided along the
boundary. This effect is clear around the central part of
the plates in Figs. 5 and 6. However the energy seems
to pass material boundaries if these are perpendicular
to the energy vector. This is particularly clear in the left
part of Fig. 5.
As stated earlier artificial damping is used in order
to penalize intermediate density elements. However it
is clear from the two examples shown that some grey
elements still exist. This means that the plate is more
damped than what is applied as material or external
damping. In the next example we therefore omit this
penalization and only include material damping.
In the final example the objective is to guide the
energy in a circular path in order to create a ring
Fig. 7 Sketch of optimization problem for creating a ring wave
in a simply supported plate. The black parts indicate fixed steel
regions and the arrows indicate in which direction the energy
transport is optimized. The two small steel areas indicate where
the forces are applied