WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

3.4 Maximizing wave dissipation 25
I
R
R
scatter
absorber
Figure 3.8 Basic setup for the problem of maximizing the dissipation of an incoming
wave by optimizing the distribution of dissipating material (absorber) and possibly an
extra reflecting material (scatter) in the background material. Incident wave power is I,
transmitted and reflected powers T and R, and the dissipated power is D. From paper
[6].
phenomenon is noticed if the dissipation of vibrations or waves is computed for a
bandgap structure with added material damping. High energy dissipation occurs for
frequencies near (but outside) bandgap frequency ranges where the eigenfrequencies
are closely spaced in the frequency domain. This numerical observation leads to the
question if periodic-like structures are also optimal for maximizing dissipation or, if
not, which kind of structures are?
Fig. 3.8 shows the setup for studying maximization of the dissipation of elastic
waves propagating through a two-dimensional section of material. The distribution
of a background material and up to two additional materials is optimized. One of
these materials is absorbing (and also slightly reflecting) and the other is a reflecting
material (which is both denser and stiffer than the background material). Special
focus is on investigating whether optimizing the distribution of both the absorbing
and a reflecting material can increase the possible dissipation in the structure
compared to distributing only absorbing material.
Fig. 3.9 depicts two examples of optimized material distributions. In Fig. 3.9a
two materials are available: background (white) and absorbing (black) and in Fig.
3.9bthe extra reflecting material is present (gray isthe absorbing material andblack
isthereflecting material). Fig.3.9ashows acleardistributionoftheabsorbing material.
The structure isclearly not periodic-like, however, the distribution of inclusions
inside the domain causes a large amount of wave reflection that leads to increased
dissipation in the inclusions. Additionally, the thin strip of absorbing material near
the inlet (left) causes an impedance match and a high transmission of the wave into
the domain. The thicker material strip at the outlet (right) creates maximum wave
reflection back into the domain. If the additional reflecting material is available (as
shown in Fig. 3.9b), it replaces the outlet strip to increase the reflection of waves
back into the absorbing domain. The distribution of the absorbing material inside
the domain is different, too.
Fig. 3.9c illustrates the corresponding dissipation curve that depicts the fraction
D
T
T