WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

712 J.S. Jensen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715
1.5 s
t
design
domain
1 m
Fig. 14. Design domain for the space–time optimization problem.
Fig. 15. Optimized space–time structure for minimum transmission of the wave
pulse.
Fig. 16. Instantaneous material distribution and wave motion at t = 5.0 s and
t = 5.4 s for the structure in Fig. 15.
reflection (as appears from the input point response). Actually, the
reflected energy is reduced to only about 1% of the input compared
x
Fig. 17. Response for the optimized structure shown in Fig. 15. Top: displacement
of input point, bottom: displacement of output point.
to about 17% for the static bandgap structure. Instead the main part
of the energy (about 75%) is extracted from the system via the time
dependent force that is needed to change the stiffness in time (cf.
the discussion on energy conservation in Section 3). Only a small
fraction (about 1%) is dissipated by the stiffness proportional
damping. However, without the presence of damping the optimization
procedure becomes unstable after only a few iterations.
The output time response in Fig. 17 indicates that the frequency
content of the wave has changed. This is confirmed by an FFT-analysis
shown in Fig. 18. In addition to the main frequency component
x0 ¼ 15:7rad=s, higher order components at 3x0 and 5x0
(and more further up in the spectrum) are seen as well. This might
seem surprising since the system is completely linear. However, as
mentioned, the material properties can only be changed in time
with a time dependent external force acting on the system, and
this force contributes to the signal with these higher order frequency
components.
Similar structures as the one shown, can be obtained with different
values of the material contrast E0. Only the wave speed in
the layered medium changes and thereby also the speed of the
bandgap material front which leads to a different ‘‘slope” of the
spatio-temporal laminates. Additionally, the ratio of the two materials
in the structure depends on the choice of E0 in such a way that
the inclusion layers become thicker with increasing contrast.
5.3. Realizable structures – patches and checkerboards
The optimized structure in the previous section would be a
challenge if it came to a practical realization. In order to generate