WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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714 J.S. Jensen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715<br />
propagating in the material with c ¼ 1m=s. Thus, the design domain<br />
had a spatial extent corresponding to 2.5 wavelengths.<br />
If the design domain length is short compared to the main<br />
wavelength, a spatially layered structure is no longer efficient for<br />
minimizing the wave transmission. Fig. 22 shows two optimized<br />
structures for k ¼ 2:5 m and k ¼ 10 m, respectively. The temporal<br />
design intervals have been increased accordingly to 5 s and 10 s.<br />
As appears, longer wavelengths result in structures that are spatially<br />
more homogeneous at any given time. In the long wavelength<br />
limit, the optimized structure approaches that of a pure<br />
temporal laminate with instant simultaneous switching of the<br />
material properties in the entire rod (cf. the analysis in Section<br />
3.1). For a discussion of temporal laminates see [36,24].<br />
5.5. Stability of structural response – shifted wave pulses<br />
The optimized dynamic bandgap structures have been demonstrated<br />
to be effective in prohibiting the propagation of waves.<br />
But a significant drawback compared to the static structures is that<br />
they are sensitive to the operating conditions and rely on a timely<br />
activation of the spatio-temporal material variation.<br />
This is illustrated by simulating the response of the structure<br />
with an input pulse that is delayed compared to the reference<br />
pulse for which the structure is optimized. Fig. 23 shows the<br />
instantaneous material distribution and wave motion as in<br />
Fig. 16(top), but with the wave pulse delayed 0.1 s, so that the<br />
wave peaks and valleys follow the rear side of the inclusions instead<br />
of the inclusion fronts. The effect on the response is significant<br />
and the resulting output wave pulse is actually magnified<br />
compared to the input pulse.<br />
A shifted (delayed or advanced) pulse will in general cause the<br />
structure to perform worse than optimal. However, the deteriora-<br />
Fig. 22. Optimized structures for long wavelengths. Top: k ¼ 2:5 m and bottom:<br />
k ¼ 10 m.<br />
Fig. 23. Instantaneous material distribution and wave motion at t ¼ 5:0 s for the<br />
optimized structure in Fig. 15 with the input pulse delayed 0.1 s.<br />
tion occurs smoothly with small perturbations only resulting in<br />
small changes in the performance and with maximum deterioration<br />
for this example occurring with the 0.1 s shift shown in<br />
Fig. 23. If the pulse is delayed another 0.1 s (0.2 s in total) the<br />
waves will again follow the inclusion fronts and the optimal situation<br />
is reestablished.<br />
6. Summary and conclusions<br />
This paper describes an extension of the topology optimization<br />
method to facilitate optimization of material distributions in space<br />
and time.<br />
The established topology optimization formulation is extended<br />
with design variables in the temporal domain that allows the<br />
point-wise optimized material properties to change in time. The<br />
extended method is described for one-dimensional wave propagation<br />
in elastic rods with time dependent Young’s modulus and subjected<br />
to transient loading.<br />
A gradient-based optimization algorithm is applied based on<br />
explicit time integration of the discretized model equation. Gradients<br />
are obtained using the adjoint method which requires just one<br />
additional transient problem to be solved irrespectively of the<br />
number of design variables.<br />
A simulation study of two simple problems illustrate the rich<br />
behavior of structures with dynamic material distributions. Instant<br />
changes in Young’s modulus or a moving interface between two<br />
material phases, result in phenomena such as increase or decrease<br />
in the total mechanical wave energy and shifts in the wavenumber/<br />
frequency of waves. The shortcomings of the numerical integration<br />
scheme are outlined and a simple remedy in form of stiffness proportional<br />
damping is proposed.<br />
These dynamic phenomena are reflected in the optimization<br />
example. The objective is to design a structure that minimizes<br />
the transmission of a sinusoidal Gauss-modulated pulse. The best<br />
static structure is a bandgap structure with periodically placed<br />
inclusion layers of the stiffer material. It is demonstrated that<br />
structures in which the inclusion layers move with the propagating<br />
waves, so-called spatio-temporal laminates, can be much more<br />
effective in minimizing the transmission. The improvement is facilitated<br />
by removal of mechanical energy via the external force that<br />
is required to change the stiffness.<br />
Additionally, it is demonstrated that if we allow the stiffness to<br />
change less often, the optimized design change qualitatively and<br />
attain a checkerboard appearance. A special design parametrization<br />
was constructed that ensures a checkerboard structure and<br />
the resulting design was compared to the other designs. The performance<br />
of the checkerboard structure is not a good as the laminated<br />
structure but still significantly better than the static bandgap<br />
structure.<br />
The sensitivity of the optimized designs were analyzed with respect<br />
to changes in the operating conditions. It was demonstrated<br />
that the performance depends on a timely activation of the space–<br />
time material pattern and if the pulse is delayed w.r.t. the optimization<br />
conditions the performance is significantly deteriorated.<br />
The ability of supply/extract energy and change the frequency<br />
contents of the wave opens up for more advanced pulse manipulation<br />
than demonstrated in this relatively simple example. Currently,<br />
the possibility to compress pulses is being investigated<br />
and will be reported elsewhere.<br />
Acknowledgements<br />
The work was supported by the Danish Research Council for<br />
Technology and Production Sciences (Grant 274-05-0498). The<br />
author wishes to thank Jon J. Thomsen and Boyan S. Lazarov (both