WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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