WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

The parameters used in this example are given as
T ¼ 10 s; u0 ¼ 1m; t0 ¼ 2:5s; N ¼ 150;
x0 ¼ 15:7 rad=s; Dt ¼ 0:06 s; b ¼ 0:1ðkg mÞ 1 ;
d ¼ 1:5 s 2 ; E0 ¼ 1:25 N=m 2 :
To be able to clearly separate input and reflected waves in the time
series, inlet and outlet sections of 2 m with constant material properties
are added on each side of the design domain of length 1 m.
The material properties of the inlet and outlet sections are normalized
to E ¼ 1N=m 2 and q ¼ 1kg=m 3 , so that the incoming wave
propagates with the speed c ¼ 1m=s. In order to simulate wave
propagation in this finite structure, fully absorbing boundaries are
added at the input and output points by appropriate viscous
dampers.
Two materials can be distributed in the design domain: the normalized
material used for the the inlet and output sections, and a
material with a slightly higher stiffness E0 ¼ 1:25. The relatively
small stiffness contrast is used in order to avoid problems with
spurious oscillations (cf. Section 3.2). The presence of oscillations
leads to incorrect sensitivity calculations that destabilize the optimization
process. A small stiffness contrast combined with added
stiffness proportional damping (b ¼ 0:1 used throughout this
example) eliminate these problems while still displaying the main
qualitative features.
5.1. Static bandgap structure
For comparison the structure is optimized for the static case, i.e.
a spatial material distribution is obtained which cannot change in
time. Similar design problems were considered recently for transient
loading in [13]. Fig. 12 shows the optimized design and
Fig. 13 shows the nodal displacements as a function of time at
the input and the output points. In the time plots one can easily
identify the input and reflected waves at the input point (top figure)
and the transmitted wave at the output point (bottom figure).
The optimized structure in Fig. 12 is a bandgap structure [8]
with periodically layered inclusions of the stiffer material. Such a
structure reflects the waves maximally and reduces the objective
function to 76% compared to the undisturbed wave. For comparison
it can be mentioned that in the case in which the design domain
is completely filled with the stiffer material the objective
function is only reduced to 93%. These computations have been
performed with stiffness proportional damping corresponding to
b ¼ 0:1. Without damping (b ¼ 0) the transmissions are 81% and
99%, respectively.
With a static structure the only way to further reduce U would
be to either increase the material contrast or to increase the length
of the design domain relative to the wavelength of the pulse. The
latter choice would results in more inclusion layers that lead to
an increased reflection of the wave.
5.2. Space–time bandgap structure
The design is now allowed to change in time as well as in space.
The optimized static bandgap structure in Fig. 12 is used as a starting
point for the dynamic structure. This is illustrated in Fig. 14
where the space–time design domain is indicated. A temporal design
interval of DT ¼ 1:5 s is chosen, which is sufficiently long to
Fig. 12. Optimized structure for the ‘‘static” wave propagation problem.
J.S. Jensen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715 711
Fig. 13. Response for the optimized structure shown in Fig. 12. Top: input point
displacement showing the input and reflected wave, bottom: output point
displacement showing the transmitted wave.
significantly modify the wave motion but still keeps the total number
of design variables at a manageable level. The start and finish
point for the optimization is chosen as T1 ¼ 4:25 s and
T2 ¼ 5:75 s and the number of sub-intervals is M ¼ 225, thus
the material properties in each element are allowed to change
150 times per second. The spatial discretization is unchanged from
the static case with 150 elements in the design domain. Thus, the
two-dimensional design grid is composed of ‘‘square” elements
1
with the dimensions 150 s 1 m and a total of 33750 design
150
variables.
The initial value of all design variables is chosen as xe ¼ 0:5,
implying that the stiffness in all design elements is initially
E ¼ 1:125 N=m2 . The optimized design is obtained after about
100 iterations and is shown in Fig. 15. The structure is seen to be
a kind of spatio-temporal laminate with space–time layered inclusions.
Properties of spatio-temporal laminates are discussed, e.g. in
[22,35,24].
At each time instance the structure is layered in a similar way as
the static bandgap structure. However, the inclusion layers move
with a constant speed corresponding to the wave speed in the layered
medium so that the wave peaks and valleys actually move together
with the front of the inclusions. This is illustrated in Fig. 16,
in which the material distribution is shown together with the wave
motion at the two time instances t ¼ 5:0s and t ¼ 5:4s.
Fig. 17 shows the input and output point time responses. The
objective is reduced to 23% relative to the undisturbed input signal
– thus, a significant reduction is noted compared to the static case
(76%). The large reduction is not a consequence of an increased