WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

5.3 Space-time topology optimization 49
input
2m
1m
design
domain
2m
output
Figure5.5 Design problemforthespace-time topology optimization example. Thetransmission
of a Gauss-modulated sinusoidal pulse is to be minimized by finding an optimized
distribution of two materials in a space-time design domain. From paper [20].
1.5s
t
design
domain
1m
Figure 5.6 Left) Indication of the position-time design domain for the optimization
problem, right) Optimized distribution of the two materials. From paper [20].
a one-dimensional bandgap structure (a so-called Bragg grating).
The optimized static structure is used as a basis for the space-time optimization
as illustrated in Fig. 5.6(left). The static bandgap structure is retained in the
simulation before the pulse enters the design domain and after the main part of
the signal is transmitted through the domain. For a time interval of 1.5s for which
the main part of the pulse propagates through the design domain the material is
allowed to change in space and time using a corresponding two-dimensional design
grid. Fig. 5.6(right) shows the material distribution in a space-time design variable
plot. The black (stiffer) material is still distributed in layers but now arranged with
a slope in space-time.
Fig. 5.7 illustrates the type of dynamic bandgap structure which is created. The
plots show the instantaneous material distribution along with the wave motion for
two separate time instances. The layered structure moves together with the wave
with a speed matching the effective speed of the wave pulse (this speed can be
computed from the slope in the space-time design plot in Fig. 5.6). The transmitted
wave energy is reduced by approximately a factor three by this dynamic design
x