WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1622 J. S. JENSEN
Response, ln(u tip u - tip )
2
0
-2
-4
-6
-8
-10
-12
-14
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency, Ω
Fig. 6a
Fig. 6c
Fig. 6d
Figure 8. Response in point A for the optimized structures in Figure 6(a), (c), (d). Dashed lines
indicate the optimization frequency interval.
in larger frequency ranges. Additionally, it is evident that the optimized material distribution for
the single frequency optimization is dominated by intermediate material properties. Not all of the
allowed reinforcement material is used in this case. The designs in Figure 6(b)–(d) show a much
more well-defined material distribution, and all of the allowed reinforcement material is utilized
to create the best possible performance.
All optimized designs have been obtained from an initial design with a uniform material distribution.
Different optimized designs (local optima) appear with other initial designs. Actually, by
using the structures in Figures 6(c) and (d) as the initial design for the optimization problem with
the frequency interval = 0.9–1.1, designs with a slightly better performance were obtained.
The same optimization set-up is now used to minimize the response in the entire body instead
of just in a single output point. The second objective function
c2 = 1
N
N k=1
u T ū (73)
is obtained by setting L to be the unit matrix. Figure 9 shows the optimized designs for two
different optimization frequency intervals and Figure 10 shows the responses for the two designs
as well as for the initial design. In this case, the anti-resonance mechanism cannot be used since
this is a local phenomenon. Instead, a low response is created by moving resonances away from
the optimization interval. For the small interval it is seen that all resonances have been removed
and a very low response is obtained in the entire interval. For the large interval, not all resonances
are removed and two response peaks remain near = 1. However, all other nearby resonances are
moved either below or above the interval and consequently a low average response is obtained.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme