WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Phononic band-gap optimization
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Figure 10. Optimization of structure for minimum response at 2 = 63 kHz. (a) Structural
domain and periodic boundary loading in the optimization procedure; (b) optimized topology;
(c) average response at the right boundary when subjected to loading at the left boundary.
Dashed line, a = 0, 3 = 50 x 103; solid line, a = = 0 0. High-contrast case.
(corresponding to the centre frequency of the optimized gap of figure 7b). In order
to obtain square symmetry of the resulting structure, we this time solve a multi-load
problem. The structure is subjected to periodic excitation at the four edges independently,
and we then minimize the sum of responses at the opposing edges. The
optimization is performed with strong damping added (a = 0,/3 = 50 x 103) in order
to remove resonance peaks and ensure stable convergence.
Figure lOb shows the optimized structure. The topology is quasi-periodic and
closely resembles the periodic band-gap structure in figure 4c. The largest difference
between the two topologies is seen at the boundaries, where the finite structure
has inclusions touching the boundaries and with a different periodicity. This prevents
waves from propagating along the boundaries: a phenomenon that cannot be
Phil. Trans. R. Soc. Lond. A (2003)
1015
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