WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

30 Chapter 3 Bandgap structures as optimal designs
Relations to recent work
Paper [4] was to the author’s best knowledge the first that applied a numerical
material distribution algorithm to the design of mechanical (or phononic) bandgap
structures. Thisworkhaslater beenfollowedbyanumber ofoptimizationstudies on
elasticstructuresforwavereflection, guidingandcontrolpurposessuchasbyHussein
et al. (2007b) for 1D banded materials, Hussein et al. (2007a) for 2D phononic
crystals, by Rupp et al. (2007) who designed surface waveguides using 3D modeling,
by Evgrafov et al. (2008) who optimized 2D tunable waveguides and by Du and
Olhoff (2007a,2010)who mininimized andcontrolledsound radiationfromvibrating
structures. The separation of eigenfrequencies was also recently considered by Du
and Olhoff (2007b,c) using topology optimization.
The conference paper [9] dealing with the acoustic design problem was later extended
to a journal publication in cooperation with Maria Dühring (Dühring et al.,
2008) where also 3D problems were analyzed as well the planar problem of optimization
of sound barriers. The acoustic problem has later been studied using topology
optimization by Wadbro and Berggren (2006) who considered optimal design of an
acoustic horn, by Lee and Kim (2009) who optimized holes in a cavity partition
in order to control eigenfrequencies. Duhamel (2006) used a genetic algorithm to
design sound barriers. Related to this problem is also the study on topology optimization
of acoustic-structure interaction by the author, Ole Sigmund and Gil Ho
Yoon (Yoon et al., 2007).
In optimization problems where a large reflection of waves is required, well defined
structures seem to appear ”automatically”. Thus there is no need for penalization
of intermediate design variables (between 0 and 1). In most other applications
of topology optimization such a penalization is essential. This phenomenon is related
to the fact that intermediate design variables reduce the contrast between the
materials and thus lead to a reduced wave reflection. Mathematically, this issue has
been studied by Bellido and Donoso (2007) who proved that ”classical solutions”
(pure 0–1 designs) are optimal under certain conditions.