WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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10.1098/rsta.2003.1177<br />
r4m THE ROYAL<br />
!.IU SOCIETY<br />
Systematic design of phononic band-gap<br />
materials and structures by<br />
topology optimization<br />
BY OLE SIGMUND <strong>AND</strong> JAKOB S0NDERGAARD JENSEN<br />
Department of Mechanical Engineering, Section for Solid Mechanics,<br />
Nils Koppels Alle Building 404, 2800 Kgs. Lyngby, Denmark<br />
Published online 26 March 2003<br />
Phononic band-gap materials prevent elastic waves in certain frequency ranges from<br />
propagating, and they may therefore be used to generate frequency filters, as beam<br />
splitters, as sound or vibration protection devices, or as waveguides. In this work we<br />
show how topology optimization can be used to design and optimize periodic mate-<br />
rials and structures exhibiting phononic band gaps. Firstly, we optimize infinitely<br />
periodic band-gap materials by maximizing the relative size of the band gaps. Then,<br />
finite structures subjected to periodic loading are optimized in order to either min-<br />
imize the structural response along boundaries (wave damping) or maximize the<br />
response at certain boundary locations (waveguiding).<br />
Keywords: phononic band gaps; topology optimization; materials; structures<br />
1. Introduction<br />
A new application of the topology-optimization method is design of materials and<br />
structures subject to wave propagation. The wave may be elastic, acoustic or electro-<br />
magnetic, but the phenomenon is the same: for some frequency bands it is possible to<br />
construct periodic materials or structures that hinder propagation. The phenomenon<br />
is a band gap.<br />
The phenomenon of band gaps may be illustrated by the following example. Fig-<br />
ure la, b shows a two-dimensional square domain subjected to a periodic loading<br />
at the left edge and with absorbing boundary conditions along all the edges. The<br />
frequency of excitation of the structure in figure la is lower than that for figure lb.<br />
It is seen that waves propagate unhindered through the structures from left to right,<br />
damped only slightly at the top and bottom due to the absorbing boundary con-<br />
ditions. Now, if we introduce a periodic distribution of inclusions with different<br />
propagation speeds, the situation changes. For the structure subjected to a lower<br />
excitation frequency (figure Ic), there is still propagation, although the waves are<br />
significantly distorted by the reflection and refraction from the inclusions. However,<br />
for the structure subjected to a higher excitation frequency (figure ld), there seems<br />
to be no propagation at all. This illustrates the band-gap phenomenon. For elastic<br />
and acoustic waves the materials are called phononic band gap materials, and for<br />
electromagnetic wave propagation the materials are called photonic band gap mate-<br />
rials.<br />
One contribution of 12 to a Theme 'Micromechanics of fluid suspensions and solid composites'.<br />
Phil. Trans. R. Soc. Lond. A (2003) 361, 1001-1019<br />
1001<br />
? 2003 The Royal Society