WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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