WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Phononic band-gap optimization
However, unlike photonic crystals, phononic band-gap materials and structures have,
to the authors' knowledge, not yet directly led to industrial applications, but have
potentially rewarding applications in frequency filters with control of pass or stop
bands, as beam splitters, as sound or vibration protection devices, as acoustic lasers
(phasers), as perfect acoustic mirrors, or as elastic waveguides.
A comprehensive and frequently updated reference list for work on photonic and
phononic band gaps can be found on http://www.pbglink.com.
Little work has been done on the systematic design of band-gap materials and
structures. In the literature, the 'optimal' band-gap material is usually found by
parameter studies based on fixed inclusion shapes (e.g. circular inclusions). A
promising method for systematic design of band-gap materials and structures is
the topology-optimization method (see Bends0e & Sigmund (2003) and references
therein). In topology optimization one discretizes the design domain (e.g. the peri-
odic base cell) by a large number of elements (typically coinciding with the finite-
element mesh) and allows the material density or type in each element to be a design
variable. By defining proper objective functions and constraints, efficient and entirely
new topologies defined as raster pictures may be obtained. Since its introduction in
the late 1980s (Bends0e & Kikuchi 1988), the topology-optimization method has been
applied to a myriad of design problems ranging from simple compliance minimization
for elastic structures over design of extremal materials with negative thermal expan-
sion coefficients (Sigmund & Torquato 1997) to design of microelectromechanical
systems (Sigmund 2001a, b). Topology-optimization methods have been applied to
the design of photonic band-gap materials considering scalar fields in Cox & Dobson
(1999, 2000), and preliminary results of the topology-optimization method applied
to the design of phononic band-gap materials considering both scalar and coupled
problems have been published in Sigmund (2001c) and Sigmund & Jensen (2002). To
the authors' knowledge, no papers on topology optimization of band gap structures
have been published, apart from some preliminary results presented in Sigmund &
Jensen (2002).
The paper is organized as follows: after defining the basic equations, we briefly
discuss the analysis of materials and structures in a standard finite-element-method
(FEM) formulation and provide examples showing the typical behaviour of periodic
materials and structures (? 2). We then introduce the topology-optimization tech-
nique and apply it to design materials with maximum relative band-gap size and
structures with optimized wave-damping or waveguiding properties (? 3). Finally, we
make some conclusions (? 4).
2. Propagation of elastic waves
Vibrations and wave propagation in a three-dimensional elastic inhomogeneous
medium are governed by
02 =
Phil. Trans. R. Soc. Lond. A (2003)
1003
, (AV .u) +V ( v+ )), (2.1)
POt2 a ax
Pt2 = 0(V u) + V Vv + (2.2)
P2 = (V ) + IV
POt2 - a(X V Vw + (2.3)
u + '