WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

4 Chapter 1 Introduction
structures.
The first paper of this thesis (paper [1]) intends to couple the analysis of the
bandgapphenomenon forelastic waves in aperiodic material withthe corresponding
dynamic behavior of a finite structure made from this periodic material. Furthermore,
this paper attempts to provide an increased understanding of the applicability
of bandgap structures in mechanical structures by studying the influence of boundaries,
damping and disorder. Paper [1] was followed by other papers that investigate
different aspects of the bandgap phenomenon.
Optimal material distribution – topology optimization
In past research bandgap materials in 2D and 3D are almost exclusively found in
the form of circular or spherical inclusions. For the static behavior of mechanical
structures it has been long known that circular or spherical holes are rarely optimal
with respect to structural performance, but similar conclusions for elastic or optical
wavesarenotwidelyknown. Intwopapers, Cox&Dobsonusedanumericalmaterial
distribution method to optimize the distribution of air and dielectric material in a
photonic bandgap material and maximized the size of the gap (Cox and Dobson,
1999, 2000). They found that circular holes were not optimal. Sigmund (2001)
used topology optimization to maximize phononic bandgaps for elastic materials
and arrived at a similar conclusion.
The method of topology optimization is widely applied in this thesis. Topology
optimization of mechanical structures was introduced by Bendsøe and Kikuchi
(1988) in order to optimize the material distribution and obtain maximum stiffness.
Paper [4] 2 attempts to apply the topology optimization method to design phononic
bandgap structures, i.e. to find the distribution of two elastic materials that optimizes
the performance of the structure. This could be to minimize the vibrational
response in a certain part of the structure. The basic hypothesis is that the optimal
material distribution in certain cases should have a periodic appearance – a bandgap
structure.
The work in paper [4] was later extended and topology optimization applied
to design a number of different bandgap structures. The design methodology used
in this thesis follows the outline schematically illustrated in Fig. 1.3. The top left
figure shows a part of an original structure with two different materials (here the
darkest color represents air holes and the two shades of gray represent a dielectric
material). The top right figure shows a corresponding finite element (FE) model
which is parameterized with a single continuous design variable xe assigned to each
element. Withasuitableinterpolationscheme, seee.g.BendsøeandSigmund(1999),
it is ensured that xe = 0 corresponds to air (illustrated as light gray) and xe = 1
corresponds to dielectric material (black). A gradient-based iterative algorithm is
thenappliedtofindtheset ofdesignvariablesthatoptimizesaspecified performance
2 In the author’s contribution to the paper.