Wave Manipulation by Topology Optimization - Solid Mechanics

Chapter 3
Topology optimization of wave problems
As briefly mentioned in the previous chapter the spatial placement and distribution
of the material properties determines how a wave behaves in a medium. Control
of these material properties allows us to manipulate wave propagation for various
purposes. The material as well as material layout needed in wave devices like Bragg
gratings[15], invisibility cloaks[16] and Fresnel zone plate lenses[17] can be derived
directly by analytic means. Such applications were developed by ingenious physicists
with great physical insight. However, in some cases the derived material properties
are very challenging to realize especially in the optical regime (e.g. invisibility
cloaks[16]), and in other cases the geometrical layout rule is known not to be optimal
(e.g. Fresnel zone plate lenses[18]). Optimization methods may be used as an
efficient design tool for such problems. Topology optimization is a gradient based
optimization method that work by varying the distribution of materials within a
bounded design domain. The strength of this method is that it can change the
shape topology without any geometrical constraints on the design. Originally the
method was developed by Kikuchi and Bendsøe using a homogenization technique
to minimize the compliance of continuum structures[19]. Since then the method
has been extended to other fields of engineering such as heat transfer (e.g. [20]),
fluid dynamics (e.g. [21]), micro electro-mechanical systems (e.g. [22, 23]), photonics
(e.g. [24, 25]) and acoustics (e.g. [6]). Today implementations[26, 27] of the topology
optimization method for structural problems are available free of charge intended
for educational purposes and even commercial software packages such as OptiStruct
by Altair and TOSCA by FE-Design exist. The purpose of this chapter is to give
a brief introduction to topology optimization and how it is applied to the acoustic
and electromagnetic wave manipulation problems covered in this thesis. A thorough
description of the topology optimization method can be found in the monograph by
Bendsøe and Sigmund[28].
3.1 Topology optimization of acoustic and electromagnetic
wave propagation problems
The behavior of electromagnetic waves is determined by the distribution of dielectric
material, magnetic material, air and/or metal. The first material distribution
technique concerning electromagnetic waves was reported more than a decade
ago[29, 30]. In the study, dielectric material was distributed in order to maximize
the band gap in 2D photonic crystals. Not long after the initial study, the transmission
through photonic crystal wave guide bends[24] and splitters[31] was maximized
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