Wave Manipulation by Topology Optimization - Solid Mechanics

Chapter 4
Topology optimized electromagnetic and acoustic
cloaks
Rendering objects invisible to the human eye is a popular topic in science-fiction novels
and movies, but it has been considered impossible by most people to achieve in
the physical world. However, in 2006 two independent papers laid out the path of designing
real-world cloaking devices for ray optics[64] and electromagnetic waves[65].
Roughly speaking an invisibility cloak is a device, which when wrapped around
or placed nearby an object, renders the object and cloak invisible to an outside
observer. An incident electromagnetic or acoustic wave is manipulated by the material
of the cloak, such that the incident wave effectively is undisturbed outside
the cloak and object. This way no scattering from the object (or cloak) are to be
detected by the outside observer. Design of electromagnetic and acoustic cloaks by
topology optimization constitute the first wave manipulation problem in this thesis.
The chapter is a summary of publications [P1], [P2] and [P3].
4.1 Transformations Optics
The material layout of the aforementioned initial cloaks are determined by transformation
optics[64, 65]. Transformation optics exploits that Maxwells’ equations and
thereby also the Helmholtz’ equation are invariant under any coordinate transformation,
as long as the material parameters (μ and ɛ) are changed appropriately by
the transformation[66]. General coordinate transformations can be derived which
compress, expand, bend or twist space. Such transformation can be utilized in wide
range of electromagnetic problems. One approach to cloaking is to expand an infinitely
small hole into the shape of the object, which should be made invisible.
As the system is transformed it carries with it all the associated fields and electromagnetic
waves are thereby excluded in a region of same size as the object. To
give a flavor of how the transformation optic method works we demonstrate by an
example adopted from [67]. In a cylindrical coordinate system (ρ,θ,z) weexpand
an infinitely small hole into a cylinder with the radius R1 by compressing the ringshape
space between R1 and an outer cylinder with radius R2, and get the following
transformation
ρ ′ =(R2 − R1)ρ/R2 + R1, θ ′ = θ, z ′ = z (4.1)
where ′ denotes the transformed coordinates. With an appropriate change of the
material properties Maxwell’s equations are invariant to the coordinate transform.
The appropriate change leads the transformed material parameters ɛ ′ and μ ′ to
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