Wave Manipulation by Topology Optimization - Solid Mechanics

Chapter 5
Backscattering cloak
Encouraged by the results from the topology optimized cloaks reported in the previous
chapter we looked for further ways to extend the cloak analysis. Here we
relax the full cloaking problem by only requiring elimination of backscattering in
a limited angular range. The less restricted case of cloaking enables us to concentrate
our study to omnidirectional designs, based on concentric rings of isotropic,
low-contrast, all-dielectric materials readily available in nature. Optimized annular
cloaks have previously been proposed in the literature (cf. Ref. [105]), however the
optimized anisotropic material properties are not easily realizable. Although relaxed
compared to the full cloaking problem, backscattering cloaks are highly relevant in
e.g. radar-setups where source and detector are coincident. The results presented
here are examples of the findings given in [P4].
With very little reformulation the optimization approach reported in the last
chapter can easily be extended to the backscattering problem. However, the symmetry
of omnidirectional ring design allows us to compute the electromagnetic field
analytically[106, 107, 108]. Derivations of e.g. the far-field on an arc line is much
faster using an analytic expression compared to a FEM solution and ring positions
are not restricted to an underlying mesh. On the other hand the analytic expression
can only be implemented in a sizing optimization of e.g. ring position and thickness
thereby loosing some of the design freedom provided by topology optimization.
For the above reasons we choose a two step approach in order to benefit from the
advantages of each method. First we use topology optimization to provide us with
basic understanding of the optimized topology. Next a parametrization of the topology
optimized designs is used in the optimization approach based on the analytic
expression.
5.1 Topology optimized backscattering designs
The setup for the topology optimization problem is almost identical to the one outlined
in the previous chapter. Here we minimize the norm of the Ez polarized scattered
field in a circular trapezoidal domain enclosed at ± 10 o from the backscattering
direction and with inner and outer radii given as 1.75 and 5 free space wavelengths
from the obstacle, respectively (see Ωout in Fig. 5.1(a)). The permittivity is allowed
to vary between ɛmin r =1andɛmax r = 2 with free space being the background material.
The results of the optimization process are surprisingly simple annular Bragglike
grating structures with layer dimensions that depend on the obstacle radius
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