Wave Manipulation by Topology Optimization - Solid Mechanics

4.4 Effect of polarization and background material[P2] 27
Figure 4.6 Polarization-dependent discrete cloak designs and the associated objective
values. The bottom panels depict the total fields while the top panels show the optimized
material layouts as linear gray-scale plots, where white and black color correspond to
ɛr =1andɛr = 2. The designs in panels (a) and (c) are optimized for Ez-polarization
only whereas the deigns in panels (b) and (d) are optimized for Hz-polarization. Even
though the cloaks (c) and (d) with ɛBG r = 2 only is required to operate for one polarization,
significant scattering is generated.
manufacturing defects. In this second study we vary the relative permittivity between
ɛmin r =1andɛmax r = 2 in the cloak domain wrapped around the cylinder. The
permittivity of the background material, ɛ BG
r , (cf. figure 4.1) is either set to 1 or 2,
depending on wether we are distributing dielectric material having a higher or lower
permittivity than the background material, respectively. A symmetry constraint is
imposed on the design with two symmetry lines around which the 4 equally sized
sub-domains can be mirrored. That is, the optimized cloaks only work for two incoming
wave angles. A min/max formulation for ΦE and ΦH is employed to make
the cloak equally effective for both Ez and Hz polarization.
Graded and discrete designs that minimize the scattering from both Ez and
Hz polarization simultaneously are presented in figure 4.5 (top panels). The total
fields for Ez and Hz polarization are shown in the middle and bottom panels, respec-
tively. In case of ɛ BG
r
= 1 the graded design result in scattering less than 1% of the
non-cloaked cylinder and even for the more restrictive case with a discrete design
scattering is reduced to less than 5% for both polarizations. The Hz polarization for
the discrete design yielding less scattering than the Ez polarization, due to the less
restrictive boundary condition for the Hz polarization. Contrary to simple intuition,
considerable scattering is produced by all designs with ɛBG r =2andgradeddesigns
with ɛBG r = 2 produce more scattering than the discrete counter-parts. To analyze
the cause for the degeneration in cloaking performance for designs with ɛBG r =2we
relax the problem and investigate whether or not effective cloaks with ɛBG r =2can
be designed individually for either the Ez or Hz polarization.
Discrete design profiles optimized individually for either Ez or Hz polarization
and the corresponding total fields are shown in figure 4.6. Even though we relax