Wave Manipulation by Topology Optimization - Solid Mechanics

4.3 Topology optimized all-dielectric cloaks[P1] 23
z
y
Ez
x
H
k
Ωout
Ωdes
ɛr(ρ)
ɛ BG
r
ΓABS
ΓPEC
Figure 4.2 Computational Domain. The subfigures (a)-(e) show the design domains
equipped with 1, 2, 3, 4 and 8 symmetry lines, respectively (PEC shown in red). The
design domain with rotational symmetry is given in subfigure (f).
minated by an Ez polarized plane wave are shown in figure 4.3(a) and (b), respectively.
The difference between the incident field and the total field is the scattered
field generated by the scattering object. The interference pattern from the scattered
field is very notable for the non-cloaked object especially as a shadow region behind
the cylinder. The objective for the cloaking problem is to achieve zero scattering
from the cloaked object in the surrounding space, Ωout. Hence, the figure of merit
in this thesis for evaluating the cloaking performance is given by the norm of the
scattered field. The formulation of the optimization problem thus takes the form
min
γ
ΦE = 1
|E
WE Ωout
s z |2 subjected to
dr
1
Ωdes
objective function (4.5)
dr
γ(r)dr − Vmax ≤ 0
Ωdes
volume constraint (4.6)
0 ≤ γ(r) ≤ 1 ∀r ∈ Ωdes design variable bounds (4.7)
where Vmax is the maximum allowed volume fraction of material 2 and the subscript,
E, reflect the polarization and W is the norm of the scattered field in the outer
domain when no cloak is present. The reference value, W , is included in order to
make the objective dimensionless as well as easy to interpret. By definition Φ is
zero for the incident field (cf. figure 4.3(a)) and unity for the non-cloaked object (cf.
figure 4.3(b)). Thus, if Φ is less than unity in the design process the object with a
cloak wrapped around generate less scattering than the non-cloaked object.
4.3 Topology optimized all-dielectric cloaks[P1]
In the first study [P1] the evolution of the optimized designs along with the corresponding
cloaking properties are investigated for increasing number of incident