META 2018 CONFERENCE, ROUND-TRIP MARSEILLE CRUISE, JUNE 24 – JULY 1, 2018

CMA-ES based topology optimization for external cloaks

Garuda Fujii1

1Institutute of Engineering, Shinshu University, Nagano, Japan*corresponding author, E-mail: g fujii@shinshu-u.ac.jp

Abstract

This paper presents topology optimization for externalcloaks which can render a scattering object at a distanceinvisible. To design external cloaks, the intensity of thescattered electric field is minimized as the objective func-tion and external cloaks are transformed in a fixed designdomain which is set at a distance from a PEC scattering ob-ject. A level set method is used to express external cloakingstructures made of dielectrics without any grayscales thatare intermediate materials between air and dielectrics. Ob-tained optimal configurations prevent scattering caused byPEC.

1. IntroductionOptical invisibility [1, 2] has been actively studied in thepast decade and spread, by analogies, to acoustic cloaks,thermal cloaks, direct current electric cloaks, and fluid flowcloaks. External cloaks [3, 4] are the optical devices thatcloak a scattering object at a distance outside the cloaks byreproducing the electromagnetic field when no scatteringobject exists.

Topology optimizations are known as a powerful ap-proaches to enhance the performance of devices in engi-neering. The design methodology provided optimal designof cloaking devices having superior performances and areapplied to the designs of optical cloaks [5], carpet cloaks[6], thermal cloaks [7], and DC electric cloaks [8].

In this work, we presents topology optimization for ex-ternal cloaks expressed by level set functions explored bycovariance matrix adaptation evolution strategy (CMA-ES).

2. Topology optimization for external cloaksThe performance of external cloaks is evaluated by theamount of scattering caused by scattering objects. Then,the objective function Ψ is set to be integrated intensity ofscattered electric field as

Ψ =1

Ψn

∫Ωout

|Es|2 dΩ,

where Es represents a scattered electric field, Ψn is thevalue for normalization of the objective function as

Ψn =

∫Ωout

|Ebare|2 dΩ,

ΩD

Ωout

ΩPEC

ΩPML

ΩPML

ΩPM

L

ΩPM

L

θ

RPEC

Lout

0

0

ΩS

ΩDΩS

RD

ΩPEC

RPEC

x

y

z

(a)

(b)

(c)

Figure 1: (a) Scheme of topology optimization for externalcloaks. (b) Level set function and structural boundaries. (c)Cloaked PEC and each axis.

where Ebare represents a scattered electric field when thebare scattering object presents in the absence of cloak. TheEs follows Helmholtz equation as

∇2Es +ω2

c2ϵ(x)Es = −ω2

c2[ϵ(x)− ϵair]Ei,

where Ei denotes the incident electric field satisfying Ez =Es+Ei, ω the circular frequency of light, and c the velocityof light in vacuum.

Topology optimization is an ill-posed problem and theoptimization must employ the relaxation of design spaceor the regularization by the geometrical constraint. In thisstudy, we employ a perimeter constraint that is a kind ofgeometrical constraints and a minimize fitness as follows:

infϕ

Ftop = Ψ + τLp,

where τ is regularization parameter, and Lp is the length ofΓS which is the boundary between ΩS and ΩD.

With ϵair and ϵS denoting the relative permittivities ofair and dielectric material of designed structures, respec-tively, the position-dependent relative permittivity ϵ(x) isdefined as

ϵ(x) =

ϵair + χ(ϵS − ϵair) x ∈ ΩD

ϵair x ∈ Ωout ,

where χ is the characteristic function,

χ(ϕ(x)) =

1 if x ∈ ΩS

0 if x ∈ ΩD\ΩS ,

and ϕ(x) denotes a level-set function whose isosurfacesprovide clear structural boundaries such that −1 ≤ ϕ(x) < 0 ∀x ∈ ΩD\ΩS

ϕ(x) = 0 ∀x ∈ ΓS

0 < ϕ(x) ≤ 1 ∀x ∈ ΩS\ΓS .

The level set function is discretized to the grid points asshown in Fig. 1(b) and is expressed as the vector ϕ =ϕ1, · · ·ϕj , · · ·ϕN where N is the number of discretizedlevel set functions. The optimal vector ϕ minimizing Ftop

is explored by CMA-ES employing a box constraint [9].

2.1. Results

Figure 2(a) shows the fixed design domain ΩD withoutcloaks and Fig. 2(b) shows light scattering caused by barePEC. Light waves are scattered at the surface of the PECand interference between incident and scattered waves areobserved in Fig. 2(b).

(b) Electric field(a) No cloak

Figure 2: Fixed design domain without cloak and electricfield scattered by PEC.

Figure 3(a) represents obtained optimal configurationunder τ = 1×10−3. The blue color represents the domainsof dielectric material given by ϵ = 2.0 and the light yellowrepresents air domains given by ϵ = 1.0. Figure 3(b) showsthe scattered light when the obtained optimal configurationsexist at left side of the PEC domain. The light scattering isprevented by the designed external cloaks.

3. ConclusionsThis paper presents optimal design of external cloaks ob-tained by topology optimization based on CMA-ES. Ob-tained optimal configuration present light scattering causedby PEC.

AcknowledgementThis work is supported by JSPS KAKENHI [Grant No.

17K17778].

References

[1] U. Leonhardt, Optical conformal mapping, Science312: 1777–1780, 2006.

(a) Optimal configuration (b) Electric field

Figure 3: Result of topology optimization obtained underτ = 1 × 10−3. The blue color represents the domains ofdielectric material given by ϵ = 2.0 and the light yellowrepresents air domains given by ϵ = 1.0.

[2] J.-B. Pendry, D. Schurig and D.-R. Smith, Control-ling Electromagnetic Field, Science, 312: 1780–1782,2006.

[3] Y. Lai, H. Chen, Z.-Q. Zhang, and C. T. Chan, Com-plementary media invisibility cloak that cloaks objectsat a distance outside the cloaking shell, Phys. Rev.Lett., 102: 093901, 2009.

[4] T. Han, X. Tang, and F. Xiao, External cloak with ho-mogeneous material, J. Phys. D: Appl. Phys., 42(23):235403, 2009.

[5] G. Fujii, H. Watanabe, T. Yamada, T. Ueta andM. Mizuno, Level set based topology optimizationfor optical cloaks, Appl. Phys. Lett. 102(25): 251106,2013.

[6] G. Fujii and T. Ueta, Topology-optimized carpetcloaks based on a level-set boundary expression, Phys.Rev. E, 94(4): 043301, 2016.

[7] G. Fujii, Y. Akimoto, and M. Takahashi, Explor-ing optimal topology of thermal cloaks by CMA-ES,Appl. Phys. Lett, (accepted).

[8] G. Fujii, Y. Akimoto and M. Takahashi, Achievingdirect current electric invisibility through a topologyoptimization based on CMA-ES, J. Appl. Phys., (sub-mitted).

[9] G. Fujii, M. Takahashi and Y. Akimoto, CMA-ES-based structural topology optimization using a levelset boundary expression―Application to optical andcarpet cloaks, Computer Methods in Applied Mechan-ics and Engineering, 332, 624 – 643, 2018, (submit-ted).

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