# Coupling Shape Optimization and Topological Derivative for Maxwell Equations

Authors: SY Alassane
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

The paper deals with a coupling algorithm using shape and topological derivatives of a given cost functional and a problem governed by nonstationary Maxwell’s equations in 3D. To establish the shape and topological derivatives, an adjoint method is used. For the topological asymptotic expansion, two examples of cost functionals are considered with the perturbation of the electric permittivity and magnetic permeability. We combine the shape derivative and topological one to propose an algorithm. The proposed algorithm allows to insert a small inhomogeneity (electric or magnetic) in a given shape.


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## Body

## 1. Introduction

Shape optimization is a minimization problem where the unknown variables run over a class of admissible domains; then, every shape optimization problem can be written in the form(1)minΩ∈ΘjΩ,uΩ,where Θ is the class of admissible domains and j is the cost functional; generally, uΩ is the solution of a given PDE. Generally, the existence and uniqueness of the solution of such problems are not guaranteed even for simple cases. However, the literature about the subject is abundant, and the techniques used are numerous (homogenization, domain parametrization, geometric shape derivative, topological optimization, …), see [1] and references therein.In this paper, we focus on geometric shape optimization and topological asymptotic expansion methods, which we briefly recall the principles to propose an algorithm coupling them that we apply to Maxwell’s equations.Shape optimization problem is a minimization problem where the unknown variables run over a class of domains; then, every shape optimization problem can be written in the form(2)minjΩ:Ω∈A,where A is the class of admissible domains and j is the cost functional.The underlying principle is the following. LetΩ⊂ℝ3 be an open bounded domain whose boundary ∂Ω is a Ck manifold oriented by the normal vector field ν outgoing to Ω, a family Ok of open bounded domains of D and t=0,T, the time interval. We note that Q=Ω×0,T is the cylinder evolution domain, and Σ=Γ×0,T is the lateral boundary associated to any Ω⊂Ok. Let ξk be the set of V∈C0,T,CkD,ℝ3 with V,ν∂D=0 for all V∈ξk; we consider the follow mapping TsV, such that, for each x,V as the form
(3)Vsx=∂∂sTs∘Tsx.For eachs∈0,S,Ts is a one-to-one mapping from D to D such that
(i)
T0=I(ii)
s,x⟶Ts belongs to C0,S,CkD,D with Ts∂D=∂D(iii)
s,x⟶Ts−1 belongs to C10,S,CkD,DSuch familyOk is stable under the perturbation Ω↦Ωs=TsVΩ. We denote by Qs, the perturbed cylinder Ωs×0,T,Γs=∂Ωs and Σs=Γs×0,T, the perturbed lateral boundary.To each elementΩ∈Ok, we associate uΩ=EΩ,HΩ, the solution of (11). For any V∈ξk, and s∈0,S, we set that us=uΩs∈L2Qs is said to be shape differentiable in L2I,HmD if there exists U∈C10,S,L2I,HmD such that
(4)Us,.,.Qs=uΩs,Us−U0s−∂sU0⟶0,inL2I,HmD,s⟶0.Definition 1.
The shape derivative ofuΩ,V to the direction V in the unique element u′Ω,V verifying
(5)u′Ω,V=∂∂sUs=0,t,x∈Q.In the same way, we define the boundary shape derivative and the material derivative.Definition 2.
The element(6)gΓ′Γ,V=∂∂sGs=0,t,x∈Σis the boundary shape derivative of g∈L1I,Hp, where G∈C10,S,L1Hp+1/2D such that
(7)∂∂νsGs=0onΣ,Gs,.,.Σs=gΓsonΣs.Definition 3.
The material derivative ofuΩ,V in the direction V noted u̇Ω,V∈L2I,HmD is defined as the limit in L2I,HmD when s goes to zeros of
(8)1suΩs∘Ts−uΩ.For more details on this approach, see [2].Remark 4.
To compute the shape derivatives, we can also use the approach of Simon and Murat in [3], recalled by Henrot and Pierre in [1] and Allaire and Jouve in [4].We consider a perturbation of the domainΩ in the following sense, for θ∈W1,∞ℝn,ℝn,Ωθ=Id+θΩ. It is well known that for θ, sufficiently small Id+θ is a diffeomorphism from ℝn.Definition 5.
The shape derivative ofuΩ at Ω is defined as the Frechet derivative in W1,∞ℝn,ℝn at 0 of the application θ⟶uId+θΩ; that is,
(9)uId+θΩ=uΩ+u′Ωθ+oθ,where u′Ω is a continuous and linear form on W1,∞ℝn,ℝn.In classical shape optimization, it is the boundary of the initial domain (or a part of the boundary) which moves for reaching the optimal shape. Thus, the optimal shape has the same topology as the initial one (for example, if the initial domain is simply connected, the optimal one will be also connected). Unlike the case of classical shape optimization, the topology of the design may change during the optimization process, for example, the inclusion of holes. The physical meaning of holes depends on the nature of the design. In the case of structural optimization, the insertion of the holes means simply removing some material, see [4]. In the case of fluid dynamics, creating a hole means inserting a small obstacle, see [5].Topological sensitivity analysis aims at providing an asymptotic expansion of a shape functional acting on the neighborhood of a small hole created inside the domain. The underlying principle is the following, see also [6]: For a criterion jΩ=JΩuΩ,Ω⊂ℝn and uΩ is the solution of a boundary value problem defined over Ω (generally a PDE), the asymptotic expansion of the cost function jΩ can be generally written in the form:
(10)jΩ\x0+εω¯−jΩ=ρεgx0+oρε,limε⟶0ρε=0,ρε>0.gx0 is called topological derivative (or topological sensibility) and provides an information for creating small hole located at x0. Hence, the function g can be used like a descent direction in the optimization process.The algorithm that we propose combines the creation of holes (insertion of a dielectric object) using the topological gradient on the one hand and on the other hand the update of the edges by making evolve using the derivative of shape.In Section2, we present Maxwell’s equations and an existence and uniqueness result and partial regularity under some conditions. The calculous of shape derivatives and topological asymptotic expansion (under small perturbations of the magnetic or electric fields) of the adjoint state is presented in Section 3. Two examples of shape functionals are considered. The main contribution of this paper is the proposition of the algorithm coupling between shape and topological derivative, which is presented in Section 4. Section 5 gives a conclusion and some possible extensions.


## 2. Maxwell’s Equations and the Preliminary Results

LetD be a fixed domain of ℝ3 and Ω⊂D be an open bounded regular domain with Lipshitz boundary ∂Ω=Γ. Let εx,t be the electric permittivity and μx,t be the magnetic permeability, which are both positive definite Hermitian 3×3 matrices.Letf1 and f2 be the electric density through Ω and g be the electric density through Γ.The evolution of the electric fieldEx,t and the magnetic field Hx,t in the space-time cylinder Q=Ω×0,T is given by the Maxwell equation:
(11)∂tεE−∇×H=f1inQ,(12)∂tμE+∇×E=f2inQ,(13)ν×E−αHτ=ginΣ.Here,ν denotes the normal exterior vector along Γ,Hτ is the tangential component of H, and α=αx,t is a positive function. For t=0, the initial conditions are given by Ex,0=E0x and Hx,0=H0x.Let(14)A0=ε00μ,Aj∂j=0−∇×∇×0,u=EH,f=f1f2.The Maxwell system (15) can be written as
(15)∂jA0u+Aj∂ju=f.Furthermore, let(16)HΩ=E,H∈L2Ω6:∇.εE=∇.μH=0,where ∇.εE=∂jεjkEk. If A is a Hermitian positive definite matrix, its transpose is denoted by AT.The well posedness of the Maxwell equations with given boundary and initial data is widely studies in the literature. We refer the literature to [7] for modeling and for the existence and unicity and regularity of the solution, see [8]. However, we recall here on a result which established the existence, unicity, and partial regularity of the Maxwell equations.Theorem 6.
Leε,μ,∂tE,∂tμ, and ∈L∞Q be such that ε,μ≥C>0 almost everywhere on Q and α≥C>0 almost everywhere on Σ. Given f,E0,H0∈L2Ω6 and g∈L2Σ3 with ν.g=0, then there exists a unique weak solution E,H∈C0,T,L2Ω6. Moreover, there exists a constant β0, such that
(17)e−βTE,HTΩ2+e−βTE,HtQ2+e−βTEτ,HτΣ2≤E0,H0Ω2+1βe−βTfQ2+e−βTgΣ2,for β≥β0.For the proof, see [2].


## 3. Shape Derivatives and Topological Asymptotic Expansion for Maxwell’s Equations

### 3.1. The Adjoint Method

To compute the shape derivative, we use the adjoint method. The fundamental propriety of the adjoint method is to provide the variation of a shape functionalJ with respect to a parameter using the solution uΩ and an adjoint state vΩ which do not depend on the chosen parameter. Numerically, it means that the two systems must be solved for obtaining an approximation of the shape derivative DJuΩ,∀x∈Ω. The principle is as follows. We consider the minimization problem
(18)minαjα=Jα,uα∈ℝn,with constraints Aαuα=Bα, α∈ℝp,uα∈ℝn,Aα∈Mn×nℝ,B∈ℝn.∀u,v, the Lagrangian of the system, is defined by
(19)Lα,uα,v=Jα,uα+Aαuαv−Bαv.Its derivative with respect toα is given by
(20)DLα,uα,vh=DαLα,uα,vh+DuLα,uα,vhDuα=DαJα,uα,vh+DuJα,uαhDuα+DαAαuαvh=AαDαuαvh−DαBαv.We setv=vα as the adjoint state to obtain
(21)Aαvαω=−DuJα,uα,∀ω∈ℝn,as an adjoint equation. It follows
(22)DLα,uα,vα=DαLα,uα,vα+DαAαvαuα−DαBαvα.


### 3.2. Shape Derivative of the Cost Function

In the sequel, we turn our attention to the minimization of the cost functional defined by(23)JΩ=12∫QEΩ−EdTεEΩ−ED+HΩ−HdTμHΩ−Hddxdtover a collection of open bounded sets Ω∈ℝ3, with fixed boundaries. EΩ,HΩ is the solution of the initial boundary value problem (11). The following result gives the shape derivative of (23) under constraints (11).Theorem 7.
Assume thatε,μ are constants of Hermitian definite positive matrices, let α be a positive constant, f=0,g=0 and E0,H0∈H1D6∩HD and EΩ,HΩ∈H1D×0,T∩HD×0,T. The shape functional (23) is Frechet differentiable at Ω in the direction V with the Frechet derivative
(24)DJΩ,V=R∫Σq.∂νEΩ×ν+α∂νHΩτ+CurlΓEΩν.q−αdivΓHΩν.qVνdxdt+12∫ΣEΩ−EdTεEΩ−ED+HΩ−HdTμHΩ−HdVνdxdt,where Vν=V0.ν,Eν=E.ν,Hν=H.ν, and divΓ are the surface divergences, CurlΓ is the surface curl, R. is the real part of a complex number, and p,q is the solution of the adjoint initial boundary value problem:
(25)ε∂tp−∇×q=0inQ,μ∂tq+∇×q=0inQ,pt=0=0qt=0=0;inΩ,ν×p+αqτ=0inΣ.
AsV, equation (23) is true for all V, and we have
(26)DJΩ=Rq.∂νEΩ×ν+α∂νHΩτ+CurlΓEΩν.q−αdivΓHΩνq+12EΩ−EdTεEΩ−ED+HΩ−HdTμHΩ−Hd.Sketch of the proof.
The Gâteaux derivation ofJΩ in the direction V is defined by
(27)DJΩ,V=lims⟶01sJΩs−JΩ=R∫QEΩ−EdTεEΩ′dtdΓ+R∫QHΩ−HdTμH′ΩdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt,where EΩ′,HΩ′ is the shape derivative of (15). Let p,q be the adjoint state, and then we have
(28)DJΩ,V=R∫Qε∂tp−∇qEΩ′dxdt+R∫Qμ∂tq−∇pHΩ′dxdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σν×pHΩ,τ′+q.ν×EΩ′dtdΓ.+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σq.Vν∂νEΩ,τ+EΩ,ν∇τVν×+αVν∂νHΩ,τ+HΩ,ν∇τVνdΓdt.12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.
To achieve the proof, we have to remove the derivativeVν, by integration by parts.
(29)DJΩ,V=R∫Σq.Vν∂νEΩ,τ+α∂νHΩ,τ+∇τVν.ν×EΩ,νq+αHΩ,νq∇τVνdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σq.∂νEΩ,τ+α∂νHΩ,τ−divΓν×EΩ,νq−αdivΓHΩ,νqVνdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σq.∂νEΩ,τ+α∂νHΩ,τ+CurlΓν×EΩ,νq−αdivΓHΩ,νqVνdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt,which finishes the proof.For the complete proof, we refer the reader to [8].


### 3.3. Generalized Adjoint Method

LetV be a complex Hilbert space. For all ε>0, let aε be a sesquilinear and continuous form on V and lε be a semilinear and continuous form on V, such that the following problem has one and only one solution:
(30)uε∈V,aεuε,v=lεv,∀v∈V.Hypothesis 1.
Assume that forε>0, there exists ρε>0, with limε⟶0+ρε=0,δa and δl two complex numbers such that
(31)uε−u0V=Oρε,aε−a0uε,v=ρεδa+Oρε,lε−l0uε,v=ρεδl+Oρε.LetJΩ,uε=JΩuε be the cost functional. We denote by jε=JΩuε,∀ε>0.Hypothesis 2.
For allu∈V, there exists a linear and continuous form, note Lu, such that
(32)JΩu+h=JΩu+δJΩu+RLuh+OhV.
. designs the norm on V.For allε>0, let vε the solution of the so-called adjoint problem
(33)vε∈V,aεvε,w=Lu0w,∀w∈V.u0 is the solution of (30) for ε=0. Under Hypotheses 1 and 2, we have the following result, which gives the asymptotic expansion for jε.Theorem 8.
If Hypotheses 1 and 2 are satisfied, the asymptotic expansion to the cost functionjε=JΩuε is given by
(34)jε=j0+ρεδj+Oρε,where δj=δa+δl+δJ.The proof of this theorem is standard in topological optimization, see, for example, [6, 9].The functionδjx0 is called topological derivative (or topological sensitivity) and provides an information for creating a small hole located at x0. Hence, the function δj can be used like a descent direction in the optimization process.Remark 9.
In the following, for all approximations, we use Laplace exterior problem(35)ΔEΩ=0inℝ3Ω,EΩx⟶0at+∞,∂EΩ∂ν=−EΩ00onΓ.The functionEΩ can be explicited by the help of a single layer potential
(36)Ex2+∫∂ω∇yUy−xEx.νydsy=ϕy,∀y∈∂ωandE∈H1/2∂Ω,with U being the fundamental solution of the Laplace operator, which is given in 3D by
(37)Ux=14πx.


### 3.4. Problem Formulation and Topological Asymptotic Expansion

The domain perturbation corresponds to the perturbation of electric permittivity and the magnetic permeability. The corresponding perturbed problem writes (for the electric field)(38)∂tEΩε−∇εε∇EΩε+μεEΩε=0inQ,(39)EΩεx,0=E0xinΩ,(40)∂EΩε∂ν=σonΣ,where
(41)εε=ε0inQ/Dεε1inDεand
(42)με=μ0inQ/Dεμ1inDε,Forε=0, the corresponding problem is to find EΩ0 such that
(43)∂tEΩ0−∇ε0∇EΩ0+μ0EΩ0=0inQ,(44)EΩ0x,0=E0xinΩ,(45)∂EΩ0∂ν=σonΣ.The variational problem assisted with (38) writes the following: find EΩε such that
(46)aεEΩε,F=lεF,where
(47)aεEΩε,F=∫Q∂EΩε∂tε,Fdxdt+∫Qεε∇.EΩε.R∇Fdxdt+∫QμεEΩεRFdxdt,(48)lεF=∫ΣσFdsdt.And the variational problem associated to (43) is to find EΩ0 such that
(49)aεEΩε,F=lεF,where
(50)a0EΩ,F=∫Q∂EΩ0∂t,Fdxdt+∫Qε0∇EΩ0.R∇Fdxdt+∫Qμ0EΩ0RFdxdt.Lemma 10.
Problems (47) and (50) admit one and only one solution. Moreover, there exists a function ρε>0 which goes to zeros with ε, such that
(51)EΩε−EΩ0V=Oρε.Proof.
For the proof, we refer the reader to [6].


#### 3.4.1. Variation of the Sesquilinear Form

Proposition 11.
LetEΩε (resp EΩ0) solution of (47) (resp. (50)), and then there exists a real number δa, and a function ρε>0 tending to zero as ε tends to zero such that
(52)aεEΩε,F−a0EΩ0=ρεδa+Oρε.Proof.
(53)aεEΩε,F−a0EΩ,F=∫Q∂EΩε∂tε,Fdxdt+∫Qεε∇.EΩε.R∇Fdxdt+∫QμεEΩεRFdxdt−∫Q∂EΩ0∂t,Fdxdt+∫Qε0∇.EΩ0.R∇Fdxdt+∫Qμ0EΩ0RFdxdt=∫Q∂EΩε∂t,Fdxdt+∫Qεε0∇.EΩε.R∇Fdxdt+∫Qεμ0EΩεRFdxdt+∫Dεε1∇.EΩε.R∇Fdxdt+∫Dεμ1EΩεRFdxdt−∫Q∂EΩ0∂t,Fdxdt+∫Qεε0∇.EΩ0.R∇Fdxdt+∫Qεμ0EΩ0RFdxdt+∫Dεε0∇.EΩ0.R∇Fdxdt+∫Dεμ0EΩ0RFdxdt=∫Q∂EΩε∂t−∂EΩ0∂t,Fdxdt+∫Qεε0∇.EΩε−EΩ0.R∇Fdxdt+∫Qεμ0EΩε−EΩ0RFdxdt+∫Dεε1∇.EΩεR∇Fdxdt+∫Dεμ1EΩεRFdxdt−∫Dεε0∇.EΩ0R∇Fdxdt+∫Dεμ0EΩ0RFdxdt.
LetXΩε=EΩε−EΩ0, and then XΩε is the solution of
(54)∂tXΩε−∇.εε∇XΩε+μεXΩε=0inQ,XΩεx,0=0inΩ,∂xΩε∂ν=0onΣ,aεEΩε,F−a0EΩ,F=∫Dεε0∇.XΩε.R∇Fdxdt+∫Dεε1−ε0∇.EΩ0R∇Fdxdt+∫Dεμ0XΩεRFdxdt+∫Dεμ1+μ0EΩ0RFdxdt.
The rest of the proof follows from the following lemmas.Lemma 12.
(55)∫Dεε0∇.XΩεR∇Fdxdt=ODε,∫Dεμ0XΩεRFdxdt=ODε.Proof.
For the proof, see [6] for the example.Lemma 13.
(56)∫Dεε1−ε0∇.EΩ0R∇Fdxdt=ε1−ε0∇.EΩ00R∇F0Dε,∫Dεμ1+μ0EΩ0RFdxdt=μ1−μ0EΩ00R∇F0Dε.Proof.
For the proof, see [6, 9] for the examples.Remark 14.
According to the definition oflε,
(57)δl=0.


#### 3.4.2. Variation of the Cost Function

We consider two examples of cost functions. Each one corresponds to a specific analysis.Example 1.
The cost functionJ defined by
(58)JΩEΩ=∫QεεEΩ−Ed2dxdt,εε=ε0inQDε,ε1inDε.Proposition 15.
LetJΩε be the functional associated to the perturbed problem, and then there exists a real δJ and a linear operator Lε such that
(59)JΩεEΩε−JΩ0EΩ0=ρεδJ+LεE+Oρε.Proof.
(60)JΩεEΩε−JΩ0EΩ0=∫QεεEΩε−Ed2dxdt−∫Qε0EΩ0−Ed2dxdt=∫Q\Dεε0EΩε−Ed2−EΩ0−Ed2+∫Dεε0−ε1EΩ0−Ed2,∫Q\Dεε0EΩε−Ed2−EΩ0−Ed2dxdt=∫Q\Dεε0EΩε−EΩ0EΩε+EΩ0−2Eddxdt=∫Q\Dεε0EΩε−EΩ02dxdt+2ε0∫Q\DεEΩε−EΩ0EΩ0−Eddxdt,∫Q\Dεε0EΩε−EΩ02dxdt≤ε0EΩε−EΩ02Q=Oρε,∫Dεε0−ε1EΩ0−Ed2dxdt=ε0−ε1EΩ00−Ed02Dε+Oρε.
We achieve the proof by settingLεE=2ε0∫Q\DεEEΩ0−Eddxdt.Example 2.
Here, we focus on the function(61)JΩEΩ=12∫Qαε∇EΩ−Ed2dxdt,αε=α0inQDε,α1inDε.Proposition 16.
LetJΩε be the functional associated to the perturbed problem, and then there exists a real δJ and a linear operator Lε, such that
(62)JΩεEΩε−JΩ0EΩ0=ρεδJ+LεE+Oρε.Proof.
(63)JΩεEΩε−JΩ0EΩε=12∫Qαε∇EΩε−Ed2dxdt−12∫Qα0∇EΩ0−Ed2dxdt=12∫Q\Dεα0∇EΩε−Ed2−∇EΩ0−Ed2dxdt⏟I+12∫Dεα1∇EΩε−Ed2dxdt⏟II−12∫Dεα0∇EΩ0−Ed2dxdt,(64)I=12∫Q\Dεα0∇EΩε−Ed2−∇EΩ0−Ed2dxdt=12∫Q\Dεα0∇EΩε−EΩ0∇EΩε+EΩ0−2Eddxdt,(65)I=12∫Q\Dεα0∇EΩε−EΩ0R∇EΩε−EΩ0+∫Q\Dεα0∇EΩε−EΩ0∇EΩ0−Ed.In settingXΩε=EΩε−EΩε, equation (63) writes
(66)I=12∫Q\Dεα0∇XΩεR∇XΩεdxdt+∫Q\Dεα0∇XΩε∇EΩ0−Eddxdt,II=12∫Dεα1∇EΩε−Ed2dxdt=12∫Dεα1∇EΩε−EΩ0+EΩ0−Hd2dxdt,=12∫Dεα1∇EΩε−EΩ02+12∫Dεα1∇EΩ0−Ed2+∫Dεα1∇EΩε−EΩ0∇EΩ0−Ed,=12∫Dεα1∇XΩε2dxdt+12∫Dεα1∇EΩ0−Ed2dxdt+∫Dεα1∇XΩε∇EΩ0−Eddxdt.Consequently,(67)JΩεHΩε−JΩ0EΩε=12∫Q\Dεα0∇XΩεR∇XΩεdxdt+12∫Dεα1∇XΩεR∇XΩεdxdt+∫Q\Dεα0∇XΩε∇EΩ0−Ed+∫Dεα1∇XΩε∇EΩ0−Ed−12∫Dεα0−α1∇EΩ0−Ed2dxdt.Setting(68)LεH=∫Qαε∇E.R∇EΩ0−EddxdtandρεδJ=12∫Dεα0−α1∇EΩ0−Ed2dxdt.As(69)12∫Q\Dεα0∇XΩεR∇XΩεdxdt+12∫Dεα1∇XΩεR∇XΩεdxdt=12∫Qαε∇XΩεR∇XΩεdxdt=Oρε,we obtain the desired result.The following result gives the asymptotic expansion of the cost function for the perturbation of the electric field.Theorem 17.
Letjε=JΩεEΩε be the cost functional defined by (58), where EΩε is the solution of (47) and EΩ0 is the corresponding solution for ε=0. Let Dε=εB0,1⊂ℝ3.j has the following asymptotic expansion:
(70)jε−j0=−43πε3μ0−μ1EΩ00RF0+ε0−ε1∇×EΩ00R∇F0+δJ+Oε3,where F is the solution of the so-called adjoint problem: find F∈V such that
(71)∂tF−∇.ε0∇F+μ0F=−LεEinQ,Fx,0=0inΩ,∂F∂ν=0onΣ.Proof.
The LagrangianL of the problem minVJΩEΩ0,F, EΩ0, solution of (43) is defined by
(72)LEΩ,F=JΩEΩ+aEΩ,F−lF.
Its variation with respect toε(73)LεEΩε,F=JΩεEΩε+aεEΩε,F−lεF.It follows form Propositions11 and 15 (or 16) that Hypotheses 1 and 2 are satisfied. We use the fact that the variation of the Lagrangian is equal to the variation of the cost function; that is,
(74)jε−j0=LεEΩε,F−LεEΩ0,F,and we use Theorem 8 to conclude.


## 3.1. The Adjoint Method

To compute the shape derivative, we use the adjoint method. The fundamental propriety of the adjoint method is to provide the variation of a shape functionalJ with respect to a parameter using the solution uΩ and an adjoint state vΩ which do not depend on the chosen parameter. Numerically, it means that the two systems must be solved for obtaining an approximation of the shape derivative DJuΩ,∀x∈Ω. The principle is as follows. We consider the minimization problem
(18)minαjα=Jα,uα∈ℝn,with constraints Aαuα=Bα, α∈ℝp,uα∈ℝn,Aα∈Mn×nℝ,B∈ℝn.∀u,v, the Lagrangian of the system, is defined by
(19)Lα,uα,v=Jα,uα+Aαuαv−Bαv.Its derivative with respect toα is given by
(20)DLα,uα,vh=DαLα,uα,vh+DuLα,uα,vhDuα=DαJα,uα,vh+DuJα,uαhDuα+DαAαuαvh=AαDαuαvh−DαBαv.We setv=vα as the adjoint state to obtain
(21)Aαvαω=−DuJα,uα,∀ω∈ℝn,as an adjoint equation. It follows
(22)DLα,uα,vα=DαLα,uα,vα+DαAαvαuα−DαBαvα.


## 3.2. Shape Derivative of the Cost Function

In the sequel, we turn our attention to the minimization of the cost functional defined by(23)JΩ=12∫QEΩ−EdTεEΩ−ED+HΩ−HdTμHΩ−Hddxdtover a collection of open bounded sets Ω∈ℝ3, with fixed boundaries. EΩ,HΩ is the solution of the initial boundary value problem (11). The following result gives the shape derivative of (23) under constraints (11).Theorem 7.
Assume thatε,μ are constants of Hermitian definite positive matrices, let α be a positive constant, f=0,g=0 and E0,H0∈H1D6∩HD and EΩ,HΩ∈H1D×0,T∩HD×0,T. The shape functional (23) is Frechet differentiable at Ω in the direction V with the Frechet derivative
(24)DJΩ,V=R∫Σq.∂νEΩ×ν+α∂νHΩτ+CurlΓEΩν.q−αdivΓHΩν.qVνdxdt+12∫ΣEΩ−EdTεEΩ−ED+HΩ−HdTμHΩ−HdVνdxdt,where Vν=V0.ν,Eν=E.ν,Hν=H.ν, and divΓ are the surface divergences, CurlΓ is the surface curl, R. is the real part of a complex number, and p,q is the solution of the adjoint initial boundary value problem:
(25)ε∂tp−∇×q=0inQ,μ∂tq+∇×q=0inQ,pt=0=0qt=0=0;inΩ,ν×p+αqτ=0inΣ.
AsV, equation (23) is true for all V, and we have
(26)DJΩ=Rq.∂νEΩ×ν+α∂νHΩτ+CurlΓEΩν.q−αdivΓHΩνq+12EΩ−EdTεEΩ−ED+HΩ−HdTμHΩ−Hd.Sketch of the proof.
The Gâteaux derivation ofJΩ in the direction V is defined by
(27)DJΩ,V=lims⟶01sJΩs−JΩ=R∫QEΩ−EdTεEΩ′dtdΓ+R∫QHΩ−HdTμH′ΩdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt,where EΩ′,HΩ′ is the shape derivative of (15). Let p,q be the adjoint state, and then we have
(28)DJΩ,V=R∫Qε∂tp−∇qEΩ′dxdt+R∫Qμ∂tq−∇pHΩ′dxdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σν×pHΩ,τ′+q.ν×EΩ′dtdΓ.+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σq.Vν∂νEΩ,τ+EΩ,ν∇τVν×+αVν∂νHΩ,τ+HΩ,ν∇τVνdΓdt.12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.
To achieve the proof, we have to remove the derivativeVν, by integration by parts.
(29)DJΩ,V=R∫Σq.Vν∂νEΩ,τ+α∂νHΩ,τ+∇τVν.ν×EΩ,νq+αHΩ,νq∇τVνdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σq.∂νEΩ,τ+α∂νHΩ,τ−divΓν×EΩ,νq−αdivΓHΩ,νqVνdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt.=R∫Σq.∂νEΩ,τ+α∂νHΩ,τ+CurlΓν×EΩ,νq−αdivΓHΩ,νqVνdΓdt+12∫ΣEΩ−EdTεEΩ−E0+HΩ−HdTμHΩ−H0VνdΓdt,which finishes the proof.For the complete proof, we refer the reader to [8].


## 3.3. Generalized Adjoint Method

LetV be a complex Hilbert space. For all ε>0, let aε be a sesquilinear and continuous form on V and lε be a semilinear and continuous form on V, such that the following problem has one and only one solution:
(30)uε∈V,aεuε,v=lεv,∀v∈V.Hypothesis 1.
Assume that forε>0, there exists ρε>0, with limε⟶0+ρε=0,δa and δl two complex numbers such that
(31)uε−u0V=Oρε,aε−a0uε,v=ρεδa+Oρε,lε−l0uε,v=ρεδl+Oρε.LetJΩ,uε=JΩuε be the cost functional. We denote by jε=JΩuε,∀ε>0.Hypothesis 2.
For allu∈V, there exists a linear and continuous form, note Lu, such that
(32)JΩu+h=JΩu+δJΩu+RLuh+OhV.
. designs the norm on V.For allε>0, let vε the solution of the so-called adjoint problem
(33)vε∈V,aεvε,w=Lu0w,∀w∈V.u0 is the solution of (30) for ε=0. Under Hypotheses 1 and 2, we have the following result, which gives the asymptotic expansion for jε.Theorem 8.
If Hypotheses 1 and 2 are satisfied, the asymptotic expansion to the cost functionjε=JΩuε is given by
(34)jε=j0+ρεδj+Oρε,where δj=δa+δl+δJ.The proof of this theorem is standard in topological optimization, see, for example, [6, 9].The functionδjx0 is called topological derivative (or topological sensitivity) and provides an information for creating a small hole located at x0. Hence, the function δj can be used like a descent direction in the optimization process.Remark 9.
In the following, for all approximations, we use Laplace exterior problem(35)ΔEΩ=0inℝ3Ω,EΩx⟶0at+∞,∂EΩ∂ν=−EΩ00onΓ.The functionEΩ can be explicited by the help of a single layer potential
(36)Ex2+∫∂ω∇yUy−xEx.νydsy=ϕy,∀y∈∂ωandE∈H1/2∂Ω,with U being the fundamental solution of the Laplace operator, which is given in 3D by
(37)Ux=14πx.


## 3.4. Problem Formulation and Topological Asymptotic Expansion

The domain perturbation corresponds to the perturbation of electric permittivity and the magnetic permeability. The corresponding perturbed problem writes (for the electric field)(38)∂tEΩε−∇εε∇EΩε+μεEΩε=0inQ,(39)EΩεx,0=E0xinΩ,(40)∂EΩε∂ν=σonΣ,where
(41)εε=ε0inQ/Dεε1inDεand
(42)με=μ0inQ/Dεμ1inDε,Forε=0, the corresponding problem is to find EΩ0 such that
(43)∂tEΩ0−∇ε0∇EΩ0+μ0EΩ0=0inQ,(44)EΩ0x,0=E0xinΩ,(45)∂EΩ0∂ν=σonΣ.The variational problem assisted with (38) writes the following: find EΩε such that
(46)aεEΩε,F=lεF,where
(47)aεEΩε,F=∫Q∂EΩε∂tε,Fdxdt+∫Qεε∇.EΩε.R∇Fdxdt+∫QμεEΩεRFdxdt,(48)lεF=∫ΣσFdsdt.And the variational problem associated to (43) is to find EΩ0 such that
(49)aεEΩε,F=lεF,where
(50)a0EΩ,F=∫Q∂EΩ0∂t,Fdxdt+∫Qε0∇EΩ0.R∇Fdxdt+∫Qμ0EΩ0RFdxdt.Lemma 10.
Problems (47) and (50) admit one and only one solution. Moreover, there exists a function ρε>0 which goes to zeros with ε, such that
(51)EΩε−EΩ0V=Oρε.Proof.
For the proof, we refer the reader to [6].


### 3.4.1. Variation of the Sesquilinear Form

Proposition 11.
LetEΩε (resp EΩ0) solution of (47) (resp. (50)), and then there exists a real number δa, and a function ρε>0 tending to zero as ε tends to zero such that
(52)aεEΩε,F−a0EΩ0=ρεδa+Oρε.Proof.
(53)aεEΩε,F−a0EΩ,F=∫Q∂EΩε∂tε,Fdxdt+∫Qεε∇.EΩε.R∇Fdxdt+∫QμεEΩεRFdxdt−∫Q∂EΩ0∂t,Fdxdt+∫Qε0∇.EΩ0.R∇Fdxdt+∫Qμ0EΩ0RFdxdt=∫Q∂EΩε∂t,Fdxdt+∫Qεε0∇.EΩε.R∇Fdxdt+∫Qεμ0EΩεRFdxdt+∫Dεε1∇.EΩε.R∇Fdxdt+∫Dεμ1EΩεRFdxdt−∫Q∂EΩ0∂t,Fdxdt+∫Qεε0∇.EΩ0.R∇Fdxdt+∫Qεμ0EΩ0RFdxdt+∫Dεε0∇.EΩ0.R∇Fdxdt+∫Dεμ0EΩ0RFdxdt=∫Q∂EΩε∂t−∂EΩ0∂t,Fdxdt+∫Qεε0∇.EΩε−EΩ0.R∇Fdxdt+∫Qεμ0EΩε−EΩ0RFdxdt+∫Dεε1∇.EΩεR∇Fdxdt+∫Dεμ1EΩεRFdxdt−∫Dεε0∇.EΩ0R∇Fdxdt+∫Dεμ0EΩ0RFdxdt.
LetXΩε=EΩε−EΩ0, and then XΩε is the solution of
(54)∂tXΩε−∇.εε∇XΩε+μεXΩε=0inQ,XΩεx,0=0inΩ,∂xΩε∂ν=0onΣ,aεEΩε,F−a0EΩ,F=∫Dεε0∇.XΩε.R∇Fdxdt+∫Dεε1−ε0∇.EΩ0R∇Fdxdt+∫Dεμ0XΩεRFdxdt+∫Dεμ1+μ0EΩ0RFdxdt.
The rest of the proof follows from the following lemmas.Lemma 12.
(55)∫Dεε0∇.XΩεR∇Fdxdt=ODε,∫Dεμ0XΩεRFdxdt=ODε.Proof.
For the proof, see [6] for the example.Lemma 13.
(56)∫Dεε1−ε0∇.EΩ0R∇Fdxdt=ε1−ε0∇.EΩ00R∇F0Dε,∫Dεμ1+μ0EΩ0RFdxdt=μ1−μ0EΩ00R∇F0Dε.Proof.
For the proof, see [6, 9] for the examples.Remark 14.
According to the definition oflε,
(57)δl=0.


### 3.4.2. Variation of the Cost Function

We consider two examples of cost functions. Each one corresponds to a specific analysis.Example 1.
The cost functionJ defined by
(58)JΩEΩ=∫QεεEΩ−Ed2dxdt,εε=ε0inQDε,ε1inDε.Proposition 15.
LetJΩε be the functional associated to the perturbed problem, and then there exists a real δJ and a linear operator Lε such that
(59)JΩεEΩε−JΩ0EΩ0=ρεδJ+LεE+Oρε.Proof.
(60)JΩεEΩε−JΩ0EΩ0=∫QεεEΩε−Ed2dxdt−∫Qε0EΩ0−Ed2dxdt=∫Q\Dεε0EΩε−Ed2−EΩ0−Ed2+∫Dεε0−ε1EΩ0−Ed2,∫Q\Dεε0EΩε−Ed2−EΩ0−Ed2dxdt=∫Q\Dεε0EΩε−EΩ0EΩε+EΩ0−2Eddxdt=∫Q\Dεε0EΩε−EΩ02dxdt+2ε0∫Q\DεEΩε−EΩ0EΩ0−Eddxdt,∫Q\Dεε0EΩε−EΩ02dxdt≤ε0EΩε−EΩ02Q=Oρε,∫Dεε0−ε1EΩ0−Ed2dxdt=ε0−ε1EΩ00−Ed02Dε+Oρε.
We achieve the proof by settingLεE=2ε0∫Q\DεEEΩ0−Eddxdt.Example 2.
Here, we focus on the function(61)JΩEΩ=12∫Qαε∇EΩ−Ed2dxdt,αε=α0inQDε,α1inDε.Proposition 16.
LetJΩε be the functional associated to the perturbed problem, and then there exists a real δJ and a linear operator Lε, such that
(62)JΩεEΩε−JΩ0EΩ0=ρεδJ+LεE+Oρε.Proof.
(63)JΩεEΩε−JΩ0EΩε=12∫Qαε∇EΩε−Ed2dxdt−12∫Qα0∇EΩ0−Ed2dxdt=12∫Q\Dεα0∇EΩε−Ed2−∇EΩ0−Ed2dxdt⏟I+12∫Dεα1∇EΩε−Ed2dxdt⏟II−12∫Dεα0∇EΩ0−Ed2dxdt,(64)I=12∫Q\Dεα0∇EΩε−Ed2−∇EΩ0−Ed2dxdt=12∫Q\Dεα0∇EΩε−EΩ0∇EΩε+EΩ0−2Eddxdt,(65)I=12∫Q\Dεα0∇EΩε−EΩ0R∇EΩε−EΩ0+∫Q\Dεα0∇EΩε−EΩ0∇EΩ0−Ed.In settingXΩε=EΩε−EΩε, equation (63) writes
(66)I=12∫Q\Dεα0∇XΩεR∇XΩεdxdt+∫Q\Dεα0∇XΩε∇EΩ0−Eddxdt,II=12∫Dεα1∇EΩε−Ed2dxdt=12∫Dεα1∇EΩε−EΩ0+EΩ0−Hd2dxdt,=12∫Dεα1∇EΩε−EΩ02+12∫Dεα1∇EΩ0−Ed2+∫Dεα1∇EΩε−EΩ0∇EΩ0−Ed,=12∫Dεα1∇XΩε2dxdt+12∫Dεα1∇EΩ0−Ed2dxdt+∫Dεα1∇XΩε∇EΩ0−Eddxdt.Consequently,(67)JΩεHΩε−JΩ0EΩε=12∫Q\Dεα0∇XΩεR∇XΩεdxdt+12∫Dεα1∇XΩεR∇XΩεdxdt+∫Q\Dεα0∇XΩε∇EΩ0−Ed+∫Dεα1∇XΩε∇EΩ0−Ed−12∫Dεα0−α1∇EΩ0−Ed2dxdt.Setting(68)LεH=∫Qαε∇E.R∇EΩ0−EddxdtandρεδJ=12∫Dεα0−α1∇EΩ0−Ed2dxdt.As(69)12∫Q\Dεα0∇XΩεR∇XΩεdxdt+12∫Dεα1∇XΩεR∇XΩεdxdt=12∫Qαε∇XΩεR∇XΩεdxdt=Oρε,we obtain the desired result.The following result gives the asymptotic expansion of the cost function for the perturbation of the electric field.Theorem 17.
Letjε=JΩεEΩε be the cost functional defined by (58), where EΩε is the solution of (47) and EΩ0 is the corresponding solution for ε=0. Let Dε=εB0,1⊂ℝ3.j has the following asymptotic expansion:
(70)jε−j0=−43πε3μ0−μ1EΩ00RF0+ε0−ε1∇×EΩ00R∇F0+δJ+Oε3,where F is the solution of the so-called adjoint problem: find F∈V such that
(71)∂tF−∇.ε0∇F+μ0F=−LεEinQ,Fx,0=0inΩ,∂F∂ν=0onΣ.Proof.
The LagrangianL of the problem minVJΩEΩ0,F, EΩ0, solution of (43) is defined by
(72)LEΩ,F=JΩEΩ+aEΩ,F−lF.
Its variation with respect toε(73)LεEΩε,F=JΩεEΩε+aεEΩε,F−lεF.It follows form Propositions11 and 15 (or 16) that Hypotheses 1 and 2 are satisfied. We use the fact that the variation of the Lagrangian is equal to the variation of the cost function; that is,
(74)jε−j0=LεEΩε,F−LεEΩ0,F,and we use Theorem 8 to conclude.


## 3.4.1. Variation of the Sesquilinear Form

Proposition 11.
LetEΩε (resp EΩ0) solution of (47) (resp. (50)), and then there exists a real number δa, and a function ρε>0 tending to zero as ε tends to zero such that
(52)aεEΩε,F−a0EΩ0=ρεδa+Oρε.Proof.
(53)aεEΩε,F−a0EΩ,F=∫Q∂EΩε∂tε,Fdxdt+∫Qεε∇.EΩε.R∇Fdxdt+∫QμεEΩεRFdxdt−∫Q∂EΩ0∂t,Fdxdt+∫Qε0∇.EΩ0.R∇Fdxdt+∫Qμ0EΩ0RFdxdt=∫Q∂EΩε∂t,Fdxdt+∫Qεε0∇.EΩε.R∇Fdxdt+∫Qεμ0EΩεRFdxdt+∫Dεε1∇.EΩε.R∇Fdxdt+∫Dεμ1EΩεRFdxdt−∫Q∂EΩ0∂t,Fdxdt+∫Qεε0∇.EΩ0.R∇Fdxdt+∫Qεμ0EΩ0RFdxdt+∫Dεε0∇.EΩ0.R∇Fdxdt+∫Dεμ0EΩ0RFdxdt=∫Q∂EΩε∂t−∂EΩ0∂t,Fdxdt+∫Qεε0∇.EΩε−EΩ0.R∇Fdxdt+∫Qεμ0EΩε−EΩ0RFdxdt+∫Dεε1∇.EΩεR∇Fdxdt+∫Dεμ1EΩεRFdxdt−∫Dεε0∇.EΩ0R∇Fdxdt+∫Dεμ0EΩ0RFdxdt.
LetXΩε=EΩε−EΩ0, and then XΩε is the solution of
(54)∂tXΩε−∇.εε∇XΩε+μεXΩε=0inQ,XΩεx,0=0inΩ,∂xΩε∂ν=0onΣ,aεEΩε,F−a0EΩ,F=∫Dεε0∇.XΩε.R∇Fdxdt+∫Dεε1−ε0∇.EΩ0R∇Fdxdt+∫Dεμ0XΩεRFdxdt+∫Dεμ1+μ0EΩ0RFdxdt.
The rest of the proof follows from the following lemmas.Lemma 12.
(55)∫Dεε0∇.XΩεR∇Fdxdt=ODε,∫Dεμ0XΩεRFdxdt=ODε.Proof.
For the proof, see [6] for the example.Lemma 13.
(56)∫Dεε1−ε0∇.EΩ0R∇Fdxdt=ε1−ε0∇.EΩ00R∇F0Dε,∫Dεμ1+μ0EΩ0RFdxdt=μ1−μ0EΩ00R∇F0Dε.Proof.
For the proof, see [6, 9] for the examples.Remark 14.
According to the definition oflε,
(57)δl=0.


## 3.4.2. Variation of the Cost Function

We consider two examples of cost functions. Each one corresponds to a specific analysis.Example 1.
The cost functionJ defined by
(58)JΩEΩ=∫QεεEΩ−Ed2dxdt,εε=ε0inQDε,ε1inDε.Proposition 15.
LetJΩε be the functional associated to the perturbed problem, and then there exists a real δJ and a linear operator Lε such that
(59)JΩεEΩε−JΩ0EΩ0=ρεδJ+LεE+Oρε.Proof.
(60)JΩεEΩε−JΩ0EΩ0=∫QεεEΩε−Ed2dxdt−∫Qε0EΩ0−Ed2dxdt=∫Q\Dεε0EΩε−Ed2−EΩ0−Ed2+∫Dεε0−ε1EΩ0−Ed2,∫Q\Dεε0EΩε−Ed2−EΩ0−Ed2dxdt=∫Q\Dεε0EΩε−EΩ0EΩε+EΩ0−2Eddxdt=∫Q\Dεε0EΩε−EΩ02dxdt+2ε0∫Q\DεEΩε−EΩ0EΩ0−Eddxdt,∫Q\Dεε0EΩε−EΩ02dxdt≤ε0EΩε−EΩ02Q=Oρε,∫Dεε0−ε1EΩ0−Ed2dxdt=ε0−ε1EΩ00−Ed02Dε+Oρε.
We achieve the proof by settingLεE=2ε0∫Q\DεEEΩ0−Eddxdt.Example 2.
Here, we focus on the function(61)JΩEΩ=12∫Qαε∇EΩ−Ed2dxdt,αε=α0inQDε,α1inDε.Proposition 16.
LetJΩε be the functional associated to the perturbed problem, and then there exists a real δJ and a linear operator Lε, such that
(62)JΩεEΩε−JΩ0EΩ0=ρεδJ+LεE+Oρε.Proof.
(63)JΩεEΩε−JΩ0EΩε=12∫Qαε∇EΩε−Ed2dxdt−12∫Qα0∇EΩ0−Ed2dxdt=12∫Q\Dεα0∇EΩε−Ed2−∇EΩ0−Ed2dxdt⏟I+12∫Dεα1∇EΩε−Ed2dxdt⏟II−12∫Dεα0∇EΩ0−Ed2dxdt,(64)I=12∫Q\Dεα0∇EΩε−Ed2−∇EΩ0−Ed2dxdt=12∫Q\Dεα0∇EΩε−EΩ0∇EΩε+EΩ0−2Eddxdt,(65)I=12∫Q\Dεα0∇EΩε−EΩ0R∇EΩε−EΩ0+∫Q\Dεα0∇EΩε−EΩ0∇EΩ0−Ed.In settingXΩε=EΩε−EΩε, equation (63) writes
(66)I=12∫Q\Dεα0∇XΩεR∇XΩεdxdt+∫Q\Dεα0∇XΩε∇EΩ0−Eddxdt,II=12∫Dεα1∇EΩε−Ed2dxdt=12∫Dεα1∇EΩε−EΩ0+EΩ0−Hd2dxdt,=12∫Dεα1∇EΩε−EΩ02+12∫Dεα1∇EΩ0−Ed2+∫Dεα1∇EΩε−EΩ0∇EΩ0−Ed,=12∫Dεα1∇XΩε2dxdt+12∫Dεα1∇EΩ0−Ed2dxdt+∫Dεα1∇XΩε∇EΩ0−Eddxdt.Consequently,(67)JΩεHΩε−JΩ0EΩε=12∫Q\Dεα0∇XΩεR∇XΩεdxdt+12∫Dεα1∇XΩεR∇XΩεdxdt+∫Q\Dεα0∇XΩε∇EΩ0−Ed+∫Dεα1∇XΩε∇EΩ0−Ed−12∫Dεα0−α1∇EΩ0−Ed2dxdt.Setting(68)LεH=∫Qαε∇E.R∇EΩ0−EddxdtandρεδJ=12∫Dεα0−α1∇EΩ0−Ed2dxdt.As(69)12∫Q\Dεα0∇XΩεR∇XΩεdxdt+12∫Dεα1∇XΩεR∇XΩεdxdt=12∫Qαε∇XΩεR∇XΩεdxdt=Oρε,we obtain the desired result.The following result gives the asymptotic expansion of the cost function for the perturbation of the electric field.Theorem 17.
Letjε=JΩεEΩε be the cost functional defined by (58), where EΩε is the solution of (47) and EΩ0 is the corresponding solution for ε=0. Let Dε=εB0,1⊂ℝ3.j has the following asymptotic expansion:
(70)jε−j0=−43πε3μ0−μ1EΩ00RF0+ε0−ε1∇×EΩ00R∇F0+δJ+Oε3,where F is the solution of the so-called adjoint problem: find F∈V such that
(71)∂tF−∇.ε0∇F+μ0F=−LεEinQ,Fx,0=0inΩ,∂F∂ν=0onΣ.Proof.
The LagrangianL of the problem minVJΩEΩ0,F, EΩ0, solution of (43) is defined by
(72)LEΩ,F=JΩEΩ+aEΩ,F−lF.
Its variation with respect toε(73)LεEΩε,F=JΩεEΩε+aεEΩε,F−lεF.It follows form Propositions11 and 15 (or 16) that Hypotheses 1 and 2 are satisfied. We use the fact that the variation of the Lagrangian is equal to the variation of the cost function; that is,
(74)jε−j0=LεEΩε,F−LεEΩ0,F,and we use Theorem 8 to conclude.


## 4. A Coupling Algorithm between Shape and Topological Derivative for Maxwell Equations

In this section, we propose an algorithm coupling the shape and topological derivative for a given shape functional under Maxwell equation as constraints. For the numerical approximation of Maxwell’s equation, one can use the finite differences method, the finite elements method, the Finite Element-Finite Difference Hybrid Methods, or the block pseudospectral method and XFEM method.The optimization algorithm is summarized in Figure1. (see also [10] in the case of linear elasticity).Figure 1
Proposed algorithm.Based on the boundary of the domain being an unknown of the problem, we introduceΩ~ fixed (in general rectangular or parallelepiped) domain which includes all potential domains Ω. The numerical method approximation requires the introduction of two finite element spaces V~h⊂H1Ω~;ℝn and W~h⊂L2Ω~;ℝn on the fictitious domain Ω~. As Ω~ can be a rectangular or parallelepiped domain, the ones can be defined on the same structured mesh Th. Next, we shall suppose that
(75)V~h=vh∈CC~¯,ℝnvΩh∈PTn∀T∈Th,where PT is a finite dimensional space of regular functions such that PT⊇PkT for some k≥1 integer. The mesh parameter h stands for h=maxT∈ThhT where hT is the diameter of T.Since we use the topological gradient to create holes (inhomogeneities) during the optimization process, it is possible to start with a shape containing some initial holes or not. A very small penalization is used when solving the direct problem and the adjoint one to avoid the indeterminacy of the rigid motions of the eventual isolated part. Concerning step 4, a new hole of a given radius is created by the simple operation on the level-set function, which can be written on each finite element nodexi(76)ψ¯xi≔maxψxi,r2−xi−c22r,where ψx is the level-set function, ψ¯xi is its new value, r is the radius of the created hole, and c is its center.In step 6, the update of the level set is done directly thanks to the shape derivative applying the following evolution equation for the level-set function:(77)∂ψ∂t=gx,inΩ~,where gx corresponds to the function in front of θ.ν in the integral of (24). This evolution equation integrated on a small time interval. In our simulations, the gradient is extended by zero to the complementary of Ω in Ω~. However, a smoother extension could be considered. This method is simpler than the classical way which is to integrate a Hamilton-Jacobi equation (see [4]). It seems also to be numerically more robust.Note that it is convenient to apply a threshold on the gradient to avoid some incoherent values where the shape gradient may have a singularity (corners, transition from Dirichlet to Neumann condition).To regularize the level-set function, the reinitialization step 7 is considered. It consists classically in solving(78)∂ψ∂t+signψ0∇ψ−1=0inΩ~×ℝ+,ψ0,x=ψ0xinΩ~,whose stationary solution is a signed distance. This Hamilton-Jacobi equation is known to admit multiple nonsmooth solutions. Classically, a smooth solution is computed thanks to an upwind scheme. Since the fictitious domain Ω~ can be a rectangular/parallelepiped domain, it is possible to use a classical upwind scheme on a Cartesian grid. However, to keep the possibility of having a nonstructured mesh, for instance, to proceed to a local refinement, we use a different strategy. Equation (78) is solved on a small time interval 0,Δt integrating the following equation where the nonlinearity is made explicit:
(79)∂ψ¯∂t+signψ0∇ψn∇ψn∇ψ¯=signψ0inΩ~×0,Δt,ψ¯0,x=ψnx.Hereψn is the level-set function at the previous time step, and ψn+1 is given by ψ¯Δt,·. The problem (79) is a pure convection one. This problem can be solved, for instance, with the simple Galerkin-Characteristic scheme proposed in [11]. This scheme is unconditionally stable but rather dissipative. The effect is that the level sets are a little smoothed.


## 5. Conclusion and Extensions

In this paper, we used the adjoint method (respectively, generalized adjoint method) to compute shape derivatives (respectively, topological asymptotic expansions) associated with a given shape functional and Maxwell’s equations. The obtained derivatives (shape and topological) allow us to construct an algorithm, which can permit simultaneously to insert a small dielectric objects (holes) with topological derivative and to control its boundary by using the shape derivative and level set method.In the forthcoming work, we will intent to apply this algorithm for some applications, for example, for the reconstruction of metallic buried objects.



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*Source: 2425990-2022-11-18.xml*