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Optimal control for electromagnetic cloaking metamaterial parameters design
Computers and Mathematics with Applications xxx (xxxx) xxx
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Optimal control for electromagnetic cloaking metamaterial parameters design✩ ∗
Zhiwei Fang a , Jichun Li a , , Xiang Wang b a b
Department of Mathematical Sciences, University of Nevada Las Vegas, NV 89154-4020, USA Department of Mathematical Sciences, Nanchang University, Nanchang, China
article
info
Article history: Received 2 December 2018 Received in revised form 2 May 2019 Accepted 24 August 2019 Available online xxxx Keywords: Maxwell’s equations Invisibility cloak Optimal control method Discontinuous Galerkin method
a b s t r a c t In this paper, we develop an optimal control problem to optimize the permittivity and permeability of the metamaterial which can have the invisible cloaking effect. The inverse problem is calibrated as an optimization problem constrained by the time-harmonic Maxwell’s equations. Proper objective functional is introduced with coefficients of the Maxwell’s equations as control variables. We adopt the discontinuous Galerkin method to solve the state equation. Several two-dimensional cloaking benchmark problems are used to test our optimal control method. Numerical results demonstrate that the optimized permittivity and permeability of the metamaterial can achieve the cloaking effectively. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In 2006, Pendry, Schuring and Smith [1] and Leonhardt [2] independently proposed some theoretical ideas of designing invisibility cloaks with negative-index metamaterials. Since then, the study of electromagnetic cloaks has attracted great attentions due to its many potential applications in both military and civic life. In the past decade, mathematicians, physicists and engineers made great efforts in investigating various ways in designing invisibility cloaks and other applications (e.g., [3–13]). The main idea of Pendry’s cloaking design is to use the so-called transformation optics to calculate the permittivity and permeability of the metamaterial. But this technique is impractical for general geometries and also the calculated permittivity and permeability are difficult to be manufactured in practice. Hence searching for the proper permittivity and permeability parameters with cloaking property by robust mathematical methods has very important role in invisibility cloak design. In this paper, we initiate our effort in designing the permittivity and permeability of the metamaterial through solving an optimal control problem with control as the coefficients of time-harmonic Maxwell’s equations. The challenge is to come up with a proper objective functional to be minimized: for any given incident/detecting wave, we aim at estimating the permittivity and permeability in the cloaking region such that the incident wave keeps invariant outside the cloaking region. Due to the challenge of cloaking design, we limit to the 2D modeling in this paper since this is our first effort in this direction. ✩ Work partially supported by NNSF of China with Nos. 11671340 and 11961048, NSF of Jiangxi Province with No.20181ACB20001 and No.20161ACB21005. ∗ Corresponding author. E-mail addresses: [email protected] (Z. Fang), [email protected] (J. Li), [email protected] (X. Wang). https://doi.org/10.1016/j.camwa.2019.08.023 0898-1221/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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In literature, there are many existing excellent works devoted to optimal control problems constrained by partial differential equations (PDEs), such as optimal control of elliptic problems [14–18], optimal control of a free boundary problem [19], optimal control of parabolic problems [20,21], optimal control of the scalar wave equation [22], and optimal control of magnetohydrodynamics (MHD) [23,24] etc. More details on optimal controls of PDEs can be found in monographs [25–30] and references therein. As for optimal control of Maxwell’s equations, there are many excellent results for the optimal control of parabolic eddy current equations (e.g., [31,32]), optimal control of linear time-harmonic eddy current equations (e.g., [33–36]), and optimal control of nonlinear Maxwell’s equations (e.g., [37,38]). But most papers are for control of source terms, the numerical analysis of optimal control problems in the coefficients of time-harmonic eddy current equations was first investigated by Yousept [35, p. 880]. To the best of our knowledge, this paper is the first attempt at solving the optimal control problem for coefficients of time-harmonic Maxwell’s equations with applications to invisibility cloak design. In recent years, there are some publications devoted to the theoretical analysis of the cloaking problem by using the optimization method, e.g., [39] considered the cloaking problem governed by the 2D Helmholtz equation with impedance boundary condition; [40] studied an approximate cloaking problem for the 3D time-harmonic Maxwell equations; [41] analyzed the 2D magnetic scattering by a bilayer shell. Due to some nice features of the discontinuous Galerkin (DG) methods (such as flexibility in h–p adaptivity, and efficiency in local solvability and parallel implementation), in recent years, DG methods have been widely used to solve various PDEs, including Maxwell’s equations in both time domain [42–44] and frequency domain [45–47]. More details on DG methods and applications can be found in books [48–50] and references therein. Of course, other popular numerical methods such as FDTD methods [51–53] and spectral methods [13] can also be considered here. In this paper, we will adopt the DG method for solving the state equation, i.e., the time-harmonic Maxwell’s equations. The rest of the paper is organized as follows. In Section 2, we introduce the 2D time-harmonic Maxwell’s equations, which is the state equation or the PDE constrain in our optimal control problem. In Section 3, we first formulate our optimal control problem for metamaterial cloaking design, then derive the first order optimality conditions through the Lagrange multiplier methodology. Finally, we describe the DG method for solving the state equation. Numerical results demonstrating the effectiveness of our optimal control method for invisibility cloak design are presented in Section 4. We conclude the paper in Section 5. 2. Formulation of the optimal control problem for cloaking In order to design the two dimensional (2D) invisibility cloaks with metamaterials, we have to solve the Maxwell’s equations, which in the frequency domain are given as follows:
∇ × E(x) + jωµH (x) = 0, ∇ × H(x) − jωϵ E(x) = J (x) in Ω , (1) √ where j = −1, x = (x, y)⊤ , E(x) = (Ex (x), Ey (x))⊤ and H(x) are the electric and magnetic fields in the frequency domain, J (x) is the applied current density, ω is the general wave frequency, and the 2 × 2 tensor ϵ (x) and the scalar µ(x) are the permittivity and permeability of the underlying material, respectively. Furthermore, here we adopt the 2D curl operators:
∂ Ex ∂ Ey − , ∇ ×E = ∂x ∂y
( and
∇ ×H =
∂H ∂H ,− ∂y ∂x
)⊤
.
Finally, we assume that Ω is a bounded domain contained in R2 with boundary ∂ Ω and unit outward normal vector n. Eliminating H from (1), we obtain the so-called time-harmonic Maxwell’s equations [54–56]:
∇ × (µ−1 ∇ × E) − ω2 ϵ E = −jωJ
in Ω .
(2)
To make the problem complete, we usually impose (2) with the perfectly conducting (PEC) boundary condition n×E =0
on ∂ Ω .
(3)
To design an invisibility cloak device, we consider solving the Maxwell’s equations with heterogeneous media in
Ω = Ω0 ∪ Ω1 ∪ Ω2 as shown in Fig. 1. Here Ω0 is the cloaked region where any objects to be hided are put, Ω1 is the cloaking region formed by a metamaterial with special permittivity and permeability to be found by the optimal control method, and Ω2 is the vacuum region. Moreover, Ωi , i = 1, 2, 3, are mutually disjoint. For simplicity, we define Ω ′ = Ω1 ∪ Ω2 . Our goal is to find the proper permittivity and permeability of the metamaterial in Ω1 such that any objects located inside Ω0 is cloaked in the sense that any given incident or detecting wave Ew (x) in Ω2 shall bypass Ω1 region without any change. By introducing the relative permittivity ϵr := ϵ/ϵ0 and relative permeability µr := µ/µ0 of the underlying material, we rewrite the governing equation (2) as follows: 1 2 ∇ × (µ− r ∇ × E) − k0 ϵr E = f
in Ω ′
(4)
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 1. The domain setup for the optimal control problem.
subject to the PEC boundary condition on ∂ Ω0 : n × E = 0,
(5)
√ where k0 = ω ϵ0 µ0 := ω/c0 denotes the wave number, ϵ0 and µ0 are the permittivity and permeability in vacuum, ω is the general wave frequency, and c0 is the speed of light in free space. Since our goal is to keep Ew invariant in Ω2 , we impose the following boundary condition on the most outside boundary ∂ Ω of Ω : on ∂ Ω .
n × E = n × Ew
(6)
Note that ϵr = µr = 1 in vacuum, which leads to the source term f being given as follows:
{ f =
0
in Ω1
∇ × ∇ × Ew − k20 Ew
in Ω2 .
To formulate the optimal control problem, we define
ϵ (x) ϵr (x) = 1 ϵ2 (x)
ϵ2 (x) ϵ3 (x)
[
]
in Ω1 ,
and identity matrix in Ω2 . Similarly, we set µr (x) = µ1 (x) in Ω1 and 1 in Ω2 . Then the optimal control problem for cloaking reads: 1
min −1
ϵ1 ,ϵ2 ,ϵ3 ,µ1 ∈L2 (Ω1 )
2
∫ Ω2
|E − Ew |2 dx +
1
∫
2
Ω2
|∇ × (E − Ew )|2 dx +
∫ 3 ∑ β1 i=1
2
Ω1
|ϵi |2 dx +
β2 2
∫ Ω1
2
1 |µ− 1 | dx
(7)
subject to
⎧ 2 −1 ⎨∇ × (µr ∇ × E) − k0 ϵr E = f n×E =0 on ∂ Ω0 ⎩ n × E = n × Ew on ∂ Ω ,
in Ω ′ (8)
where β1 , β2 > 0 are regularization parameters. Namely, we are going to find ϵ1 , ϵ2 , ϵ3 and µ1 by minimizing the cost functional defined above subject to (8). 1 Note that since the µ1 appears in the problem as µ1−1 and is nonzero, we treat the unknown function µ− directly. 1 In the definition of state equation (forward problem) (4), the permittivity and permeability to be designed are ϵr and µr , respectively. Since they are identity matrix and constant 1 in Ω2 respectively, we only consider their components in Ω1 , 1 i.e., ϵ1 , ϵ2 , ϵ3 and µ− 1 , as unknowns. The weak formulation of the state equation (8) reads: find E ∈ HE (curl; Ω ′ ) such that 1 2 (µ− r ∇ × E , ∇ × v) − k0 (ϵr E , v) = (f , v)
∀v ∈ H0 (curl; Ω ′ )
(9)
where the Sobolev space HE (curl; Ω ) is defined as: ′
HE (curl; Ω ′ ) = E ∈ H(curl; Ω ′ ) : n × E = 0 on ∂ Ω0 , n × (E − Ew ) = 0 on ∂ Ω ,
{
}
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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and is equipped with norm
( ) 12 ∥v ∥H(curl;Ω ′ ) = ∥v ∥2(L2 (Ω ′ ))2 + ∥∇ × v ∥2(L2 (Ω ′ ))2 , { } where H(curl; Ω ′ ) := v ∈ (L2 (Ω ′ ))2 : ∇ × v ∈ L2 (Ω ′ ) . To accommodate the PEC boundary condition, we denote the Sobolev space H0 (curl; Ω ′ ) = v ∈ H(curl; Ω ′ ) : n × v = 0 on ∂ Ω ′ .
}
{
Moreover, the notation (·, ·) denotes the usual inner product on L2 (Ω ′ ). 3. Finite element method for the optimal control problem 3.1. The first order optimality conditions Following optimal control theory [29], we can convert the constrained problem (7)–(8) to an unconstrained one: 1 L((ϵ1 , ϵ2 , ϵ3 , µ− 1 ), E , p),
min
1 2 ∈L (Ω1 ) ϵ1 ,ϵ2 ,ϵ3 ,µ− 1
(10)
where the Lagrangian functional L : (L2 (Ω1 ))4 × H(curl, Ω ′ ) × H(curl, Ω ′ ) ↦ → R is given as follows: L
((
∫
) ) 1 1 ϵ1 , ϵ2 , ϵ3 , µ− , E, p = 1 +
∫ 3 ∑ β1 i=1
2
Ω1
|ϵi |2 dx +
2
Ω2
β2
∫
2
|E − Ew |2 dx +
1
∫
2
Ω2
|∇ × (E − Ew )|2 dx (11)
2
Ω1
1 −1 2 |µ− 1 | dx + (µr ∇ × E , ∇ × p) − k0 (ϵr E , p) − (f , p).
We derive the first order optimality conditions by the Lagrange multiplier methodology. It is not difficult to see that the Gaˆ teaux derivative of L with respective to (w.r.t) E in direction ˜ E ∈ H0 (curl; Ω ′ ) is: LE
((
ϵ1 , ϵ2 , ϵ3 , µ1
−1
)
)( ) , E, p ˜ E =
∫
∫ (E − Ew ) · ˜ Edx +
∇ × (E − Ew ) · ∇ × ˜ Edx ) ( ) + µr ∇ × ˜ E , ∇ × p − k20 ϵr˜ E, p . Ω ( 2−1
Ω2
So, the adjoint problem reads: find p ∈ H0 (curl; Ω ′ ) such that LE
((
) )( ) 1 , E, p ˜ E =0 ϵ1 , ϵ2 , ϵ3 , µ− 1
∀˜ E ∈ H0 (curl; Ω ′ )
Similarly, for i = 1, 2, 3, we can derive the Gaˆ teaux derivative of L with respective to ϵi in direction ˜ ϵi ∈ L2 (Ω1 ) as: Lϵ i
((
) ) 1 ϵ1 , ϵ2 , ϵ3 , µ− , E , p (˜ ϵ i ) = β1 1
∫ Ω1
∫
ϵi˜ ϵi dx − k20
Ω1
ϵt ,i E · pdx,
where the matrices ϵt ,i are given as follows:
ϵt ,1 =
[ ˜ ϵ1 0
]
0 , 0
ϵt ,2 =
[
0
˜ ϵ2
˜ ϵ2
]
0
,
ϵt ,3 =
[
0 0
0
]
˜ ϵ3
.
˜ 1 −1 2 Finally, we have the Gaˆ teaux derivative of L with respective to µ− 1 in direction µ1 ∈ L (Ω1 ): Lµ−1
((
1
) ) ) (˜ 1 −1 = β2 ϵ1 , ϵ2 , ϵ3 , µ− , E , p µ 1 1
∫ Ω1
1˜ −1 µ− 1 µ1 dx +
∫ Ω1
˜ −1 µ 1 ∇ × E · ∇ × pdx.
1 Hence, the gradients of the loss functional L with respect to the unknowns µ− 1 and ϵi , i = 1, 2, 3, are given by
) ) 1 , E , p (˜ ϵi ) , ϵ1 , ϵ2 , ϵ3 , µ− 1 ) (( ) ) (˜ −1 1 Lµ−1 ϵ1 , ϵ2 , ϵ3 , µ1 , E , p µ− , 1 Lϵi
((
1
∀˜ ϵi ∈ L2 (Ω1 ), i = 1, 2, 3, ˜ −1 ∀µ ∈ L2 (Ω ). 1
1
3.2. The discontinuous Galerkin method for the optimality conditions To solve the state equation (9), we adopt the discontinuous Galerkin method. We consider a shape-regular affine mesh Th that partitions the domain Ω ′ into triangles {K }, where hK is the diameter of element K ∈ Th and h = maxK ∈Th hK is the mesh size of Th . We denote by FhI the set of all element interior faces of Th , by FhB the set of all boundary faces, and denote Fh = FhI ∪ FhB . For a piecewise smooth vector-valued function v, we introduce the following trace operators. Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 2. Cylindrical cloak: The mesh used and the exact solution.
Let f ∈ FhI be an interior face shared by two neighboring elements K + and K − with unit outward normal vectors n± , respectively. Denoting by v ± the traces of v taken from within K ± , respectively. We also define the tangential jump and average across face f by
[[v ]] = n+ × v + + n− × v − ,
{{v }} =
v+ + v− 2
,
(12)
respectively. On a boundary face f ∈ FhB , we set [[v ]] = n × v and {{v }} = v. For the given partition Th of Ω ′ and any integer l ≥ 0, we choose the following discontinuous finite element space Vh = vh ∈ (L2 (Ω ′ ))2 : vh |K ∈ (P l (K ))2 , ∀K ∈ Th ,
{
}
where P l (K ) denotes the space of polynomials of total degree at most l on K . Now we can define the DG scheme for solving the state equation (9): finding Eh ∈ Vh such that 1 2 ah (Eh , vh ; µ− r ) − k0 (ϵr Eh , vh ) = F (vh )
∀vh ∈ Vh ,
(13)
where 1 −1 ah (Eh , vh ; µ− r ) := (µr ∇h × Eh , ∇h × vh ) −
∫
1 [[Eh ]] · {{µ− r ∇h × vh }}ds ∫ ∫ α −1 − [[vh ]] · {{µr ∇h × Eh }}ds + [[Eh ]] · [[vh ]]ds
Fh
Fh
Fh
hf
and F (vh ) := (f , vh ) −
∫
α
∫
∂Ω
n × Ew · ∇h × vh ds +
∂Ω
hf
n × Ew · n × vh ds,
where hf denotes the diameter of face f and is a function of x, ∇h × denotes the element-wise application of curl operator ∇×, the penalty parameter α∫ > 0 is a constant independent of mesh size and wave number, and the integral on Fh is ∫ ∑ defined as F ϕ ds = ϕ ds. f ∈F f h
h
To formulate the DG discretization of adjoint and optimality conditions, we first introduce the following bilinear form bh (uh , vh ; D) =
∫
∫ ∇h × uh · ∇h × vh dx −
∫D
[[uh ]] · {{∇h × vh }}ds Fh ∩D
∫ [[vh ]] · {{∇h × uh }}ds +
− Fh ∩D
Fh ∩D
α hf
[[uh ]] · [[vh ]]ds,
where D ⊂ Ω ′ . Thus, the DG scheme for the adjoint problem can be written as: find ph ∈ Vh , such that for ∀˜ Eh ∈ Vh ,
∫ Ω2
1 2 ˜ (Eh − Ew ) · ˜ Eh dx + bh (Eh − Ew , ˜ Eh ; Ω2 ) + ah (˜ Eh , ph ; µ− r ) − k0 (ϵr Eh , ph ) = 0,
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 3. Cylindrical cloak: The permeability and permittivity parameters obtained from optimal control.
holds true. The Lϵi , i = 1, 2, 3, and Lµ−1 can be approximated by 1
Lϵ i , h
Lµ−1 ,h 1
(( ) )( ) 1 ϵ1,h , ϵ2,h , ϵ3,h , µ− ϵi,h = β1 1,h , Eh , ph ˜
((
∫ Ω1
ϵi,h˜ ϵi,h dx − k20
∫ Ω1
ϵt ,i,h Eh · ph dx,
∀ϵi,h ∈ Uh , i = 1, 2, 3, ∫ ( ) ( ) ) ) ˜ ˜ 1 −1 −1 ˜ −1 −1 ϵ1,h , ϵ2,h , ϵ3,h , µ− , E , p µ = β µ µ dx + a E , v ; µ h h 2 h h h 1,h 1,h 1,h 1,h 1 ,h , Ω1
˜ −1 ∀µ 1,h ∈ Uh , Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 4. Cylindrical cloak: the field ∇h × Eh obtained after 500 iterations without the curl term in the objective functional.
Fig. 5. The physical domain for the carpet cloaking.
Fig. 6. Carpet cloak: The mesh and the exact solution.
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 7. Numerical results of the optimal control for carpet cloaking.
respectively. Here Uh denotes the DG finite element space for scalar functions: Uh = {vh ∈ L2 (Ω1 ) : vh |K ∈ P l (K ), ∀K ∈ Th }, and the definition of ϵt ,i,h is the same as ϵt ,i but replacing ˜ ϵi by ˜ ϵi,h . Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 8. The mesh for the oval cloaking.
4. Numerical results In this section, we present three numerical examples to show the effectiveness of finding the optimized material coefficients for the invisibility cloak design by solving the optimal control problem (7)–(8). All the numerical results in this section are obtained by solving the optimization problem with the limited memory BFGS (L-BFGS) method [29] implemented under the FEniCS [57] adjoint package. The stopping rule used in the optimization algorithm is |L| ≤ 10−2 , where L is the loss functional defined in (11). 4.1. Example 1: Cylindrical cloak In this subsection, we solve a cylindrical cloaking problem originally proposed by Pendry et al. [1], and later simulated by Li et al. [58,59]. We set the domain Ω = [−2, 2]2 , and put any cloaked objects inside a cylinder centered at the original with radius R1 = 0.3 and perfectly conducting boundary condition so that no wave can enter into the cloaked region. The cloaked cylinder is wrapped by a cylindrical metamaterial region with thickness R2 − R1 , where R2 = 0.6. Hence in this simulation, Ω0 = {x : ∥x∥R2 ≤ 0.3}, Ω1 = {x : 0.3 < ∥x∥R2 ≤ 0.6} and Ω2 = {x ∈ Ω : ∥x∥R2 > 0.6}, where ∥ · ∥R2 denotes the Euclidean norm in R2 . In this example, we apply an initial triangulation on Ω ′ with 14 400 elements, then refine the mesh on Ω1 , as shown in Fig. 2(a). We use the first order discontinuous basis function to approximate E, and piecewise constant to approximate the control variables ϵ1 , ϵ2 , ϵ3 and µ. We choose the regularization parameters β1 = β2 = 10−8 in the Lagrangian functional and the penalty parameter α = 15 in the DG method. The wave source for this simulation is Ew = (0, − cos(k0 x)) with 1 k0 = 11. The initial guesses for all ϵi and µ− 1 are ∥x∥R2 . The optimal parameters recovered from solving this optimal control problem are shown in Fig. 3. Since the solution Eh is a vector, in Fig. 3 we present the solution ∇h × Eh , which is actually a multiple of H due to (1). For comparison, we also show the analytical result in Fig. 2(b) obtained with the perfect permeability and permittivity parameters (see [1] and [60]). We like to point out that the curl term in the objective functional (7) is necessary from our numerical tests. Solving the same optimization problem without the curl term for 500 iterations, we only produced unsatisfactory cloaking as shown in Fig. 4. 4.2. Example 2: Carpet cloak In this subsection, we consider the carpet cloaking model, which was first designed by Chen et al. [61] and was later simulated by Li et al. [62] using a finite element time domain solver. The model setup is shown in Fig. 5, where the subdomain Ω0 is a triangle formed by vertices (−0.5, 0), (0, 0.5) and (0.5, 0), while the subdomain Ω1 is a quadrilateral formed by vertices (−0.5, 0), (0, 0.5), (0.5, 0) and (0, 1). The computational domain is Ω = [−1, 1] × [0, 2] and Ω2 = Ω \ (Ω0 ∪ Ω1 ). In the simulation, we apply an initial triangulation of Ω ′ with 14 400 elements, then refine the partition of Ω1 one 1 more time. The initial guesses for this problem are ϵ1 = ϵ3 = µ− 1 = 1 and ϵ2 = −1 if x1 ≥ 0 while ϵ2 = 1 if x1 < 0. The resulting mesh used for the simulation is shown in Fig. 6(a). We used the same finite element spaces, regularization and penalty parameters and wave source as the last example. Instead of just assuming that the permeability and permittivity being in L2 (Ω1 ), we want them to be piecewise constants since the original carpet cloak design has piecewise Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 9. Numerical results of the optimal control for oval cloaking.
constant permittivity and constant permeability (see [61,62]). To this end, we add two more penalty terms in the original object functional (7): min
1
∫
1 2 ϵi ,µ− ∈L (Ω1 ) 2 1
+
Ω2
|E − Ew |2 dx +
∫ 3 ∑ γ1 i=1
2
∂S
1
∫
2
[[ϵi ]]2 ds +
Ω2
γ2 2
|∇ × (E − Ew )|2 dx +
∫ 3 ∑ β1 i=1
∫ ∂S
2
Ω1
|ϵi |2 dx +
β2 2
∫
2
Ω1
1 |µ− 1 | dx
1 2 [[µ− 1 ]] ds,
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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and also in (11): 3 (( ) ) (( ) ) ∑ γ1 1 1 L˜ ϵ1 , ϵ2 , ϵ3 , µ− , E , p := L ϵ1 , ϵ2 , ϵ3 , µ− , E, p + 1 1 i=1
2
∫
2
∂S
[[ϵi ]] ds +
γ2 2
∫ ∂S
1 2 [[µ− 1 ]] ds,
where ∂ S denotes the set of element edges in = Ω1 ∩ {x : |x| > 0.2h}, where h is the maximal mesh size, and [[·]] denotes the jump across the element boundaries defined in (12). In our simulation, we choose the penalty parameters γ1 = γ2 = 105 . The numerical results of inverse problem are shown in Fig. 7. As a comparison, we also present the perfect cloaking obtained with the exact permeability and permittivity in Fig. 6(b). The obtained numerical results are presented in Fig. 7. Fig. 7(b) shows that the difference between the numerical field and the source wave ∇h × (Eh − Ew ) is almost zero out of the cloaking region, which justifies the excellent cloaking effect. In Fig. 7(c), we present the obtained optimal parameter ϵ1 ranging from the smallest value 2.00617032 to the largest value 2.00996683 in the cloaking region, which is very accurate compared to the exact value ϵ1 = 2 (cf. [62, p. 1137]). The calculated optimal parameters ϵ2 , ϵ3 and µ−1 and their minimal and maximal values are presented in Fig. 7(d), (e) and (f), respectively. The results show that the optimal cloaking parameters obtained from the optimal control method are all very accurate to the exact solution, which justifies the effectiveness of our optimal control method. 4.3. Example 3: Oval cloak In this subsection, we carry out a new cloaking experiment to test our optimal control method. Here we consider the physical domain Ω = [−2, 2]2 , the cloaked region is a disc Ω0 = {x : ∥x∥R2 ≤ 0.3}, the cloaking region is an ellipse Ω1 = {x ∈ Ω \ Ω0 : 0.5x2 + y2 ≤ 0.6} and the vacuum region Ω2 = Ω \ (Ω1 ∪ Ω2 ). Unlike the previous two examples, there is no exact formula for the permittivity and permeability for this cloaking. For this example, we use an initial triangulation of Ω ′ with 14 400 elements, then refine the mesh on Ω1 once, which leads to the mesh shown in Fig. 8. We keep all the computational parameters and setups same as Example 1 as well as the initial guesses. The numerical results are presented in Fig. 9, which shows that we can obtain excellent cloaking phenomenon with proper permittivity and permeability given in Fig. 9(c)–(f). 5. Conclusion In this paper we propose to use the optimal control method to search for the optimized permittivity and permeability of the metamaterial which can lead to the invisibility cloaking phenomenon. Proper objective functional is proposed and the first order optimality conditions are derived by using the Lagrange multiplier methodology. The state equation is solved by the discontinuous Galerkin method. Our optimal control method is tested with several benchmark cloaking problems. Numerical results demonstrate the effectiveness of our method. In the future, we hope to develop and establish the rigorous analysis of optimal control problems in the coefficients of both time-harmonic and time-dependent Maxwell’s equations. We will focus on existence analysis of the optimal control, the first-order necessary optimality conditions, numerical analysis of the optimal control problems including proper discretization for the state equation and the control, and the convergence of the numerical method. References [1] J.B. Pendry, D. Schurig, D.R. Smith, Controlling electromagnetic fields, Science 312 (2006) 1780–1782. [2] U. Leonhardt, Optical conformal mapping, Science 312 (2006) 1777–1780. [3] H. Ammari, H. Kang, H. Lee, M. Lim, S. Yu, Enhancement of near cloaking for the full maxwell equations, SIAM J. Appl. Math. 73 (6) (2013) 2055–2076. [4] G. Bao, H. Liu, J. Zou, Nearly cloaking the full maxwell equations: cloaking active contents with general conducting layers, J. Math. Pures Appl. 101 (2014) 716–733. [5] S.C. Brenner, J. 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Optimal control for electromagnetic cloaking metamaterial parameters design✩ ∗
Zhiwei Fang a , Jichun Li a , , Xiang Wang b a b
Department of Mathematical Sciences, University of Nevada Las Vegas, NV 89154-4020, USA Department of Mathematical Sciences, Nanchang University, Nanchang, China
article
info
Article history: Received 2 December 2018 Received in revised form 2 May 2019 Accepted 24 August 2019 Available online xxxx Keywords: Maxwell’s equations Invisibility cloak Optimal control method Discontinuous Galerkin method
a b s t r a c t In this paper, we develop an optimal control problem to optimize the permittivity and permeability of the metamaterial which can have the invisible cloaking effect. The inverse problem is calibrated as an optimization problem constrained by the time-harmonic Maxwell’s equations. Proper objective functional is introduced with coefficients of the Maxwell’s equations as control variables. We adopt the discontinuous Galerkin method to solve the state equation. Several two-dimensional cloaking benchmark problems are used to test our optimal control method. Numerical results demonstrate that the optimized permittivity and permeability of the metamaterial can achieve the cloaking effectively. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In 2006, Pendry, Schuring and Smith [1] and Leonhardt [2] independently proposed some theoretical ideas of designing invisibility cloaks with negative-index metamaterials. Since then, the study of electromagnetic cloaks has attracted great attentions due to its many potential applications in both military and civic life. In the past decade, mathematicians, physicists and engineers made great efforts in investigating various ways in designing invisibility cloaks and other applications (e.g., [3–13]). The main idea of Pendry’s cloaking design is to use the so-called transformation optics to calculate the permittivity and permeability of the metamaterial. But this technique is impractical for general geometries and also the calculated permittivity and permeability are difficult to be manufactured in practice. Hence searching for the proper permittivity and permeability parameters with cloaking property by robust mathematical methods has very important role in invisibility cloak design. In this paper, we initiate our effort in designing the permittivity and permeability of the metamaterial through solving an optimal control problem with control as the coefficients of time-harmonic Maxwell’s equations. The challenge is to come up with a proper objective functional to be minimized: for any given incident/detecting wave, we aim at estimating the permittivity and permeability in the cloaking region such that the incident wave keeps invariant outside the cloaking region. Due to the challenge of cloaking design, we limit to the 2D modeling in this paper since this is our first effort in this direction. ✩ Work partially supported by NNSF of China with Nos. 11671340 and 11961048, NSF of Jiangxi Province with No.20181ACB20001 and No.20161ACB21005. ∗ Corresponding author. E-mail addresses: [email protected] (Z. Fang), [email protected] (J. Li), [email protected] (X. Wang). https://doi.org/10.1016/j.camwa.2019.08.023 0898-1221/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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In literature, there are many existing excellent works devoted to optimal control problems constrained by partial differential equations (PDEs), such as optimal control of elliptic problems [14–18], optimal control of a free boundary problem [19], optimal control of parabolic problems [20,21], optimal control of the scalar wave equation [22], and optimal control of magnetohydrodynamics (MHD) [23,24] etc. More details on optimal controls of PDEs can be found in monographs [25–30] and references therein. As for optimal control of Maxwell’s equations, there are many excellent results for the optimal control of parabolic eddy current equations (e.g., [31,32]), optimal control of linear time-harmonic eddy current equations (e.g., [33–36]), and optimal control of nonlinear Maxwell’s equations (e.g., [37,38]). But most papers are for control of source terms, the numerical analysis of optimal control problems in the coefficients of time-harmonic eddy current equations was first investigated by Yousept [35, p. 880]. To the best of our knowledge, this paper is the first attempt at solving the optimal control problem for coefficients of time-harmonic Maxwell’s equations with applications to invisibility cloak design. In recent years, there are some publications devoted to the theoretical analysis of the cloaking problem by using the optimization method, e.g., [39] considered the cloaking problem governed by the 2D Helmholtz equation with impedance boundary condition; [40] studied an approximate cloaking problem for the 3D time-harmonic Maxwell equations; [41] analyzed the 2D magnetic scattering by a bilayer shell. Due to some nice features of the discontinuous Galerkin (DG) methods (such as flexibility in h–p adaptivity, and efficiency in local solvability and parallel implementation), in recent years, DG methods have been widely used to solve various PDEs, including Maxwell’s equations in both time domain [42–44] and frequency domain [45–47]. More details on DG methods and applications can be found in books [48–50] and references therein. Of course, other popular numerical methods such as FDTD methods [51–53] and spectral methods [13] can also be considered here. In this paper, we will adopt the DG method for solving the state equation, i.e., the time-harmonic Maxwell’s equations. The rest of the paper is organized as follows. In Section 2, we introduce the 2D time-harmonic Maxwell’s equations, which is the state equation or the PDE constrain in our optimal control problem. In Section 3, we first formulate our optimal control problem for metamaterial cloaking design, then derive the first order optimality conditions through the Lagrange multiplier methodology. Finally, we describe the DG method for solving the state equation. Numerical results demonstrating the effectiveness of our optimal control method for invisibility cloak design are presented in Section 4. We conclude the paper in Section 5. 2. Formulation of the optimal control problem for cloaking In order to design the two dimensional (2D) invisibility cloaks with metamaterials, we have to solve the Maxwell’s equations, which in the frequency domain are given as follows:
∇ × E(x) + jωµH (x) = 0, ∇ × H(x) − jωϵ E(x) = J (x) in Ω , (1) √ where j = −1, x = (x, y)⊤ , E(x) = (Ex (x), Ey (x))⊤ and H(x) are the electric and magnetic fields in the frequency domain, J (x) is the applied current density, ω is the general wave frequency, and the 2 × 2 tensor ϵ (x) and the scalar µ(x) are the permittivity and permeability of the underlying material, respectively. Furthermore, here we adopt the 2D curl operators:
∂ Ex ∂ Ey − , ∇ ×E = ∂x ∂y
( and
∇ ×H =
∂H ∂H ,− ∂y ∂x
)⊤
.
Finally, we assume that Ω is a bounded domain contained in R2 with boundary ∂ Ω and unit outward normal vector n. Eliminating H from (1), we obtain the so-called time-harmonic Maxwell’s equations [54–56]:
∇ × (µ−1 ∇ × E) − ω2 ϵ E = −jωJ
in Ω .
(2)
To make the problem complete, we usually impose (2) with the perfectly conducting (PEC) boundary condition n×E =0
on ∂ Ω .
(3)
To design an invisibility cloak device, we consider solving the Maxwell’s equations with heterogeneous media in
Ω = Ω0 ∪ Ω1 ∪ Ω2 as shown in Fig. 1. Here Ω0 is the cloaked region where any objects to be hided are put, Ω1 is the cloaking region formed by a metamaterial with special permittivity and permeability to be found by the optimal control method, and Ω2 is the vacuum region. Moreover, Ωi , i = 1, 2, 3, are mutually disjoint. For simplicity, we define Ω ′ = Ω1 ∪ Ω2 . Our goal is to find the proper permittivity and permeability of the metamaterial in Ω1 such that any objects located inside Ω0 is cloaked in the sense that any given incident or detecting wave Ew (x) in Ω2 shall bypass Ω1 region without any change. By introducing the relative permittivity ϵr := ϵ/ϵ0 and relative permeability µr := µ/µ0 of the underlying material, we rewrite the governing equation (2) as follows: 1 2 ∇ × (µ− r ∇ × E) − k0 ϵr E = f
in Ω ′
(4)
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 1. The domain setup for the optimal control problem.
subject to the PEC boundary condition on ∂ Ω0 : n × E = 0,
(5)
√ where k0 = ω ϵ0 µ0 := ω/c0 denotes the wave number, ϵ0 and µ0 are the permittivity and permeability in vacuum, ω is the general wave frequency, and c0 is the speed of light in free space. Since our goal is to keep Ew invariant in Ω2 , we impose the following boundary condition on the most outside boundary ∂ Ω of Ω : on ∂ Ω .
n × E = n × Ew
(6)
Note that ϵr = µr = 1 in vacuum, which leads to the source term f being given as follows:
{ f =
0
in Ω1
∇ × ∇ × Ew − k20 Ew
in Ω2 .
To formulate the optimal control problem, we define
ϵ (x) ϵr (x) = 1 ϵ2 (x)
ϵ2 (x) ϵ3 (x)
[
]
in Ω1 ,
and identity matrix in Ω2 . Similarly, we set µr (x) = µ1 (x) in Ω1 and 1 in Ω2 . Then the optimal control problem for cloaking reads: 1
min −1
ϵ1 ,ϵ2 ,ϵ3 ,µ1 ∈L2 (Ω1 )
2
∫ Ω2
|E − Ew |2 dx +
1
∫
2
Ω2
|∇ × (E − Ew )|2 dx +
∫ 3 ∑ β1 i=1
2
Ω1
|ϵi |2 dx +
β2 2
∫ Ω1
2
1 |µ− 1 | dx
(7)
subject to
⎧ 2 −1 ⎨∇ × (µr ∇ × E) − k0 ϵr E = f n×E =0 on ∂ Ω0 ⎩ n × E = n × Ew on ∂ Ω ,
in Ω ′ (8)
where β1 , β2 > 0 are regularization parameters. Namely, we are going to find ϵ1 , ϵ2 , ϵ3 and µ1 by minimizing the cost functional defined above subject to (8). 1 Note that since the µ1 appears in the problem as µ1−1 and is nonzero, we treat the unknown function µ− directly. 1 In the definition of state equation (forward problem) (4), the permittivity and permeability to be designed are ϵr and µr , respectively. Since they are identity matrix and constant 1 in Ω2 respectively, we only consider their components in Ω1 , 1 i.e., ϵ1 , ϵ2 , ϵ3 and µ− 1 , as unknowns. The weak formulation of the state equation (8) reads: find E ∈ HE (curl; Ω ′ ) such that 1 2 (µ− r ∇ × E , ∇ × v) − k0 (ϵr E , v) = (f , v)
∀v ∈ H0 (curl; Ω ′ )
(9)
where the Sobolev space HE (curl; Ω ) is defined as: ′
HE (curl; Ω ′ ) = E ∈ H(curl; Ω ′ ) : n × E = 0 on ∂ Ω0 , n × (E − Ew ) = 0 on ∂ Ω ,
{
}
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
4
Z. Fang, J. Li and X. Wang / Computers and Mathematics with Applications xxx (xxxx) xxx
and is equipped with norm
( ) 12 ∥v ∥H(curl;Ω ′ ) = ∥v ∥2(L2 (Ω ′ ))2 + ∥∇ × v ∥2(L2 (Ω ′ ))2 , { } where H(curl; Ω ′ ) := v ∈ (L2 (Ω ′ ))2 : ∇ × v ∈ L2 (Ω ′ ) . To accommodate the PEC boundary condition, we denote the Sobolev space H0 (curl; Ω ′ ) = v ∈ H(curl; Ω ′ ) : n × v = 0 on ∂ Ω ′ .
}
{
Moreover, the notation (·, ·) denotes the usual inner product on L2 (Ω ′ ). 3. Finite element method for the optimal control problem 3.1. The first order optimality conditions Following optimal control theory [29], we can convert the constrained problem (7)–(8) to an unconstrained one: 1 L((ϵ1 , ϵ2 , ϵ3 , µ− 1 ), E , p),
min
1 2 ∈L (Ω1 ) ϵ1 ,ϵ2 ,ϵ3 ,µ− 1
(10)
where the Lagrangian functional L : (L2 (Ω1 ))4 × H(curl, Ω ′ ) × H(curl, Ω ′ ) ↦ → R is given as follows: L
((
∫
) ) 1 1 ϵ1 , ϵ2 , ϵ3 , µ− , E, p = 1 +
∫ 3 ∑ β1 i=1
2
Ω1
|ϵi |2 dx +
2
Ω2
β2
∫
2
|E − Ew |2 dx +
1
∫
2
Ω2
|∇ × (E − Ew )|2 dx (11)
2
Ω1
1 −1 2 |µ− 1 | dx + (µr ∇ × E , ∇ × p) − k0 (ϵr E , p) − (f , p).
We derive the first order optimality conditions by the Lagrange multiplier methodology. It is not difficult to see that the Gaˆ teaux derivative of L with respective to (w.r.t) E in direction ˜ E ∈ H0 (curl; Ω ′ ) is: LE
((
ϵ1 , ϵ2 , ϵ3 , µ1
−1
)
)( ) , E, p ˜ E =
∫
∫ (E − Ew ) · ˜ Edx +
∇ × (E − Ew ) · ∇ × ˜ Edx ) ( ) + µr ∇ × ˜ E , ∇ × p − k20 ϵr˜ E, p . Ω ( 2−1
Ω2
So, the adjoint problem reads: find p ∈ H0 (curl; Ω ′ ) such that LE
((
) )( ) 1 , E, p ˜ E =0 ϵ1 , ϵ2 , ϵ3 , µ− 1
∀˜ E ∈ H0 (curl; Ω ′ )
Similarly, for i = 1, 2, 3, we can derive the Gaˆ teaux derivative of L with respective to ϵi in direction ˜ ϵi ∈ L2 (Ω1 ) as: Lϵ i
((
) ) 1 ϵ1 , ϵ2 , ϵ3 , µ− , E , p (˜ ϵ i ) = β1 1
∫ Ω1
∫
ϵi˜ ϵi dx − k20
Ω1
ϵt ,i E · pdx,
where the matrices ϵt ,i are given as follows:
ϵt ,1 =
[ ˜ ϵ1 0
]
0 , 0
ϵt ,2 =
[
0
˜ ϵ2
˜ ϵ2
]
0
,
ϵt ,3 =
[
0 0
0
]
˜ ϵ3
.
˜ 1 −1 2 Finally, we have the Gaˆ teaux derivative of L with respective to µ− 1 in direction µ1 ∈ L (Ω1 ): Lµ−1
((
1
) ) ) (˜ 1 −1 = β2 ϵ1 , ϵ2 , ϵ3 , µ− , E , p µ 1 1
∫ Ω1
1˜ −1 µ− 1 µ1 dx +
∫ Ω1
˜ −1 µ 1 ∇ × E · ∇ × pdx.
1 Hence, the gradients of the loss functional L with respect to the unknowns µ− 1 and ϵi , i = 1, 2, 3, are given by
) ) 1 , E , p (˜ ϵi ) , ϵ1 , ϵ2 , ϵ3 , µ− 1 ) (( ) ) (˜ −1 1 Lµ−1 ϵ1 , ϵ2 , ϵ3 , µ1 , E , p µ− , 1 Lϵi
((
1
∀˜ ϵi ∈ L2 (Ω1 ), i = 1, 2, 3, ˜ −1 ∀µ ∈ L2 (Ω ). 1
1
3.2. The discontinuous Galerkin method for the optimality conditions To solve the state equation (9), we adopt the discontinuous Galerkin method. We consider a shape-regular affine mesh Th that partitions the domain Ω ′ into triangles {K }, where hK is the diameter of element K ∈ Th and h = maxK ∈Th hK is the mesh size of Th . We denote by FhI the set of all element interior faces of Th , by FhB the set of all boundary faces, and denote Fh = FhI ∪ FhB . For a piecewise smooth vector-valued function v, we introduce the following trace operators. Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 2. Cylindrical cloak: The mesh used and the exact solution.
Let f ∈ FhI be an interior face shared by two neighboring elements K + and K − with unit outward normal vectors n± , respectively. Denoting by v ± the traces of v taken from within K ± , respectively. We also define the tangential jump and average across face f by
[[v ]] = n+ × v + + n− × v − ,
{{v }} =
v+ + v− 2
,
(12)
respectively. On a boundary face f ∈ FhB , we set [[v ]] = n × v and {{v }} = v. For the given partition Th of Ω ′ and any integer l ≥ 0, we choose the following discontinuous finite element space Vh = vh ∈ (L2 (Ω ′ ))2 : vh |K ∈ (P l (K ))2 , ∀K ∈ Th ,
{
}
where P l (K ) denotes the space of polynomials of total degree at most l on K . Now we can define the DG scheme for solving the state equation (9): finding Eh ∈ Vh such that 1 2 ah (Eh , vh ; µ− r ) − k0 (ϵr Eh , vh ) = F (vh )
∀vh ∈ Vh ,
(13)
where 1 −1 ah (Eh , vh ; µ− r ) := (µr ∇h × Eh , ∇h × vh ) −
∫
1 [[Eh ]] · {{µ− r ∇h × vh }}ds ∫ ∫ α −1 − [[vh ]] · {{µr ∇h × Eh }}ds + [[Eh ]] · [[vh ]]ds
Fh
Fh
Fh
hf
and F (vh ) := (f , vh ) −
∫
α
∫
∂Ω
n × Ew · ∇h × vh ds +
∂Ω
hf
n × Ew · n × vh ds,
where hf denotes the diameter of face f and is a function of x, ∇h × denotes the element-wise application of curl operator ∇×, the penalty parameter α∫ > 0 is a constant independent of mesh size and wave number, and the integral on Fh is ∫ ∑ defined as F ϕ ds = ϕ ds. f ∈F f h
h
To formulate the DG discretization of adjoint and optimality conditions, we first introduce the following bilinear form bh (uh , vh ; D) =
∫
∫ ∇h × uh · ∇h × vh dx −
∫D
[[uh ]] · {{∇h × vh }}ds Fh ∩D
∫ [[vh ]] · {{∇h × uh }}ds +
− Fh ∩D
Fh ∩D
α hf
[[uh ]] · [[vh ]]ds,
where D ⊂ Ω ′ . Thus, the DG scheme for the adjoint problem can be written as: find ph ∈ Vh , such that for ∀˜ Eh ∈ Vh ,
∫ Ω2
1 2 ˜ (Eh − Ew ) · ˜ Eh dx + bh (Eh − Ew , ˜ Eh ; Ω2 ) + ah (˜ Eh , ph ; µ− r ) − k0 (ϵr Eh , ph ) = 0,
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 3. Cylindrical cloak: The permeability and permittivity parameters obtained from optimal control.
holds true. The Lϵi , i = 1, 2, 3, and Lµ−1 can be approximated by 1
Lϵ i , h
Lµ−1 ,h 1
(( ) )( ) 1 ϵ1,h , ϵ2,h , ϵ3,h , µ− ϵi,h = β1 1,h , Eh , ph ˜
((
∫ Ω1
ϵi,h˜ ϵi,h dx − k20
∫ Ω1
ϵt ,i,h Eh · ph dx,
∀ϵi,h ∈ Uh , i = 1, 2, 3, ∫ ( ) ( ) ) ) ˜ ˜ 1 −1 −1 ˜ −1 −1 ϵ1,h , ϵ2,h , ϵ3,h , µ− , E , p µ = β µ µ dx + a E , v ; µ h h 2 h h h 1,h 1,h 1,h 1,h 1 ,h , Ω1
˜ −1 ∀µ 1,h ∈ Uh , Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 4. Cylindrical cloak: the field ∇h × Eh obtained after 500 iterations without the curl term in the objective functional.
Fig. 5. The physical domain for the carpet cloaking.
Fig. 6. Carpet cloak: The mesh and the exact solution.
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 7. Numerical results of the optimal control for carpet cloaking.
respectively. Here Uh denotes the DG finite element space for scalar functions: Uh = {vh ∈ L2 (Ω1 ) : vh |K ∈ P l (K ), ∀K ∈ Th }, and the definition of ϵt ,i,h is the same as ϵt ,i but replacing ˜ ϵi by ˜ ϵi,h . Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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9
Fig. 8. The mesh for the oval cloaking.
4. Numerical results In this section, we present three numerical examples to show the effectiveness of finding the optimized material coefficients for the invisibility cloak design by solving the optimal control problem (7)–(8). All the numerical results in this section are obtained by solving the optimization problem with the limited memory BFGS (L-BFGS) method [29] implemented under the FEniCS [57] adjoint package. The stopping rule used in the optimization algorithm is |L| ≤ 10−2 , where L is the loss functional defined in (11). 4.1. Example 1: Cylindrical cloak In this subsection, we solve a cylindrical cloaking problem originally proposed by Pendry et al. [1], and later simulated by Li et al. [58,59]. We set the domain Ω = [−2, 2]2 , and put any cloaked objects inside a cylinder centered at the original with radius R1 = 0.3 and perfectly conducting boundary condition so that no wave can enter into the cloaked region. The cloaked cylinder is wrapped by a cylindrical metamaterial region with thickness R2 − R1 , where R2 = 0.6. Hence in this simulation, Ω0 = {x : ∥x∥R2 ≤ 0.3}, Ω1 = {x : 0.3 < ∥x∥R2 ≤ 0.6} and Ω2 = {x ∈ Ω : ∥x∥R2 > 0.6}, where ∥ · ∥R2 denotes the Euclidean norm in R2 . In this example, we apply an initial triangulation on Ω ′ with 14 400 elements, then refine the mesh on Ω1 , as shown in Fig. 2(a). We use the first order discontinuous basis function to approximate E, and piecewise constant to approximate the control variables ϵ1 , ϵ2 , ϵ3 and µ. We choose the regularization parameters β1 = β2 = 10−8 in the Lagrangian functional and the penalty parameter α = 15 in the DG method. The wave source for this simulation is Ew = (0, − cos(k0 x)) with 1 k0 = 11. The initial guesses for all ϵi and µ− 1 are ∥x∥R2 . The optimal parameters recovered from solving this optimal control problem are shown in Fig. 3. Since the solution Eh is a vector, in Fig. 3 we present the solution ∇h × Eh , which is actually a multiple of H due to (1). For comparison, we also show the analytical result in Fig. 2(b) obtained with the perfect permeability and permittivity parameters (see [1] and [60]). We like to point out that the curl term in the objective functional (7) is necessary from our numerical tests. Solving the same optimization problem without the curl term for 500 iterations, we only produced unsatisfactory cloaking as shown in Fig. 4. 4.2. Example 2: Carpet cloak In this subsection, we consider the carpet cloaking model, which was first designed by Chen et al. [61] and was later simulated by Li et al. [62] using a finite element time domain solver. The model setup is shown in Fig. 5, where the subdomain Ω0 is a triangle formed by vertices (−0.5, 0), (0, 0.5) and (0.5, 0), while the subdomain Ω1 is a quadrilateral formed by vertices (−0.5, 0), (0, 0.5), (0.5, 0) and (0, 1). The computational domain is Ω = [−1, 1] × [0, 2] and Ω2 = Ω \ (Ω0 ∪ Ω1 ). In the simulation, we apply an initial triangulation of Ω ′ with 14 400 elements, then refine the partition of Ω1 one 1 more time. The initial guesses for this problem are ϵ1 = ϵ3 = µ− 1 = 1 and ϵ2 = −1 if x1 ≥ 0 while ϵ2 = 1 if x1 < 0. The resulting mesh used for the simulation is shown in Fig. 6(a). We used the same finite element spaces, regularization and penalty parameters and wave source as the last example. Instead of just assuming that the permeability and permittivity being in L2 (Ω1 ), we want them to be piecewise constants since the original carpet cloak design has piecewise Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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Fig. 9. Numerical results of the optimal control for oval cloaking.
constant permittivity and constant permeability (see [61,62]). To this end, we add two more penalty terms in the original object functional (7): min
1
∫
1 2 ϵi ,µ− ∈L (Ω1 ) 2 1
+
Ω2
|E − Ew |2 dx +
∫ 3 ∑ γ1 i=1
2
∂S
1
∫
2
[[ϵi ]]2 ds +
Ω2
γ2 2
|∇ × (E − Ew )|2 dx +
∫ 3 ∑ β1 i=1
∫ ∂S
2
Ω1
|ϵi |2 dx +
β2 2
∫
2
Ω1
1 |µ− 1 | dx
1 2 [[µ− 1 ]] ds,
Please cite this article as: Z. Fang, J. Li and X. Wang, Optimal control for electromagnetic cloaking metamaterial parameters design, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.023.
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11
and also in (11): 3 (( ) ) (( ) ) ∑ γ1 1 1 L˜ ϵ1 , ϵ2 , ϵ3 , µ− , E , p := L ϵ1 , ϵ2 , ϵ3 , µ− , E, p + 1 1 i=1
2
∫
2
∂S
[[ϵi ]] ds +
γ2 2
∫ ∂S
1 2 [[µ− 1 ]] ds,
where ∂ S denotes the set of element edges in = Ω1 ∩ {x : |x| > 0.2h}, where h is the maximal mesh size, and [[·]] denotes the jump across the element boundaries defined in (12). In our simulation, we choose the penalty parameters γ1 = γ2 = 105 . The numerical results of inverse problem are shown in Fig. 7. As a comparison, we also present the perfect cloaking obtained with the exact permeability and permittivity in Fig. 6(b). The obtained numerical results are presented in Fig. 7. Fig. 7(b) shows that the difference between the numerical field and the source wave ∇h × (Eh − Ew ) is almost zero out of the cloaking region, which justifies the excellent cloaking effect. In Fig. 7(c), we present the obtained optimal parameter ϵ1 ranging from the smallest value 2.00617032 to the largest value 2.00996683 in the cloaking region, which is very accurate compared to the exact value ϵ1 = 2 (cf. [62, p. 1137]). The calculated optimal parameters ϵ2 , ϵ3 and µ−1 and their minimal and maximal values are presented in Fig. 7(d), (e) and (f), respectively. The results show that the optimal cloaking parameters obtained from the optimal control method are all very accurate to the exact solution, which justifies the effectiveness of our optimal control method. 4.3. Example 3: Oval cloak In this subsection, we carry out a new cloaking experiment to test our optimal control method. Here we consider the physical domain Ω = [−2, 2]2 , the cloaked region is a disc Ω0 = {x : ∥x∥R2 ≤ 0.3}, the cloaking region is an ellipse Ω1 = {x ∈ Ω \ Ω0 : 0.5x2 + y2 ≤ 0.6} and the vacuum region Ω2 = Ω \ (Ω1 ∪ Ω2 ). Unlike the previous two examples, there is no exact formula for the permittivity and permeability for this cloaking. For this example, we use an initial triangulation of Ω ′ with 14 400 elements, then refine the mesh on Ω1 once, which leads to the mesh shown in Fig. 8. We keep all the computational parameters and setups same as Example 1 as well as the initial guesses. The numerical results are presented in Fig. 9, which shows that we can obtain excellent cloaking phenomenon with proper permittivity and permeability given in Fig. 9(c)–(f). 5. Conclusion In this paper we propose to use the optimal control method to search for the optimized permittivity and permeability of the metamaterial which can lead to the invisibility cloaking phenomenon. Proper objective functional is proposed and the first order optimality conditions are derived by using the Lagrange multiplier methodology. The state equation is solved by the discontinuous Galerkin method. Our optimal control method is tested with several benchmark cloaking problems. Numerical results demonstrate the effectiveness of our method. 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