DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2015.8.419DYNAMICAL SYSTEMS SERIES SVolume 8, Number 3, June 2015 pp. 419–434

RADAR CROSS SECTION REDUCTION OF A CAVITY IN THE

GROUND PLANE: TE POLARIZATION

Gang Bao

Department of Mathematics, Zhejiang UniversityHangzhou 310027, China

and

Department of Mathematics, Michigan State UniversityEast Lansing, MI 48824, USA

Jun Lai

Courant Institute of Mathematical Sciences, New York UniversityNew York, NY 10012, USA

Abstract. The reduction of backscatter radar cross section(RCS) in TE po-larization for a rectangular cavity embedded in the ground plane is investigated

in this paper. It is established by placing a thin, multilayered radar absorb-

ing material(RAM) with possibly different permittivities at the bottom of thecavity. A minimization problem with respect to the backscatter RCS is for-

mulated to determine the synthesis of RAM. The underlying scattered field

is governed by a generalized Helmholtz equation with transparent boundarycondition. The gradient with respect to the material permittivity is derived by

the adjoint state method. A fast solver for the Helmholtz equation is presented

for the optimization scheme. Numerical examples are presented to show theefficiency of the algorithm for RCS reduction.

1. Introduction. Radar cross section(RCS) is an important measure for radarsystem to characterize the scattering from a target[19]. Reducing the RCS for acavity structure is of great interest for civil and military applications due to its broadexistence in practice, examples including jet engine inlet ducts, vehicles with groovesand cavity-backed antennas. One of the effective approaches for RCS reduction isto use multilayered radar absorbing material(RAM), which is presumably thin andlight, to absorb the energy. The mechanism underlying the RAM is to convertelectromagnetic energy into heat and consequently the reflected energy is reduced.Our aim in this paper is to study the optimal design of a cavity with multilayeredRAM, so that the RCS is reduced over a range of incident angles or frequencies.

More specifically, this paper focuses on the backscatter RCS reduction for a cavityembedded in the infinite ground plane. The cavity is assumed to be invariant inthe z direction(Figure 1) and illuminated by a time harmonic plane wave. A thin,multilayered RAM is coated at the bottom of the cavity. Assume the material hasthe same magnetic permeability as the empty space, but different permittivities at

2010 Mathematics Subject Classification. 35Q93, 35J05.Key words and phrases. Optimal design, RCS reduction, adjoint state method, radar absorbing

materials.The research was supported in part by the NSF grants DMS-0908325, DMS-0968360, DMS-

1211292, the ONR grant N00014-12-1-0319, a Key Project of the Major Research Plan of NSFC

(No. 91130004), and a special research grant from Zhejiang University.

419

420 GANG BAO AND JUN LAI

each layer. The permittivity for each layer is determined through the optimizationthat minimizes the backscatter RCS. Based on the coevaporation technique[18],the admissible set for the permittivity is assumed to be a continuous variable in aspecified interval.

Lots of optimization techniques are proposed to reduce the RCS for an objectwith multilayered RAM [12, 23, 25]. When the material is chosen from a discreteset, genetic algorithm(GA) is considered as a practical optimization method, sincenumerically it leads to an output that is close to the global minimum. However, GAis a stochastic based method and generally requires a large amount of simulationsto achieve convergence. For our problem, the admissible set for the material is notdiscrete. Meanwhile, for a given material, solving the forward scattering problemneeds a considerable amount of time. Therefore, it is inappropriate to apply GAto the optimal design problem. Instead, a gradient based approach is desired. Inparticular, the method we employ is sequential quadratic programming(SQP), whichis effective for nonlinear optimization problem with continuous design variables [24].Although the gradient based method generally leads to a local minimum, numericalexperiments show these local minimums are satisfactory with a good initial guess.

As one of the essential components in our optimization method, the gradient forthe RCS with respect to the permeability is evaluated through the adjoint statemethod. The adjoint state method is used widely in the nonlinear optimizationproblem especially when the state variable is taken from the solution of a differentialequation [13]. Although the gradient can also be simply evaluated by numericaldifference, the accuracy will be lost given the sensitivity of the problem. Meanwhile,the time to evaluate the numerical gradient depends linearly on the number of layers,while the gradient based on the adjoint state method is independent of the amountof layers.

Since our problem is essentially two dimensional, there exist two fundamentalpolarizations for Maxwell’s equation: TM (Transverse magnetic) polarization andTE (Transverse electric) polarization. This paper focuses on the TE polarization,which continues our study for TM polarization in [7]. Two difficulties arise fromTE case compared to TM case. The first is that TE polarization is governed by ageneralized Helmholtz equation, where the permittivity is included in a differentialoperator. The derivation for the gradient via the adjoint state method would bemore involved. The second is the fast algorithm used in [8] for the forward scatteringproblem does not directly apply to TE polarization when the material has layeredstructure. Both of these issues will be addressed in this paper. In particular,we extend the algorithm proposed in [8] to the layered medium for TE case byemploying the continuity condition. The resulted algorithm is fast and accurate,which exactly fits the needs for an optimization algorithm.

It is worth mentioning that the cavity problem has been extensively studied interms of both mathematical analysis and numerical simulation. In [1, 2, 3], theexistence and uniqueness for the forward scattering problem of a cavity were es-tablished in a rather general setting. The inverse scattering problem was studiedin [5], where the uniqueness and the stability results were investigated. In [9],the authors considered the scattered field of a rectangular cavity illuminated by ahigh frequency wave. The explicit dependence on the wavenumber for the energyat the aperture were given. Numerical simulation to evaluate the RCS for largeand deep cavities is considered as a great challenge over the last decades. Lots ofnumerical methods have been proposed to this problem[4, 10]. Standard methods

RCS REDUCTION FOR A CAVITY IN TE POLARIZATION 421

include finite difference[8], finite element with boundary integral[16, 21], method ofmomentum[15]. High frequency techniques include Gaussian beam asymptotics[11],shooting and bouncing rays(SBR)[20], iterative physical optics[14]. A comprehen-sive review for all the computational techniques developed over the last two decadescan be found in [4]. Recently, we developed a fast mode matching method for threedimensional rectangular cavity in [6].

The paper is organized as follows. In the next section, the optimal design of acavity with multilayered RAM is formulated as a minimization problem, with theconstraint as a generalized Helmholtz equation with artificial boundary condition.The objective function and design variables are also defined in this section. Thegradient based on the adjoint state method is derived in Section 3, following aremark on the finite dimensional counterpart. Section 4 gives a brief introductionon the SQP optimization technique followed by a description of the minimizationalgorithm. The solver for the forward scattering problem is presented in Section5. Numerical experiments are provided in Section 6 to show the efficiency of themethod on RCS reduction. Section 7 concludes the whole discussion.

2. Problem formulation. Consider a rectangular cavity Ω = [0, a] × [−b, 0] em-bedded in the ground plane illuminated by a time harmonic plane wave, as illus-trated in Figure 1. The problem is formulated in 2-D by assuming that the cavityand the material are invariant in the z direction. Above the ground plane, themedium is assumed to be homogeneous with dielectric permittivity ε0. The surfaceof the ground plane Γc and the boundary S of the cavity are assumed to be perfectconductors. The bottom of the cavity is coated by a thin, layered medium withpermittivities given by εj , j = 1, 2, · · · , L, where L is the number of layers. Abovethe coating material, the cavity is filled with the same material as the upper halfspace. For convenience, denote ε as the permittivity function in terms of variable yand k = ω

√εµ0 as the wavenumber in the whole space. In this paper, we consider

TE(transverse electric) polarization and assume the magnetic permeability µ0 isthe same everywhere. The goal is to determine the permittivities of the coatingmaterial within a presumably small thickness h, such that the backscatter RCS isminimized.

θui

usur

Γ Γc

Ω

y

xz

S

(0,0)

εL−1

εL

ε2

ε1

Figure 1. Geometry of the problem

Assume the incident field is given by a plane wave: ui = ei(αx−βy), where α =k0 cos θ, β = k0 sin θ, θ is the incident angle with respect to the x-axis and k0 =

422 GANG BAO AND JUN LAI

ω√ε0µ0 is the wavenumber in the upper half space R2+. Denote u(x, y) and us(x, y)

as the total field and the scattered field respectively. Following the formulation in[17], the total field u(x, y) is the summation:

u(x, y) = us(x, y) + ui + ei(αx+βy) (1)

where ei(αx+βy) is identified as the reflected field ur. u(x, y) satisfies the followinggeneralized Helmholtz equation, which can be easily derived from the time harmonicMaxwell equation:

∇ · ( 1k2∇u) + u = 0 in Ω ∪ R2+

∂u∂n = 0 on Γc ∪ S

(2)

We now apply Fourier Transform method to derive a transparent boundary con-dition on Γ, by which Eq (2) is reduced to a bounded domain problem. Ideas oftransparent boundary condition have been applied broadly in many wave propaga-tion problems. One resorts to [22] and reference therein for the applications.

Let H1loc(R2+) be the Sobolev space for functions that are locally integrable in

R2+. The exterior scattered field us(x, y) ∈ H1loc(R2+) satisfies the equation:

∆us + k20us = 0 in R2+

∂us

∂n = φ(x) on Γ∂us

∂n = 0 on Γc(3)

together with the radiation condition:

limρ→∞

ρ1/2

(∂us

∂ρ− ik0u

s

)= 0 (4)

Here ρ =√x2 + y2 and φ(x) is assumed to be known. Let φ(x) denote the zero

extension of φ(x) on y = 0. Taking a Fourier Transform for us with respect to xand denoting it as us(ξ, y), we have:

∂2us

∂y2 + (k20 − ξ2)us = 0 for y > 0,

∂us

∂n =φ(ξ) on y = 0.

(5)

Using the outgoing radiation condition (4), we easily obtain the solution for (5):

us(ξ, y) =φ(ξ)

i√k2

0 − ξ2ei√k20−ξ2y, if ξ 6= k0 (6)

Taking an inverse Fourier Transform with respect to ξ and letting y = 0 yields:

us(x, 0) =1

2πi

∫R

1√k2

0 − ξ2

φ(ξ)eiξxdξ (7)

Observing that φ(x) = ∂us

∂n = ∂u∂n on Γ, we therefore obtain the transparent bound-

ary condition for u:

u(x, 0) = T

(∂u

∂n

)+ g, on Γ (8)

where

T

(∂u

∂n

)=

1

2πi

∫R

1√k2

0 − ξ2

∂u

∂neiξxdξ,

g = 2eiαx,

RCS REDUCTION FOR A CAVITY IN TE POLARIZATION 423

and ∂u∂n denotes the zero extension of ∂u

∂n on y = 0.Equipped with the transparent boundary condition, Eq. (2) is reduced to a

bounded domain problem:∇ · ( 1

k2∇u) + u = 0, in Ω∂u∂n = 0, on S

u(x, 0) = T

(∂u∂n

)+ g, on Γ

(9)

Eq. (9) is uniquely solvable for u ∈ H1g (Ω), where

H1g (Ω) =

φ ∈ H1(Ω)

∣∣∣∣ ∂φ∂n ∈ H−1/2(Γ), φ = T

(∂φ

∂n

)+ g on Γ

, (10)

and H−1/2(Γ) is the dual space of H1/2(Γ).The proof for the existence can be found in [2]. The normal derivative of u at

the aperture Γ can be found through the numerical solution of (9) and then usedto evaluate the backscatter RCS. The formula of RCS is given in the next section.

2.1. Objective function. This paper aims to reduce the backscatter RCS, sincein general, larger RCS of an object indicates easier detection by the radar. Thedefinition of RCS in the two dimension is given by:

σ(ϕ, θ) = limρ→∞

2πρ|us(φ, θ)|2

|ui(θ)|2, (11)

where θ is the incident angle and φ is the observation angle. When φ = θ, itis called backscatter RCS or monostatic RCS. The reason we consider reducingbackscatter RCS only is because most radar systems are monostatic in practice[19].Since the upper half space is homogeneous, the backscatter RCS of the cavity inTE polarization can also be evaluated through[17]:

σ :=4

k0

∣∣∣∣12∫

Γ

∂u

∂ne−ik0x cos θdx

∣∣∣∣2 (12)

which is defined as the objective function in this paper. If the RCS reduction overa set of incident angles or frequencies is desired, the cost function may be changedto the following:

σtotal :=

r∑l=1

wlσ(tl) (13)

where tl represents the dependence on different angles or frequencies, and wl > 0,l = 1, 2, · · · , r are the corresponding weights. Furthermore, depending on whichangle is important, these weights may be chosen accordingly.

Since the permittivity ε is the design variable and u depends on ε from the model(9), the optimization problem can be rewritten as:

minε∈Λ

σ(ε) (14)

or

minε∈Λ

σtotal(ε) (15)

where the admissible set Λ for electrical permittivity ε satisfies the following twoconditions:

424 GANG BAO AND JUN LAI

• Layered medium: ε(y) in Λ is constant for each layer, either lossy or lossless.

In other words, ε(y) = εj at the jth layer, where εj = ε′j+iε′′

j , j = 1, 2, · · · , L,L is the number of layers of the coating material.

• Boundedness: We assume there exist εmin = ε′min + iε′′min, εmax = ε′max +

iε′′max such that ε′min < ε′j < ε′max and ε′′min < ε′′

j < ε′′max for j =1, 2, · · · , L. Namely, the permittivity ε(y) is bounded between εmin and εmax.

Due to the finite number of layers, one can easily prove the admissible set Λ iscompact in L∞(Ω). Since the total field u depends continuously on ε ∈ L∞(Ω), bythe standard argument, it follows:

Theorem 2.1. The minimization problem (14) or (15) admits at least one solutionin Λ.

Remark 1. It is possible to extend Thm. 2.1 to non-layered medium, in whichcase Λ is only weak∗ compact in L∞(Ω). This, however, requires u to be boundedin H1

g (Ω) independent of the variation of ε and the proof is far from trivial.

We intend to use a gradient based optimization algorithm for the RCS reduction.It is therefore very important to evaluate the gradient of the objective functionaccurately and efficiently, which will be shown in the next section.

3. Gradient by the adjoint state method. To determine the descent directionof the backscatter RCS (12) with respect to the permittivity, we need to evaluatethe gradient along with the PDE constraint (9). It can be approximated by thenumerical derivative, namely, finite difference. However, given the sensitive depen-dence of RCS on the permittivity and the amount of computational cost to evaluate(9), it is generally undesirable to use such a simple scheme. Our approach here isto derive the gradient for TE polarization based on the adjoint state method. Thismethod is well-known for its effectiveness in evaluating the gradient when the statevariable comes from a solution of a forward problem. Lots of nonlinear optimiza-tion problems have been studied based on this method; for example, the design ofantireflective structure in [13]. Here we provide a formal derivation for the gradientbased on the adjoint state method with proof omitted.

For convenience, let us first define the far field coefficient:

Pθ(ε) =1

2

∫Γ

∂u

∂ne−ik0x cos θdx, (16)

Comparing with (12), we have

σ(ε) =4

k0|Pθ(ε)|2

Let δε be a small perturbation to the permittivity ε. We denote the linearizationof σ(ε) with respect to δε by δσ, which is:

δσ = 2Re

(4

k0δPPθ(ε)

)(17)

where δP denotes the linearization of Pθ(ε) and Pθ is the complex conjugate of Pθ.From (16), it is easy to see that

δP =1

2

∫Γ

∂δu

∂ne−ik0x cos θdx (18)

RCS REDUCTION FOR A CAVITY IN TE POLARIZATION 425

where δu solves the linearized problem∇ ·(

1k2∇δu

)+ δu = ∇ ·

(ω2µ0δεk4 ∇u

)in Ω

∂δu∂n = 0 on S

δu = T

(∂δu∂n

)on Γ

(19)

Let u∗ solve the adjoint state equation:∇ ·(

1

k2∇u∗

)+ u∗ = 0 in Ω

∂u∗

∂n = 0 on S

u∗ = T ∗(∂u∗

∂n

)+ 1

2k2eik0x cos θPθ(ε) on Γ

(20)

where k2 is the complex conjugate of k2 and T ∗(·) is the adjoint operator of T (·),given by

T ∗(∂u∗

∂n

)= − 1

2πi

∫R

1√k2

0 − ξ2

∂u

∂neiξxdξ

Given equations (19) and (20), Green’s theorem implies that∫Ω

u∗∇ ·(ω2µ0δε

k4∇u)dx =

∫∂Ω

1

k2u∗∂δu

∂ndx−

∫∂Ω

1

k2

∂u∗

∂nδudx (21)

Applying the integration by parts yields:∫Ω

u∗∇ ·(ω2µ0δε

k4∇u)dx = −

∫Ω

∇u∗ ·(ω2µ0δε

k4∇u)dx+

∫∂Ω

u∗(ω2µ0δε

k4

∂u

∂n

)dx

(22)Recalling the homogeneous Neumann boundary condition, the second term on theright hand side of (22) is zero since the permittivity is constant near the apertureΓ. Hence, combining (21) and (22) together with the property of adjoint operator,we obtain:

−∫

Ω

∇u∗ ·(ω2µ0δε

k4∇u)dx =

1

2Pθ(ε)

∫Γ

∂δu

∂ne−ik0x cos θdx (23)

Comparing with (17), we then have

δσ = −2Re

(4

k0

∫Ω

∇u∗ ·(ω2µ0δε

k4∇u)dx

)(24)

Up to a constant multiple, the gradient of the cost function σ(ε) is the function g(ε)such that

δσ(ε) = Re

∫ 0

−bδεg(ε)dy (25)

Comparing (24) with (25), we arrive at the formula for the gradient:

g(ε) = − 8

k0ω2µ0

∫ a

0

1

k4

(∇u∗

)·(∇u)dx (26)

In other words, to find the gradient at a particular point ε ∈ Λ, we need to solvethe forward scattering problem (9) as well as the adjoint state problem (20).

426 GANG BAO AND JUN LAI

Remark 2. The gradient given in (26) is in fact L2 gradient, which is different fromthe usual vector form. In the numerical simulation, we need the finite dimensionalcounterpart of the gradient, which points to the maximal increase of RCS in the ad-missible set. Since the change δε is a constant for each layer, the finite discretizationfor (25) implies the gradient can be found through the following vector:

(

∫ −b+h−b

g(ε)dy,

∫ −b+2h

−b+hg(ε)dy, · · · ,

∫ −b+Lh−b+(L−1)h

g(ε)dy, ) (27)

where h is the thickness of each layer and L is the number of layers. In practice,we can also split each element in (27) into real and imaginary part, which gives thecorresponding change for the real and imaginary part of δε, respectively.

For the cost function (13), since it is a linear combination of (12), the gradientcan be easily deduced from (26). It can be seen that the adjoint problem shares thesame structure as the original scattering problem, hence the same solver may beused for the adjoint problem. This is a huge advantage given the level of difficultiesfor solving the scattering problem from a cavity.

4. Optimization method: Sequential Quadratic Programming. SequentialQuadratic Programming(SQP) is a gradient based, iterative optimization method.Each iteration is composed by solving a quadratic subproblem. Each subproblem isbased on the quadratic approximation of the Lagrangian function. In general, theHessian matrix used in the quadratic subproblem is not obtained analytically, butapproximated by some quasi-Newton methods. SQP provides a superlinear conver-gence near the region of the optimal point under some appropriate assumptions.Here we give a brief description of SQP to the RCS reduction problem. Detailedimplementation can be found in [24].

Define the Lagrangian function:

L(ε, λ) = σ(ε) +

4L∑j=1

λjhj(ε), (28)

where λj is the Lagrangian multiplier, and hj(ε) gives the constraint on the per-mittivity:

h4(j−1)+1 = −Re(εj) + ε′min

h4(j−1)+2 = Re(εj)− ε′maxh4(j−1)+3 = −Im(εj) + ε′′min

h4(j−1)+4 = Im(εj)− ε′′maxj = 1, 2, · · · , L

(29)

Here we consider the real and imaginary part of the permittivity separately. Theboundedness of ε is simply expressed as

hj(ε) ≤ 0 (30)

Following the theory of SQP, the m-th subproblem during the iteration is givenby:

mind∈CL

σ(εm) + g(εm)T (d) +1

2dTHmd (31)

RCS REDUCTION FOR A CAVITY IN TE POLARIZATION 427

with the linear constraint (29) satisfied by εm + d, where g(εm) is the gradient ofσ(ε) at εm, Hm is the Hessian of the Lagrangian function (28), which in this paperis approximated by Broyden-Fletcher-Goldfarb-Shanno(BFGS) method.

Solution of the subproblem (31) provides an update for ε along the direction d

εm+1 = εm + αd (32)

The step length α ∈ (0, 1) can be determined by an appropriately defined meritfunction. We use the following L1 merit function

ϕ(ε) = σ(ε) +

4L∑j=1

µj max(0, hj(ε)) (33)

for sufficiently large µj > 0, which gives penalty for solutions outside the admissibleset.

We are now ready to state the minimization algorithm:Data: Initialize permittivity ε0, tolerance ε, penalty coefficients µj , for

j = 1, · · · , 4LLet m = 0, α = 1;

while α > ε doEvaluate g(εm), Hm, h(εm);

Solve (31) to obtain d;

Set εm+1 ← εm + αd;

while ϕ(εm+1) > ϕ(εm) doα← α/2;

Update εm+1 ← εm + αd;

end

end

Remark 3. Since the problem (14) we consider is generally not convex with respectto ε, the solution from the algorithm above may not converge to the global one. Inpractice, we either expect a good initial guess is provided, or try different initialvalues in the parameter space and choose the most optimal one. More complicatedstrategy is possible, but we will not pursue it here.

Throughout the algorithm, it is required to provide the objective function, con-straint, gradient and a direct solver for the forward scattering problem (9). Wealready discussed how to find the gradient in Section 3. In the next section, wepresent an accurate and efficient direct solver for the optimization algorithm.

5. Direct solver for the forward scattering problem. Numerical solution forthe electromagnetic scattering from a cavity has long been known as a difficultproblem [21], especially when the size of the cavity is large or the wavenumber ishigh. Here we extend the finite difference method developed in [8] to be the directsolver of the optimization problem. The idea to solve the resulting linear system in[8] is to use the discrete cosine transform in the horizontal direction and a Gaussianelimination process in the vertical direction. The resulting system becomes anM×M (M is number of mesh points at the opening of the cavity) linear system,which is much smaller than the original one. However, the original algorithm in [8]can not directly apply to the design problem here since it requires the permittivityε of the medium inside the cavity to be differentiable in the vertical direction forTE polarization, which is not satisfied for layered medium. Therefore, we modify

428 GANG BAO AND JUN LAI

the algorithm to make it applicable for our case. The idea is simple: we applyfinite difference in each layer and connect each layer by the continuity condition,while still keep the resulting system suitable for the discrete cosine transform in thehorizontal direction.

Assume Ω is composed by L layers with each layer denoted by Ωl, l = 1, · · · , Land the associated permittivity is given by εl. Similar to [8], the interior and the

aperture of the cavity is uniformly partitioned by the mesh points xm, ynM+1,N+1m=1,n=1

with hx = xm+1 − xm and hy = yn+1 − yn. We further assume there exist meshpoints on each interface of two layers with n = n1, n2, · · · , nL−1. The algorithmwill not lose the generality by this assumption since the extension to nonuniformmesh in y direction is straightforward. Define um,n as the corresponding numericalsolution at (xm, yn). Since the material is homogeneous in each layer, the secondorder finite difference scheme in the l-th layer is simply given by the following:

um+1,n − 2um,n + um−1,n

h2x

+um,n+1 − 2um,n + um,n−1

h2y

+ ω2µ0εlum,n = 0,

where (xm, yn) ∈ Ωl (34)

Equations at different layers are coupled together via the continuity condition:

u+ = u−, (35)

1

ε+

∂u+

∂y=

1

ε−∂u−

∂y, (36)

where ‘+’ and ‘−’ denote the upper and lower side of an interface, respectively.Condition (35) implies it is unnecessary to distinguish the value on the two sides

of an interface. Condition (36) can be approximated by the finite difference withpoints on the the same vertical line. Here we use the second order approximationin order to be consistent with (34). For n = n1, n2, · · · , nL−1, the approximation isgiven by:

1

εl+1

1/2um,n+2 − 2um,n+1 + 3/2um,nhy

=1

εl

−1/2um,n−2 + 2um,n−1 − 3/2um,nhy

(37)

For simplicity, the first order approximation is applied to the homogenous Neu-mann boundary condition at ∂Ω. It is also used to discretize the normal deriv-ative and the improper integral at the aperture Γ, which results in the followingdiscretization[17]:

um,N+1 =

N∑n=1

Gm,n

(um,N+1 − um,N

hy

)+ gm, m = 1, 2, · · · ,M (38)

where

Gm,n =

i2

[1 + 2i

π ln(0.1638k0hx)]hx if n = m

i2H

(2)0 (k0|xn − xm|)hx if n 6= m

(39)

gm = 2eiαxm . (40)

For compactness, the scheme can be rewritten in the matrix form:

(Ax ⊗ I∗N + IM ⊗A∗y + IM ⊗D∗)u+ (IM ⊗ aN+1)uN+1 = 0

GuN + (hyIM −G)uN+1 = hyg (41)

RCS REDUCTION FOR A CAVITY IN TE POLARIZATION 429

where ⊗ denotes the tensor product(Kronecker product), IM is the M ×M identitymatrix, I∗N is similar to the N ×N identity matrix but the entry at (nl, nl) is zerofor l = 1, 2, · · · , L− 1, and G = (Gm,n)Mm=1.n=1.

For the other matrices, their forms are:

Ax =1

h2x

−1 11 −2 1

. . .. . .

. . .

1 −2 11 −1

M×M

aN+1 =1

h2y

00. . .

01

(N+1)×1

g =

g1

g2

. . .

gM−1

gM

(M+1)×1

(42)

A∗y =1

h2y

−1 11 −2 1

. . .. . .

. . .

1 −2 112εl+1

εl−2 εl+1

εl32

(1 + εl+1

εl

)−2 1

2

−1 −2 1. . .

. . .. . .

1 −2 11 −2

N×N

· · ·nlth

(43)

and

D∗ = diag(ε1, · · · , ε1, 0, ε2, · · · , εL−1, 0, εL, · · · , εL), (44)

u = (u1,1, · · · , u1,N , u2,1, · · · , u2,N , · · · , uM,1, · · · , uM,N )T ,

un = (u1,n, · · · , uM,n)T

where D∗ is a diagonal matrix with zero appearing at the nlth row, l = 1, 2, · · · , L.We then apply the discrete cosine transform to Ax and the rest of the algorithm

is completely analogy to the method introduced in [8], which we will omit and referthe readers to [8] for details. We also omit the proof for the unique solvability ofthe resulting linear system (41) for conciseness. As the algorithm highly utilizesthe special structure of the cavity problem, it is extremely fast and efficient, whichsatisfies the requirement for a computation engine of the optimization method.

6. Numerical experiments. In this section, we apply our algorithm to the RAMdesign through three numerical examples. Throughout, we assume the dimension ofthe cavity is 1m in width and 0.3m in depth. In other words, Ω = [0, 1]× [−0.3, 0].We partition Ω uniformly by a 512×512 mesh. The thickness of the coating materialfor each layer is 2.3mm, with four layers in total. Assume the relative permittivityis a continuous variable between 1 + 0i and 100 + 100i. Note that these settings areonly used for illustration purpose. Our algorithm applies to any finite number oflayers and any finite bounds of the permittivity. All the numerical experiments arecarried out on a laptop with 2.1Ghz Intel dual core and 4GB memory. For physical

430 GANG BAO AND JUN LAI

interest, the RCS is expressed in terms of decibel(dB), which is

RCS: = 10 log10 σ dB.

Initially, we assume the layered RAM has a uniform permittivity, namely, 5 + 5i.As we mentioned before, the resulted permittivity of the material after the optimiza-tion may highly depend on the initial value. We certainly have tried other initialvalues, but no significant improvement was observed in terms of RCS reduction.Therefore, all the examples shown here are based on the same initial value.

Example 1. Consider the scattering from a cavity by the incidence of a plane wavewith wavenumber k0 = 16π. Figure 2(a) gives the result for the optimization atθ = 3π

5 . RCS is reduced by more than 100dB at θ = 3π5 . The reduction is not only

confined around θ = 3π5 . It also extends to the other angular sector with an average

reduction around 10dB, which can be considered as a large improvement. Similareffect has been shown in Figure 2(b), which gives the RCS when the optimizationis conducted at θ = 5π

6 . The deep well in the graph shows a large amount of

reduction at θ = 5π6 . Meanwhile, the RCS at the other angular sector also gets

reduced. Figure 2(c) shows the RCS reduction when the optimization is conductedby the combination of θ = 3π

5 and θ = 5π6 . The weights for these two angles are

set to be equal. Compared to the optimization at a single angle, optimization withmultiple angles avoids the sharp reduction at a particular angle and thus provides asmoother reduction for all the angles. The resulted relative permittivities are listedin Table 1 with the corresponding number of iterations and CPU time. Generally,the computational time for optimization with two angles are much longer than thetime spent for a single angle. These observations are consistent to the optimizationresult in TM polarization[7].

Example 2. Consider the incidence of a plane wave with wavenumber k0 = 32πon the cavity. Similar to Example 1, the optimization are performed at 3π

5 ,5π6 and

both. Figure 3(a) gives the result for the optimization at θ = 3π5 . RCS are reduced

by more than 100dB at θ = 3π5 . At the other angular sectors, RCS also gets reduced,

although the performance is not as good as θ = 3π5 . Figure 3(b) shows the result

for RCS optimization at θ = 5π6 . The deep well in the graph shows a large amount

of reduction at θ = 5π6 . Meanwhile, the reduction is also obtained at the other

angles. Figure 3(c) gives the optimized RCS when the optimization is conductedby the combination of θ = 3π

5 and 3π5 . The sharp reduction at a particular angle

disappears but the overall reduction gets improved. Table 1 contains all the resultedrelative permittivities with the corresponding number of iterations and CPU time.It again shows more time is needed for the optimization over the combination ofmultiple angles. As shown in [7], the first few steps during the iteration achievesmost of the reduction for multiple angles. The computational time therefore canbe significantly reduced by terminating the algorithm if only minor improvementappears.

Example 3. In the last example, we test our algorithm by showing a RCS reductionover a range of frequencies. In particular, we consider the reduction between 16πand 32π, which implies the frequency ranges from 2.4Ghz to 4.8Ghz. Take theobjective function as a combination of RCS at k = 16π and k = 32π. The formulais given in (13) with equal weights for the two wavenumber. Results are shown inFigure 4 for both θ = π

2 and θ = 3π5 and the corresponding CPU time are given in

Table 1. From Figure 4(a), we see some reduction at the two end. However, the

RCS REDUCTION FOR A CAVITY IN TE POLARIZATION 431

(a)

100 120 140 160 180−100

−50

0

50

θ (Degree)

RC

S (

dB

)

Without coatingWith coating

(b)

100 120 140 160 180−120

−100

−80

−60

−40

−20

0

20

40

θ (Degree)

RC

S (

dB

)

Without coatingWith coating

(c)

100 120 140 160 180−30

−20

−10

0

10

20

30

θ (Degree)

RC

S (

dB

)

Without coatingWith coating

Figure 2. RCS reduction for k0 = 16π. (a)Optimized at θ = 3π5 .

(b)Optimized at θ = 5π6 . (c)Optimized with the combination of

θ = 3π5 and θ = 5π

6 .

overall effect is merely some kind of horizontal shift of RCS. Generally it implies areduction at one frequency accompanies an enhancement at another frequency fornormal incidence. The situation is changed for oblique incidence, as shown in 4(b),which gives an overall reduction for all the frequencies. As mentioned in [7], thedifference between these two can be explained by ray tracing. For normal incidence,most of the rays are directly reflected back to the same direction. While for obliqueincidence, rays are bouncing back and forth inside the cavity, so more energy canbe absorbed by the RAM as compared to normal incidence. In addition, most ofthe rays shooting out of the cavity are not in the same direction as they come from.Therefore, more reduction is observed for oblique incidence.

7. Conclusion. In this paper, the design of a multilayered RAM for reducing RCSof a cavity is formulated as a minimization problem. The descent direction for thecost function is evaluated through the adjoint state method. Subsequently, SQPis integrated with the gradient to obtain the optimal absorbing materials for thecavity. The algorithm is implemented with a fast and accurate direct solver for thescattering problem. Numerical results show that RCS is reduced significantly with

432 GANG BAO AND JUN LAI

(a)

100 120 140 160 180−150

−100

−50

0

50

θ (Degree)

RC

S (

dB

)

Without coatingWith coating

(b)

100 120 140 160 180−150

−100

−50

0

50

θ (Degree)

RC

S (

dB

)

Without coatingWith coating

(c)

100 120 140 160 180−40

−30

−20

−10

0

10

20

30

θ (Degree)

RC

S (

dB

)

Without coatingWith coating

Figure 3. RCS reduction for k0 = 32π. (a)Optimized at θ = 3π5 .

(b)Optimized at θ = 5π6 . (c)Optimized with the combination of

θ = 3π5 and θ = 5π

6 .

2.5 3 3.5 4 4.5−40

−30

−20

−10

0

10

20

30

40

Frequency (GHz)

RC

S (

dB

)

Without coatingWith coating

2.5 3 3.5 4 4.5−40

−30

−20

−10

0

10

20

Frequency (GHz)

RC

S (

dB

)

Without coatingWith coating

Figure 4. RCS reduction for k between 16π and 32π.(a)Optimized at θ = π

2 . (b)Optimized at θ = 3π5 .

RCS REDUCTION FOR A CAVITY IN TE POLARIZATION 433

3)

Example 1 Example 2 Example 3k 16π 16π 16π 32π 32π 32π 16π-32π 16π-32πθ 3π

55π6

3π5 ,

5π6

3π5

5π6

3π5 ,

5π6

π2

3π5

1st(Re) 4.669 5.374 1.000 4.809 5.004 1.000 1.000 93.0981st(Im) 6.737 5.066 3.293 4.925 4.936 0.000 0.009 62.3672nd(Re) 6.786 7.005 1.000 3.659 4.793 54.316 1.000 20.880

2nd(Im) 6.341 4.837 2.662 4.370 4.775 25.262 0.000 18.1163rd(Re) 11.107 10.176 29.271 2.892 4.684 1.000 1.000 6.7693rd(Im) 6.767 4.931 0.000 2.732 4.384 5.669 0.000 0.042

4th(Re) 16.573 13.985 3.937 4.788 5.141 16.503 8.587 8.3384th(Im) 9.981 6.613 10.149 1.017 4.035 0.506 0.177 0.000

Iter. 15 16 100 10 11 129 24 81

CPU 48.9s 52.7s 1113s 50.5s 49.1s 2404s 33.8s 1420s

Table 1. Relative permittivities after the optimization, numberof iterations and CPU time for all the three examples.

a thin RAM coated at the bottom of the cavity. It is also observed that differ-ent weights in the cost function generally lead to different RAM design and RCSreduction. Future work includes considering RCS reduction for three dimensionalcavities.

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Received October 2013; revised March 2014.

E-mail address: drbaogang@gmail.com

E-mail address: lai@cims.nyu.edu