Multiple Cavities Scattering Problem

# Analysis of Time-domain Electromagnetic Scattering Problem by Multiple Cavities

Yang Liu School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, P.R.China Yixian Gao School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, P.R.China  and  Jian Zu School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, P.R.China
###### Abstract.

Consider the time-domain multiple cavity scattering problem, which arises in diverse scientific areas and has significant industrial and military applications. The multiple cavity embedded in an infinite ground plane, is filled with inhomogeneous media characterized by variable dielectric permittivities and magnetic permeabilities. Corresponding to the transverse electric or magnetic polarization, the scattering problem can be studied for the Helmholtz equation in frequency domain and wave equation in time-domain, respectively. A novel transparent boundary condition in time-domain is developed to reformulate the cavity scattering problem into an initial-boundary value problem in a bounded domain. The well-posedness and stability are established for the reduced problem. Moreover, a priori estimates for the electric field is obtained with a minimum requirement for the data by directly studying the wave equation.

###### Key words and phrases:
Helmholtz equation, wave equation, multiple cavities, stability, a priori estimates
The research of YG was supported in part by NSFC grant 11871140, JJKH20180006KJ, JLSTDP 20190201154JC, 20160520094JH and FRFCU2412019BJ005. The research of JZ was supported in part by NSFC grand 11571065,11671071.

## 1. Introduction

This paper is concerned with the mathematical analysis of the time-domain electromagnetic scattering problem of multiple cavities, which is embedded in a conducting ground planes. The cavity scattering problem arises in diverse scientific areas and has significant industrial and military applications, including the design of cavity-backed conformal antennas for civil and military use, and the characterization of radar cross-section (RCS) of vehicles with grooves, especially to design RCS. It is used to detect airplanes in a wide variation of ranges. For instance, a stealth aircraft will have design features that give it a low RCS, as opposed to a passenger airliner that will have a high RCS. RCS is integral to the development of radar stealth technology, particularly in applications involving aircraft and ballistic missiles. The cavity RCS caused by jet engine inlet ducts or cavity-backed antennas can dominate the total RCS. A thorough understanding of the electromagnetic scattering characteristic of a target, particularly a cavity, is necessary for successful implementation of any desired control of its RCS.

The descriptions of cavity scattering problem were centered on methods developed in the time-harmonic and time-domain. For the time-harmonic problems were introduced firstly by engineers [17, 16, 18, 20, 29]. The mathematical analysis of the cavity scattering problem was given by three fundamental papers [1, 2, 3], where the existence and uniqueness of the solutions were obtained based on a non-local transparent boundary condition on the cavity opening. A large amount of information was available regarding their solutions for both the two-dimensional Helmholtz and the three-dimensional Maxwell equations[8, 4, 5, 7, 22, 26, 25, 28]. A good survey to the problem of cavity scattering can be found in [23]. The time-domain scattering problems have recently attracted considerable attention due to their capability of capturing wide-band signals and modeling more general material and nonlinearity[9, 19, 21, 30], which motivates us to tune our focus from seeking the best possible conditions for those physical parameters to the time-domain problem. Comparing with the time-harmonic problems, the time-domain problems are less studied due to the additional challenge of the temporal dependence. The analysis can be found in [12, 13, 14, 6, 15, 31] for the time-domain acoustic, elastic and electromagnetic scattering problems in different structures including bounded obstacles, periodic surfaces, and unbounded rough surfaces. Inspired by the one open cavity structure in [24], we extends the results to the multiple cavity scattering problem. It appears more complicated because of the unbounded nature of the domain and the novel transparent boundary condition on multiple apertures. Utilizing the Laplace transform as a bridge between the time-domain and the frequency domain, we develop an exact time-domain transparent boundary condition (TBC) and reduce the scattering problem equivalently into an initial boundary value problem in a bounded domain. Using the energy method with new energy functions, we can show the well-posedness and stability of the time-domain multiple cavity scattering problem.

The paper is organized as follow. In section 2, we introduce the model problem of one cavity scattering problem and establish a time-domain TBC. Section 3 is concentrated on the analysis of two cavities scattering problem, while the well-posedness and stability are addressed in both the frequency and time-domain. The multiple cavity problem is proposed in section 4, while a priori estimates with explicit time dependence for the quantities of electric filed is obtained with a minimum requirement for the data by directly studying the wave equation. We conclude the paper with some remarks in section 5.

## 2. one cavity scattering problem

In this section, we shall introduce the mathematical model for a single cavity scattering problem and develop an exact TBC to reduce the scattering problem from an unbounded domain into a bounded domain.

### 2.1. Problem formulation

Consider a simpler model for the open cavity scattering problem by assuming that the medium and material are invariant along the -axis. Let be the cross section of a -invariant cavity with a Lipschitz continuous boundary , as seen in Figure 1. The cavity is filled with some inhomogeneous medium, characterized by the variable dielectric permittivity and magnetic permeability . The exterior region is filled with some homogeneous material with a constant permittivity and a constant permeability . Here the cavity wall is assumed to be a perfect electric conductor and the cavity opening is aligned with the perfectly electrically conducting infinite ground surface . An open cavity , enclosed by the aperture and the wall , is placed on a perfectly conducting ground plane .

The electromagnetic wave propagation is governed by the time-domain Maxwell equations

 {∇×E(r,t)+μ∂tH(r,t)=0,∇×H(r,t)−ε∂tE(r,t)=0, (2.1)

where is the electric field, is the magnetic field, and are the dielectric permittivity and magnetic permeability, respectively, and satisfy

 0<εmin≤ε≤εmax<∞,0<μmin≤μ≤μmax<∞,

while are constants. The system is constrained by the initial conditions

 E∣∣t=0=0,H∣∣t=0=0. (2.2)

Since the structure is invariant in the -axis, the problem can be decomposed into two fundamental polarizations: transverse electric (TE) and transverse magnetic (TM). The three-dimensional Maxwell equations can be reduced to the two-dimensional wave equation.

(i) TE polarization: the magnetic field is transverse to the -axis, the electric and magnetic fields are

 E(r,t)=[0,0,u(ρ,t)]⊤,H(r,t)=[H1(ρ,t),H2(ρ,t),0]⊤,

where Eliminating the magnetic field from (2.1), we get the wave equation for the electric field

 ε∂2tu−∇⋅(μ−1∇u)=0in   Ωe∪Ω,  t>0. (2.3)

By the perfectly conducting boundary condition on the ground plane and cavity wall we can get

 u=0on     S∪Γc,    t>0.

It follows from the initial condition (2.2) that satisfies the homogeneous initial conditions

 u(ρ,t)∣∣t=0=0,∂tu(ρ,t)∣∣t=0=0in  Ωe∪Ω.

(ii) TM polarization: the electric field is transverse to the -axis, the electric and magnetic fields are

 E(r,t)=[E1(ρ,t),E2(ρ,t),0]⊤,H(r,t)=[0,0,u(ρ,t)]⊤.

We may eliminate the electric field from (2.1) and obtain the wave equation for the magnetic field

 μ∂2tu−∇⋅(ε−1∇u)=0in    Ωe∪Ω,    t>0. (2.4)

It also follows from the perfectly conducting boundary condition on the ground plane and cavity wall that

 ∂νu=0on  S∪Γc,    t>0,

where is the unit outward normal vector on The initial conditions for the TM is

 u(ρ,t)∣∣t=0=0,∂tu(ρ,t)∣∣t=0=0in   Ωe∪Ω.

It is clear to note from (2.3) and (2.4) that TE and TM polarizations can be handled in a unified way by formally exchanging the roles of and . We will just discuss the results in detail by using (2.3) (TE case) as the model equation in the rest of the paper. The method can be extended to the TM polarization with obvious modifications.

Let an incoming plane wave be incident on the cavity from above, where is a smooth function and its regularity will be specified later, and Clearly, the incident field satisfies the wave equation (2.3) with The total field can be split into the incident field, the reflected field and the scattered field:

 u=uinc+ur+usc,

where (or ) is the reflected field in TE (or TM) case . To impose the initial conditions, we assume that the total field, the incident field and the reflected field vanish for , so that the scattered field for . Moreover, the scattered field is required to satisfies the Sommerfeld radiation condition:

 1√ε0μ0∂rusc+∂tusc=o(r−1/2)as  r=|ρ|→∞,   t>0. (2.5)

To analyze the problem, the open domain needs to be truncated into a bounded domain. Therefore, a suitable boundary condition has to be imposed on the boundary of the bounded domain so that no artificial wave reflection occurs. We shall present a transparent boundary condition on the open domain enclosing the inhomogeneous cavity.

#### 2.1.1. Laplace transform and some notation

We first introduce the Laplace transform and present some identities for the transform. For any with , define by the Laplace transform of the function , i.e.,

 ˘u(s)=L(u)(s)=∫∞0e−stu(t)dt.

Using the integration by parts yields

 ∫t0u(τ)dτ=L−1(s−1˘u(s)),

where is the inverse Laplace transform. One verify form the formula of the inverse Laplace transform that

 u(t)=F−1(es1tL(u)(s1+is2)),

where denotes the inverse Fourier transform with respect to

Recalling the Plancherel or Parseval identity for the Laplace transform (cf. [10, (2.46)])

 12π∫∞−∞˘u(s)˘v(s)ds2=∫∞0e−2s1tu(t)v(t)dt,∀ s1>σ0>0, (2.6)

where and is abscissa of convergence for the Laplace transform of and

Hereafter, the expression stands for , where is a positive constant and its specific value is not required but should be always clear from the context.

The following lemma (cf.[27, Theorem 43.1]) is an analogue of Paley–Wiener–Schwarz theorem for Fourier transform of the distributions with compact support in the case of Laplace transform.

###### Lemma 2.1.

Let denote a holomorphic function in the half-plane , valued in the Banach space . The two following conditions are equivalent

1. there is a distribution whose Laplace transform is equal to ,

2. there is a real with and an integer such that for all complex numbers with it holds that ,

where is the space of distributions on the real line which vanish identically in the open negative half line.

Next, we introduce some function space notation. Let be a bounded Lipschitz domain with boundary Denote the Sobolev space: . To describe the boundary operator and transparent boundary condition in the formulation of the boundary value problem, we define the trace functional space

 Hν(R)={u∈L2(R):∫R(1+ξ2)ν|^u|2dξ<∞},

whose norm is defined by

 ∥u∥Hν(R)=(∫R(1+ξ2)ν|^u|2dξ)1/2,

where is the Fourier transform of defined as

 ^u(ξ)=∫Ru(x)eixξdx.

It is clear to note that the dual space of is under the inner produce

 ⟨u,v⟩=∫Ru¯vdx=∫R^u¯^vdξ.

#### 2.1.2. Transparent boundary condition

We introduce a time-domain TBC to formulate the cavity scattering problem into an equivalent initial-boundary value problem in a bounded domain. The idea is to design a Dirichlet–to–Neumann (DtN) operator which maps the Dirichlet data to the Neumann data of the wave field. More precisely, we will address the reduced initial-boundary value problem

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩ε∂2tu−∇⋅(μ−1∇u)=0in   Ω,   t>0,u∣∣t=0=0,∂tu∣∣t=0=0in   Ω,u=0on   S,    t>0,∂nu=T u+gon   Γ,    t>0, (2.7)

where is a time-domain boundary operator and will be given later. In what follows, we derive the formulation of the operator and analyze its important properties.

Since in the equations (2.3) and (2.4) together with the radiation condition (2.5) implies the scattered field satisfies

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩Δusc−ε0μ0∂2tusc=0in   Ωe,   t>0,usc∣∣t=0=0,∂tusc∣∣t=0=0in     Ωe,usc=−(uinc+ur)on   Γc,   t>0,(ε0μ0)−1/2∂tus+∂rus=o(r−1/2)as  r=|ρ|→∞,   t>0. (2.8)

Let be the Laplace transform of with respect to . Recalling that

 L(∂tu)=s˘u(⋅,s)−u(⋅,0),L(∂2tu)=s2˘u(⋅,s)−su(⋅,0)−∂tu(⋅,0).

Taking the Laplace transform of (2.8) with the initial conditions, we can get the time-harmonic Helmhlotz equation for the scattered field with the complex wave number

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩Δ˘usc−s2c2˘usc=0in Ωe,˘usc=−(˘uinc+˘ur)% on   Γc,sc˘usc+∂r˘usc=o(r−1/2)%as  r=|ρ|→∞, (2.9)

where is the light speed in the free space.

By taking the Fourier transform of the first equation in (2.9) with respect to , we have an ordinary differential equation with respect to :

 (2.10)

It follows form the radiation condition in (2.9), we deduce that the solution of (2.10) has the analytical form

 ^˘usc=^˘usc(ξ,0)eβ(ξ)y, (2.11)

where

 β(ξ)=(ξ2+s2c2)1/2 with  Re(β(ξ))<0. (2.12)

Taking the inverse Fourier transform of (2.11), we find that

 ˘usc(x,y)=∫R^˘usc(ξ,0)eβ(ξ)ye−ixξdξin Ωe.

Taking the normal derivative on and evaluating at yields

 ∂n˘usc(x,y)∣∣y=0=∫Rβ(ξ)^˘us(ξ,0)e−ixξdξ, (2.13)

where is the unit outward normal on , i.e.

For any with , define the boundary operator

 Bw:=∫Rβ(ξ)^w(ξ,0)e−ixξdξ, (2.14)

which leads to a transparent boundary condition for the scattered field on :

 ∂n˘usc=B˘usc.

From , we can get an equivalent transparent boundary condition for the total field

 ∂n˘u=B˘u+˘gon  Γc∪Γ, (2.15)

where .

Taking the inverse Laplace transform of (2.15) yields the TBC in the time-domain

 ∂nu=Tu+g on   Γc∪Γ,    t>0, (2.16)

where and .

Since is defined on and the transparent boundary condition above is derived for . In order to derive the transparent boundary condition for the total field on , we make the zero extension as follows: for any given on , define

 ~u(x)={ufor  x∈Γ,0for  x∈Γc.

Since the cavity is placed on a perfectly conducting ground plane , i.e. the total filed is required to be zero on , it is obviously that above zero extension is consistent with the problem geometry. Based on the extension and the transparent boundary conditions (2.15) and (2.16), we have the transparent boundary conditions for the total field on the opening

 ∂n˘u=B˘~u+˘gon  Γ;∂nu=T~u+gon  Γ, t>0.

Define a dual paring by

 ⟨u,v⟩Γ=∫Γu¯vdγ.

By the definition of extension, this dual paring for and is equivalent to the scalar product in for their extension, i.e.,

 ⟨u,v⟩Γ=⟨~u,~v⟩.

The following lemmas are useful in the proof of the well-posedness of the reduced problem.

###### Lemma 2.2.

The boundary operator is continuous, i.e.,

 ∥Bu∥H−1/2(R)≤C∥u∥H1/2(R),∀u∈H1/2(R).
###### Proof.

For any , it follows from the definitions (2.14) that

 ⟨Bu,v⟩=∫RBu ¯vdξ=∫Rβ(ξ)(1+ξ2)1/2(1+ξ2)1/4u⋅(1+ξ2)1/4¯vdξ.

To prove the lemma, it is required to estimate

 |β(ξ)||(1+ξ2)|1/2,−∞<ξ<∞.

Let

 s2c2=a+ib,a:=s21−s22c2,b:=2s1s2c2.

Denote

 β2(ξ)=s2c2+ξ2=ϕ+ib,

where A simple calculation gives

 |β(ξ)||(1+ξ2)|1/2=[ϕ2+b2(1+ϕ−a)2]1/4.

Define

 F(t)=t2+b2(1+t−a)2,t≥a.

It follows

 F′(t)=2(1+t−a)(t(1−a)−b2)(1+t−a)4.

We consider it in two cases:
(i) . It can be verified that the function decreases for and increase for Thus

 F(ϕ)≤max{F(a)=a2+b2,  F(+∞)=1}.

(ii) It is easy to verify that decreases for Thus, we have

 F(ϕ)≤F(a)=a2+b2.

Combining above estimates and using the Cauchy–Schwarz inequality yield

 |⟨Bu,v⟩|≤C∥u∥H1/2(R)∥v∥H1/2(R),

where

 C=max{(a2+b2)1/4,1}.

Thus we have

 ∥Bu∥H−1/2(R)≤supv∈H1/2(R)|⟨Bu,v⟩|∥v∥H1/2(R)≤C∥u∥H1/2(R).

It follows from Lemma 2.1 and Lemma 2.2 that the inverse Laplace transform in (2.16) is make sense.

###### Lemma 2.3.

It holds that

 −Re⟨(sμ)−1Bu,u⟩≥0,u∈H1/2(R).
###### Proof.

By the definition (2.14), we find

 −⟨(sμ)−1Bu,u⟩=−∫R(sμ)−1β(ξ)|u|2dξ=−∫R¯sβ(ξ)μ|s|2|u|2dξ.

Let with . Taking the real part of the above equation gives

 −Re⟨(sμ)−1Bu,u⟩=−∫Rs1ς+s2ϱμ|s|2|u|2dξ. (2.17)

Recalling , we have

 ς2−ϱ2=ξ2+c−2(s21−s22),ςϱ=c−2s1s2. (2.18)

Using (2.18), it gives

 s1ς+s2ϱ=s1ς(ς2+c−2s22). (2.19)

Substituting (2.19) into (2.17), we have

 −Re⟨(sμ)−1Bu,u⟩=−∫R1μ|s|2s1ς(ς2+c−2s22)|u|2dξ≥0,

which completes the proof. ∎

###### Lemma 2.4.

For any with initial value it holds that

 −Re∫T0⟨Tu(⋅,t),∂tu(⋅,t)⟩Γdt≥0.
###### Proof.

Let be the extension of with respect to in such that outside the interval , and be the Laplace of . By the Parseval identity (2.6) and Lemma 2.3, we get

 −Re∫T0e−2s1t⟨Tu,∂tu⟩Γdt= −Re∫T0e−2s1t∫Γ(Tu)∂t¯udγdt = −Re∫Γ∫∞0e−2s1t(T~u)∂t¯~udtdγ = −12π∫∞−∞Re⟨B˘~u,s˘~u⟩Γds2 = −12π∫∞−∞|s|2μRe⟨(sμ)−1B˘~u,˘~u⟩Γds2≥0,

which completes the proof after taking . ∎

The following trace theorem are useful in the following reduced problem, the proof can be found in (cf. [11]).

###### Lemma 2.5.

(trace theorem) Let be a bounded Lipschitz domain with boundary For the interior trace operator

 T0:Hν(Ω)→Hν−1/2(Γ)  is ~{}~{} % bounded,∀w∈Hν(Ω),

where .

### 2.2. The reduced one cavity scattering problem

In this section, we will present the well-posedness of the reduced problem by a variation method, and given the stability of one cavity scattering problem.

#### 2.2.1. well-posedness in the s-domain

Taking the Laplace transform of (2.7) and using the transparent boundary condition, we may consider the following reduced boundary value problems

 ⎧⎪⎨⎪⎩sε˘u−∇⋅((sμ)−1∇˘u)=0in    Ω,˘u=0on    S,∂r˘u=B˘u+˘gon    Γ, (2.20)

where , with .

By multiplying a test function and integrating by parts, we arrive at the variational formulation of (2.20): find such that

 a1(˘u,v)=⟨˘g,v⟩Γ,∀v∈H1S(Ω), (2.21)

where the sesquilinear form

 a1(˘u,v)=∫Ω((sμ)−1∇˘u⋅∇¯v+sε˘u¯v)dρ−⟨(sμ)−1B˘u,v⟩Γ. (2.22)
###### Theorem 2.6.

The variational problem (2.21) has a unique solution which satisfies

 ∥∇˘u∥L2(Ω)2+∥s˘u∥L2(Ω)≲s−11∥s˘g∥H−1/2(Γ). (2.23)
###### Proof.

It suffices to show the coercivity of the sesquilinear form of . The continuity of sesquilinear form follows directly from the Cauchy–Schwarz inequality, Lemma 2.2 and Lemma 2.5

 |a1(˘u,v)|≤ 1|s|μmin∥∇˘u∥L2(Ω)2∥∇v∥L2(Ω)2+|s|εmax∥˘u∥L2(Ω)∥v∥L2(Ω) +1|s|μmin∥B˘u∥H−1/2(Γ)∥v∥H1/2(Γ) ≲ ∥˘u∥H1(Ω)∥v∥H1(Ω).

Letting in (2.22), we get

 a1(˘u,˘u)=∫Ω((sμ)−1|∇˘u|2+sε|˘u|2)dρ−⟨(sμ)−1B˘u,˘u⟩Γ. (2.24)

Taking the real part of (2.24) and using Lemma 2.3, yields

 Re(a1(˘u,˘u))≥C1s1|s|2(∥∇˘u∥2L2(Ω)2+∥s˘u∥2L2(Ω)), (2.25)

where .

It follows from the Lax–Milgram lemma that the variational problem (2.21) has a unique solution Moreover, we have from (2.21) that

 |a1(˘u,˘u)|≤|s|−1∥˘g∥H−1/2(Γ)∥s˘u∥L2(Ω). (2.26)

 ∥∇˘u∥2L2(Ω)2+∥s˘u∥2L2(Ω)≲s−11∥s˘g∥H−1/2(Γ)∥s˘u∥L2(Ω),

which gives estimate of (2.23) after applying the Cauchy–Schwarz inequality.

#### 2.2.2. well-posedness in the time-domain

Using the time-domain transparent boundary condition, we consider the reduced initial-boundary value problem

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩ε∂2tu−∇⋅(μ−1∇u)=0in   Ω,   t>0,u∣∣t=0=0,∂tu∣∣t=0=0in   Ω,u=0on   S,    t>0,∂ru=Tu+gon   Γ,    t>0. (2.27)
###### Theorem 2.7.

The initial-boundary problem (2.27) has a unique solution , which satisfies

 u∈L2(0,T;H1S(Ω))∩H1(0,T;L2(Ω)),

and the stability estimate

 maxt∈[t,T](∥∂tu∥L2(Ω)+∥∂t(∇u)∥L2(Ω)2)≲∥g∥L1(0,T;H−1/2(Γ))+maxt∈[t,T]∥∂tg∥H−1/2(Γ)+∥∂2tg∥L1(0,T;H−1/2(Γ)). (2.28)
###### Proof.

First, we have

 ∫T0(∥∇u∥2L2(Ω)2+∥∂tu∥2L2(Ω))dt ≤∫T0e−2s1(t−T)(∥∇u∥2L2(Ω)2+∥∂tu∥2L2(Ω))dt =e2s1T∫T0e−2s1t(∥∇u∥2L2(Ω)2+∥∂tu∥2L2(Ω))dt ≲∫∞0e−2s1t(∥∇u∥2L2(Ω)2+∥∂tu∥2L2(Ω))dt.

Hence it suffices to estimate the integral

 ∫∞0e−2s1t(∥∇u∥2L2(Ω)2+∥∂tu∥2L2(Ω))dt.

Let By Theorem 2.6, we have

 ∥∇˘u∥2L2(Ω)2+∥s˘u∥2L2(Ω)≲s−21|s|2∥˘g∥2H−1/2(Γ)≲s−21|s|2∥˘uinc+˘ur∥2H1(Ω).

It follows from (cf.[27, Lemma 44.1]) that is a holomorphic function of on the half plane where is any positive constant. Hence we have from Lemma 2.1 that the inverse Laplace transform of exists and is supported in

One may verify from the inverse Laplace transform that

 ˘u=L(u)=F(e−s1tu),

where is the Fourier transform in . Recalling the Plancherel or Parseval identity for the Laplace transform in (2.6), it follows

 ∫∞0e−2s1t (∥∇u∥2L2(Ω)2+∥∂tu∥