Towards a Unified Approach to Electromagnetic Analysis of Objects Embeddedin Multilayers

Xiaochao Zhoua, Zekun Zhua, Shunchuan Yanga,∗

aBeihang University, Xueyuan Road 37, Haidian District, Beijing, China

Abstract

According to the surface equivalence theorem [1], any enclosed surface with an electric and magnetic currents en-forced on it can generate exactly the same fields as the original problem. The problem is how to construct the currentsources. In this paper, an efficient and accurate unified approach with an equivalent current density enforced on theoutermost boundary is proposed to solve transverse magnetic (TM) scattering problems by objects embedded in mul-tilayers. In the proposed approach, an equivalent current density is derived after the surface equivalence theoremis recursively applied on each boundary from innermost to outermost interfaces. Then, the objects are replaced bythe background medium and the equivalent electric current density only on the outermost boundary is derived. Thescattering problems by objects embedded in multilayers can be solved with the electric field integral equation (EFIE).Compared with the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) and other dual source formulations, theproposed approach shows significant benefits: only the surface electric current density instead of both the electric andmagnetic current densities is required to model the complex objects in the equivalent problem. Furthermore, the elec-tric current density is only enforced on the outermost boundary of objects. Therefore, the overall count of unknownscan be significantly reduced. At last, several numerical experiments are performed to validate its accuracy and effi-ciency. Although overhead is required to construct the intermediate matrixes, the overall performance improvement isstill significant as numerical results are shown. Therefore, it shows great potential useful in the practical engineeringapplications.

Keywords: electromagnetic scattering, multilayer, single-source formulation, surface equivalence theorem, surfaceintegral equation, transverse magnetic

1. Introduction

The objects embedded in multilayers are widely used in the practical engineering applications, such as multilayerintegrated circuits [2], thin layer coated fibers [3], power cables [4], coating aircrafts [5], and so on. There are variousnumerical methods proposed to model such objects. One of the powerful numerical tools is the method of moment(MOM) [6] based on the surface integral equations (SIEs), which is widely used due to its unknowns residing on theinterfaces of different homogenous media. Therefore, the overall count of unknowns is significantly less than those ofpartial differential equation (PDE) based methods, like the finite element method (FEM) [7] and the finite-differencetime-domain (FDTD) method [8], which requires volumetric meshes.

There are various SIEs, like the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) formulation [9], the com-bined tangential formulation (CTF) [10], can model objects embedded in multilayers. In these formulations, boththe surface equivalent electric and magnetic current densities are introduced and several two-region problems arerequired to be solved simultaneously. Therefore, as the layer number increases, the overall count of unknowns willgreatly increase.

To mitigate the problem above, a number of efforts are made to introduce single-source formulations. In [11], asingle-source surface-volume-surface formulation is proposed to model penetrable objects. In this formulation, the

∗Corresponding authorEmail address: scyang@buaa.edu.cn (Shunchuan Yang )

Preprint submitted to xxxx January 23, 2020

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volumetric integral operator is mapped from surface to volume operator and then back from volume to surface opera-tor. Therefore, significant efficiency improvement can be obtained. However, compared with the SIEs, it still requiresto evaluate the volume integral operator. Various other single-source formulations based on the surface equivalencetheorem are proposed in [12, 13, 14, 15]. Only single current source is required in those formulations. Therefore, moreefficient formulations can be obtained compared with their dual source counterparts. Through combing the magneticfield integral equation (MFIE) inside the conductor and electric field integral field (EFIE) outside the conductor andmathematically eliminating the magnetic current, a generalized impedance boundary condition (GIBC) is proposedto model the interconnects [16]. In [17], a single-source formulation based on the differential surface admittanceoperator (DSAO) to model high-speed interconnects is proposed. The DSAO is derived through the surface equiva-lent theorem and electric fields in the equivalent problem are enforced to equal to those in the original problem. Itscapability is further enhanced to model arbitrarily shaped interconnects [18], circular solid and hollow cables [19],circular hole cables [20], three dimensional scattering [21]. There are several equivalence principle algorithms (EPAs)proposed in [23], [25], [26], in which a fictitious surface is constructed and both the equivalent surface electric andmagnetic currents are enforced on the fictitious boundary. Those show better conditioning and possible less numberof knowns compared with direct solution of the MOM. Especially, in [22], each element in a large antenna array isreplaced by a fictitious enclosed surface and a single equivalent surface current density is introduced to ensure thefields unchanged, which is similar to the EPA [23]. This formulation shows better conditioning in the coefficientmatrix, therefore, better convergence properties and great performance improvement compared with the direct MOM.However, in those formulations, they are only applicable for penetrable objects and those two-regions problems arerequired to be solved simultaneously as the PMCHWT formulation. They still suffer from the problem stated above.

In this paper, we proposed a unified single-source SIE incorporated the DSAO to solve two dimensional transversemagnetic (TM) scattering problems by objects embedded in multilayers to address the problem above. The equivalenttheorem is recursively applied from the innermost to outermost boundaries. An equivalent object is then derived inthe proposed formulation, which is filled with the background medium, and an electric current density enforced onthe original outermost boundary. It should be noted that the proposed approach is significantly different from theapproach in [27], which is based on the definition of recursive Green function. In [27], the recursive Green functionis defined through recursively combing unit fictitious cells and then is used to solve the entire problem. However, theproposed approach directly uses the surface equivalent theorem on each interface and derives the equivalent currentdensity. Compared with the PMCHWT formulation [9], other two-region formulations [10], and the EPA basedtechniques [23, 25, 26], the overall count of unknowns of the proposed approach can be significantly reduced. Theformulation shows three features: (1) Only the single-source electric current is required. (2) The surface equivalentelectric current density is only enforced on the outermost boundary rather than all the boundaries between differenthomogenous media. (3) If the multilayered cylindrical objects are involved, the proposed method can avoid thetroublesome multilayer Green function evaluation, the proposed approach only requires to evaluate the free spaceGreen function. Compared with those single-source formulations [12, 13, 14, 15] and GIBC [16], the proposedapproach can model objects embedded in multilayers and only apply the single current source on the outermostboundary. The approach proposed in this paper can significantly simplify the analysis and provide a unified approachto analyze objects embedded in multilayers and reduce the overall number of unknowns for complex objects. Wereported some preliminary results upon it in [31] and this paper extensively and rigorously analyzed the proposedapproach.

The main contributions in this paper are summarized into three aspects as follows. (1) A novel and general SIE isproposed to model objects embedded in multilayers. In this formulation, the surface equivalence theorem is recursivelyapplied from the innermost to outermost interfaces. Only surface equivalent sources are enforced on the outermostboundary. Compared with other SIE based techniques, which solve two-region problems simultaneously, each tworegion problem is solved separately. Therefore, several small problems rather than a large one including all theunknowns residing on all the interfaces are solved in the proposed formulation. In addition, the proposed formulationis better conditioned than the PMCHWT formulation since unknowns only reside on the outermost boundary and anypossible geometry fine details are only implicitly included in the final system. Therefore, the proposed approach canimprove the conditioning and thus the convergence property of the final linear system. (2) The proposed approachis incorporated with the DSAO to further enhance the performance. Only an electric current density is required tobe enforced on the outermost boundary of objects, which retain the fields the same as those in the original problem.Unknowns involved only the electric current density in the proposed SIE only reside on the outermost boundary.

2

(a)

nJ

n

0 0 0, ,

0 0 0, ,

(b)

Figure 1: (a) The original model and (b) the equivalent model with inner media replaced by its background medium and enforcing the surfaceelectric current density ~Jn on the outermost boundary

Therefore, significant efficiency improvement can be obtained. (3) Two special scenarios of the proposed approachare derived. One is that perfectly electric conductor (PEC) scatters embedded in multilayers. The other is that afictitious boundary resides in the same medium. These are commonly used in the practical engineering problems, likethe antenna array modeling [22] and circuit modeling [16].

This paper is organized as follows. In Section 2, the configuration of objects embedded in multilayers is describedin detail. We proposed a new unified approach to model these objects in Section 3. We introduced a single sourceformulation incorporated with the DSAO for a single penetrable object in the free space. Then, we proposed arecursive approach for two layered embedded objects and further extended the proposed approach to model objectsembedded in multilayers. Two special scenarios including PEC objects embedded in multilayers and extension in thesame medium are derived based on the proposed approach. In Section 4, some remarks upon the performance uponthe proposed approach are made. Then, in Section 5, its performance in terms of convergence property, accuracy andefficiency are numerically validated through three numerical examples. At last, we draw some conclusions in Section6.

2. Methodology

2.1. The Problem ConfigurationIn this paper, a two-dimensional TM scattering problem by objects embedded in multilayers is considered as shown

in Fig. 1(a). To derive a unified approach to solve it, we resort to the equivalence theorem to obtain an equivalentmodel, in which objects are replaced by the background medium and the exterior fields are exactly the same as thoseof the original model. Our goal is to introduce only one single surface equivalent electric or magnetic current densityon the outermost boundary γn as shown in Fig. 1(b) in the equivalent model to keep the fields in outermost regionunchanged. We keep this goal in mind and will derive an equivalent current density enforced on the outmost boundaryγn from inner to exterior layer by layer. The original objects embedded in multilayers are shown in Fig. 1(a), where εi,µi denote the permittivity and the permeability of the ith layer medium, which is bounded by two adjacent boundaries,γi−1 and γi, respectively.

Before we derive the detailed formulations, let’s make more remarks on the single source formulations. Our goal isto derive an equivalent model as shown in the Fig. 1(b), in which the equivalent electric current density Ji is introducedon the fictitious boundary, γi. According to the equivalent theorem [29], the equivalent electric and magnetic currentdensities can be expressed as

Ji(~r) = Hti (~r) − Hti (~r), (1)

Mi(~r) = Eti (~r) − Eti (~r), (2)

where ~r ∈ γi, Eti (~r), Eti (~r), Hti (~r), Hti (~r) are the surface tangential electric and magnetic fields in the original andequivalent model, respectively. All quantities with denote their values in the equivalent model.

3

1 1 1, ,

(a)

1J

(b)

Figure 2: (a) The original model for a single penetrable object and (b) the equivalent model

Since the electric and magnetic fields in the equivalent model can be arbitrary, we can obtain the single electriccurrent source by enforcing that Eti (~r) = Eti (~r) and (1) and (2) are rewritten as

Ji(~r) , 0, Mi(~r) = 0. (3)

On the other hand, we can have the single magnetic current source by enforcing that Hti (~r) = Hti (~r) and (1) and (2)are expressed as

Mi(~r) , 0, Ji(~r) = 0. (4)

Both (3) and (4) can give us a single-source formulation. In this paper, we use (3) to obtain the single sourceintegral formulation and derive the surface equivalent electric current density using the contour integral method forobjects embedded in multilayers. In the proposed approach, we will derive the DSAO Ysn according to the surfaceequivalence theorem, which relates the outermost electric fields and currents of objects on γn.

2.2. The Single-source SIE for A Penetrable Object

Let us first consider a single penetrable object with the permittivity ε1, the permeability µ1 and the conductivityσ1, respectively, as shown in Fig. 2(a). Its boundary is denoted as γ1. The permittivity, the permeability and theconductivity of the background medium are ε0, µ0 and σ0, respectively. According to the equivalence theorem [29],an equivalent model, in which the object is replaced by its surrounding medium and a surface equivalent electriccurrent density is introduced on γ1 as shown in the Fig. 2(b), can be obtained. Interested readers are referred to [18]for more details.

Let’s consider that the penetrable object does not include any sources, electric fields must satisfy the followingscalar Helmholtz equation inside γ1

∇2E1 + k21E1 = 0, (5)

subject to the boundary conditionE1(~r)

∣∣∣~r∈γ1

= E1(~r)∣∣∣∣~r∈γ1

, (6)

where E1 and E1 denote the electric field inside γ1 for the original and equivalent models, respectively, and E1(~r)∣∣∣~r∈γ1

and E1(~r)∣∣∣∣~r∈γ1

denote their values on the inner side of γ1.

(5) can be solved through the second scalar Green theorem [30]. Then, the electric field E1 inside γ1 can beexpressed in terms of E1 and its normal derivative on γ1 as

T E1(~r) =

∮γ1

[G1(~r,~r′)

∂E1(~r′)∂n′

−∂G1(~r,~r′)

∂n′E1(~r′)

]dr′, (7)

where the constant T = 1/2 when the source and observation points are located on the same boundary, otherwise, T = 1and G1(~r,~r′) is the Green function expressed as G1(~r,~r′) = − jH(2)

0 (k1ρ)/4, where j =√−1, k1 is the wavenumber in

4

the penetrable object and H(2)0 (·) is the zeroth-order Hankel function of the second kind. In addition, the tangential

magnetic field relates to the electric field on γ1 through the Poincare-Steklov operator [17]

H1(~r) =1

jωµ1

∂E1(~r)∂n

∣∣∣∣∣∣~r∈γ1

, (8)

where µ1 is the permeability of the object. We discretize γ1 into m1 segments and use the pulse basis function toexpand E1 and H1 on γ1 in (7) and (8) as

E1(~r) =

m1∑n=1

en fn(~r), (9)

H1(~r) =

m1∑n=1

hn fn(~r), (10)

where fn(~r) denotes the nth basis function. We use the Galerkin scheme to test (7) and (8) at each segment of γ1 andcollect all E1 and H1 expansion coefficients into two column vectors E1 and H1 as

E1 =[

e11 e12 ... e1m1

]T, (11)

H1 =[

h11 h12 ... h1m1

]T. (12)

Then, (7) can be rewritten into the matrix form as

12

L1E1 = P(1)1 H1 + U(1)

1 E1, (13)

where L1 is a diagonal matrix with the length of each segment as its entities on γ1, the subscript 1 denotes that thetesting procedure is applied on γ1, and the superscript (1) denotes the equivalence theorem applied on γ1 in the nextsubsection, and elements of P(1)

1 and U(1)1 are expressed as

[U(1)

1

]m,n

=

∫γ1m

∫γ1n

k1

~dm · n′

dmG′1(~r,~r′)dr′dr, (14)

[P(1)

1

]m,n

=

∫γ1m

∫γ1n

ωµ1 jG1(~r,~r′)dr′dr. (15)

where G′

1 = − jH(2)1 (k1ρ) /4. Then, through inversing the square matrix P(1)

1 , we obtain the surface admittance operator(SAO) Y1 [18] as

H1 = [P(1)1 ]−1

(12

L1 − U(1)1 )︸ ︷︷ ︸

Y1

E1. (16)

When all the parameters are replaced by those of its surrounding medium, the equivalent model is obtained asshown in Fig. 2(b). The H1(~r) is expanded through the pulse basis function as

H1(~r) =

m1∑n=1

hn fn(~r). (17)

With the similar procedure in the original model, for the equivalent model we can obtain

H1 = [P(1)1 ]−1

(12

L1 − U(1)1 )︸ ︷︷ ︸

Y1

E1, (18)

5

!γ

!γ

! ! !" "ε µ σ

! ! !" "ε µ σ

ε0, µ0,σ0�k�Ein

(a)

!γ

!γ

! ! !" "ε µ σ

!!

!!"

�k�Ein

(b)

!γ

!γ

ε0, µ0,σ0

�J2�k

�Ein

(c)

Figure 3: (a) The original two layered model, (b) the equivalent model with the innermost medium replaced by its surrounding (the second) mediumand enforcing the surface current density ~J1 on γ1, and (c) the equivalent model with the inner medium replaced by its surrounding medium andenforcing the surface current density ~J2 on γ2

whereH1 =

[h11 h12 ... h1m1

]T(19)

and H1 denotes the discretized magnetic field expansion coefficient vector in the equivalent model.The surface equivalent current density on γ1 is expanded through pulse functions, and the coefficients are extracted

into the column vector J1,J1 =

[j11 j12 ... j1m1

]T. (20)

Since (3) is enforced, only single equivalent electric current density J1 is required. By substituting (16) and (18) into(3), J1 is obtained as

J1 = Ys1 E1, (21)

where Ys1 is the DSAO [18] and can be expressed as

Ys1 = Y1 − Y1 =[P(1)

1

]−1(

12

L1 − U(1)1

)−

[P(1)

1

]−1(

12

L1 − U(1)1

). (22)

2.3. The Single-source SIE for Objects Embedded in Two Layered Media

2.3.1. The original modelWe now consider a slightly more complex scenario in which a layered medium with the permittivity ε2, the

permeability µ2 and the conductivity σ2 encloses a penetrable object as shown in Fig. 3(a). For this problem, wederive an equivalent model similar to that in previous subsection, in which objects are replaced by its surroundingmedium with the permittivity ε0, the permeability µ0 and the conductivity σ0, and an equivalent current density ~J1on γ1 as shown in Fig. 3(b) is enforced to ensure the fields in the outer region unchanged. By comparing with Fig.2(a) and Fig.3 (b), it is easy to find that this problem is similar to each other. The only difference is that there is anequivalent current density ~J1, which is derived in the previous subsection, inside the new equivalent object. It shouldbe careful to handle this current density. In γ2, the Helmholtz equation is used to calculate electric fields in the innerregion, and note that the inhomogeneous Helmholtz equation is required due to the existence of the equivalent surfacecurrent density ~J1 introduced in previous subsection to represent the penetrable object. Therefore, the inhomogeneousHelmholtz equation can be expressed as

∇2E2 + k22E2 = jωµ2J1, (23)

where E2 is the electric field inside the boundary γ2, and J1 is the equivalent current density on γ1 after the firstequivalent theorem applied in Section 2.2. Through solving (23) with the second scalar Green function theorem [29],we obtain

T E2(~r)

=

∮γ2

[G2

(~r,~r′

) ∂E2(~r′

)∂n′

−∂G2

(~r,~r′

)∂n′

E2(~r′

)]dr′ −

∫γ1

jωµ2G2J1ds. (24)

It should be noted that T = 1/2 when the source and observation points are located on γ2, otherwise, T = 1.

6

We expand the electric field E2 and the magnetic related field ∂E2(~r′)/∂n′ in (24) using the pulse basis functionsand then test it using the Galerkin scheme on γ1. Then, the electric field on the left side of (24) is the electric field onγ1. We collect the expansion coefficient into column vectors and write it into compact form as follows

L1E1 = U(2)1 E2 + P(2)

1 H2 + G(2)1 J1. (25)

Then, we further test (24) on γ2, and obtain the following compact form as

12

L2E2 = U(2)2 E2 + P(2)

2 H2 + G(2)2 J1, (26)

where E1 and E2 denote the electric field expansion coefficients on γ1 and γ2, respectively, expressed as E1 =[e11 e12 ... e1m1

]T, E2 =

[e21 e22 ... e2m2

]T, and H2 is the magnetic field expansion coefficients on

γ2 expressed as H2 =[

h21 h22 ... h2m2

]T. L1 and L2 are diagonal matrixes with the length of each segments as

their entries on γ1 and γ2, respectively. J1 is the surface equivalent current density expansion vector, defined in (21),on γ1 obtained in Section 2.2. The elements of U(1)

2 , U(2)2 , P(2)

1 , P(2)2 G(2)

1 and G(2)2 are expressed as

[U(2)

1

]m,n

=

∫γ1m

∫γ2n

k2

~dm · n′

dmG′

2(~r,~r′

)dr′dr, (27)

[U(2)

2

]m,n

=

∫γ2m

∫γ2n

k2

~dm · n′

dmG′

2(~r,~r′

)dr′dr, (28)

[P(2)

1

]m,n

=

∫γ1m

∫γ2n

ω2µ2 jG2(~r,~r′

)dr′dr, (29)

[P(2)

2

]m,n

=

∫γ2m

∫γ2n

ω2µ2 jG2(~r,~r′

)dr′dr, (30)

[G(2)

1

]m,n

= −

∫γ1m

∫γ1n

ω2µ2 jG2(~r,~r′

)dr′dr, (31)

[G(2)

2

]m,n

= −

∫γ2m

∫γ1n

ω2µ2 jG2(~r,~r′

)dr′dr, (32)

where ~dm = ~r′ − ~r, dm =∣∣∣~r′ − ~r∣∣∣, G2 = −

j4 H(2)

0 (k2ρ), G′

2 = −j4 H(2)

1 (k2ρ).After substituting (21) into (25) and making some mathematical manipulations, E1 is expressed in terms of E2 and

H2 asE1 =

(L1 −G(2)

1 Ys1

)−1︸ ︷︷ ︸V(2)

1

(U(2)

1 E2 + P(2)1 H2

). (33)

Then, after substituting (21) and (33) into (26), the relationship of E2 and H2 on γ2 is obtained as

H2 = Y2E2. (34)

where

Y2 =[P(2)

2 + F(2)2 P(2)

1

]−1(

12

L2 − U(2)2 − F(2)

2 U(2)1

), (35)

where F(2)2 = G(2)

2 Ys1 V(2)1 .

7

! ! !" "ε µ σ

! ! !" "ε µ σ

! ! !" "ε µ σ

! !! ! !ε µ σ

�k�Ein

εi, µi,σi

γi

γ1

γ2

γn

(a)

! ! !" "ε µ σ

�k�Ein

�Jnγi

γ1

γ2

γn

(b)

Figure 4: (a) The original object embedded into multilayer dielectric media and (b) the equivalent model with the inner medium replaced by itsbackground medium and enforcing the surface electric current density ~Jn on the outermost boundary γn

2.3.2. The equivalent model for two layered objectsAccording to the equivalence theorem, the equivalent model is shown in Fig. 3(c), where the inner medium is

replaced by its surrounding medium with parameters ε0, µ0, σ0. In the equivalent model, there is no current sourcesexisting inside the equivalent object. However, another surface equivalent current density, ~J2, on the boundary γ2 isintroduced to enforce the fields exactly the same as those in the original model.

With the similar procedure in Section 2.2 for the equivalent model of a penetrable object, the relationship betweenthe electric field E2 and the magnetic field H2 on γ2 in equivalent model is obtained as

H2 =[P(2)

2

]−1(

12

L2 − U(2)2

)︸ ︷︷ ︸

Y2

E2. (36)

After substituting (35) and (36) into (3), we can obtain the equivalent current density J2 on γ2 as

J2 = H2 − H2 = (Y2 − Y2)︸ ︷︷ ︸Ys2

E2. (37)

As shown in Fig. 3(c), the objects embedded in multilayers are replaced by its background medium and only asingle surface electric current density is introduced on γ2. The PMCHWT formulation has unknowns on both γ1 andγ2. However, the proposed approach only has unknowns residing on the outermost boundary γ2, which will greatlyreduce the overall number of unknowns. To derive a unified approach for any objects embedded in multilayers, wefurther generalize the proposed approach in the next subsection.

2.4. Generalization of the Proposed Approach to Arbitrary Objects Embedded in Multilayers

Once we have presented the approach to model the two layered embedded objects as shown in the previous twosubsections, it is straightforward to extend the approach to model any objects embedded in multilayers. Similarly,for objects embedded a n layered medium, we can apply the proposed approach recursively and obtain the surfaceequivalent current on the outmost boundary. The first equivalent procedure is similar to that presented in the Section2.2, and the subsequent equivalent procedure is similar to that in the Section 2.3. The surface equivalent currentdensity induced from the (i − 1)th medium is expressed as

Ji−1 = Ysi−1 Ei−1. (38)

For the original object with the ith (i ≥ 2) layer media, it is required to test (24) on the (i − 1)th layer as

Li−1Ei−1 = U(i)i−1Ei + P(i)

i−1Hi + G(i)i−1Ji−1, (39)

8

and on the boundary of the ith layer as

12

LiEi = U(i)i Ei + P(i)

i Hi + G(i)i Ji−1, (40)

where Ei, Hi denote the electric field on γi and the entities of other three matrices are denoted as follows

[U(q)

p

]m,n

=

∫γpm

∫γqn

kq

~dm · n′

dmGq

(~r,~r′

)dr′dr, (41)

[P(q)

p

]m,n

=

∫γpm

∫γqn

ωqµqGq(~r,~r′

)dr′dr, (42)

[G(q)

p

]m,n

= −

∫γpm

∫γ(q−1)n

ωqµqGq(~r,~r′

)dr′dr. (43)

Through some mathematical manipulations using (38), (39) and (40), we obtain

Hi =[P(i)

i + F(i)i P(i)

i−1

]−1(

12

Li − U(i)i − F(i)

i U(i)i−1

)Ei, (44)

where F(i)i = G(i)

i Ysi−1 V(i)i−1, V(i)

i−1 =(Li−1 −G(i)

i−1Ysi−1

)−1. Then, it can be rewritten into a more compact form as

Hi = YiEi. (45)

For the equivalent model, the procedure is similar to that in the equivalent model as shown in Section 2.3. Thesurface discretized magnetic and electric fields can be expanded as

Hi =[P(i)

i

]−1(

12

Li − U(i)i

)︸ ︷︷ ︸

Yi

Ei. (46)

Therefore, the equivalent current density Ji on γi is expressed as

Ji = Hi − Hi = (Yi − Yi)︸ ︷︷ ︸Ysi

Ei. (47)

Then, the equivalent model obtained is shown in Fig. 4(b), and the single surface equivalent current Jn = Ysn En onthe outermost layer boundary γn is obtained. The original object is replaced by its surrounding medium along with asurface equivalent current density Jn on γn as shown in Fig. 4(b).

2.5. Multiple Scattering Objects

When there are multiple scattering objects involved on γi, we first compute the equivalent current density for eachobject according to the proposed approach in Section 2.4 as

J(p)i = Y(p)

si E(p)i , (48)

where the superscript (p) represents the pth object, J(p)i , E(p)

i denote the electric current density and electric field ofthe pth object, respectively.

Then, we collect all the surface equivalent current together and obtain

Ji = Ysi Ei, (49)

9

where the surface admittance operator Ysi of the ith layer is a diagonal block matrix assembling from the surfaceequivalent operator Y(p)

si of each object and can be expressed as

Ysi =

Y(1)

si

Y(2)si

...

Y(p)si

. (50)

Ei is a column vector assembling all electric field coefficients of each object and is expressed as

Ei =[

E(1)1 ... E(1)

m1 ... E(p)1 ... E(p)

mp

]T, (51)

and Ji is expressed asJi =

[J(1)

1 ... J(1)m1 ... J(p)

1 ... J(p)mp

]T. (52)

2.6. Scattering Modeling of the Exterior Problem

The electric field ~E outside the object is the superposition of the incident field ~Ei and the scattered field ~E s. Sincethe surface equivalent current ~Jn exists on the outermost layer of the equivalent object, the induced electric field andmagnetic field by ~Jn are expressed as

E s (~r) = − jωµ∫γn

Jn(~r′

)G0

(~r,~r′

)ds′, (53)

where ~r′ on the outermost boundary γn of the object, G0 is the Green function expressed as G0 = − jH(2)0 (k0ρ)/4,

where k0 is the wavenumber in the background medium. Obviously, Jn is the equivalent current on γn. Therefore, theelectric and magnetic fields can be simplified as

En = G(n+1)n Jn + Ei

n, (54)

where the elements in E and Ei are the integral of the total electric field and the incident electric field over eachsegment, respectively.

We substitute (47) into (54) and obtain the electric field En on the outermost boundary γn as

En =(I + G(n+1)

n Ys

)−1Ei

n, (55)

where I is an identity matrix. Once the electric field En on γn is obtain, we can easily calculate the surface equivalentcurrent Jn and other interested parameters.

3. Two Special Cases

3.1. PEC Embedded Objects

In this section, we derive the formulation in the proposed approach for the PEC objects embedded multilayers.

3.1.1. A single PEC objectFor a single PEC object, there is a current density ~J1 on γ1 as shown in Fig. 5(a) and fields inside γ1 vanish. This

scenario is different from that present in the Section 2.2. Therefore, for a single PEC object, we do not need to applythe equivalence theorem to obtain the surface equivalent current on its boundary. The current ~J1 is the true surfacecurrent density flowing on γ1.

10

!γ

!γ

!!

!!"

�k�Ein

!"#

! ! !" "ε µ σ

ε0, µ0,σ0

(a)

!γ

!γ

ε0, µ0,σ0

�J2�k

�Ein

(b)

Figure 5: (a) Original PEC embedded object and (b) the equivalent model with the inner medium replaced by its surrounding medium and enforcingthe surface currents density on γ2

3.1.2. A PEC object with a layer of mediumThe PEC object with a layer of medium is shown in Fig. 5(a). It can be seen from Section 3.1 that the object can

be equivalent to Fig. 5(b). The permittivity, the permeability and the conductivity of medium inside γ2 are ε1, µ1 andσ1, and the current ~J1 exists on γ1.

According to the inhomogeneous Helmholtz equation, we can solve the electric field on γ2 as

T E2(~r) =

∮γ2

[G2(~r,~r′)

∂E2(~r′)∂n′

−∂G2(~r,~r′)

∂n′E2(~r′)

]dr′ −

∫γ1

jωµ2G2J1ds. (56)

Then, we test the above equation on γ1. Since the electric field vanishes on γ1, we obtain the following formulation

0 = U(2)1 E2 + P(2)

1 H2 + G(2)1 J1. (57)

Next, we test (56) on γ2 and obtain

12

L2E2 = U(2)2 E2 + P(2)

2 H2 + G(2)2 J1. (58)

The entries of each matrix are the same as those in (27)-(32).According to the Section 2.3, by using (57) and (58) along with some mathematical manipulations, we can obtain

the relationship between the equivalent current density J2 on γ2 and E2, which can be expressed as

J2 = Ys2 E2. (59)

where

Ys2 =

[P(2)

2 −G(2)2

(G(2)

1

)−1P(2)

1

]−1[12

L2 − U(2)2 + G(2)

2

[G(2)

1

]−1U(2)

1

]−

[P(2)

2

]−1(

12

L2 − U(2)2

). (60)

3.1.3. A PEC object embedded into multilayered mediaWhen the PEC object is embedded in multiple layered media, we can obtain the equivalent current ~J2 on the

boundary γ2 through Section 2.4. Then, according to the equivalence theorem and the method proposed in Section2.4, we can derive the equivalent current ~Jn on γn layer by layer and then obtain the final equivalent model.

3.2. Surface Extension in the Same Medium

We further extend the proposed approach to model the fictitious extension boundary in the same medium. Accord-ing to the method proposed in the Section 2.4, we have obtained the equivalent current on the outermost boundary ofa object embedded in multilayers as

Jn = Ysn En, (61)

11

! ! !" "ε µ σ

! ! !" "ε µ σ

! ! !" "ε µ σ

! !! ! !ε µ σ

�k�Ein

εi, µi,σi

γi

γ1

γ2

γn

γn+1

(a)

! ! !" "ε µ σ

! ! !" "ε µ σ

! ! !" "ε µ σ

! !! ! !ε µ σ

�k�Ein

εi, µi,σi

γi

γ1

γ2

γn

γn+1

�Jn+1

(b)

Figure 6: (a) The original model and (b) the equivalent model with the medium replaced by its background medium and enforcing the surfacecurrent ~Jn+1 at γn+1

and the relationship between electric and magnetic fields on the outermost boundary can be expressed as

Hn = YnEn. (62)

We can make a fictitious boundary γn+1 in the background medium as shown in Fig. 6(a). Then, an equivalentmodel is obtained through enforcing an equivalent surface current at γn+1. If we treat the medium between the targetboundary γn+1 and the outermost boundary γn of the object as a new one with the same constitutive parameters asthose of the background medium, the proposed approach can be further extended to handle this objects.

The method in the section 2.4 is used to conduct the equivalence of a new layer of medium, as shown in the Fig.6(b). We can obtain the relationship between tangential magnetic field and normal electric field on γn+1 of the originalmodel as

Hn+1 = Yn+1En+1, (63)

where

Yn+1 =[P(n+1)

n+1 + F(n+1)n+1 P(n+1)

n

]−1(

12

Ln+1 − U(n+1)n+1 − F(n+1)

n+1 U(n+1)n

). (64)

Then, in the equivalent model, an surface equivalent current is enforced at γn+1 as shown in Fig. 6(b), where therelationship between tangential magnetic field and electric field on boundary γn+1 can expressed as

Hn+1 =[P(n+1)

n+1

]−1(

12

Ln+1 − U(n+1)n+1

)En+1. (65)

For the (n + 1)th layered medium, since the outermost medium is equal to the background medium, we get thefollowing equation

P(n+1)n+1 = P(n+1)

n+1 ,U(n+1)n+1 = U(n+1)

n+1 . (66)

Other procedures for this scenario is exactly the same as those in the previous proposed method. Finally, we can getthe equivalent current Jn+1 of the target surface γn+1 as

Jn+1 = Hn+1 − Hn+1 = Ysn+1 En+1. (67)

4. Remarks Upon the Proposed Approach

Let’s assume objects embedded in multilayers with m1,m2, ...,mn segments on each interface and N = m1 + m2 +

... + mn. Therefore, the PMCHWT formulation requires 2N unknowns in the final system and all the geometricalfine details are directly included in the final system. However, for the proposed approach, the dimension of the finalsystem is mn, which only dependents on the unknowns residing on the outermost boundary γn and is significantlysmaller than that of the PMCHWT formulation. However, as shown by (35), several matrix inversion and multiplying

12

(a) (b)

Figure 7: (a) RCS obtained from the proposed method, the PMCHWT formulation and the analytical solution, and (b) RCS obtained from theproposed method, the PMCHWT formulation and the Comsol.

are required to construct the final system, which implies overhead for time cost. If many small scatters are involvedas shown in [22], this overhead can be ignored compared with the overall time cost since that of the direct matrixinversion algorithm, like the LU decomposition, is small. Although overhead is required to construct the DSAO Ysn ,the computational performance improvement in terms of memory consumption and CPU time can still be significant asshown in the numerical examples in the next section. However, when large scatters are involved in the computationaldomain, direct construction of intermediate matrixes can be computationally expensive. This problem can be mitigatedusing efficient direct solvers, like hierarchical matrix based direct algorithm [32], [33], Hierarchically Semi-Separable(HSS) solver [34], [35]. Acceleration of the proposed approach with those mentioned techniques is beyond the scopeof this paper. We will report related results in the future. Another merit of the proposed approach is that the conditionof the final system can be improved compared with that of the PMCHWT formulation and can be similar to that ofthe EPA based techniques [23], [22] since the geometry fine details are implicitly incorporated into the final systemof the proposed approach.

5. Numerical Results and Discussion

All numerical examples in this paper are performed on a computer with Intel i7-7700 3.6 GHz CPU and 32Gmemory. All approaches except the Comsol are coded through the Matlab. All the codes of different approaches arerunning using one thread to make a fair comparison.

5.1. A Single Dielectric ObjectWe first consider two objects, an infinite long dielectric cylinder and cuboid with the relative permittivity εr = 4.

The radius of the cylinder and the side length of the cuboid are both 1λ0, where λ0 is the wavelength in the free space.The incident plane wave is along the x-axis with f = 300MHz and the averaged segment length is used as 0.05m todiscretize the contour, which corresponds to λ/10, where λ is the wavelength inside dielectric objects.

Fig. 7 shows RCS of the two dielectric objects obtained from the proposed method, the PMCHWT formulation andthe analytical method or the Comsol. The reference RCS of the cuboid is obtained from the Comsol with extremelyfine mesh. It is easy to find that the results obtained from the proposed approach for both objects agree well with thosefrom the Comsol, the analytical method and the PMCHWT formulation. Therefore, the proposed approach can obtainaccurate results for both smooth and non-smooth objects.

Then, we check the convergence property of the proposed method with mesh sizes. The relative error (RE) isdefined as

RE =

∑i

∥∥∥RCScal (φi) − RCSref (φi)∥∥∥2∑

i

∥∥∥RCSref (φi)∥∥∥2 , (68)

13

0.1 0.08 0.06 0.04 0.02Mesh size (m)

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Rel

ativ

e er

ror

10-3

ProposedPMCHWT

(a)

0.1 0.08 0.06 0.04 0.02Mesh size (m)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Rel

ativ

e er

ror

ProposedPMCHWT

(b)

Figure 8: Relative error of the proposed method and the PMCHWT formulation for (a) the dielectric cylinder and (b) the dielectric cuboid.

0.1 0.08 0.06 0.04 0.02

Mesh size (m)

0

500

1000

1500

2000

2500

Con

dit

ion

Nu

mb

er

Proposed

GIBC

PMCHWT

0.1 0.08 0.06 0.04 0.02

Mesh size (m)

0

500

1000

1500

2000

2500

Con

dit

ion

Nu

mb

er

Proposed

GIBC

PMCHWT

0.1 0.08 0.06 0.04 0.02

Mesh size (m)

0

500

1000

1500

2000

2500

Con

dit

ion

Nu

mb

er

Proposed

GIBC

PMCHWT

0.1 0.08 0.06 0.04 0.02

48

48.5

49

0.1 0.08 0.06 0.04 0.02

48

48.5

49

0.1 0.08 0.06 0.04 0.02

48

48.5

49

(a)

0.1 0.08 0.06 0.04 0.02

Mesh size (m)

0

1000

2000

3000

4000

5000C

on

dit

ion

Nu

mb

er

Proposed

GIBC

PMCHWT

0.1 0.08 0.06 0.04 0.02

Mesh size (m)

0

1000

2000

3000

4000

5000C

on

dit

ion

Nu

mb

er

Proposed

GIBC

PMCHWT

0.1 0.08 0.06 0.04 0.02

Mesh size (m)

0

1000

2000

3000

4000

5000C

on

dit

ion

Nu

mb

er

Proposed

GIBC

PMCHWT

0.02 0.04 0.06 0.08 0.1

363840

0.02 0.04 0.06 0.08 0.1

363840

0.02 0.04 0.06 0.08 0.1

363840

(b)

Figure 9: The condition number of the final linear equations for the proposed approach, the GIBC formulation, and the PMCHWT formulation for(a) the cylinder and (b) the cuboid.

where RCScal (φi) denotes results calculated from the proposed approach or the PMCHWT formulation and RCSref (φi)is the reference obtained from the analytical method or the Comsol. As shown in Fig. 8, it can be found that as meshsize decreases, the relative errors of both the PMCHWT formulation and the proposed approach decrease. Whenmesh size is relative large, say larger than 0.04m, the relative error of the proposed approach is smaller than that of thePMCHWT formulation. Then, as we further decreases mesh size, both approaches reaches the same level of accuracy.

In addition, we investigate the condition number of the final linear equations. For comparison purpose, the GIBCformulation is also included in our study. As shown in Fig. 9, the condition number of the three approaches slightlyincreases as mesh size decreases. However, the condition number of the proposed formulation and the GIBC for-mulation is much smaller than that of the PMCHWT formulation. Take mesh size 0.1m for example. The conditionnumber of the proposed formulation, the GIBC formulation, and the PMCHWT formulation is 47.72, 47.72, and2222.41, respectively. It can be found that the proposed formulation is much better conditioning and then better con-vergence property than the PMCHWT formulation. It should be noted that only a single dielectric object is involvedin each simulation. Therefore, only a half of count of unknowns for the proposed formulation compared with that ofthe PMCHWT formulation is required.

14

(a)

0 6 16 26 36Distance (mm)

3.2

3.4

3.6

3.8

4

4.2

4.4

Rel

ativ

e er

ror

10-4

(b)

Figure 10: (a) RCS obtained from the Comsol, the PMCHWT formulation, the proposed approach for a cylindrical conductor coated with a layerdielectric medium, and the proposed approach for the object with a fictitious boundary introduced in the exterior region, and (b) Relative errorobtained from the proposed method with different d.

Table 1: COMPARISON OF THE CONDITION NUMBER BETWEEN THE PROPOSED APPROACH AND THE PMCHWT FORMULATION

Condition NumberProposed

P(1)1 P(1)

1 V(2)1 P(2)

2 + F(2)2 P(2)

1 P(2)2

L2 + P(2)2 Ys2 P(3)

3 + F(3)3 P(3)

2 P(3)3

L3 + P(3)3 Ys3

d m3

(= V(3)

2

)(final matrix)

0 176 1 25.09 25.08 26.51 64.43 131.36 - - -6 252 1 25.09 25.08 26.51 64.43 131.36 50.50 41.32 69.2616 380 1 25.09 25.08 26.51 64.43 131.36 87.35 47.11 11.4426 504 1 25.09 25.08 26.51 64.43 131.36 51.19 51.19 65.0736 628 1 25.09 25.08 26.51 64.43 131.36 67.86 330.67 497.02PMCHWT 7436.56

5.2. A Single Layer Coated Cylindrical Conductor

An infinitely long cylinder conductor coated with a layer dielectric medium is then considered. As shown in Fig.10, the radius of the conductor and the coated medium are 10 mm and 14 mm. The conductivity of the conductor is5.6 ×107 S/m and the relative permittivity εr of the coated medium is 2.3. The background medium is air. A planewave incidents from the x-axis with the frequency of 30 GHz. The radar cross section (RCS) is compared with theproposed approach, the PMCHWT formulation, and the Comsol. The reference solution is obtained from the Comsolwith extremely fine mesh.

Two scenarios are considered for the proposed approach: the object with or without an extended fictitious boundaryin the background medium as shown in Fig. 10. d is the distance between the outermost boundary and the fictitiousboundary. The RCS with d = 0 mm, which corresponds to the case that results obtained without the fictitious boundary,d = 6 mm, the PMCHWT formulation and the Comsol are shown in Fig. 10(a). This fictitious boundary can be selectedarbitrarily. There is only one implication that the boundary should be enclosed and include the whole object. Resultsobtained from the proposed approach with d = 0 mm, 6 mm agree well with the reference solution. It demonstratesthat the proposed approach can accurately model the multilayer embedded objects.

Then, to study the numerical stability of the proposed method with respect to d, we calculated the relative errorswith different d. As shown in Fig. 10(b), the relative errors for all cases are quite small, which implies that thenumerical stability of the proposed approach is quite good. Table 1 shows the number of unknowns and the conditionnumber required by our proposed approach and the PMCHWT formulation with different d. The table lists all the

15

copper copper

copper

2.5m

2m

3.5m1 2.3 =

2 4 =

3 2.3 =air

Ein

Figure 11: Configurations of the multilayer embedded objects and RCS from Comsol, PMCHWT and the proposed method.

Table 2: COMPARISON OF THE OVERALL COUNT OF UNKNOWNS, THE COUNT OF FLIP-FLOP OPERATIONS AND TIME COSTBETWEEN THE PROPOSED APPROACH AND THE PMCHWT FORMULATION

Metric PMCHWT Proposed Ratio

Count of unknowns 4188 736 0.18

Count of flip-flop operations 7.347 × 1010 5.107 × 109 0.07

Filling matrix [s] 43.52 36.62 0.84

solving matrix [s] 1.6 0.43 0.27

Overall time cost [s] 45.12 37.05 0.82

condition numbers of the intermediate matrixes requiring inversion in our proposed approach. As shown in Table I,all the condition numbers of intermediate matrixes are quite small, which means that quite low cost is required tocalculate their matrix inversion when iterative algorithms are used. As d increase, the condition number of the finalsystem seems to grow larger. Similar to the previous numerical example, the final condition number of the linearequation of the proposed approach is much smaller than that of the PMCHWT formulation. It should be noted thatas d increases, the number of unknowns on the fictitious boundary increases as shown in first column of Table 1.Therefore, a balance with respect to the count of unknowns should be made for d.

5.3. A Complex Structure

We finally consider an object with a complex geometry as shown in Fig. 11. The innermost objects are threesector-shaped copper conductors with an angle of 120 degrees and a radius of 1 m. The relative permittivity of thesecond, third and fourth layer dielectric cylindrical media are 2.3, 4.0, 2.3, respectively. This object contains both amultilayer structure and multiple scatters inside the layered media.

The RCS is calculated from the proposed approach, the PMCHWT formulation, the Comsol as shown in the Fig.11. Results from the three approaches show excellent agreement with each other. It demonstrates that the proposedapproach can indeed accurately model the complex objects embedded in multilayers.

Table 2 shows the overall count of unknowns, the count of flip-flop operations, time cost of the proposed approachand the PMCHWT formulation. The ratio in the last column of Table 2 is defined as the ratio of the quantity ofthe proposed approach to that of the PMCHWT formulation. As shown in the second row of Table 2, the proposed

16

Table 3: COMPARISON OF THE CONDITION NUMBER BETWEEN THE PROPOSED APPROACH AND THE PMCHWT FORMULATION

Condition NumberProposed PMCHWT

The 1st Layer P(1)11

1.02 P(1)12

1.02 P(1)13

1.02

24745.34

P(1)11

41.03 P(1)12

40.81 P(1)13

41.03The 2nd Layer V(2)

1 75.75 P(2)2 + F(2)

2 P(2)1 51.40 P(2)

2 42.45The 3rd Layer V(3)

2 32.65 P(3)3 + F(3)

3 P(3)2 62.67 P(3)

3 55.48The 4th Layer V(4)

3 72.61 P(4)4 + F(4)

4 P(4)3 87.89 P(4)

4 82.57Solving E4 L4 + 1

2 P(4)4 Ys4 (final matrix) 198.65

approach only needs 736 unknowns to solve this complex problem compared with 4188 unknowns for the PMCHWTformulation. Its ratio is 0.18, which means only quite a small fraction of amount of unknowns is required in theproposed approach. This is becasue that the PMCHWT formulation requires both the electric and magnetic currentdensities on all the interfaces of different homogenous media. However, in the proposed formulation only singleelectric current density on the outermost boundary of objects is required. Therefore, the overall count of unknownscan be significantly reduced. Moreover, we count the overall flip-flop operations for the two approaches as shown inthe third row of Table 2. The count of flip-flop operations of our proposed approach is only 7% of that of the PMCHWTformulation. The time cost for both approaches are shown from the fourth to sixth rows of Table 2. The time cost forthe PMCHWT formulation requires 45.12 seconds to solve this problem. However, only 37.05 seconds are requiredfor the proposed approach. Its ratio is 0.82. The improvement for the time cost and the count of unknowns is not alinear scale. This is because that overhead is required to construct the DSAO Yn. However, as shown in Table 2, thetime cost of matrix filling and solving linear equations is reduced compared with that of the PMCHWT formulation.As the number of interfaces increases, the performance improvement in terms of time cost and count of unknownswould be more significant.

As shown in Table 3, the condition numbers required by our proposed approach for all intermediate matrixeswhen we construct Yn. Similar observations with the previous numerical example can be obtained in this example.The condition number of the final linear equation in the PMCHWT formulation is 24,745. However, the conditionnumber for the proposed formulation is only 198. Therefore, the proposed approach shows much better conditioningand then convergence property compared with the PMCHWT formulation.

6. Conclusion

In this paper, we proposed a novel and unified single-source SIE to model multilayer embedded objects. Theproposed approach only needs a single electric current density on the outermost boundary of objects, which can bederived by recursively applying the equivalent theorem on each boundary from inner to exterior regions. The surfaceelectric field is related to the magnetic field on the outermost boundary through a DSAO. Then, with combining theEFIE with the equivalent current density, we can accurately solve the scattering problems by objects embedded inmultilayers. Numerical results demonstrate that the proposed approach can significantly improve the performancein terms of the count of unknowns and time costs, which implies great potential for electromagnetic analysis ofcomplex multilayer embedded objects. Currently, development of the proposed approach into three dimensionalgeneral scenarios is in progress. We will report more results on this topic.

Acknowledgement

This work was supported in part by the National Natural Science Foundation of China through Grant 61801010,in part by Beijing Natural Science Foundation through Grant 4194082.

17

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