1

Generalized SS-SIE for Electromagnetic Analysis ofArbitrarily Connected Penetrable and PEC Objects

with Non-Conformal MeshesZekun Zhu, Graduate Student Member, IEEE, Aipeng Sun, Xiaochao Zhou,Shunchuan Yang, Member, IEEE, and Zhizhang (David) Chen, Fellow, IEEE

Abstract—We proposed a simple and efficient modular single-source surface integral equation (SS-SIE) formulation for elec-tromagnetic analysis of arbitrarily connected penetrable andperfectly electrical conductor (PEC) objects. In this formulation,a modular equivalent model for each penetrable object consistingof the composite structure is first independently constructedthrough replacing it by the background medium, no matterwhether it is surrounded by the background medium, othermedia, or partially connected objects, and enforcing an equivalentelectric current density on the boundary to remain fields inthe exterior region unchanged. Then, by combining all themodular models and any possible PEC objects together, anequivalent model for the composite structure can be derived. Thetroublesome junction handling techniques are not needed andnon-conformal meshes are intrinsically supported. The proposedSS-SIE formulation is simple to implement, efficient, and flexible,which shows significant performance improvement in termsof CPU time compared with the original SS-SIE formulationand the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT)formulation. Several numerical examples including the coateddielectric cuboid, the large lossy objects, the planar layereddielectric structure, and the partially connected dielectric andPEC structure are carried out to validate its accuracy, efficiencyand robustness.

Index Terms—Arbitrarily connected objects, composite struc-tures, method of moment, non-conformal meshes, single-sourcesurface integral formulation

I. INTRODUCTION

The method of moment (MOM) is widely used to solvevarious electromagnetic problems, such as scattering [1], ra-diation [2], parameter extraction in integrated circuits [3],due to its unknowns residing on the boundaries of differentpiecewise homogeneous objects, which can significantly re-duce the overall count of unknowns compared with the finite-

Manuscript received xxx; revised xxx.This work was supported in part by the National Natural Science Foundation

of China through Grant 61801010, Grant 62071125, Grant 61631002, andFundamental Research Funds for the Central Universities. (Correspondingauthor: Shunchuan Yang)

Z. Zhu, A. Sun, X. Zhou are with the School of Electronic and Infor-mation Engineering, Beihang University, Beijing, 100083, China. (e-mail:zekunzhu@buaa.edu.cn, sap1997@163.com, zhouxiaochao@buaa.edu.cn)

S. Yang is with the Research Institute for Frontier Science and the School ofElectronic and Information Engineering, Beihang University, Beijing, 100083,China. (e-mail: scyang@buaa.edu.cn)

Z. Chen was with College of Physics and Information Engineering, FuzhouUniversity and on leave from the Department of Electrical and ComputerEngineering, Dalhousie University, Halifax, NS, Canada B3J 2X4. (e-mail:z.chen@dal.ca)

difference time-domain (FDTD) method [4] and the finiteelement method (FEM) [5].

To solve the challenging electromagnetic problems inducedby arbitrarily connected objects, many efforts have been madein the MOM, such as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation [6], the combined tangen-tial formulation (CTF) [7], the Muller formulation [8], theequivalence principle algorithms (EPAs) [9]-[12], and so on.In those formulations, boundary conditions of the electricand magnetic fields are required to be enforced to guaran-tee correct solutions. Especially, junctions of multiple mediaintersection are required to be paid special attention to. Manytechniques, such as the modified rooftop basis function [13]-[15], the current continuous boundary conditions [16], [17],are proposed to handle the boundary conditions on junctions.For those EPAs, fictitious boundaries are assumed not to betouched to avoid troublesome boundary issues [9], [10]. Amacromodeling approach based on the PMCHWT formulationis recently proposed in [18] to model large inhomogeneousantenna arrays, in which fictitious interfaces are touchedwith each other through carefully handling various boundaryconditions. However, those are bookkeeping formulations ofthe boundary conditions in different scenarios. A general for-mulation, which can handle arbitrarily connected objects, canhardly be obtained. In addition, both the equivalent electric andmagnetic current densities are required in those formulations.

Various single-source (SS) formulations, such as the surfaceintegral equation (SIE) with the differential surface admittanceoperator (DSAO) [19]-[25], the single-source surface-volume-surface (SS-SVS) formulation [26]-[29], with only electriccurrent density are developed. Those formulations show ef-ficiency improvement in terms of CPU time and memoryconsumption compared with their dual source counterparts.In [22], a single-source surface integral equation (SS-SIE)formulation is proposed to extract the electrical parametersof solid and hollow conductors. The media surrounding andinside the hollow conductors are assumed to be the same,and fictitious interfaces are not allowed to be touched in thisformulation. In [30], through carefully enforcing boundaryconditions and eliminating all the interior unknowns, an SS-SIE formulation is proposed to model partially connectedpenetrable objects. However, it suffers from troublesome math-ematical manipulations.

Domain decomposition methods (DDMs) can also effi-ciently model electrically large and multiscale composite

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(a) (b)

Fig. 1. (a) The general structure including arbitrarily connected penetrableand PEC objects, (b) the equivalent model with the physics and equivalentelectric current densities on the boundaries of the composite structure.

structures. In [31], [32], the SIE-DDMs are proposed to solvethe electromagnetic problems induced by complex compositestructures. In those formulations, an approximation boundarycondition, e.g., the transmission conditions, between adjacentdomains is required to ensure electric current density to becontinuous. However, those boundaries impose challenges forpractical implementations if non-conformal meshes are used.

In this paper, we proposed a generalized, simple and ef-ficient modular SS-SIE formulation in transverse magnetic(TM) mode to model arbitrarily connected penetrable andperfectly electrical conductor (PEC) objects as shown in Fig.1(a). In this formulation, an equivalent model with only theelectric current density on the boundaries of the originalstructures is derived as shown in Fig. 1(b). First, throughreplacing each object by the background medium and enforc-ing an equivalent current density on the boundary, modularequivalent models for objects consisting of the compositestructure are independently constructed no matter whetherthey are surrounded or partially connected with other objects.Only the surface equivalent electric current density is enforcedon its corresponding boundary to keep fields unchanged inthe exterior region by incorporating with the DSAO. Second,those modular equivalent models are combined into a singleequivalent model for the composite structure, and the surfaceequivalent current density in the final equivalent model is equalto the summation of all the current densities on the interfacesof penetrable and PEC objects.

In [33], a modular approach to handle the compositestructure is proposed. After each object is replaced by theequivalent current density, additional tangential boundary con-ditions of the magnetic fields are required. However, nosuch conditions are required and non-conformal meshes aresupported in our proposed SS-SIE formulation. Comparedwith our previous work [30], in which continuous boundaryconditions are explicitly enforced on the shared interface be-fore the surface equivalence theorem is applied, the proposedSS-SIE formulation in this paper can avoid the troublesomemathematical derivation and is suitable to model arbitrarilyconnected penetrable and PEC objects. Although slightly moreunknowns compared with the SS-SIE formulation in [30] arerequired, significant performance improvement in terms ofCPU time and high flexibility to model complex structures canbe obtained as shown in the numerical examples. Therefore,it is much preferred in the practical simulations.

(a) (b)

Fig. 2. (a) The partially connected structure for penetrable objects, (b) thepenetrable and PEC partially connected objects.

There are three obvious merits of the proposed SS-SIEformulation over other existing techniques.

1) Boundary conditions are implicitly enforced during theconstruction of the modular equivalent models for thepenetrable objects, and no extra requirements are neededin the final equivalent model for the composite struc-tures. Troublesome junction handling techniques are notneeded any more. Therefore, the derivation of the SIEformulation for the general composite structures is sig-nificantly simplified and a general SS-SIE formulationfor arbitrarily connected penetrable and PEC objects isobtained.

2) Non-conformal meshes are intrinsically supported sincethe equivalent model is modularly constructed andboundary conditions are automatically satisfied. Eachobject can be discretized independently based on itsown material parameters and geometric details. It isextremely flexible and useful to model multiscale andelectrically large structures. Furthermore, much higherefficiency than that of the original SS-SIE formulation[19]-[25] can be obtained through decomposing the largestructures into small units.

3) Only the single electric current density is required inour proposed SS-SIE formulation through incorporatingwith the DSAO. Shorter CPU time and less memoryconsumption are required compared with the dual sourceformulations, such as the PMCHWT formulation.

This paper is organized as follows. In Section II, config-urations and preliminary notations are defined. In addition,the proposed equivalent SS model for arbitrarily connectedpenetrable and PEC objects is presented. In Section III,detailed implementations for the proposed SS-SIE formulationare shown. In Section IV, we present how to solve thescattering problems with non-conformal meshes, near fieldcalculation, and some discussion upon the proposed SS-SIEformulation are also presented. In Section V, several numericalexamples are carried out to validate its accuracy, efficiency androbustness. At last, we draw some conclusions in Section VI.

II. THE PROPOSED EQUIVALENT MODEL BASED ON THESURFACE EQUIVALENCE THEOREM FOR COMPOSITE

OBJECTS

A. Configurations and Preliminary Notations

To make the derivation concise, two typical scenarios asshown in Fig. 2 are selected to illustrate the proposed SS-

3

(a) (b)

Fig. 3. (a) A penetrable object, (b) the equivalent model with the equivalentelectric current densities for the penetrable object.

SIE formulation. Then, we demonstrate how the formulationcan be extended to model arbitrarily connected penetrable andPEC objects.

Fig. 2(a) presents a composite structure including twopartially connected penetrable objects denoted by Ω1 and Ω2.γ1 and γ2 are the boundaries of Ω1 and Ω2, respectively.γc is the shared part of γ1 and γ2. The permittivity andpermeability of Ω1 and Ω2 are ε1, µ1 and ε2, µ2, respectively.The permittivity εi is εi = ε0(εri + jσi/(ε0ω)) with i = 1, 2,where εri , σi are the relative permittivity and the conductivityof the penetrable object, respectively, and ω is the angularfrequency. The background medium is denoted by Ω0 withconstant parameters ε0, µ0. Similar notations are used forobjects in Fig. 2(b). Ω3 denotes the penetrable object withconstant parameters of ε3, µ3, and Ω4 for the PEC object. Thehollow character with a subscript, e.g., Ei, denotes a matrixassociated to γi and the bold character with a subscript, e.g.,Ei, denotes a column vector on γi. A quantity with , e.g., E,is used for the equivalent model.

B. The Equivalent Model with the Single Electric CurrentDensity for A Penetrable Object

Let’s first consider a single penetrable object in Fig. 3(a).According to the surface equivalence theorem [45, Ch. 12,pp. 653-658], an appropriate equivalent electric and magneticcurrent density on an enclosed surface can reproduce exactlythe same fields in the equivalent configuration as those in theoriginal model. The surface equivalent electric and magneticcurrent densities 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 surfacetangential electric and magnetic fields in the original andequivalent model, respectively. The unit normal vector nipointing into the interior region of Ωi is selected in this paper,and all the tangential fields are obtained through applying ni×operator to the corresponding fields.

When Eti(r) = Hti(r) = 0, it becomes the Love’sequivalence theorem [45, Ch. 12, pp. 653-658]. Based onthem, many SIE formulations [6], [7], [18] are derived to solvevarious electromagnetic problems. In this paper, we use themto derive the SS-SIE formulation.

(a) (b)

Fig. 4. (a) A composite structure with two penetrable objects, (b) theequivalent model for the composite structure, and a small gap is added forbetter visualization.

Since fields in the equivalent model can be arbitrary, weenforce Eti(r) = Eti(r) [35], and (2) becomes

Mi(r) = 0. (3)

Then, the magnetic current density vanishes and only theelectric current density exists in the equivalent model for apenetrable object as shown in Fig. 3(b). When Eti(r) = 0, (1)and (3) correspond to the PEC objects, and Eti(r) 6= 0 is forthe penetrable objects. Therefore, they are applicable for boththe PEC and penetrable objects. In the next subsections, wewill derive an equivalent model with only the electric currentdensity for partially connected penetrable and PEC objectsusing (1) and (3).

C. The Proposed Equivalent Model with the Single ElectricCurrent Density for Penetrable Composite Objects

Let’s consider the scenario in Fig. 4(a). According to thesurface equivalence theorem [45, Ch. 12, pp. 653-658] andthe SS formulation in Section II-B, to support exactly thesame fields in the exterior region as those in the originalconfiguration, the surface equivalent electric current densitiesare required to be enforced on γ1 and γ2, which can beexpressed as

J1s(r) = Ht1s(r)−Ht1s

(r), r ∈ γ1, r /∈ γc, (4)

J2s(r) = Ht2s(r)−Ht2s

(r), r ∈ γ2, r /∈ γc, (5)

Jc(r) = Ht2c(r) + Ht1c

(r), r ∈ γc, (6)

subject to the boundary conditions

Et1s (r) = Et1s (r), r ∈ γ1, r /∈ γc, (7)

Et2s (r) = Et2s (r), r ∈ γ2, r /∈ γc, (8)

Et1c (r) = −Et2c (r) = Et1c (r) = −Et2c (r), r ∈ γc, (9)

where the subscript 1s and 2s denote that the quantities aredefined on γ1 and γ2 except for γc, and the subscript 1c and 2cdenote quantities defined on γc. Since the unit normal vectorspoint into the interior region of each object, plus sign in (6)and minus sign in (9) should be used.

To obtain a general SIE formulation for arbitrary compositestructures, fields on γc need special treatments. By taking

4

Ht1c(r) = −Ht2c

(r) into consideration in the original model,(6) can be modified as

Jc(r) = Ht2c(r) + Ht1c

(r)

=(Ht1c

(r)−Ht1c(r))

︸ ︷︷ ︸J1c (r)

+(Ht2c

(r)−Ht2c(r))

︸ ︷︷ ︸J2c (r)

.

(10)As shown in (10), Jc is split into two surface electric currentdensities, namely J1c , J2c on γc. Therefore, we can assumetwo γc exist. One γc is on γ1 and the other is on γ2. Jc is thesummation of two surface electric current densities, J1c on γcof Ω1 and J2c on γc of Ω2, as shown in Fig. 4(b). Then, aftercombining J1c and J1s , J2c and J2s , (4)-(6) are rewritten as

J1(r) = Ht1(r)−Ht1(r), r ∈ γ1, (11)

J2(r) = Ht2(r)−Ht2(r), r ∈ γ2, (12)

subject to the boundary conditions in (7)-(9). Up to this point,J1 and J2 are defined on the whole boundaries of Ω1 and Ω2,respectively. Therefore, the original partially connected objectsare separated and the equivalent model for the compositestructure can be constructed through combining the modularmodels for each penetrable object.

As shown in the previous subsection, (11) and (12) are theequivalent electric current densities enforced on the boundariesof Ω1 and Ω2, respectively, when they are replaced by thebackground medium. It is also true for arbitrarily connectedpenetrable objects. Therefore, we can obtain the followingprocedure to derive the equivalent model for any penetrablecomposite objects.

1) A modular equivalent model for each penetrable objectof the composite structure in Fig. 4(a) is first derivedby using (1) and (3), no matter whether the object issurrounded by the background medium, other media, orconnected with other objects as shown in Fig. 1(a).

2) All the modular equivalent models are combined to-gether to construct the equivalent model for the com-posite structure. The equivalent current density in thefinal equivalent model is equal to the summation of allthe modular equivalent current densities.

Unlike the PMCHWT formulation and similar others, inwhich the current densities are treated as one set of variableson the shared interfaces through the boundary conditions, themodular electric current densities are used as independentvariables in the proposed SS-SIE formulation. The novel use ofit makes the proposed SS-SIE formulation intrinsically supportnon-conformal meshes, which will be shown in Section V.

D. The Proposed Equivalent Model with the Single ElectricCurrent Density for Penetrable and PEC Connected Objects

Let’s consider the partially connected PEC and penetrableobjects in Fig. 2(b). Compared with the previous scenarioin Fig. 2(a), one physics rather than the equivalent electriccurrent density J4 exists on the surface of the PEC object.Therefore, the surface equivalent theorem is not needed to be

(a) (b)

Fig. 5. (a) A composite structure consisting of a penetrable and PEC partiallyconnected object, (b) the equivalent model for the composite structures, anda small gap is added for better visualization.

applied to those PEC objects. The electric current density onthe boundary can be expressed as

J4s(r) = Ht0(r), r ∈ γ4, r /∈ γc, (13)J4c(r) = Ht3c

(r), r ∈ γc, (14)

where Ht0(r) is the tangential magnetic fields on γ4 exceptfor γc and Ht3c

(r) denotes the tangential magnetic fields inΩ3 on γc.

For the penetrable object Ω3, the modular equivalent modelcan be derived through the procedure in the previous sub-section. Similarly, the penetrable object is replaced by thebackground medium, and the electric current density in thefinal equivalent configuration in Fig. 5(b) can be expressed as

J3s(r) = Ht3s(r)−Ht3s

(r), r ∈ γ3, r /∈ γc, (15)

Jc(r) = Ht3c(r), r ∈ γc, (16)

where the subscript 3s denotes the quantities are defined onγ3 except for γc. Similarly, (16) can be modified as

Jc(r) = Ht3c (r)

= Ht3c(r)−Ht3c

(r) + Ht3c(r)

=(Ht3c

(r)−Ht3c(r))

︸ ︷︷ ︸J3c (r)

+Ht3c(r)︸ ︷︷ ︸

J4c (r)

.(17)

Therefore, the summation of J3s in (15) and J3c in (17) isthe modular equivalent electric current density on γ3. Thesummation of J4s in (13) and J4c in (17) is equal to thephysics electric current density on γ4. It is also true forthe arbitrarily connected penetrable and PEC objects. Theequivalent model for the composite structure can be alsoobtained through the similar procedure in previous subsection.

In the following section, we present detailed implementa-tions of the proposed equivalent model in the MOM and howit can support non-conformal meshes.

III. DETAILED IMPLEMENTATIONS OF THE PROPOSEDSS-SIE FORMULATION

A. The Modular Equivalent Model for Each Penetrable Object

In this paper, we consider the TM mode to demonstratethe implementations of the proposed SS-SIE formulation.However, it can also be used to solve more general vectorEM problems. In the appendix, a numerical example in the

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vector TE mode is provided to demonstrate the applicabilityof the proposed SS-SIE formulation to solve the vector TEelectromagnetic problems.

Without loss of generality, there are no sources inside thepenetrable region Ωi. The electric fields on γi can be expressedas

TEi(r) =

∮γi

[Gi (r, r′)

∂Ei (r′)

∂n′

− ∂Gi (r, r′)

∂n′Ei (r′)

]dr′, (18)

where T = 1/2 when the source point r′ and the observa-tion point r are located on the same boundary, otherwise,T = 1. Gi (r, r′) is the Green’s function with Gi (r, r′) =

−jH(2)0 (ki|ρ|) /4, j =

√−1, ρ = r − r′, where ki is the

wavenumber inside the object, H(2)0 (·) is the zeroth-order of

Hankel function of the second kind. According to the Poincare-Steklov operator [35], we have

Hti(r) =1

jωµi

∂Ei(r)

∂n

∣∣∣∣r∈γi

, (19)

where µi is the permeability of the penetrable object. Bysubstituting (19) into (18), we have

TEi(r) =

∮γi

[jωµiGi (r, r′)Hti(r

′)

− ∂Gi (r, r′)

∂n′Ei (r′)

]dr′, (20)

After γi is discretized into mi segments, the tangential electricand magnetic fields can be expanded as

Ei(r) =

mi∑n=1

enfn(r), Hti(r) =

mi∑n=1

hnfn(r), (21)

where fn is the pulse basis function, en and hn are the ex-pansion coefficients, respectively. We collect all the expansioncoefficients into column vectors as Ei and Hi, which can beexpressed as

Ei =[e1 e2 · · · emi

]T,

Hi =[h1 h2 · · · hmi

]T. (22)

By substituting (21) into (20) and using the Galerkin schemeon (20), we get the following matrix equation

T LiEi = PiHi + UiEi, (23)

where Li is a diagonal matrix, in which entities are equal tothe length of segments. The entities of matrix Pi and Ui aregiven by

[Ui]m,n =

∫γim

∫γin

kiρ · n′

|ρ|Gi (r, r′) dr′dr, (24)

[Pi]m,n =

∫γim

∫γin

jωµiGi (r, r′) dr′dr. (25)

To handle the nearly singular and singular integrals in (24)and (25), the approach in [34] is used. Interested readers arereferred to it for more details.

Then, by moving the second terms on the right hand side(RHS) of (23) to its left hand side (LHS) and inverting thesquare matrix Pi, we obtain

Hi = [Pi]−1 (T Li − Ui)︸ ︷︷ ︸Yi

Ei, (26)

where Yi is the surface admittance operator (SAO) [35].By applying the surface equivalence theorem to the pene-

trable object, it is replaced by the background medium andan equivalent surface electric current density is enforced onγi with Eti(r) = Eti(r). In this configuration, the tangentialmagnetic fields can be expanded as

Hti(r) =

mi∑n=1

hnfn(r). (27)

We collect all the expansion coefficients into a column vectorHi, which can be expressed as

Hi =[h1 h2 · · · hmi

]T. (28)

With the similar procedure in the original model, we obtain

Hi =[Pi]−1 (

T Li − Ui)

︸ ︷︷ ︸Yi

Ei. (29)

The equivalent current density can be expanded by the pulsebasis function, and the expansion coefficients are written intoa column vector Jei as

Jei =[j1 j2 · · · jmi

]T. (30)

By substituting (26) and (29) into (1), Jei can be expressedas

Jei = Hi −Hi = YsiEi, (31)

where Ysi is the DSAO [35] given by

Ysi = Yi − Yi =[Pi]−1 (

T Li − Ui)

− [Pi]−1 (T Li − Ui) . (32)

B. Expansion of the Electric Current Density for PEC Objects

If the object in Ωi is PEC, the surface electric currentdensity on the interface of PEC object can be expanded as

Jpi(r) =

mi∑n=1

jnfn(r), (33)

where Jpi is the physics current densities on the boundary ofΩi, and jn is the corresponding expansion coefficient. Then,we collect all the expansion coefficients into a column vectorJpi as

Jpi =[j1 j2 · · · jmi

]T. (34)

Once all the penetrable objects are replaced by the back-ground medium, the equivalent current densities in (30) andthe physics current densities in (34) are obtained. Then, we canconstruct the equivalent model for the composite structures.

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C. The Electric Current Density in the Equivalent Model forthe Composite Structure

As stated in the previous section, the electric current densityin the final equivalent model is obtained through collecting allthe modular equivalent electric current densities in (30) andthe physics current density in (34), which is expressed as

J =

[JeJp

], (35)

Assume that there are M penetrable objects and N PECobjects, and the unknowns on the boundaries of penetrableand PEC objects are collected together for a more intuitiverepresentation. Je and Jp are expressed as

Je =[Je1 · · · JeM

]T, (36)

Jp =[Jp1 · · · JpN

]T. (37)

When the electric current density in the final equivalentmodel is defined in (35), we can solve the electromagneticproblem through combining the electric field integral equation(EFIE) in the exterior region.

IV. SCATTERING MODELING

A. Scattering Modeling

When a plane wave incidents from the exterior region, thetotal electric fields in TM mode can be expressed as

E(r) = −jωµ0

∫γ

G0 (r, r′) J (r′) dr′ + Einc(r), (38)

where Einc denotes the plane wave, γ is the union of allenclosed boundaries, and J is the electric current densitiesconsisting of the equivalent and physics current densities.When the observation points are fixed on the boundaries, bysubstituting (35) into (38) and testing it on γ, we obtain thefollowing matrix equation

L[

E0

]= P

[JeJp

]+ Einc, (39)

where the entities of E with dimension of∑Mi=1mi are the

expansion coefficients of the tangential electric fields and the

entities of 0 with dimension of∑M+Ni=M+1mi are all zeros. L

is a diagonal matrix, which is given by

L =

Le︷ ︸︸ ︷L1

. . .LM

0

0

LM+1

. . .LM+N︸ ︷︷ ︸

Lp

. (40)

P is given by (41) at the bottom of this page. In (41), thesubscript (i, j) denotes the testing and source boundaries areγj , γi, respectively.

It should be noted that L is a diagonal matrix. As statedin Section II, we treat E on the shared boundary as twoindependent variables. Although they are placed on the samelocation, only the main diagonal entity of L is nonzero. Asshown in (38), the two variables represent electric fields onthe shared boundaries. Therefore, these two electric fields arealways equal to each other, which implies that the boundaryconditions for electric fields are automatically satisfied in theproposed SS-SIE formulation. In addition, it is true when themeshes are non-conformal. This is one of the key factors thatthe proposed SS-SIE formulation can support non-conformalmeshes.

By moving the first term on the RHS of (39) to its LHS,and inverting the square coefficient matrix, we obtain[

EJp

]=

[Le − P(e,e)Ys −P(e,p)

−P(p,e)Ys −P(p,p)

]−1Einc, (42)

where Ys is the DSAO of the composite structure assemblingfrom the modular DSAO for all penetrable objects, which isexpressed as

Ys =

Ys1

Ys2. . .

YsM

. (43)

Through solving (42), the tangential electric fields andthe physics electric current density can be obtained. Then,other interested parameters, such as near fields, the radarcross section (RCS), can be calculated. In the following

P =

P(e,e)︷ ︸︸ ︷P(1,1) · · · P(1,M)

.... . .

...P(M,1) · · · P(M,M)

P(e,p)︷ ︸︸ ︷P(1,M+1) · · · P(1,M+N)

.... . .

...P(M,M+1) · · · P(M,M+N)

P(M+1,1) · · · P(M+1,M)

.... . .

...P(M+N,1) · · · P(M+N,M)︸ ︷︷ ︸

P(p,e)

P(M+1,M+1) · · · P(M+1,M+N)

.... . .

...P(M+N,M+1) · · · P(M+N,M+N)︸ ︷︷ ︸

P(p,p)

(41)

7

subsection, we introduce how near fields can be calculatedthrough the tangential electric fields and the physics electriccurrent density.

B. Near Field Calculation

There are three scenarios for near field calculation. First, theelectric fields on the boundary of each dielectric objects havebeen computed through (42). Second, after all equivalent andphysics current densities are calculated from (42), the electricfields in the exterior regions of the original structures can becalculated according to (38). Third, through substituting Eiinto (26), Hi can be obtained, and then the electric fieldsin the interior regions of dielectric objects can be calculatedthrough (20), where the constant parameters in the originalstructures should be used.

C. Discussion

There are several obvious merits compared with otherexisting techniques to handle complex composite structuresin the proposed SS-SIE formulation. First, each object ismodularly modeled to derive the equivalent current density.It is easy to implement the proposed SS-SIE formulation.Second, when the equivalent model for the composite struc-tures is constructed, the equivalent current density is thesummation of each modular current densities enforced on thepenetrable object. Furthermore, they are treated as independentquantities. There are no requirements upon meshes on the con-nected interfaces, which implies that non-conformal meshesare intrinsically supported. As other non-conformal DDMsusually require to enforce the boundary conditions, it leadsto challenges for implementation in the practical engineeringproblems. However, since the boundary conditions for theelectric fields are automatically satisfied, no additional require-ments in the proposed SS-SIE formulation are needed. It isextremely easy to implement the proposed SS-SIE formulationand useful to model multiscale and electrically large structures.Third, since the DSAO is incorporated into the proposed SS-SIE formulation, only the electric current density is required.

One main bottleneck for the original SS-SIE formulations[20]-[24] incorporated the DSAO to model multiscale andelectrically large objects is that the construction of the DSAOis computationally intensive since matrix inversion is requiredas shown in (32). Therefore, those formulations are mainlyused to model small periodic and quasi-periodic scatters in[18], [44]. However, the proposed SS-SIE formulation in thispaper does not suffer from such issue as we can divide thosestructures into small units with the same constant parametersas those of the original object. We only needs to calculatethe DSAO for small units. Therefore, its construction canbe significantly accelerated. In addition, the large structurescan be partitioned into many identical units. Since only oneDSAO is required for each type of units, the efficiencycan be further improved. Although there are slightly moreunknowns required in the proposed SS-SIE formulation com-pared with those in [20], [44], the overhead can be ignoredfor challenging electromagnetic simulations compared with theoverall computational cost in terms of CPU time and memory

consumption. Significant less CPU time and high flexibilityto model complex structures can be obtained as shown in thenumerical examples in the later section.

V. NUMERICAL RESULTS AND DISCUSSION

Our in-house codes are developed in Matlab and all sim-ulations in the following subsection are carried out on aworkstation with a 3.2 GHz CPU and 256 G memory. Inaddition, all the codes are vectorized in Matlab. To makea fair comparison, only a single thread without any parallelcomputation is used to carry out the simulations.

A. A Coated Dielectric Cuboid

In this subsection, a coated dielectric cuboid is consideredas shown in Fig. 6. The side length of the inner and outer crosssection is 0.5 m and 1 m, respectively. The relative permittivityof the inner and outer dielectric object is εr1 = 9 and εr2 = 4.A TM polarized plane wave with 300 MHz incidents from thex-axis. The averaged length of segments used to discretize allthe boundaries is λ0/20, where λ0 is the wave length in thefree space, which corresponds to 6.7 and 10 samples in theouter and inner dielectric objects, respectively.

Fig. 6. Geometric configurations of the coated dielectric cuboid and itsconstant parameters.

Fig. 7 shows the RCS calculated from the COMSOL, thePMCHWT formulation, and the proposed SS-SIE formulation.The same meshes are used in both the PMCHWT formulationand the proposed SS-SIE formulation. As shown in Fig. 7,results obtained from the three approaches show excellentagreement. Fig. 8(a) and (b) show the magnitude of electricfields near the dielectric cuboid obtained from the proposedSS-SIE formulation and the COMSOL. It can be found thatfield patterns from the two approaches are almost exactly thesame and show excellent agreement with each other.

Fig. 7. The RCS of the coated dielectric cuboid obtained from the COMSOL,the PMCHWT formulation, and the proposed SS-SIE formulation.

8

(a) (b) (c)

Fig. 8. Near fields obtained from (a) the proposed SS-SIE formulation and (b) the COMSOL, (c) the relative error of near fields obtained from the proposedSS-SIE formulation.

To quantitatively measure the error, we calculated the rela-tive error of near fields, which is defined as

RE =

∣∣Ecal − Eref∣∣

max |Eref|, (44)

where Eref are the reference solutions obtained from theCOMSOL, Ecal are the solutions obtained from the proposedSS-SIE formulation, and max|Eref| is the maximum magnitudeof the reference solutions, which can guarantee the relativeerror well defined in the whole computational domain.

As shown in Fig. 8(c), the relative error is less than 2%in the most of computational domain and slight large nearthe boundary and corners of the inner dielectric object. It isexpected since in those regions fields changes sharply due tomedia discontinuity. In the current mesh configuration, only6.7 and 10 sample segments per wavelength for the innerand outer dielectric object are used, respectively. The relativeerror can be further reduced if finer mesh is used in thesimulation. Therefore, the proposed SS-SIE formulation canobtain accurate both near and far fields induced by dielectricstructures with non-smoothing boundaries.

To finish this simulation, the overall count of unknownsin the PMCHWT formulation is 408. For the proposed SS-SIE formulation, only 272 is used to model this compositestructure. In the PMCHWT formulation both the electric andmagnetic current densities are required. However, only theelectric current density is required in the proposed SS-SIE for-mulation. Therefore, much fewer unknowns are required in theproposed SS-SIE formulation compared with the PMCHWTformulation. The PMCHWT formulation uses 2.4 seconds forthis model, and the proposed SS-SIE formulation only takes1.7 seconds. To further test the capability of the proposedSS-SIE formulation, more challenging examples with theelectrically large objects and partially connected objects areconsidered in the next subsection.

B. Domain Decomposition for Large Objects

The SS-SIE formulation incorporated with the DSAO showssignificant performance improvements to model structureswith multiple repetitions in large arrays [24], [44] comparedwith the PMCHWT formulation. This is because the DSAOfor each type of units is only calculated once and then it canbe reused to assemble the global DSAO. When identical units

Fig. 9. Geometric configurations of nine large lossy objects.

Fig. 10. The nine large lossy objects are decomposed into 137 identical smallunits. The dimension of each unit is 1.5λ×9λ.

Fig. 11. The RCS obtained from the COMSOL, the PMCHWT formulation,the SS-SIE formulation in [44], and the proposed SS-SIE formulation.

9

(a) (b) (c)

Fig. 12. Near fields obtained from (a) the proposed SS-SIE formulation, (b) the COMSOL, (c) the relative error of near fields obtained from the proposedSS-SIE formulation.

are separated with no touched surfaces, the proposed SS-SIEformulation becomes exactly the same as those formulationsin [44]. Therefore, the proposed SS-SIE formulation inheritsthis merit. Furthermore, another useful merit is that a largestructure can be decomposed into many small identical unitsto enhance the efficiency.

As shown in Fig. 9, nine lossy objects with 117λ in lengthand 52.5λ in width are considered, where λ is the wave lengthin the objects. The relative permittivity is εr = 9 and theconductivity is 0.1 S/m for those objects. A TM polarizedplane wave with a frequency of 90 GHz incidents from thex-axis.

For the SS-SIE formulation in [44], each object is firstreplaced by the background medium and an equivalent currentdensity is constructed through inverting a large square matrixP, which is computationally intensive and leads to efficiencydegeneration. However, the objects are decomposed into 137identical small units in the proposed SS-SIE formulation asshown in Fig. 10. Each unit is 1.5λ in width and 9λ in length.Since only one type of units exists in the example, only oneDSAO is calculated and then is reused to construct the equiva-lent model for the large objects. Therefore, it can significantlyaccelerate the construction of the DSAO. The averaged meshsize used to discretize all boundaries is about λ0/30, whichcorresponds to 10 sample segments per wavelength in the lossydielectric objects.

Fig. 11 shows the RCS calculated by the COMSOL, thePMCHWT formulation, the SS-SIE formulation in [44], andthe proposed SS-SIE formulation. It can be found that the re-sults obtained from the four approaches are in good agreementin most regions. We can observe some discrepancies in the farfields between the COMSOL and other three formulations,such as around 22, and 99. However, all the three integral-equation-based formulations agree well with each other.

Fig. 12(a) and (b) show the near fields obtained from theproposed SS-SIE formulation and the COMSOL. It is easy tofind that fields pattern obtained from the two approaches arealmost the same. In addition, we calculated the relative errorof near fields obtained from the proposed SS-SIE formulationin Fig. 12(c). In most regions, the relative error is less than5% and slightly large in a few points, but only around 7%.Therefore, the proposed SS-SIE formulation can accuratelycalculate both near and far fields from electrically largeobjects.

Table I shows the comparison of computational consump-tion among the PMCHWT formulation, the SS-SIE formula-

tion in [44] and the proposed SS-SIE formulation. We onlycalculate time costs without preprocessing and postprocessingtime since they are almost the same for the three formulations.In the proposed SS-SIE formulation, the nine large objectsare decomposed into 137 small identical units to constructthe DSAO, the time for the generation of DSAO is only 4.9seconds and much less than that of the SS-SIE formulation in[44], which requires 5,350.6 seconds. Time costs for matricesfilling and matrices inversion are 6,518.2 seconds and 780.3seconds for the proposed SS-SIE formulation, respectively,which are more compared with the formulation in [44], whichrequires 5,835.4 seconds and 555.2 seconds, respectively. Thereason is that more unknowns exist on the interfaces betweentwo adjacent units in the proposed SS-SIE formulation. How-ever, the overall time cost for the proposed SS-SIE formulationis only 7,303.4 seconds, which shows significant performanceimprovement in terms of CPU time compared with 11,741.2seconds for the SS-SIE formulation in [44] and 22,988.9seconds for the PMCHWT formulation. In addition, memoryconsumption of the proposed SS-SIE formulation, 63,322 MB,is slightly more compared with that of the formulation in[44], 52,751 MB, because more unknowns are needed to bestored. However, in comparison with that of the PMCHWTformulation, 92,297 MB, the proposed SS-SIE formulation stillshows significant performance improvement.

TABLE ICOMPARISON OF COMPUTATIONAL COST FOR THE PMCHWT

FORMULATION, THE SS-SIE FORMULATION IN [44], AND THE PROPOSEDSS-SIE FORMULATION

PMCHWT Formulation in [44] Proposed

Total Time [s] 22,988.9 11,741.2 7,303.4Time for Ys - 5,350.6 4.9

Generation [s]Time for Ma-

21,563.0 5,835.4 6,518.2trices Filling [s]Time for Ma-

1,425.9 555.2 780.3trix solving [s]

Memory Con-92,297 52,751 63,322

sumption [MB]Overall Counts

60,000 30,000 39,456of Unknowns

C. Layered Penetrable Objects

An 8-layered dielectric structure is considered in this sub-section. The relative permittivity of each layer is 7.1, 3, 2.77,

10

4.2, 8, 2.77, 8 and 11.7 from the top to bottom layers asshown in Fig. 13. The electrical size of this structure is around6.7λ in length and 1.5λ in width. A plane wave with thefrequency of 250 GHz incidents from the y-axis. The meshsize is chosen based on the wave length in the correspondinglayer, which is λ0/27, λ0/17, λ0/16, λ0/20, λ0/28, λ0/16,λ0/28 and λ0/34, respectively. Fig. 14 shows the meshes usedin our simulation. A small gap is added and the triangle edgesare presented to better visualize the non-conformal meshes. Itshould be noted that only the boundary segments are used inour simulations. The numbers marked on the LHS indicate thecount of segments on the shared boundaries.

Fig. 13. Geometric configuration of the 8-layered structure, which includeseight partially connected dielectric objects with different permittivity asmarked in the figure.

Fig. 14. The non-conformal meshes for the layered objects. The numbers onthe left of the structure indicate the count of segments on the top and bottomboundaries used in our simulations. Note that a small gap is added, and thetriangle edges are presented to better visualize the non-conformal meshes.Only the boundary segments are used in our simulations.

Fig. 15 shows the RCS obtained from the COMSOL, thePMCHWT formulation, and the proposed SS-SIE formula-tion. It can be found that numerical results from the threeapproaches are in good agreement. Therefore, the proposedSS-SIE formulation can obtain accurate far fields for layeredcomposite objects with non-conformal meshes.

Fig. 16 shows near fields obtained from the proposed SS-SIE formulation and the COMSOL. It can be found thatnear fields obtained from the two formulations show goodagreement in Fig. 16(a) and (b). We also calculated therelative error of near fields obtained from the proposed SS-SIE formulation. As shown in Fig. 16(c), the relative error inmost regions is less than 3%, which shows that the proposedSS-SIE formulation can obtain accurate near fields like othernumerical techniques.

To finish the simulation, the PMCHWT formulation uses401.1 seconds and 7,572 unknowns, where the averaged lengthof segments is λ0/34, and only the conformal meshes canbe used. However, the proposed SS-SIE formulation only

Fig. 15. The RCS obtained from the COMSOL, the PMCHWT formulation,and the proposed SS-SIE formulation.

takes 115.2 seconds and 2,600 unknowns since the non-conformal meshes are used, and only the electric currentdensity is required. Therefore, it shows significant performanceimprovement.

D. A Composite Structure with Partially Connected Penetrableand PEC Objects

A composite structure with five different objects is con-sidered in Fig. 17. There are four dielectric objects withthe relative permittivity of εr1 = 6.25, εr2 = 4, εr3 = 9,εr4 = 2.25 and a PEC object. The radii of inner dielectricobject is 0.5 m and 1.5 m for the four outer quarter con-centric objects. The averaged length of segments to discretizethe five objects is λ0/25, λ0/20, λ0/30, λ0/15 and λ0/10,respectively. Similar to previous example, a small gap is addedand the triangle edges are presented to better demonstrate thenon-conformal meshes. The numbers of segments to discretizethe corresponding boundary are marked in Fig. 18, which areselected based on the constant parameters. A plane wave withthe frequency of 300 MHz incidents from the x-axis.

Fig. 19 shows the RCS obtained from the COMSOL, thePMCHWT formulation, and the proposed SS-SIE formulation.It is easy to find that the results from the three approachesare again in good agreement. To finish this simulation, thePMCHWT formulation uses 21.0 seconds and 1,406 un-knowns, where the averaged length of segments is λ0/30in the conformal meshes. However, the proposed SS-SIEformulation only takes 509 unknowns and 7.7 seconds, sincenon-conformal meshes are used and only the electric currentdensity is required.

The near fields obtained from the proposed SS-SIE for-mulation and the COMSOL are shown in Fig. 20(a) and(b). It can be found that the fields obtained from the twoapproaches show good agreement. In addition, the relativeerror of near fields between the two approaches is illustrated inFig. 20(c). As shown in the numerical results, the relative erroris less than 4%. Therefore, the proposed SS-SIE formulationis accurate to calculate complex composite structures withpartially connected penetrable and PEC objects.

E. Discussion

As shown in the four numerical examples, the potential ofthe proposed SS-SIE formulation is demonstrated by modeling

11

(a) (b) (c)Fig. 16. (a) Near fields obtained from the proposed SS-SIE formulation, (b) the COMSOL, (c) the relative error of near fields obtained from the proposedSS-SIE formulation.

Fig. 17. Geometric configuration of the composite structure consisting offour concentric dielectric and one PEC objects.

Fig. 18. The non-conformal meshes for the composite object with fourpartially connected dielectric and one PEC objects, and the numbers indicatethe count of the segments on the shared boundaries used in our simulations.

Fig. 19. The RCS obtained from the COMSOL, the PMCHWT formulation,and the proposed SS-SIE formulation.

simple dielectric objects with non-smoothing boundaries, theelectrically large lossy objects, the planar layered objectsand the partially connected dielectric and PEC objects. Bothnear and far fields are accurately and efficiently calculatedfrom the proposed SS-SIE formulation. Although our currentimplementations are in the TM mode, and our numericalexamples are simplified from practical engineering problems,like metallic traces in integrated circuits, the planar multilayermedia, the proposed SS-SIE formulation shows great potentialin solving practical engineering problems.

However, since additional unknowns will be added on theshared boundaries, the proposed SS-SIE formulation mayshow efficiency degeneration to model objects with large crosssection, like objects with large circular cross sections. Thisissue can be mitigated through the fast direct solver, such asthe H-matrix [46], HSS methods [47], and so on.

VI. CONCLUSION

An efficient and simple SS-SIE formulation for electromag-netic analysis of arbitrarily connected penetrable and PEC ob-jects is developed. Through modularly constructing the equiv-alent model incorporating with the DSAO for each penetrableobjects, and combining the equivalent current densities andphysics current densities on the PEC boundaries, an equivalentmodel for the composite structures with only the electriccurrent densities is derived. The proposed SS-SIE formula-tion shows many significant advantages over other existingtechniques, like implicitly enforced boundary conditions, easyimplementation, intrinsically non-conformal mesh support andonly the single electric current density. Those merits arequite useful for challenging electromagnetic simulations, likescattering from multiscale and electrically large objects, pa-rameter extraction for high density integrated circuits. As ournumerical results shown, significant performance improvementin terms of the CPU time and memory consumption is obtainedcompared with the traditional PMCHWT formulation. It ismuch more flexible than the original SS-SIE formulation tosolve the challenging electromagnetic problems.

Extension of current work into three dimensional generalscenarios is in progress. We will report more results upon thistopic in the future.

APPENDIX

An extensional example is presented to demonstrate thecapability of the proposed SS-SIE formulation to solve the

12

(a) (b) (c)Fig. 20. Near fields obtained from (a) the proposed SS-SIE formulation, (b) the COMSOL, (c) the relative error of near fields obtained from the proposedSS-SIE formulation.

vector TE mode in Fig. 21. There are three dielectric quartercylinders with the relative permittivity of εr1 = 5, εr2 = 8,εr3 = 2 and a PEC quarter cylinder. The radii of the concentricquarters is 1 m. The basis function used to discretize thevector electric fields and electric current density is the rooftopbasis function, which mimics the RWG basis function in threedimensional space [48, Ch. 2, pp. 92]. In Fig. 22, four basisfunctions are marked at the junctions of the four quartercylinder intersection. The averaged mesh size of the fourboundaries is λ0/22, λ0/28, λ0/14, λ0/10, respectively. Itis obvious that the mesh is non-conformal and the count ofsegments for each region is shown in Fig. 22. Similar to ourprevious two numerical examples, a small gap is added and thetriangle edges are added to better visualize the non-conformalmeshes. In this example, a plane wave with a frequency of300 MHz incidents from the x-axis.

Unlike the traditional SIE formulations [13]-[17], in whichspecial attention should be paid to the boundary conditions atthe junction, the proposed SS-SIE formulation does not needany special treatments at the junction, which is quite easyto handle this structures. Fig. 23 presents the RCS obtainedfrom the COMSOL and the proposed SS-SIE formulation.Results obtained from the proposed SS-SIE formulation showexcellent agreement with those of the COMSOL. Therefore,the proposed SS-SIE formulation is also applicable to solvethe general electromagnetic problems in the vector TE mode,which can significantly simplify the implementations on thejunctions of multiple media intersection.

Fig. 21. A composite structure consisting of four quarters including threedielectric and one PEC objects.

Fig. 22. The non-conformal meshes for the composite object includingthree dielectric and one PEC objects, and the numbers indicate the countof segments on the shared boundaries used in our simulations.

Fig. 23. The RCS obtained from the COMSOL and the proposed SS-SIEformulation.

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