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A Surface Integral Formulation for ScatteringModeling by 2D Penetrable Objects

Xiaochao Zhou, Graduate Student Member, IEEE, Shunchuan Yang, Member, IEEE,and Donglin Su, Member, IEEE

Abstract—In this paper, we proposed a single-source surfaceintegral formulation to accurately solve the scattering problemsby 2D penetrable objects. In this method, the objects are re-placed by their surrounding medium through enforcing a surfaceequivalent electric current to ensure fields exactly the same asthose in the original scattering problem. The equivalent electriccurrent is obtained through emitting the equivalent magneticcurrent by enforcing that the electric fields in the original andequivalent problems equal to each other. Through solving theHelmholtz equation inside objects by the scalar second Greentheorem, we could accurately model arbitrarily shaped objects.Then, we solve the exterior scattering problems through thecombined integral equation (CFIE) with the equivalent electriccurrent. The proposed formulation only requires a single electriccurrent source to model penetrable objects. At last, two numericalexperiments are carried out to validate its accuracy, stability andcapability of handling non-smoothing objects.

Index Terms—Penetrable, surface integral equation, surfaceadmittance, scattering

I. INTRODUCTION

THE method of moment (MOM), which is based onsurface integral equations, is widely used to solve scat-

tering problems [1] and extract the electrical parameters oflarge-scale integrated circuits (ICs) [2]. Compared with othervolumetric methods, like the finite-difference time-domain(FDTD) method [3] and the finite element method (FEM) [4],it has much smaller count of unknowns.

To model the piecewise homogenous penetrable media,various formulations, like the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) [5], the combined tangential field (CTF)[6], are proposed in the last decades. In those formula-tions, a two-region problem is solved by introducing boththe surface equivalent electric and magnetic currents throughthe equivalence theorem. Therefore, it would be desirableto achieve single-source formulations to model penetrableobjects. When considering the conducting media, the single-source formulation could be obtained through incorporatingwith the impedance boundary condition [7] or the general-ized impedance boundary condition [8]. Further investigationsfound that they suffer from accuracy issue in the low frequency

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

dation of China through Grant 61801010, Grant 61727802, the “13th Five-Year” Equipment Pre-research Fund through Grant 61402090602 and BeijingNatural Science Foundation through Grant 4194082. (Corresponding author:Shunchuan Yang)

All authors are with the School of Electronic and Information En-gineering, Beihang University, Beijing, 10083, China (e-mail: zhouxi-aochao@buaa.edu.cn, scyang@buaa.edu.cn, sdl@buaa.edu.cn).

range or inner resonance problem. Recently, new surface-volume-surface single-source formulations through mappingvolume integral operator to its surface counterpart were pro-posed to model penetrable objects [9], [10]. They couldsignificantly improve the efficiency compared with the volumeintegral equation methods. However, they still involve thevolume integral operator.

In this paper, we proposed another single-source formu-lation to solve the 2D transverse magnetic (TM) scatteringproblems for penetrable objects. The proposed formulation isbased on the combined integral equation (CFIE) incorporatedwith the equivalent surface electric current obtained from thedifferential surface admittance operator, which was introducedin [11] to model high-speed interconnects. Many other effortsare made to extend its capabilities such as modeling circularcables [12], arbitrarily shaped interconnects [13], 3D scatteringproblems [14] and antenna array [15]. In this paper, wefurther extend the differential surface admittance operator tosolve the 2D TM scattering problems by arbitrarily shapedpenetrable objects. In the proposed formulation, only thesurface equivalent electric current source is required.

The paper is organized as follows. In Section II, we demon-strate the proposed formulation with the single equivalentelectric current source in detail. In Section III, numericalexperiments are carried out to validate its accuracy, stabilityand capability of handling non-smoothing objects. At last, wedraw some conclusions.

II. FORMULATION

A. The Equivalent Surface Current

We consider a penetrable object denoted its boundary asγ and interior domain as S, with the permittivity, the per-meability and the conductivity as ε1, µ1, σ1, respectively.It is surrounding by a background medium with ε0, µ0, σ0and illuminated by a plane wave as shown in Fig. 1(a).According to the surface equivalence theorem, we could obtainan equivalent problem, in which fields are exactly the same asthose in the original problem through replacing the objects bytheir surrounding medium and enforcing a surface equivalentelectric and magnetic current at γ as shown in Fig. 1(b) [11].The exterior fields in the original problem are exactly thesame as those in the equivalent problem, denoted as E0 = E0

and H0 = H0. These two surface currents are nothing but thedifference between the tangential fields in the original andequivalent problems. If we carefully select the electric andmagnetic fields in the equivalent problem, a single source

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(b)Fig. 1. (a) Original scattering problem, where E0 and H0 are exterior fieldsand Ez , Ht are interior fields and (b) the equivalent problem with the objectreplacing by its surrounding medium and enforcing the surface electric andmagnetic currents at γ, where E0, H0, Ez and Ht are exterior and interiorfields, respectively.

formulation could be obtained. As stated in [11], a singlesource formulation by carefully choosing the electric fields inthe equivalent problem is achieved. In this paper, we considerthe single surface equivalent electric current, denoted as Js.However, we could get its dual single source formulation forthe magnetic source using the same manner.

According to the equivalence theorem [16], the surfaceequivalent electric current and magnetic current could beexpressed as

Js(~r′) = Ht(~r

′)− Ht(~r′), Ms(~r

′) = Et(~r′)− Et(~r

′) (1)

where Et(~r′), Et(~r′), Ht(~r′), Ht(~r

′) are the surface tangentialelectric and magnetic fields in the original and equivalentproblems, respectively and all quantities with a hat denotetheir values in the equivalent problem.

Since the electric and magnetic fields in the equivalentproblem could be arbitrary, we could obtain the single electriccurrent source by enforcing that Et(~r′) = Et(~r

′) and have

Js(~r′) 6= 0, Ms(~r

′) = 0. (2)

On the other hand, we could have the single magnetic currentsource by enforcing that Ht(~r

′) = Ht(~r′) and have

Ms(~r′) 6= 0, Js(~r

′) = 0. (3)

Both of them could give us a single-source formulation. In thispaper, we select the first to obtain the single-source integralformulation and then derive the surface equivalent electriccurrent using the contour integral method [18].

Considering the penetrable objects do not include anysources, the electric fields must satisfy the following scalarHelmholtz equation in S,

∇2Ez + k2Ez = 0, (4)

subject to the boundary condition,

Ez(~r) = Ez(~r), ~r ∈ γ. (5)

(4) could be solved through the second scalar Green theorem,and we obtain the relationship between the electric field Ezand its normal derivative of Ez at γ [18],

1

2Ez(~r) =

∮γ

[G(~r, ~r′)

∂Ez(~r′)

∂n′− ∂G(~r, ~r′)

∂n′Ez(~r

′)

]dr′,

(6)

where G is the Green function expressed as G = − i4H

(2)0 (kρ),

where k is the wavenumber in the 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 in 2D-TM at γ through the Poincare-Steklov operator [11] as

Ht(~r) =1

jωµ

[∂Ez(~r)

∂n

]~r∈γ

, (7)

where µ is the permeability of the object. We use the pulsebasis function to expand Ez and Ht and point-matchingscheme to test (6) and (7) at midpoints of each segment ofγ. Then, after substituting (7) into (6), we collect all Ez andHt coefficients into two column vectors, E, H, and write (6)into matrix form as

PH = UE. (8)

Next, through inversing the square matrix P, we obtain thesurface admittance operator Y as

H = P−1U︸ ︷︷ ︸Y

E. (9)

E, H are denoted as the discretized Ez , Ht at the interior of γin the original problem. When all the parameters are replacedby its surrounding medium, we obtain the equivalent problem.With similar procedure, for the equivalent problem, we couldobtain

H = P−1U︸ ︷︷ ︸Y

E, (10)

where H represents the discretized magnetic field Ht in theequivalent problem, as shown in Fig. 1(b). Since we haveenforcing (2), we have

Js = YsE, (11)

where Ys is the differential surface admittance operator andcould be expressed as

Ys = Y − Y = P−1U− P−1U. (12)

When multiple scatters are involved, the Ys is a block diag-onal matrix assembling from all differential surface admittanceoperator for each object.

B. The Scattering Modeling

The scattering fields induced by Js at γ in the exteriorregion of the equivalent problem [17] could be expressed as

Es(~r) = −jωµ0

∫γ

Js(~r′)G0(~r, ~r

′)ds′, (13)

Hs(~r) = −n× Js2−∫γ

Js ×∇G0 (~r0, ~r′) d~s′, (14)

where ~r′ ∈ γ, ω, µ0 and n denote the angular frequency, thepermittivity of the background medium, and the unit normalvector pointing out of the object, respectively, G0(~r, ~r

′) is theGreen function expressed as G0(~r, ~r

′) = − j4H(2)0 (k0ρ), where

k0 is the wavenumber in the background medium. Unlike forthe interconnect problems, where a 2D static Green functionis used with the quasi-static assumption, the full wave Greenfunction should be used for scattering modeling.

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Therefore, the total fields at the exterior region could beexpressed as the superposition of the scattering and inci-dent fields, E0 = Es +Einc and H0 = Hs +Hinc. We couldrewrite them as

Eo = L(Js) +Einc, (15)

Ho = K(Js) +Hinc, (16)

where

L(Js) = −jωµ0

∫γ

Js(~r′)G0(~r, ~r

′)ds′, (17)

K(Js) = −n×Js2−∫γ

Js ×∇G0 (~r0, ~r′) d~s′. (18)

To avoid possible internal resonance, we select to use thelinear combination of (15) and (16). Therefore, we have thefollowing formulation,

αEo + (1− α) n× ηHo = α(Es +Einc

)+(1− α) n× η

(Hs +Hinc

),

(19)

where α is a real constant in [0, 1] and η denotes the waveimpedance of the background medium. α is selected as 0.5 inall simulations.

We expand Eo and Ho with the pulse basis function anduse the point-matching scheme at each segment of γ. Since thesurface electric field at γ in the equivalent problem has beenenforced to equal to that in the original problem, Ys relatesJs and E through (12), where Ys is introduced in [11], [13]for interconnect modeling. It should be noted that [19] explorethe possibility to use the surface admittance operator to solvethe scattering problems. However, the method in [19] is onlyapplicable for the canonical objects and possibly suffers frominternal resonance issues. In this paper, the proposed methodis free from internal resonance and applicable for arbitrarilyshaped objects.

With the definition of Y in (9) and Ys in (12), we couldsolve the electric field E at γ through (19) as

E =[α(I+ LYs

)+ (1− α) η

(Y + KYs

)]−1[αEinc + (1− α) n× ηHinc

],

(20)

where I is identity matrix. Readers should be careful that Yin (20) rather than Y should be used. Since no currents at γexist, the magnetic fields at inner and exterior side of γ in theoriginal problem equal to each other as shown in Fig. 1(a), i.eHt(~r

′) = H0(~r′), ~r′ ∈ γ. In the equivalent problem, we have

H0(~r′) = H0(~r

′). Therefore, Y should be used.

C. Electric Field Computation

There are three domains, the exterior region of objects, theboundary and inner region of objects in the computationaldomain. When we require the electric fields in the exteriorregion of objects, they could be obtained through (13) and(11). At γ, we already have the fields at the middle pointsof each segment. Therefore, the fields at other locations of γcould be easily interpolated through calculated values. In S,we obtain fictions fields since we use the equivalence theorem.

-1.5 -1 -0.5 0 0.5 1 1.5x(m)

-1.5

-1

-0.5

0

0.5

1

1.5

y(m

)

-1

-0.5

0

0.5

1

1.5

(a)

-1.5 -1 -0.5 0 0.5 1 1.5x(m)

-1.5

-1

-0.5

0

0.5

1

1.5

y(m

)

0.005

0.01

0.015

0.02

0.025

(b)

2 4 6 8 10 12 14Relative permittivity

0.01

0.02

0.03

0.04

0.05

0.06

0.07

RM

S

(c)

20 40 60 80 100 120 140Frequency (MHz)

-10

-5

0

5

RC

S (d

Bsm

)

the proposed methodthe analytical method

(d)Fig. 2. (a) Electric fields obtained from the proposed method, (b) relative errorbetween electric field obtained from the proposed method and the analyticalmethod, (c) relative error verse the permittivity of the object and (d) relativeerror of the RCS obtained from the proposed method and the analytical methodin a wide frequency range.

We could use the tangential electric fields at γ to calculate thefields in the interior region of objects as,

Ein = (U′ −P′Y)E, (21)

where Y is exactly the same as that in (9), P′, U′ are obtainedfrom the first and the second term of right hand side of (6),respectively.

III. NUMERICAL RESULTS AND DISCUSSION

In this section, we present two numerical examples todemonstrate the accuracy, stability and the capability of han-dling non-smoothing objects.

A. A Dielectric Cylinder Object

The first numerical experiment considered is a infinite cylin-der with εr = 4. Its radius is one wavelength, 1λ0, where λ0 isthe wavelength in the free space. The background medium isair. A plane wave with the frequency 300 MHz incidents fromthe x axis. The minimum segment length is selected as λ0/10,0.1 m to discretize the contour of the cylinder. In the followingsimulations, we will use this configuration otherwise stated.The relative error is defined as |Eref − Ecal|/max|Eref |,where Eref and Ecal denote the reference and calculatedfields. The analytical fields are selected as the reference.

We obtain the electric fields in the computational domainas shown in Fig. 2 (a) and the error pattern is shown as inFig. 2(b). It is easy to find that the maximum relative erroris below 3% compared with the analytical fields. It showsthat the proposed method could obtain accurate results for thepenetrable objects. The error could be further reduced as werefine the mesh.

4

-1 -0.5 0 0.5 1x (m)

-1

-0.5

0

0.5

1

y (m

)

0.005

0.01

0.015

0.02

0.025

(a)

0° 90° 180° 270° 360°scattered angle

-50

-40

-30

-20

-10

0

10

RC

S (d

Bsm

)

the proposed methodthe Comsol

(b)

Fig. 3. (a) relative error between electric field obtained from the proposedmethod and Comsol and (b) RCS obtained from the proposed method and theComsol.

We further investigate the accuracy and stability of theproposed method for different εr. For convenience, we fixedthe mesh size to 0.05 m for the wideband analysis. The error is

defined as RMS=√∑

|Eref − Ecal|2/∑|Eref |2. As shown

in Fig. 2(c), we find that the proposed method could getaccurate results with εr changing from 1.1 to 15. In addition,as expected, the error will rise as the permittivity of the objectincreases since in our simulations we fixed the mesh size. Forthe medium with a high permittivity, the mesh could not beenough sampled. However, the error could be further reducedas the mesh is refined. Therefore, the proposed method isaccurate and stable from the low to high εr values. The resultsshow that the proposed method has excellent performance interms of accuracy and stability for different media parametercontrast.

Fig. 2(d) illustrates the accuracy of the proposed method in awide frequency range. We fixed the mesh size at 0.05 m. TheRCS obtained from the proposed method and the analyticalsolution shows good agreement from 15 MHz to 150 MHz.Therefore, the proposed method could obtain accurate resultsin a wideband frequency range.

B. A Dielectric Cuboid Object

An infinite long cuboid with non-smoothing corners andεr = 4 is considered. Its surrounding medium is air. Theincident plane wave is along the x axis with the frequency300 MHz. The side length is 1λ0, 1 m. The minimum meshsize is 0.05 m to discretize the contour of the cuboid.

Fig. 3(a) shows the relative error between the electric fieldsobtained from the proposed method and the Comsol. It is easyto find that the relative error in the whole computational do-main is less than 3%. It implies that the proposed method couldget accurate fields compared with the Comsol. Meanwhile, theRCS obtained through both methods shows good agreementas shown in Fig. 3(b).

As shown in these numerical experiments, the proposedmethod is applicable objects with not only smoothing surfacesbut also non-smoothing surfaces and shows good performancein terms of accuracy and stability.

IV. CONCLUSION

In this paper, we proposed a surface integral formulation forscattering modeling by 2D penetrable objects. The proposed

method only requires a single electric current source derivedfrom the surface equivalence theorem. We related the surfaceelectric current and surface electric field through the differen-tial surface admittance operator. However, we could obtain itsdual magnetic single source formulation with a similar manner.Then, with combining the CFIE, we could solve the scatteringproblems by arbitrarily shaped objects and the fields also couldbe reconstructed from the computed surface electric fields atany location. Numerical results show that the proposed methodis robust, accurate to handle smoothing and non-smoothingobjects with the low to high permittivity in a wide frequencyrange.

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