A Macromodeling Approach to Efficiently Compute Scattering ... · to be a daunting task. Unlike volumetric methods such as the ﬁnite element method (FEM) [3] and the ﬁnite difference

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2018.2866509, IEEETransactions on Antennas and Propagation

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 1

A Macromodeling Approach to Efficiently ComputeScattering from Large Arrays of Complex Scatterers

Utkarsh R. Patel, Student Member, IEEE, Piero Triverio, Senior Member, IEEE, and Sean V. Hum, SeniorMember, IEEE

Abstract—Full-wave electromagnetic simulations of electricallylarge arrays of complex antennas and scatterers are challenging,as they consume large amount of memory and require longCPU times. This paper presents a new reduced-order modelingtechnique to compute scattering and radiation from large arraysof complex scatterers and antennas. In the proposed technique,each element of the array is replaced by an equivalent electriccurrent distribution on a fictitious closed surface enclosing theelement. This equivalent electric current density is derived usingthe equivalence theorem and it is related to the surface currentson the scatterer by the Stratton-Chu formulation. With theproposed approach, instead of directly solving for the unknownsurface current density on the scatterers, we only need to solve forthe unknowns on the equivalent surface. This approach leads toa reduction in the number of unknowns and better conditioningwhen it is applied to problems involving complex scattererswith multiscale features. Furthermore, the proposed approach isaccelerated with the adaptive integral equation method to solvelarge problems. As illustrated in several practical examples, theproposed method yields speed up of up to 20 times and consumesup to 12 times less memory than the standard method of momentsaccelerated with the adaptive integral method.

Index Terms—surface integral equation method, macromodel,equivalence theorem, adaptive integral equation method, multi-scale problems, reduced-order modeling

I. INTRODUCTION

Accurate numerical methods are needed to design andoptimize large arrays of antennas and scatterers such asphased arrays, frequency selective surfaces, metasurfaces, andreflectarrays. Currently, most of these arrays are designed andanalysed with simulation tools that apply periodic boundaryconditions [1], [2], [3], neglecting effects of finite array sizeand dissimilar array elements. Despite recent advances in com-putational hardware capabilities, the full-wave electromagneticsimulation of large arrays of antennas and scatterers continuesto be a daunting task. Unlike volumetric methods such as thefinite element method (FEM) [3] and the finite difference (FD)method [4], the surface integral equation (SIE) method [5]only requires discretization of the surfaces composing thescatterer(s), which makes it an appealing technique to solvemany scattering problems. However, as the electrical sizeof the problem increases, even the SIE method requiresprohibitive amounts of memory and CPU time. Multiscale

Manuscript received ...; revised ...U. R. Patel, P. Triverio, and S. V. Hum are with the Edward S. Rogers

Sr. Department of Electrical and Computer Engineering, University ofToronto, Toronto, M5S 3G4 Canada (email: [email protected],[email protected], [email protected]).

features, commonly found in reflectarrays and metasurfaces,can further hinder the performance of traditional SIE methods.

Electrically large problems can be solved with the SIEmethod using either acceleration methods or reduced-ordermodeling methods. Acceleration and reduced-order modelingtechniques achieve scalability in different ways. In acceler-ation techniques, the far-field interactions between the basisand testing functions are accelerated using efficient matrix-vector product routines. The most commonly used accelera-tion techniques found in the literature are the fast multipolemethod (FMM) [6], [7], multi-level fast multipole method(MLFMM) [8], [9], adaptive integral method (AIM) [10],pre-corrected fast Fourier transform (pFFT) [11]–[13], andconjugate gradient fast Fourier transform (CG-FFT) [14]–[17].

In FMM and MLFMM, the spherical wave expansion isused to approximate far-field interactions. Alternatively, thefast Fourier transform may be used to accelerate far-fieldcomputations as done in AIM, pFFT, and CG-FFT. While theaforementioned acceleration techniques allow simulating largestructures, memory consumption and computation time maystill increase dramatically in presence of multiscale features.Multiscale features also require long times to compute near-field interactions which, even in accelerated methods, continueto be a bottleneck.

The goal of reduced-order modeling techniques is to de-crease the number of unknowns to be solved in the linearsystem. In these techniques, the original set of basis functionsis projected onto a new set of basis functions that canwell approximate the solution space with a fewer number ofbasis functions. The new set of basis functions is obtainedmathematically using either the eigenvalue decomposition orthe singular value decomposition. Common reduced-ordermodeling techniques in the literature include macro-basisfunctions [18], characteristic basis functions [19], syntheticbasis functions [20], and eigencurrent basis functions [21].

The equivalence principle algorithm (EPA) [22] is a comple-mentary approach to acceleration and reduced-order modelingtechniques for tackling multiscale electromagnetic problems.The EPA is based on the Love’s equivalence principle [23]. Inthe EPA, the scatterer is first enclosed by a fictitious closedsurface. The field interactions are characterized by the scat-tering matrix, which relates the incident and scatterered tan-gential electric and magnetic fields on the equivalent surface.Furthermore, the coupling between two equivalent surfaces iscaptured by translation operators which relate the tangentialelectric and magnetic fields on all equivalent surfaces. Thismethod allows computing scattering by solving for unknowns




only on the equivalent surfaces. The accuracy of the EPA hasbeen improved with higher order basis functions [24] andtangential surface integral equations [25]. Furthermore, theEPA has been accelerated with MLFMM [26] and combinedwith reduced-order basis functions such as characteristic basisfunctions [27]–[30] as well as eigencurrent basis functions [31]to further reduce the number of unknowns.

In this paper, we present a novel reduced-order modelingscheme based on the Stratton-Chu formulation and the equiv-alence theorem to efficiently compute scattering from largearrays of scatterers made up of perfect electric conductors(PECs) in free space. This work is an extension of a similaridea proposed for a 2D transmission line problem [32]. Inour method, each element of the array is modeled by a so-called macromodel that compactly represents the scatteredfield from the element. The macromodel is made up of anequivalent electric current density introduced on a fictitiousclosed surface surrounding the element and a linear transferoperator to relate the equivalent electric current density tothe actual surface currents on the scatterer. Unlike most SIEmethods where both the tangential electric and magnetic fieldsare expanded with RWG basis functions, we employed RWGand dual RWG basis functions [33] to expand the tangentialmagnetic and electric fields, respectively. By using RWG anddual RWG basis functions all integral operators in the Stratton-Chu formulation can be well-tested, which ultimately allows usto derive a macromodel that is robust. The proposed method,like the EPA, is faster than traditional SIE methods for threereasons. First, the proposed method has fewer unknowns thanthe original problem. In the original problem, the unknownsare coefficients of electric surface current density on thescatterer. On the other hand, in the equivalent problem, theunknowns are only on the equivalent surface. Hence, forcomplex scatterers with multiscale features, the number ofunknowns on the equivalent surface could be significantlylower than the number of unknowns on the scatterer. Second,the macromodel approach improves the condition numberof the linear system since the unknowns are only on theequivalent surface, which has no fine features. Finally, theproposed approach exploits repeatability of elements in thearray. That is, the macromodels generated for one elementcan be reused for other identical elements in the array, whichleads to significant computational and memory savings.

In contrast to the EPA [22], which requires both theequivalent electric and magnetic current densities to modelthe scatterer, the proposed method applies the equivalenceprinciple in a unique way such that the scatterer can bemodeled by only an equivalent electric current density [34].The absence of an equivalent magnetic current density leadsto solving fewer unknowns overall. A single-source versionof the EPA has been presented where half of the unknownsare eliminated mathematically from the original EPA byrelating the discretized equivalent magnetic current densityto the discretized electric current density by the generalizedimpedance boundary conditions [35], [36]. While this elimi-nates the magnetic current density from the EPA in the discretedomain, magnetic current density still exists in the continuousdomain. In the proposed formulation, however, the equivalent

Ŝm

Sm

(a) Original

Ŝm

(b) Equivalent

Fig. 1. (a): Original configuration: Sample unit cell made up of metallicscatterers enclosed by the equivalent surface. (b): Equivalent configurationwhere the metallic scatterers are removed from the equivalent surface. Torestore the fields outside Ŝm, an equivalent electric current density (shownwith blue arrows) is introduced on Ŝm.

magnetic current density is not required whatsoever. As aconsequence, the exterior problem in the proposed formulationis simpler with fewer integral operators than even the single-source EPA [35], [36]. Furthermore, both the EPA and theproposed method derive a different macroscopic quantities tomodel interactions with the scatterer. While the EPA derivesa scattering matrix to relate the tangential incident electricand magnetic fields to the scatterered tangential electric andmagnetic fields, the proposed method derives an admittancematrix to relate the equivalent electric current density to thetangential electric field to model the scatterer.

The paper is organized as follows. First, we provide themathematical framework to create the macromodel for asingle element of an array in Sec. II. Then, in Sec. III, wereplace all elements of the array by their macromodels. Oncemacromodels are generated, the coupling between them iscaptured by the electric field integral equation in Sec. IV. Totackle electrically large problems, the proposed approach isaccelerated with AIM in Sec. V. In Sec. VI, we present threenumerical examples to show the accuracy and efficiency ofthe proposed method. Finally, Sec. VII presents concludingremarks on this work.

II. MACROMODEL OF A SINGLE ELEMENTWe consider the problem of computing scattering from

an M -element array of complex scatterers. In this section,we derive a macromodel for one element of the array. Themacromodel derived in this section is based on the equiva-lence theorem and it is therefore exact, except for numericalerrors introduced by discretization of fields and currents. Thederived macromodel efficiently describes the electromagneticbehaviour of the original element using fewer unknowns,reducing memory consumption and computation time. Forsimplicity, we assume that the structure is excited by anelectric field incident on the scatterer. Results in Sec. VI,however, show that the proposed idea is also applicable todriven antenna elements.

A. Fields and Currents Discretization

We consider the m-th element of the array. This elementconsists of several PEC surfaces, which are denoted by Sm.




We enclose the element by a fictitious closed surface, whichis denoted by Ŝm. A sample scatterer and the enclosingequivalent surface are shown in Fig. 1a.

1) Discretization of Sm: The electric surface current den-sity on the metallic scatterers is expanded as

Jm(r) =

Nm∑n=1

Jm,nΛm,n(r) , (1)

where Λm,n(r) is the n-th RWG basis function [37] on them-th element. The coefficients of Jm(r) in (1) are collectedinto vector

Jm =[Jm,1 Jm,2 . . . Jm,Nm

]T. (2)

2) Discretization of Ŝm: Like the surface current densityon the element, the tangential magnetic field on the equivalentsurface Ŝm is also expanded with RWG basis functions

n× Ĥm(r) =N̂m∑n=1

Ĥm,nΛ̂m,n(r) , (3)

where n is the normal vector pointing into the surface Ŝm.Note that we use ̂ to denote all quantities on the equivalentsurface. The tangential electric field on Ŝm is, instead, ex-panded with dual RWG basis functions [33], [38], [39]

n× Êm(r) =N̂m∑n=1

Êm,nΛ̂′m,n(r) . (4)

The n-th dual RWG basis function Λ̂′m,n is approximatelyorthogonal to the n-th RWG basis function Λ̂m,n. The use ofboth RWG functions and their duals is necessary to achieve awell-conditioned formulation and high robustness, as will bediscussed in detail in the next sections. As in (2), we collectthe coefficients of the tangential electric and magnetic fieldsin (3) and (4) into vectors

Ĥm =[Ĥm,1 Ĥm,2 . . . Ĥm,N̂m

]T(5)

Êm =[Êm,1 Êm,2 . . . Êm,N̂m

]T. (6)

B. Equivalence Theorem

As seen from Fig. 1a, a unit cell of a typical metasurface orreflectarray can be quite complex. In order to handle complexunit cells efficiently, we apply the equivalence theorem [23]to Ŝm. As shown in Fig. 1b, we replace all PECs inside thesurface with free space and introduce on Ŝm an equivalentelectric current density [23]

Ĵm(r) = n×[H̃m(r)− Ĥm(r)

](7)

and an equivalent magnetic current density [23]

M̂m(r) = −n×[Ẽm(r)− Êm(r)

]. (8)

In (7) and (8), H̃m(r) and Ẽm(r) are the electric and magneticfields on Ŝm in the equivalent problem. According to theequivalence theorem, these currents will produce the sameelectric and magnetic fields outside Ŝm as the actual currents

V

Sn

H(r), E(r)

J(r)µ0, ε0

Fig. 2. Sample boundary value problem considered in Sec. II-C. Electric andmagnetic fields are defined on the boundary of a closed surface S. There isalso an additional electric current density J(r) inside V .

on the PEC elements, allowing us to compute the radiationfrom the array.

Most SIE formulations are based on the Love’s equivalencetheorem [23], which sets H̃m and Ẽm to zero, and requireboth Ĵm(r) and M̂m(r) to restore the electromagnetic fieldsoutside the scatterer. However, in this work, we enforce thatẼm(r) is equal to Êm(r) [40], [41]. Therefore, the magneticequivalent current in (8) is zero and only a single equivalentcurrent source is required to model the scatterer. This single-source equivalence approach has been successfully applied tomodel 3D dielectrics [34], [42] and conductors [43].

We expand the tangential magnetic field n× H̃m(r) on Ŝmin the equivalent problem using RWG basis functions

n× H̃m(r) =N̂m∑n=1

H̃m,nΛ̂m,n(r) , (9)

and collect its expansion coefficients into vector

H̃m =[H̃m,1 H̃m,2 . . . H̃m,N̂m

]T. (10)

Similarly, the equivalent electric current density is also ex-panded with RWG basis functions as

Ĵm(r) =

N̂m∑n=1

Ĵm,nΛ̂m,n(r) , (11)

and its expansion coefficients are collected into vector

Ĵm =[Ĵm,1 Ĵm,2 . . . Ĵm,N̂m

]T. (12)

Typically, Ŝm needs to be meshed with 6-8 triangles perwavelength to accurately capture the inter-element coupling.By substituting, (11), (9) and (3) into (7) we obtain

Ĵm = H̃m − Ĥm (13)

in the discrete domain.Next, we simplify (13) by applying the Stratton-Chu formu-

lation to two problems: the original problem and the equivalentproblem.

C. Stratton-Chu Formulation

On a generic closed surface S enclosing volume V filledwith a homogeneous material (Fig. 2), the Stratton-Chu for-mulation [44] reads

jωµ0n× n×([LJ](r) +

[L (n×H)

](r))

+ n× n×[K (−n×E)

](r) = −1

2n× n×E(r) . (14)




Integral operators L and K in (14) are defined to be [5]

[LX] (r) =[1 +

1

k20∇∇·

] ∫V

G(r, r′)X(r′)dr′ , (15)

[KX] (r) = p.v.[∇×

∫V

G(r, r′)X(r′)dr′], (16)

where p.v. stands for principle value, the wavenumber k0 =ω√µ0ε0 and the Green’s function

G(r, r′) =e−jk0|r−r

′|

4π |r− r′|, (17)

are associated with the material inside V . It is important tonote that (14) is independent of material and sources outsideS.

D. Stratton-Chu Formulation Applied to the Original Problem

If the Stratton-Chu formulation is applied to the originalproblem shown in Fig. 1a, then we obtain

jωµ0n× n×([LJm] (r) +

[L(n× Ĥm

)](r))

+ n× n×[K(−n× Êm

)](r) = −1

2n× n×E(r) .

(18)

We evaluate this equation twice, first assuming r ∈ Sm, andthen assuming r ∈ Ŝm. For r ∈ Sm, the right hand side of (18)is zero because Sm is a PEC, and for r ∈ Ŝm the right handside is − 12n× n× Êm(r).

1) Surface Integral Equation on Sm: We substitute theexpansion of fields and currents in (1), (3), and (4) into (18).Next, we test the resulting equation with RWG basis functionsΛm,n on Sm. The resulting system of equations is writtencompactly as

G(E,J)m Jm +G(E,Ĥ)m Ĥm +G

(E,Ê)m Êm = 0 , (19)

where entry (n, n′) of matrices G(E,J)m , G(E,Ĥ)m , and G(E,Ê)mis given by[

G(E,J)m](n,n′)

= jωµ0

〈Λm,n(r),n× n× [LΛm,n′ ] (r)

〉(20)[

G(E,Ĥ)m](n,n′)

= jωµ0

〈Λm,n(r),n× n×

[LΛ̂m,n′

](r)〉

(21)[G(E,Ê)m

](n,n′)

=〈Λm,n(r),−n× n×

[KΛ̂′m,n′

](r)〉

(22)

and the inner product is defined as〈f(r),g(r)

〉=

∫∫S

f(r) · g(r)dr . (23)

2) Surface Integral Equation on Ŝm: Now, we re-evaluate (18) on Ŝm. We again substitute (1), (3), and (4)into (18), and test the resulting equation with the RWG basisfunctions on Ŝm, i.e. Λ̂m,n. The resulting equations can becompactly written as

G(Ê,J)m Jm +G(Ê,Ĥ)m Ĥm +G

(Ê,Ê)m Êm = 0 , (24)

where entry (n, n′) of G(Ê,J)m , G(Ê,Ĥ)m , and G

(Ê,Ê)m is[

G(Ê,J)m](n,n′)

= jωµ0

〈Λ̂m,n(r),n× n× [LΛm,n′ ] (r)

〉(25)[

G(Ê,Ĥ)m](n,n′)

= jωµ0

〈Λ̂m,n(r),n× n×

[LΛ̂m,n′

](r)〉

(26)[G(Ê,Ê)m

](n,n′)

=〈Λ̂m,n(r),−n× n×

[KΛ̂′m,n′

](r)〉

+1

2

〈Λ̂m,n(r),n× Λ̂′m,n′(r)

〉. (27)

It is important to note that the novel usage of the dual basisfunctions to expand n × Êm(r) ensures that both L and Koperators in (18) are well-tested because RWG and dual RWGbasis functions are approximately orthogonal [45]. Hence,all three discretized matrices in (24) are well-conditioned.In particular, if we had expanded n × Êm(r) with RWGbasis functions, then matrix G(Ê,Ê)m would have been poorlyconditioned.

We want to use the discretized Stratton-Chu formula-tion (24) to eliminate Ĥm from (13). Therefore, since G

(Ê,Ĥ)m

is a well-conditioned matrix, we rewrite (24) as

Ĥm = −[G(Ê,Ĥ)m

]−1 [G(Ê,Ê)m Êm +G

(Ê,J)m Jm

]. (28)

Equation (28) requires the LU factorization of G(Ê,Ĥ)m , whichis a dense matrix. However, this matrix is only of size N̂m ×N̂m, and thus the cost of this LU factorization will be relativelysmall compared to the total time to solve the entire problem.

Equation (28) allows us to eliminate Ĥm from (19) andobtain[

G(E,J)m −G(E,Ĥ)m

[G(Ê,Ĥ)m

]−1G(Ê,J)m

]︸︷︷︸

Am

Jm+

[G(E,Ê)m −G

(E,Ĥ)m

[G(Ê,Ĥ)m

]−1G(Ê,Ê)m

]︸︷︷︸

Bm

Êm = 0 , (29)

where we have introduced matrices Am and Bm. From (29)we have

Jm = −A−1m BmÊm , (30)

which relates the current density on the PEC scatterers to thetangential electric field on Ŝm.

E. Stratton-Chu Formulation Applied to the Equivalent Prob-lem

We now apply the Stratton-Chu formulation (14) to theequivalent problem shown in Fig. 1b. Since there is no electriccurrent distribution inside Ŝm, the Stratton-Chu formulationfor r ∈ Ŝm reads

jωµ0n× n×[L(n× H̃m

)](r)

+ n× n×[K(−n× Êm

)](r) = −1

2n× n× Êm(r) ,

(31)




which is similar to (18), except there is no contribution fromJm. By testing (31) with RWG basis functions on Ŝm, weobtain

G(Ê,Ĥ)m H̃m +G(Ê,Ê)m Êm = 0 , (32)

where entries of matrices G(Ê,Ĥ)m and G(Ê,Ê)m are given

in (26)-(27). Similarly to (28), (32) can be rewritten as

H̃m = −[G(Ê,Ĥ)m

]−1G(Ê,Ê)m Êm , (33)

to obtain the tangential magnetic field on Ŝm in the equivalentproblem in terms of the tangential electric field on Ŝm.

F. Equivalent Current

We can now simplify the expression for the equivalentelectric current density on Ŝm in (13). We substitute (28)and (33) into (13) and simplify the resulting equation to obtain

Ĵm =[G(Ê,Ĥ)m

]−1 [G(Ê,J)m

]︸︷︷︸

Tm

Jm , (34)

where Tm is the transfer matrix that relates the electric currentdensity on Sm to the equivalent electric current density onŜm. By substituting (30) into (34) we can relate Êm to Ĵm byan admittance operator, which is analogous to the differentialsurface admittance operator in [34]. In proposed formulation,Ĵm(r) has a different value than the equivalent electric currentdensity in the EPA [22]. The equivalent current densities inthe EPA are equal to the tangential electric and magneticfields on Ŝm. However, in this proposed formulation, theequivalent electric current density is equal to the contrastin tangential magnetic fields with and without the scatterer.Since the proposed formulation does not require an equivalentmagnetic current density [34], it is simpler to implement andrequires testing fewer integral operators than the EPA [22].

III. MACROMODEL FOR EACH ELEMENT IN AN ARRAYThe macromodel procedure presented in Sec. II for a single

element is then applied to each element of the array. Wereplace the array of scatterers with an array of equivalentelectric current densities (equivalent surfaces). An implicitassumption made here is that the PEC traces of none of thearray elements touch one another, which is the case in manypractical arrays of interest.

The electric current density coefficients on all elements arecollected into a vector

J =[JT1 JT1 . . . JTM

]T(35)

of size N × 1 where N =∑Mm=1Nm. Likewise, the coeffi-

cients of Ĵm(r) on all equivalent surfaces are collected into avector

Ĵ =[ĴT1 ĴT1 . . . ĴTM

]T(36)

of size N̂ × 1 where N̂ =∑Mm=1 N̂m. The two current

densities, as presented in (34), are related by

Ĵ = TJ , (37)

where

T =

T1

T2. . .

TM

(38)is a block-diagonal transfer matrix that relates the equivalentelectric current density on Ŝm to the current density on Smfor all elements. Similar to (38), we introduce matrices A andB which are block diagonal matrices made up of blocks Amand Bm, respectively. In most array problems, many elementsare identical. Therefore, matrices Tm, Am, and Bm only needto be calculated once for each distinct element.

IV. EXTERIOR PROBLEM

After applying the macromodeling technique, we havesimplified the original problem to an equivalent problemcomposed of an array of equivalent electric current densities.We apply the electric field integral equation to capture thecoupling between the macromodels.

According to the electric field integral equation, the totaltangential electric field on the m-th equivalent surface is

n× n× Êm(r) = −jωµ0M∑

m′=1

n× n×[LĴm′

](r)

+ n× n×E(inc)(r) , (39)

where the right hand side is the sum of the total scattered fieldproduced by the equivalent electric currents Ĵm′(r′) and theincident electric field E(inc)(r). In contrast to the EPA [22],[35], the exterior problem in (39) does not have the K integraloperator, which leads to a simpler exterior problem than theEPA.

We substitute (4) and (11) into (39) and test the resultingequation with RWG basis functions Λ̂m,n for m = 1, . . . ,M .The resulting equations can be compactly written as

DÊ = −G(Ê,Ĵ)Ĵ+ V̂ , (40)

where Ê and V̂ are vectors of electric field coefficients andthe excitation vector, respectively, and read

Ê =[ÊT1 ÊT2 . . . ÊTM

]T, (41)

V̂ =[V̂T1 V̂T2 . . . V̂TM

]T. (42)

Vector V̂m in (42) is a vector of size N̂m×1 whose n-th entryis

V̂m,n =〈Λ̂m,n(r),n× n×E(inc)(r)

〉, (43)

which is the projection of incident electric field on RWG basisfunctions. In (40), G(Ê,Ĵ) is a block matrix of the form

G =

GÊ,Ĵ1,1 G

Ê,Ĵ1,2 . . . G

Ê,Ĵ1,M

GÊ,Ĵ2,1 GÊ,Ĵ2,2 . . . G

Ê,Ĵ2,M

......

. . ....

GÊ,ĴM,1 GÊ,ĴM,2 . . . G

Ê,ĴM,M

, (44)




where the (n, n′) entry is[GÊ,Ĵm,m′

]n,n′

= jωµ0

〈Λ̂m,n(r),n× n×

[LΛ̂m′,n′

](r)〉.

(45)Finally, matrix D in (40) is block diagonal and reads

D =

D1 . . .DM

, (46)where the (n, n′) entry of Dm is given by

[Dm](n,n′) =〈Λ̂m,n(r),n× Λ̂′m,n′(r)

〉. (47)

It is important to note that D is well-conditioned and itis diagonally dominant because Λ′m,n(r) is approximatelyorthogonal to Λm,n(r) [33].

Next, we subsitute (37) and (30) into the outer problem (40)to obtain the final system of equations(

D−G(Ê,Ĵ)TA−1B)Ê = V̂ , (48)

which is only in terms of the electric field coefficients onthe surface of the equivalent box. After solving for Ê, thecurrent distribution J on the original scatterer, if desired, canbe computed very inexpensively using (30) for each element.

Notice that solving the original problem with the standardMoM would have required solving for N unknowns. How-ever, (48) involves only N̂ unknowns. For many problemswith complex, multiscale scatterers, N̂ is much smaller thanN , and therefore the proposed method results in faster solutiontimes and lower memory consumption. Furthermore, (48) isusually better conditioned than the standard MoM formulationbecause the equivalent surface can have a coarser mesh thanthe original scatterer due to the absense of any fine featuresin the equivalent problem.

V. ACCELERATION WITH AIM

Even after the reduction in the number of unknowns, thecomputation cost of the proposed approach can grow quickly.Therefore, we solve the linear system (48) iteratively using thegeneralized minimal residual (GMRES) algorithm [46]. Thisalgorithm requires an efficient way to compute(

D−G(Ê,Ĵ)TA−1B)Y , (49)

where Y is an arbitrary vector of size N̂ × 1. A trivial wayto compute (49) is to first generate D, G(Ê,Ĵ), T, A, and B,and then carry out the required matrix-vector multiplicationsand vector additions. Since matrices T, A, and B are block-diagonal matrices with M blocks, computing matrix-vectorproducts with these matrices is inexpensive. Matrix D in (48)is also a block-diagonal matrix with M sparse blocks, and somatrix-vector products with this matrix is also inexpensive.However, G(Ê,Ĵ) is a dense matrix, and so generating thismatrix and storing its values requires a lot of computationaltime and memory.

For this reason, we accelerate the computation of G(Ê,Ĵ)Yusing AIM [10], [12]. This section briefly describes how

the standard AIM can be used to accelerate the proposedmacromodeling technique. In AIM, the problem domain isdiscretized using a 3D Cartesian grid. Each basis function isassigned to a stencil. Each stencil is made up of NO + 1grid points in each direction, where NO is the stencil order.Furthermore, we define the NNF stencils surrounding the basisfunction in each direction to be the near-field region associatedwith each basis function. Figure 3 shows a sample 2D AIMgrid with NO = 3 and NNF = 1.

For AIM, G(Ê,Ĵ) is decomposed into two parts: the near-field matrix G(Ê,Ĵ)NF and the far-field matrix G

(Ê,Ĵ)FF . The near-

field matrix is a sparse matrix whose elements are computedby (45) with some pre-correction [10], [12]. The far-fieldmatrix, on the other hand, is factorized as a product of threematrices as

G(Ê,Ĵ)FF =∑

ψ={Ax,Ay,Az,φ}

IψHψPψ , (50)

where Iψ , Hψ , and Pψ are the interpolation, convolution, andprojection matrices, respectively. These matrices, in general,need to be calculated for the scalar potential φ and vectorpotentials in the three principal directions x, y, and z.

Given Y, the matrix-vector product G(Ê,Ĵ)FF Y is computedin AIM by the following steps:

(i) Compute equivalent grid charges and grid currents (inthe x, y, and z directions) on the projection stencilusing interpolation polynomials of order NO. Theseequivalent grid charges and currents produce the samefields as the original source basis functions in the farfield. Mathematically, this operation can be expressed as

Y(1)ψ = PψY , (51)

where Pψ is a sparse matrix.(ii) Compute grid scalar and vector potentials from the grid

charges and currents. Mathematically, this operation canbe expressed as

Y(2)ψ = HψY(1)ψ . (52)

Hψ is a dense matrix with size equal to the number ofgrid points, hence storing Hψ can be very expensive.However, due to the position-invariance property of theGreen’s function, the fast Fourier transform (FFT) isapplied to compute the matrix-vector product in (52).

(iii) From the grid potentials calculated in step (ii), computethe electric field and its projection on the testing func-tions. To find the electric field on the basis function fromthe grid potentials, we use an interpolation polynomialof order NO. Mathematically, this operation can beexpressed as

Y(3)ψ = IψY(2)ψ , (53)

where ψ ∈ {φ,Ax, Ay, Az}. The interpolation matrix Iψis a sparse matrix.

Finally, the result of matrix-vector product is G(Ê,Ĵ)FF Y =∑ψ={Ax,Ay,Az,φ}Y

(3)ψ . Readers interested to learn more de-

tails about AIM are referred to [10], [12].




projection stencil

near-field region

test basis 1

source basis

interpolation stencil

test basis 2

Fig. 3. Sample AIM grid with one source RWG basis function and two testingfunctions. The inner product between the source function and test function1 is calculated directly. The inner product between source function and testfunction 2 is computed via AIM.

3.89 m

3.89m

9.5 cm

x

y

z

Fig. 4. Layout of the 20 × 20 array of spherical helix antennas consideredin Sec. VI-A. The array is uniformly spaced along the x− and y-directionswith interelement spacing of dx = dy = 0.2 m.

VI. NUMERICAL RESULTS

In this section, three examples are presented to demonstratethe accuracy and performance of the proposed method com-pared against an in-house standard MoM solver acceleratedwith AIM. All computational codes were developed usingPETSc [47], [48], [49] and FFTW3 [50] libraries. The numeri-cal tests were performed on a machine with an Intel Xeon E5-2623 v3 processor and 128 GB of RAM. All simulations wererun on a single thread without exploiting any parallelization.

A. Array of Spherical Helix Antennas

Spherical helix antennas, like many other electrically smallantennas, have complex geometries with electrically fine fea-tures. Therefore, simulating an array of such antennas requiresa long computation time and large memory. In the proposedtechnique, a macromodel is created for each small antenna. Asdemonstrated by this example, the macromodel can accuratelycapture radiation from the antenna using fewer unknowns,which leads to significant savings in computation time andmemory. This example also demonstrates that the proposedmacromodel approach can be applied to fed antenna arrays.

−150 −100 −50 0 50 100 150

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

AIMProposed

(a) φ = 0◦

−150 −100 −50 0 50 100 150

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

AIMProposed

(b) φ = 90◦

Fig. 5. Directivity of the 20×20 array of spherical helix antennas consideredin Sec. VI-A calculated with AIM and the proposed technique.

TABLE ISIMULATION SETTINGS AND RESULTS FOR THE 20× 20 ARRAY OF

SPHERICAL HELIX ANTENNAS CONSIDERED IN SEC. VI-A

AIM ProposedAIM Parameters

Number of stencils 120× 120× 2Interpolation order 3Number of near-field stencils 4

Memory ConsumptionTotal number of unknowns 260,800 62,400Memory used 19.53 GB 3.60 GB

Timing ResultsMacromodel generation N/A 18 sMatrix fill time 1.12 h 336 sPreconditioner factorization 414 s 23 sIterative solver 28 s 4 sTotal computation time 1.25 h 6.35 min

We consider the 20 × 20 array of identical spherical helixantennas shown in Fig. 4. Each element of the array is excitedwith a uniform delta-gap voltage source at the center ofeach helix operating at 300 MHz. With this excitation, thearray radiates with main beam in the broadside direction. Wecomputed the radiation pattern from the antenna array usingthe standard MoM and the proposed method, both acceleratedwith AIM and solved iteratively using GMRES with an ILU-2preconditioner [48]. In the proposed method, we first createdthe macromodel for a spherical helix antenna by computingTm, Am, and Bm using a sphere of radius 5 cm as theequivalent surface. This macromodel was then reused for allelements of the array, as described in Sec. III. The AIMparameters, computational times, and memory requirements




w

w

h

h

(a) Original (b) Equivalent

Fig. 6. (a): Original unit cell of the two-layer reflectarray considered inSec. VI-B with w = 3.75 mm and h = 0.76 mm. (b): Equivalent unit cellobtained after applying the image theorem.

78.75 mm = 7.875λ

78.75

mm

x

y

z

Fig. 7. Top view of the 21 × 21 reflectarray considered in Sec. VI-B. Topand bottom layers of the reflectarray are shown in blue and red, respectively.The reflectarray is uniformly spaced along the x- and y-directions withinterelement spacing of 3.75 mm (0.375λ).

to simulate this problem are given in Tab. I. As seen fromTab. I, the proposed method reduces the number of unknownsby a factor of approximately four because N = 260, 800 andN̂ = 62, 400. This reduction in the number of unknowns leadsto a simulation that is 12 times faster and requires 5 times lessmemory than the AIM-accelerated MoM solver. As evidentfrom Tab. I, the proposed method is faster than AIM becauseit takes less time to assemble all the matrices and factorize thepreconditioner, since it has fewer unknowns than the standardMoM formulation. Figure 5 shows the radiation pattern ofthe array in the two principal plane cuts. Results in Fig. 5confirm that the proposed macromodeling approach providesan excellent accuracy compared to the standard MoM code.

B. Two-layer Reflectarray with Jerusalem Cross Elements

1) Design and Simulation Setup: Next, we consider a two-layer dual-polarized reflectarray with 21× 21 elements madeup of Jerusalem crosses [51]. The unit cell of the reflectarray

TABLE IISIMULATION SETTINGS AND RESULTS FOR THE 21× 21 REFLECTARRAY

CONSIDERED IN SEC. VI-B

AIM ADF ProposedAIM Parameters

Number of stencils in x dir. 126 - 84Number of stencils in y dir. 126 - 84Number of stencils in z dir. 2 - 2Interpolation order 3 - 3Number of near-field stencils 4 - 4

Memory ConsumptionTotal number of unknowns 324,420 324,420 111,132Memory used 40 GB 37 GB 15 GB

Timing ResultsMacromodel generation N/A N/A 0.054 hMatrix fill time 1.48 h 2.26 h 0.42 hPreconditioner factorization 1.25 h 1.02 h 0.31 hIterative solver 0.22 h 0.11 h 2.80 minTotal computation time 3.30 h 3.40 h 0.82 h

is shown in Fig. 6a. In this example, the reflectarray iselectrically large with dimensions of 7.875λ × 7.875λ at30 GHz. It also includes sub-wavelength features and strongmutual coupling between the unit cells. In practice, due tosimulation difficulties, reflectarrays of this size and complexityare rarely simulated with full-wave electromagnetic solvers.One of the motivations of this work is to enable an efficientsimulation of such problems.

Since the proposed method currently does not support mul-tilayer dielectrics, we assume that all layers of the reflectarrayhave permittivity ε0 and permeability µ0. Furthermore, weapply the image theory [23] to model the ground plane at thebottom of the reflectarray. According to the image theory, thetwo Jerusalem crosses in each unit cell are duplicated belowthe image plane as shown in Fig. 6b. Thus, each unit cell haseffectively four Jerusalem crosses.

In this example, the reflectarray is designed to produce themain beam of the scattered field in the broadside directionwhen the reflectarray is excited by a dipole feed antennaoperating at 30 GHz1. The feed is placed 40 mm along theaxis of the reflectarray at the prime focus position, so that thefocal length to diameter ratio is 0.51. The top view of thefinal reflectarray design is shown in Fig. 7. The final designcontains 441 total elements with eight unique elements usedto discretize the reflectarray phase curve.

2) Scattered Field: We calculated the scattered field fromthe reflectarray using three tools: an in-house AIM acceleratedMoM code, the Antenna Design Framework (ADF) [52] –an AIM-accelerated commercial SIE solver, and the proposedtechnique. For the proposed method, we enclosed each elementwith an equivalent box of dimensions 3.60 mm× 3.60 mm×4.00 mm. We generated macromodels for the array by firstcomputing Tm, Am, and Bm for the eight unique elements inthe array, and then generating T, A, and B. Figure 8 shows thedirectivity in three planes generated with the proposed method,the in-house MoM code, and the ADF solver. An excellentmatch between all three methods validates the accuracy of

1A dipole feed antenna was chosen due to its simplicity, although it is notan optimal feed model for reflectarrays.




−80 −60 −40 −20 0 20 40 60 80

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

ADFAIMProposed

(a) φ = 0◦

−80 −60 −40 −20 0 20 40 60 80

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

ADFAIMProposed

(b) φ = 90◦

−80 −60 −40 −20 0 20 40 60 80

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

ADFAIMProposed

(c) φ = 45◦

Fig. 8. Directivity of the 21 × 21 two-layer reflectarray considered inSec. VI-B calculated with ADF, AIM, and the proposed technique.

the proposed method. In particular, despite of the unit cellsbeing very close to one another, the macromodel approachaccurately predicts the mutual coupling between them. Sim-ulation settings, memory consumption, and timing results ofthe simulations run with the in-house MoM code, ADF, andthe proposed method are given in Tab. II. It is seen that the in-house AIM-accelerated MoM code performs on-par with thecommercial AIM-accelerated MoM code. For this simulation,the proposed method requires 4 times less computational timeand 2.7 times less memory, which is a substantial savings.In this example, the number of basis functions to expandthe surface current density was N = 324, 420 and thenumber of basis functions to expand the equivalent electriccurrent density was N̂ = 111, 132. In general, for reflectarrayexamples, the proposed method converges monotonically withthe number of iterations. Furthermore, the number of iterationsrequired for a simulation to converge increases approximatelylinearly with the number of array elements.

66 mm = 3.1λ

66 mm

x

y

z

Fig. 9. Top view of the 11 × 11 reflectarray considered in Sec. VI-C. Thearray is uniformly spaced along the x− and y-directions with interelementspacing of dx = dy = 6 mm (0.28 λ)

TABLE IIISIMULATION SETTINGS AND RESULTS FOR THE 11× 11 REFLECTARRAY

CONSIDERED IN SEC. VI-C

AIM ProposedAIM Parameters

Number of stencils in x dir. 126 84Number of stencils in y dir. 126 84Number of stencils in z dir. 2 2Interpolation order 3 3Number of near-field stencils 4 4

Memory ConsumptionTotal number of unknowns 262,616 27,588Memory used 67 GB 5.4 GB

Timing ResultsMacromodel generation N/A 14.2 minMatrix fill time 5.31 h 7.8 minPreconditioner factorization 3.83 h 4.15 minIterative solver 2.04 h 26 sTotal computation time 11.18 h 27 min

C. Reflectarray Composed of Elements with Fine Features

As a final example, we consider a single-layer reflectarraywith very fine meander-line features [53]. The top view of thisreflectarray is shown in Fig. 9. The mesh size of such elementsis electrically very small, which causes conditioning issueswith the standard MoM. However, with the proposed approachthe structure can be simulated faster due to fewer unknownsand better conditioning of the equations to be solved.

All elements in this example are suspended in free space0.75 mm above a PEC ground plane. As in Sec. VI-B, we usethe image theory to model the ground plane. The reflectarrayhas a total of 11 × 11 elements, out of which eight elementsare unique. The structure is designed to scatter fields withthe main beam in the broadside direction. The reflectarray isexcited by a dipole antenna operating at f = 14 GHz that isplaced 43 mm along the axis of the reflectarray so that focallength to diameter ratio is 0.65. We simulated the problemwith the in-house MoM code and the proposed macromodelingtechnique, both accelerated with AIM. In the proposed method,an equivalent box of dimensions 5.5 mm×5.5 mm×3.0 mmwas introduced to enclose each element.




−80 −60 −40 −20 0 20 40 60 80−40

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

AIMProposed

(a) φ = 0◦

−80 −60 −40 −20 0 20 40 60 80−40

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

AIMProposed

(b) φ = 90◦

−80 −60 −40 −20 0 20 40 60 80−40

−20

0

20

θ [degrees]

Dir

ectiv

ity[d

Bi]

AIMProposed

(c) φ = 45◦

Fig. 10. Directivity of the 11 × 11 reflectarray considered in Sec. VI-Ccalculated with AIM and the proposed technique.

Fig. 10 shows the directivity of the reflectarray calculatedwith both techniques. The agreement between the resultsobtained with the proposed method and the standard MoMcode confirms that proposed method can accurately capturethe strong coupling between the elements and fine features ofthe reflectarray unit cell.

Tab. III shows AIM parameters, storage statistics, and tim-ing statistics to simulate this problem. As seen from Tab. III,the proposed method is 24 times faster and consumes 12 timesless memory than the standard MoM solver. The proposedmethod solves the problem in 1/2 h as opposed to 11 hrequired with the standard MoM formulation, which is asignificant savings. The proposed method is faster becauseit requires solving a problem with 9 times less number ofunknowns with N = 262, 616 and N̂ = 27, 588. Furthermore,the proposed formulation converges significantly faster thanthe standard MoM formulation due to a significantly smallercondition number. We computed, in PETSc, the condition

number of the proposed formulation and the standard MoMformulation for a smaller sized reflectarray with 3 × 3 ele-ments2. It was found that the condition number of the proposedformulation was 129.31, which was significantly smaller thanthe condition number of the standard MoM equations whichwas 1.013× 105.

This example demonstrates that the proposed method canbe very efficient, in terms of computation time and memoryconsumption, to simulate arrays with complex elements.

VII. CONCLUSIONSIn the classical circuit theory, complexity of a large elec-

trical network of linear elements is often reduced using theNorton equivalent circuit models. With the Norton equivalentcircuit model, a complex portion of a large electrical net-work can be replaced by an equivalent current source andimpedance. By doing this, the equivalent electrical networkcan be greatly simplified, as it will have fewer nodes, branches,and elements. In this paper, we explored whether this ideacan be extended to modeling electromagnetic scatterers. Asdemonstrated in this work, this is indeed possible by com-bining the equivalence theorem and Stratton-Chu formulation.It was also found that both RWG and Dual RWG basisfunctions were necessary for robust numerical implementation.We presented an equivalent model, which we referred to asa macromodel, through which a complex scatterer is replacedby an equivalent electric current source, which is analogous tothe Norton equivalent current source, and a transfer operator,which is analogous to the Norton impedance. Like the Nortontheorem, the macromodel approach presented in this papermakes no heuristic approximations and is exact.

In this paper, we applied this macromodeling technique toefficiently simulate large arrays of complex scatterers. Theproposed method is more efficient than the standard MoMfor three reasons. First, it significantly reduces unknownscount, which results in lower solution time and matrix filltime. Second, the proposed method results in a linear systemwith better conditioning than the standard MoM, leadingto faster convergence of iterative methods when applied tomultiscale problems. Third, the proposed method exploits therepeatability of elements in large arrays. The proposed methodwas applied to compute scattering from an array of sphericalhelix antennas and reflectarrays. It was shown that the methodis up to 20 times faster and consumes up to 12 times lessmemory than the standard MoM simulation, while givingaccurate results.

Although in this work we investigated array problems only,the proposed macromodel approach can be applied to manyother multiscale problems. For example, it can be applied tomodel antennas for channel modeling or model antennas onlarge aircraft or ships. The proposed approach can also beadopted to model uncertainty, when many runs are neededto capture statistics of a problem. Currently, we are workingto extend the proposed method to include dielectric substrates.Through this extension, we will be able to simulate many otherpractical metasurfaces and reflectarrays.

2Computing the condition number of the 11 × 11 reflectarray was notfeasible due to its prohibitive computational cost.




VIII. ACKNOWLEDGMENT

The authors thank IDS Corporation for providing a licensefor the Antenna Design Framework (ADF). The authors alsothank Mr. Ciaran Geaney for his help with the reflectarray testcases.

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Utkarsh R. Patel (S’13) received the B.A.Sc. andM.A.Sc. degrees in Electrical Engineering from theUniversity of Toronto in 2012 and 2014, respec-tively. Currently, He is pursuing the Ph.D. degreein Electrical Engineering at the same institution.His research interests are computational and ap-plied electromagnetics. He received the Best PaperAward at the IEEE Topical Meeting on ElectricalPerformance of Electronic Packaging and Systemsin 2017 and the Best Student Paper Award at theIEEE Workshop on Signal and Power Integrity in

2017. He was also awarded the NSERC Alexander Graham Bell CanadaGraduate Student Scholarship (CGS-D) in 2016.

Piero Triverio (S’06 – M’09 – SM’16) received theM.Sc. and Ph.D. degrees in Electronic Engineeringfrom Politecnico di Torino, Italy in 2005 and 2009,respectively. He is an Associate Professor at theUniversity of Toronto, where he is affiliated to theDepartment of Electrical and Computer Engineeringand to the Institute of Biomaterials and BiomedicalEngineering. He holds the Canada Research Chairin Computational Electromagnetics. His researchinterests include signal integrity, computational elec-tromagnetism, model order reduction, and computa-

tional fluid dynamics applied to cardiovascular diseases. He received severalinternational awards, including the 2007 Best Paper Award of the IEEETransactions on Advanced Packaging, the 2010 EuMIC Young EngineerPrize, the Ontario Early Researcher Award, and the Best Paper Award atthe IEEE Topical Meeting on Electrical Performance of Electronic Packagingand Systems in 2008 and 2017.

Sean Victor Hum (S’95-M’03-SM’11) received theB.Sc., M.Sc., and Ph.D. degrees in electrical andcomputer engineering from the University of Cal-gary, Calgary, AB, Canada, in 1999, 2001, and 2006,respectively. He joined the Edward S. Rogers SeniorDepartment of Electrical and Computer Engineering,University of Toronto (UofT), Toronto, ON, Canada,in 2006, where he currently serves as a Professor.He leads the Reconfigurable Antenna Laboratoryat UofT and, along with his students, conductsresearch in the areas of reconfigurable/multifunction

antennas, space-fed arrays, electromagnetic surfaces, and antennas for spaceapplications.

Prof. Hum received the Governor General’s Gold Medal for his work onradio-on-fiber systems in 2001, and an IEEE Antennas and Propagation Soci-ety Student Paper Award for his work on electronically tunable reflectarraysin 2004. In 2006, he received an ASTech Leaders of Tomorrow Award for hiswork in this area. In 2012, he also received an Early Researcher Award by theGovernment of Ontario. He was twice the co-recipient of the IEEE Antennasand Propagation Society R. W. P. King Award, in 2017 and 2015. On theteaching side, he has received five UofT Departmental Teaching Awards since2007, and received a faculty Early Career Teaching Award in 2011. He servedon the Steering Committee and Technical Program Committee of the 2010IEEE AP-S International Symposium on Antennas and Propagation. He servedas an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS ANDPROPAGATION from 2010-2017. He was the TPC Co-Chair of the 2015IEEE AP-S International Symposium on Antennas and Propagation.

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