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Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)

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Abstract—An efficient algorithm for fast field

evaluation, namely multilevel Green’s function

interpolation method (MLGFIM), has been successfully

applied to solve electromagnetic problems for both low

frequency and full-wave. Further refinements, including

new interpolation scheme, QR factorization, and FFT

technique, have been applied to accelerate this algorithm

and extend the problem scale that MLGFIM can solve. In

this paper, we present a review of different techniques in

MLGFIM for the fast evaluations of large-scale

electromagnetic problems. The applications of simulating

periodic structures and layered mediums have been

accelerated by MLGFIM as well. Some numerical

examples are given in this paper to demonstrate the

accuracy and efficiency of this algorithm.

Index Terms—Method of Moments (MoM), Green’s

Function Interpolation, Fast Solver, Parallel Computing,

FFT.

I. INTRODUCTION

Calculation of antenna radiation pattern, analysis of radar

cross section (RCS), and extraction of circuit parameters are

often performed using integral equation-based computational

techniques, e.g., the method of moment (MoM) [1]. However,

the computational requirements for MoM are very high, and

various fast numerical methods have been proposed to

accelerate the computation, for instance, multilevel fast

multipole algorithm (MLFMA) [2]-[4], pre-corrected fast

Fourier transform (FFT) [5],[6], adaptive integral method

(AIM) [7],[8], and sparse matrix canonical grid (SMCG)

[9],[10]. The first method uses multipole expansion to

approximate the far field interactions, while the later three

methods apply FFT for fast field evaluation. Although these

methods calculate the matrix-vector multiplication indirectly

and thus significantly reduce the computational complexity,

each of the methods has its own weaknesses. The FFT-based

methods are most suitable for densely-packed structures, in

which the unknowns are associated with most of the grid

points. The PFFT approach employs the bi-linear

interpolation method, and it is not suitable for analysis of

full-wave electromagnetic problems as bi-linear interpolation

is not accurate for the rapidly-varying kernels in full-wave

applications. On the other hand, the multipole-based

algorithms depend on the type of Green’s function used for

the specific problems and they become cumbersome when

dealing with structures embedded in a multilayered medium.

When the simulated structure changes, for instance, insert an

additional layer of dielectric substrate, the problem has to be

reformulated.

Some interpolation-based fast algorithms have the potential

to address many of the aforementioned weaknesses. In [11],

Brandt developed a multilevel interpolation approach, in

which a softened kernel and phase compensation is employed

when the kernel is oscillatory. Non-uniform grid (NG) [12],

[13] and multilevel NG [14],[15] algorithms, as kernel

independent algorithms, are proposed for fast field evaluation.

With the technique of phase and amplitude compensations,

NG-based algorithms sample the compensated field using a

non-uniform polar grid. The NG and multilevel NG methods

achieve computational complexities of O(N3/2) and O(NlogN),

respectively. Adaptive cross approximation (ACA) [16],[17]

employs interpolation approach directly on low-rank

sub-matrices denoting the far field interaction, and thus this

method is not difficult to implement. The H2 matrix-based

method is proposed in [18],[19]. It applies multilevel

interpolation with Lagrange interpolation function and Tartan

grid.

Another kernel-independent interpolation algorithm, viz.

multilevel Green’s function interpolation method (MLGFIM),

has been developed to solve quasi-static [20]-[22] and

full-wave electromagnetic problems [23],[24]. The

application of an octa-tree structure and non-requisite of a

softened kernel distinguish the MLGFIM with the algorithm

in [11]. For full-wave problems, instead of using Lagrange

Multilevel Green’s Function Interpolation

Method: An Interpolation-based Solver for

Efficient Analysis of Large-scale

Electromagnetic Problems

Peng Zhao(1,2), Da Qing Liu(2) and Chi Hou Chan(2)

(1) Electronics and Information College, Hangzhou Dianzi University, Hangzhou, China(2) State Key Laboratory of Millimeter Waves, City University of Hong Kong, Hong Kong

SAR, China

(Email: [email protected])

*This use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author.*


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interpolation as in the quasi-static problems, radial basis

function (RBF) interpolation is employed. Lagrange

interpolation method can not be directly applied for the

approximation of the full-wave Green’s function, unless

phase compensation technique is incorporated [25]. The RBF

interpolation has excellent approximation properties [26], and

we find that if the object is not extremely large, the full-wave

Green’s function can be interpolated without the

compensation technique, which is different from the

NG-based algorithm. Although ACA is flexible for solving

problems with a complex medium, it is difficult to enhance

the efficiency by using the translational invariance

characteristics of the Green’s functions in some problems, e.g.

the multilayered planar problem. In addition, the multilevel

structure of MLGFIM also differs from ACA. Compared with

the H2 matrix-based method, the applications of RBF

interpolation and staggered Tartan grid (STG) make the

number of interpolation points in MLGFIM increases more

slowly. This is because the interpolation points at the coarser

levels do not coincide with those at the finer levels in the

MLGFIM, viz., the volumetric density of interpolation point

at the coarser levels is lower than that at the finer levels.

Moreover, some novel techniques used in MLGFIM, e.g. QR

factorization [27] and hybrid quasi-2D/3D multilevel

partitioning approach [24], distinguish itself from other

interpolation-based methods. In addition, the applications of

peer-level and lower-to-upper-level interpolations make the

MLGFIM very efficient for the analysis of large-scale

problems, and the computational complexity of 3-D full-wave

MLGFIM is O(NlogN) [23].

Although so many fast algorithms have been developed,

there are still many challenges in the electromagnetic

simulations. For instance, electromagnetic enhancement of

surface enhanced Raman scattering (SERS) attributed to the

enhanced E-fields in the vicinity of plasmonic nanostructures

is the major approach of SERS enhancement [28],[29]. When

the exciting electromagnetic radiation approaches the

resonance frequency, the enhancement factor which is

proportional to the fourth power of the gain in the

electromagnetic field becomes very difficult to evaluate. To

accurately calculate the electromagnetic field, the

nanoparticles must use deep sub-wavelength discretization.

Due to interpolation-based algorithm, MLGFIM is capable of

dealing with the low-frequency and multibody structure. In

addition, some thin strip with high dielectric constant has been

used to promote the magnetic emission in quantum emitters

[30],[31], and we found when the emitter is close to the

scatters, the near-field is difficult to be accurately computed

as well. The high-contrast material makes the whole structure

extremely in-homogenous, and thus deep sub-wavelength

mesh is also required. Moreover, the analysis of metasurfaces,

e.g. holographic metasurfaces [32],[33] which excites surface

waves with the desirable field distributions, is challenging.

For many of the holographic structures, the size and shape of

each sub-scatter which stuck on a substrate may be different,

so the computational burden is very large. This problem can

be released by adopting multilayered medium Green’s

function, and MLGFIM is able to be implemented based on

this complex Green’s function.

In this paper, various techniques for the MLGFIM to solve

electrically large electromagnetic problems are reviewed. The

MLGFIM used for accelerating the matrix-vector

multiplication is introduced first. QR factorization technique

is employed to compress the matrix storage. In order to make

the MLGFIM more efficient, different radial basis functions

(RBFs) are compared and interpolation grid patterns are

designed so that the MLGFIM could be accelerated by

applying newly proposed interpolation schemes. Moreover,

the technique of fast Fourier transform (FFT), which has been

combined with MLFMA [34],[35], could also be applied to

reduce the translation time in MLGFIM. The parallelized

MLGFIM versions based on OpenMP and message passing

interface (MPI) techniques have been realized to make this

fast solver more efficient for the large-scale electromagnetic

problems. Since the MLGFIM is a kernel-independent

method, it is easy to use MLGFIM for the analysis of periodic

structures and multilayer problems. Numerical results

demonstrate the good accuracy and efficiency of the above

implementations of the MLGFIM.

II. DESCRIPTION OF MLGFIM

For the 3-D free-space aperiodic problems, it is always

described by electric-field integral equation (EFIE),

magnetic-field integral equation (MFIE) and combined-field

integral equation (CFIE). With boundary conditions imposed

on the surface of the body, another integral equation, i.e.

Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT)

integral equations, could be used to analyze 3-D composite

metallic and dielectric bodies with surface discretization.

Using Galerkin method, the aforementioned integral

equations are converted to N×N dense matrix equations, viz.,

* *A x b (1)

where * can represent EFIE, MFIE, CFIE or PMCHWT. For

simplicity and clarity, we use EFIE as an example. The entries

in N×N matrix EFIEA and N×1 vector

EFIEb are written as:

2

0 0

[ ( ) ( )

1( ) ( )] ( , )

i jij i j

s s

i j

A ds ds

G

j r j r

j r j r r r

(2)

and

0

4( ) ( )

i

inc

i is

ib ds

j r E r (3)

where r , r , , 0, 0, ( )j r , ( , )G r r and ( )inc

E r are the

observation point, source point, angular frequency, free space

permittivity and permeability, basis function, interaction

function, and incident electric field, respectively.

To apply MLGFIM, (2) is decomposed into scalar form as:

x y z d

ij ij ij ij ijA A A A A (4)

where

# # #( ) ( ) ( , )

# , or,

i jij i j

s sA ds ds G

x y z

j r j r r r (5)


3

and

2

0 0

1[ ( ) ( )] ( , )

i j

d

ij i js s

A ds ds G

j r j r r r (6)

Each scalar part ijA in (5) and (6) has the same form as:

( ) ( ) ( , )i j

ij i js s

A ds ds G r r r r (7)

where ( )i r and ( )j r are related to the weighting function

and the basis function, respectively.

n3

n2

m2 m2 m2 m2

m3 m3

m3 m3

m3 m3

m3 m3

m3 m3

m3 m3

m2 m2 m2 m2

m2 m2

m2 m2

(a)

m2

n2

rr

r r

o

(b)

Fig. 1. (a) 2-D pictorial representation of 3-D multilevel division in which

the interaction list of box n2 at level 2 are marked with m2, and the interaction

list of box n3 at level 3 are marked with m3. (b) 2-D pictorial representation of

3-D locations of the interpolation points (white circles) and interpolated

points (black circles) are in the boxes m2 and n2.

In [24] and [36], it is proven that PMCHWT and CFIE

which are associated with both EFIE and MFIE can also be

decomposed into the same form as (7). The interaction

functions ( , )G r r corresponding to the EFIE and MFIE can

be expressed as:

( )

( )

3

(for EFIE)

( , ')

1 (for MFIE)

ik

ik

e

Ge

ik

r r

r r

r rr r

r rr r

(8)

For the far-field interactions, instead of directly calculating

the function ( , )G r r , MLGFIM use interpolation method in a

multilevel tree to quickly approximate this function. The

multilevel tree is constructed by enclosing the object into a

box (level 0) and recursively dividing the large box into

sub-boxes. Box ml is the neighbor/interaction list of box nl at

level l if ml is near/well separated from nl, and their parents are

neighbors of each other, as shown in Fig. 1(a). If box ml is the

interaction list of box nl at level l, the interaction function

between these two boxes could be obtained by interpolation

method, as shown in Fig. 1(b), viz.:

, ,

1 1

, ,

;

, ,

l l

l ll l

l l l l

K K

m i n j

i j

m i n j

T

m m n n

GG

G

r r r r r r

r r

(9)

where,lm i

r and ,ln jr are the i-th and j-th interpolation points in

field box ml and source box nl, ,lm i r and ,ln j

r are the i-th

and j-th interpolation functions, respectively. Kl is the number

of interpolation points at level l.

Moreover, using the interpolation method, 1 ,l lm nG

and

1,l lm nG

can also be interpolated using ,l lm nG [20], viz.:

1 1

1 1

; ; ;

; ; ;

l l l l l l

l l l l l l

m n m m m n

T

m n m n n n

G C G

G G C

(10)

where 1l lm m ,

1l ln n , and matrix 1;l lm mC

is named as

lower-to-upper-level interpolation matrix. If the box ml is

interaction list of box nl are at level l, ( , )G r r at the finest

level L can be obtained using (9) and (10), viz.:

1 11 1

1 1 11

; ; ;

; ; ; ; ;

( , ) ( ) ( )

( ) ... ... ( )

l l l l l ll l

l l l l l lL L L LL L

TT

m m m n n nm n

T TT

m m m m m n n n n nm n

G C G C

C C G C C

r r r r

r r

(11)

Substituting (11) into (7) gives:

1 1 11

1 1 11

; ; ; ; ;

; ; ; ; ;, ,

[ ( ) ( )] ... ...

[ ( ) ( )]

= ( ) ... ... ( )

l l l l l lL L L LLi

Lj


T TT

m m m m m n n n n nij i ms

j ns

T TT

m m m m m n n n n nm i n j

A ds C C G C C

ds

T C C G C C Z

r r

r r

r r

(12)

It should be mentioned that if box mL is the interaction list

of box nL at level L, there is no lower-to-upper-level

interpolation matrix in (12), and it becomes the conventional

peer-level interpolation. Subsequently, the sub-matrix

L Lm nA can be rewritten as:

1 1 11

11

,1 1

,2 2; ; ; ; ;

,

,1 1 ,2 2 ,

; ; ; ;

( )

( )... ...

( )

( ) ( ) ( )

...

L

L


L m mL L

L L L n nL L

l l l l l lL LL

T

m

TT T

mm m m m m n n n n nm n

T

m N N

n n n N N

m m m m m n n nm

T

TA C C G C C

T

Z Z Z

T C C G C

r

r

r

r r r

1 1 ;...L L L

T TT

n n nC Z

(13)

whereLm

N andLnN denote the number of unknowns in boxes

mL and nL, respectively. From (13), it is observed that only the


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interpolation matrixes LmT and

LnZ at level L should be

calculated. Assuming there are PL boxes at level L, the

MLGFIM algorithm can be finally shown as Fig. 2.

:

( 1,2, , )

LL L

T

nn n

L L

S Z x

n P

1 1

1

;

( 1, 2, , 2)

l l l l

l l

n n n n

n n

S C S

l L L

2 2 2 2

2 2

;

interaction list of

:m m n n

n m

B G S

11

;

interaction list of

; ( 3, , )

l l l l

l l

ll l

m m n n

n m

T

mm m

B G S

C B l L

1

( 1,2, , )

L Lm m

L L

b T B

m P

Step 2:

Step 1:

Step 3:

Step 4:

Step 5: Step 6:

Upward Pass

Upward PassDownward Pass

Downward Pass

1 ;

neighbors of

:L L L

L L

m n n

n m

b b A x

Fig. 2. Flow chart of the implementation of MLGFIM.

Considering the interaction function matrix ;l lm nG

represents interactions between the interpolation points of two

well separated boxes, it is in low rank and it could be

factorized with QR factorization technique as:

;; ; l ll ll l

m nm n m nG Q R (14)

where ;l lm n

Q and ;l lm nR are the Kl × rl unitary matrix and the rl

× Kl upper triangular matrix. rl is the numerical rank of ;l lm nG ,

and the condition rl << Kl always holds. Instead of directly

store the Green’s function matrix ;l lm nG which is a Kl × Kl

full-matrix, we use (14) to compress the Green’s function

matrix and it also brings a great advantages in the computation

of step 3 and 4 of Fig. 2 [23].

III. THE INTERPOLATION SCHEME FOR MLGFIM

The accuracy and efficiency of MLGFIM tightly depend on

the interpolation scheme for the approximation of interaction

function. RBF interpolation method has been proven to be an

appropriate approach for the interpolation of interaction

function [23]. A lot of efforts have been made on searching a

better RBF interpolation scheme to improve the efficiency of

MLGFIM. Generally, an RBF i iR r r is a

continuous function that depends only on the distance

between the data (interpolated) point r and the center

(interpolation) point ri. Considering an approximation

function f (r) in an influence domain that has a set of K

arbitrarily distributed nodes with corresponding

values 1{ }K

i if r , the RBF interpolation reads:

1

1 2

1 2

( )

K

T

K

f R R R

f f f

r

r r r

(15)

where the entries of matrix are expressed as

ij j i r r .

There are many types of RBFs that can be used as the

interpolation function. Table 1 shows three types of RBFs that

have been used in MLGFIM. Parameter c in these expressions

is named as the shape parameter, and it determines the shape

of the interpolation function. In [37], it has proven that GA

RBFs can obtain better interpolation accuracy than IMQ

RBFs which have been used in [23] with the same number of

interpolation points, because they have better convergence

behavior. As a new class of oscillatory RBFs, BE RBFs have

shown their good interpolation performance in [38], and have

been applied to further enhance the efficiency of MLGFIM as

well [39].

Table 1. Definitions of Some Typical Radial Basis

Functions

Infinitely smooth functions R

Inverse multiquadric (IMQ) 2 2

1

1 c R

Gaussian (GA) 2exp c R

Bessel (BE)

/2 1

/2 1, 1, 2,

v

v

J cRv

cR

grid element

x

yz

(a)

grid element

x

yz

(b)

x

yz

grid element

(c)

Fig. 3. 2-D pictorial representation of (a) original STG, (b) modified STG

and (c) Boundary clustered STG (square points, hollow and solid dots are

volume center, surface center and vertex points in a grid element,

respectively).

Besides, the distribution of interpolation grid also affects

the interpolation efficiency of Green’s function. Tartan grid,

as a conventional grid pattern which is always used in the

Lagrange interpolation, just distributes interpolation points

evenly spaced in the source/field boxes, so the total number of


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interpolation points is K = K1D3. Since the RBF method allows

us to use scattered point distribution, STG which outperforms

Tartan grid in accuracy is used in MLGFIM. Three types of

STGs have been applied as the grid patterns which are shown

in Fig. 3. The original STG (Fig. 3(a)) consists of Tartan grids

and the points in the volume center of grid element of which

vertexes are formed by adjacent Tartan grids [23]. And the

modified STG, as shown in Fig. 3(b), consists of Tartan grids

and the points in the center of the sides of the grid element

[37]. Hence the total number of interpolation points for the

original STG and modified STG are K1D3+(K1D-1)3 and K1D

3+3K1D(K1D-1)2, respectively. It has been proven that with

similar number of interpolation points, the results using the

modified STG are more accurate than those using the original

STG [37]. This is because the condition number of matrix

generated by the modified STG is better than the original

STG. However, it is pointed out that an accurate method on

high-degree uniform grid sets leads to large errors near the

boundaries [40],[41]. So if the boundary errors can be

suppressed, the interpolation accuracy will be further

enhanced. Boundary clustered STG is generated by making

the points denser near the boundary [42]. In 1-D, the

Chebyshev-Lobatto nodes in the closed interval [-1, 1] are

expressed as:

1

1

2cos , 0,1, 1

1 2s D

D

sh s K

K

(16)

where hs is the location of s-th interpolation point. However, it

has been proven that the 3-D non-uniform grid generated by

simply clustering the interpolation points toward boundary

using (16) in x, y and z directions cannot improve the

interpolation accuracy [42]. This is because this grid helps to

reduce the boundary interpolation errors but it induces large

interior errors as well. In order to balance the boundary and

interior errors and make interpolation points still symmetric to

the center of interval, the 1-D grid pattern in the closed

interval [0, l] is generated as:

1

1

1 11

1

121 cos , 0,1,

2 1 2 2

2 2 ( 1) / 2 11 cos , , 1

2 1 2 2

D

D

s

D DD

D

Kl ss

K

h

s K Kls K

K

(17)

In function (17), the parameter is introduced to adjust the

degree of boundary clustering. The interpolation points, as

approaches 0, are distributed at l/2. Increasing will shift the

interpolation points toward the boundaries, and it improves

the resolution at the boundary but lowers the resolution in the

interior box. By adopting function (17) in x, y and z directions,

we obtain a boundary clustered Tartan grid with K1D3

interpolation points. Then, the boundary clustered STG is

generated by adding 3K1D(K1D-1)2 interpolation points to the

center of the sides of the rectangular block (grid element), as

shown in Fig. 3(c).

The maximum interpolation errors using three different

interpolation schemes are compared in Table 2. The first two

schemes employ the same interpolation grid (viz., original

STG), and the threshold of the maximum error for the second

scheme is set as 0.05. With the same number of interpolation

points, the maximum errors using IMQ RBFs are always

larger than those using BE RBFs, as shown in Table 2. And as

the edge length of the box increases, the method using BE

RBFs becomes more accurate than IMQ RBFs. The third

scheme adopts a new interpolation grid pattern, viz., boundary

clustered STG, to further enhance the accuracy and efficiency

of the interpolation. With the same error threshold, the

method using boundary clustered STG requires less number

of interpolation points than the original STG. Moreover, the

interpolation accuracy using the third scheme, even with less

number of points, is better than the other two. And it is also

observed that if we employ the third scheme, much more

number of interpolation points is saved as the size of box

increases. The efficiency of MLGFIM can be dramatically

enhanced by adopting the third method.

Table 2. Comparison of accuracy using different kinds of interpolation schemes.

Interpolation

scheme

Box

length ()

Number of

interpolation

points

Optimized

in (17)

Optimized

shape

parameter c

Maximum

error

IMQ RBFs

and original

STG

1 189 2.10 0.00681

2 559 0.80 0.0731

3 1241 0.69 0.131

4 2331 0.51 0.296

BE RBFs and

original STG

1 189 6.00 0.00632

2 559 3.20 0.0174

3 1241 2.35 0.0295

4 2331 1.705 0.0498

BE RBFs and

boundary

clustered STG

1 172 0.75 6.90 0.00439

2 365 0.97 3.21 0.0125

3 666 0.96 2.10 0.0291

4 1099 0.83 1.60 0.0465


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IV. MLGFIM FOR PERIODIC STRUCTURES AND

MULTILAYERED PROBLEMS

As a kernel independent algorithm, MLGFIM can also be

used for the fast analysis of other complex applications, e.g.,

periodic structures and multilayer problems. The Floquet

theorem allows limiting the analysis of periodic structures to

that of a unit cell; however, the number of unknowns can still

be large with there are fine features exist in the unit cell.

For periodic structures, instead of interpolating free-space

Green’s function, we should interpolate periodic Green’s

function. Ewald’s transformation [43],[44], which

efficiently combines both spatial and spectral formulations

of the periodic Green’s function, is applied to speed up the

evaluation of periodic Green’s function. Besides, since the

unit cells are actually connected together, the periodic

boundary condition must be taken into consideration. Half

basis functions at the connection boundary between two

adjacent unit cells have to be introduced to ensure the

boundary currents continuous.

After that, we can use MLGFIM to accelerate the analysis

of periodic structures. There are some modifications in the

definition of neighbor and interaction list when applying

MLGFIM for these structures, and they are defined as [45]:

box m is the neighbor/interaction list of box n when box m is

near/well separated from box n and the images of box n, and

parent box of box m is also near parent boxes of box n and

images of box n simultaneously. Fig. 4 shows the neighbor

and interaction lists of box n at level 3. In order to implement

MLGFIM easier, the box of interaction list which is not

around box n (but around the image of box n), e.g. box m1 in

Fig. 4, should be moved to the place of box m2 (image of m1)

which is around box n.

m1

n

image

n

image

n

image

n

m2

ra

rb

Fig. 4. 2-D representation of a 3-D multilevel division in which the

neighbor (white colored) and interaction (slash) lists of box n at level 3 are

shown.

In order to address the problem of the value of optimum

shape parameter c shift during the interpolation, RBF-QR

method is applied for the interpolation of periodic Green’s

function [46]. As the periodicity of one structure changes,

the interpolated function (viz., periodic Green’s function)

changes, and consequently the optimum value of shape

parameter c in RBF should be retested. Because the RBF-QR

method is insensitive to the value of c, it is not required to

retest the shape parameter when the interpolated function

changes. By using expansion with Chebyshev polynomials

and QR-factorization to the Gaussian RBFs, a better

conditioned interpolation matrix with its condition number

insensitive to c is generated [46],[47]. With RBF-QR

interpolation, MLGFIM becomes more robust for the

analysis of periodic structures.

By interpolating the multilayered Green’s function,

MLGFIM has been applied to solve the problems of

multilayered medium structures. An improved discrete

complex image method (DCIM) which uses a new sampling

path directly goes through the branch point is adopted for the

evaluation of the multilayered Green’s function [48]. This

method helps to capture the contribution to the lateral wave

more accurately. The property of planar structures of

multilayered problems demands only 2-D interpolation

scheme (viz., the interpolation points are only distributed at

the interfaces of every two layers), while 1-D interpolation

scheme is used for the vertical via holes only at the finest

level [49]. We will later show that this interpolation

approach is very efficient for the analysis of multilayered

medium problems.

V. OTHER APPROACHES TO COMBINE WITH MLGFIM

Given that the translation occupies most of the

computation time in the interpolation procedure of

lower-to-upper-level, it is possible to further enhance the

efficiency of MLGFIM by reducing the translation time.

Although the interpolation performance of 3-D Tartan grid is

not as good as the STG-based grid patterns, it generates a

3-D Toeplitz translation matrix G which could use FFT to

accelerate the matrix-vector multiplication. Moreover, the

difference in the interpolation efficiency between 2-D

Tartan grid and STG-based grid patterns is smaller than that

of the 3-D case, and hence the MLGFIM-FFT can be used to

accelerate the solution, especially the solution of multilayer

problems where 2-D interpolation scheme is adopted.

For the multilevel tree structure, there are two ways to

implement the FFT-accelerated translation procedure. In

[50],[51], the multilevel tree will be truncated at certain

level, and the pre-corrected FFT approach is applied at this

level. A global Toeplitz matrix is generated and the

interactions among all boxes are efficiently calculated by

performing only one FFT. However, a pre-corrected near

field evaluation is required for this method. So the

translation (also shown in Fig. 2) is rewritten as a

convolution form, viz.:

1;

neighbors of

l l l l

l l

m m n n

n m

B FFF FFT G FFT S G S

(18)

where G is comprised of Green’s function matrix ;l lm nG

and S consists of vectorlnS . Below the truncated level, the

lower-to-upper-level interpolation as shown in Fig. 2 could

be carried out as well [50].

The second way is based on local boxes, so it generates a

sub-domain Toeplitz matrix. Consequently, sub-domain

FFT is applied to accelerate the translation. Pre-correction is


7

not required because the sub-domain Toeplitz Green’s

function matrix is formed by two well separated boxes [52].

Hence, instead of (18), the translation could be expressed as:

;

interaction list of

1;

interaction list of

l l l l

l l

l l l

l l

m m n n

n m

m n n

n m

B G S

FFF FFT G FFT S

(19)

where ;l lm nG is a vector extracted from the Toeplitz matrix

;l lm nG . The FFT is not implemented only at one level, but at

many high levels unless the number of interpolation points is

too small and direct matrix-vector multiplication will be

more efficient than the FFT. With the second method, no

truncated level exists, so the lower-to-upper-level

interpolation is implemented from the top level to the bottom

level to make the MLGFIM-FFT more efficient.

In addition, parallel computing could also be applied to

accelerate the solution. Parallelized MLGFIM using OpenMP

[50] and MPI [36] have been developed. For achieving high

efficiency, we must make workload balanced and that could

be accomplished by assigning approximately the same

number of boxes at each level into each processor. Different

from the OpenMP parallelization on a share-memory

computer system, the MPI parallelization on a

distributed-memory computer must take into account of the

communications among different processors as well. If the

upward or downward pass shown in Fig. 2 is implemented

through different processors, a large amount of data

communication is required. Hence, each root box (at level 2)

and all its descendants should be assigned to the same

processor [36]. Besides, in order to reduce the data

communication in the computation of near field, the

neighbors of each finest-level box are duplicated if they are

not assigned to the same processor.

Parallelized MLGFIM on a distributed-memory computer

system can be developed by modifying the steps of the

original MLGFIM. First, the matrix-vector multiplication

LL Lnn nS Z x is calculated at the finest level l = L with its own

sub-tasks. Second, the vectors 1 1

1

;l l l l

l l

n n n n

n n

S C S

for each

box assigned to this processor are computed from level l = L -

1 to 2. Then the translation from the source box to the

interaction list is computed. Data communication is required

in this step, and we use MPI_Allreduce command to collect

the vectors lnS from each processor, and distribute it to all

processors. After that, the vectors

11; ;

interaction list of

l ll l l l l

l l

T

m mm n n m m

n m

B G S C B

in the corresponding

boxes for the processor from level l = 3 to L are computed.

Finally, the matrix-vector multiplication L L

L

T

m mm nearb b T B is

performed and the vector Lmb is also collected and distributed

to all processors.

Certain factors should be considered when examining the

efficiency of parallel computing. Intuitively, a parallel

program leads to a shorter execution time for a given

computation task, which can be expressed by a speedup and

efficiency factors as:

p s ps n t t (20)

and

100%sp

p p

te n

n t

(21)

where ts and tp are serial execution time using one processor

and parallel execution time using np processors, respectively.

If the efficiency factor approaches 100%, this indicates that

the parallel computing is very efficient and the

communication/computation ratio is very small.

VI. NUMERICAL EXAMPLES

In order to demonstrate good performance of the proposed

method, some examples are given as follows. The generalized

minimal residual (GMRES) iteration method with a relative

error norm of 10-3 is employed, and the inner loop of GMRES

contains 100 matrix-vector multiplications.

x

y

o

w1

dd

w2

t

x

yz

o

(a) (b)

0 20 40 60 80 100 120 140 160 180

(degree)

-110

-100

-90

-80

-70

-60

-50

-40

-30

RC

S(d

Bs

m)

MLGFIMMoM

(c)

Fig. 5. Plane wave impinges normally on a fishnet-type array. (a)

Geometry. (b) Unit cell. (c) Bistatic RCS with polarization.

In the first example, MLGFIM is applied to plane wave

scattering from a fishnet-type array with 10×10 elements, as

shown in Fig. 5(a). The geometrical parameters of the array

are as follows: w1 = 10 m, w2 = 17.5 m, d = 115m, t = 9 m.

The patterns on the top and bottom surfaces of the substrate

are made of the same shaped PEC, and the relative

permittivity of the substrate is 2.25. A plane wave with

electric field parallel to the x-axis propagates along –z-axis,

and the center frequency is 0.5 THz. PMCHWT integral

equation is used for this structure to obtain the surface current.

The bistatic RCS with polarization is calculated using the

5-level MLGFIM, and compared with MoM, as shown in Fig.

5(b). From this figure, it is observed that these two results

agree very well.


8

x y

z

Improved MLGFIM

Original MLGFIM

0

45

90

18

0

13

5

27

0

22

531

5

4 1 -2 -5 -8

-27 -21 -15 -9 -3 3 9 15 21 27

y (cm)

-27

-21

-15

-9

-3

3

9

15

21

27

z (

cm

)

Poyting vector (kW/m^2)

1000+

875 to 1000

750 to 875

625 to 750

500 to 625

375 to 500

250 to 375

125 to 250

0 to 125

(a) (b) (c)

Fig. 6. A dipole in front of an 11×11 dielectric sphere array. (a) Geometry. (b) Polar plot of the directivity versus in dBi when= 90o. (c) Distribution of

Poynting vector in yz plane.

Table 3. Comparison of CPU Time and Memory Requirement for Simulating the 11×11 Dielectric Sphere Array

Method

Maximum

electrical

length(d)

The

number of

levels

The number

of

unknowns

CPU time for per

matrix–vector

multiplication

Memory

requirement

Original MLGFIM 16.87 6 144,020

96.3 s 8.7 GB

Improved MLGFIM 21.4 s 3.1 GB

In the second example, a radiation problem that a dipole in

front of an 11×11 dielectric sphere array (r = 2.25) is

investigated, as shown in Fig. 6(a). The radius of the sphere is

2 cm and the distance between the centers of two adjacent

spheres is 5 cm. A Hertzian dipole which works at the

frequency of 6.25 GHz is located 10 cm in front of the array

center and placed along the z-axis. In this example, ten grids

per wavelength d (where 0d r

) are used to discretize

this structure, so that the number of unknowns for equivalent

electric and magnetic currents is 144,020. The polar plot of

the directivity versus when= 90o is shown in Fig. 6(b).

Two methods, i.e., original MLGFIM (IMQ RBF and original

STG) and improved MLGFIM (BE RBF and boundary

clustered STG), are used for the simulation. Good agreement

can be observed between these two results. The distribution of

Poynting vector in the plane of x = –20 cm is also calculated,

as shown in Fig. 6(c). The computational time and memory

requirement for this example are shown in Table 3. Compared

with the original method, only 22.2% CPU time and 35.6%

memory are required for the improved one.

(a)

0

45

90

135

180

225

270

315

-35

-20

-20

-5

-5

10

10

25

25

Double Positive Medium

Single Negative Medium

Double Negative Medium

(b)


9

104

105

106

Number of unknowns (N)

1

10

100

CP

U t

ime

(s

)

1

10

100

Impoved MLGFIM

Original MLGFIM

2.7e-5*NlogN

7 levels

6 levels

5 levels

(c)

104

105

106

Number of unknowns (N)

0.1

1

10

Me

mo

ry r

eq

uir

em

en

t (G

B)

0.1

1

10

Improved MLGFIM

Original MLGFIM

1.6e-2*N

7 levels

6 levels5 levels

(d)

Fig. 7. Plane wave scattering from a dielectric prolate spheroid. (a)

Geometry. (b) Bistatic RCS with polarization at 300 MHz. (c) CPU time.

(d) Memory requirement.

Next, an example is provided to show the CPU time and

memory requirement with different number of unknowns. A

dielectric prolate spheroid is considered, as shown in Fig.

7(a). The long axis of the prolate spheroid is 6 m, whereas the

short axis is 1.5 m. A plane wave with a frequency of 300 MHz

impinges on the prolate spheroid along –x-axis. The plane

wave scattered by prolate spheroids composed of double

positive (r = 2.25, r = 1.0), single negative (r = –2.25, r =

1.0) and double negative (r = –2.25, r = –1.0) medium are

studied. Fig. 7(b) compares the bistatic RCS with

polarization for these three cases. The forward scattering is

enhanced significantly due to negative permittivity and

permeability. The CPU time for performing each

matrix-vector multiplication and the corresponding memory

requirement are shown in Figs. 7(c) and (d). In these two

figures, about 20 grids per wavelength are used to discretize

the prolate spheroid, and the frequency varies from 150 MHz

to 660 MHz. Three types of numbers of levels are used for the

calculation of the structure, and the original MLGFIM cannot

simulate the structure when the number of unknowns is larger

than 5×105 because of the memory limitation. From Fig. 7(c)

and (d), it is observed that the CPU time and memory

requirement of the improved MLGFIM are always smaller

than those of the original one. And when the number of

unknowns increases to 1,089,224, the CPU time for

performing one matrix-vector multiplication and the memory

requirement for the improved MLGFIM are 128.7 seconds

and 17.68 GB, respectively. In addition, it is also observed

that the improved MLGFIM approximately obeys a

computational complexity of O(N log N) and memory

complexity of O(N).

x

yz

3 mm

1.8 mm

0.6 mm

3 mm

1.8 mm

(a) (b)

-200 -150 -100 -50 0 50 100 150

Bistatic angle (degree)

-120

-100

-80

-60

-40

-20

0

RC

S (

dB

sm

)

Parallelized MLGFIM

Serial MLGFIM

(c)

Fig. 8. Plane wave scattering from a metallic scatterer. (a) Geometry. (b)

Unit. (c) Bistatic RCS with polarization.

Table 4. Comparison of CPU Time and Memory

Requirement

Number

of levels

Number of

processors Speed-up

Efficiency

(%)

Save

time (%)

7

1 1.0 100 0

2 1.98 99.2 49.6

4 3.66 91.5 72.4

8 6.51 81.3 84.6

To show the effect and accuracy of parallelized MLGFIM,

we consider the plane wave scattering from a complicated

metallic scatterer, as shown in Fig. 8(a). This structure is

composed of orthogonal square rings, and four square rings

orthogonally connect with one ring to generate a unit, as

shown in Fig. 8(b). The outer length, inner length and height

of each ring are 3 mm, 1.8 mm and 0.6 mm, respectively. On

the boundary of this structure, there exists only half rings, and

hence the total structure contains 6 orthogonal rings in each

direction. The structure is investigated at 300 GHz and

illuminated by a plane wave propagating along the

–x-direction, with the electric field polarized in the

z-direction. After the structure is discretized, the total number

of unknowns obtained is 1,134,768. The bistatic RCS result


10

calculated by the serial MLGFIM and parallelized MLGFIM

using 8 processors are compared in Fig. 8(c). Good agreement

is observed from this figure. Furthermore, the speed-up factor

and efficiency factor are demonstrated, as shown in Table 4.

According to this table, we observe that the computational

time is reduced significantly with the increase of the number

of processors. The efficiencies of the parallelized MLGFIM

are larger than 80% for all cases. The CPU time is reduced by

84.6% after calculating this problem with eight processors,

simultaneously.

r=3.0

4.83 mm

5 m

m

5 mm

d

0.17 mm

0.17 mm

0.027 mm

x

y

(a)

10 15 20 25 30 35 40

Frequency (GHz)

-30

-25

-20

-15

-10

-5

0

Re

fle

cti

on

Co

eff

icie

nt

MLGFIMReference [47]Reference [48]

d=3.0 mm

d=2.5 mm

(b)

Fig. 9. FSS comprising two centered square loops on a dielectric substrate.

(a) Side and front view of the unit cell. (b) TM reflection coefficient with

different values of d.

In the fifth example, MLGFIM is used to analyze a periodic

structure, in which two centered square loops are etched on a

dielectric layer (r = 3). Fig. 9(a) shows the geometry of the

frequency selective surface (FSS), and it is illuminated by a

uniform plane wave incident from the normal direction.

Periodic Green’s function interpolation and periodic

boundary conditions are used for this structure. The results for

different dimensions d of the inner loop are calculated by

MLGFIM and compared with the simulation results in [53]

and experimental results in [54]. RBF-QR method is adopted

as the interpolation approach, so that the MLGFIM becomes

more robust for the analysis of periodic problems. From Fig.

9(b), it can be seen that our results are closer to the

experimental results compared with the simulation results in

[53]. And we also find that the second resonance shifts toward

a lower frequency band with the increase of the dimension d.

wl

ra

rb

x

y

(a)

0 10 20 30 40 50 60 70 80 90-70

-60

-50

-40

-30

-20

-10

0

E-f

ield

(d

B)

8 by 8

16 by 16

32 by 32

64 by 64

(b)

10000 100000 1E+006

Number of unknowns

0.001

0.01

0.1

1

10C

PU

tim

e (

se

co

nd

)

0.01

0.1

1

10

Me

mo

ry (G

B)

MLGFIM

MLGFIM_FFT

MLGFIM

MLGFIM_FFT

(c)

Fig. 10. Plane wave scattering from a patch array. (a) Geometry. (b)

Normalized E-field on xz plane. (c) CPU time and memory requirement.

In the sixth example, MLGFIM is applied to analyze some

patch arrays with planar multilayer substrate, in which

sub-domain FFT is combined to accelerate the calculation. A

plane wave with the electric field parallels to the y axis

incidents from the normal direction. The parameters of the

microstrip patch array are l = 49 mm, w = 41 mm, ra = 90

mm,rb = 82 mm, as shown in Fig. 10(a). The thickness and

dielectric constant of the substrate are 1.59 mm and 2.2,

respectively. The working frequency is set as 2.2GHz, and the

normalized electric fields on xz plane for 8×8, 16×16, 32×32,

and 64×64 arrays are shown in Fig. 10(b). In Fig. 10(c), the

efficiencies, using MLGFIM and MLGFIM-FFT, for

analyzing these arrays with different number of unknowns are

compared. For this example, 2-D interpolation method is

applied, and hence the CPU time and memory requirement are

both much smaller than those of the above examples which

use 3-D interpolation. From Fig. 10(c), it is observed that the

CPU time and memory requirement using MLGFIM-FFT are

always smaller than those using MLGFIM alone. With the


11

increase of the array size, the reductions of time and memory

become more significant.

x

y

Feeding

line

(a)

ra

rb

wl

t1

t2

(b)

-100 -80 -60 -40 -20 0 20 40 60 80 100

(degree)

-60

-50

-40

-30

-20

-10

0

Ra

dia

tio

n (

dB

)

co-pol

cr-pol

(c)

-100 -80 -60 -40 -20 0 20 40 60 80 100

(degree)

-60

-50

-40

-30

-20

-10

0

Ra

dia

tio

n (

dB

)

co-pol

cr-pol

(d)

Fig. 11. Radiation of a 32×32 parallel fed antenna array. (a) Geometry. (b)

Unit. (c) Radiation pattern on H-plane (= 0º). (d) Radiation pattern on

E-plane (= 90º).

Finally, the MLGFIM-FFT method is used to calculate the

radiation pattern from a 32×32 patch array with planar

multilayer substrate. The structure of this array is shown in

Fig. 11(a). And this structure is composed by eight 4×4 unit in

one direction which is given in Fig. 11(b). The parameters of

the microstrip patch array are l = 41 mm, w = 49 mm, ra = 100

mm, rb = 100 mm, t1 = 5.0 mm, t2 = 1.5 mm, and the thickness

and relative dielectric constant of the substrate are 1.59 mm

and 2.2, respectively. The working frequency is 9.42 GHz and

the number of the unknowns of the metallic layer is about

310,000. The radiation patterns in both E- and H-plane are

plotted in Fig. 11(c) and (d). A very narrow beam is formed as

expected. The FFT will be applied at level 2 and 3. Because

the structure is electrically large and complex, the

convergence of the iteration process will be a big problem. To

ensure convergence, the number of the spanning vector in

GMRES is set to 1,000. The CPU time per iteration is about

1.8 seconds which is still very efficient, and the memory

requirement is 11 GB due to the large number of spanning

vector.

VII. CONCLUSIONS

The development and improvement of the efficient fast

algorithm, namely MLGFIM, is reviewed. Objects in

free-space, periodic and multilayer structures are analyzed

using this fast simulation algorithm. Various techniques,

including: BE RBF interpolation functions, boundary

clustered STG pattern, FFT technique and parallelization

approaches, are used to enhance the efficiency of the

MLGFIM. Seven examples are given to validate the accuracy,

and demonstrate the improvement of the MLGFIM after using

the above new techniques.

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http://jpier.org/PIER/pier.php?paper=07102101



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Peng Zhao received his B.Eng degree

and M.Phil degree in electronic

engineering department both from

Zhejiang University, China, in 2006 and

2008, respectively. He earns his Ph.D.

degree in electronic engineering from

City University of Hong Kong, China, in

2014. He is currently a faculty member in

the CAD laboratory at Hangzhou Dianzi

Univeristy. His research interest focuses on computational

electromagnetics.

Da Qing Liu received the B.S. degree in

optical science from Wuhan University of

Science and Technology, Wuhan, China,

in 2008 and the M.S. degrees in electronic

engineering from Zhejiang University,

Hangzhou, China, in 2011. He is

currently pursuing his Ph.D. degree in

Electronic Engineering at City University

of Hong Kong, China. His research

interests include computational electromagnetic and antenna

design.

Chi Hou Chan received the Ph.D.

degree in electrical engineering from the

University of Illinois, Urbana, IL, USA,

in 1987.

From 1989 to 1998, Dr. Chan was a

faculty member in the Department of

Electrical Engineering at the University

of Washington, Seattle, WA, USA. In

1996, he joined the Department of Electronic Engineering,

City University of Hong Kong, and was promoted to Chair

Professor of Electronic Engineering in 1998. From 1998 to

2009, he was first Associate Dean and then Dean of College of

Science and Engineering. He also served as Acting Provost of

the university from July 2009 to September 2010. He is

currently the Director of State Key Laboratory of Millimeter

Waves, Partner Laboratory in the City University of Hong

http://dx.doi.org/10.1137/09076756X

http://dx.doi.org/10.1137/09076756X

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6192310




14

Kong. His current research interests include computational

electromagnetics, millimeter-wave circuits and antennas, and

terahertz science and technology.

Dr. Chan received the U.S. National Science Foundation

Presidential Young Investigator Award in 1991 and the Joint

Research Fund for Hong Kong and Macao Young Scholars,

National Science Fund for Distinguished Young Scholars,

China, in 2004. He received outstanding teacher awards from

the Department of Electronic Engineering, CityU in 1998,

1999, 2000, and 2008. He is the General Co-Chair of ISAP

2010, iWAT2011, and iWEN 2013.

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