transformation-based metamaterials to eliminate the staircasing error in the finite difference time...

Transformation-Based Metamaterials to Eliminatethe Staircasing Error in the Finite Difference TimeDomain Method

Ozlem Ozgun,1 Mustafa Kuzuoglu2

1 Department of Electrical and Electronics Engineering, TED University, Ankara, Turkey2 Department of Electrical and Electronics Engineering, Middle East Technical University, 06531,Ankara, Turkey

Received 30 June 2011; accepted 4 January 2012

ABSTRACT: A coordinate transformation technique is introduced for the finite difference

time domain method to alleviate the effects of errors introduced by the staircasing approxi-

mation of curved geometries that do not conform to a Cartesian grid. An anisotropic meta-

material region, which is adapted to the Cartesian grid and designed by the coordinate

transformation technique, is constructed around the curved boundary of the object, and the

region occupied between the curved boundary and the inner boundary of the anisotropic

metamaterial layer is discarded. The technique is validated via several numerical simula-

tions. VC 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE 00:000–000, 2012.

Keywords: finite difference time domain (FDTD); method; staircasing error; anisotropic metama-

terials; coordinate transformation; transformation media

I. INTRODUCTION

Finite difference time domain (FDTD) method is a power-

ful numerical technique for solving time-dependent prob-

lems in computational electromagnetics [1, 2]. It uses a

leap-frog scheme for marching in time, where the electric

and magnetic fields are staggered on a Cartesian coordi-

nate grid. Despite its several advantages, the main disad-

vantage of the FDTD method is the reduced accuracy and

efficiency in modeling curved geometries that do not con-

form to a Cartesian grid. A simple and common approach

to modeling such structures is to use a staircase approxi-

mation of the curved surface, which indeed introduces

errors known as staircasing errors in the FDTD commu-

nity. Several techniques have been proposed in the litera-

ture to overcome these errors [3–9]. Two straightforward

ways that have been reported in the literature are the use

of local Cartesian subgridding close to curved boundaries

or the use of irregular nonorthogonal grids [3, 4], both of

which make the simple structure of the conventional

FDTD algorithm much more complicated and increase the

time and memory consumption. More efficient techniques,

known as locally conformal FDTD, have also been pro-

posed to eliminate the staircasing errors [7–9].

In this study, we introduce a coordinate transformation

technique to alleviate the effects of errors introduced by the

staircasing approximation. The principal idea in the pro-

posed approach is to construct an anisotropic metamaterial

region (called a transformation medium), which is adapted

to the Cartesian grid, around the curved boundary of the

object, and to discard the region between the curved bound-

ary and the inner boundary of the anisotropic metamaterial

layer. The anisotropic material parameters are designed by

using a specially defined coordinate transformation tech-

nique, which maps the region inside the layer to the region

between the curved boundary and the outer boundary of the

layer. Thus, this approach reduces the approximation errors

without the need for changing simple Cartesian grids. Basi-

cally, the concept of ‘‘transformation medium’’ refers to ar-

tificial medium whose constitutive parameters are designed

by using the form invariance property of Maxwell’s equa-

tions under coordinate transformations so as to mimic the

field behavior in the modified coordinate system. The coor-

dinate transformation technique provides the duality

between geometry and material parameters in the sense that

Maxwell’s equations preserve their form in the modified

coordinate system, but the medium turns into an anisotropic

medium to convey the action of the coordinate

Correspondence to: O. Ozgun; e-mail: [email protected]

VC 2012 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20642Published online in Wiley Online Library

(wileyonlinelibrary.com).

1

transformation to the electromagnetic fields. The concept of

the invisibility cloak as a transformation medium initiated

the widespread use of the coordinate transformation tech-

nique as an intuitive design tool [10]. However, the range

of applications of coordinate transformations goes well

beyond cloaking and varies from the design of perfectly

matched layers (PMLs) for the purpose of mesh truncation

in finite methods [11, 12], to the design of electromagnetic

reshapers for both objects and waveguides, concentrators,

rotators, lenses, etc. [13–21]. Almost all recent applications

deal with the material itself, i.e., constitutive parameters,

physical realization, etc. However, the approach in this arti-

cle aims at using such materials for obviating the staircasing

errors in numerical modeling, and hence, the physical real-

ization of these ‘‘virtual’’ materials is beyond the scope of

this article.

The organization of this article is as follows: In Sec-

tion II, the proposed coordinate transformation technique

is presented together with the discussion on the form-

invariance property of Maxwell’s equations under a gen-

eral coordinate transformation. In Section III, the FDTD

update equations are derived with extensions to handle ar-

bitrary material tensors. In Section IV, various numerical

simulations are demonstrated to validate the accuracy of

the proposed technique in the context of two-dimensional

(2D) electromagnetic scattering problems. Finally, the

conclusions are presented in Section V.

II. COORDINATE TRANSFORMATION TECHNIQUE

The design procedure of the proposed coordinate transfor-

mation technique is illustrated in Figure 1, wherein a dia-

mond-shaped object (in fact, a rotated square) is consid-

ered. Figure 1a shows the standard modeling of the object

by a Cartesian grid. It is evident that the geometry of the

object is not conformal to the Cartesian grid, and its stair-

case approximation causes errors because the conventional

FDTD cannot capture the field variations across the

boundary (especially, through the corner regions in Fig.

1a). In the proposed technique in Figure 1b, an equivalent

problem is designed by locating an anisotropic metamate-

rial region, which is adapted to the Cartesian grid, around

the boundary of the object. The region occupied between

the object’s boundary and the inner boundary of the aniso-

tropic metamaterial layer is discarded. In designing the

metamaterial layer, each point P inside XA is mapped to~P inside the transformed region ~X ¼ X [ XA, by using the

following coordinate transformation T : XA ! ~X

~r ! ~~r ¼ T ~rð Þ ¼ ~rb �~rck k~rb �~rak k ~r �~rað Þ þ~rc; (1)

where ~r and ~~r are the position vectors of the points P and~P in the original and transformed coordinate systems,

respectively. In addition, ~ra, ~rb, and ~rc are the position

vectors of the points Pa, Pb, and Pc, through the unit vec-

tor at, denoting the direction of transformation and origi-

nating from a point inside the innermost domain (such as

the center-of-mass point) in the direction of the point Pinside the metamaterial layer. Herein, ||�|| denotes the Eu-

clidean norm.

The transformation in (1) represents a general tech-

nique and can be used for any arbitrarily shaped object in

Figure 1 FDTD modeling of diamond-shape object: (a) Staircased modeling by a Cartesian grid and (b) modeling by locating an aniso-

tropic metamaterial region.

2 Ozgun and Kuzuoglu

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 000, No. 000, Month 2012

a straightforward manner. The analytical expression of the

transformation depends on the geometry at hand and

should be performed for each different geometry. To be

more precise in the analytical implementation of the trans-

formation, we consider three cases, which will be used for

validation purposes in Section IV. For the diamond-shaped

object in Figure 1b, the position vectors ~ra, ~rb (u ¼ a if

~ru ¼~ra, and u ¼ b if ~ru ¼~rb) and ~rc can be obtained for

each region as follows:

~ru ¼ � uxy ax þ uay

� �in X1 ðþÞ and X3 ð�Þ

� uax þ uyx ay

� �in X2ð�Þ and X4 ðþÞ

(;

(2:a)

~rc ¼

axyþ xj j ax þ ay

yþ xj j ay in X1

� axx� yj j ax � ay

x� yj j ay in X2

� axy� xj j ax � ay

y� xj j ay in X3

axxþ yj j ax þ ay

xþ yj j ay in X4

8>>><>>>:

: (2:b)

Herein, a and b are the edge half-lengths of the inner

and outer boundaries of the metamaterial layer,

respectively.

For the geometry in Figure 2, the position vectors ~raand ~rb are defined in the same manner, but only in regions

X1, X2, and X3 shaded by the light-gray color. The posi-

tion vector ~rc can be determined as follows:

~rc ¼ axffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p ax þ ayffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p ay: (3)

Note that, in Figure 2, only the light-gray region corre-

sponds to the anisotropic metamaterial layer due to the na-

ture of the transformation. This is expected because other

parts of the object are conformal to the Cartesian grid,

and the transformation yields original coordinates in

regions surrounding those parts (i.e., if ~ra ¼~rc, then

~r ¼ ~~r ; 8~r 2 X). Another important observation is that the

coordinate transformation in (1) is continuous in the sense

that two closely located points ~r and ~r� in XA are mapped

to also closely located points ~~r and ~~r � in ~X. That is, givene > 0, there exists d > 0 (d depends on e) such that

T ~rð Þ � T ~r �ð Þk k < e; whenever ~r �~r �k k < d. Otherwise,

considerable deviations in the entries of the permittivity

and permeability tensors corresponding to contiguous

points result in unreliable results in the FDTD simulation.

In addition, it is useful to emphasize that the transformed

coordinates and the original coordinates are continuous

along the outer boundary of the metamaterial region (i.e.,

if ~r ¼~rb, then ~r ¼ ~~r ¼~rb on @XA). This guarantees the

field continuity along the boundary.

We also note that this approach transforms the inner

boundary of the metamaterial layer to the boundary of the

object. Hence, the boundary conditions that must be

imposed on the boundary of the conducting object (i.e.,

tangential component of the electric field must be zero on

the boundary) must be imposed on the inner boundary of

the metamaterial layer. This approach can be extended to

dielectric objects in a straightforward manner. This can be

achieved by letting ~rc ¼ 0, and by obtaining the material

parameters with respect to the dielectric constant of the

object when the transformed point falls into the object.

The computation of the material parameters will be dis-

cussed below in conjunction with the form invariance

property of Maxwell’s equations.

A. Form Invariance property of Maxwell’s EquationsUp to this point, we have considered only how to define

the coordinate transformations. Now, we will answer this

question: What is the relation between the transformed

coordinates and the constitutive parameters of the aniso-

tropic material?

The effect of the coordinate transformation in (1) is

reflected to the electromagnetic fields by replacing the

original medium with an anisotropic medium so that trans-

formed fields satisfy the original forms of Maxwell’s

equations. This is known as the form-invariance property

of Maxwell’s equations under coordinate transformations.

Permittivity and permeability tensors of the new aniso-

tropic medium can be obtained by [22]

��e ¼ e ��K; ��l ¼ l ��K; (4:a)

where

��K ¼ dJðJT � JÞ�1; (4:b)

where dJ denotes the determinant of J, and

J ¼@ ~x; ~y; ~zð Þ=@ x; y; zð Þ is the Jacobian tensor. If the origi-

nal medium is an arbitrary anisotropic medium with pa-

rameters ( ��e0, ��l0), then the parameters of the new aniso-

tropic material in the transformed space are determined by

Figure 2 Illustration of proposed coordinate transformation

technique for half-circle (example scenario in Fig. 7).

Transformation-Based Metamaterials to Eliminate FDTD Error 3

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

��e ¼ dJ J�1� �T � ��e0 � J�1

� �; (5:a)

��l ¼ dJ J�1� �T � ��l0 � J�1

� �: (5:b)

Note that the constitutive parameters of the metamate-

rial are all real-valued, and hence, the anisotropic meta-

material is lossless if the original medium is ideally

lossless.

Under the coordinate transformation in (1), the fields and

sources (i.e., the current density function ~J, if any) within

the transformed space are transformed as follows [22]:~~E ~r; tð Þ ¼ JT � ~Eð~~r; tÞ, ~~H ~r; tð Þ ¼ JT � ~Hð~~r; tÞ, and~~J ~r; tð Þ ¼ dJ J�1 � ~Jð~~r; tÞ. In 2D transverse magnetic (TM)zcase where ~E ~r; tð Þ ¼ azEz x; y; tð Þ, the equivalence of the

fields is expressed as

Ez ~x; ~y; tð Þ ¼ ~Ez x; y; tð Þ; (6:a)

Jz ~x; ~y; tð Þ ¼ dJ ~Jz x; y; tð Þ; (6:b)

due to the special form of the Jacobian tensor [viz.,

@~z=@z ¼ 1, and all other off-diagonal z-dependent terms

are zero]. The expressions in (6) reveal that the original

fields in transformed coordinates [i.e., ~E ~~r; t� �

] and the

transformed fields in the original coordinates [i.e.,~~E ~r; tð Þ]

are inter-related. Namely, the original fields can be recov-

ered at all points inside the transformed space, after the

fields are calculated inside the anisotropic material. This

principle of field equivalence is important because it

allows us not to lose the near-field information inside the

discarded region around the object.

As Maxwell’s equations are form-invariant under coor-

dinate mapping, Maxwell’s equations in original coordi-

nates are equivalent to those in transformed coordinates as

follows (assuming that the original medium is free-space):

r� ~~E ~r; tð Þ ¼ ��l � @~~H ~r; tð Þ@t

$

~r� ~E ~~r; t� �

¼ �l0@ ~H ~~r; t

� �@t

;

(7:a)

r� ~~H ~r; tð Þ ¼ ��e � @~~E ~r; tð Þ@t

þ ~~J ~r; tð Þ $

~r� ~H ~~r; t� �

¼ e0@~E ~~r; t

� �@t

þ ~J ~~r; t� �

;

(7:b)

where ~Eð~~r; tÞ and ~Hð~~r; tÞ are the transformed fields (i.e.,

mapped versions of ~E ~r; tð Þ and ~H ~r; tð Þ to transformed

space, respectively), and ~r ¼ ½J�1�T � r is the nabla oper-

ator in transformed space.

III. FDTD EQUATIONS IN ANISOTROPIC MATERIAL

The FDTD method was first introduced by Yee [1] to

solve the time-domain Maxwell’s equations directly and

was extensively used in the literature to solve various

electromagnetic radiation/scattering problems. Although

the FDTD equations for isotropic materials are very well

known and easier to implement, the equations for aniso-

tropic materials require special treatment. In addition,

there are different approaches in handling anisotropic

materials whose constitutive parameters are defined by

tensors. Therefore, it will be useful, for the sake of com-

pleteness, to derive the FDTD equations that are used in

this study to simulate the TMz case, where Ex ¼ 0, Ey ¼0, and Hz ¼ 0.

In the FDTD algorithm, both space and time are dis-

cretized, and the spatial and the temporal derivatives

appearing in Maxwell’s equations are approximated by

difference equations. Afterward, the difference equations

are solved by marching the fields in time. The spatial

space is discretized in such a way that each field quantity

is available only at a single location, as described by the

Yee cell. We assume that the electric displacement field

Dz is only found at the following location:

Dzjni;j ¼ Dz iDx; jDy; nDtð Þ (8)

Herein, (Dx, Dy) represents the grid size corresponding

to the coordinate variables, and Dt is the step size in time.

Similarly, the magnetic flux density quantities are only

found at the following locations:

Bxjnþ12

i;jþ12

¼ Bx iDx; jþ 1

2

� �Dy; nþ 1

2

� �Dt

� �(9:a)

By

��nþ12

iþ12;j¼ By iþ 1

2

� �Dx; jDy; nþ 1

2

� �Dt

� �(9:b)

Note that the electric displacement field is available

only at integer values of Dt in time, while the magnetic

flux density is available a half time step away.

Our purpose is to solve Maxwell’s equations defined in

the anisotropic material, assuming source-free and lossless

medium ( ~J ¼ 0). Just for the purpose of clarity, we drop

the ‘‘tilda’’ on top of the field quantities in (7), and

express Maxwell’s equations as follows:

r� ~E ¼ � @~B

@tand r� ~H ¼ @~D

@t(10)

The constitutive relations are ~D ¼ ��e � ~E and ~B ¼ ��l � ~H,where the permittivity and permeability tensors have com-

ponents whose off-diagonal z-related terms are zero.

We note that if the constitutive relations are substituted

into (10) and Maxwell’s equations are expressed only in

terms of the E- and H-fields, which is indeed a typical

approach in the literature, the simple FDTD update equa-

tions for isotropic materials cannot be used and some

additional effort is needed to express the update equations

in terms of the entries of the arbitrary material tensors.

Instead, we will first express the update equations for the

B- and D-fields using (10) in terms of the E- and H-fields,which will be similar to the standard FDTD equations

used in an isotropic medium. Next, the equations corre-

sponding to the E- and H-fields will be derived by

expressing the constitutive relations as follows:



~E ¼ ��e�1 � ~D and ~H ¼ ��l�1 � ~B. For the sake of clarity, we

will denote the inverses of the material tensors by intro-

ducing new variables, i.e., ��e ¼ ��e�1 and ��l ¼ ��l�1. This

approach allows us to use certain field interpolation

schemes in an easier manner if the field quantities are not

available in the FDTD grid, which will be clear below.

First, we consider Maxwell’s curl-H equation in (10),

which can be expressed as follows:

@Hy

@x

��nþ1

2

i0 ;j0�@Hx

@y

��nþ1

2

i0;j0¼ Dzjnþ1

i0;j0 �Dzjni0;j0Dt

(11)

It is important to note that this is the general expression

that is valid at any arbitrary location (i0Dx, j0Dy). If the fieldsare not defined at the proper locations, given in (8) and (9),

they are expressed by interpolating the fields from

Figure 3 (Scenario 1) Field maps of original and equivalent problems at different time instants. (Distance between the object and the

inner metamaterial layer ¼ 0.3k, Thickness of the metamaterial layer ¼ 1.3k). [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]



neighboring values, which will be discussed later. As Dz is

defined at (iDx, jDy), and because the field quantities at time

n þ 1 are unknown and must be computed by using the

fields at the previous time steps, (11) can be expressed as

follows:

Dzjnþ1i;j ¼ Dzjni;jþ Dtð Þ @Hy

@x

��nþ1

2

i;j

�@Hx

@y

��nþ1

2

i;j

" #(12)

The derivatives@Hy

@x

��nþ12

i;jand @Hx

@y

��nþ12

i;jare obtained from

the FDTD grid as follows:

@Hy

@x

��nþ1

2

i;j

¼ 1

DxHy

��nþ12

iþ12;j�Hy

��nþ12

i�12;j

� �(13)

@Hx

@y

��nþ1

2

i;j

¼ 1

DyHxjnþ

12

i;jþ12

�Hxjnþ12

i;j�12

� �(14)

By using the constitutive relation, the E-field can be

expressed in terms of the D-field (i.e., Ezji;j¼ ezzDzji;j),and then, the update equation for the E-field can be

derived as follows:

Ezjnþ1i;j ¼ Ezjni;jþ Dtð Þezz

� 1

DxHy

��nþ12

iþ12;j�Hy

��nþ12

i�12;j

� �� 1

DyHxjnþ

12

i;jþ12

�Hxjnþ12

i;j�12

� � ð15Þ

Herein, the material parameter is obtained at the loca-

tion where the E-field is defined, namely ezz ¼ ezzji;j.

TABLE I (Scenario 1) Error Values Comparing the Fields Outside the Metamaterial Region

Distance Between

Object and Inner

Metamaterial Layer

Thickness of

Metamaterial Layer

Time-Step n ¼ 900 Time-Step n ¼ 800 Time-Step n ¼ 700

Err1 Err2 Err1 Err2 Err1 Err2

0.2k 1.5k 2.25 E �3 8.68 E �4 1.88 E �3 7.86 E �4 8.46 E �4 5.22 E �4

1.4k 2.76 E �3 9.64 E �4 2.22 E �3 8.56 E �4 1.49 E �3 6.94 E �4

1.3k 2.25 E �3 8.72 E �4 2.05 E �3 8.24 E �4 9.46 E �4 5.54 E �4

1.2k 6.49 E �3 1.48 E �3 4.85 E �3 1.27 E �3 2.97 E �3 9.81 E �4

1.1k 1.10 E �2 1.93 E �3 8.18 E �3 1.65 E �3 4.91 E �3 1.26 E �3

0.1k 1.3k 1.45 E �3 7.01 E �4 1.08 E �3 5.98 E �4 6.64 E �4 4.64 E �4

0.2k 2.25 E �3 8.72 E �4 2.05 E �3 8.24 E �4 9.46 E �4 5.54 E �4

0.3k 4.96 E �3 1.29 E �3 4.09 E �3 1.16 E �3 2.87 E �3 9.63 E �4

TABLE II (Scenario 1) Error Values Comparing the Fields Inside the Entire Region After Transforming the Fields Insidethe Metamaterial Region

Distance Between

Object and Inner

Metamaterial Layer

Thickness of

Metamaterial Layer

Time-Step n ¼ 900 Time-Step n ¼ 800 Time-Step n ¼ 700

Err1 Err2 Err1 Err2 Err1 Err2

0.2k 1.5k 4.49 E �3 1.26 E �3 4.44 E �3 1.23 E �3 4.39 E �3 1.21 E �3

1.4k 5.98 E �3 1.45 E �3 5.65 E �3 1.39 E �3 5.30 E �3 1.32 E �3

1.3k 3.67 E �3 1.14 E �3 3.86 E �3 1.15 E �3 3.88 E �3 1.13 E �3

1.2k 1.01 E �2 1.89 E �3 9.04 E �3 1.76 E �3 7.37 E �3 1.56 E �3

1.1k 1.46 E �2 2.27 E �3 1.24 E �2 2.06 E �3 9.18 E �3 1.74 E �3

0.1k 1.3k 3.06 E �3 1.04 E �3 2.88 E �3 9.93 E �4 2.71 E �3 9.48 E �4

0.2k 3.67 E �3 1.14 E �3 3.86 E �3 1.15 E �3 3.88 E �3 1.13 E �3

0.3k 1.32 E �2 2.16 E �3 1.27 E �2 2.09 E �3 1.17 E �2 1.97 E �3

Figure 4 (Scenario 1) Field profiles along the path joining the

source and the object. (Distance between the object and the inner

metamaterial layer ¼ 0.2k, Thickness of the metamaterial layer ¼1.3k). [Color figure can be viewed in the online issue, which is




Second, Maxwell’s curl-E equation in (10) is approxi-

mated in a similar manner. Together with the proper field

locations in (9), the update equations for the x- and

y-components of the magnetic flux field can be derived,

respectively, as follows:

Bxjnþ12

i;jþ12

¼ Bxjn�12

i;jþ12

� DtDy

Ezjni;jþ1�Ezjni;jh i

(16)

By

��nþ12

iþ12;j¼ By

��n�12

iþ12;jþ DtDx

Ezjniþ1;j�Ezjni;jh i

(17)

The update equations for the H-field can be computed

from (16) and (17), with the use of the constitutive rela-

tion, as follows:

Hxji;jþ12¼ lxxBxji;jþ1

2þlxyBy

��i;jþ1

2

(18)

Hy

��iþ1

2;j¼ lyxBxjiþ1

2;jþlyyBy

��iþ1

2;j

(19)

Note that we dropped the time index for generality,

because these equations are valid at any given time

instant, which is a half time step away from the integer

values of the time step.

In (18), although Bxji;jþ12is available in the FDTD gri

d, the term lxyBy

��i;jþ1

2

is unknown due to the field

Figure 5 (Scenario 2) Field maps: (a) Original problem (without staircasing error), (b) rotated original problem (staircased), and (c)

equivalent problem with metamaterial layer. (Thickness of the metamaterial layer ¼ 1.085k). [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com.]

TABLE III (Scenario 2) Error Values Comparing theFields Along the Path Between the Source and theObject (Time Step n 5 900)

Err1 Err2

Rotated Original problem (staircased) 0.0516 0.0181

Thickness of metamaterial layer ¼ 1.285k 0.0160 0.0101




The values are with respect to the original problem.



locations defined in (9). This term can be obtained by

interpolation in terms of Byjiþ12;j, Byji�1

2;j, Byjiþ1

2;jþ1 and

Byji�12;jþ1, and the material parameter lxy. Different inter-

polation schemes can be developed, but by referring to

the discussion in [23], the following interpolation

scheme is chosen to ensure that the FDTD algorithm is

stable.

lxyBy

��i;jþ1

2

¼ lxy��i;jþ1

By

��iþ1

2;jþ1þBy

��i�1

2;jþ1

2

þ lxy��i;j

By

��iþ1

2;jþBy

��i�1

2;j

2ð20Þ

Similarly, in (19), lyxBxjiþ12;j

must be calculated

through interpolation by using Bxji;jþ12, Bxji;j�1

2, Bxjiþ1;jþ1

2

and Bxjiþ1;j�12. For a stable FDTD algorithm, the term is

interpolated as follows:

lyxBxjiþ12;j¼ lyx

��iþ1;j

Bxjiþ1;jþ12þBxjiþ1;j�1

2

2

þlyx��i;j

Bxji;jþ12þBxji;j�1

2

2

(21)

We also mention that lxx and lyy in (18) and (19),

respectively, are determined at the positions where the cor-

responding H-fields are defined, viz., lxxji;jþ12and lyy

��iþ1

2;j.

IV. NUMERICAL SIMULATIONS

In this section, we present the results of a number of nu-

merical experiments to validate the performance of the

proposed technique in 2D TMz electromagnetic scattering

problems involving infinitely long cylindrical conducting

objects. All simulations are performed by means of our

FDTD code using uniform Cartesian grid. The computa-

tional domain is terminated with a PML absorber. The ex-

citation is driven with a sine wave at 3 GHz.

In each scenario given below, we simulate two cases:

(i) The original problem, (ii) The equivalent problem hav-

ing metamaterial region. To measure the performance of

the proposed method, we introduce two error criteria: (i) a

mean-square percentage error criterion as

Err1 ¼PX

Eez � Eo

z

�� 2PX

Eoz

�� 2 (22)

and, (ii) a root-mean square error criterion as

Err2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPX

Eez � Eo

z

�� 2N

vuut(23)

where Eze and Eo

z are the z-component of the electric field

calculated in equivalent and original problems, respec-

tively, in the computational domain X. Here, N is the

number of field points inside the domain of interest.

A. Scenario 1The objective of the first scenario is to measure the per-

formance of the equivalent problem, which is constructed

by a metamaterial region, over the original problem that

does not exhibit any staircasing error. We choose this sce-

nario to show the extent to which a metamaterial region

introduces errors in problems that are free of staircasing

errors.

In the original problem, we consider the problem of

scattering from a square object that conforms to a Carte-

sian grid (see Fig. 3). The edge length of the object is 2k.In the equivalent problem, we realize the same problem

with a metamaterial layer constructed conformally over

the object. The grid size is k/10 in all simulations. In

Tables I and II, we tabulate the error values at different

time instants by varying the location and thickness of the

metamaterial layer. In Table I, we compare the field val-

ues outside the metamaterial region, whereas in Table II,

we compare the field values inside the entire

Figure 6 (Scenario 2) Field profiles along the path joining the

source and the object. (Thickness of the metamaterial layer ¼1.085k). [Color figure can be viewed in the online issue, which is


TABLE IV (Scenario 3) Error Values for the Geometry inFigure 2

Figure 2b

Err1 Err2

Time step n ¼ 800 1.40 E �4 3.91 E �4

Time step n ¼ 1000 9.61 E �4 9.30 E �4

Time step n ¼ 1200 1.31 E �3 1.19 E �3



computational domain after transforming the field values

inside the metamaterial region by using the field equiva-

lence in (6). In Figure 3, we plot the field maps assuming

that the distance between the object and inner metamate-

rial layer is 0.3k, and the thickness of metamaterial layer

is 1.3k. In Figure 4, we plot the field values along the

path joining the source and the object as shown in Figure

3, assuming that the distance between the object and inner

metamaterial layer is 0.2k, and the thickness of metamate-

rial layer is 1.3k. This figure indeed illustrates the field

equivalence, in a way such that the desired fields (shown

by the black and red curves) can be recovered from the

fields inside the metamaterial region (shown by the blue

curve) after transformation.

The good agreement between the equivalent and origi-

nal problems illustrates the validity of the proposed

approach. Small error values observed in the simulations

are due to coarse grid size and FDTD modeling errors. We

observe that the results are improved by increasing the

thickness of the layer, because of better handling of spatial

variations of the constitutive parameters. In addition, as the

distance between the object and the metamaterial layer

increases for fixed thickness, the error values increase

because of faster spatial variations of the material parame-

ters. Furthermore, the error values increase as time pro-

gresses because of error accumulation occurring due to the

discretization of Maxwell’s equations at each time step. We

also note that the error values in Table II are slightly larger

than those in Table I, because of interpolating the field

points that do not match one-to-one in original and equiva-

lent problems after transformation. That is, to be able to

compare the fields of both cases at exact locations, we

interpolated the fields from neighboring available points of

the equivalent problem in transformed coordinates. In any

case, the error values are in acceptable levels.

B. Scenario 2The objective of the second scenario is to demonstrate

that the equivalent problem reduces the staircasing error

in a problem that does have staircasing error. For this

Figure 7 (Scenario 3) Field maps of original and equivalent problems at different time instants for the geometry in Figure 2. [Color

figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]



purpose, we simulate three cases: (i) Original problem

(free of staircasing error), involving a square object with

edge-length 2k, which conforms to the Cartesian grid (see

Fig. 5a), (ii) Rotated original problem (staircased), involv-

ing a square object that does not conform to the Cartesian

grid (see Fig. 5b), and (iii) Equivalent problem, which is

a realization of the rotated original problem with a meta-

material layer (see Fig. 5c). To compare these three cases,

we choose a path between the source and the object, as

shown in Figure 5. We expect that the equivalent problem

provides better results than the rotated original problem,

with respect to the original problem that does not possess

staircasing error. In Table III, we list the error values of

the staircased original problem and the equivalent problem

designed by the metamaterial layer. In all cases, the grid

size is set to k/10. We plot the field maps and the field

values along the path in Figures 5 and 6, respectively, to

compare the aforementioned three cases. These results

reveal the success of the proposed technique in decreasing

the staircasing error.

C. Scenario 3The objective of the last scenario is to measure the per-

formance of the equivalent problem over the original

problem, which is staircased on a ‘‘fine’’ Cartesian grid

(i.e., the staircasing error is reduced to a certain value by

adjusting the grid size sufficiently small). In this section,

we deal with the geometry in Figure 2, where the edge

lengths of the inner and outer layers of the matematerial

are set to 1k and 2k, respectively. In the original problem,

we simulate the object over a fine Cartesian grid whose

size is set to k/40. In the equivalent problem, we simulate

the same object covered by a metamaterial layer. In Table

IV, we list the error values at different time instants.

Finally, we plot the field maps in Figure 7. The results

demonstrate that the proposed approach can reliably be

used without the need for changing simple Cartesian grids.

Note that as the grid gets coarser, the difference between

the original and equivalent problems increases because the

original problem yields larger staircasing errors.

V. CONCLUSIONS

We have presented a specially defined coordinate trans-

formation technique to eliminate staircasing errors occur-

ring in the FDTD modeling of curved geometries that

are not conformal to the standard Cartesian grid. This is

achieved by embedding an anisotropic metamaterial

region, whose parameters are obtained by the coordinate

transformation, around the object to map the fields in the

close vicinity of the curved boundary to the metamaterial

region that is conformal to the Cartesian grid. We have

numerically explored the functionality of the technique

in various configurations with the aid of numerical simu-

lations and observed good agreements between the theo-

retical predictions and the numerical simulations. It is

also useful to note that the proposed coordinate transfor-

mation technique can be applied to transverse electric

(TE)z case and three-dimensional problems in a straight-

forward manner. In fact, in the TEz case, the implemen-

tation of the transformation will be the same, but the

dual equations will be formulated in the FDTD method

(i.e., Ez is replaced by Hz, etc.).

ACKNOWLEDGMENT

This work has been supported by the Scientific and Techni-

cal Research Council of Turkey (TUBITAK) (Project no:

109E169).

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BIOGRAPHIES

Ozlem Ozgun received the Ph.D.

degree in electrical engineering from

Middle East Technical University

(METU), Ankara, Turkey, in 2007.

She has been with the same univer-

sity since 2008. She will join the

TED University, Ankara, Turkey, in

February 2012. Her main research

interests are computational electromagnetics, finite ele-

ment method, domain decomposition, electromagnetic

propagation and scattering, metamaterials, and stochastic

electromagnetic problems.

Mustafa Kuzuoglu received the

B.Sc., M.Sc., and Ph.D. degrees in

electrical engineering from the Mid-

dle East Technical University

(METU), Ankara, Turkey, in 1979,

1981, and 1986, respectively. He is

currently a Professor in the same uni-

versity. His research interests include

computational electromagnetics, inverse problems, and

radars.



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