accuracy improved adi-fdtd methods

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS

Int. J. Numer. Model. 2007; 20:35–46Published online 20 November 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jnm.628

Accuracy improved ADI-FDTD methods

Iftikhar Ahmedz and Zhizhang (David) Chen*,y

Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, Canada

SUMMARY

FDTD method plays an important role for simulation of different structures in various fields ofengineering, such as RF/microwaves, photonics and VLSI. However, due to the CFL stability constraint,the FDTD time step is still small and the related CPU time is still large for modelling fine geometry wheresmall cell sizes are required to resolve fields. As a result, the unconditionally stable CFL-condition-freeADI-FDTD method is becoming a popular alternative to the FDTD method. The ADI-FDTD methodallows the use of larger time steps; however, it comes at the cost of larger errors. To mitigate the problem ofthese larger errors, in this paper we propose to modify the conventional ADI-FDTD algorithm. Themodifications are based on the fact that because the ADI-FDTD is a truncated form of the Crank–Nicolson (CN) method, the truncated terms can be re-introduced approximately into the ADI algorithmsto improve accuracy. Two accuracy-improved ADI-FDTD algorithms are derived and then validated fortwo-dimensional cases. Unfortunately, in the three-dimensional case the proposed methods are not foundto be unconditionally stable. Copyright # 2006 John Wiley & Sons, Ltd.

KEY WORDS: FDTD; unconditional stability; splitting error; CN-FDTD; ADI-FDTD

1. INTRODUCTION

Since the FDTD method was proposed in 1966 by Yee [1], significant advances have beenachieved using the method. Currently, several FDTD software packages are commerciallyavailable and many more are being developed. In spite of the progress, computational efficiencyof the FDTD method is still bounded by the well-known CFL stability condition that sets anupper limit for the FDTD time step. Such a limit is related to spatial steps or FDTD cell sizes.The smaller the cell sizes, the smaller the time step. Therefore, for fine structures where small cellsizes are required to resolve fields with a high degree of variance, time step is small and CPUtime is large. To overcome this problem, subgridding techniques in FDTD were developed, butthey face the late-time instability problem. Many other methods [2, 3] were introduced to make

*Correspondence to: Zhizhang (David) Chen, Department of Electrical and Computer Engineering, DalhousieUniversity, Halifax, NS, Canada.yE-mail: [email protected]: [email protected]

Copyright # 2006 John Wiley & Sons, Ltd.

FDTD computation more efficient, but the CFL condition persisted. The condition waseventually removed after the development of ADI-FDTD [4, 5], which is unconditionally stable.Because of the unconditionally stable nature of ADI-FDTD method, it is becoming a popularalternative to the conventional FDTD method. However, although the ADI-FDTD method iswithout the stability problem, its error also becomes large with large time steps. To reduce theerrors, improvements have been made [6–9]. In particular, other forms of ADI related FDTDschemes have been proposed [8, 9], but these methods take comparatively longer simulation timethan the conventional ADI-FDTD method.

In this paper, we propose two modified ADI-FDTD algorithms with reduced errors and withlittle additional computational burdens. They are based on the fact that the ADI-FDTD is atruncated form of the unconditionally stable Crank–Nicolson (CN) method [9], which has betteraccuracy than the pure ADI-FDTD method. Therefore, by re-introducing the truncated termsinto the ADI-FDTD method in an approximating manner, the modified ADI-FDTD methodswill have accuracy similar to that of the CN methods while maintaining the computationalefficiency of the ADI methods. In other words, the proposed methods are the approximate formof the CN method but their computation procedure is like that for the conventional ADI-FDTD method. The accuracy of these proposed methods will then be better than theconventional ADI-FDTD, but slightly less than that of the CN-FDTD method.

This paper is divided into following sections: in Section 2 formulations and numerical resultsof the 2D accuracy improved-ADI-FDTD methods are presented, in Section 3 formulations andnumerical results of the 3D accuracy improved-ADI-FDTD methods are given, and in Section 4conclusions are stated.

2. 2D ACCURACY-IMPROVED ADI-FDTD METHODS

The proposed methods are first formulated in two dimensions. For simplicity, a TE-to-z mode isconsidered and the related Maxwell’s equations are:

@Ex

@t¼

1

e@Hz

@y

� �ð1Þ

@Ey

@t¼ �

1

e@Hz

@x

� �ð2Þ

@Hz

@t¼

1

m@Ex

@y�@Ey

@x

� �ð3Þ

Equations (1)–(3) can be written in a matrix form as

@

@t

Ex

Ey

Hz

2664

3775 ¼

0 0@

e@y

0 0 �@

e@x@

m@y�

@

m@x0

266666664

377777775

Ex

Ey

Hz

2664

3775 ð4Þ

I. AHMED AND Z. CHEN36

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2007; 20:35–46

DOI: 10.1002/jnm

or simply

@U

@t¼ ½C�U ð5Þ

where U ¼ ½Ex;Ey;Hz�T

and

½C� ¼

0 0@

e@y

0 0 �@

e@y

@

m@y@

e@y0

2666666664

3777777775

ð6Þ

Equation (6) can be split into two matrices, [A] and [B], such that

@U

@t¼ ½A�U þ ½B�U ð7Þ

with

½A� ¼

0 0@

e@y

0 0 0

@

m@y0 0

26666664

37777775and ½B� ¼ �

0 0 0

0 0@

e@y

0@

m@y0

26666664

37777775

Replacement of derivatives with corresponding central finite differences and use of the averagingapproximation for (7) lead to [8]

Unþ1 �Un ¼ Dtð½A� þ ½B�ÞUnþ1 þUn

2

� �

or

½I � �Dt2½A� �

Dt2½B�ÞUnþ1 ¼ ð½I � þ

Dt2½A� þ

Dt2½B�ÞUn

� �ð8Þ

where [I] is the identity matrix.Equation (8) can be further factorized as

½I � �Dt2½A�

� �½I � �

Dt2½B�

� �Unþ1 ¼ ½I � þ

Dt2½A�

� �½I � þ

Dt2½B�

� �Un þ

Dt2

4½A�½B� Unþ1 �Un

� �

ð9Þ

Equation (9) can be computed in two steps with the introduction of the intermediate quantity,Utmp

Step 1:

½I � �Dt2½A�

� �Utmp ¼ ½I � þ

Dt2½B�

� �Un þ

Dt2


� �ð10Þ

ADI-FDTD METHODS 37


DOI: 10.1002/jnm

andStep 2:

½I � �Dt2½B�

� �Unþ1 ¼ ½I � þ

Dt2½A�

� �Utmp þ

Dt2


� �ð11Þ

Equations (10) and (11) are essentially the Crank–Nicolson formulations [8] that have betteraccuracy than the conventional ADI-FDTD method [6]. However, Equations (10) and (11) areimplicit and not easy to solve (because quantities to be updated also appear on the right-handsides). In the following paragraphs, we propose two approximate computational methods.

Method # 1: To make the computation explicit on the right-hand side of Equations (10) and(11), the term Dt2=4 ½A�½B�ðUnþ1 �UnÞ in Equation (10) and (11) is modified with the followingapproximating equations Un ¼ Unþ1=2 þUn�1=2=2 and Unþ1=2 ¼ Unþ1 þUn=2; respectively.Here we suppose that tmp ¼ nþ 1

2and Utmp ¼ Unþ1=2:

As a result, (10) and (11) are approximated with the following equations:Step 1:

½I � �Dt2½A�

� �Unþ1=2 ¼ ½I � þ

Dt2½B�

� �Un þ

Dt2

4½A�½B� Un �Un�1=2

� �ð12Þ

andStep 2:

½I � �Dt2½B�

� �Unþ1 ¼ ½I � þ

Dt2½A�

� �Unþ1=2 þ

Dt2

4½A�½B� Unþ1=2 �Un

� �ð13Þ

Method # 2: In this method, to make the computation explicit on the right-hand side ofEquations (10) and (11), the term Dt2=4 ½A�½B�ðUnþ1 �UnÞ in equation (9) is modified with thefollowing approximating equation:

Un ¼Unþ1 þUn�1

2ð14Þ

Here we suppose that tmp=n+1. Now Equations (10) and (11) are modified asStep 1:

½I � �Dt2½A�

� �Utmp ¼ ½I � þ

Dt2½B�

� �Un þ

Dt2

8½A�½B� Un �Un�1� �

ð15Þ

Step 2:

½I � �Dt2½B�

� �Unþ2 ¼ ½I � þ

Dt2½A�

� �Utmp þ

Dt2

8½A�½B� Utmp �Un

� �ð16Þ

In the context of TE modes, the above two methods can be more specific. For instance, withMethod # 1, the field equations are:

Step1:

Enþ1=2x ¼ En

x þDt2e

@

@yHnþ1=2

z �Dt2

4me@

@x@yðEn

y � En�1=2y Þ ð17Þ

Enþ1=2y ¼ En

y �Dt2e

@

@xHn

z ð18Þ



DOI: 10.1002/jnm

Hnþ1=2z ¼ Hn

z þDt2m

@

@yEnþ1=2x �

Dt2m

@

@xðEn

yÞ ð19Þ

andStep 2:

Enþ1x ¼ Enþ1=2

x þDt2e

@

@yHnþ1=2

z �Dt2

4me@

@x@yðEnþ1=2

y � Eny Þ ð20Þ

Enþ1y ¼ Enþ1=2

y �Dt2e

@

@xHnþ1

z ð21Þ

Hnþ1z ¼ Hnþ1=2

z þDt2m

@

@yEnþ1=2x �

Dt2m

@

@xðEnþ1

y Þ ð22Þ

In the ADI-FDTD format, the above equations can be discretized and shown asStep 1:

Exnþ1=2iþ1=2;j

��¼ Exjniþ1=2;jþ

Dt2eDy

Hzjnþ1=2iþ1=2;jþ1=2�Hzj

nþ1=2iþ1=2;j�1=2

� ��

Dt2

4meDxDy

� Ey

��niþ1;jþ1=2�Ey

��niþ1;j�1=2�Ey

��ni;jþ1=2þEy

��ni;j�1=2�Ey

��n�1=2iþ1;jþ1=2þEy

��n�1=2iþ1;j�1=2�Ey

��n�1=2i;jþ1=2þEy

��n�1=2i;j�1=2

� �

ð23Þ

Ey

��nþ1=2i;jþ1=2¼ Ey

��ni;jþ1=2�

Dt2eDx

Hzjniþ1=2;jþ1=2�Hzjni�1=2;jþ1=2� �

ð24Þ

Hzjnþ1=2iþ1=2;jþ1=2¼ Hzjniþ1=2;jþ1=2þ

Dt2mDy

Exjnþ1=2iþ1=2;jþ1�Exj

nþ1=2iþ1=2;j

� ��

Dt2mDx

Ey

��niþ1;jþ1=2�Ey

��ni;jþ1=2

� �ð25Þ

andStep 2:

Exjnþ1iþ1=2; j

¼ Exjnþ1=2iþ1=2;jþ

Dt2eDy

Hzjnþ1=2iþ1=2;jþ1=2�Hzj

nþ1=2iþ1=2;j�1=2

� ��

Dt2

4meDxDy

� Ey

��nþ1=2iþ1;jþ1=2�Ey

��nþ1=2iþ1;j�1=2�Ey

��nþ1=2i;jþ1=2þEy

��nþ1=2i;j�1=2�Ey

��niþ1;jþ1=2þEy

��niþ1;j�1=2�Ey

��ni;jþ1=2þEy

��ni;j�1=2

� �

ð26Þ

Ey

��nþ1i;jþ1=2¼ Ey

��nþ1=2i;jþ1=2�

Dt2eDx

Hzjnþ1iþ1=2;jþ1=2�Hzjnþ1i�1=2;jþ1=2

� �ð27Þ

Hzjnþ1iþ1=2;jþ1=2¼ Hzjnþ1=2iþ1=2;jþ1=2þ

Dt2mDy

Exjnþ1=2iþ1=2;jþ1�Exj

nþ1=2iþ1=2;j

� ��

Dt2mDx

Ey

��nþ1iþ1;jþ1=2�Ey

��nþ1i;jþ1=2

� �ð28Þ

ADI-FDTD METHODS 39


DOI: 10.1002/jnm

It can be seen from Equations (17) and (20) that each contains one extra term as compared tothe conventional ADI-FDTD method [4], where it is truncated. These two terms are not exactlythe same as the truncated terms described in [6] but approximate them. Therefore, they areresponsible for the accuracy improvement of the proposed methods. For Method #2, similarequations and results can be obtained.

2.1. Numerical results for the two dimensional case

In the following paragraphs, numerical results of the proposed Methods #1 and #2 arepresented and compared with the conventional ADI-FDTD and FDTD methods. The structureunder consideration consisted of two parallel plates of zero thickness in free space. Each platewas 2m long and had a distance of 0.2m in between the two plates. The geometry is shown inFigure 1. The plates were enclosed by perfect magnetic walls (PMW) on all four sides. The cellsizes for both methods in each direction were 0.2m. This structure is the same as that used in [6].The raised cosine waveform with frequency 750 kHz was used as a source.

Figure 2 shows the computed electric field Ey along the x-axis with different values of CFLfactor ‘s’ (which is the ratio of time step to the CFL limit). As can be seen, for s415, the resultsobtained with Method # 1 are the same as those obtained with the conventional ADI-FDTDmethod at s ¼ 0:5: The results with Method # 2 are similar but have much larger error.

Figure 3 shows the relative errors of the three methods (the conventional ADI-FDTD,Method #1 and Method #2). In computing the relative error, a separate simulation of thestructure using the conventional FDTD was run and the results were used as the reference. Ascan be seen from Figure 3, all the methods have similar errors at s40.5. However, with theincrease of the CFL factor (or time step), the errors of Method #2 increase but it is less than thatof the conventional ADI-FDTD method. On the other hand, the errors of Method #1 are sosmall that they are invisible. In other words, Method #1 presents the same results as thereference method (conventional FDTD), but it used a time step much larger than the CFL limitand as a result saved simulation time.

Table I shows the computer resources used by the conventional ADI-FDTD method and theproposed methods. It shows that simulation time used by Method #1 and Method #2 isapproximately the same as that used by the conventional ADI-FDTD with slightly largermemory requirements for the proposed methods.

Figure 1. Geometry under studies: parallel conducting plates.



DOI: 10.1002/jnm

Figure 4 shows simulation time vs the CFL factors. It is clear from this plot that thesimulation time taken by the proposed methods is almost the same as that of the conventionalADI-FDTD for different CFL factors.

3. 3D ACCURACY-IMPROVED ADI-FDTD METHODS

By following the procedure presented in Section 2, the formulations of the 3D accuracy-improved ADI-FDTD methods can be derived. In a linear and lossless medium, the

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

Position along x-axis (m)

Ey

(v/m

)

FDTD s=0.5Conv.ADI-FDTD s=0.5Method#2 s=0.5Method#1 s=0.5Conv.ADI-FDTD s=5Method#2 s=5Method#1 s=5Conv.ADI-FDTD s=15Method#2 s=15Method#1 s=15

Figure 2. Electric field Ey obtained with FDTD, ADI-FDTD and proposed methods.

0

5

10

15

20

0 5 10 15 20

25

30

CFL factor 's'

Rel

ativ

e E

rror

Method#1

Method#2

Conv.ADI-FDTD

Figure 3. Relative errors vs CFL factors.

ADI-FDTD METHODS 41


DOI: 10.1002/jnm

corresponding 3D Maxwell’s equations can be written in a matrix form as

@

@t

Ex

Ey

Ez

Hx

Hy

Hz

2666666666664

3777777777775

0 0 0 0 �@

e@z@

e@y

0 0 0@

e@z0 �

@

e@x

0 0 0 �@

e@y@

e@x0

0@

m@z�

@

m@y0 0 0

�@

m@z0

@

m@x0 0 0

@

m@y�

@

m@x0 0 0 0

266666666666666666666664

377777777777777777777775

Ex

Ey

Ez

Hx

Hy

Hz

2666666666664

3777777777775

ð29Þ

or

@U1

@t¼ ½A1�U1 þ ½B1�U1 ð30Þ

Table I. Computational resources memory used by the conventional ADI-FDTD and the proposedADI-FDTD methods.

ConventionalADI FDTD

ProposedMethod #1

ProposedMethod #2

Time(s)

Memory(MB)

Time(s)

Memory(MB)

Time(s)

Memory(MB)

CFLfactor ‘s’

No. ofiterations

43 1.336 43 1.344 43 1.344 0.5 20004 1.336 4 1.344 4 1.344 5 2001 1.336 1 1.344 1 1.344 10 10051 1.336 51 1.344 51 1.344 15 67

0

10

20

30

40

50

0 5 10 15 20CFL Factor ('s')

Tim

e (s

ec)

Method#1 & #2

Conv.ADI-FDTD

Figure 4. Simulation time vs CFL factors.



DOI: 10.1002/jnm

where U1 ¼ ½Ex;Ey;Ez;Hx;Hy;Hz�T

A1½ � ¼

0 0 0 0 0@

e@y

0 0 0@

e@z0 0

0 0 0 0@

e@x0

0@

m@z0 0 0 0

0 0@

m@x0 0 0

@

m@y0 0 0 0 0

266666666666666666666664

377777777777777777777775

and B1½ � ¼

0 0 0 0 �@

e@z0

0 0 0 0 0 �@

e@x

0 0 0 �@

e@y0 0

0 0 �@

m@z0 0 0

�@

m@z0 0 0 0 0

0 �@

m@x0 0 0 0

266666666666666666666664

377777777777777777777775

Equation (30) is like Equation (7) for the 2D formulations with a difference in the ranks of thematrices. In the 3D case, the matrices are of 6� 6 instead of 3� 3 of the 2D case. Formulationsfor Method #1 and Method #2 in the 3D case can then be obtained by following the sameprocedure as for the 2D case.

For example, the formulations for Method #2 are as follows.Step 1:

Etmpx ¼ En

x þDt2e@Htmp

z

@y�

Dt2e

@Hny

@z�

Dt2

8me@2

@x@yðEn

y � En�1y Þ ð31aÞ

Etmpy ¼ En

y þDt2e@Htmp

x

@z�

Dt2e@Hn

z

@x�

Dt2

8me@2

@y@zðEn

z � En�1z Þ ð31bÞ

Etmpz ¼ En

z þDt2e

@Htmpy

@x�

Dt2e@Hn

x

@y�

Dt2

8me@2

@x@zðEn

x � En�1x Þ ð31cÞ

Htmpx ¼ Hn

x þDt2m

@Etmpy

@z�

Dt2m@En

z

@y�

Dt2

8me@2

@x@zðHn

z �Hn�1z Þ ð31dÞ

Htmpy ¼ Hn

y þDt2m@Etmp

z

@x�

Dt2m@En

x

@z�

Dt2

8me@2

@x@yðHn

x �Hn�1x Þ ð31eÞ

Htmpz ¼ Hn

z þDt2m@Etmp

x

@y�

Dt2m

@Eny

@x�

Dt2

8me@2

@y@zðHn

y �Hn�1y Þ ð31fÞ

ADI-FDTD METHODS 43


DOI: 10.1002/jnm

andStep 2:

Enþ2x ¼ Etmp

x þDt2e@Htmp

z

@y�

Dt2e

@Hnþ2y

@z�

Dt2

8me@2

@x@yðEnþ1

y � EnyÞ ð32aÞ

Enþ2y ¼ Etmp

y þDt2e@Htmp

x

@z�

Dt2e@Hnþ2

z

@x�

Dt2

8me@2

@y@zðEnþ1

z � Enz Þ ð32bÞ

Enþ2z ¼ Etmp

z þDt2e

@Htmpy

@x�

Dt2e@Hnþ2

x

@y�

Dt2

8me@2

@x@zðEnþ1

x � EnxÞ ð32cÞ

Hnþ2x ¼ Htmp

x þDt2m

@Etmpy

@z�

Dt2m@Enþ2

z

@y�

Dt2

8me@2

@x@zðHnþ1

z �Hnz Þ ð32dÞ

Hnþ2y ¼ Htmp

y þDt2m@Etmp

z

@x�

Dt2m@Enþ2

x

@z�

Dt2

8me@2

@x@yðHnþ1

x �HnxÞ ð32eÞ

Hnþ2z ¼ Htmp

z þDt2m@Etmp

x

@y�

Dt2m

@Enþ2y

@x�

Dt2

8me@2

@y@zðHnþ1

y �Hny Þ ð32fÞ

0 200 400 600 800 1000 1200 1400 1600 1800 2000-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1x 10306

t (ps)

Ele

ctric

fie

ld E

y

Figure 5. Results for Ey obtained with the 3D formulations.



DOI: 10.1002/jnm

With replacements of the derivatives with their corresponding finite differences and use ofaveraged field quantities, the accuracy-improved ADI-FDTD formulations can be obtained ina form similar to Equations (23)–(28).

As can be seen from Equations (31) and (32), in comparison with the conventionalADI-FDTD method, all twelve equations have one extra term that is proportional to square ofthe time step and the difference between field values of the two previous time steps. They are themodified terms that improve the accuracy of the methods. Similar processes can be followed forMethod #1.

3.1. Numerical results for 3D case

To test the 3D formulations, a cavity with dimensions of 9mm� 6mm� 15mm was considered.Numerical cell sizes were taken to be 0.6mm in all three directions. A Gaussian pulse was usedas a source. Figure 5 shows the simulation results for the 3D case. Unfortunately, the 3Dsolutions become unstable after a few hundred iterations. At the time of this report, we have notbeen able to find the reasons and a way to obtain stable solutions.

4. CONCLUSIONS

In this paper, we proposed the accuracy-improved ADI-FDTD methods. They not only retainthe same numerical computational efficiency as the conventional ADI-FDTD method, but alsoachieve accuracy similar to that with the CN-FDTD. The simulation time and the memorytaken by the proposed methods are approximately the same as those for the conventionalADI-FDTD with only minor increases in memory usage. In particular, the first method presentssuperior performance with the large time steps. Unfortunately, when applied in threedimensions, the proposed methods become unstable. In spite of this, the work presented inthis paper has presented a direction to further improve and extend the usages of the ADI-FDTDmethod.

REFERENCES

1. Yee KS. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media.IEEE Transactions on Antennas and Propagation 1966; AP-14:302–307.

2. Taflove A. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House: Boston,1995.

3. Taflove A, Hagness SC. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House:Boston, 2000.

4. Zheng F, Chen Z, Zhang J. Toward the development of a three-dimensional unconditionally stable finite-differencetime-domain method. IEEE Transactions on Microwave Theory and Technology 2000; 48(9):1550–1558.

5. Namiki T. A new FDTD algorithm based on alternating-direction implicit method. IEEE Transactions onMicrowave Theory and Technology 1999; 47(10):2003–2007.

6. Garcia SG, Lee T-W, Hagness SC. On the accuracy of the ADI-FDTD method. IEEE Antennas and WirelessPropagation Letters 2002; 1:31–34.

7. Wang S. On the current source implementation for the ADI-FDTD method. IEEE Microwave and WirelessComponents Letters 2004; 14(11):513–515.

8. Wang S, Teixeira FL, Chen J. An iterative ADI-FDTD with reduced splitting error. IEEE Microwave and WirelessComponents Letters 2005; 15(2):92–94.

9. Sun G, Trueman CW. Approximate Crank–Nicolson schemes for the 2-D finite difference time domain method forTEz waves. IEEE Transactions on Antennas and Propagation 2004; 52(11):2963–2972.

ADI-FDTD METHODS 45


DOI: 10.1002/jnm

AUTHORS’ BIOGRAPHIES

Iftikhar Ahmed received the BSc Electrical Engineering degree from the University ofEngineering and Technology Taxila, Pakistan, in 1995, the MSc ElectricalEngineering degree from University of Engineering and Technology, Lahore,Pakistan, in 1999, and the PhD degree in Electrical Engineering from the DalhousieUniversity, Halifax, NS, Canada in 2006. His research interests include computa-tional electromagnetics, RF circuit design, numerical modelling of RF/Microwavestructures, and wireless communication system design.

Zhizhang (David) Chen received the BEng degree from Fuzhou University, Fuzhou,China in 1982, the MASc degree from Southeast University, Nanjing, China in 1992,and the PhD degree from the University of Ottawa, Ottawa, ON, Canada in 1992.From January of 1993 to August of 1993, he was a Natural Science and EngineeringResearch Council (NSERC) Post-Doctoral Fellow with the Department of Electricaland Computer Engineering, McGill University, Montreal, Quebec, Canada. In 1993,he joined the Department of Electrical and Computer Engineering, DalhousieUniversity, Halifax, NS, Canada, where he is presently a full Professor and a KillamChair in Wireless Technology. He was the recipient of the 2005 Association ofProfessional Engineers of Nova Scotia (APENS) Engineering Award and the 2006Dalhousie Student Union (DSU) Teaching Excellence Award in the Category of

Faculty of Graduate Studies. He has authored and coauthored over 120 journal and conference papers aswell as industrial reports in the areas of computational electromagnetics and RF/microwave electronics forwireless communications. His current research interests include numerical modelling and simulation, RF/microwave electronics, smart antennas, and wireless transceiving technology.



DOI: 10.1002/jnm

accuracy improved adi-fdtd methods

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