Proceedings CEEM'2009/Xi'an

PML Implementation for WLP-FDTD inCylindrical Coordinates

Yin Qin, Chen Bin, Xiong Run, Yang Xiaoshuan, Zhou BihuaLab ofElectromagnetics, Nanjing Engineering Institute, No.1 Haifuxiang, Nanjing 210007, China

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Abstract: In this paper, a perfectly matchedlayer (PML) implementation for anunconditionally stable finite-differencetime-domain scheme based on weightedLaguerre polynomials(WLP-FDTD) incylindrical coordinates is presented. Anumerical example is introduced to validatethe proposed formulations.

Key Words: unconditionally stable FDTD,weighted · Laguerre polynomials(WLP),Cylindrical coordinates, PML

II Mathematical Formulation

Using the complex co-ordinate stretchingtechnique, the axis-symmetrical TM modelformulation in cylindrical coordinates is:

aHJ'01&E =__<fJ

r az. H", aH",

J01&Ez =---+--r ar

. H aEz aErJ01p ", =-----ar az

Where ~ and ~are complex co-ordinatevariables:

q vEzr + CTr E =..!..... vB" (9)r at 8

0zr 8

0ar

.; vB"r + CTr B =l- vEz (10)r at &0 ",r Po ar

P ~ p = Po +JP sp(p~dp'Po

z ~ Z = Zo +Jz sz(z~dz'Zo

From (1)~(3),we can get:aH

j01&E =--'" (4)r szaz

H aHj01&E =-'"+-'" (5)

z s", srar

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Where: j: a,s, =~r+-.-

J01&o

j: O'zSz =~z +-.-

J01&o

0'S =~ +-"'-", ", jtoe,

Splitting the field E and H, using theinverse Fourier transform relation, (4)~(6)can be written in the time domain as:

q vEr + CTz E =_i vB" (7)z at &0 r e az

ee; O'ip Hip (8)~",-+-Ezip-

at &0 &0

I Introduction

In many applications, we have to confrontthe cylindrical structures such as in opticalfiber communications, integrated optics,defense industry, and geophysicalexplorationl'l, If we adopt the conventionalFDTD method to discretize the cylindricalstructure with a uniform Cartesian grid, asignificant staircasing error appears. So, theFDTD algorithm with cylindrical grid todiscretize the computational domain isdeveloped'<'. To over come the difficulty ofCFL stability condition in fine structures, anunconditionally stable FDTD with weightedLaguerre polynomials (WLP-FDTD) wasproposed'<'. For open-region problems,efficient absorbing boundary conditions(ABCs) are needed to truncate thecomputational domains. The perfectlymatched layer (PML) is very efficient andpopular for grid truncation of open-regionproblemsi".

In this paper, a new split-field PMLformulations for the WLP-FDTD incylindrical coordinates is derived. Tovalidate the proposed method, a numericalexample in cylindrical is introduced.

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Proceedings CEEM'2009/Xi'an

.; fJHrpz + CTz H =_.£ fJEr (11)z at 8

0ipZ Po az

where: s, =Ezip +Ezr' Hip =Hipr +Hipz

With reference to Chung et a1.[3], usingthe basis weighted Laguerre polynomial, thetemporal coefficients in can be expanded as:

Er(r,t) = fE;(r)rp/i) (12)p=o

00

Ezr(r,t) ='LE~(r)rpp(f) ( 13)p=o

00

Eztp(r,t) = 'LE:tp(r)rpp(f) ( 14)p=o

-HI 2itl - 2 (T, i,j = (l + 2(T,(j) )r i,j - SPo;, (i)tlr(l + 2(T,(i~ ) , S&o;,(j)

S&O;r(l)

-HI 2 -HI 2D, i,j = SPo;z(j)!!z(l + 2(Tz(j~) a, i,j = (l + 2(Tz(j~)

S&o;z (J) , S&O;z(J)

Equations (17}···{21) are implicitdifference equations, inserting ( 20 ) and (21 )into (17)-{19), we get equations (22)" (23) .

The singularities in z axis can be modified

by the Ampere's law[5].

Hrpz(r,t) = 'fH;z(r)rp/T)p=o

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q I -HI ( ql ql)H 1 1 =-Dz . E 1 -E 1ipz i+-,j+- i+.!. J'+.!. r i+-,j+l r i+-,j2 2 2' 2 2 2 (21 )

Iq - l

-H I- 0'z . 1 . i ' L H;z i+.!. '+.!.

1+'2')+'2 k=O,q>O 2') 2

Htpr(r,t) = fH;'(r)rp/T) (16)p=o

Then, we can get:

Eqll =-15:1 .(Hqll 1-Hqll I).r i+"2,j i+~,j 9' i+"2,j+"2 9' i+"2,j-"2 ( 17 )

Then, we can get:

(8 S +_1_)Eql __1_Eql16 psb.r2 z i,j+f psb.r2

Z i+J,j+f

+ 1 Eql 1 Eql (25)ps~~ r i+f,j+J ps~~ r i+-t,j

=-irl - &8 ~ Eql _.£ ~ Hq!8 Z i,j+f 8 k=O,q>O z i,j+-f ~ k=O,q>O ip i+-f,j+-f

III Numerical Results

In this section, numerical examples areimplemented to validate the proposedsplit-PML formulations. A sinusoidallymodulated Gaussian pulse is used",

t-TJi(t) =exp(-(_C)2)sin(21rfc(t- ~» (26)

t;Where t; =o.5fc,~=3Td ,!c =lGHz.

And we choose:Tf =10ns,B=5GHz,NL =150,s=3x10Jo

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The computational domain is subdividedinto a 30 x 30 lattice along the rand z

direction respectively, with~ = tJ.r = & = O.Olm. The electric current ofsinusoidally modulated Gaussian pulse islocated on the Z axis. The computationaldomain is truncated by four additional PMLlayers. Fig.1 shows the electric fields andmagnetic fields at measurement points. Theagreement between the first order Mur ABCmethod and the proposed method is verygood. Thus the proposed method is valid.Fig.2 shows the relative reflectioncoefficients of the proposed formulation.

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Iq - l-E

-0' . E k

rp i,j+i k=f.;>o zrp t,j+i

I -EI (I I)Eq 1 =Dr · Hq 1 1 - H". 1 1zr i,j+- . . 1 ip i+- j+- ip i-- j+-

2 ")+"2 2' 2 2' 2

I -EI ( )E: 1 = Dip . Hq 1 1 +Hq 1 1ip i )'+- .. 1 rp t+- )'+- rp t-- )'+-

, 2 ")+"2 2' 2 2' 2

1

q - l-E I

-O'r . L E:,. 1, . 1 1 +-")+"2 k=O,q>O ,) 2

I -HI (I I)H': 1 1 = Dr . E" 1 - Eq 1ipr i+-,j+- i+.!. J'+.!. z i+l,j+- z i,j+-2 2 2' 2 2 2

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Proceedings CEEM'2009/Xi'an

References

[1] Q. H. Liu, "Diffraction of non axisymmetric

waves in cylindrically layered media by horizontal

discontinuities," Radio Sci., vol. 27, pp.569-58l,

1992.

[2] Jiang-Qi He, "A Nonuniform Cylindrical

FDTD Algorithm with Improved PML and

Quasi-PML Absorbing Boundary Conditions"

IEEE TRANSACTIONS ON GEOSCIENCE

AND REMOTE SENSING, YOLo 37, NO.

2,pp:1066-l0n,MARCH 1999

[3] Y. S. Chung, T. K. Sarkar, B. H. Jung, and M.

Salazar-Palma, "An unconditionally stable scheme

for the finite-difference time-domain method,"

IEEE Trans. Microw. Theory Tech., vol. 51, no. 3,

pp. 697-704, Mar. 2003.

[4] J. P. Berenger, "A perfectly matched layer for

the absorption of electromagnetic waves," J.

Comput. Phys., vol. 114, pp. 185-200, Oct. 1994.[5] Zhang Xun-li, Hang Hong-xin and Song

Feng-hong, " Implementation of a perfectly

matched layer(PML) in three complex

coordinates", Journal of Harbin Institute of

Technology,Yol.37,

No.11,pp:1549-1 551,Nov.2005

8

........M.Jr

3 6

Time(ns)

(;:'\ t!\.:'\! ; \r""'; )! \:'

, !;i:~

o

Fig.2 Reflection Coefficient

-1

Fig.l Numerical Results

time

o.

-20

-120

-100

Acknowledgements

This work is supported by NationalScience Foundation of China(No.6067I 007) .

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-EI+D", .. 1 •I,j+"2

-EI+D", .. 1 •I,j+"2

-EI+Dr .. 1·I,j+"2

-EI«D, .. 1·I,j+"2

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