research article frequency-dependent fdtd algorithm...

Research ArticleFrequency-Dependent FDTD Algorithm UsingNewmark’s Method

Bing Wei, Le Cao, Fei Wang, and Qian Yang

School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China

Correspondence should be addressed to Le Cao; xidian [email protected]

Received 1 July 2014; Accepted 23 August 2014; Published 13 October 2014

Academic Editor: Tat Soon Yeo

Copyright © 2014 Bing Wei et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

According to the characteristics of the polarizability in frequency domain of three commonmodels of dispersivemedia, the relationbetween the polarization vector and electric field intensity is converted into a time domain differential equation of second orderwith the polarization vector by using the conversion from frequency to time domain. Newmark 𝛽𝛾 difference method is employedto solve this equation.The electric field intensity to polarizability recursion is derived, and the electric flux to electric field intensityrecursion is obtained by constitutive relation. Then FDTD iterative computation in time domain of electric and magnetic fieldcomponents in dispersive medium is completed. By analyzing the solution stability of the above differential equation using centraldifference method, it is proved that this method has more advantages in the selection of time step. Theoretical analyses andnumerical results demonstrate that this method is a general algorithm and it has advantages of higher accuracy and stability overthe algorithms based on central difference method.

1. Introduction

In 1966, the finite difference time domain (FDTD) method[1] is put forward by Yee. With the further development, theFDTD method probably becomes the most popular numeri-cal technique for solving complex electromagnetic problems[1–8]. FDTD is a time domain algorithm but the constitutiverelation of dispersivemedium is given by a frequency domainexpression. Hence, it is difficult to use FDTDmethod directlyin analyzing electromagnetic characteristics of dispersivemedium.According to the different treatments of constitutiverelation, the FDTD methods used in dispersive medium canbe classified as RC method [5, 6], JEC method [7], PLJERCmethod [8], ADE method [9], 𝑍 transform method [10–12],SO method [13–15], SARC method [16], and so on.

In the case of nondispersive medium, the spatial dis-cretization of Maxwell curl equation can be determined bythe requirement from numerical dispersion to stability, andthe time discrete can be determined by Courant stabilitycondition. So the calculation involves just the first orderpartial differential equation. The time domain expression of

constitutive relation often can be written as a second orderpartial differential equation in the case of dispersive medium.To obtain an accurate solution of the partial differentialequation, time step must be very small. In this paper, thegeneral differential equation of dispersive medium is givenand the Newmark method [17–19] is introduced to thetreatment of the constitutive relation in dispersive medium.This new method has a higher accuracy and stability thanthe central difference method. The detailed process is asfollows.

According to the polarizability characteristics in fre-quency domain of three common models (Debye, Drude,and Lorentz models) of dispersive medium, the relation infrequency domain betweenP andE is converted into a secondorder differential equation (has a general form to the threecommon models) by using the conversion 𝑗𝜔 → 𝜕/𝜕𝑡 fromfrequency to time domain. Then the Newmark 𝛽𝛾 methodwhich is used in FETD is employed to solve the equation.Hence the E → P recursion is derived, and the D → Erecursion is obtained by constitutive relation.

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014, Article ID 216763, 6 pageshttp://dx.doi.org/10.1155/2014/216763

2 International Journal of Antennas and Propagation

2. The General Newmark-FDTD Method

2.1. Maxwell Equation in Dispersive Medium and ConstitutiveRelation. Maxwell curl equation in linear isotropic dispersivemedium is

𝜕D𝜕𝑡

= ∇ ×H,

𝜕H𝜕𝑡

= −1

𝜇0

∇ × E.(1)

The electric constitutive relation is

D (𝜔) = 𝜀 (𝜔)E (𝜔) = 𝜀0[𝜀∞+ 𝜒 (𝜔)]E (𝜔) , (2)

where 𝜀(𝜔) and 𝜒(𝜔) are permittivity coefficient and polar-izability in frequency domain, respectively. 𝜀

∞is the relative

permittivity when frequency is infinite.The FDTD iterative computation of E → H → D can

be accomplished by the discretization of (1) according to thestanding Yee cell. And it is needed to deal with constitutiverelation in the iterative computation ofD → E.

2.2. The United Expression of Polarizability Function of Com-mon Dispersive Medium Models. Three common dispersivemediummodels are Debyemodel, Drudemodel, and Lorentzmodel [2]. Their polarizabilities can be written as [20]

𝜒Debye (𝜔) =𝐿

∑𝑙=1

𝜒Debye𝑙

(𝜔) =𝐿

∑𝑙=1

Δ𝜀𝑙

1 + 𝑗𝜔𝜏0,𝑙

,

𝜒Drude (𝜔) =𝐿

∑𝑙=1

𝜒Drude𝑙

(𝜔) =𝐿

∑𝑙=1

−𝜔2𝑙

𝜔2 − 𝑗𝜔]𝑐,𝑙

,

𝜒Lorentz (𝜔) =𝐿

∑𝑙=1

𝜒Lorentz𝑙

(𝜔) =𝐿

∑𝑙=1

Δ𝜀𝑙𝜔20,𝑙

𝜔20,𝑙+ 2𝑗𝜔]

𝑐,𝑙− 𝜔2

,

(3)

where 𝐿 is numbers of poles, 𝑙 represents the 𝑙th pole, Δ𝜀𝑙=

𝜀𝑠,𝑙− 𝜀∞, 𝜀𝑠,𝑙

is the relative permittivity at zero frequency,𝜏0,𝑙

is the relaxation time of pole, ]𝑐,𝑙

is collision frequency,𝜔𝑙is Drude frequency, and 𝜔

0,𝑙is the natural frequency of

oscillator.Let

𝜉𝑙(𝜔) =

1

𝜒𝑙 (𝜔)

. (4)

𝜉𝑙(𝜔) of the three dispersive medium models in (3) can be

written as a united form:

𝜉𝑙(𝜔) = 𝑀

𝑙⋅ (𝑗𝜔)

2+ 𝐶𝑙⋅ (𝑗𝜔) + 𝐾

𝑙, (5)

where

𝑀𝑙= 0, 𝐶

𝑙=𝜏0,𝑙

Δ𝜀𝑙

, 𝐾𝑙=

1

Δ𝜀𝑙

Debye

𝑀𝑙=

1

𝜔2𝑙

, 𝐶𝑙=]𝑐,𝑙

𝜔2𝑙

, 𝐾𝑙= 0 Drude

𝑀𝑙=

1


, 𝐶𝑙=

2]𝑐,𝑙


, 𝐾𝑙=

1

Δ𝜀𝑙

Lorentz.

(6)

2.3. Constitutive Relations in Time Domain and Its Newmark’sSolution. Polarization vector P is introduced:

P (𝜔) = 𝜒 (𝜔)E (𝜔) . (7)

The constitutive relation equation (2) can be rewritten as

D (𝜔) = 𝜀0𝜀∞E (𝜔) + 𝜀0P (𝜔) . (8)

Let

P𝑙(𝜔) = 𝜒

𝑙(𝜔)E (𝜔) . (9)

Then from (3) and (7) we know that

P (𝜔) =𝐿

∑𝑙=1

P𝑙(𝜔) . (10)

From (4) and (9), we have

E (𝜔) = 𝜉𝑙(𝜔)P𝑙(𝜔) . (11)

After substituting (5) into (11) and using conversion 𝑗𝜔 →𝜕/𝜕𝑡 from frequency to time domain, we have

E (𝑡) = 𝑀𝑙

𝑑2

𝑑𝑡2P𝑙(𝑡) + 𝐶

𝑙

𝑑

𝑑𝑡P𝑙(𝑡) + 𝐾

𝑙P𝑙(𝑡) . (12)

E(𝑡) in the above differential equation with second order timederivative is equivalent to a excitation source. In the year of1959, a time-stepping algorithm for solving the differentialequations like (12) is given by Newmark according to thetime derivative difference approximate solution of Taylorseries and the mean value theorem. This algorithm provideshigher accuracy and stability [17]. By the deriving method ofZienkiewicz, solution of (12) is [18]

P𝑛+1𝑙

= 𝑤1,𝑙P𝑛𝑙+ 𝑤2,𝑙P𝑛−1𝑙

+ 𝑢0,𝑙E𝑛+1 + 𝑢

1,𝑙E𝑛 + 𝑢

2,𝑙E𝑛−1, (13)

where

𝑤1,𝑙=2𝑀𝑙− (1 − 2𝛾) Δ𝑡𝐶

𝑙− ((1/2) + 𝛾 − 2𝛽) (Δ𝑡)2𝐾𝑙𝑆𝑙

,

𝑤2,𝑙=−𝑀𝑙− (𝛾 − 1) Δ𝑡𝐶

𝑙− ((1/2) − 𝛾 + 𝛽) (Δ𝑡)2𝐾𝑙𝑆𝑙

,

𝑢0,𝑙=(Δ𝑡)2𝛽

𝑆𝑙

, 𝑢1,𝑙=(Δ𝑡)2 ((1/2) + 𝛾 − 2𝛽)

𝑆𝑙

,

𝑢2,𝑙=(Δ𝑡)2 ((1/2) − 𝛾 + 𝛽)

𝑆𝑙

,

𝑆𝑙= 𝑀𝑙+ 𝛾Δ𝑡𝐶

𝑙+ 𝛽(Δ𝑡)

2𝐾𝑙.

(14)

Equation (13) is called two-step or Newmark 𝛽𝛾method [19].The calculation ofP𝑛+1

𝑙involves the values ofP𝑛

𝑙andP𝑛−1

𝑙.The

coefficients in (13) include two variables 𝛽 and 𝛾.The stabilityand accuracy of (12) when selecting different 𝛽 and 𝛾 are alsodiscussed in [19] in detail. Usually, 0 ≤ 𝛾 ≤ 1 and 0 ≤ 𝛽 ≤ 1/2.

International Journal of Antennas and Propagation 3

To assure the accuracy, it is needed to choose an appropriate𝛾 and 𝛽 ≥ 0.25(0.5 + 𝛾)2. According to [19], unconditionalconvergence can be obtained by taking 𝛽 = 0.25, 𝛾 = 0.5. Thetime-stepping calculation from E𝑛+1 to P𝑛+1

𝑙can be finished

by (13).After converting the constitutive relation equation (8)

into discrete time domain, we have

D𝑛+1 (𝑡) = 𝜀0(𝜀∞E𝑛+1

(𝑡) +𝐿

∑𝑙=1

P𝑛+1𝑙

(𝑡)) . (15)

Substituting (13) into (15), we know that

(𝜀∞+𝐿

∑𝑙=1

𝑢0,𝑙)E𝑛+1 = D𝑛+1

𝜀0

−𝐿

∑𝑙=1

(𝑤1,𝑙P𝑛𝑙) −𝐿

∑𝑙=1

(𝑤2,𝑙P𝑛−1𝑙

)

−𝐿

∑𝑙=1

𝑢1,𝑙⋅ E𝑛 −

𝐿

∑𝑙=1

𝑢2,𝑙⋅ E𝑛−1.

(16)

Then the time-stepping calculation fromD𝑛+1, P𝑛𝑙to E𝑛+1 can

be accomplished.In conclusion, the general Newmark-FDTD method in

dispersive medium can be reduced as follows.

(1) Use the difference discrete expressions in curl (1) toget E → H → D.

(2) Use (16) to getD,P → E.(3) Use (13) to get E → P.(4) Back to step 1.

3. The Advantages of Algorithm in This Paper

For the second order differential equations like

𝑀𝑑2

𝑑𝑡2𝜙 + 𝐶

𝑑

𝑑𝑡𝜙 + 𝐾𝜙 = 0. (17)

After using central difference method, we have

(𝑀 +𝐶

2Δ𝑡) 𝜙𝑛+1 − [2𝑀 − 𝐾(Δ𝑡)

2] 𝜙𝑛

+ [𝑀 −𝐶

2Δ𝑡] 𝜙𝑛−1 = 0.

(18)

The time domain solution stability of the above differen-tial equation can be analyzed by using the 𝑍 transform. To atime series of 𝑥(𝑛𝑇) (𝑛 = 0, 1, 2...), its 𝑍 transform is definedas

𝑋 (𝑧) =∞

∑𝑛=0

𝑥 (𝑛𝑇) 𝑧−𝑛. (19)

𝑋(𝑧) is the 𝑍 transform of the sampling sequence 𝑥(𝑛𝑇).𝑍 transform is proposed to (18); we know that

(𝑀 +𝐶

2Δ𝑡)𝑍2 − [2𝑀 − 𝐾(Δ𝑡)

2] 𝑍 + [𝑀 −𝐶

2Δ𝑡] = 0.

(20)

According to the final value theorem, time domainfunction 𝑥(𝑛𝑇) (𝑛 = 0, 1, 2...) has time stable solutionswhen the poles of 𝑋(𝑧) located in a unit circle on complexplane in 𝑍 domain. An analysis of the solution stability ofsecond order homogeneous differential equationwithout firstorder derivative is made in [21]. In this paper, the case ofincluding first order derivative is analyzed. Equation (20) isconsidered as a quadratic equation with one unknown using𝑍 as dependent variable. The equation can be solved and let−1 ≤ 𝑍 ≤ 1; we know that

𝐾𝐶(Δ𝑡)3 + 2𝑀𝐾(Δ𝑡)

2 − 4𝑀𝐶Δ𝑡 − 8𝑀2 ≤ 0. (21)

The corresponding cubic equation with one unknown tothe above inequality has three solutions:

Δ𝑡1= −2

𝑀

𝐶, Δ𝑡

2= −2√

𝑀

𝐾, Δ𝑡

3= 2√

𝑀

𝐾. (22)

Let𝑀,𝐶,𝐾 be all positive quantities.The derivative of thecubic equation with one unknown is 8𝑀𝐶+8𝑀√𝑀𝐾 atΔ𝑡

3,

which is an increasing function satisfying the requirementsof the inequality. Hence, when central difference method isused to solve (17), the time-stepping is limited to

Δ𝑡 ≤ 2√𝑀

𝐾. (23)

According to (23), the requirements of time-stepping whencentral difference method is used to solve the second ordertime domain differential equations corresponding to thethree models of dispersive medium in (6) are

Δ𝑡Debye ≤ 0,

Δ𝑡Drude ≤ ∞,

Δ𝑡Lorentz ≤2

𝜔0,𝑙

.

(24)

That is to say, for Debye medium, corrected results cannotbe obtained whatever the value of the time-stepping takeswhen central difference method is used to solve (17). ForDrude medium, corrected results can be obtained whateverthe value of the time-stepping takes. But for Lorentzmedium,the solution stability is relevant to its resonance frequency atpoles.

However, the solution of (12) is unconditionally stablewhen using Newmark difference algorithm. Hence, the selec-tion of time-stepping is more flexible. Meanwhile, a higheraccuracy is obtained because the Newmark 𝛽𝛾method has acombination of Taylor expansions and means value theoremis employed.

To analyze the difference between using Newmark 𝛽𝛾method and central difference method, the results obtainedfrom these two methods and Runge-Kutta solutions [20] aregiven. Figure 1 is the absolute error comparing the classicalRunge-Kutta solutions of using these two methods, in whichthe solid line and the circle are the result from Newmark 𝛽𝛾and central difference method, respectively. As it is shown in


0 1 2 3 4 5 6 7

0.0

T (ns)

Central difference method

Abso

lute

erro

r

−6.0

−4.0

−2.0

2.0

4.0

6.0× 10−10

Newmark 𝛽𝛾 method

Figure 1: Comparison of stability and calculation error betweencentral difference and Newmark method.

the figure, the time it takes to get steady state for Newmarkmethod is shorter than central difference method. Let

𝐾 = 7.96 × 107, 𝑀 = 2.95 × 10−14,

𝐶 = 2.65 × 10−3, 𝑓 = sin (4 × 109𝑡)(25)

in (17).The theoretical analysis and numerical results show that

the algorithm presented in this paper has higher steady stateand calculation accuracy than central difference method.

4. Numerical Results

To demonstrate the correctness of our algorithm, the back-scatter of three different dispersive media sphere is given andcompared with Mie series solution. Let 𝛽 = 0.25, 𝛾 = 0.5,Δ𝑡 = 𝛿/2𝑐, where 𝛿 is spatial discrete grid-scale of Yee cell.Consider

𝐸𝑖(𝑡) = exp[−

4𝜋(𝑡 − 𝑡0)2

𝜏2] , (26)

where 𝜏 = 60Δ𝑡 and 𝑡0is the pulse peak time.

Example 1 (the backscatter of Debye sphere). Let the radiusof the sphere be 0.25m and filled with single pole Debyemedium.The complex relative permittivity is

𝜀𝑟(𝜔) = 𝜀

∞+ 𝜒 (𝜔) = 𝜀

∞+𝜀𝑠,1− 𝜀∞

1 + 𝑗𝜔𝜏0,1

, (27)

where 𝜀𝑠,1

= 1.16, 𝜀∞

= 1.01, 𝜏0,1

= 6.497 × 10−10 s. Thebackscatter RCS is shown in Figure 2.The solid line and circleare the results from this method and Mie series solution,respectively. It is shown that these two results are in good

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6−80

−70

−60

−50

−40

−30

−20

RCS

(dBs

m)

f (GHz)

Newmark-FDTDMie

Figure 2: The backscatter RCS of Debye sphere.

0 20 40 60 80 100 120−110

−100

−90

−80

−70

−60

−50

−40

Newmark-FDTDMie

RCS

(dBs

m)

f (GHz)

Figure 3: The backscatter RCS of plasma sphere.

agreement with each other. Consider 𝛿 = 3.3 × 10−3minFDTD calculation.

Example 2 (the backscatter of plasma sphere). The radiusof the sphere is 3.75mm and filled with single pole Drudemedium.The complex relative permittivity is


∞+ 𝜒 (𝜔) = 1 +

−𝜔21

𝜔2 − 𝑗𝜔]𝑐,1

, (28)

where 𝜔1= 1.8 × 1011 rad/s, ]

𝑐,1= 2.0 × 1010Hz. The

backscatter RCS is shown in Figure 3. The solid line andcircle are the result from this method andMie series solution,respectively. It is shown that these two results are in goodagreement with each other. Consider 𝛿 = 5.0 × 10−2mm inFDTD calculation.

International Journal of Antennas and Propagation 5

Newmark-FDTDMie

−220

−200

−180

−160

−140

RCS

(dBs

m)

f (GHz)

0.0 0.5 1.0 1.5 2.0

× 107

Figure 4: The backscatter RCS of Lorentz sphere.

Example 3 (the backscatter of Lorentz sphere). The radius ofthe sphere is 15.0 × 10−9m and filled with single pole Lorentzmedium.The complex relative permittivity is


∞+ 𝜒 (𝜔) = 𝜀

∞+

(𝜀𝑠,1− 𝜀∞) 𝜔20,1

𝜔20,1+ 2𝑗𝜔]

𝑐,1− 𝜔2

, (29)

where 𝜀𝑠,1

= 2.25, 𝜀∞

= 1.0, ]𝑐,1

= 0.28 × 1016Hz, 𝜔0,1

=

4.0 × 1016Hz.The backscatter is shown in Figure 4. The solidline and circle are the result from this method and Mie seriessolution, respectively. It is shown that these two results are ingood agreement with each other. Consider 𝛿 = 3.0 × 10−10min FDTD calculation.

5. Conclusions

In this paper, a general Newmark-FDTD algorithm is givento deal with the electromagnetic problems in dispersivemedium.This newmethod combines theNewmark 𝛽𝛾 differ-ence method which is widely used in FETD calculationin dispersive medium. Theoretical analyses and numericalresults demonstrate that this algorithm has advantages ofhigher accuracy and stability over the algorithms based oncentral difference method.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This project is supported by the National High Tech-nology Research and Development Program of China(2012AA01308), the National Basic Research Program,

and the National Natural Scientific Foundation of China(61231003 and 61401344).

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VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


research article frequency-dependent fdtd algorithm...

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