512 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 58, NO. 2, APRIL 2016

A Corner-Free Truncation Strategy inThree-Dimensional FDTD Computation

Yuxian Zhang, Student Member, IEEE, Naixing Feng, Student Member, IEEE, Hongxing Zheng, Member, IEEE,Hai Liu, Member, IEEE, Jinfeng Zhu, Member, IEEE, and Qing Huo Liu∗, Fellow, IEEE

Abstract—The corner-free truncation (CFT) strategy is pro-posed to improve the efficiency of numerical simulation in com-putational electrodynamics. The spherical boundary, namely theCFT strategy, is built up in the Cartesian coordinate system, whichhas been implemented via making use of the impedance-matchedlayer. The proposed CFT strategy is used for terminating compu-tational domain of the conventional finite-difference time-domain(FDTD) to absorb the outward electromagnetic waves. Moreover,based on the proposed boundary, the computation becomes sim-pler. The original FDTD computational domain is reduced to benearly a half, and the computational resource is saved significantly,both of which are due to the fact that no curvilinear model is appliedin truncating the boundary. The numerical simulations of targetscattering problems have been achieved to validate the proposedalgorithm.

Index Terms—Corner-free truncation strategy, finite-differencetime-domain, impedance-matched layer (IML), spherical bound-ary, target scattering.

I. INTRODUCTION

R ECENTLY, resembling SATIMO [1] antenna test systemkeeps circular appearance on two-dimensional (2-D) cross

section, as shown in Fig. 1. The system, consisting of an un-reflected microwave chamber, is used to measure antenna andtarget scattering. When tested object is located at the centerof the chamber, it can make electromagnetic waves transmit toabsorber around boundary in 0° incidence. This appearance illu-minates stratagem of truncation boundary selected in the com-putational electrodynamics. The truncation boundary must be

Manuscript received July 27, 2015; revised October 30, 2015; accepted De-cember 15, 2015. Date of publication January 8, 2016; date of current versionMarch 8, 2016. This work was supported by the National Natural ScienceFoundation of China (NSFC) under Grant 61371043, Grant 61307042, Grant41504111, Grant 41390453, and Grant 11501481, the Advanced Technology ofTianjin Municipality of China under Grant 12JCYBJC10500, the Fundamen-tal Research Funds for the Central Universities under Grant 10120131072, theScientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry, and the Fundamental Research Funds for the CentralUniversities under Grant 20720150083. (Corresponding author: Qing Huo Liu.)

Y. Zhang, H. Liu, and J. Zhu are with the Department of ElectronicScience, Institute of Electromagnetics Acoustics, Xiamen University, Xia-men 361005, China (e-mail: yxzhang_tute@126.com; liuhai8619@xmu.edu.cn;nanoantenna@hotmail.com).

N. Feng is with the Department of Electronic Science, Institute of Elec-tromagnetics Acoustics, Xiamen University, Xiamen 361005, China, and alsowith the Department of Electrical and Computer Engineering, Duke University,Durham, NC 27708 USA (e-mail: fengnaixing@gmail.com).

H. Zheng is with the Institute of Antenna and Microwave Techniques, Tian-jin University of Technology and Education, Tianjin 300222, China (e-mail:hxzheng@tute.edu.cn).

Q. H. Liu is with the Department of Electrical and Computer Engineering,Duke University, Durham, NC 27708 USA (e-mail: qhliu@duke.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2015.2510297

Fig. 1. SATIMO test system for antenna measurement.

suitable for matching or absorbing-boundary condition (ABC)in order to make electromagnetic waves transmit onto bound-ary without reflection. For the published numerical methods, itis well known that the finite-difference time-domain (FDTD)method [2] is one of the most powerful tools in microwavecommunity. However, the conventional FDTD method is al-ways based on the cubic truncation boundary. Those boundaryconditions have been researched by Liao et al. [3], Mur [4], Meiand Fang [5], Berenger [6], Gedney [7], etc. In particular, theGedney’s perfectly matched layer (PML) is more popular andhas overcome reflection from cubic corners, but it takes muchtime to execute codes on corners when simulation is running.According to the SATIMO’s illumination, if we use curvedboundary to truncate the simulation region in an identical way,the process can be equivalent to the SATIMO system. Bothmeasurement and simulation results can be obtained under thecircumstance of boundaries in the same form. In this way, wecan compare simulated results with measurement. The orthogo-nal curvilinear coordinate system is used before [8]–[11], but itis difficult to process on the truncation boundary. Besides, morenumerical dispersion error occurs. To overcome the limitation ofthe orthogonal FDTD method, the nonorthogonal FDTD basedon the conventional cubic boundary is used to model the curvesurface of the complex target scattering [12], [13]. Therefore,accurate results have been obtained. Although the program codehas been simplified for the boundary processing, it takes us muchmore time to tackle corners. If those corners without curvilineargrids can be omitted, much more memory and computationaltime can be saved. Assuming that v1 and v2 can be defined asvolume of cube and sphere in the three-dimensional (3-D) situ-ation, respectively, where the edge length of the cube is equal tothe diameter of the sphere, as shown in Fig. 2(a), then we have

1 − v2

v1= 1 − π

6= 47.64%. (1)

It can be seen in (1) that volume can be reduced to be ap-proximately a half. Using the sphere boundary in the numerical

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ZHANG et al.: CORNER-FREE TRUNCATION STRATEGY IN THREE-DIMENSIONAL FDTD COMPUTATION 513

Fig. 2. Spherical CFT strategy: (a) Geometric structure with radius r in the Cartesian coordinate system; (b) cross section for cubic grids in the FDTD region.

simulation, the computational region can be reduced dramati-cally and much more computer resource can be saved efficiently.In the same way, over one fifth computer’s resource can alsobe reduced in 2-D case. This stratagem is referred as corner-free truncation (CFT) strategy. Compared with the square PMLboundary, the CFT strategy in 2-D case has been successfullyexplained in [14] and applied in the far-field extrapolation fromthe near-field results.

In this paper, the CFT strategy without curvilinear grid isinvestigated. The principle of optical film with increasing trans-mittance [15] is applied to truncating the boundary in an FDTDtechnique. Imaginary thin layer can meet impedance-matchedcondition, and absorb the electromagnetic waves efficiently.This concept is applied around the computational region. Con-sequently, the iterations of those corners are vanished. The pro-posed CFT strategy has been verified by using the examples oftarget scattering. Results show that computational efficiency hasbeen enhanced obviously.

In Section II, the principle of the CFT strategy is discussed.In Cartesians’ coordinate system, an impedance-matched layer(IML), as shown in Fig. 2(b), is considered to implement thespherical CFT strategy around the computation region, but thePML technique is not included in the computational region sincethe projections or split components [16] are too difficult in theCFT strategy. Sometimes, computational region with the CFTstrategy may be larger than one with cubic boundary when weuse the IML. Some special cases, such as interpolation trun-cation [17]–[19] and the Mur’s ABC [20] are discussed. Thecritical size of suitable simulation region has been found. Theabsorption performance of the truncation boundary is related toincident angle according to Engquist–Majda’s theory [21]. Forexample, large incident angle makes more obvious reflection.However, the reflection of the CFT strategy from maximum in-cident angle is much less than the ones of other cubic truncation.It has been proved strictly in this section. Afterward, the numer-ical experiments have been done in Section III. Computationalparameters of the IML have been obtained by proper selection.After these parameters of absorbing layer have been obtainedfrom one-dimensional (1-D) IML, they are applied in the FDTDsimulation. Then Gaussian pulse, sinusoidal wave and electricdipole have been used as incident sources. Absorption abilityand stability of the CFT strategy also have been verified innumerical experiments. Computer memory and CPU time are

Fig. 3. Three different media (εi , μi , σei , σm i , i = 1, 2, 3) with two bound-aries, medium 2 is as the IML.

saved much more compared to cubic PML [22]–[24] and Mur’s[25] truncation boundaries. In Section IV, two numerical exam-ples are discussed to verify the accuracy and efficiency of theproposed technique. A metal sphere with lossy isotropic coatingand a conductor cone-spheroid have been calculated by using theapproximate operator expansion (AOE)-FDTD [26] method, inthe CFT strategy and cubic PML truncations, respectively, andmeasured by the SATIMO test system. The radar cross section(RCS) of above targets has been obtained from simulations andexperiments. Finally, this approach displays great superiorityand efficiency, and the conclusion is given in Section V.

II. REVIEW OF THE CFT STRATEGY

A. Implementation With IML in 1-D Condition

In Cartesians’ coordinate system, the IML is considered toimplement the CFT strategy around the computation region.Medium 2 of three media (εi, μi , i = 1, 2, 3) with two bound-aries, as shown in Fig. 3 for simplicity, is regarded as a match-ing layer. When the electromagnetic waves from medium 1pass through it, waves’ attenuation happens. The medium 2is with complex permittivity ε and permeability μ, then wecan define them as εi = ε0(ε′i + jε′′i ), μi = μ0(μ′

i + jμ′′i ), i =

1, 2, 3, where imaginary part ε′′ = σe/ωε0 and μ′′ = σm /ωμ0 .σe, σm , ω, ε0 , and μ0 are, respectively, defined as conductivity,equivalent magnetic loss, angular frequency, permittivity, andpermeability in the vacuum. The imaginary part ε′′ and μ′′ arerepresented as the relative physical parameters, which are the

514 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 58, NO. 2, APRIL 2016

dimensionless quantities [27] corresponding to those of the realpart ε′ and μ′.

Due to the fact that medium 3 is perfectly electric con-ductor (PEC), the most energy of electromagnetic waves hasbeen reflected back to the medium 2. For the shortest thicknessd = 0.25λ2 , λ2 is center wavelength in the medium. Becauseof the impedance-matched condition Z1 = Z2 [14], the absorp-tion and attenuation of wave energy can be implemented validly.Hence, not only does the medium 2 satisfy impedance match-ing with the medium 1, but also it can ensure electromagneticwaves keep transmission into one direction in the medium 1.Since the medium 2 is attached on the PEC directly, we have toconsider its property when simulation is started. Assumed thatthe medium 1 is general; the impedance-matched condition canbe represented by

μ1ε2 −σm1

ω1μ0· σe2

ω2ε0+ j

[σm1

ω1μ0ε2 +

σe2

ω2ε0μ1

]

= μ2ε1 −σm2

ω2μ0· σe1

ω1ε0+ j

[σm2

ω2μ0ε1 +

σe1

ω1ε0μ2

]. (2)

Then, ε′′2 and μ′′2 are easily given by

ε′′2 =

∣∣∣∣∣∣ε′2 μ′

2

(ε′1)2 + (ε′′1)

2 ε′1μ′1 + ε′′1μ

′′1

∣∣∣∣∣∣/ ∣∣∣∣∣∣

ε′1 μ′1

ε′′1 μ′′1

∣∣∣∣∣∣ (3)

μ′′2 =

∣∣∣∣∣∣∣μ′

2 ε′2

(μ′1)

2 + (μ′′1)

2 ε′1μ′1 + ε′′1μ

′′1

∣∣∣∣∣∣∣/ ∣∣∣∣∣∣

μ′1 ε′1

μ′′1 ε′′1

∣∣∣∣∣∣ . (4)

Generally, the medium 1 is lossless, and then we have

ε′2μ′

2=

ε1

μ1Z2

0 =σe2

σm2Z2

0 . (5)

This result is similar to the condition of Berenger’s PML [6]and Gedney’s PML [7], but the field-splitting methods and theanisotropic property do not occur in the CFT strategy. However,(3) and (4) have strongly connected to numerical electric con-ductivity σe2 and equivalent magnetic loss σm2 . As a result, wecan find a suitable and static value to make the electromagneticwave energy absorbed on the boundary.

B. Necessary Size Review

Theoretically, to absorb electromagnetic waves completely,the thickness of IML should be with 0.25λ2 [14]. It is supposedthat some special cases, respectively, with interpolation trunca-tion or Mur’s absorbing boundary, are discussed in the conven-tional FDTD to make comparisons. Sometimes, computation re-gion with CFT strategy may be larger than one with cubic bound-ary when we use IML, as shown in Fig. 4(a), where each one ofcubic surfaces is a inner tangential plane of the sphere (withoutIML). We have to review critical size to find suitable region be-fore simulation. Assuming that r is a distance between the centreand boundary of computational region. The IML region is di-vided into n discrete grids. S represents the whole computationregion. The volume of cube and sphere is 8r3 and 4π(r + n)3/3,

Fig. 4. Spherical CFT strategy’s project plane in 3-D case: (a) Inner tangentialplane and (b) diagonal section in a cubic region.

Fig. 5. Representation for ΔS in different radius r and layer number n:(a) 3-D spherical CFT strategy and (b) 2-D circular CFT strategy.

respectively. ΔS is expressed as the difference between the CFTstrategy and conventional boundary, and we have

ΔS3D = 8r3 − 43π(r + n)3 . (6)

The ΔS3D will be different with the change of the r and then, which is depicted in Fig. 5(a). It is found that the curveskeep triple acceleration from figure. ΔS3D = 0 relates the samecomputer resource occupied between the CFT strategy and thecubic truncation. The critical value rc is its corresponding radius.When r > rc , computational region will be reduced, while theCFT strategy is used. For the different n, the critical value rc isgiven in Table I. Beyond all doubt, the CFT strategy is suitablefor large-size cases.

ZHANG et al.: CORNER-FREE TRUNCATION STRATEGY IN THREE-DIMENSIONAL FDTD COMPUTATION 515

Fig. 6. Schematic of maximum incident angle for (a) spherical, (b) square,and (c) circular truncation.

In addition, if the cubic PML set the same thickness as theIML, as shown in Fig. 4(b), those corners from the cube andΔS3D > 0 will exist all the time. Computational region, com-puter memory, and CPU time will be further reduced, while weuse the CFT strategy.

In the same way for 2-D case, the area of square and circle is4r2 and π(r + n)2 , respectively, then we obtain

ΔS2D = 4r2 − π(r + n)2 . (7)

Quadratic enhancement curves are obtained to depart from thepoint in ΔS2D = 0. Fig. 5(b) depicts the relationship betweenΔS2D and r with different n. The critical value rc is also givenin the Table I. When satisfying r > rc , computational regionis also reduced. Therefore, r can be chosen as suitable size toimprove the computation efficiency. Compared to rectangularPML in 2-D case, the CFT stratagem is better than before.

C. Reflection Coefficient

Generally, absorbing performance of the boundary is relatedto the incident angle according to Engquist–Majda’s theory [21].For example, larger incident angle makes reflection more obvi-ous. If we implement the CFT strategy via using the IML, suchreflection has to be reviewed. In the conventional cubic FDTDsimulation [27], the hard-source technique is always applied.Either the Gaussian pulse or the sinusoidal wave is projected toconnective boundary in the computation region, which is relatedto the incident angle. The reflection from truncation boundaryis also related to it. Assuming that location of the source (P), asshown in Fig. 6(a), is changed from center (O) to the boundaryto analyze the maximum reflection, we consider the angle of ef-fected reflection coefficient. From figure, angles ϕ and θ are theparameters for cube and sphere truncation, respectively. Here,we name both the ϕ and the θ are incident angles, each of themis from source point to the boundary, and then reflected back tothe center of computational region. For the cubic truncation, we

TABLE ICRITICAL Rc FOR DIFFERENT DISCRETE GRIDS n IN THE CFT STRATEGY

n 5 6 7 8 9 10

3-D 20.77 24.92 29.08 33.23 37.39 41.542-D 38.94 46.73 54.52 62.32 70.1 77.89

Fig. 7. Incident angle versus d/r for CFT strategy and square boundary.

have

ϕ = tan−1(√

2 +d

r

)(8)

where d is distance between P and O, r is a half of the edgelength of a cubic boundary. Then, for spherical truncation withthe same source location above, we have

sinθ =d

r

√1 − cos2ξ

1 + (d/r)2 + 2 (d/r) cosξ(9)

where

cosξ =

√1 + (d/r)4 −

[1 + (d/r)2

]2 (d/r)

(10)

which is derived in Appendix, and we can find the maximum ofθ to explain the minimum reflection.

Incident angles ϕ and θ changed with d/r are given in Fig. 7. Itis greatly obvious that the θ is much less than the ϕ for the samepositions of P. We can obtain the maximum of the ϕmax = 67.5◦

and θmax = 53.51◦ from figure. In most d/r cases, ϕ is greaterthan 60°. The larger the angle is, the more reflection rises. Thisresult tells us about the CFT strategy will not increase morereflection from incident angle. It can be proved analytically asfollows.

For the sake of clarity, a half of computational region isprojected to 2-D for both cubic and spherical truncation cases,respectively, as shown in Fig. 6(b) and (c). From these figures,the incident angle ϕ′ can be given by

ϕ′ = tan−1(

1 +d

r

). (11)

The ϕ′ will change with d/r which is also given in the Fig. 7.We can obtain the ϕ′

max = 63.43◦ when the source moves to

516 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 58, NO. 2, APRIL 2016

Fig. 8. Reflection rate changing with (a) incident angle and (b) d/r.

boundary. The θ′ expression for the circular CFT strategy is thesame as the one for the spherical CFT strategy in 3-D case.

For analytical simplicity, we consider reflection coefficients,RE and RC for ϕ and θ, respectively. In the second-orderEngquist–Majda ABC [21], [28], the RE is expressed by

RE =−cosϕ + 1 − 0.5sin2ϕ

cosϕ + 1 − 0.5sin2ϕ. (12)

When ϕ = 60◦, RE = 1/9. This is not what we accept. In orderto reduce reflection, incident angle has to be smaller than 60°.Obviously, the computation range on the incident angle has beenlimited. Fortunately, using the CFT strategy, incident angle doesnot tend to be so large. It is found θmax < ϕmax for the samesource positions, and the θmax = 53.51◦ if and only if P movesto boundary. This proves that the reflection can be reducedeffectively when the CFT strategy is applied. The RC can becalculated by the boundary condition, we have

RC =

∣∣∣∣∣∣cosθ −

√(ε′2 + jε′′2) − sin2θ

cosθ +√

(ε′2 + jε′′2) − sin2θ

∣∣∣∣∣∣ . (13)

As shown in Fig. 8(a), the CFT strategy does not exist reflec-tion when θ > θmax = 53.1◦, but second-order Engquist–MajdaABC still has larger reflection when ϕ > θmax . For differentd/r, obviously from Fig. 8(b), maximum reflection of second-order Engquist–Majda ABC is greater than the CFT strategyall the time. In most cases, it will be always less than 0.08when we set d/r = 0.8. The reflection error [29] can be de-fined as Er = (Rc)2 . Then, reflection error in the CFT strategyis always lower than 0.7%. The numerical verification will bediscussed in the next section.

III. NUMERICAL VERIFICATION FOR THE CFT STRATEGY

To verify mentioned above, the numerical experiments havebeen implemented for practical application. Computational pa-rameters of the IML have to be determined first. Gaussian pulse,sinusoidal wave, and dipole have been used as incident sourcesin the computation region, respectively. Absorbing performanceof IML has to be checked in the corner-free FDTD domain.Computational efficiency of the CFT strategy will be comparedwith one of other truncations in this section.

Fig. 9. CFT strategy’s reflection rate changing with εr .

A. Selection of Parameters σe2 and σm2

According to definition of wave impedance, Z = E/H , theE and H must be located at the same point in the space. Thisimpedance concept is not suitable for the conventional 3-DFDTD simulation because there are twelve edges and eight ver-tices on a cubic truncation boundary, where the wave impedanceis singularity, as well as 2-D case. However, those extraordinarypoints do not exist for the CFT strategy, where the layered-absorbing media is located. The property of this media is thesame as the IML in 1-D condition. Therefore, it is applied toimplementation of the CFT strategy.

Moreover, governing parameters of the IML are σe2 and σm2 .For the sake of valuable selection of them, we must find properparameters ε′2 and μ′

2 . For (3) and (4), it occurs that both numer-ator and denominator are equal to zero. According to limit the-ory, nonzero result can be found. Assuming that μ′

1 = ε′1 + C1dand μ′

2 = ε′2 + C2d, let δ = 0.0001, and C1 = C2 = 1.0. Sub-stituting them into (3) and (4), we obtain σe2 = 0.9512, andσm2 = 13519.31. To look for the suitable ε′2 , we can calculatethe reflection coefficient (Rc ) from medium 1 to medium 2.When the εr changed from 1.0 to 9.0, result is shown in Fig. 9.It can be found that the proper value ε′2 = μ′

2 = 1.875 fromfigure at the minimum of reflection coefficient. Obviously, thisvalue must satisfy (5).

ZHANG et al.: CORNER-FREE TRUNCATION STRATEGY IN THREE-DIMENSIONAL FDTD COMPUTATION 517

Fig. 10. Computation parameters determining from experimental formula, (a) σ∗e varied with s, (b) σ∗

e varied with p, (c) CFT strategy’s reflection rate variedwith s, and (d) varied with p.

TABLE IIPARAMETERS OF IML FOR THE CFT-FDTD SIMULATION WHEN λd = 40

n 1 2 3 4 5

σ ∗e 0.8823 1.0260 1.1207 1.1931 1.2525

σ ∗m 125289 145697 159137 169421 177854

n 6 7 8 9 10

σ ∗e 1.3032 1.3476 1.3874 1.4234 1.4564

σ ∗m 185055 191370 197014 202130 206820

In many situations, numerical value of the σe2 and σm2are considered as gradually changed with grids. They can bedetermined by an experimental formula, we have

σ∗w = [i(0.25λd − s)−1 ]1/pσw (w = e,m) (14)

where s = λd/n, and according to exponential fitting, relationbetween p and λd can be given by

p = 3.195 × 1016e−

(λd −1 9 5 . 6

2 3 . 5 3

)2

+ 5.165e−

(λd −4 6 . 0 8

1 3 . 0 8

)2

+ 0.4(15)

where 0 < p < 10. Three segments of p are greater than zeros. Ifp is great enough, σ∗

e will be close to σe . By proper selection of p,the reflection rate Rc can be reduced availably. Then, the σ∗

e andσ∗

m are changed in terms of the FDTD grids n. If s increases,variation curve of σ∗

e , shown in Fig. 10(a), will be risen. If pincreases, the variation curve of σ∗

e , shown in Fig. 10(b), will berotated clockwise around the same point. As shown evidentlyin Fig. 10(c) and (d), reflection is the lowest when s = 4.0 andp = 4.15. Therefore, the better parameters are set to the Table II

Fig. 11. Gaussian pulse Ex propagating to the CFT strategy’s bound-ary at different timesteps n, (a) n = 225, (b) n = 285, (c) n = 350, and(d) n = 1000.

for different FDTD truncation grids. They are used in the CFT-FDTD simulation afterward.

B. Gaussian Pulse Used for Verifying the CFT Strategy

The Gaussian pulse is used as an incident source, whichpropagates into truncation boundary, we have

Ex(i, t) = e−4π

(t−T cT d

)2

(16)

where Td = 10 ns, Tc = 0.8 Td . The central wavelength λ isdivided by numerical wavelength λd = λ/Δz. Let λ = 0.01 m,λd = 40. The wave propagation procedure runs in the 1-DFDTD simulation, as shown in Fig. 11. When the pulse marches

518 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 58, NO. 2, APRIL 2016

Fig. 12. Simulation of central source at 1050th timestep in the circular CFTstrategy’s boundary: (a) field distribution, (b) contour profile.

Fig. 13. Simulation of central source at 1050th timestep in the squaretruncation: (a) Field distribution and (b) contour profile.

onto the IML, it is absorbed obviously for different timesteps.The wave energy is attenuated in the CFT region with theincrease of timesteps. It can be seen from figures that thereflection of amplitude from interface of the CFT region isabout 0.1%–0.2%. This result shows the reflection from theCFT region can be negligible.

C. Point Sinusoidal Excitation in 2-D CFT-FDTD

To verify the stability of the CFT strategy, the Ez point sourceof sinusoidal waves is placed in the center of the 2-D FDTDcomputation region. Same parameters as above, and in Table II,are adopted in the circular truncation layer. The incident wavearrives at the CFT strategy boundary. It is absorbed obviouslywhen timestep runs to 1050. Results are shown in Fig. 12. Bothamplitude distribution and contour line are distortionless.

However, for the square truncation with the same parameter incomputation field, distribution and contour line are distorted at1050th timestep. Results are shown in Fig. 13. We have knownthat reflected amplitude is about 10%, while incident angle is45° at the corner, which is maximum value. Without corner, theincident angle is much less than 45°. Therefore, reflection isvery small.

It can be verified further that point source is not placed at thecenter of region. We set it at (80δ, 80δ) in the region, and area is300 × 300δ2. Other computational parameters are not changed.The timestep runs to 1050, we can obtain field distribution andcontour line distortionless, as shown in Fig. 14. These resultstell us about the CFT strategy shows very stable characteristics.Obviously, compared to the square boundary, if we use the CFT

Fig. 14. Simulation of noncentral source at 1050th timestep in the circularCFT strategy’s boundary: (a) Field distribution and (b) contour profile.

Fig. 15. Distortionless concentric circle in the CFT strategy for the Ez at timen = 30000.

Fig. 16. Near-field distribution of Ez ’s display. (a) Amplitude and (c) phasefor the CFT strategy; (b) amplitude and (d) phase for the square PML.

strategy with the same computation parameters and timestep,computer memory will be saved much more.

The proposed CFT strategy is aim to apply in large FDTDregion and long time running. If larger region is considered andcomputed, the advantage in time saving and high accuracy willbe performed. Area of the region and timestep is π × 500δ2 and30 000, respectively. A point sinusoidal wave source is put on theposition deviating from the center; other computing parametersare set as same as above. After 10729.74 s (corresponding to the30 000 timesteps), as shown in Fig. 15, the Ez is expressed by

ZHANG et al.: CORNER-FREE TRUNCATION STRATEGY IN THREE-DIMENSIONAL FDTD COMPUTATION 519

TABLE IIIEFFICIENCY COMPARISON OF THE MUR’S ABC, THE CUBIC PML, AND THE CFT STRATEGY

Region (45δ )3 (65δ )3

Truncation Time (s) Grid Errorm a x (dB) Time (s) Grid Errorm a x (dB)

Mur’s ABC 9.26 91125 −57.7815 38.2 274625 −74.868Cubic PML 24.39 166375 −58.9593 67.47 421875 −62.1229CFT Strategy 9.17 87113 −56.1994 28.92 220893 −73.1049

Fig. 17. Comparison of dipole radiation in Ez (10δ, t), (a) (45δ)3 and (b) (65δ)3.

concentric circle without any distortion. The computation resultindicates that the CFT strategy is still much stable.

The square column is considered and simulated to verify the2-D CFT-FDTD in the near-field region. The plane waves arechosen with the 45° incidence in the total-field region. Fig. 16demonstrates Ez ’s amplitude and phase. The results for theCFT strategy are shown in Fig. 16(a) and (c), while those for thesquare PML are exhibited in Fig. 16(b) and (d). As comparedwith the square PML, the CFT strategy avoids the four corners’computation surrounding zero points in the fringe amplitudesand phases. Little difference between them can be occurred andviewed in the main region, because the evanescent waves donot stay on the CFT boundary. There are the same phases inFig. 16(c) and (d) to explain the forward propagation withoutthe backward phase deviation. Due to the advantage in 21.5%resource saving via the equation from [14], computational timeof the CFT strategy is 11.65 s, which has about 23% reductionas compared with 15.16 s of the square PML.

D. Dipole Source Excitation in 3-D CFT-FDTD

To further investigate the CFT strategy, an electric dipoleis located at the center of the 3-D FDTD region. Relationshipbetween current density J and electric moment p is representedby

∫JdV = dp/dt . To simplify the computation, the region

size is set as 125cm3 with Yee’s-cell grids. If the electric dipoleis excited by Gaussian pulse, we have

p(t) = ez10−10exp

[−

(t − 3T

T

)2]

(17)

when θ = 90°, radiation field is represented by

E (r, t) =μ

4πr

(∂2

∂t2+

c

r

∂

∂t+

c2

r2

)p

(t − r

c

). (18)

For the FDTD computation, we have

En+1z = En

z +Δt

ε0(∇× H)|n+1/2

z − Δt

ε0 δ3

[dp

dt

]n+1/2

.

(19)Numerical results, Ez at (10, 0, 0) point, are shown in

Fig. 17. In addition, analytical solution, Mur’s ABC [4], andcubic PML truncations in conventional FDTD results are alsogiven in the figure with the different number of cells. Theseresults are compared with analytical solution, as shown inTable III. The maximum errors are not almost changed for threetruncations. Obviously, the PML technique need much moretime than other two methods because of complex algorithmand larger computing region. With larger radius r, it can beseen that the CFT strategy can save more execution time andcomputing resource. Simultaneously, it can also be developedin other numerical techniques of electrodynamics.

IV. APPLICATION FOR THE CFT STRATEGY

Here, are two examples discussed to verify the accuracy andefficiency of the proposed numerical method. The SATIMOsystem is applied for measuring the targets scattering in ourlaboratory. The standard target is a metal sphere with radius29.48 cm. It is used as calibration accuracy of the system at9.0 GHz first. We obtain near-fields distribution, then transforminto the far-field region [2]. The RCS of the targets has beenobtained. Dipole source, as discussed above, with differentialGaussian pulse excitation was used in both measurement andsimulation. A 2.8-GHz Pentium PC with 2GB memory wasused for the calculations. The values in Table II are applied forcomputation parameters in the CFT strategy. As comparison,unsplit PML ABC [30] is applied in the conventional cubicFDTD truncation.

520 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 58, NO. 2, APRIL 2016

Fig. 18. Bistatic RCS from a coated metal sphere. Incident wave is (a) VV-polarized and (b) HH-polarized.

Fig. 19. Measured and computed RCS of a metal cone-spheroid in case of (a) VV-polarization and (b) HH-polarization.

TABLE IVCOMPARISON OF COMPUTER RESOURCES USING THE CUBIC PML AND THE

CFT STRATEGY

TruncationCoated Sphere Cone Spheroid

CPU Time (s) Memory (MB) CPU Time (s) Memory (MB)

Cubic PML 318 650 555 486CFT Strategy 111 351 167 267

One example is concentrated about a metal sphere withlossy isotropic coating. εr = 5.6 and σ = 1.71 S/m are coatedmaterial parameters. The size of the sphere is kr = 2π. Ther = 50 Δs, and coating thickness is d = 8 Δs for FDTD mod-eling. For the FDTD simulation, sphere surface is modeled bythe AOE method [26]. Both the cubic PML and the sphericalCFT strategy have been used as truncation boundary. BistaticRCS of this coated sphere has been obtained from simulationand experiment. Results are depicted in Fig. 18. As comparison,the analytical Mie [31] result is also shown in the same figure.It can be seen that the simulation results are satisfactory forboth the VV- [see Fig. 18(a)] and HH-polarized [see Fig. 18(b)]cases. However, the proposed method can save 46% memoryand 65% computation time compared to the cubic PML trun-cation because those cubic corners are not required calculation.Table IV lists the computer resources of two numerical methods.It can be seen from table that the CFT strategy is more efficient.Our code runs very stable, and no divergence is observed evenwhen 7000 timesteps run.

Another example is a conductor cone-spheroid (from USAEMCC’s Benchmark). This is a new problem for the FDTDcommunity because of the large size ratio between length andsphere’s radius. The geometry size of the cone-spheroid body isgiven by⎧⎨

⎩y = 0.12278(x + 605.0657)cosϕ

z = 0.12278(x + 605.0657)sinϕ, −605.0657 < x < 0 (20)

⎧⎪⎪⎨⎪⎪⎩

y = 74.8506√

1 −(

x−9 .12274 .8506

)2 cosϕ

z = 74.8506√

1 −(

x−9 .12274 .8506

)2 sinϕ

, 0 < x < 83.9726 (21)

where −π < ϕ < π, semicone angle α = 7◦, cone lengthL = 605.0657, and sphere radius R = 74.8506. We use theAOE-FDTD method [26] on the cone-spheroid surface, andboth the cubic PML [32] and the proposed spherical CFT strat-egy have been utilized as the truncation boundary. A nonuniformgrid was adopted to locate electric and magnetic field compo-nents on the cone surface as much as possible. The size of gridsvaried between 0.8 and 1.8 mm, and the minimum grid size ofΔs = 0.8 mm was used at the jut. Simulations, as well as exper-iment results are shown in Fig. 19. As can be observed from thefigure, all the bistatic RCS results from the AOE-FDTD methodgive better agreement with the measured result for both theVV- [see Fig. 19(a)] and HH-polarized [see Fig. 19(b)] cases.Both the CFT strategy and the cubic unsplit PML boundarieshave been used to terminate the AOE-FDTD region. The com-puter resources involved for this example are also listed in the

ZHANG et al.: CORNER-FREE TRUNCATION STRATEGY IN THREE-DIMENSIONAL FDTD COMPUTATION 521

Table IV. Compared with the cubic unsplit PML ABC, 70%of the CPU time and 45% of the memory have been savedefficiently and evidently by using the proposed CFT strategy.Therefore, our code performs and works very well.

V. CONCLUSION

The CFT strategy has been applied to the FDTD technique.Especially, this truncation boundary has been presented to im-plement a sphere in 3-D case. More importantly, it can reduce thewhole computation region to be nearly a half, avoid computingthose unnecessary corner, and save large computer resources,memory, and time, significantly. Via corresponding combinationof the proposed approach with the AOE-FDTD technique, nei-ther truncation corner nor inside the object are required to calcu-late. Therefore, the computational time and memory are reducedsubstantially. Moreover, based on the impedance-matched prop-erties of this imaginary medium, it can also be suitable for othercomputational methods, and we can design different and reason-able convex shape, not constrained in circular in 2-D and spherein 3-D, to implement the CFT strategy to absorb the outwardelectromagnetic waves.

APPENDIX

From Fig. 6(c), assume p is a length between P and Q,according to cosine and sine principle, we have

p2 = r2 + d2 − 2rdcos(π − ξ) (A1)

p−1sin(π − ξ) = d−1sinθ. (A2)

From (A2), we have

sinθ =d

psinξ =

dsinξ√r2 + d2 + 2rdcosξ

. (A3)

In order to find maximum θ, partial differentiation operationof θ for ξ can be written by

∂θ

∂ξ=

dcosξ(r2 + d2 + rdcosξ) + rd2

(r2 + d2 + 2rdcosξ)√

(r2 + d2 + 2rdcosξ)2 − (dsinξ)2(A4)

and ∂θ∂ξ = 0 must be satisfied, then we have

rdcos2ξ + (r2 + d2)cosξ + rd = 0. (A5)

Resolve this equation, we have

cosξ =−[1 + (d/r)2 ] +

√1 + (d/r)4

2(d/r). (A6)

Therefore, θmax can be found by

sinθmax =d

r

√1 − cos2ξ

1 + (d/r)2 + 2(d/r)cosξ. (A7)

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Yuxian Zhang (S’16) received the B.S. and M.S.degrees from Tianjin University of Technology andEducation, China, in 2012 and 2015, respectively.He is currently working as a postgraduate studenttoward his Ph.D. degree in Radio Physics at Xia-men University, Xiamen. His current research inter-est is computational electromagnetics, especially inthe FDTD method and unconditionally-stable time-domain method, imaging algorithms in the ground-penetrating radar systems and the microwave circuitdesign. He has published 15 papers in refereed jour-

nals and conference proceedings. He has got Chinese National Scholarship fortwo times and participated in Chinese Graduate Mathematical Contest in Mod-eling for five times since 2011 with national awards and Chinese MathematicsCompetition of Chinese University Students in 2009 with national third prize.

Naixing Feng (S’16) received the B.S. degree inElectronic Science and Technology and the M.S. de-gree in Micro-Electronics and Solid-State electronicsfrom Tianjin Polytechnic University, Tianjin, China,in 2010 and 2013, respectively. He is currently work-ing toward his Ph.D. degree in Radio Physics at Xia-men University, Xiamen, China, and as a Joint Ph.D.student with the department of Electrical and Com-puter Engineering, Duke University, Durham, NC,USA, under the financial support from the ChinaScholarship Council. His current research interests

include computational electromagnetics, acoustics, and meta-materials. He haspublished over 25 papers in refereed journals and conference proceedings. Hehas served as a reviewer of more than 5 journals.

Hongxing Zheng (M’01) was born in Yinchuan,Ningxia Hui Autonomous Region, China. He re-ceived the B.S. degree in physics from Shaanxi Nor-mal University, Xi’an, Shaanxi, China, in 1985, andthe M.S. degree in physics and the Ph.D. degree inelectronic engineering from Xidian University, Xi’an,in 1993 and 2002, respectively.

From 1985 to 1989 and 1993 to 1998, he wasa Lecturer with the Ningxia University, Yinchuan.From 2001 to 2002 and 2004 to 2005, he was a Re-search Assistant and Research Fellow with the De-

partment of Electronic Engineering, City University of Hong Kong, Kowloon,Hong Kong, respectively. In 2003, he was a Postdoctoral Research Fellow withthe College of Precision Instrument and Opto-Electronics Engineering, Tian-jin University. He is currently a Professor with the Institute of Antenna andMicrowave Techniques, Tianjin University of Technology and Education, Tian-jin, China. He has authored six books and book chapters and more than 200journal papers and 50 conference papers. He holds 34 China patents issued in2014. His recent research interests include wireless communication, modeling ofmicrowave circuit and antenna, and computational electromagnetics.

Dr. Zheng is a Senior Member of the Chinese Institute of Electronics. Hereceived the University Distinguished Teacher Award of the Tianjin Universityof Technology and Education in 2012, he also received the 2008 Young Scien-tists Awards presented by the Tianjin Municipality, China, and was listed in theWho’s Who in the World and Who’s Who in the Science and Engineering in theWorld.

Hai Liu (S’11–M’13) received the B.S. and M.S.degrees in civil engineering from Tongji University,Shanghai, China, in 2007 and 2009, respectively, andthe Ph.D. degree in environmental studies from To-hoku University, Sendai, Japan, in 2013.

He joined the Center for Northeast Asian Studies,Tohoku University, as a Research Fellow in 2013.He is currently an Assistant Professor with the In-stitute of Electromagnetics and Acoustics, XiamenUniversity, Amoy, China. His current research inter-ests include the development of ground-penetrating

radar systems and imaging algorithms for a wide variety of applications,such as nondestructive testing in engineering, environmental monitoring, andarcheological investigation.

Dr. Liu received the Young Researcher Award at the 14th International Con-ference on Ground Penetrating Radar in 2012 and the Excellent Paper Award ofthe IET International Radar Conference 2013.

Jinfeng Zhu (M’15) received the B.S. degree in elec-tronic communication science and technology and thePh.D. degree in physical electronics from the Univer-sity of Electronic Science and Technology of China,Chengdu, Sichuan, China, in 2006 and 2012, respec-tively.

From November 2009 to November 20011, hewas a Visiting Researcher with the Department ofElectrical Engineering, University of California, LosAngeles, CA, USA, under the financial support fromthe China Scholarship Council. Since July 2012,

he has been with Xiamen University, Xiamen, Fujian, China, where he iscurrently an Associate Professor of electrical engineering. He has authoredand coauthored more than 30 peer-reviewed journal and conference papers.His research interests include microwave metamaterials, nanophotonics, andnanotechnology.

Qing Huo Liu (F’05) received the B.S. and M.S. de-grees in physics from Xiamen University, Xiamen,China, in 1983 and 1986, respectively, and the Ph.D.degree in electrical engineering from the Universityof Illinois at Urbana-Champaign, Champaign, IL,USA, in 1989.

He was with the Electromagnetics Laboratory,University of Illinois at Urbana-Champaign, as a Re-search Assistant from September 1986 to December1988, and as a Postdoctoral Research Associate fromJanuary 1989 to February 1990. He was a Research

Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield,CT, USA, from 1990 to 1995. From 1996 to May 1999, he was an AssociateProfessor with New Mexico State University. Since June 1999, he has beenwith Duke University, Durham, NC, USA, where he is currently a Professorof electrical and computer engineering. He has published more than 500 pa-pers in refereed journals and conference proceedings. His research interestsinclude computational electromagnetics and acoustics, inverse problems, geo-physical subsurface sensing, biomedical imaging, electronic packaging, and thesimulation of photonic and nanodevices.

Dr. Liu is a Fellow of the Acoustical Society of America, a Member of PhiKappa Phi, Tau Beta Pi, a Full Member of the U.S. National Committee of URSICommissions B and F. He is currently the Deputy Editor in Chief of the Progressin Electromagnetics Research, an Associate Editor for the IEEE TRANSACTIONS

ON GEOSCIENCE AND REMOTE SENSINg, and an Editor for the ComputationalAcoustics. He was a Guest Editor in Chief of Proceedings of the IEEE for aspecial issue on large-scale computational electromagnetics published in 2013.He received the 1996 Presidential Early Career Award for Scientists and En-gineers from the White House, the 1996 Early Career Research Award fromthe Environmental Protection Agency, and the 1997 CAREER Award from theNational Science Foundation.