A split-field iterative ADI method for simulating transverse-magnetic waves in lossy media

2196 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 5, MAY 2006

A Split-Field Iterative ADI Method for SimulatingTransverse-Magnetic Waves in Lossy Media

Shumin Wang and Jeff H. Duyn

Abstract—A novel split-field alternating-direction implicitmethod is proposed to simulate two-dimensional transverse-magnetic waves in lossy media. By splitting the electric field withrespect to the two transverse-magnetic field components, an effi-cient implementation scheme is obtained. Stability and numericaldispersion are further examined. Finally, numerical examplesdemonstrate the validity of the proposed method.

Index Terms—Finite-difference method, iterative method, lossymedia.

I. INTRODUCTION

THE alternating-direction implicit (ADI) method is a finite-difference time-domain (FDTD) method suitable for spa-

tially oversampled problems [1]–[6]. Due to its implicit natureand the fact that only tri-diagonal matrices are generated, it isan efficient unconditionally stable method that does not requiresolving general sparse linear systems.

Although the time-step size of the ADI method is no longerbound by the Courant–Friedrichs–Lewy (CFL) condition [7],the traditional ADI method suffers a large splitting error in ad-dition to a numerical dispersion error [2], [8]. Unlike the numer-ical dispersion error, which mainly affects electrically large orhighly resonant problems, the splitting error is associated withrelatively large field variations such as near-field sources, metaledges, corners, tips, etc. Since these structures are common insimulations, the splitting error has a broad range of impact onthe accuracy in practice.

Recently, the iterative ADI method has been proposedto reduce the splitting error [3]. This method interprets thetraditional ADI method as a special relaxation solver of theCrank–Nicolson (CN) scheme [8]. By applying a few iterationsat each time step, the splitting error can be effectively reduced.Based on the above interpretation, more efficient approachescan be developed. For example, the pre-iterative scheme isproposed in [4], which provides a better initial guess in regionswith a large splitting error. In [5], the geometrical multigridscheme was applied to further increase the convergence rate.

Although the fundamental theory of the iterative ADI methodremains the same, the efficiency (and sometimes the accuracy

Manuscript received July 26, 2005; revised December 1, 2005. This work wassupported by The National Institute of Neurological Disorders and Stroke underthe Intramural Research Program.

The authors are with the Laboratory of Functional and Molecular Imaging,The National Institute of Neurological Disorders and Stroke/National Institutesof Health, Bethesda, MD 20892 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMTT.2006.873634

[6]) depends on the implementation of lossy media. The appro-priate treatment of conductive terms in the transverse-electric(TE) case can be achieved by relatively simple matrix manipula-tion [4]. However, this is not true for lossy transverse-magnetic(TM) waves, which are common in bioelectromagnetic applica-tions [9]. In fact, this problem has more profound consequencesin general three-dimensional (3-D) simulations where TE andTM waves usually coexist. A proper treatment of conductiveterms in 3-D simulations should at least be applicable to TMwaves. To retain the efficiency of the ADI method, it is neces-sary to properly treat lossy TM waves.

In this paper, we present a novel scheme by employing asplit-field formulation. This scheme splits the electric field ac-cording to the two magnetic field components. As a result, asystem of equations that can be solved much more efficientlyby the iterative ADI method is obtained. Before introducing theproposed method, we stress the difference between the split-ting error and split-field formulation. The splitting error is origi-nated from splitting the CN operators [8], while both the electricand the magnetic fields are the original ones. In the split-fieldmethod, the electric field is split to construct a system suitablefor the iterative ADI method.

In this paper, we first review the conventional treatment oflossy media and the iterative ADI method in Section II. Theproposed treatment is introduced in Section III. Its stability andnumerical dispersion are analyzed in Section IV. In Section V,numerical examples demonstrate the validity of the proposedmethod. Finally, concluding remarks are drawn in Section VI.

II. REVIEW OF THE CONVENTIONAL TREATMENT

AND THE ITERATIVE ADI METHOD

Maxwell’s equations for lossy waves can be written inmatrix form as follows:

(1)

where . Instead of using the concept ofconsecutive forward/backward difference introduced in [1], aniterative ADI scheme can be generally obtained in three steps[3]. First, the right-hand-side matrix splits into two matrices.

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WANG AND DUYN: SPLIT-FIELD ITERATIVE ADI METHOD FOR SIMULATING TM WAVES IN LOSSY MEDIA 2197

According to the conventional treatment of lossy media, theyare

(2)

(3)

By applying the CN scheme at time step , which essen-tially employs a central difference to approximate time-deriva-tive terms and the average of and time-step values toapproximate nontime-derivative terms [8], we have

(4)

where is the identity matrix. Finally, the above equation canbe solved in two sub-steps, i.e.,

(5)

in the first step and

(6)

in the second step, where denotes the th iteration. The solu-tion of the traditional ADI method was shown as the first itera-tive solution with a special initial guess [3], whichis equivalent to an omission of the last terms in (5) and (6). Thisomission eventually introduces an error to the solution of (4),which manifests itself as the splitting error in the traditional ADImethod. By fully exercising the iterative solving procedure, theiterative ADI method provides a systematic way to obtain moreaccurate results. This has been demonstrated in [3] and [4].

According to (2) and (3), the splitting error of the conven-tional treatment of waves in lossy media is represented by

(7)

In (7), we notice three entries in the above expression containingconductivity . Thus, the splitting error is not only proportionalto the field variation as in the free-space and lossy TE cases [3],[4], but also proportional to conductivity. This implies that thesplitting error is no longer local to structures such as near-fieldsources and metal discontinuities. Furthermore, four nonzeroterms exist in (7). Compared with the free-space and cases[3], [4], which only contain one nonzero term, the iterative ADIscheme based on (5) and (6) is more difficult to implement andmore computationally expensive.

III. SPLIT-FIELD APPROACH

The proposed treatment starts from splitting into twoparts, i.e.,

(8)

where and satisfy

(9)

and

(10)

respectively. When combining the above two equations, weobtain

which is equivalent to the traditional Maxwell’s equation. Theother two Maxwell’s equations are

(11)

(12)

We notice that neither , nor has physical mean-ings. Only their combination represents the physical field.The main purpose of splitting is to obtain a new system ofequations that is more appropriate for the iterative ADI method.We also notice the difference between the proposed split-fieldand Berenger’s split-field perfectly matched layer (PML) [10].In that approach, and are associated with and

, respectively. In the proposed approach, andare associated with and , respectively.


Equations (9)–(12) are rewritten as

(13)

where . An iterative ADI schemeis obtained following the three steps outlined in Section II. Inthe proposed treatment, the right-hand side matrix in (13) splitsinto two matrices as follows:

(14)

(15)

As a result, the splitting error becomes

(16)

Expression (16) is much simpler than (7). Only one nonzeroentry exists, which is proportional to the spatial variation offield. As a result, the splitting error is again local to regions withlarge field variations. Furthermore, the simple format of (16) in-volves less computational cost compared to (7). With the pro-posed approach, we are able to achieve the same splitting-errorproperty and computational efficiency as the free-space and thelossy TE cases [3], [4].

With (14) and (15), (4)–(6) are followed to develop the corre-sponding iterative ADI scheme. Multiple choices exist in termsof which unknown is solved implicitly. One can derive the fol-lowing two sets of equations:

(17)

(18)

(19)

(20)

in the first step and

(21)

(22)

(23)

(24)

in the second step. In each step, a tri-diagonal matrix is con-structed in the same way as introduced in [1].

IV. STABILITY AND THE NUMERICAL DISPERSION ERROR

A. Stability

We study numerical stability of the proposed method in a loss-less case, as lossy media naturally introduces wave dissipation.Following the standard procedure, and are expanded intoa discrete set of Fourier modes. For each mode,

(25)

(26)

(27)

(28)

By substituting the discrete Fourier modes into (13), the systemamplification matrix [8] is derived from (4) as

(29)


where and are given by

(30)

(31)

and , , and . In an uniformmesh, where , the dominate eigenvalues of thesystem amplification matrix are

where

With the help of Mathematica,1 it was found that .Thus, the proposed method is stable and nondissipative in a loss-less case.

1[Online]. Available: http://www.wolfram.com

Fig. 1. Numerical dispersion error of the proposed and traditional ADI.

Fig. 2. Relative error of the proposed method with different number ofiterations.

B. Numerical Dispersion Error

To derive the numerical dispersion error, we add anterm to the Fourier modes, where is the angular frequency,is the time-step index, and represents the time-step size. Bysubstituting the Fourier modes into (13), we obtain the followingequation:

(32)

where is the identity matrix and is given by (29). Thenumerical dispersion relation is obtained from (32) by forcingits determinant to be zero, i.e.,


Fig. 3. Two-dimensional (2-D) head/coil system. The outer circle represents the RF shield. The dots inside the outer circle represents the legs of a birdcage coil,which were modeled as current sources in the simulation. The gray scale with respect to the background represents the phase and magnitude of the excitations.

The result is

(33)

where is the theoretical phase velocity in the medium, is theCFL number defined by , and is the numericalphase velocity, which is the unknown to be solved. Equation(33) is a transcendental equation and can be solved numerically.

In Fig. 1, we compare the largest dispersion error (of all prop-agation angles) of the proposed method with that of the tra-ditional ADI scheme. In this figure, the spatial sampling rateis denoted by points-per-wavelength (PPW). Although the dis-persion relations take different forms, the numerical dispersionerror of the proposed ADI method is nearly identical to the tra-ditional one.

V. NUMERICAL EXAMPLES

To verify the proposed method, we simulated the field distri-bution of an infinite long electric current source in lossy media( , ). The source is at the center of the computa-tional domain and is excited at 1 GHz. A uniform mesh was con-structed with 1-mm cells. The boundary of the computationaldomain is sufficiently far away from the source to eliminate re-flection errors. field at each observation point was obtainedby the discrete Fourier transform.

The proposed ADI method was tested with and adifferent number of iterations at each time step (note that the it-erative ADI method with one iteration recovers the traditional

ADI method). Yee’s FDTD method is used to provide the refer-ence solution [7]. The time-step size of Yee’s method is 2.36 psand that of the proposed ADI method is 47.2 ps. Fig. 2 shows therelative error of the proposed method with a different number ofiterations. The relative error is defined by

where represents the numerical solution and repre-sents the reference solution. Fig. 2 clearly indicates that the rel-ative solution error is reduced successively as we increase thenumber of iterations per time step. We notice that the iterativeADI method does not provide better results at each spatial lo-cation. Instead, it reduces the maximum solution error, whichcorresponds to the infinite norm of the solution error of[11]. Furthermore, the reduction of solution error tends to leveloff around the skin depth of the lossy media, which is 5 cm (or0.16 ) in this example.

In the second example, we calculate the electric field dis-tribution of a 7.0-T magnetic resonance imaging (MRI) coilsystem. The coil is a shielded and linearly polarized 16-strutbirdcage coil [9] operating at 300 MHz. The head/coil system isillustrated in Fig. 3, where the head model is adapted from theBrooks’ man model.2 The diameter of the coil is 28.8 cm andthat of the shield is 31.2 cm. The coil is modeled as 16 elec-tric current sources surrounding the head. The magnitude of thecurrent follows a cosine distribution in the angular direction [9].The shield is modeled as copper with . The con-stitutive properties of some tissues in the head model are listedin Table I.

2[Online]. Available: http://www.brooks.af.mil


TABLE ICONSTITUTIVE PROPERTIES OF THE HEAD MODEL AT 300 MHZ

Fig. 4. Comparison ofE -field distribution in the head model along the centralvertical line.

A uniform mesh was constructed with 2-mm cells, whichcorresponds to 500 PPW at 300 MHz in free space. Again,Yee’s FDTD method is used to provide the reference solution.The proposed ADI method was tested with and a dif-ferent number of iterations at each time step. The time-stepsize of Yee’s method is 4.72 ps and that of the proposed ADImethod is 236 ps. From Fig. 1, the corresponding numericaldispersion error is approximately . Since the largest elec-trical size of this example is 0.36 , it is not considered asan electrically large one and the numerical dispersion is not aspecial concern.

Fig. 4 illustrates the computed -field distribution in thehead along the central vertical line (the -direction), where thetraditional ADI method shows large error. By using the pro-posed iterative ADI method with three iterations per time step,the accuracy is greatly improved. Fig. 5 shows the relative errorof different schemes, where the effect of the number of itera-tions is clearly indicated.

According to (16), the splitting error is a function of the spa-tial variation rate of field in the -direction (the direction alongwhich the observations are). From Fig. 4, there exists large fieldvariation in the middle of the head. Thus, the splitting error is

Fig. 5. Relative error of E -field in the head model along the central verticalline.

TABLE IICPU TIME COMPARISON

relatively more prominent in that region, as shown in Fig. 5.We notice that electric current sources are positioned next to thehead along the -direction. Due to the singularity of near-fieldcurrent sources, the splitting error is also large on the air/tissueinterface, as shown in Figs. 4 and 5.

Finally, a comparison of the CPU time is provided in Table II,where I-ADI denotes the proposed iterative ADI method, thefirst number in the parenthesis is the CFL number, and thesecond number in the parenthesis is the number of iterationsper time step. It is clear that the proposed iterative ADI methodis more efficient. As an iterative solver, the actual performanceof the proposed method depends on the accuracy require-ment. Table II and Fig. 5 can be used as a guideline for otherapplications.

VI. CONCLUSIONS

We have presented an iterative ADI method with specialtreatment of conductive medium for TM waves. The proposedmethod splits the electric field with respect to the two magneticfield components. Stability and the numerical dispersion errorwere also examined analytically. Its effectiveness and efficiencywere demonstrated by simulating a line source in lossy mediaand a 16-strut loaded MRI coil system. Future research prioritywill be given to the implementation of the iterative ADI methodin 3-D lossy media.

REFERENCES

[1] T. Namiki, “A new FDTD algorithm based on alternating-direction im-plicit method,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 10, pp.2003–2007, Oct. 1999.


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[5] ——, “A multigrid ADI method for two-dimensional electromagneticsimulations,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp.715–720, Feb. 2006.

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[7] A. Taflove, Ed., Advances in Computational Electrodynamics: The Fi-nite-Difference Time-Domain Method. Boston, MA: Artech House,1998.

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Shumin Wang received the B.S. degree in physicsfrom Qingdao University, Qingdao, China, in 1995,the M.S. degree in electronics from Peking Univer-sity, Beijing, China, in 1998, and the Ph.D. degree inelectrical engineering from The Ohio State Univer-sity, Columbus, in 2003.

He is currently a Staff Scientist with the Na-tional Institutes of Health (NIH), Bethesda, MD.His research interests include time-domain differ-ential-equation-based methods, integral-equationmethods, high-frequency asymptotic methods and

their applications to biomedical problems, very large scale integration (VLSI)packaging, geo-electromagnetics, and electromagnetic scattering.

Jeff H. Duyn received the M.S. and Ph.D. degreesfrom Delft University of Technology, Delft, TheNetherlands, in 1984 and 1988, respectively, both inphysics.

In 1989, he was a Post-Doctoral Fellow withthe Atomic Physics Department, University ofTrento, Trento, Italy, where he performed positronlifetime measurements. From 1991 to 1992, hewas a Post-Doctoral Fellow with the University ofCalifornia at San Francisco, where he was involvedwith magnetic resonance spectroscopy. In 1992, he

joined the National Institutes of Health (NIH), Bethesda, MD, where he isinvolved with the development of MRI and spectroscopy, initially as a ResearchFellow, and currently as an Independent Investigator. His laboratory specializesin the improvement of sensitivity and image resolution through the use ofmultichannel detectors and high magnetic field scanners.

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A split-field iterative ADI method for simulating transverse-magnetic waves in lossy media