A 3-D Perfectly Matched Medium from Modi�edMaxwell's Equations with Stretched CoordinatesyWeng Cho Chew and William H. WeedonElectromagnetics LaboratoryDepartment of Electrical and Computer EngineeringUniversity of Illinois, Urbana, IL 61801Key Terms: Maxwell's equations, coordinate stretching, perfectly matchedlayer, �nite di�erence time domain, massively parallel computer.AbstractA modi�ed set of Maxwell's equations is presented that includes complex coordinatestretching along the three cartesian coordinates. The added degrees of freedom in themodi�ed Maxwell's equations allow the speci�cation of absorbing boundaries with zerore ection at all angles of incidence and all frequencies. The modi�ed equations are also re-lated to the perfectly matched layer that was presented recently for 2-D wave propagation.Absorbing material boundary conditions are of particular interest for �nite di�erence timedomain (FDTD) computations on a single-instruction multiple-data (SIMD) massively par-allel supercomputer. A 3-D FDTD algorithm has been developed on a Connection MachineCM-5 based on the modi�ed Maxwell's equations and simulation results are presented tovalidate the approach.1. IntroductionThe �nite di�erence time domain method [1, 2] is widely regarded as one of the mostpopular computational electromagnetics algorithms. Although FDTD is conceptually verysimple and relatively easy to program, the method is actually quite e�cient since it involvesO(N1:5) computational complexity in 2-D and O(N1:33) computational complexity in 3-D[3]. In fact, FDTD can be considered an optimal algorithm since O(N�) numbers arey This work is supported by the O�ce of Naval Research under grant N000-14-89-J1286, the ArmyResearch O�ce under contract DAAL03-91-G-0339, the National Science Foundation under grant NSF-ECS-92-24466, and NASA under grant NASA-NAG-2-871. Computer time is provided by the NationalCenter for Supercomputer Applications at the University of Illinois, Urbana-Champaign.File: pml.tex, January 12, 1995Appeared in Micro. Opt. Tech. Lett. , vol. 7, no. 13, pp. 599-604, September 1994.1

produced in O(N�) operations.FDTD is also ideally suited for implementation on a single-instruction multiple-data(SIMD) massively parallel computer. The reason is that the stencil operations that mustbe computed at each node of the space grid involve only nearest-neighbor interactions andmay be implemented at a minimum communication cost [4]. A major challenge, however,is in implementing absorbing boundary conditions (ABCs) at the edges of the FDTDgrid. On scalar and vector computers, these boundary conditions are typically computedusing methods such as the Engquist-Majda [5], Mur [6], Liao [7] or Higdon [8] ABC.However, these methods are not ideal for parallel supercomputers since they all involvecommunication with many elements normal to the grid boundary. Such communicationcan easily surpass the time spent computing core FDTD operations in the grid interior,especially for higher-order boundary conditions, and hence can become a bottleneck in theFDTD code. Also, they do not allow for SIMD operation on a parallel machine withoutthe use of masking.An alternate method of implementing an ABC is to use a conventional absorbingmaterial boundary [4, 9{14] . For SIMD parallel computation, these methods have theadvantage that the ABC may be implemented with the same FDTD stencil operation asthe interior nodes by modifying the conductivity material parameter at the edge of theFDTD grid. The disadvantage is that the re ection coe�cient at the absorbing border iszero only at normal incidence and is both angle and frequency dependent. Consequently,the absorbing material border region must be made quite large|typically 20{100 gridpoints along each edge in order to minimize re ections.Recently, Berenger [15] suggested a more general method of implementing an absorbingmaterial boundary condition. Berenger proposed a procedure for 2-D wave propagationwhereby Maxwell's equations are generalized and added degrees of freedom are introduced.The added degrees of freedom allow the speci�cation of absorbing borders with zero re- ection coe�cient at all angles of incidence and all frequencies. Moreover, the generalized2

Maxwell's equations reduce to the familiar Maxwell's equations as a special case and hencethe same generalized equations can be used to propagate �elds in both the interior regionand absorbing region. Although the interface between the interior region and the absorb-ing boundary is re ectionless, there is still a re ection from the edge of the grid. Theadvantage of using Berenger's procedure is that much larger conductivity values may bespeci�ed in the absorbing region, leading to a drastic reduction in the number of gridpoints required for the absorbing boundary.In the present paper, a formulation similar to the Berenger idea is derived for 3-D wavepropagation from �rst principles using a coordinate stretching approach. The advantageof the new method for SIMD parallel computation is stressed. The method is validatedwith 3-D FDTD numerical computations on a Thinking Machines Corporation ConnectionMachine CM-5.2. Modi�ed Maxwell's EquationsFor a general medium, we de�ne the modi�ed Maxwell's equations in the frequencydomain, assuming e�i!t time dependence, asre �E = i!�H (1)rh �H = �i!�E (2)rh � �E = � (3)re � �H = 0 (4)where re = x 1ex @@x + y 1ey @@y + z 1ez @@z (5)rh = x 1hx @@x + y 1hy @@y + z 1hz @@z : (6)In the above, ei, hi, i = x; y; z are coordinate-stretching variables that stretch the x; y; zcoordinates for re and rh. It shall be shown later that when ei and hi are complex3

numbers, the medium can be lossy. Note that (3) and (4) are derivable from (1) and (2).A general plane wave solution to Equations (1) { (4) has the formE = E0 eik�r (7)and H =H0 eik�r (8)where k = xkx+ yky+ zkz. Substituting Equations (7) and (8) into Equations (1) and (2)above gives ke �E = !�H (9)kh �H = �!�E; (10)where ke = xkxex + y kyey + z kzez and kh = xkxhx + y kyhy + z kzhz . Combining the above, we have�!2��H = ke � kh �H= kh(ke �H)� (ke � kh)H: (11)But from Equation (9), ke �H = 0 for a homogeneous medium. This gives the dispersionrelation !2�� = ke � kh (12)or �2 = 1exhxk2x + 1eyhy k2y + 1ezhz k2z (13)where �2 = !2��. Equation (13) is the equation of an ellipsoid in 3-D and is satis�ed bykx = �qexhx sin � cos�; (14)ky = �qeyhy sin � sin �; (15)and kz = �qezhz cos �: (16)4

Note that when ei, hi, i = x; y; z are complex, the wave in the x, y, and z directions areattenuative and can be independently controlled. Under the matching condition, ex = hx,ey = hy, and ez = hz, we have jkej2 = jkhj2 = �2. The wave impedance is then given by� = jEjjHj = jkhj!� = r�� ; (17)irrespective of the values for ei; i = x; y; z and the direction of propagation.3. Single Interface ProblemAssume that a plane wave is obliquely incident on the interface z = 0 in Figure 1.Furthermore, we may assume that the plane wave is of arbitrary polarization. The incident�eld may be decomposed into a sum of two components, one with electric �eld transverseto z (TEz) and the other with magnetic �eld transverse to z (TMz). We will examinethese two components individually.x

y

z

transmittedwave

reflectedwave

waveincident

z=0plane

region 1

region 2

Figure 1: Plane wave with arbitrary polarization incident on the planez = 0.In the (TEz) case, we let the incident �eld in region 1 be given asEi = E0 eiki�r: (18)In the above, khi � E0 = 0, and E0 is in the xy plane. Similarly, we de�ne the re ected�eld in region 1 as Er = RTEE0r eikr�r (19)5

and the transmitted �eld in region 2 asEt = TTEE0t eikt�r: (20)Phase matching requires that kix = krx = ktx and kiy = kry = kty. Hence, we cande�ne E0r = E0t = E0 since they all point in the same direction. Applying the boundarycondition that the tangential electric �eld must be continuous across the plane z = 0*, wehave 1 + RTE = TTE: (21)The magnetic �eld may be determined using Equation (9) for regions 1 and 2 asH1 = kie �E0!�1 eiki�r + RTE kre �E0!�1 eikr�r (22)and H2 = TTE kte �E0!�2 eikt�r (23)where kie = xkixex +y kiyey +z kizez and similarly for kre and kte. We also de�ne k1z = kiz, k2z =ktz and note that krz = �k1z. Then equating the tangential components of Equations (22)and (23), we have k1ze2z�2 h1� RTEi = TTE k2ze1z�1: (24)Combining Equations (21) and (24), we haveRTE = k1ze2z�2 � k2ze1z�1k1ze2z�2 + k2ze1z�1 (25)and TTE = 2k1ze2z�2k1ze2z�2 + k2ze1z�1 : (26)Applying a similar procedure to the TMz component, we haveRTM = k1zh2z�2 � k2zh1z�1k1zh2z�2 + k2zh1z�1 (27)* This boundary condition follows from the modi�ed Maxwell's equation (1).6

and TTM = 2k1zh2z�2k1zh2z�2 + k2zh1z�1 : (28)4. A Perfectly Matched InterfaceThe phase matching condition requires that k1x = k2x and k1y = k2y, or�1qe1xh1x sin �1 cos�1 = �2qe2xh2x sin �2 cos�2 (29)and �1qe1yh1y sin �1 sin �1 = �2qe2yh2y sin �2 sin �2 (30)where �1 = !p�1�1 and �2 = !p�2�2. For a perfectly matched medium, we choose�1 = �2, �1 = �2, ex = hx and ey = hy. Equations (29) and (30) becomee1x sin �1 cos�1 = e2x sin �2 cos�2 (31)and e1y sin �1 sin �1 = e2y sin �2 sin�2: (32)If we now choose e1x = e2x and e1y = e2y, then �1 = �2, �1 = �2 and we can show thatboth RTE = 0 and RTM = 0 for all angles of incidence and all frequencies.If region 1 is a vacuum, then � = �0, � = �0, and(e1x; e1y; e1z; h1x; h1y; h1z) = (1; 1; 1; 1; 1; 1): (33)In order to have a lossy region 2 with no re ections at the region 1/region 2 interface, wechoose (e2x; e2y; e2z; h2x; h2y; h2z) = (1; 1; s2; 1; 1; s2) (34)where s2 is a complex number. In this case,k1x = k2x = �0 sin � cos� (35)k1y = k2y = �0 sin � sin � (36)7

k1z = �0 cos � (37)k2z = �0s2 cos � (38)where �0 = !p�0�0. If s2 = s02+ is002 , the wave will attenuate in the z direction. This kindof interface is useful for building material ABCs in a FDTD simulation.5. Modi�ed Equations in the Time DomainFor the general case of a matched medium, we let ex = hx = sx, ey = hy = sy andez = hz = sz. Then, re = rh = x 1sx @@x + y 1sy @@y + z 1sz @@z . In Equation (1), we write thecurl as re �E = 1sx @@xx�E + 1sy @@y y �E + 1sz @@z z �E: (39)Then, de�ning Hsx, Hsy , and Hsz in terms of the components of Equation (39), we leti!�Hsx = 1sx @@xx�E (40)i!�Hsy = 1sy @@y y �E (41)and i!�Hsz = 1sz @@z z �E (42)where H =Hsx +Hsy +Hsz . Similarly, we can write Equation (2) as�i!�Esx = 1sx @@xx�H (43)�i!�Esy = 1sy @@y y �H (44)and �i!�Esz = 1sz @@z z �H: (45)where E = Esx +Esy +Esz . Note that Hsi, Esi, i = x; y; z are two-component vectors.We now let sx = 1+ i�x=!�, sy = 1+ i�y=!� and sz = 1+ i�z=!�. Writing Equations(40) { (42) and (43) { (45) in the time domain, we have�@Hsx@t + �x�� Hsx = � @@xx�E (46)8

�@Hsy@t + �y�� Hsy = � @@y y �E (47)�@Hsz@t + �z�� Hsz = � @@z z �E (48)and �@Esx@t + �xEsx = @@xx�H (49)�@Esy@t + �yEsy = @@y y �H (50)�@Esz@t + �zEsz = @@z z �H: (51)Equations (46) { (51) described 3-D wave propagation in a perfectly matched medium.The wave propagation phenomenon described by these equations is very similar to thatdescribed by Maxwell's equations with the exception that attenuation may be controlledthrough the �x, �y and �z variables. The FDTD implementation of these equations on aYee FDTD grid is straightforward. Absorbing boundaries at the edges of the simulationregion may be created by choosing appropriate values of �x, �y and �z. Equations (46) {(51) may be seen to include Berenger's equations [15] as a subset for the 2-D TE or TMcase.The above equations involve 12 components of electromagnetic �elds. For a free-space/lossy medium interface, a scheme may be devised using only 10 �eld componentsfor the 3-D case, and only 3 components for the 2-D case. However, this is achieved at theloss of SIMD operation on a parallel machine.6. Computer Simulation ResultsIn order to demonstrate the new method, a 3-D orthogonal grid FDTD algorithm wasdeveloped based on Equations (46) { (51). The FDTD algorithm was implemented as aSIMD code on the Thinking Machines Corporation Connection Machine CM-5. The al-gorithm operates very e�ciently on the CM-5 because the FDTD stencil operations thatneed to be computed at each node involve only nearest-neighbor interactions. The com-munication operations resulting from the nearest-neighbor interactions are at a minimum9

cost since the neighboring processors are for the most part at the bottom of the fat-treecommunication network, where communication bandwidth is maximum.To validate our 3-D FDTD algorithm, we solved a simple problem of computing the�eld radiated from an in�nitesimal electric dipole in free space. An analytic solution wasalso computed in the frequency domain for many excitation frequencies. The frequencydomain solution was then multiplied by the spectrum of FDTD source pulse and inverseFourier transformed to yield a time-domain analytic solution for comparison with theFDTD solution.The FDTD solution was solved in a cubic region of dimension (Nx; Ny; Nz) =(128; 128; 32) grid points. The grid parameters chosen were �x = �y = �z = 2:5 mm,�t = 4:5 ps and Nt = 512 time steps were computed.The in�nitesimal electric dipole was simulated by exciting the Ey �eld in a single gridcell with the source pulseJy(t) = 1�x�y�z h4(t=�)3 � (t=�)4i e�t=� (52)where � = 1=4�f0 and a value of f0 = 1:0 GHz was chosen. The dipole source was locatedat grid location (nx; ny; nz) = (91; 64; 16). The Ex and Ey �elds were obtained by samplingthe �elds at grid location (nx; ny; nz) = (37; 91; 16).The absorbing boundaries used for the FDTD simulation consisted of planar layers ofthickness 8 grid points on all surfaces. Along the borders parallel to x axis, the value of �xwas speci�ed, while �y and �z were speci�ed on the borders parallel to the y and z axis,respectively. The conductivity values were chosen with a parabolic taper decreasing fromthe maximum value towards the center of the grid such that the re ection coe�cient atnormal incidence was R0 = :0001,The Ex �eld computed using both the analytic formulation and the FDTD algorithmare overlaid in Figure 2. The curves due to the analytic and numerical solutions are barelydistinguishable, indicating excellent agreement. Similarly, the Ey �eld due to the analyticand numerical solutions are overlaid in Figure 3. Again, we see excellent agreement.10

Any di�erence between the analytic and numerical solutions in Figures 2 and 3 may beattributed to modeling errors such as the �nite size of the dipole source and the discreteapproximation of Maxwell's equations in addition to re ections due to imperfections in theabsorbing boundaries.

0 100 200 300 400 500 600-10000

-8000

-6000

-4000

-2000

0

2000

4000

Time step

Am

plitu

de o

f E

x fi

eld

Figure 2: Analytic and numerical FDTD solution overlaid for the Ex �eldresulting from an in�nitesimal electric dipole.The CM-5 machine used to solve the FDTD problem is located at the National Centerfor Supercomputing Applications (NCSA) at the University of Illinois. The program waswritten in CM Fortran and compiled using CMF version 2.1. The CM-5 at the NCSA has512 nodes with vector units. CPU times were determined by running the problem on 32,64, 128 and 256 node partitions. For this problem, a total of 0.5 million unknown �eldquantities (128� 128� 32 grid) were determined for 512 time steps. The CPU times areshown in Table 1.7. Conclusions 11

0 100 200 300 400 500 600-1.5

-1

-0.5

0

0.5

1x 10

4

Time step

Am

plitu

de o

f E

y fi

eld

Figure 3: Analytic and numerical FDTD solution overlaid for the Ey �eldresulting from an in�nitesimal electric dipole.Table 1: CPU times for FDTD Problem on CM-5Nodes

CPU sec (Run 1, Run 2, Run 3, Avg.)

32 50.5, 50.2, 50.6; 50.4

64 29.9, 30.0, 30.0; 30.0

128 17.9, 18.4, 18.4; 18.2

256 12.4, 13.2, 12.7; 12.8A modi�ed set of Maxwell's equations have been introduced using complex coordinatestretching factors along the three cartesian coordinate axis. This modi�cation introducesadditional degrees of freedom in Maxwell's equations such that absorbing boundaries maybe speci�ed with zero re ection coe�cient at all frequencies and all angles of incidence.12

The formulation was shown to be related to the perfectly matched layer that was recentlyderived by Berenger for 2-D wave propagation. A 3-D FDTD algorithm was developed fromthe modi�ed Maxwell's equations that uses the re ectionless absorbing interface propertyto implement radiation boundary conditions at the edges of the FDTD grid. The accuracyof the algorithm was validated by computing the �eld radiated from an in�nitesimal elec-tric dipole and comparing against a known analytical expression. The FDTD algorithmwas implemented on the Connection Machine CM-5 and timing results were presented.This breakthrough in absorbing material boundary conditions allows EM scattering to becomputed very e�ciently on SIMD parallel computers.AcknowledgementThe authors wish to thank J. P. Berenger for sending us a preprint of his work and toA. Ta ove for bringing to our attention J. P. Berenger's work.

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References[1] K. S. Yee, \Numerical solution of initial boundary value problems involving Maxwell'sequations in isotropic media," IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302{307, 1966.[2] A. Ta ove, \Review of the formulation and applications of the �nite-di�erence time-domain method for numerical modeling of electromagnetic wave interactions witharbitrary structures," Wave Motion, vol. 10, pp. 547{582, 1988.[3] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand,1990.[4] W. H. Weedon, W. C. Chew, and C. M. Rappaport, \Computationally e�cient FDTDsimulation of open-region scattering problems on the connection machine CM-5," inIEEE Antennas and Propagation Society International Symposium Digest, (Seattle,WA), June 19{24, 1994.[5] B. Engquist and A. Majda, \Absorbing boundary conditions for the numerical simu-lation of waves," Math. Computation, vol. 31, pp. 629{651, 1977.[6] G. Mur, \Absorbing boundary conditions for the �nite-di�erence approximation ofthe time-domain electromagnetic �eld equations," IEEE Trans. Electromag. Compat.,vol. EMC-23, pp. 377{382, 1981.[7] Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan, \A transmitting boundary fortransient wave analysis," Scientia Sinica. (Series A), vol. 27, no. 10, pp. 1063{1076,1984.[8] R. L. Higdon, \Numerical absorbing boundary conditions for the wave equation,"Math. Comput., vol. 49, pp. 65{90, 1987.14

[9] I. Katz, D. Parks, A. Wilson, M. Rotenberg, and J. Harren, \Non-re ective free spaceboundary conditions for SGEMP codes," Systems, Science and Software, vol. SSS-R-76-2934, 1976.[10] R. Holland and J. W. Williams, \Total-�eld versus scattered-�eld �nite-di�erencecodes: A comparative assessment," IEEE Trans. Nuclear Sci., vol. NS-30, pp. 4583{4588, 1983.[11] J.-P. Berenger in Actes du Colloque CEM, (Tregastel, France), 1983.[12] C. Cerjan, D. Koslo�, R. Koslo�, and M. Reshef, \A nonre ecting boundary conditionfor discrete acoustic and elastic wave equations," Geophysics, vol. 50, pp. 705{708,1985.[13] R. Koslo� and D. Koslo�, \Absorbing boundaries for wave propagation problems," J.Computational Physics, vol. 63, pp. 363{376, 1986.[14] C. M. Rappaport and L. Bahrmasel, \An absorbing boundary condition based onanechoic absorber for EM scattering computation," J. Electromag. Waves Appl., vol. 6,no. 12, pp. 1621{1634, 1992.[15] J.-P. Berenger, \A perfectly matched layer for the absorption of electromagneticwaves," J. Computational Physics, accepted for publication, 1994.

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