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1

Chapter 1

INTRODU TION

f the four forces in nature - strong, weak, electromagnetic, and gravitational - the electromagnetic force is the most technologically pervasive. fthe three methods of predicting electromagnetic effects - experiment, analysis, and computation - computation is the newest and fastest-growing approach. f the many approaches to electromagnetic computation, includingmethod of moments, finite difference time domain, finite element, geometrictheory of diffraction, and physical optics, the finite difference time domainFDTD) technique is applicable to the widest range of problems.

This text covers the FDTD technique. Emphasis is placed on the separatefield formalism, in which the incident field is specified analytically and onlythe scattered field is determined computationally. This approach is only slightlymore complex in its basic implementation than the total field approach andmore readily allows for the absorption of scattered fields at the limits of theproblem space. The total field can be easily obtained from the combination ofthe scattered and incident fields. Though slight advantages may be found ineither approach, they are very similar in concept and capabilities. The FDTDtechnique treats transients e.g., pulses) in the time domain, and it is applicableover the computationally difficult-to-predict resonance region in which a wave

length is comparable to the interaction object size.As a forro of computational engineering, FDTD is part of a three-tier

hierarchy consisting of:

Computer Science Stresses the mathematics underlying algorithms as well as the structure

and development of the algorithmComputer Engineering Hardware based on and concemed with hardware architecture and capa-

bilities including parallelism and fault toleranceComputational Engineering Explores various engineering problems via numerical solutions to sys-

tems of equations describing the phenomenon or process in q u s t i o ~

Computational engineering relies on computer science and engineering, but isnot hardware, language, or operating system specific. It requires a computerpowerful enough to accomodate the problem in question, running withinacceptable times and costs while producing the desired accuracy.

Electromagnetic computational engineering encompasses the electromagnetic modeling, simulation, and analysis of the electromagnetic responses ofcomplex systems to various electromagnetic stimuli. t provides an understand-ing of the system response that allows for the better design or modification ofthe system.

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2 The Finite ijference Time omain Merhodfor Electromagnetics ntroduction 3

The PDTD technique offers many advantages as an electromagnetic mod- within an order of magnitude of each other. Thus, lho broadband

' ' d u d o o . . '' ''Hybrid techniques employing the geometrical theory of diffraction (GTD). Broadband response predictions centered about the system resonances or physical optics (PO) along with FDTD can in principie provide predictions

. . . < ' . m v t . . . . . . .Interaction with an object of any conductivity from that of a perfect such as Prony's method, allow arbitrarily long-time response pre

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4 The Finite ifference Time oma n Method for Electromagnetics Jnrroduction 5

It should alsobe noted that at least six kindsof electromagnetic computa- p e l f ~ tc o n d u c t o ~ s ~The r u d i ~ ~..ntary c m ~ p u t ~ rcode r e q ~ i r e m e n t sand p

presented as well. Much more detail about these issues is given in latero Generation (power, devices, klystrons, etc.)

etc.) c h a p t e r ~ . , _,..,..,. T no ... n H waVe11Uideso Reception/detection/radiation (antennas) formula domof Chapter2. This includes limitationson cell and time stepo Coupling shielding/penetration sizes, specifying the incident tield and the objectto be analyzed, estimat-

Scattering ing the computer resources required, and applying an outer boundaryo Switchinglnonlinearities conditionat the extremitiesof the FDTD computation space.

Ali but the firstcan be treated using FOTD.The first area either~ u i r e s The second section treats the basic applicationsof the basic fonnulation o fas for the klystron,the additionof charged particles~ ra 60-Hz calculauonf ~ ~ FOTO given in the first section:power trequenctes wntcnI billion) for analysis. o Chapter 4: Coupling Effects.The scatteredfi.eld fonnulationof FOTD

WhileFDTD is most suited to computing transient responses,FDTD may was fi.rst applied to ao F-111 aircraft to calculate the induced surfacebe the computational approachof choice e v e ~' :hen a ~ i n g l efrequencyor currents and charges from a simulatedEMP field. This brought togethercontinuous wave (CW) response is sought. Thts ts especmlly thec a ~ ew ~ e n ali the elements representativeof FOTO modeling, and for this reasoncomplex geometriesor difficult environments,_ suchs an _antenna~ a t~ sbuned and for some senseof history the modeling effort is discussed in detail.or dielectrically clad, are considered. lnteresnngly,mtenor couphngm t ~me- ~ n l ye ~ t e r i ~ ~c o u ~ ~ ~ gi ~ t r e ~ t e dw i t ~ t h i se x ~ g i e~ . . , d t o_ c o m p l e t ~the ~ n ; ~ - - ~ < : nl1 ~ i n m t i n nwhereinFOTD is the methodof chmce. ACW analysis, using the methodof moments, for example, will most likely fail penetratedby an aperture and comaining an interior wire is presented.to capture the highly resonant behaviorof a metallic enclosure, even when The response is examined above and below aperture cutoff with restmant

_ _ . ~ , +;. ;nt

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6 The Finite Difference Time Domain Me thodf or Electromagnetics

The third section treats extensions to the basic formulation that provide~ ~ 1 , Th, ~ arP. nnite varied.

They represent in most cases the fruition of past research efforts to extend tltebasic capabilities of FDTD.

Chapter 7: Near to Far Field Transformation. Many scattering problems,in panicular RCS problems, require the far fields. In addition, radiationfrom antennas and inadvertent antennas or radiators such as transmissionines require far fields for the radiation pattern. While these prohlems

typically irwolve perrect conuuctors m e t ~ Itney cana so mvo vemocomplex materiais as for a stripline antenna. This chapter develops abroadband time domain near to far field transformation and shows howit may be applied to 2 or 3-u pro[}Jems.Chapter 8: Frequency Dep endent Materiais. An advantage of the FDTDmethod is its capability to produce wide frequency band results from onecomputation with pulse excitation. lf materiais with constitutive parameters that vary with frequency are involved, the FDTD formulation ofChapter 2 must be extended to include this variation if more than an

approximate result is desired .. i r _

ing the fields interior to a volumetric scatterer can be avoided by specifying the impedance relating the electric and magnetic fields at the

~

materiais with both constant and frequency-dependent constitutive pa

rameters.Chapter lO: Subcellular Extensions. Often an FDTD calculation includesa structural element much sma ler than one cell size for calculation.Using a finer mesh throughuut the prob1em space is typtcally toocomputationally "expensive". In this case, a subcellular reduction of themesh on and ~ s s i h l y )about the element is needed. Examples treated inthis chapter include thin wires for antennas with diameters well belowa cell size, lumped circuit elements, and the expansiun technique forregridding a sub volume more finely, as in the interior o f an aircraft whereimportant structural details would otherwise be lost.Chapter 11: Nonlinear Loads and Materiais. Nonlinear materiais aremore easily accomodated in FDTD than in frequency domain methods.Some examples, including transients in antennas with nunlinear loads,

~ ~ a ~ ~ : ; ~ 2 :Visualization. Immense amounts of data can be generatedwith FDTD, in the terabyte range in some instances. Only ~ y app1ying

hensible. This chapter discusses the progress made in the visualization o felectromagnetic fields. It is noted that this area ofvisualizati on is in itselfvcry computationally intensive and demanding.

ntroduction 7

The three chapters of the fourth section treat advanced applications made' ' . . . o .

Chapter 13: Far Zone Scattering. The scattered field FDTD fonnulation .

Severa of the special capabilities of Part 3 are combined to providescattering cross-section results, including the far zon e transformation andfrequency-dependent materiais. Scattering examples are also given inChapters 7 and 9 in association with the develop ment of far zone trans-ormat10n ana sunace tmpeoance.

Chapter 14: Antennas. A basic approach to using Fffi 'D to determineantenna self and mutual impedance, efficiency, and gain are presentedor Wlre antennas, tollowed by resu ts or more challengmg geometnes.

Total fields are directly computed, and far zone transformation and subcell methods from previous chapters are utilized.Chapter 13: Gyrotropic Media. Gyrotropic media possess both strongfrequency dependence oftheir constitutive parameters and anisotropy. Inthis chapter the frequency-dependent FDTD methods of Chapter 8 areextended to include these materiais. Both magnetiT.ed plasm as a nd fenites

The final part of the book provid es the detailed mathematical foundations ~ -~ n . ~ . .

Chapter 16: Difference Equations in General. The curl equations, fromwhich a wave equation may be derived and vice versa, are a set ofhyperbolic equations for which a number of differencing schemes arepossib1e. In the interest of mathematical completeness, the differenttypes of possible differencing schemes are presented. The advantages,even necessity in a practical sense, of the leapfrog method is stressed.Higher order formulations of the leapfrog method are also discussed.Chapter 17: Stability, Dispersion, Accuracy. The stability requirementfor the leapfrog method (and for several other methods), the Courantstability condition, is discussed in detail here along with general stabilityconsiderations. Numerical dispersion, a source of error that exists inFDTD computations cxcept under special conditions, spreads or disperses the scattered field leading to time domain envelope enors and

Chapter 18; Outer Radiation Boundary Conditions. A finite prohlemspace is subject to reflections of scattered fields at the faces of the

radiation boundary condition that absorbs much of the scattered wave,simulating energy scattering into infinite space. A number of approachesexist, and a general discussion uf the different approaches is given.

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8 The Finite ifference Time omain Metho dfor Etectromagnetics

Emphasis is placed on the widely employed Mur absorbing boundary

. Chapter 19: Altemate Formulations. While the scattered field FDTDformulation is extremely flexible and effective, other formulations have

r .., onl en ountered is the totalfield furmalism, an approach. that can be obtained quite simply in manycases) from the scattered field fonnalism. Less commonly encounteredare implicit, as opposed to the explicit, F T formulations given in tlteprior discussions. The sought-after advantage in implicit schemes isarmtramy 1 ng t1me steps. Anomer vana on ts e vecformulation. Botb low and high frequency extensions of the FDTDtechnique are also possible.

Appendix A gives F T equations in other coordinate systems and reduceddimensions. Whilel ess useful than the generdi3-D results given in the previoustext, they have application in special situations. PART 1:Appendix B contains FORTRAN listings ofhasic FDTD codes and assoei-ated compu er codes including a fast Fourier transfonnat ion FFT) code. These FUNDAMENTALislings pruvide precise documentation of the hasic FDTD equations given in

While providing a mathematical basis for FDTD, this book is intended more ~ V l ' I I \ . . . . L l1

as a practical g u i d ~~ o rstudenls and r ~ s e a r ; h ~ r s .i n t e r ~ s t ~in a ~ ~ y i ~ gl ~ T D

organization of the book, with practical fundamentais preceding the math-ematicaJ basis.

Because FDTD is an area of ongoing research and development, this bookcan provide only the fundamentais and some example applications, with moreadvanced topics available only from the current literature.

L

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Chapter 2

SCATTERED FIELD FDTD FORMULATION

, ~ ~ ~ ~ ~ - - ~~ - . : ~ ~ ~

We begin by examining lhe differentiaili me domain Maxwell equations ina linear medium:

V x = - d B d t 2.1)

VxH = dD/ 11 + J 2.2)

V o p 2.3)

V B O 2.4)

where

D = eE 2.5)

.,

This is ali the infonnation needed for linear isotropic materiais to completeiy~ l - y I O . . ~ ; ;specif1ed and satisfies the Maxwell equations. Conveniently, the field andsources are sei to zero ai lhe initial time, often taken as time zero. The lwodivergence equations are in fact redundant as lhey are cuntained within the curlequations and the initial boundary conditions.

Thus, the slar ting point for the FDTD fon nulati ons is lhe curl equations.They can be recasl inlo the form used for FDTD:

aHtilt o Ia

" - ( V x E ) " - H 2.7)

aEtilt o"a

( V X H 2.8)EE E

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I

12 The Finite ifference Time omain Metho dfor Electromagnetics Scattered Field FDTD F ormulatio n 13

where we have Iet J = aE to allow for lossy dielectric material and have E _ Brotai =: EiDCident Escancred (2.9)

enn aThc fonnulation only treats the electromagnetic fields, E and H, and not the

H HtDial Hincident Hscatten:d (2.10)

so that any linear isotropic material property can be specified. In the formula- The rationaJe for the separate field approach is that the incident field compo-tion developed here we do not attempt to simplify the calculations by normal-

nents can be specified analytically throughout the problem space while theizing to a unity speed of light, or letting the permittivity and permeability of scattered fields are found computationally and only the scattered fields need tofree space to be nonnalized to I, as proposed by some practitioners. We feel h h ' .. ~ . ,_ .that this removes lhe intuitive and physical basis of lhe ca1culations for very imponant one. The scattered fields, emanating from a scattering or interactionlittle, i any, gain in computationaJ accuracy or speed. object, can be more readily absorbed than a total field by an outer radiation

It is easily shown that we need only consider the curl equations as the h ' ~ oh. ~ h , ; . . : .divergence equations are contained in them. To do this simply take the diver- especiaJly imponant in situations in which the scattered fields are desired andgence of lhe curl equations (2.1 and 2.2) to obtain are of u ~lower amplitude than lhe total fields.

On a more philosophicallevel, lhis SCf, aration allows further insigltt int o tlteinteraction process.

V (VxE = -JB/Jt) -> O = -J(VB)/d t-> V . B = constamThe scattered wave arises on and within the interaction object in response

to the incident field so as to satisfy the appropriate boundary conditions on orv (VxH = d D/d t+ l ) -> O= d(VD)/dt V J within lhe interaction object. These boundary conditions are tlte Maxwell

d ( V D ) / d t - p / t ( f romcon t inu i tyVJ+p / t = O ) equatlons themselves, which in lhe limit of a perfect conductor require E ' ' -= Einci

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14 The Finite Dijference Time Domain Me thodf or ElectromagneticsScattered Field F T F ormulation 15-

V X H'otol = e d E t o ~ a '/dt + crE 'l (2.12) ' V X (Hinc + Hscat) = eod(Eino + E'c"') dt (2.22)

wltile the incident fields traversing the media satisfy free space conditionsNow, subtracting the incident fields we obtain the equations goveming thescattered fields in free space:

V X Emc - l

0H 'Jdt (2.13) v X E 1 = - .L0dH'c"'/dt (2.23)

V x H i n c = 0Einc/dt (2.14)

V X H a ~= E0iiE'ca1/0t (2.24)

as expected. Note that these equations could have been found from theRewriting lhe total field behavior as equations for the scattered f1elds in a media hy letting the media hecome free

space; i.e., Equations 2.17 and 2.18 become Equations 2.23 and 2.24 when

V X ( E ' ~ +E ') = - ~ a ( w+ H'g')tat

_ cr*(Hinc+H'ct)(2.15) ~ - - > ~ o

E --> Eoo --> o

V X (Hinc + HICal) = ea{Einc + Esc"')ldt o --> o(2.16)

+ 0\D + IIn summary, only one set of equations is needed for the separate field

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16 The Finite Difference Time Domain ethodfor ElectromagneticsScartered Field F T Formulation - 17

'For a perfect conductor u = -. and for this situation Equation 2.29riH cr cr--

' ~ ~, \ ,,., (2.25) Escat - E i n c (2.30)'

~ dt- ~ ( V X h J

Inside the perfect conductor we apply Equation 2.30, rather than 2.26 with

a =

Thus, if only free space and perfect conductor are present. only aspccification of the incident field, the free space equa tions 2.27 and 2.28 for theiiR cr cr scattered field, plus the relation Equation 2.30 are needed to apply FDTD. A

dt E E further simplification is to note that Equation 2.30 need only be applied at lhe(2.26) surface of the perfect conductor. Interior portions of the perfect conductor, if

(,-E_) iiE' ' I oresent, are completely isolated from the rest of the problem space.

'dl T ; : \ > i We now differcnce the free space scattercd field equations. In essence finite

differencing replaces derivatives with differences:

We could difference this set of scauered field equations, but it is moreinstructive to difference these equations in the limito a perfect con ductor first ar . r(x,t,) - r(x,t,) r(x,t,) - r(x,t,)and then the equations as presented here. This allows one to see lhe essentials = hm = (2.31)uf the differencing scheme in the perfect conductor case, as this is the most

dt < i t ~ O At A;

~ i h l WP wiil then re fnm to the more e:eneral case of a

scatterer with finite , Jl, o, and o. ar Iim f ~ ~ ~ ~ ~ ~= f(x 2,tJ f[x 1,t) (2.32)dx

-11-->0 & &

' .. n T n ' ' 'NOutside the scatterer the scattered fields satisfy the free space conditions where in the above approximation

1:1 and x are finite rathe r than infinitesimal.In short, calculus becomes algebra.

where o u O, jl '= llo and e == .,. so that Equations 2.25 and 2.26 Some criticai issues aside from this algebraic replacement include:reduce to

What form the differencing takes

aHct 1- V X Escat) 12.271: ~.. s e an explicit central difference scheme here that only retains first

- t enn , . The E nd H ield' ru< I

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-18 The Finite Difference Time Domain Methodfor Electromagnetics Scattered Field FDTD Formulation 19

OE 1 1 (dH' ' OHscatjThis completes the perfect conductor separate field formulation. We next

' ' 0 l dz - dx ) L . , O jdt code using the FORTRAN language.O A r n N n l l r T n D ' '' rn

OE:cal( dH d H ~

lWilh a little more work, we can recast the above formula into the forrn used= _1_ - - ' - - - - ' - {2.33c) in a perfectly conducting version o f an FDTD code. We quantize space, lettingdt E

0dx iJy

x = I 6x, y = J 6y, and z = K 6z, and time, letting t = n 6t. We can defineuniform cells in the problem space and locate them by the I, J, K indices.Within each cell we choose to loca e the field components at offsets (Figure 2-l) as given by Y ee. 1This Yee cell , as il is called, results in spatially centered

'S = ~ l ~ ,_o; _J (2.33d) differencing. In Yee notation E,n (I,J,K) represents the z component of theiJt J.l0

z dy electric field at time t = n6t and at spatiallocation x = I6x, y = J6y, and z =(K+l/2)6z, as can be seen in Figure 2.1. Other field components will havedifferent offsets as can be seen from the figure.

As a mnemonic aid we sha11 write E'cot,n in the l,J,Kth Yee cell as lhe

dHscat rdE: ' d E : ~ ~1FORTRAN subscripted array variable EXS(l,J,K), with the time step deter-, = (2.33e) mined by an index (integer variable) N in lhe code itself. Similarly H;at, n+}; ; ; - - - ; ; ; -

o .. . : .for EXS, is detennined by the H 1 location in the Yee cell. The temporal offsetis also understood, so that lhe time index variable N in the computer code

-

dH (dE dE ) magnetic fields (this oJercan be reversed with no loss o f generality). We writez 1 X Yi i t = ll o ---a;-- ---a;- {2.330 E me as EXI(I,J,K), and for the lossy dielectric version of the code very natu-ra{Iy we shall write O inc / dt = E inc as DEXI(I,J,K).

With minimal a\geb:aic m a n i o ~ l a t i o nthen, lhe FDTD eouations for scat-For simplicity, we will only treat the pairE.'' '; and Hy' ', lhe other components tered fields propagating in free space are recast in the form of FORTRANfollow naturally. (Note that E.. ' and Hy ' ' could be used alone in a 1-D statements (remember these are FORTRAN assignments, not equalities)transmissio n line analvsis with propaRation in the z direction i f only E and Hfield oomponents ex.ist.)

EXS(I.l,K) = EXS(I,J,K)Replacing derivatives with differences we find+ ~ [ H Z S ( I , J , K ) -HZS(I,J -l ,K)

' 'eo 6Y

_ E ~ a t . n - 1scat.n-- AH scat.n-2 HYS(I,l,K) -;;Y S(J,J, K - I ) l (2.36)E:cat,n I Ml, 2= - , -

At E Av A' (2.34)

UV

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20 The Finite ifference Time omain Method for Electromag netics Scattered Field F T F ormulation 21

t we p ~ v i o u s l yderived the equations for scatlered field propagation in lossyI IQ

/ ... A((. ''" 't f ~ t I.. IPI ) H"' H't ~ ~

a(l..lol,. ~ { l t t. . . . . ( ~ - ~ J a H ; , ,t(v E"") (2.25)o ; ; a . - ;; X

' oL U . ~li

l t ~ 1 1/ ::___ = - " E' ' - " E' ' - \' :c - + f e t. l l l (2.26). . . ,111 lfiJ (vxH''"')

'AGURE Z-1. Convention for imposing lhe (I,J,K) indices on lhe (x,y.z) problem space. andlocation of tbe six-field evaluation poinu in a typical cell. These are equivalent to the equations in the original scattered field pape r 2. We

where it is understood that terms such as llZ, .6.Y, e 0 , etc. would ctr;., - . ct - R ' ' l ' l 2 beca - actually be stored as FORTRAN variables. The b o v ~ ~ o t t i o nshows E at time

1 erencmg use m e erence , use 1t ts stmpler and (as implemented

h l ~ . ; _ N lhere) wi_ll yi_eld a m o ~stable formulation for higb values of conductivity.

and the curl of H ai time n - N 1/2, where t t. Next, H is evaluated '"" n g 'at n =N + 1 1 from its earlier value at n =N - 1/2 and the curl of E at n =N. "This interleaves E and H tempora1ly and results in a centered differenceor "leapfrog in time" approach. After each update of E and H the index f.dE' - ' - -"' .

,aE; ~ \ 'is increased by 1 and the process 1s repeaiea. l1 ie spauar m01ces m t t

the curl ca1culations are detennined by the Yee cell geometry. The curlcalculations are a1so center differenced, and represent nearest-neighbor inter- which is annroximaterl usinP centererl finite differenc "ctions.

The above equalions apply to cells containing free space. In electric fieldcell locations I,J,K containing perfect electrical conductur the FORTRAN e(E' - E'n-l) + crdtE'" :o - crtE'" - (r-Eo}AtE;,,assignment EXS(I,J,K) = -EXI(I,J,K) is made at each time step rather thanFORTRAN assignment Equation 2.36. + ( v x H n ~ } ~ t (2.39)

2,5 LOSSY MATERIAL FORMULATION

Usingand can be reexpressed as

~ E + E'Cft ' (e+ o-Lu)E'" - eE'n-1 o-dtEi.n- e - e0

}.1tEi,n

+ VX Hs,n-i ).11 (2.40)H' = H' < + H cat (2.10)

L_

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22 The Finite Difference Time Domain Method for ElectromaRnetics Scattered Field FDTD Formulation 23

or finally n. Let us. now _ on_sideri m p l e ~ e n t i n gEquation 2.42 inFORTRAN.

E' = ( ' = ) E ' ' - ( c r ~ , ; ;)E'constant multiplying tenns, but would prefer to store and reuse them.Onthe other hand, storing allof the multiplying terms for each field component

-[(e-eJ'']il'" +( v x w ~ J ( ~

J(2.41)

reduce the storage required significantly for most problems, we recognize thattypically only a few different typesof materiais will exist in a particular

e+olu e+cr6.t FDTD space. The space may have millionsof cells, but perltaps only afew different materiais with different pennittivities, conductivities,andpenneabi Jttes will be modeled in the space.The multiplying constants

The useof E'" in the expressions involvinga using the most recent valueof need to be evaluated for each typeof material rather than for each cell,electric field to determine the current density, is the keyto obtaining stability with a pointer array used to designate which material is in a given I,J,Kfor large conduct1v1ty values. ote that asa becomes mttmte hquatJonl .41 ocat10n. We let IDUNE(I,J,K) be the poimer array designating thecorrectly givesEcat"" -E' '. type of material interacting with the xcomponent of electric field at

We can now write theFORlRAN-Iike ex:pression for updating the electric Yee celllo cation I,J,K. We have foundit convenientto let IDONE( ,J,K)""Ofield in a lossy dielectric medium a ~ mean that in cell I,JK lhe x component o f electric field is in free space,

IDONE(I,J,K)= l for a perfect conductor interacting with this electric field,and values> l for user-defined valuesof permittivity and conductivity at this

' ' v ' . ' 'v,-i( ' _ I loc:ti? ' ' \.E+U.l)

( ( e - , ) ~ 1 . , , , .' n lossy dielectriccontained in the FDm space, we can express the ahove

(. ""' lu i v' constant multipliers for this material as:\e+crt) ' l e + a t ~

' ' ECRLY(M) = DT ':(I.J,K) -2 - H: (l .J -l,K) -2 ( l (EPS(M) + SIGMA(M)'DT) ' DY =+ &.. e + a t (2.42) (e+a6 t )6Y' ' ECRLZ(M)

DT ';(l,J,K) -2 H ~ ( I , J , K- l ) - ~r ~ "c)

=(EPS(M)+ SIGMA(M)'DT)' DZ = ( e + a t ) , 2

+

ESCTC(M) = EPS(M)/(EPS(M)+SIGMA(M)'DT)= 'e+ at

In a similar way the corresponding equations for updating the other electric EINCC(M)field components can be obtaincd. Should we wish to consider lossy magnetic = SIGMA(M)'DT /(EPS(M)+ SIGMA(M)'DT)media the corresponding magnetic field equations canbe derived using lhe cr"tsame approach. =

e+ at

2.6 LOSSY D ::-FDTif r u rcuJJE EDEVCN(M) DT'(EPS(M) EPSO) (EPS(M) + SIGMA(M)' DT)

We now considerextendin >" the nrevious oerfect conductor FDTD code to (e-eJ"'lossy dielectric materiais. The only addition to the previously described perfect e+ atconductor FDTD implementation is that Equation 2.42 must be used to updateelectric fields at spatial locations wheree # fv Because we are assuming thatno magnetic materiais are present, Equation 2.37 canbe used throughout the Using these definitionsth_e F O ~ T R A Nexpression for Equation 2.42 forE ; in

'

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24 The Finite ifference Time omain ethodfor Electromngnetics Scattered Field F T Formulation 25

EXS l J K\ o EXSIJ,J, K) ESCTC(IDONE(l,l, K))In addition to the time-stepped field subroutines we must also have outer

. . - ~ . . ' ' '- EINCC(IDONE(l,J,K))' EXI(l,l,K) outennost portion of the problem space

. PnPVCNIIDQNEII J K\)'DEXHI,J ,K\When time ~ p p i nis completed we must have a way of outputting

. . .+ (HZS(l,J,K)- HZS(l,l - , K ) ) ' ECRLY(IDONE(l,l,K))

(2.43) time step and then at lhe end of the run "dump" it uut as a listing or as a

- (HYS(I,J, K ) - HYS(l,J,K - ) ) ' ECRLZ( DONE(l,J,K))file for postprocessing. This output process may involve transformation ofthe near zone FDTD fields lo the far zone for radiation or scattering calcula-tions.

In a lossy dielectric FDTD code the FORTRAN assignment (2.36) is1nese coae reqmrements are summan7.eu as:

executed for cells containing free space, (2.43) is executed for cells containingDriver

JOSSyuteleCt

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26 The Finite Difference Time Domain Melhodf or Electromagneti( S Scattered Field FDTD Formulation 27

Test object detinillonf ieM . .,

O I J ~ M I

other more complicated material; it is usually convenient to set he default material to free space; lhe array of "flags" can be readfor a preprocessing check of the geometry and composition of the UPIIJI.TE E FIEU>Sobject. I

E,H field algorithms I APPLY ORBC. Calculate the response of a component from its own prior time valueInd that of the nearest-neighbor field quantifies (Es around Hs and Hs

around Es) according to the type of material present at that component UPDJ\TE I FIILDS

location: Iree spaceSilVE IIUII 20 11: P IJ:LDS

Lossy dielectricIoss y magnetic

Peifect conductor SAVII: FAli ZOifE FIELDG

Outer radlation boundary condion ' .

the problem space ~Data saver WRI'I'II OU'I'P I'l'

quantities in the FDTD computation space in arrays at chosen time steps FIGURE 2.2. fU TD flnw chart.Far zone transformationEvaluates tangential electric and magnetic currents on a closed surfacesurrounding the obiect and computes the corresponding scattered orradiated fields in the far zone REFERENCES

The architecture of the code is quite straightforward given the modular v v onature of each subroutine. FORTRAN COMMON blocks can be used to pass equa ions in isolmpic media, IEEE Trans. Anl. Prop.,l4 3), 302 1966data between subroutines, with the main Driver program orchestrating events. 2 Holland, R., Simpson, L,, and Kunz, K. S., Finire-difference analysis of EMP couplingA simplified flow chart of the code appears in Figure 2-2. A computer code 10 lossy dielectric structures, IEEE Trans. EMC. 22(3), 203, 1980,

liSting is included in Appendix R.

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9

Chapter 3

FDTD BASICS

3 1 INTRODUCTION

In this chapter, the practical considerations o f FDTD calculations are treated:

Cell size Time step size lncident field specification Scattering object construction Direct calculation o total fields

Outer radiation boundary condition Resource requirements

The choice o cell size is criticai in applying FDTD. lt must be small enoughto permit accurate results at the highest frequency o interest, and yet be largeenough to keep resource requirements manageable. Cell size is directly affected by the materiais present. The greater the permittivity or conductivity, theshorter the wavelength at a given frequency and the smaller the cell sizerequired.

Once the cell size is selected, the maximum time step is determined by theCourant stability condition. Smaller time steps are permissible, but do notgenerally result in computational accuracy improvements except in specialcases. A larger time step results m mstabthty.

When using the scattered field FDTD formulation the incident field mustbe analytically specified. An infinite variety o waveforms are possible, butexpenence fias led to the Gausstan pulse as the mctdent wavetorm otchoice. The exception to this is when frequency-dependent materiais areincluded, in which case a smoothed cosine pulse has advantages, as de-

scnbed m Chapter 8FDTD is capable o providing very large dynamic ranges, which can be

in excess o 120 dB. Accuracies o 0.1 dB or less can also be obtaineddependmg on the cell stze, frequency, and shape ot the obJects bemg constdered.

The scattering object can be constructed for the FDTD calculation usinginteger arrays for each fteld component. Dtfterent mteger vlues mdtcate adifferent material and determine which FDTD field equations are used withwhat multiplying constants (depending on material) to update the field compo-

nem. In lhe scheme described later m thts chapter, free space FDTD equauonscorrespond to a value o zero stored in the integer array, while perfectlyconducting field equations are used for E field components corresponding toa value o 1 Numbers >1 may be used for lossy dielectric and/or magnetic

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30 The Finite Difference Time Donwin Metho dfor ElectronwKnetics F T asics 31

materiais. Any shape and material can be i n c l ~ d e dwithin the cunstraints of the material we must use the wavelength in the material to detennine the maximum. ' ,., . . . ~ ~ . '' . .For problems which do not have an incidem field, antenna radiation calcu- cells in the material that are much smaller than if only free space and perfect

lations, for examplc: an FDTD computer c o d ~based on the s c ~ t , t e r e dfie d ~ n d u c t o r s:::e e ~ ~ g - ~ ~ s i d e - ~ ~ -I a uniform e l ~ ~ ~ z ~is u s e d . ~ ~ : o ~ ~ ~ ~ tthis. .,Section 3.6. greatly increase the number o cells needed. Possible measures to deal with this

Whether total or scattered fields are computed, Mur's first or second order include nonunifonn cells (smaller cells in the dense material, larger cells

absorbing boundaries provide a relatively reflection-free and easily imple- outside) or surface or sheet impedance methods, which are considered in

mented tennination for the FDTD space. Other absorbing boundary condi- Chapter 9.

tions are available which provide better absorption with fewer cells reqmreu ~ unocrstana wny n ~ceu s1ze must oe mucn smaucr tnan onc wave.cngm,

between the object and the outer boundary, but at the ex pense of added constder that at any parttcular time step the FDTD grid is a discrete spatial

comnlexitv. These are discussed in Chapters 14 and 18, although for most sample of the field distribution. From the Nyquist sampling theorem, there

FOTO users, the Mur absorbing boundaries are adequare and re atiVCIY m u s ~ at east two samples per spatial period (waveJengtnJ m oroer or tne

simple to apply. spatJ.al mfonnation to be adequately sampled. Because our sampling is not

After the user determines the cell size, a problem space Iarge enough to e,.act, and our smallest wavelength is not precisely detennined, more than two

encompass the scattering ohject, plus space between the object and the absorb- samples per wavelength are required. Another related consideration is grid

ing boundary, is detennined. Also, a number of time steps sufficiently large to dispersion errar. Oue to the approximations inherent in FDTD, waves of

allow a full characterization of the interaction of objcct and fields, most different frequencies will propagate at slightly different speeds through the

importantly any resonant e ~ a v i o r ,is estim:UCd. From lhe n u ~ b e rof Yee cells grid. This difference in propagation speed also depends on the direction of. ; :- ' : ~ ' ' ' 'ee an ~ P " . ' ~ ~ , . . .

closely estimated. These resources mclude CPU t1me (detenn1n ed m part by the dispersion error must be reduced to an acceptable levei, which can be readily

speed of the computcr being used) and the amount of solid s ~ a t ememory accomplished by reducing the cell size.: _.,'. . . . ~ ~ _,'~ K f \ : V Janu e x ~ e n u c ~ ~

problem geometry must be accurately modeled. Nonnaliy this will be metneeded for the calculation and for storing the results.

Upon completing this chapter, the reader should be able to make FDTO automatically by making the cells smaller than 1/10 . or so, unless some

calculations with an appreciation of the general constraints and requirements special geometry features smaller than this are factors in detennining the

' ' response of interest. An ex ample is thin wire antennas, in which a change inthe wire thickness from 1/10 to 1/20 (and even smaller) will affect the

3.2 DETERMINING THE CELL SIZEantenna impedance. An other ex.ample is co mputation of low levei scatteringfrom smooth targets in which the "staircase" effects of modeling a smoothsurface with rectangular cells may cause significant errors. Good results in

The fundamental constraint is that the cell size must be much l c ~ sthan the these and similar situations may require extremely small cells, or alternative

smallest wavelength for which accurate results are desired. The obvious ques- measures such as sub-cell modeling or special grids which approximate thetion is How much less?", and to this must be added the question, How actual geometry better than Yee cells of the same size. These situations are also

accurate do you want the results to bc?" An oftenquoted constraint is 10 cells discussed ]ater in this book.per wavelength", meaning that the side of each cell should be l/lO or less at Once the cell size has been detennincd, thc number o f cells nccdcd to modclthe highest frequency (shortest wavelength) of interest. For some situations, the object and a reasonable amount of free space between the object and the

sucn as a very accurrue 1 01 ng - ,FDTD space ~ cells is detennined. Assuming a 3-D problem, a total numberor smaller cells may be necessary. On the other hand, reasonable results have

been obtained with as few as four ceils per wavelength . I th e cell size is made o f cells from a few hundred tho usand up to severa million can be accomodated

mucn smauer 1uan ts ' Y'-1 ' Y - closely for reasonable results to be obtained and significant aliasing is possible supercomputers. We now consider other basic asp ects o f FOTO calculations in

for signal components above the Nyquist limit. the following sections o f this chapter, with the final sectio n devoted to estimat-A word of caution here is that FOTO is a volumetric computational method, ing the compu er resources required for an FOTO problem once the number o f

' h ;r me nortiun of the comoutational space is filled with penetrable cells has been detennined.

L

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32 The Finite Difference Time Domain Methodfor ElectronuJgnetics FDTD asics 33

3.3 TIME STEPSIZE FOR STABILITY 3.4 SPECIFYING THE INCIDENT FIELD

Once the cell size is detennined, the maximum sizeof the time stepL .t Akey advantage of using the scattered field fonnulation is that the incidemli. te follows from the Courant condition.To understand the basis for field is specified analytically. In this section the practical considerations for

the Courant condition, consider a plane wave propagating through anl direction, an incident plane wave from this direction can be specified as

for stability.

E= [E,l+E,.Y(t+(i'i)/o+R/o) (33)_ ~ ; - - ~ Iv .t ~ I / - - , - - , + "{;:-:\2 (3,2), ..., .. ,

Experience has indicated that for actual computations theLit valu_e givenby the equality in(3.1) or (3.2) ': 'illprovide~ c u r t er ~ s u l t sa n ~ ~ ~ ~ nmost j j

= [ ~ -~ } ( t + ( i1)/o+R/o) (3.4)SUUa JOnS IIIVIC < l U.

where and$ are the spherical coordinate system unit vectors,Tj is thef t.t. In fact, when thc equality holds, the discretized wave most closelyapproximates the actual wave propagation, and grid dispersion errors are i ~ ~ d a n c eof free space, cis the speedof light, andf is the vector from themtmmtze.

e v ~ ~ a t ethe. ~ ~ i d e n tfield. The function f(t)m:;'be any functionof time, aowever, exceptions to this occur.One situationin which thetime_ s t ~ pmust be rcduced rclative to(3.2) is when lhe conductivityof the matenalts sine wave or a pulse, for instance. R is an arbitrary reference distance. Formuch greater than zero. For conducting materiais(a> 0), stable calculations transient calculations the pulse must propagate into theFDTDspace ratherrequirc time steps smaller than lhe Courant limiL Thisis usuallynota n : > b l e ~ than suddenly appear at the scattering object, and the valueof Ris chosenbecause in most calculationsthe time step size is set by the speedof hght m accordingly.free space. As the velocity in the conducting material will be smaller than in We can easily obtain the amplitudesof the Cartesian componentsof the

rho me ste in anFDTDcalculation that includes both free space incident fields asand conducting materiais willbe such that the Courant limit will~ s a t i s f i ~everywhere. However, the short wavelength inside highly conductmg matcn-

E, = E9 cose cos4l - E, sinl lnl" reauire much smallerFOTO cells than in surrounding frce spaceregions. ~ LJ&C s sm ''Another situation where the time step must be reduced below the Courant E,= -E& smBJimit occurs for nonlinear materiais. Chapter11 contains discussions of stabil- H,= (Ee sirujl + E cose cos$)/TJity concerns forb a ~nonlinear (Sections 11.2 and 11.3) and conducting Y ~tE9 c o s ~ + E cosB sin4l)l TJH E s i n / n1,3CCL On >J '

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34 The Finite Difference Time omain Merhodfor ElatromaRnetics FDTD asics 35

Next consider specifying a particular field component for a Gaussian pulse. . ' ' ' ' .

let the function f(t) be a Gaussian pulse and express the x component of the 1.0\lectric field of the plane wave as

E ~ I . J , K tE p p ( - a ( ( t - ~ ~ t ) ' ) ) (3.5) O.B

I \R, O I \At + f' T c + R/c (3.6) .t " S nand incorporates the relative time delay for the Ex component in cell I,J,K at \ime step nAt. For our plane wave incident from the 6,4fl direction we llave 0.2'f ' . f = ((I -1) + 0.5)t.x cos$ sin8 0.0

+ J - l )Ay sincjl sin8 1.7\0.00 0.25 0.50 0 .75 _\ 1.00 1.25 1.50

+ (K - I ) coseFlGURI : 3-1. Gaussian pulse for l-em cubicaJ FDTD cells for a time step ai Courant stabilitylimit and = 32.

Note the l/2 cell offset in (3.7) corresponding to the location in the tf,J,K.) _reecell of the E, component. Corresponding offsets must be included for each field is -120 dB (six significam decimal digits) this value of a resu lts in truncatio ncomponent. frequencies well below this levei. As a ma,.imum a 32-bit world length allows

We now define a and and constrain t soas to provide a suitable pulse as at most a liUle over 190 dB dynamic range.

pulse extends infinitely i ~ Y t i m eours must be t r u ~ c t e din our calculations, and often used for other examples in this book, a 3-D cubic cell with l-em sides.the effects ofthis must be wnsidered. We must also specify the time duration ~ ~ ~ ~ ~ ~ ~t . h ~ r ~ o u r a ~ t~ t a b i l i t yconditio? we obtain a 6.t o ~ I.924E-1l.s or

To start, determine tlle duration of the Gaussian pulse. We select ~ = 32, the Gaussian pulse of Figure 3-1. The approximation to tlle pulse is mostwhere is the number o f time steps in the Gaussian pulse from lhe peak value noticeable near the peak value where the curve of the actual pulse is awroxi-to the truncation value. The pulse will exist from t = O until 't = 2 ~ . 1 . t ; mated with straight line segments between the discrete values.approximated as zero outside this range, with peak value at t = ~ 6 . 1 . The fast Fourier transfonnruion (FFf) of this pulse, shown in Figure 3-2,

The value at truncation (at t =O, 2f36.t) is detennined by a , andas seen from mak.es it clear that the truncation of the pulse did not introduce unwanted high(3.5) the Gaussian pulse at truncation will have a value e x p - a . ~ i l l ) 2 )down frequencies. At ten cells per wavelength we would hope to get accurate resultsfrom the maximum value. We now need to detemrine a so that this truncation from our FDTD calculalio ns for freauencies unto 3 GHz, and from Fiure 3-does not introduce unwanred high frequencies into our spectrum, and yet does 2 it is ele ar thal our Gaussian pul sei s providing relativ ely high sign allev els outnot waste computation time on detenn ining values o f the incident field that are to this frequency. On lhe other hand, we may be concemed about noise andessentially zero. instability if we have appreciable enere:y in the incident wave for wavelenl :ths

There is no correct answer, but a practical solution is to let a change with in which our cell size is less than four cells per wavelength. This correspondsf3 sothat at truncation the amplitude of the pulseis always redu cedb y the same to a frequency of 7.5 GHz, and from Figure 3-2 we see that our spectrum isvalue. We let o:= (4/(fill,t)f. Thus, a1 truncation the pulseis down by exp(-16), down by approximately 120 dB at this frequency- small enough to provideor almost 140 dB. Because a reasonable goal for a single precision calculation stability.

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36 The Finite Dijference Time Domu in Methodfor Electromagnetics FDTDBasics 37

~ .............. _.,... ;.:;' / .' . maoeriao ; a< mi> =aoion. r o r

in the volumetric region containing the dielectric materia1, as the nanower example , a conte nt o f O might spe cify free space; 1, perfect conduc tor; 2 a

pulse will contain significant energy at wavelengths too small to be adequately l ~ s s ydi_elec.tric ;-ith a specific pennittivity and conductivity; 3, for a lossy

sampled inside the dielectric. Thus, the ~ aojustment ~ m r um me a vematerial type that exists in the FDID space are set once a nd the correspond-scheme for detennining the Gaussian pulse parameters ts that the value of P

must be increased inversely as the time step size is decreased below the ing multiplying tenns in the FOTO update equations are calculated and

Courant limit.stored before time stepping is begun. Note that because the materiaJ that

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38 The Finite Difference Time Domnin Methodfor Electromagnetics FDTD asics 39

7

DO 101-4 7 J 70010J-4 7 6

I ~ 1:NE.7) I D O N ~ I J , K ) 2 g g j ~ ' , j~ J 10 CiliNuE

,5 IFYJE.WDFOR/,,J,K)-11IF J NE IDFIV ,J,K -11

1 NTINUE 5

4 5 8 7 4

., 4 7I

Hy QDFIV

. i\ Iu : I= =Ex IDONE) HxQDFOR)

FIGURE 3-4. FORTRAN code and cnrresponding view of 3 x 3 ceU FDID dielectric plate in FIGUREJ-5. Anempt to build magnetic material plate with same FORTRAN logic as forlY plane at z = K . - p a W< composeo or ma enru type , m wmcn tne constttuttve param-tropic materiais with diagonal pcrrnittivity or penneability tensors is str:ught- eters of this material are set elsewhere in the FDTD computc r c ode. Using t hcforward in FDID. IOONE and IDTWO arrays, this plate can be built with the FORTRAN shown

As six separate arrays are usea tne user can comrot muepenuenuy m rJgure.>- m m1s examp e, note me two 1t' statements tnat must be me .uded

placement of dielectric and magnetic material in the Yee cells. The placeme nt so that the plate is properly formed. Merel y setting the ID arrays for nine cellsof the field components in a Yee cell is shown in Figure 3-3. For example, (3 x 3) will not result in a plate with smooth sides.setting an element of lhe IDONE array at some I,J,K location is actually To illustrate this further, consider building the corresponding magnetic

~ 1 nofelectr ic. ' oh. - .

field located at position I+ 1/2,J,K in the FDTD space. Setting an element of the magnetic plate, the object generated woul d actually be unconnected, as illus-IDFOR array at some I,J,K location is actually locati.Rg magnetic material ~ ~ ~ din : : ~ ~ e - r - ~ The correct way to build the magnetic plate is with the

'' of lccMerl ' nn

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l

40 The Finite Difference Time Domain Me thodf or Electromagnetics FDTD asics 41

comer of a corresponding m a g n e t ~ ccube, offset one ~ a i ~cell in the x / ; . ~ In FOTO calculations the zero electric fields are at the Yee locations

directions, wc wouta sel me 1oca : - ~ , ,HY(I,J+l,K), HZ(l,J,K+l) as magnetic material using the TDFOR, IDFIV, the H field locations just outside the conducting region. Thus, the FDTD

and IDSIX arrays. This example i n d i c a t ~ st h ~following correspondence "surface'' is approximated as being halfway between these electric and

between "buildmg dtetecmc ana magn ' ' '' ' : ~ . : ~

the outer electric field locations set to perfect conductor.

Dieledric object Magnetic objedSo just how big is an FDID object? It is generally a small fraction of a cell

larger than the geometry employed. We must keep in mind that FDTD is an

mnhlW' n n = IOFOR(I+l),K) approximate method, and our physical reality o f the plate is only approximatedDTWO(l,l.K) JOFIV(I,J+l,K) by FOTO.

IDTHRE(I,J K) = IOSIX(I.J,K+l) Retuming to the general topic of "building" objects, a good approach is togenerate squares (in 2 O or for thin objects in 3-0) orcubes ofmaterial, where

This correspondence agrees with the changes in the FORTRAN code necessary setting a cube o f material is not the same as setting one Y ee cell. Setting a cube

to build the plates in the preceding example. of dielectric or magnetic material requires setting 12 Yee cell field locationsa ~

Before Ieaving the topic o f plates, a per tinent quest ion might be how thick shown in the following FORTRAN, which fills a cube of space located in Yee

are the plates that are built in thc above examples, i.e., how physically thick is cell I,J,K with dielectric material type "MTYPE":

a plate that is one cell thick in FOTO space?" More generally, wc can considerwhere the "surlace" of a volumetric conductor "built" o f FDTD cells is located. IDONE(I,J,K)=MTYPE

The answers to these questions will give us some insight into how FOTO IDONE(l,J,K+l)=MTYPE

approximates structures, because no umque an w F rst consider the situation that we are comput ing the scattering by one of IDONE(l,J+ I ,K)=MTYPE

these p l ~ t e sfor a plane wave incident from El= 90, that is edge-on, and with IDTWO(l,J,K)=MTYPE

the electric field polarized in the e or eqmvatenuy un:; : ~ _ u ;.. , . . ..

E, component is present, and all IDTHRE and IDSIX array Iocatmns are IDTWO(l+l,J,K+ l)=MTYPEassumed set to O (free spll{;e; we would not set these to any other values when IDTW0(I,J,K+1)=MTYPE

building the plates), the scattered. field will be identically zero, and we con- IDTHRE(I,J,K)c::oMTYPEIDTHRE(I+l),K) MTYPE

n u ~ o w e v c r ~ ~ n s i d e rthe case in which the plane wave is incident from the l IDTHRE(l+l,J+l,K) MTYPE= oo direction, and thus nonnally incident on the plate. Boca use the plate is thin IDTHRE(I,J+l,K)=MTYPE

" t a ' a heet resistan ce

or impedance. This is discussed in Chapter 13, which shows that the corrcct This approach to modeling a solid object, a sphcre for examplc, isto determine

result is obtained when the FDTD plate is considered to b e M thick. if a particular Yee cell cube is within the sphere, and if so set ali 12 field

w e might also consider the location of the real FDTD "surface" for a more locations to the material o f the sphere as shown above. Setting individual field

general FDTD geometry. For examplc, considcr "building" a perlectly con- l o c a t i o ~ swill result in a sphere that does not have a closed surface. This may

ducting (PEC) cube with FDTD cells. At first we would s ~ u m ethat the surface not be 1mportant for penetrable materiais, but for a conducting sphcrc would

of the cube corresponds with the outer Yee celllocations, where E fields are correspond to having short "wires" sticking out of the surlace, where individual

"""' 0 rfect conductor. Then we also may consider that the FOTO "cube"field c o m p o n e ~ t sare set ~ t are not a part of a complete cube.

extends halfway betwecn these oute r E fieciTocatmns anu e a Ja e mg

locations that are in free space. Expericnce in calculating scattered field cross- ID array does not cause problems, but rather may be a useful technique.

sections from oerlectly conducting volumetric targets indicates that the surface Writing .into 10 arrays is analogous to painting on a canvas, in that the last

of the FOTO object extends about one fourt 01 me 1 ee ceu _ ,value u s ~ din the calculations. Just as an artist may paint a b a c k g r o : ~ d~ ~ ~ ~l.y, or ll.z) beyond the locations where the E fields are set to perlect conductor.

3

An explanation for this one fourth cell extension is that when a p l a ~ e then part1ally cover this with grass, and in tum paint a lake over a part of the

wave is normally incident on a planar perfect cunductor, the t a n ~ e n t l a l grass, we can consider that we can fill ali the FDTD cells with free space, then fields are maumum. fill a section o these with some other material, then f i l l a POTtiono f this section

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42 The Finite Difference Time Vomain MethodjOr Electromagnetics~

i FDTD asics 43

with some different materiaJ by overwriting the ID arrays. This approach is Other situations in which total field computalion may be desirable uccur due ' r l heo oh orl" ' oh ' . . . . . " . '"" r.:orl r c ' >h.

within the material region. ing FOTO to anterma radiation problems. I we are dealing with a wire antennaIn summary, the point of this section is not that ''building" objects in Yee and i s hto excile fields in the gap of the antenna, anaJytically detennining the

- . . . . . . ,,o ' '""'

Simple objects can be built by writing FORTRAN code specifically, as in lhe tedious and time consuming. As shown in Chapters lO and 4, it is relativelyplate example above. For more complicated structures, translation programs simple to specify a source i n this gap and directly compute the total fields thatmay be necessary. These will take geometrical data already in some database, result from this source interacting with the antenna geometry.from a computer-aided design (CAD) software package for example, and When we wish to compute in the total field mode computer codes based ontranslate it into Yee cetltocat mns. nowever, one must a ways Kecp m mmu tue scattereu uetu IOnnUJahon are simp le to use, because total field formulation isdistribution of field components in a Y ee cel , and that setting one spatial a special case of the scattered field fonnulation. We merely specify a zerolocation to a material type is equivalent to specifying the material which amplitude incident field and then insert in o the FORTRAN code the necessaryinteracts with a specific electromagnettc nela component at a spect IC 10catmn. source tenns. or examp e, consider the case in which we are modelin g a wire

antcnna and wish to locate a voltage source at the location of E, (I,J,K) in the3.6 DIRECT COMPUTATION OF TOTAL FIELDS FDID grid. We set the incident field amplitude to zero, and after each update

of the E, field values, we execute a line of FORTRAN specifying the source

In this book we stress using the scattered field fonnulation in FOTO field at this location. This process is described in Chapter I O and examples are

problems, an approach that has many advantages. Some have already been given in Chapter 14.

discussed in Chapters 1 and 2, and others will be illustrated in ater chapters In summary, for most situations, and especially for scattering problems,r l ;

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i44 The Fimte Differmce Time Domain Methodfor Electromagnetics FDTDBasics 45

number of cells needed to modcl an extent of free spacc surrounding theE"''(O,j,k + 1/2) E"-'(l,j,k + 1/2)' ' erlo rl w;th ' ; i n

computer resources by application of an ORBC.+ - ' E;''(I,j, k + 1/2) + E"-'(O,j, k + 1/2))To understand the need for an ORBC in scattering and radiation problems, c.6.t+& '

' fie> , fe ' oh 'l'h. not beupdated using the usual FDTD equaLions of Chapter 2 because some of the + (E;(o,j,k + 112)+ E"(l,j,k + 112))nearest-neighbor field components needed to evaluate the finite-difference curl c..1t+..x '

enclosing it are outside the problem space and not available. The usual basisfor ORBCs isto estimate the missing field components jus outside the problem

x(

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46 The Finite Dijference Time Domain ethodfor Electromagnetics FDTD asics 47

_ _ One o t h . ~consideration is ~ - a t~ ~ ~ e t e ~ i n i n g~ ~ r l d i s . t ~ n c e b e t w e e ~ . . J ~ h e Storage N X r Components x 4 Bytes +

is located the better the absorption of lhe outward traveling waves. This is due m, Byte lo these waves ~ ~ o m i n gmo:: hke l a n . : _ w a v e ~as they ~ ~ v ~ ~farther f r ~ ~the 6 x lCell 1D

between lhe object and the outer boumlary is limited by computer memory. Acommon criteria is a minimum of ten cells between lhe object and outer where we have neglected the relatively small number of auxiliary variablesboundary. For some situatiols more than ten will he required, especially ifhigh needed to store temporary values, index DO loops, save results for ateraccuracy is needed. Some eJ .amples are shown in Section 13.5. processing or display, and similar functions, and we are also ncglccting thc

Movmg the outer boundary too dose to the obJect may cause mstabiltttcs m memory needed to store the executable instructions. This overhead is nearlythe Mur (and other) outer absorbing boundary implementations. This may be independem of the number of cells in the problem space, so that as the totalmore of a problem for antenna and other calculations, which are exciled by a number o f cells increases it will become a smaller fract ion of the total memorysource wi lhin the space rather than by a plane wave, as the outer boundary must required.absorb total fidds. Also, some felds that are required for an accurate solution We can also e ~ t i m a t elhe computational cost in terms of the number ofmay be absorbed if the outer boundary is too dose. For example, sphere floating point operations required usingscattering includes a creeping wave that propagates around the sphere andradiates energy. The radiation from this creeping wave can easily be seen in the

Operations = N x 6 components cell I O operatio ns componen t x Ttransient backscauer rcsults (scc Chaptcr 13). If the absorbing outer boundaryis too dose to the sphere this wave will be disturbed and the scattered field

w

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48 The Finile Difference Time Domain Methodfor Electromar:neticsFDTDBasics 49

From this we see that the total number of floating point operations required is boundaries, setting up lhe problem geometry, etc., since for large calculations. ; . . ~ i - - .proportionalto the number or ceHs m me .

Now, Iet us consider how the number of floating point operations scales However, if lhe FDTD code being considered requires a significam number of

with frequency. The size of lhe FDTD cell must be scaled proportional to additional computations, such as may occur when calculating far zone fields

wavelength to maintam a certam numner ~ r c e u ~~ :"5' ,number of cells in each linear dime nsion wtll scaJe proport10n to frequency. convolutions in large numbers of cells filled with dispersive materiais (Chapter

This means the number of cells in the (3-D) problem space will be proportional8), the above estimates may be somewhatlow and actual resource requirements

to frequency to the third power, and the number of floating point operations will depend on the particular application and FDTD computer code. In any

f ~ ; ~ , . nlo 1he fourth wer. This fourth- case, lhe above guidelines will provide a basis for estimating the resources

power scali ng with frequency compa res favorably to other methods, sue h as the required for a particular f ~ r ..hl m. These

l ext, we-cm . .estimates are only approximate, as the total numher of time steps reqmred,depends on the geometry being considered. More time steps will be necessary

1or resona n ge , ;.,.hlv flamned re-

sponses.For a (65 ceJI)l problem with I024 time steps to be calculated, approxi-

mately 16.9 x 109 floating point operations are required, and about 8 Mbyteso f memory are required to store the field components and ID??? arrays. Speedsof available machines range from 1000 or more MFLOPS (Mi Jion FLoatingpoint Operations Per Second) for a supercomputer through 10 to 50 MFLOPS

r ~ \f r work stations to aoproximately 2 MFLOPS for a 32-bJt PC. Therun (CPU) times are then estimated from thc above dtscusswn as 1/ s, LIS min(for a 10-MFLOPS work station), and 141 min, respectivcly. In practice, with- ~ - ? n ~ " , . nra.dius dielectric sphere in this problem space and far zone fieldscalculated, running times on a 10-MFLOPS work stauon anu a . J . J -LVHlZbased PC are 38 and 210 min. Obtaining results for problems of this size is

feasible on ali these machines.The above estimates of the number of required operations assume basic = ~ : _ ; I in absorbinP

uw 0 ..L._