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Computer Physics Communications 72 (1992) 22 39 Computer Ph ys icsNorth-Holland Communications

SolutionofMaxwellsequations

Michael Bartscha,Micha Dehier a,MartinDohlusa,FrankEbelinga,Peter

Hahne a,ReinhardKlatt a,FrankKrawczyka,MichaelaMarx b, ZhangM m a,Thomas Prpper a,DietmarSchmitt a,Petra Schtta,BernhardSteffenC~BernhardWagner a,ThomasWeiland a,Susan G. Wjpfb and HeikeWolter aa Technische Hochschule Darmstadt, Fachbereich 18 , FachgebietTheorieElektromag netischerFelder,

SchloJigartenstraj3e8 , W-6100 Darmstadt, Germanyb Deutsches Elektronen-SynchrotronDESY, NotkestraJ3e 8 5, W-2000 Hamburg 52, Germany

CKernforschungsanlageJulichKFA, W-5170 .Jlich, Germany

A numericalapproach for the solutionofMaxwellsequations is presented. Basedon afinitedifferenceYee lattice themethod transforms eachofthe four Maxwell equations into an equivalent matrix expressionthatcan be subsequently

treated by matrix mathematics and suitable numerical methods for solvingmatrix problems. Thealgorithm, althoughderived fromintegralequations, can be consideredto beaspecialcase offinitedifference formalisms. Alarge variety oftwo- and three-dimensional fieldproblems can besolved by computer programs based on this approach: electrostaticsand magnetostatics,low-frequency eddycurrents in solidand laminated ironcores, high-frequency modes in resonators,waves on dielectric or metallic waveguides, transient fields ofantennas and waveguide transitions, transient fields of

free-movingbunches ofcharged particles etc.

1. Introduction

The fieldofacceleratorphysics largelydeals with controlled applicationofelectromagnetic forces tocharged particles. Theseforcesoccur invariousparts of an acceleratorat differentlevels ofcomplexity:Magnetostaticand electrostatic fields are used to steer, focus and accelerate particle beams. RF-fieldsin metallicresonatorsare themostcommon typeofaccelerating devices. CW-transmittersin theUHF

rangeofmore than 1

MW output power are used in large numbers, pulsed tubes are being built upto 100 MW in theS-band.Transient fields are excitedby charged particles when passing acceleratorstructures andcause difficult nonlinear problems (collective effects).

In the field ofelementary particle physics acceleratorsas largeas 27 kilometers incircumference arecommissionedand plans formachinesbeyond 180 kilometers have been presented. The cost ofthesefront-line accelerators is enormous. Thus it is extremelyimportant to ensure, prior to construction,thatsuch a devicewillworkas planned. In order topredictthe behaviour ofcharged-particlebeams in

accelerators,one has tosolve avariety offieldproblems forrealistic structures suchas those mentionedabove. This has occasioned thedevelopementof algorithmsand program systems,which have already

proved dependable anduseful in many areasofphysics and engineering.

Correspondence to: T.Weiland,TechnischeHochschule Darmstadt, Fachbereich 18 , FachgebietTheorieElektromagnetischer

Felder, Schlol3gartenstraBe 8, W-6100 Darmstadt, Germany

0110-4655/92/s 05.00 1992 Elsevier Science Publishers B.V. All rights reserved

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M. Bartsch eta !./Solution ofM a xw el ls equations 23

2. The method

In orderto avoidspecializations ofMaxwells equations prior to numerical solution it isadvanta-geous tosolve Maxwellsequationsdirectly, rather than solving apartial differential equation denvedtherefrom. Using SI units, E andHforthe electric andmagnetic field strength, Dand B for the fluxdensities and .1 forthe current density the equationsto solve read

y~E.ds=-ff~.dA~ (1)

y~H.ds=ff(~/~+J).dA~ (2)

ffB.dA=O~ (3)

jf(~+J).dA=o~ (4)

with the following relations:

D = iE, B = uH, I = ,cE + pv. (5,6,7)

A gridG isdefinedin the orthogonal coordinate system r= r(u,v,w) as

G={(uI,vJ,wk);ul

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24 M. Bartsch eta!./Solution ofMaxwe lls equations

UI U2 U 1 U 1

Fig. 1. Three-dimensional grid in theorthogonal not necessarilyCartesian system (u, v, w).

E~,~+/ v , n +/)J /

~bV/

Fig. 2. Allocation ofunknown field components in the grid G showingthe indexing.

ofeq. (1) andobtain the following algebraic expression replacing the integralalong theborder ofanelementary cell:

E. ds =+(u~+i~ + (v~+ivj)Ev,n+M~ (u1+i ui)Eu,n+M~

~ (11)

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M. Bartsch eta!./Solution ofM a xw el l s equations 25

G

~B7~{E}

{~Fig. 3. Two neighbouring mesh cells showingthe allocation oftheelectricand magnetic field components in G and G.

Fortherighthandside ofthatequation,we obtain (againby the lowest-orderintegration formula):

ff ~.dA~W,fl(uj+l_uj)(vj+l_vj)+O([(uj+1_uj)(vj+1_vj)]2). (12)

Equating the two expressions yields adiscrete replacementfor the first Maxwell equationon eachsurface of the grid cells and thus on every area composed of mesh cell surfaces.

In order to describe all these equations forall surfaces weintroduce abasic discretization matrixwith onlytwo bands and with elements taking only the values0 , +1 or 1:

(1, p=m,Pu:(Pu)m,p_~+l, p_m+M~, m,p=1,...,N. (13)

(0, otherwise,

Outofsuch simple matrices we combinethe matrix C that replaces the contour integral operatorineq. (1) for all meshpoints:

0 P~P~\

c=( Pw 0 Pa). (14)

\P~Pu 0 JAll unknown componentsofthe electric fieldE are put into avector ofdimension 3N= 3 (JfK):

e = ~ . .. ,Ew,N)t (15)

andwe replaceD by d, B by b, Hby h and J by j, respectively.The topological part ofthecontourintegral (curl) is represented by the matrix C . The actual length of the integrationpath isput into a

diagonal matrix:

=Diag(Aul,...,AuN,~v1,...,~vN,i~w1,...,z~wN), (16)

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26 M. Bartsch eta !./Solution ofM a xw el ls equations

ui +1

~={f~Oru~vJ~wk~dU1~i

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M. Bartsch eta!./Solution ofM a xw el l sequations 27

C = Ct. (34)

The diagonal matrices which take careofthe piecewisedifferent values for thematerial constants are

given as

(D~)~= 1~Un_M~+ ~~!L), (35)

2

1Uu,fl.M~

2!Lu,n

(D )2N+n = (AW,flMUEW,fl..MU +AW,n_MVEW,n_MV

+~ + AW,nMU_MV~w,nM~M~ )/(4Aw,n), (36)

(D,c)2N+n = (Aw,n_M~~w,n_M~+AW,n_MVKW,n_MV

+~ +AW,fl_MU..MVICW,fl_MU_MV)/(4AW,fl), (37)

while ~, and ic may have different propertiesin the directions u, v andw . Although not originallydefinedin thegrid, one may relate thefield strength, flux density and current density and formallydefine a magnetic fieldand an electric flux densityby

b =D,~h (B = huH), (38)

d =D~e (D =EE), (39)

j=D~e+ D,,v (I = , c E+pv). (40)

In order to complete the transformation ofMaxwells equations to thegrid space, weapproximatethe divergenceequation by integratingB oversurfaces of each mesh cell ofG.W e canthenwrite thisequation by defining the discrete div operatoron G as

S (Pu,Pv,Pw) (41)

andobtain

SDAb = 0. (42)

This equation allocatesthe (nonexistent) magneticcharges to nodes of G. Thecorrespondingmatrix

on the dual grid G is simply givenby

~Pt t t

U, tJ W -

Sothe continuityequationin the grid space reads

SDA(d+i)=O (44)

Thus electric charges aredefinedon nodes oftheoriginal grid G.

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3. Properties ofthe gridequations

O ne oftheoutstanding propertiesofMaxwells grid equations is that propertiesofanalytical solu-tions have theirexact analogues inth~grid space. The analytical identity div curl 0 reads in the gridspace as

div curl 0 ~ SC SC 0. (45)

This identity in grid space can easilybe provedby showingthat:

SC = (0,0,0) = (PVFW ~ ~ P~P~). (46)

Thusthe proofreduces to the equivalence ofinterchanged partial differentiation:

a a a a~=P~P~~ = ~~---). (47)

Thismatrix identitycan easily beverified. The analytical propertythat scalar potential fields arecurl

free is also found in the grid spaceas

curlgrad 0 ~ CtSt 0. (48)

The proofofthis identity isfound by simply transposing the identity (div curl 0) eq. (45). Thusthe source-freenessofthecurly field is in that sense equivalent to the fact that the scalar gradientfield isirrotational.

Thesepropertiesofthe Maxwellgrid equations not only offer aunique tool totestnumericalresultsfor theirphysical- correctness but also avoid the occurrenceofincorrect solutions in the calculation ofthree-dimensionaleigenmodes inresonators, orat least their identification [3].Thisis quiteimportantsince numericalerrors in the matrixalgorithms may occur andcannot generally be detected. In the

field ofaccelerator physics, oneoften investigatesunknown phenomena by using such codes, so that

an incorrect solution could lead to physical misinterpretations.

4. TheMaxwellgrid equations

Withoutspecifying anything aboutthe shape ofmaterialsnor the time dependence ofthefields, wehave obtained a set ofmatrix equationsthatapproximateMaxwellsequations on a double grid G,G

(R 3 and~ representthe physical spaces and IF L ~ . the time):

Real space E J ~ 3 ~ Grid space R3N

jE. ds = j7~.dA ~ CDe = -DAb, (49)

~H.th=ff(~+I).dA ~ CD5h=DAd+j, (50)

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ffB.dA=0 ~ SDAb=0, (51)

(52)

D=E ~ d=D~e, (53)

B=uH ~* b=D,~h, (54)

J=KE+pV ~ j=D,e+D~,v, (55)

divcurl 0 ~ SC = SC = 0, (56)

curl grad 0 ~. C~S~= CtSt = 0. (57)

5. Special cases ofMaxwellequations

In orderto solvethese equations one has onlyto perform matrix manipulationsand thensolvethe

establishedmatrix problemnumerically.

5.1. Staticfields

In the case ofstatic fields it is in generalnot necessary todescribe theproblemby vectors.Gradients

ofa scalar potentialare used to derive the Poisson equation:

E= grad~E, (58)

div(Egrad~E)= p. (59)

InsteadofnumericallysolvingMaxwells equation we definethe electricpotentialson nodes of G as

= (~E,1,~E,2,Q5E,3,. - . ~ (60)

and derive the corresponding equationsonlyby matrix manipulation:

e = StD~le, (61)

SDDADs1St~e = q. (62)

Thelatterequation is ofthe order Nand is the GridPotential Equation.The righthand side contains

all charges on the nodes in the vector q ofdimension N.For themagnetostaticfield theprocedureis similarexcept that thecurlypartofthe magnetic field

has to be takenout in orderto allowthe use ofa scalar potential:

H= H~ gradq~H, (63)

curlH~=

I, (64)div(

1ugradq~H)=div(itH~). (65)

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30 M. Bartsch et a! ./Solution ofM a xw el ls equations

Here the right hand side is given b y magnetic charges which result from thecurly field but haveotherwiseno physical meaning.Equation (65) isequivalentto (59) where ~u,Handdiv (~uH~)corre-

spond to e, Eand p. One obtainsthe samematrixequationsby allocatingHcomponentson the gridand calculatingD~similarlyto D C :

h = StDl~h (66)

CDShC=D~,j, (67)

SD,~DAD~St~h= SD~DAhc. (68)

Thisallows the use ofthesame iterative algorithm forsolvingelectro- and magnetostatic problems.

5 . 2 . Time harmonic andr e s o n a n tf i e l d s

The electric fields simultaneously have to obey the followingtwo equations,that can be directly

derivedfrom Maxwells equations

curl -~-curiE = w2eEioncE iwl, (69)

divE= -1-(div,cE+divj). (70)

Thenumerical formulationcan be derived again by matrix manipulation:

(CD5D~D~CD5+ ~WDAD,C- W

2 D A D C )e = -iWDADPV, (71)

SDADCe = (SDADKe + ~IiADPv). (72)

Equation (71) represents a 3Nx 3Nequation system,which can usually be solved unless the stimu-lation frequency w corresponds to a system resonancefrequency. The resulting matrix, however, has

a multiply vanishing eigenvaiue for w = 0 as the existenceofstatic solutions cannot be excludedby curl operationsonly.Positive frequenciesyield eigenvalues witha negative real part, which createproblems for many algorithms forthe solution ofequation systems. Thatis why the combination ofeq. (71) with derivations ofeq. (72) seems to beadvantageousso that a better distributionofeigen-values can be achieved. A variety ofcombinations is possible; e.g.withany diagonal matrixD~whichvanishesfor every pointwhereeq. (72) does not, the following3N x 3N system can bederived:

D~DVSDADCe= 0. (73)

Equation (72) vanishes for all grid-pointswhich are not on asurface where material propertieschangeandfor which the current densityis source free. In orderto convertthe algorithm to a symmetric

form we must perform adiagonal transformation, which physically replacesthe field strength by theroot ofthelocal energydensity (except from a scalar factor):

e = (DCDADS)2e, (74)

((DCD)(DCD)t + icoD,~D~- co2I)e~= -1WDDADPV, (75)

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M. Bartsch et a! ./Solution ofMaxwe lls equations 31

D~D~DvSDe= 0, (76)

with the matrices

D = (D~D5D~)

112, (77)

D =(D;D~D~)2. (78)

Solvingacombination ofthese linear systems for agiven driving currentdistribution yieldsthe fieldat alllocations in G. Further specialisation to e.g.low-frequencyeddy currentsinvokes mixing ofscalarfields andvector fields. Fora loss free medium with no driving current one obtains an eigenvaiue

equation,the eigenvalues ofwhich are the resonantfrequencies squared [3]:

(DCD)(DCD)te = w2e. (79)

The abovementioned multiple eigenvalueat zero can beavoidedby combining eq. (79) with (76).Asthetotal number ofeigenvalues remains at 3N, unphysical eigenvaluesappear, which can beidentifiedeasilyby resubstituting the eigenvectors ineqs. (79) and (76).

The analytical analogue to this combination of eqs. (75) or (79) with (76) is the equationcurl curlE= grad divEV ~E anddivE = 0 thus solving V2E .

5 . 3 . F i e l d s i n t h e t i m e domain

For transientfields with a central time difference formula, step size t, no currents and no losses,one obtainsthe Yee-algorithm [1]. In the presenceoffree moving charges the algorithmhas moreconstraintsto ensure charge conservation [5]. The upper index n denotes the time in units oft.These recursive formulae allow the easy calculation oftransient fields ofantennas, particlebeams inaccelerators [51orwave propagation problemseven in the lossy case:

= b~z~tD~CD

5e~

5, (80)

= + (I Da)D;D~CDsD~b~ (IDa)~ (81)

with

Da = exp(D~DK1~t). (82)

The detailedsetting up ofthefinal matrix problem to be solved depends too much on the specificproblem to be explained herefor all possible cases. We refer to refs. [3,4] andwill demonstrate thewide applicability ofthismethodby means ofa few examples.

6. Description oftheMAFIA programs

The MAFIAprograms are a setoftwo-and three-dimensionalcomputercodesbasedon the MaxwellGrid Equations.They were developed for use in the computer-aided designofparticle accelerators

and are now finding wider applicationsin otherfields such as tomography, filters, integratedcircuitsand resonators.See table 1 .

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32 M. Bartsch eta!./Solution ofM a xw el ls equations

Table 1Descriptionofthe MAFIA programs.

Module Description Specialfeatures Geometry

M mesh generator, translates the many predefinedshapes, interactive remeshing (x,y, z), (x, y),physicalprobleminto mesh data (r,z)

S staticsolver, electro- andmagne- open boundary, nonlinear magnetic material (x,y,z), (r,z)tostatic problems properties, advanced multigrid solver for very

largemeshes

R , E matrixgenerator, eigenvalue sol- periodic boundary conditions, resonance fre- (x,y,z), (x, y),

ver forresonator and waveguide quencies (several hundred modes),waveguide (r,z)problems modes and propagation parameters

T2, T3 time domain solver losses, open boundary, stimulation by: initial (r ,z), (x,y, z)field, current,waveguide, incidentwaves;wakepotentials

TS2, TS3 particle in cell code, calculates losses, stimulation by initial field, calculation (x,y,z), (r,z)equation ofmotion for free mov- oftrajectories, velocities,charge densities, etc.ingcharges

W3 eddy current solver, frequency losses, current stimulation (x,y,z)

domain

P post processor, displays results 1, 2 and 3Dgraphicsofscalar andvectorfields, (x,y,z), (x, y),and calculates secondary field powerloss integrals, field energy, far fieldpat- (r , z)

quantities tern, shunt impedance

I _____ _____ _____ _____ I

S R W3 T2 T3 TS2L TS3

Hi~ _

Fig. 4. The MAFIA system with its interrelationships.

The acronym MAFIA stands for the solution ofMA xwells equations by theFinite IntegrationAlgorithm. Figure 4 showsthe interconnections betweenthe various programs otthe third release [6].The second releaseofthe MAFIA Programs, comprising M3, R3,E31 , E32 and P3, has been distributedto over 120 installationsworldwide including most countries of Europe, USA,USSR, China, Japan,India and Brazil. The programshave already proved theirworth throughcomparisonwith theoreticalcalculations and by thesuccessful designofaccelerator components.The MAFIA codes are writtenin standard FORTRAN77and currently runon IBM, CRAY,VAX, APOLLO, HP, SUN, CONVEX,

AMDAHL, FUJITSU, HITACHIand STELLAR computers, among others. The distribution centerfor codes and userguides is the Technische HochschuleDarmstadt, for information contact Prof. Dr.-Ing. ThomasWeiland.

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M. Bartsch eta!./Solution ofM a xw el ls e q u a t i o n s 33

Fig. 5. Netplot ofthefield dependant ~ i n thesymmetry planeoftherelay.

Fig. 6 . Theleft ofthis figure showsa 3D perspective ofthe end region ofthe RFQ. Only one quarterofthis structure needsto be calculated, due to existing symmetries. For this structurethe powerloss in a cut plane at the and ofthestructure is

displayed.

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34 M. Bartsch et a!./Solution ofM a xw el ls equations

7. Applications

The applicability ofthis method and of the computercodes based on iti s almostunlimited. Inthisshort presentationwe can only give a few typical applications from the area ofelectrical engineeringand acceleratorphysics.

/ T 7 T ~ 1 T j ~ I : : , -,t ~ -, ~ ~I t t l f t i ~ .~ - ~4~~4~I1~1

Fig. 7. Arrowplot oftheelectric fieldoftheE

01 -mode in a circular waveguidewith an asymmetrically placed dielectric rod.(Geometry: R = 1 m, r/R= 0.5, ofIset/R = 0.1, e 1 1 =1 , ~r = 8, f=119.36 MHz).

\

Fig. 8 .Nondestructive testing ofa tubeby inducing an azimuthal eddy currentby means ofasinglecoil atthe positionz = 0.

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M.Bartsch eta !./Solution ofMaxwe lls equations 35

The first example is an application ofthe MAFIA magneto- and electrostatic moduleS [8,9]. Re-laysare electric switching devices. The design goal for such adevice might be to reduce itsweight, theneeded driving currentor to minimize the force at themoving armature. Thisnonlinear magnetostaticproblemhas been calculated with our static module and also computedwith the wellknownPROFI.

The comparison has been performed for the achievable accuracy and the computationaleffort. Theagreement in the results was reasonable. S has theadvantage that the use of open boundary condi-tionsallows adescription ofthisstructure with lessunknowns thanis necessarywhen usingstandardDirichletor Neumann boundary conditions.Here apossiblereduction in the number of unknownsof10% reduced the neededCPU time by about20%. Figure 5 displays the relayand the 1u-distributioninside a cutthroughthe central planeofthe relay. Onecan clearly se e the saturation effectat the bot-

tomofthe structure and insidethetop armature,thatindicatethe necesityofredesigning the relay inorderto obtain reasonablebehaviour.

The next example is a radiofrequencyquadrupole (RFQ)thathas been calculated with the MAFIAfrequency domainmoduleR/E [10]. The device might be tuned for a certain minimum eigenmode orin orderto minimize the losses. Three-dimensional calculations are important especiallyfor the endregion of such aRFQ. A careful design using theMAFIA meshgenarator has been carried out, usingabout 75000 unknowns.The lowest eigenmode calculated, 344.78 MHz is in good agreement withmeasurements and other calculations. Figure 6 displays the power loss. This loss is strongest on the

Table 2

Comparison offi (1 /m) ofref. [7] with our method.

Mode Solution [7] fi (1/rn) Numericsolution fi (l/m)

5.649 5.6492 5.648 5.6413 4.204 4.194 3.079 3.082

i 4 , 1

r1r~ryv4c~

7-

Fig. 9. Same tube andsame stimulationas in the abovefigure but the tube has a radial hole at z =0.

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36 M.Bartscheta!./Solution ofMaxwellsequations

..,, _ _ _ _ _

~J.. I . ____

Fig. 10. Differenceofthe electricfields ofthe tube with and without the hole.

a A a a a -, .. ., _ - - .- , I 7 ~

.~t ~ L u I I ~--.~LFig. 11. Electric field ofa ~%/2-dipolewith an unsymmetric coating. Since the geometrical variation of the fields i s small,

only a mesh sized ~ x 0.75)~had to b e computed.

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M.Bartsch eta!./Solution ofMaxwe lls equations 37

outsidewall. The powerloss can be calculated by the MAFIA-postprocessor from the field solutions.Differentspecialapproachesexist fortheanalysis oftheeigenvalues and field componentsofdielec-

trically loadedwaveguides. Itis necessary toknow thesevalues to be able to estimate theinfluence ofmechanical tolerancesand todesign filtersand resonators. Table 2 compares semianalytical results forthephase constant 4 8 ofa calculation similar to [7] with the values achieved by our method. Figure7shows theelectric vectorcomponentsofthe third mode (E

01-mode).A typical problemfor nondestructive testing (NDT)is the detectionofcracks and holesby means of

eddy currents [11]. Forexamplea tube can betested by moving acoilthrough the tube and measuring

itsimpedance. Material defects intheconductingtubearedetected by thechange in the impedance. In

Fig. 12. Far field diagram for thefielddistribution asshown in the last figure. The maximum ofthefar field amplitude is

shifted towards higher0 values.

0

Fig. 13 . Humantest body i n a three-dimensional nuclear resonance spectrometer.

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38 M. Bartsch eta!./Solution ofMaxwe lls equations

the followingexample the azimuthally orientedcoil excites an azimuthal current density field, which

is shown in fig. 8 . The z-coordinate axis is coincident with the axis ofthe tube andthe coil, the x-coordinateaxis here correspondsto theradial coordinate. In a second calculation ahole in the tubewall was modelled. The resultingcurrent density distributionin the plane normal to the hole isshownin fig.9 . The electric fields induce a voltage in the stimulation coil, the difference ofthe fields with

and without the holei s plotted in fig. 10.A far field calculation example is shown forthe electromagnetic field generated by a 2/2-dipole.The

left metallic stub is coated with amaterialwith the parameters ~r = 2, Pr = 2. The dipole is excitedby a current connecting both metallic stubs. Figure 1 1 shows an arrowplotoftheelectric field. Thefarfield characteristiccan be seen in fig. 12 .

S

I ~ I? I - ~

Fig. 14 . Arrow plot of electric field in a cross-section of the nuclear resonance spectrometer. (Due to symetry the half

structure was calculated.)

a134~4~7

a

Fig. 15 . Absorbed power density in the same cross-sectionas in the above figure.

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M.Bartsch eta !. /Solution ofMaxwe lls equations 39

The lastapplication examined the influenceofa humantest body in a three-dimensional nuclearresonance spectrometer, as seeninfig. 13.The iistimulationstructures are the twostrip lines,mounted

on dielectric supports at each end, on either side of the test person. Figures 14 and 15 show thecalculatedfield (T3 [121) fora 9 0 MHz driving current in a cut plane through the body and theantennas,first the electric field as an arrow plot, thenthe absorbedpower density as a contour plot.

The field ofa 2/2-antenna resonance can clearly be recognised. The woakfield density within thebody is aconsequence ofabsorptionand reflection caused by theconductivityand highpermeabilityofbiological tissue. Most ofthe power is absorbed atthe part of the body closestto theantennas.Thearea of the neckshows a local maximum, which indicates body resonance.

References

[I] K.S . Yee, Numerical solution ofinitialboundary value problems involvingMaxwells equationsin isotropic media, IEEETrans. Antennas Propag. 14 (1966) 302.

[2] T. Weiland, A discretization method for the solution ofMaxwells equations for 6 componentfields, AEU, Arch. frElektron. und Ubertragungstech. Electron.and Commun.31(1977) 116.

[3] T.Weiland,Onthe numericalsolutionofMaxwells equations and applications inthefield ofaccelerator physics, ParticleAccel. 15 (1984) 245,and references therein.

[4] T.Weiland,Ontheuniquesolution ofMaxwellian eigenvalue problems inthreedimensions, ParticleAccel. 1 7 (1985) 227.

[5] T.Weiland,Transientelectromagnetic fieldsexcitedby bunches ofcharged particles in cavities ofarbitraryshape,in: Proc.

Xl-thm t. Coni on High Energy A ccelerators,Geneva, 1980 (Birkhauser Verlag,Basel, 1980) p. 570.[6] M. Bartsch et al., MAFIA release 3.X, in: Proc. 1990 Linear Accelerator Conference, Albuquerque, LA12004-C, Los

Alamos,p.372.[7] K.Steinike, Calculationofthepropagation characteristics ofa circular waveguideexcentrically loadedwith a dielectric

rod, Frequenz 44 (1990) 1 .[8] F. KrawczykundTh. Weiland, A newstatic solver with open boundary conditions inthe3D-CAD system MAFIA, IEEE

Trans. Magn. 2 2 (1988) 55.[9] F.KrawczykundTh. Weiland, 3D magneto- and electrostatic calculations using MAFIA-S3, in: Proc. 1st EPAC Conf.,

Rome, 1988 (World Scientific, Singapore, 1988)Vol 1, pp. 520 522.[10]D. Schmitt and T. Weiland,IEEE Trans. Magn. 2 8 (1992) 1793.

[11]P.Hahneand T. Weiland,IEEETrans.Magn. 2 8 (1992) 1801.[12]M. Dehler,M. Dohiusand T. Weiland,IEEEtrans.Magn. 28 (1992) 1797.