Holland 1977

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  IEEE Tka nactions on Nucteoa Science, V'o.t.S-24, No.6, Decembet 1977 THREDE: A FREE-FIELD EMP COUPLING AND SCATTERING CODE Richard Holland Mission Research Corporation PostOffice Box 3693 Albuquerque, New Mexico 87108 Abstract Equations ToBe Solved THREDE is a time-domain, linear, finite-difference,three-dimensional EMP coupling and scattering code. In its present form, it can accomodate a problem spacecon- sisting of a 30x3Ox30 mesh. Differencing is linear. Problem-space boundaries are provided with a radiating condition which doesnot generate ficticious mathemati- cal echos at late times.The scatterermust be a per- fect conductor, although a nonideal groundplane(runway)may be close by. The presentarticledescribes the mathematical basis ofTHREDE,and showsthe resultsof applying it to predict the responseof an F-lll shell in the horizontally-polarized dipole  HPD EMP simulator. Amplitude agreementbetweenexperiment and prediction for this example is typically on the order of 20 ; resonant frequencies are predictedmore closely than this. THREDE costs about 1 second ofCDC7600 computertime per program cycle; most practicalanalysesrequire 500 to 1000 programcycles. Introduction Thispaper describes the three-dimensional finite- difference code THREDE, for solving time-domain Mlaxwell's equations in cartesian coordinates and threedimensions.The primaryapplication ofTHREDE is in calculatinq EMP free-fieldscattering and surface currents for a com- plicated strurture,such as an aircraft.Scattering structuresmay either be imbedded in homogeneous space (as an aircraft in flight) or located over a lossy groundplane (as an aircraft on a runway). The algorithm upon which THREDE is based was first described by Yee1 about ten years ago. At that time, computers did not exist whichcould implement thisscheme practically in threedimensions. This is no longer the case. It is now possible to mesh a problem space ofinterest into a 30x30x30 grid and have random access to the resulting 162,000 field quantities. These advances in computer technology have prompted Longmire to credit Yee's algorithm as being  the fastest numeri- cal way of solving Maxwell's equations .2 Theother significant cornerstoneupon which THREDE is based is Merewether's radiating boundary condition.3 This mathematical techniquepermits one to treat a finite problem space without being troubled at late times by numericalechos from the problem-space boun- daries. Basically, it utilizes thefact that, far from a scatteringobject, the electromagnetic scattered fieldmust behave like f(e,q)g(t- r/c)/r. In the following sections of this article, we will first summarize the Yee technique for finite-differencing Maxwell's equations. Next, we will describe theradia- ting boundary condition. Finally, we will illustrate an application ofTHREDE by showing how it is used to predict theEMPresponseof an F-111 aircraft shell in the horizontally-polarized dipole (HPD) simulator. Thisfinal section will include experimentalcomparisons withthe THREDE predictions. A more detailed description of the THREDEcode is also available,4 in a report which was not prepared un- der the brevity constraint applied to the present article. The experimental data described in thispaper was ob- tained at the Air Force Weapons Laboratory, Kirtland Air Force Base, New Mexico, under Contract F29601-76- C-0064 (EG&G, Inc., prime). Maxwell 's equations, in the presence of electricaland magneticconductivity and current, are (1) (2) M VxE = J* *H - = t - - VAH = C -+ J +  iE - at We will be dealing withproblems where the total E and H are the superposition ofincident and scattered fields: T inc s E = E n E HTinc s = H + H (3)(4) THREDE is presentlywritten in a linearformwhere u and o*donot depend on ET or HT. Subject to this linearity assumption, the scattered field alonealso satisfiesMaxwell's equations. We shallpresume the scatteringobject is perfectly electrically conducting and completely enclosed by the problem space. We shall alsopresumethereare no inde- pendentelectric or magnetic currents J or J* in the problem space. We donot require c, ui, a oro* tobe homogeneous, and wo do not require the incident field to be a plane wave. Then the boundary condition at the surface of the scattering body is that thetotal tangential E field vanish, =-Einc = -En :-t an --t a n (5) The distant boundary onthe scattered field is that the field takestheformof an outgoing radial wave s = f g,) (t- r/c) f(6 4) r (6) At t= O, the scattered field is presumed to be every- where zero. Subject to the linearity assumption, these conditions uniquely specify Es and Hs within the problem space at all time. The logic of THREDE procedes as follows: Wle assume we know the incident field (Einc, Hlnc) at all times and places;this is the field which would exist in the absence of a scatterer. The main portion of THREDE is devoted to finite-differencing Maxwell 's equations for the scatteredfield (ES, HS). Outputquantities of THREDE are usually total fields or total surfacecurrents on the scatterer. Total fields are given by (3) or (4). Total surface currents areT inc + = nx (H c+Hs , (7) where n is the outward-pointing unit normal on the scatterer. Difference Equations Figure 1 illustrates the manner of imposing the (I,J,K) index-space mesh on the problem space. It also illustrates thelocation of evaluation points forthesix scatteredfield components in a typical cell. Note thatthe Es and Hs evaluation points are interlaced al- ternatelyalong each spatial axis. This interlacinq is 2416  The other four component equations may be identically differenced to yield corresponding formulas. If the medium filling the problem space is inhomogeneous, c, uj, a and a* will depend on (I,J,K). The finite-difference equations forthe scattered fields arenow solved as follows: Let us assume the grid indices run I= 1 to Nx, J= 1 to Ny and K= 1 to Nz. Let us further assume the scatterer is modelable as an assemblage of conductive cells completely interior to the problem space. At t<0, all scattered fields are zero: EZ( H (I,J,K) yvW ~~~~~~~ S o( I) XO I Xf+ z,K Figure 1. Convention for imposing the (I,J,K) indices on the (x,y,z)problem space, and location ofthesix field evaluation points in a typical cell.also done in time: If At is a program cycle time, Hs is evaluated atnAt, and Es is evaluated at (n+12)AtY The THREDE mesh is not required to be uniform. However, if nonuniformmeshing is used, we do not recommend that adjacent cells differ in sizeby more than 30 . In componentform, two of Maxwell'sequations for the scatteredfields are 3E5 3E5 az ay= 3y az aHz 3Hs zy y ay az M s x 3Es x + s £ t x (8) (9) The other four equations are obtained from these by indexpermutation. Let us define E n+(I,J,K) to be Es evaluated at the point indicated in Figure 1, and at time (n+'2)At. Let us furtherdefine Hn(I,J,K) to be Hs at the correspondinglyindicated point, and at time nAt. The other fourfield components may be analogously spe- cified. Subject to these conventions, Eqs. (8) and (9), when finite-differenced consistently withFigure 1, yield Hn(I,J_K) Ia*/2 H - I,J -(,u/At + o*/2) At+ c*/2  I-Jx K E (I,JK+   E(Ij1K) EE2 IJK +- E/At + a/12) -1 z   ,JsK z J ,K 10( ) Y E nI,,KJK+l) En(I9J9K) Z(K)- Z(K-1 Z K)   (10) 0 0/ En+2 I J ,K) =  E/At-+a/2 En 2 I,J K llHn  I,J,K -Hn(,,j_l 9K +.  c/At   cI2) Iz Y J) -Y J-l H(I,J,K) Hn I,J ,K-1)~ 7(K Z K-1) E2 (I,J,K) = 0 E 2(I,J,K) = 0 y E2 (I,J,K) = 0 H (I,J,K)=O x H0  I,J, K) =o y Ho  I,J ,K = 0 z l<I<NX- ; l<J<Ny; l<K<Nz I < I < Nx 1 < J < N y-1 ;1 < K< N   -- y  -  z 1 <I<Nx ; 1 <J<N y; 1 < K< N -1 1 < I < Nx; I < J < Ny-- ; < K < Nz-z 1 < I< NX -1; 1 < J< N -1; 1 < K< N - I<N- I - - y - -   Nz  1 2)  1 3) (14)  1 5) (16)  1 7) Letthe incident fieldfirsttouch the scatterer at  2At, and considerwhat happens then on a surface facing -x. If the -x surfaceof Cell (I,J,K) coincides with the -x surface ofthe scatterer, E2(I,J,K), E 2(I,J,K+1), E1 I,J,K) and E(I,J+1,K) are all determined by Eq.(5). Teese four quantities, in turn,define Hl I,J,K) via Eq. (10). The same is trueat all subsequent program cycles, n>l. Likewise, on a surface facing +x, let Cell (I',J', K') have a +x surface coincident with the +x surface of the scatterer. Then E2 2(I-+1,J-,K-), En (I +l,J-,K +1) E  2 12 y Ez i(I+l,J',K'), and Ez (IV+l,J'+l,K') are all deter-mined by Eq. (5) for n>0. These four quantities subse- quently determine Hx l(I +l,JY,K') for n>0. Identical considerations apply on the scatterersurfaces facing ±y and ±z. In this way, the tangential E and normal H components ofthe scatteredfields are determinate on all the externalsurfacesof the cells which make up the conducting scatterer model. Mathematically, the difference equations pertaining to the fields atthe interior points of the scatterer thus become decoupled from the equations pertaining to the exterior points. In otherwords, if E n(I,J,K) describes the field at an Ex meshpoint in the interior ofthe scatterer, this quantity cannot at some later timehave any influence on any field at some mesh point external tothe scatterer. We can thus set all the internal mesh-point fields to zero every program cycle without affecting theexternal scattering-problem solu-tion. Knowledge ofH nl (I,J,K)for 1 < I < N , I <J < N -1, nx -x y 1 < K< N -1, of E 2(I,J,K)for < I< , 1 < J< N z zn x y 1 < K< Nz-l,andEy 2(I,J,K) for 1 <I < Nx, 1 < J < Ny-l , 1 < K< Nz now determines the new Hn(I,J,K) for 1 < K< Nx, 1 < J< Ny.l 1 < K<Nz-l by means of Eq. (10). Similarly, knowledge of all the problem-space Hn (1,J,K) Ex (I,J,K) and Ez (I,J,K) determines all the new Hn I,J,K) in the problem space. Finally, all the Hz (I,J,K), Ex (I,J,K) and En  I ,J,I) may be used to find all the new Hn(I,J,K). z 2417 YiJ  K)  1+1 ,J,K)  Theelectric fields are a bitmorecomplicated to advance than the magnetic fieldsdue to the fact that the radiationboundary condition, Eq. (6), must be ap- plied on thesefields atthe limits of the problem space. If weknow Ex (I,J,K), Hy(I,J,K) and Hz( I,J,K) at all the meshpoints in the problem space wecanuse Eq. (11) to find thenew Ex (I,J,K) everywhere except on the problem-space boundary. Thefollowing Ex 2(I,J,K) problem-soace boundary values,however, must be obtained by meansof the radiation-boundary condition, not Eq. ( 1) : En +2(I,lK) x En+2(IIN ,K) x y En+(I,J,1) Ex 2(I,J,N Z) Implementation of the next section. 1 < I < N - 1 , I < K < Nz 1 I<N-1, 1 <K<N 1 < I<N -1 ,1 <J<N I- - x i i y I < I < N- -1 1 < J < Ny this condition will be described in In a similar way,the other electric fieldcompo- nents Ey 2(I,J,K) and EZ 2(I,J,K) are advanced by finite- difference formulas at interior mesh points, and bythe radiationcondition atthe problem-spaceboundaries. Radiation BoundaryCondition The purposeof the radiationboundary condition is to permit the volumeof the problem space to be finite without generating nonphysicalreflections off the problem-space boundary. This is achieved in TyREDE by assuming the fields drop off asan outgoing r wave in Eq. (6) if one is  far enough awayfrom the scatterer. It is, of course, desirable to make  far enough as small as possible, soas to minimize thenumber ofbuf- fermeshes which must be interposed between thescattererandthe problem-space boundary.The first step towards this goal is to centerthescatterer within the problem space. Thus, we shall hereafter presume r=O coincides with some point ator near the center of thescatterer. Subject to this convention, if d is the greatestdimen-sionof thescatterer, Merewether has empirically found that far enough occurs in a two-dimensional program when r is about d/2 past thescatterer in everydirec- tion3 (see Figure 2). We shall presume this result is alsovalid in threedimensions. Actual construction of the radiation boundary is best explained byexample. One of the boundary-field quantities we need to specify is En 2(l1,J,K) for some (JK) such that 1 < J < Ny-l, I < K < {Iz. (This E mesh point lies on the -x facing surface ofthe proNlem space.) In our scheme, this quantity is determinedfrom old values of Ey(2,J,K) by three-point interpolation overtime.Let Ryx(I,J,K) be the value of r at which Ey(l,J,K) or Ey(2,J,K) is calculated. Furthermore, define e (J,K,(n+-')At - R (1,J,K)/c) = y yx Ryx (l,J,K)E 2(l ,J ,K) y y Equation (18) can also be expressed as R yx(l,J,K)(y (1,J,K) R  2,J,K) ey(J,K,(n-'2)At + At y -JsK) y ~~~~~~~~c R (1,J,K) - R (2,J,K)\   R (2,J,K) ey JK,(n-1-2)At - _Y- + ey (J,K)At) (18) (19) Figure 2. Separationrequired between scatteringobject and problem-space boundary for radiation-boundary approximation to-be valid. where R 1 yx JK) - Ryx (2,J,K) o (J,K) = 1 - yx y yxcAt In view ofthe radiation assumption, Eq. (6),it is also true that R (2,J,K)E n-2- (2,J,K) = yx y c for any Z. If we abbreviate the quantity Eq. (21) as R (2,J,K)En 12-(2,J,K) = f (-QAt) yx y yx 1 (21) indicated in 5 (22) the quantitydescribed by Eq. (19) can be written R (l,J,K)E n2(1,J,K) = f  e (J,K)At) yxy yx yx (23) We now assume that f x(u) can be represented by the firstthree terms in ; Maclaurin series, f (u) = A + Bu + Cu2 yx (24) where A, B and C are unknowns tobe determined. THREDE is programmed so that access to oldvalues of Ey(2,J,K) is available forthreeselected previouscycles. These old values are denoted to be k,l'2 and Q3 proqram cycles in thepast. In otherwords, THREDE backstores En   2 2,J,K) for =Z1'Q2'Q3S If we define f by  - ~21t3 fi fyx (-1it) = Ryx  2,JyK)E d i(2,J,K), i = 1-3 , (25) we obtainthree equations for the unknowns A, B and C intermsofthebackstored En-12-i 2,J,K): y f. = A Bk At + Ck   At2, i = 1-3  26) 2418 (20)  Simultaneous solutionofthesethreelinear equations is accomplished bythe usual use of determinants. Re- sultingformulas for A, B and C in termsoftheback- storedfields are given elsewhere,4 but in the interest of brevity, are omitted here. It is now possible to express the radiation-condition value forthe desired boundary field En+2 1,J,K) entirely in terms of previously-determinedquantities: 9 9 E +2(1,J,K) = y A + BO (J,K)At + Cox (iJ,K)AtL yx lJ yx Ry  I J,K) Radiating-conditionboundary fieldsmust also be obtained for En+2(N ,J,K) y x En+2  N ,J,K) z x En +2(I,N,K) x y E n+12 (I,N ,K) z y l <J < Ny 1 < K< Nz- 1 <J < N-1, 1 < K< Nz 1 <J < Ny, 1 < K< Nz- 1 < I < NX- 1, 1 < K< Nz I < I < Nx, 1 < K < Nz- 1 < I'< NX- 1, I < K< Nz I < I < N , 1 < K< Nz- closest interior field, En-M Zi(2,J,K), atthe present cycle  Q; = -1) andthe two mostrecent past cycles (N = 0 and +1). For uniform cubical meshing, Eq. (28) indicates the maximum stable timestep to beAt = AX/ v 3 c) 9 (29) where AX is the dimension of any ofthecubical cell edges. Thus, under these circumstances, an electromag- netic signal will propagate .58 AX during eachtimestep.At this point, it may be helpful to refer to Figure 3, and to bear in mindthat we are trying to compute the 7) new E+2 (1,J,K) from Eyh2k (2,J,K) obtained previously  Q > -1). As timeprogresses, the field at I= 2 propa- gates towards the I= 1 mesh point. At Qi = -1 (the pre- sent cycle), this propagation has just begum. By i = 0 (the mostrecent pastcycle), the propagation has pro- gressed .58 AX; by Qi = 1 (the nextmostrecent past cycle) the propagation has progressed 1.16AX, andthus passed the I= 1 mesh point. Thus, forthis case, taking Qi= -1, 0 and 1 guarantees that the new boundary field, En+2 1,J,K), is interpolatively obtained from previously determined values which have propagated to bothsides ofthe relevant I= 1 mesh point. Ey l,J,K) Ey 2,J,K) X0 l   X0(2) 0 En 2(I,J,l) En 2(I,J,l   y En+2(I,J,N   x z En +12 I,J,N   y z I < I< N -1 1 <J < N I < I < N   1 <'J < Ny I1 1 < I < N x- I, I J < Ny I < I< N , 1 <J < N y-. These eleven additional cases oftheradiation- boundary condition are all developed by straightforward, but tedious, repetitions of the derivation of Eq. (27) with appropriatepermutations ofthe subscripts. Stability Considerations The algorithm described hereinfor solving Maxwell's equations is unstable if too large a value is assigned the time step At. For stability, At must obey At < m in1X  I+1) - X  I)) -2 -1 + (Yo(J+0 0+ (Y (3+1)- y )- + (z (K+l) - Z (K))Y2(p) 1 ) (28) If c or p is inhomogeneous, their dependence on (I,J,K) must be included in evaluating the minimal timestepof Eq. (28). A somewhat moresubtle stabilitydifficulty canbe generated at the radiationboundary if one is careless in picking values forthe ki. In particular, onecan inadvertantly turn the radiation-interpolation procedure into a less stable extrapolation procedure. For thecase of uniformcubicalmeshing, this difficulty will be avoided if onetakes k. =-l, 0 and 1 as the backstored fields used to determine A, B and C from Eq. (26). Thischoice of Li corresponds to deter- mining the new boundary field En Nl1,J,K) from the y direction of propagation .4 +t i -1 4 .58AX IQ =O -  { 1. 16AX   k= 1 Figure 3. Location ofthe boundary and firstinterior E (I,J,K) mesh points. Note that by i= =1, the signal from XO(2) has passed Xo(l) for uniform cubical meshing. Thus, forthis case,the new signal at Xo(l) can be temporally interpolated from its Zi= -1, 0 and +1 values at X (2). On the other hand, if oneuses an expanding mesh near the radiation boundary in order to satisfy the previously-described  far enouah criterion with a mini- mum number of buffer meshes, selecting Qi = -1, 0 and -1 may notbe satisfactory. Suppose, for example, XO(l) and XO(2) are separated by threetimes the dimension of thesmallest mesh. Then the signal starting out from I =2 at Li = -1 will-have propagated only 1.16/3 of thedis- tance to I= 1 by Li = +1. In this case, kj = -1,0 and 1 all correspond to signals which are on the same side of I=1. In the case of nonuniform meshing, it is generally necessary to make at least one ofthe ki greater than +1. In other words, generally it is necessary to backstore information from longer ago thanthe two immediately pre- ceding cycles. Itis not, however, necessary to back- storefieldsfrom more than two previous cycles at any given time - to do otherwise in a three-dimensionalcode is very extravagant of computer-memory resources. The manner in which THREDE selects cycles forbackstorage and determines the corresponding Li for an arbitrary, nonuni- form mesh is of sufficient complexity to preclude pre- sentation here. It has been documented in the THREDEreference report, however.4 2419 I
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