IEEE Transactions on Nuclear Science, Vol. NS-29, No. 6, December 1982Treatment of Late Time Instabilities in Finite-Difference

EMP Scattering Codes*

Larry T. Simpson, Richard Holland,** and Shahnaz ArmanMission Research Corporation1720 Randolph Road, S. E.

Albuquerque, New Mexico 87106

ABSTRACT

Time-domain solutions to the finite-differencedMaxwell's equations give rise to several well-knownnonphysical propagation anomalies. In particular,when a radiative electric-field look back scheme isemployed to terminate the calculation, a high-frequency, growing, numerical instability is intro-duced. This paper describes the constraints made onthe mesh to minimize this instability, and a tech-nique of applying an absorbing sheet to damp out thisinstability without altering the early time solu-tion. Also described are techniques to extend thedata record in the presence of high-frequency noisethrough application of a low-pass digital filter andthe fitting of a damped sinusoid to the late-timetail of the data record. An application of thesetechniques is illustrated with numerical models ofthe FB-111 aircraft and the B-52 aircraft in thein-flight refueling configuration using the THREDEfinite difference computer code. Comparisons aremade with experimental scale model measurements withagreement typically on the order of 3 to 6 dB nearthe fundamental resonances.

INTRODUCTION

Three-dimensional finite-difference EMP and SGEMPcodes have found extensive use in computing electro-magnetic coupling to complex structures in the timedomain. These codes solve the differenced Maxwell'sequations in leapfrog fashion in a spatial coordinatesystem appropriate to the shape of the object beingmodeled, be it rectangular, cylindrical, orspherical. The grid size is selected to adequatelydescribe the features of the object over the fre-quency range of interest, nominally four grid pointsper wavelength at the highest frequency of interest.To keep computer costs and resources to a manageablesize, the computation is terminated in an absorbing(impedance) or radiating boundary condition. Theradiating boundary condition is based on the extra-polation of the electric and/or magnetic fields.Absorbing boundaries introduce a fictitious con-ductivity on several of the outermost cells to"absorb" the radiated fields. To maximize the dis-tance to the outer boundary, these finite-differencecodes typically use an expanding grid between theobject and the radiating or absorbing boundary.

Anomalous propagation properties of electro-magnetic waves on a spatially varing grid, that isthe trapping of high frequency noise, the dispersionof waves in the grid, and the amplitude growth ofwaves propagating through the grid, have been dis-cussed in some detail by others and will not berepeated here.1-3 The amplitude growth of high fre-quency waves through multiple reflections from theouter boundary is a severe problem when the late time

solution is desired. Figure 1 illustrates the expon-ential growth of this numerical noise, which quicklyexceeds the magnitude of the true signal. If theseresults are Fourier transformed, severe truncationerrors are introduced due to the short data record.

16

12

8

I-

zw

m0he

4

0

-4

-8

-12

-16 ' --0 .4 .8 1.2 1.6 2.0 2.4 2.8

TIME(p s)

Figure 1. Illustration of the instability.

In this paper we will describe techniques toremove or minimize the late-time instability intro-duced by spurious multiple reflections from theouter boundary. The technique employs both an ab-sorbing surface and a radiating boundary in conjunc-tion with a judicious selection of the size of thegrid space for the problem. In addition, techniquessuitable for signal extraction in the presence ofboundary reflections will be presented as a means ofextending the usable data record. This techniqueincorporates a low-pass digital filter to remove thehigh frequency noise, a linear least squares fit tothe low-frequency filtered response, and the mergingof ori gi nal, fi ltered, and fi tted data to preservethe high frequency early-time response and extend theusable late-time response. These techniques will beillustrated with computations using the THREDE4finite-difference models of the FB-111 aircraft andthe B-52 and KC-135 in the in-flight refueling con-figuration. The experimental data used for compari-son is obtained from scale model measurements made atthe University of Michigan.5

COMPUTATIONAL MODEL

* Work supported by Defense Nuclear Agency andperformed under Naval Surface Weapons CenterContract N60921-81-C-0134.

**Currently unaffiliated: address 1625 Roma NE,Albuquerque, N.M. 87106.

The computer program used in this analysis,THREDE, solves the finite-differenced Maxwell's equa-tions in Cartesian coordinates. It is based on thecentered difference algorithm first reported by Yee6

0018-9499/82/1200-1943$00.75© 1982 IEEE1943

p

and incorporates the look back electric field radiat-ing boundary condition described by D. E.Merewether.1 THREDE is a scattered field code, mean-ing that the total field is partitioned into an inci-dent and scattered component. On the surface of thescattering object, assumed to perfectly conducting,the scattered field is set equal to minus the inci-dent field. Applications using this computer codehave appeared in the literature.8-12

Currently, the code can treat mesh sizes up to90x90x90, although the two examples described in thispaper, the FB-111 aircraft and the B-52 aircraft inthe in-flight refueling configuration, are gridded ina mesh of size (43x37x47) and (65x71x88) respec-tively. These computations were made using a CDCCyber 176 computer. In order to achieve these largemesh sizes with the addressable memory available, itwas necessary to buffer the electric and magneticfields to disk storage, along with some of the elec-tric fields saved for the radiating boundary condi-tion.

Figure 2, our solid model of the FB-111 aircraftwith the wings extended, illustrates the typicalstructural detail that can be resolved in these typesof models. The wings and fuselage typically have20-30 cells in their length, and the grid exterior tothe aircraft model is rapidly expanded to move theboundary as far from the scatterer as possible. Ofcourse, it is possible to mesh the object with morepoints, but the cost of the computation rapidly be-comes prohibitive. Considering four points per wave-length at the highest frequency of interest, thisyields for the FB-111 mesh an upper frequency limitof 75 MHz on the computation. The mesh will propa-gate above these frequencies, but the solution rapid-ly degrades. Even below this limit, the nonlinearpropagation effects in the expanding portion of themesh lower the frequency limit for which scatteredwaves are accurately propagated. A more realisticupper frequency limit would be 40 MHz. Typically,this is not a severe problem when comparing calcula-tions with experimental measurements made at EMPsimulators. For an object similar to the FB-111,whose fundamental resonant frequency is on the orderof 6-7 MHz, the signal to noise ratio is not greatenough to make valid integrated external measurementsat this higher frequency limit.

y

Fi gure 2. Finite difference model of the FB-111.

GRIDDING CONSIDERATIONS WHICHMINIMIZE THE INSTABILITY

When this model of the FB-111 aircraft was placedin a mesh of size 39x34x43, the numerical instability

1944

began to dominate after about 600 ns (-810 programcycles). Figure 3a shows the normal electric fieldon the side of the aircraft near the cockpit, and itis typical of the magnitude of the instability seenover the entire surface. The mesh was of physicaldimensions 44 m x 19.3 m x 43.4 m, and had an expan-sion factor in those cells between the aircraft andthe boundary of 1.215, 1.256 and 1.275 respectivelyin the x, y, and z directions. A series of numericalexperiments were conducted in which we varied: (1)the number of mesh points in the problem space, (2)expansion factors for cells exterior to the airplanemodel, and (3) departure of the mesh from a cubicalshape (physical lengths of the sides of the cellswere not equal). While we cannot be quantitative inour conclusions, the general trends are very clear.

The larger the physical separation between theobject and the boundary, the slower the growth of theinstability, which is consistent with a reflectionphenomenon. However, if an excessively large expan-sion factor is used, there are serious distortions inthe solution due to reflections and propagationanomalies in the mesh. We have found that for expan-sions of the mesh of less than or equal to 1.20, theinter-grid reflection and trapping effects aresmall. When we used expansion factors five percentlarger, 1.25, a noticeable degradation in the solu-tion was observed.

We have also found that the instability is fairlysensitive to the proximity of the mesh to being cubi-cal. For those cases for which there is a naturaldissipater, such as the ground in the mesh of theproblem, this condition appears to be less important.

Figure 3b illustrates the results of using ajudicious choice for the mesh size. The mesh is43x37x47, having physical dimensions 62.6 m x 36 m x58.6 m. Expansion factors for the variable part ofthe grid to the boundary are respectively 1.15, 1.20and 1.15. It is clear from this figure that the in-stability is still present, and first appears aboutthe same time. However, its rate of growth has beenslowed enough to increase the data record by 60 per-cent.

In summary, the meshing considerations we havefound to yield the best results are:

. Keep the radiating boundary as far from thescattering object as is practical from thestandpoint of computer costs and run times.As a minimum, keep the boundary L/2 away fromthe scatterer, where L is the maximum lengthalong the scatterer.

* Keep expansion factors in the mesh as small aspossible, no larger than 1.20.

^ Keep the physical dimensions of the grid asnearly cubical as possible.

Keep the largest cells in the mesh less thanhalf a wavelength of the highest frequency ofinterest.

ABSORBING SHEETS

In an effort to selectively damp this growing in-stability, we introduced a two sheet absorbing layerone mesh point in from the outer radiating boundary.Our objective was to match the growth rate of the in-stability with the damping rate of our absorber, notto totally absorb the radiated fields. To do this,we introduced a small fictitious magnetic conduc-tivity on two planes of the mesh and a fictitious

electric conductivity on the other four planes of themesh. These six planes were each one mesh plane infrom the outer radiating boundary, and no attempt wasmade to match the electric and magnetic fields on agiven surface to the free-space impedance.

Figure 3c illustrates the application of theseabsorbing sheets to the FB-111 in the 43x37x47 mesh.The electric conductivity chosen was 5xlO- mhos/m.The magnetic conductivity denoted by a* is givenby:

* I0oEC0

As can be seen, this absorber completely removes theinstability with only minimal changes in the earlytime response. Again, this is an area where adynamic tradeoff is taking place - if the conduc-tivity is too high the boundary reflects too much,perturbing the solution, and if the conductivity istoo low, the instability is not damped. Based on thesmall sample of problems that we have examined atthis point, we can offer no quantitative method toselect the best absorber. It is clear that the low-est conductivity consistent with damping the insta-bility is the one of choice.

60

40

20-

0

-20

-40 -

-60

'i> 60-

40

" 20

-J 0

4

-40

-60

the original and filtered waveforms at late times.The merge was necessary because the filter also re-moves some of the high frequency information found onthe early time portion of the data record. By merg-ing the two, that is, using the original data recordat early times, and merging into the filtered recordat late times, the high frequency early time responsewas not sacrificed in removing the noise.

Since the growing instability occurs at a fre-quency above those which are usable due to mesh limi-tations, it is reasonable to employ an ideal low passfilter with cutoff frequency wo:

sinw th (t) = itt PW (w)

0(1)

where pLoo(w) is the gate function of width 2wocentered at X = 0 in the frequency domain. The cut-off frequency wo is selected to be below the fre-quency of the unwanted noise but above the usableportion of the frequency spectra.

This ideal filter cannot be realized, due to thefinite data record length available from the computermodel. An approximation to the idealfilter is given by:

sinw t

ho(t) = PT(t) 0,t Ho(X )

(A) ORIGINAL (39X34X43) GRID

WITH RADIATING OUTER

BOUNDARY ONLY.

(B) EXPANDED GRID (43X37X47)WITH RADIATING OUTER

BOUNDARY ONLY.

where ho(t) is the truncated Fourier kernel13 andPT(t) is a unit amplitude gate function of lengthT. Ho(w) can be written as the frequency domainconvolution:

w

Interm ofth sin integra fucto(STi) th

(2)

(3)

In terms of the sine integral function (Si ), thefilter derived from the truncated Fourier kernel is:

Ho(w) =1 { Si[(w + w)T] Si[(Iw - wO)T]}

Figure 4 shows the realizable filter Ho(w) withWoT equal to the first few positive integermultiples of i-.

60 -

40-

20

0-

-20

-40 -

-60 -

O 0.4 0.8 1.2

TIME CuS)

1.2

.9

(C) EXPANDED GRID (43x37x47)WITH ABSORBING SHEETS AND

RADIATING OUTER BOUNDARY

1.6

3 .60

.3

0

Figure 3. Illustration of the judicious selection ofgrid and the application of an absorbingsheet for the FB-1I1 model.

-2.4 -1.2 0 1.2 2.4

W/60

FILTERING

For those time domain data records in which itwas not possible to remove the growing instability,useful results were obtained by passing the datathrough a low-pass digital filter and then merging

Figure 4. Filter derived from the truncated Fourierkernel.

We found in practice that multiple passes throughthe filter were required to reduce the numericalnoise to a usable level. Filtered waveforms were ofthe form:

1945

N

fm+i(nAt) = A ho(Ti - nAt)fm(Ti)ATi m=0,1,2,... ,(4)i =1

where m denotes the number of passes through thefilter, and A represents the normalization constant

N

A-1 E ho(-ri - nAt) A (!i=1

where ATi was selected to be an integer multiple ofthe time interval between successive points in thedata record. Cutoff frequency wo and the Tj weredefined so that T = 80 At with woT slightly largerthan Nir. Figure 5 shows the results of multiplepasses through the filter.

The filtered data record is multiplied by f(t), andthe two are summed. Figure 6 illustrates the mergeddata record where tf - ti represents approxi-mately 130 time steps of the program or equivalently200 ns.

4

5) .

1-z'U

U

32 12 r1- ORIGINAL FILTERED

-16 6V

rr

-16- -61

-32 LL ,-120 1.2 2.4 3.6 4.8 0 1.2 2.4 3.6 4.8

TIME (ps) TIME (us)

12 i T

PASS

6-

-6

-12 -0 1.2 2.4 3.6 4.8

TIME(us)

-124.8 0 1.2 2.4 3.6 4.8

TIME(p s)

Figure 5. Application of filter derived from thetruncated Fourier kernel.

In order to merge the original and filtered datafunctions without creating discontinuities in thewaveforms, we employed a function of the form,

0

f + sin(2,rut+,)2

1

t < t.

t. < t < tt > tf

where

ti = starting time for the merge

tf = end time for the merge

l= 1./2(tf-ti)

w=--- - 2iruta

The final point of the merge tf is chosen to coin-cide with that point where the instability first isdetectable. The initial time ti is chosen to pro-vide a smooth merge between the original and filtereddata. The original data record is multiplied by g(t)where

g(t) = 1 - f(t)

-12 ---!-i--.0 1.2 2.4 3.6 4.8

TIME (ps)

Figure 6. Merging the original and filtered datarecords.

CURVE FITTING

Having filtered and merged the data record, itwas desirable to extend it in time, so that theFourier transform could be obtained with minimumtruncation error. This extension was achieved byfitting a damped sinusoid to the tail of the datarecord. As will be shown for this data record, atlate times the response is almost purely a singledamped sinusoid. The information content at otherfrequencies is very small.

Two different minimum least squares fit ap-proaches were tried. The first was a general non-linear, four-parameter least squares fit to afunction of the form:

bxf(a,b,u,4,x) = ae xsin(2irux+f)The quality of the fit obtained with this method wasinconsistent. The second approach, which yieldedconsistently better results, used a linear minimumleast squares procedure to fit the damped expo-nential. This procedure first determined the phaseand frequency of the sine wave from the zero cros-sings of the data record. From the sign of thederivative at the zero crossings and the frequency ofthe sinusoid, the maxima of the sine wave were deter-mined. From these maxima a linear least squares fitwas performed on the logarithm of the amplitude.Figure 7 is an overlay of the fit to the tail of thedata record. Figure 8 is the difference functionbetween the original and fitted function.

TIME(ps)

12

0

-6

-1 1 I20 1.2 2.4 3.6

TIME(ps)

1946

6

4

2

0

-2

-4

-6

Figure 7. Overlayrecord.

of the fitted tail

COMPARISON WITHEXPERIMENTAL MEASUREMENTS

Comparison of the THREDE calculations for themodel of the B-52 and KC-135 in the in-flight re-fueling configuration required application of all thetechniques described above for suppression of thenumerical instability and extension of the datarecord. Figure 11 shows our three dimensional modelof the two aircraft in the refueling mode connectedby the refueling boom. The boom was modeled withboth the thin-strut formalism,'' with which the dia-meter of the boom can be specified, and with a simpleboundary condition on the electric fields. Both ofthese methods yielded similar answers, as was ex-pected for the ratio of the boom diameter to the gridsize in the mesh. The airplanes are contained in a

3.6 4.8 mesh 65x71x88, having physical dimensions 181 m x147 m x 179 m. An absorbing sheet one cell from theouter boundary of the problem space is employed toreduce the rate of growth of the numerical insta-

lothedata bility. The electrical conductivity of the absorberto the data was selected to be 2.5x1O-4 mho/m.

.6

.4

.2 [.-

id OA

-.2

-.4

-.60 1.2 2.4 3.6

TIME (xis)

Figure 8. Difference between the original andfitted function.

To combine the fitted and original data recordsthe merging function was used, producing the mergeddata record shown in Figure 9. Finally, having fil-tered, fitted and merged the data, the extended datarecord suitable for transforming to the frequencydomain is shown in Figure 10. Our attempts to usethe Prony algorithm to fit the time domain waveformdid not result in as good a fit as this simple mini-mum least squares fit.

12 12

6F 6

0 1.4 2.8 4.2 5.6 0 2 4 6 8TIME (,Us) TIME (Ws)

Fi gure 9. Merge of the Fi gure 10. Extensi ons offitted tail the data recordto the data through the fit-record . ted damDed si nu-

soid.

- 39 m _

KC-135 MODEL .5 m

6.1mm--12.3 m IN LENGTH 30\ B-2MDE p .

;=3 . 7~~~~~~~~. m

47 m

Figure 11. Finite differential model of the B-52in the in-flight refueling configuration.(65x71x88 grid).

In order to decrease the computer I/O time re-quired to buffer large 400,000 decimal element arraysto and from disk storage, electric and magnetic fieldquantities were packed two quantities per word priorto buffering. This results in a slight increase inthe CP time required, but substantial savings in thewall clock time required to run the program. Infor-mation was packed so that full precision was retainedin the exponent but only about 4 or 5 digits weresignificant in the mantissa. With the accumulatedroundoff error one would expect the S/N ratio to beno worse than 30 dB.

The measurements available for comparison weremade in the frequency domain on scale models of thetwo aircraft5. These were swept frequency measure-ments made over the range of 118 to 4400 MHz andscaled down. All measurements were normalized by theincident field.

Figure 12 is an overlay of the measurement andcalculation of the normalized amplitude of the cur-rent density on the boom. The THREDE calculation isthe normalized transform of the time domain waveformshown in Figure 10. The magnitude of the fundamentalresonance of the two connected aircraft around 1.1MHz agrees very well. The fundamental fuselage reso-nance of the B-52 aircraft at 2.2 MHz appears inboth. Beyond that frequency the agreement is not asgood. Both have resonances near 3 and 5 MHz, whichare assumed to be the third and fifth harmonic of thefundamental. The discrepancy in the magnitude be-tween the two near 3 MHz is not understood. Above 10MHz the signal to noise ratio is so small that com-parisons are not valid.

1947

0-

zw()

0 1.2 2.4TIME (is)

--TI

AlA ,i\,e

_ .. ,

1.1 1.0 10.0 100.0FREQUENCY (MHz)

Figure 12. Normalized amplitude of the current

density on the boom for the B-52 modelin the in-flight refueling configuration.

Extension of the data record in time, eitherthrough a Prony analysis or the fitting of some givenfunction of time, is limited to problems in which thelate time behavior is known from physical considera-tions or examination of early-time model results.Application of these techniques to a complex finite-difference model of several aircraft yielded agre-ement of about 6 dB with empirical data over alimited bandwidth near the fundamental exterior reso-nance frequencies.

REFERENCES

1. Katz, I., D. Parks, A. Wilson and J. Harvey,Non-Reflective Free Space Boundary Conditionsfor SGEMP Codes, Systems, Science and SoftwareTopical Report, SSS-R-76-2934, May 1976.

2. Delmer, T. N., Numerical Propagation of Electro-magnetic Waves in a Varying Grid, ScienceApplications, Inc., SAI-76-522-AQ, 1976.

3. Gi l bert, J. L., Studies in Outer Boundary Conditions For Finite Difference SGEMP Codes, MissionResearch Corporation, MRC-R-296, November 1976.

Figure 13 is an overlay of the normalized amplit-ude of the normal electric field on the vertical sta-bilizer of the B-52. Here the fundamental resonanceof the overall structure and the B-52 fuselage reson-

ance are resolved. For this particular comparisonthe data record was not extended and consequentlythere is some truncation error in the computed trans-form, lowering the amplitude.

sr '

0W

i-

uJ

4 0-

a Ck

m IL

-J

I-0a

0-WU.

I-

210

10

I

I I II

U OF M MEASUREMENT(FWLI 4ORS)

- -- THREDE CALCULATION(TP20)

I~~~~~~~~~~~~~~~~~~

-111,, g11,1 1 1 i~~~~~~~~~~~~~1111 11

10 1 1.0 lO.O 100.04.

FREQUENCY (MHz)

Figure 13. Normalized amplitude of the normalelectric field on the verticalstabilizer of the B-52 model in thein-flight refueling configuration.

CONCLUSION

Data processing techniques discussed above haveproved helpful as a means of extending the usabledata records obtained from finite-difference calcula-tions in the presence of late-time numerical insta-bilities. Selection of the problem space mesh, theapplication of absorbing sheets and filtering are ap-

plicable to a wide range of finite-differencesol utions.

1948

4. Holland, R., "THREDE: A Free-Field EMP Couplingand Scattering Code," IEEE Trans. on Nuc. Sci.,Vol. NS-24, pp. 2416-2421, December 1977.

5. Liepa, V. V., Scale Model Measurements of theB-52, Air Force Weapons Laboratory InteractionAppl7ication Memos, Memo 36, December 1980.

6. Yee, K. S., "Numerical Solution of Initial Boun-dary Value Problems Involving Maxwell's Equa-tions in Isotropic Media," IEEE Trans. Ant. andProp., Vol. AP-14, pp. 302-307, May 1966.

7. Merewether, D. E., "Transient Currents Inducedon a Metallic Body of Revolution by an Electro-magnetic Pulse," IEEE Trans. Electro. Compat.,Vol. EMC-13, pp. 41-44, May 1971.

8. Kunz, K. S. and K. M. Lee, "A Three-DimensionalFinite Difference Solution of the ExternalResponse of an Aircraft to a Complex TransientEM Environment: Part 1 - The Method and itsImplementation," IEEE Trans. Electro. Compat.,Vol. EMC-20, May 1978.

9. Holland, R., L. Simpson and R. H. St. John,"Code Optimization For Solving Large 3D EMPProblems," IEEE Trans. on Nuc. Sci., Vol. NS-26,pp. 4964-4969, December 1979.

10. Holland, R., L. Simpson and K. S. Kunz, "Finite-

Difference Analysis of EMP Coupling to LossyDielectric Structures," IEEE Trans. Electro.Compat., Vol. EMC-22, pp. 203-209% August 1980.

11. Holland, R., and L. Simpson, "Implementation and

Optimization of the Thin-Strut Formalism in

THREDE," IEEE Trans. on Nuc. Sci., Vol. NS-27,

pp. 1625-1630, December 1980.

12. Kunz, K. S. and L. Simpson, "A Technique for

Increasing the Resolption of Finite-DifferenceSolutions of the Maxwell Equation," IEEE Trans.

Electro. Compat., Vol. EMC-23, pp. 419-422,November 1981.

13. Papoulis, Athanasios, The Fourier Integral and

Its Appli cations, McGraw Hill Book Company,Inc., New York, NY, 1962.

z 10I

C_

w

2OZYN

ccz 10

00w

N

O 2

Z 0

--