AIAA-95- 196 7

A Message Passing Finite Volume Algorithm for Maxwell’s Equations on Parallel Machines

Vineet Ahuja and Lyle N. Long

Department of Aerospace Engineering The Pennsylvania State University University Park, P.4 16802

26th AlAA Plasmadynamics and Lasers Conference

June 19-22,1995/San Diego, CA For permission to copy or republish, contact the American instftute of Aeronautics and Astronautics 370 CEniant Promenade, S.W., Washington, O.C. 20024

e A Message Passing Finite Volume Algorithm for Maxwell's Equations on Parallel Machines

Vineet Ahuja*and Lyle N Longt

Department of Aerospace Engineering The Pennsylvania State University

University Park, PA 16802

1 Abstract temporal resolution makes these problems suitable for parallel computation. Algorithmic development in com-

A zonal approach to solving the Mawell's etlllations for putational electromagnetics started with Yee's Leapfrog generalized body conformal curvilinear grids on paral- based algoritllm [1] a staggered uniform grid. AI- le1 computers is Presented. The 3-D finite vohme a[- though this algorithm is robust, efficient for parallel com- gorithm is explicit in nature and is especially suited for putation and lacks the propagation of spurious waves due t,he message passing Paradigm. It utilizes a four stage to non-uniform grid size, it is prone to staircasing errors Runge-Kutta time integration method. Integration of and requires the use of a huge number of grid points the Maxwell's equations is carried out on a dual grid for bodies with curvature. In order to eliminate these wherein the electric and magnetic field quantities are staircasing a host of finite difference and finite evaluated on different grids. Each zone is placed on a volume algorithms [2]-[G] have been developed for body- separate processor and inter-processor communication is coIlformal orthogonal grids. rn the present paper we carried out using the Message Pas ing Library (AtPL). make use of the finite volume approach with the Runge- The algorithm has been successfully tested on the SP-2 Kutta time integration method that has been success. in solving scattering problems of electromagnetic waves fully used in aerodynamic [7] and time dependent aero. from various targets. R c s results are presented for the acoustic type problems [8].[g]. The Finite Volume Tirne problem of scattering from a Perfectly conducting sphere Domain (FV-TD) algorithm is based on the four-stage and a perfectly conducting ogive. These results are in Runge-Kutta explicit tirne stepping method on a dual extremely good agreement with the exact solution and grid. All three electric field components are computed with results obtained with a standard finite difference at tile Same grid points and three time domain code. Qualitative results are also provided nentS are solved for on the Same dual grid points. The for scattering from metallic trapezoidal wing, and an for adopting a dual grid based approach is to aircraft engine. The formulation used i n this case is for avoid adding artificial dissipation explicitly (typical of the scattered field and the Liao boundary condition is central differencing schemes) or to make the code &pen- used a t the outer non-reflecting boundary. The far zone dent on algorithmic dissipation (upwind/characteristic transformation has also been implemented efficiently to schemes). evaluate the far zone scattering results.

v

field

In the recent past, there has been a considerable ef- fort made in utilizing both the data parallel and message

Thc evolution of computational electromagnetics has led Passing Paradigms to solving FDTD/FVTD Problems to an increasing need for computational in in computational electromagnetics. In [lo] paralleliza- solving of scattering and Radar cross set. tion issues related to the Yee algorithm are discussed for tion (RCS). The complexity in the shapes of the scat- the data Parallel Paradigm on the CM-5. Paralleliza- terers, coupled with the need for adequate spatial and tion issues have also been discussed previously by the

authors in [ll] for the CM-200/CM-5 machines. Certain drawbacks of the data parallel paradigm are the over- head costs incurred in performing the near-to-far-field

2 Introduction

'Graduate Student, Student Member, AIAA +Associate Professor, Senior Member, AIAA

u

1

transformation and computation of the outer radiation boundary condition. Shang et a1 [I21 addresses the do- main decomposition strategies for message passing type electromagnetic algorithms. Chris Rowel1 et a1 [I31 have also incorporated a zone based parallel starategy on the Intel Paragon for their flux difference algorithm. Nguyen and Hutchinson [14] have used the bicharacteristic form of the Maxwell's equations on parallel architectures sup- porting message passing. In this paper, we present a dual grid based approach where the message passing is reduced due to the fact that only the E-field variables need to be passed across processors, thus making the al- gorithm very efficient on parallel architectures that sup- port message passing. Implementation of the algorithm has been carried out on the IDM SP-2 using Message Passing Library (MPL).

3 Numerical Model Maxwell's equations can be written in the conservative vector form as

- = - V x E (1) at aB

OD - = V x H - j at

d point and all the H-field components are solved at the same dual grid point. The dual grid is generated by first using a standard grid generator to generate the grid for the E-field points for a scatterer that is a perfect con- ductor (and this grid would correspond to a H-field grid if the scatterer was composed of a magnetic material). The H-field grid is then generated from the E-field grid by taking the average of the coordinates of the corners of each E-field cell.

The system is solved using a four-stage Runge-Kutta time integration method given by

QT = Qr - y,,AtRY-' rn = 1 , 2 , 3 , 4

QY+' = QF + Qp4 The time step is denoted by n and each stage of the Runge-Kutta method by m, where the coefficients are ym = $, $, i, 1 respectively.

Q I = [ 51 Q 2 = [ $ 1 ( 5 )

The residuals Ri are defined as

1 Ri = - AV (AF + AG + AK)

L/

(7) 1

A vd

Assuming the scatterer to be a perfect conductor and rewriting the equations in their integral form for free space we have

R2 = - (AL + AM + AN)

where

Since the system of hlaxwell's equations are linear, the equations are solved only for the scattered fields since the analytical solution for the incident field is

1 P

AG = - (S!' z+1/2 , j+1 ,k+1/2 x ';+1/2,j+1,k+1/2

known. Henceforth the fields written without any su-

( E F Esco"nnd H s Hsenft), If the total or incident

noted by their respective superscripts.

perscripts will be understood to be the scattered fields -sy+1/2,j,k+1/2 x ~ : + 1 / 2 , j , X + 1 / 2 )

- Sti1/2.j+1/2,k E f + 1 / 2 , j + l l 2 , k )

fields have to be alluded to they will be explicitly de-

A finite volume formulation is used to solve t,he above system of equations. A dual grid is used in this formula- tion. Each E-field point is located at the centroid of the In the above expressions SE, s', SC represent the pro- cube whose corners are H-field points and similarly each jected surface areas of constant c,q, < faces respectively. H-field point is located at the centroid of an E-field cube. They are calculated by taking half the vector cross prod- Figure 1 depicts the combination of a E-field cell and an uct of the two diagonal vectors that join the four vertices H-field cell that form the dual grid. This formulation of a cell face. differs from the Yee cell formulation in that although It should be noted that the electric field points used the E-field points and the H-field points are not c o b in the evaluation of the above fluxes lie in the center of cated all the E-field components are solved at the same the cell faces and are not computed directly from the

1 AK = - (S< *+1/2 , j+1/2 .k+1 x E:+1/2,j+1/2,k+l

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integration process. Instead they are extrapolated from the electric field points that make up the corners of the cell faces. For example from Figure 2

- Kordulla and Vinokur [15] have proposed an efficient

method of calculating cell volumes for threc-dimensional flow predictions. The volume A V is calculated by parti- tioning the hexahedron cell into five tetrahedra and com- puting the volume of each tetrahedron separately. This decomposition helps avoid gaps and overlaps in comput- ing cell volumes and is computationally very efficient.

The residual Rz is used to evaluate the E-field com- ponents and can be computed in a similar manner:

field are equal to zero

E'Otd - - 0 'I

where n, is the unit normal to the surface of the con- ductor.

The flux at the face that lies on the surface of the body can be exactly calculated by using the above con- dition and the analytically computed incident fields (Fig- ure 3).

rneident S, x E9C"'t = -S,! x E

To compute the fluxes on the faces perpendicular t o the surface of the scatterer it can be easily seen that the above condition is not enough to compute the three components of the electric field for the grid points that lie on the surface of the scatterer. We need an additional condition in order to make the system determinate. In this case we take into Consideration Gauss's electric law which can be written in the integral form <v

nr

-Sdy+l12,j,k+l,.2 x H7+lp2,j,k+l12) We nccd to apply Gauss's electric law to each cell that lies on the boundary. Assuming that the grid points that lie on the surface of the conductor do not contribute anything to the dot product in Gauss's law for those sides of the cell that are perpendicular to the conducting

'u'

1 A N = -- (Sdi 2+1/2.j+l/a,k+1 x ~ ! + l / z , j + l / z , k + l

L -

surface (Figure 4). - sd ;+ i / z , j+ l /2 .k H!+1/2.j+l/2,k) Combininrc this condition with the fact that the tan- -

gential components of the total fields are equal t o zero we can write explicit expressions for the scattered field components of the electric fields at the boundaries.

expressions s d < , S d r ~ , ~ d c represent pro- jected surface areas of constant <, q , C faces of the dual grid respectively. It must he pointed out the subscripts

scripts on the dual grid. (10) sz (TI - i>j,k used in the calculation of the above fluxes are snb- EFnfnormol -

4 Boundary Conditions We have to consider the boundary conditions at the sur- face of the scatterer and the radiation condition at the outer boundary. In both cases we have only the E-field points located at the boundaries. The H-field points lie half a cell away from the boundaries and are consid- ered as interior points. Hence, we have to determine the boundary conditions for only the E-field components.

4.1 Surface Boundary Condition

At the surface of a perfectly conducting scatterer we know that the tangential components of the total electric

where T is the Gauss Divergence Law summation over the whole cell in accordance with the figure 4.

4.2 Far Field Boundary Condition

Most body conformal grid algorithms in electromagnet- ics [4]-[6] use a simple outer boundary condition that either solves the one-dimensional wave equation directly

3

or is based on the onedimensional method of character- istics in Cartesian coordinates. These boundary condi- tions are analogous to the hlur-type 1191 boundary con- dition that is frequently used in computational &ct.ro- magnetics. They generally work well when the outgo- ing wave is a plane wave travelling perpendicular to the boundary.

In order to improve the outer boundary condition the Lim 1201 form of the non-reflecting boundary condi- tion has been implemented. This can be written in its discrete form as

where

where: t = the elapsed time i = is the unit vector to the far zone field point r'= is the vector to the source point of integration

Assuming that the surface q = constant denotes the integration surface for the far zone transform the equa- tions can be recast in discretized form as follows:

u [t - (i. i ) /c] = (E x S")"" - (E x S"))" (17)

A t each time step the contribution of each cell surface that makes up the integration surface T ] = constant is calculated and placed in its corresponding w bin whose index is computed by subtracting the time delay factor from the elapsed time and dividing the result by the time step At.

(' - - ') Tlz = s(2 - ,y)

2

Ti3 = 7

The retarded potential w has to be computed in a slightly different manner since the magnetic field points do not lie on the surface q = constant. Instead, there

Til =

s(s - 1) '

exists a magnetic field point half a cell above and half a cell below the surface. Therefore we can take the average of the time derivatives of the eel1 above and below and

and

d c a t An s = a- tulae~e 0 5 a 5 2 compute the retarded potential w.

5 Far Zone Transform One of the most important applications of solving t,he hlaxwell's equations in the time domain is the analysis of the scattered fields from radar targets. For this purpose, it is important to compute the far zone scattered fields. In this section we have used the method developed in (181 and modified it for the FV-TD algorithm. It should he mentioned that although the method developed in [18] is quite generic, the discussion of its application was limited to the Yee algorithm. The whole concept of the far zone transform is based on the fact that there exists a closed surface around the scatterer where the electric and magnetic scattered surface currents, j, and me , can he calculated.

j , = n x H (13)

w [ t - (i. 7') /c] = (18)

(H"" - H")"++ + (H"" - H" 1,-i 2

so x

The far zone scattered fields can be computed from the above potentials as

Eg = -Zwg - E4 = -ZW+ + ug where Z is the impedance of free space. The farzone scattered fields are then Fourier trans-

formed and divided by the Fourier transform of the inci- dent pulse to obtain the scattering cross-section, which can he defined as:

m , = E x n (14)

w and u can be computed from the scattered surface In 1181 the authors show that the retarded potentials 6 Parallelization Issues and Im-

plement ation currents using the following formulas:

Some of the features that make this algorithm suitable for parallelization are it's explicit character, the narrow

(15) stencil size and the dual grid based approach. In Ill]

- w [t - ( i . i ) /c] = (A) [ 1" ( t ) d s )

4

these features were exploited in a data parallel manner tasks involvcd in setting up the parallel environment and for the CM-ZOO/CR.I-5 machines. .4lthough the imple- all parameters required for message passing. It reads in mentation was quite efficient, there was an overhead of the grid and distributes it to the various processors de- computing outer radiation boundary conditions and the pending upon t,he type of domain decompostion. Each far zone transformation. In [Ill an efficient send-with- processor then generates its own dual grid from the in- add type structure was used in the implementation of formation it has about its own grid. The grid and the the far zone transformation it was still found to be quite constructed dual grid together constitute the PD of the expensive due to the inherently serial nature of the t,rans- processor. Message passing between processors is carried formation. out by issuing explicit MPL calls. For example

Parallelizat&n of the algorithm was performed on the SP-2 using the’Message Passing Library (RIPL). MPL CALL MpaSEiq)(qeast, jmax kmax 3 r4size, uses the SingleTrogram Multiple Data (SPMD) model wherein the s&e program is executed on all proces- sors. However; each processor has it’s identifier pro- cessor number and this permits MPL to have proces- will send the array “cleast” of size ( j m w x kmax x 3)

based con&iona1 statements. Each processor has a to the processor with a processor identification number certain part of the grid located on it. For the sake of equal to (taskidfl). Similarly, the call hrevitv. let us call this the Processor Domain JPD). In

-

taskidfl, msgid)

, ,

CALL MPBRECV(qcast, jmax x kmax x 3 x r4size, most cases, the PD is synonymous with a zone of the grid. However, certain complicated scatterers may have zones that are constrained by geometry. Therefore, a taskid-1, msgid, nbytes)

subtle difference in terminology needs to be maintained between zones and PD’s although in this paper they can be frccly interchanged. Domain decomposition in allo- cating PD’s is very important and careful consideration has to be paid t o issues of load balancing, synchroniza- tion, distribution of boundaries over processors and the amount of message passing. The domain decomposi- tion strategy used in most cases presented in this paper is that of I-D parallelization. For example, for the case of the ogive the PD consisted of (ni x jmax x kmax) grid points, where ni refers to the number of i-planes in the PD. As has been pointed out in [12] this sort of partitioning achieves near perfect load balancing. Each processor has the same number of interior points and the same number of outer boundary points correponding to t,he j=jmaw boundary. Since only neighbour in. formation is required by the algorithm for flux computa- wcessors have made the corresponding call. tion, messaRe passing between two contiguous domains

will receive the array ‘cqeast” of size (jmax kmax x 3) from the processor with a processor identification rlunlher equal to (taskid-1).

The farzone potentials are sent to the control proces- sor where they are summed up and stored depending on their retardation times. Since the whole simulation is a time dependent process all processors are forced to be synchronized after each Runge-Kutta stage with a MPL call:

.d’

CALL MPSYNC(a1lgrp)

which blocks all execution on all processors till all

- . - is now only (3 x jmax x kmax) E-field variables. The dual grid formulation provides the added advantage of having to pass only the E-field variables across proces- SOTS. 4 s is evident in figure 5 the interior cell values from one processor are passed to the neighbouring pro- cessor and they act as the pseudo-boundary values for the neighbouring processor and vice versa. This leads to an overlap of one layer of cells between processors and is really the only overhead incurrcd in parallelizing the

One of the major drawbacks of using a zonal based message passing approach is that domain decompostion IS highly dependent on the type of grid system (eg. 0-H type, C-H type etc.) that is employed for the scatterer. This leads to re-writing parts of the computer code re- lated to message passing, especially those utilizing MPL, thus adversely affecting the generality and portability of the computer code. This has been countered by using connectivitv arravs between zones. verv much like those . ” cocie. used in unstructured grid solvers to identify neighbour-

Figure 6 illustrates the SPMD mode and the man- ing cells. This also permits the code to be used without ner in which the algorithm is implemented in MPL. A any modifications for grids that have zones constrained prc-determined processor (with processor ID=O) acts as by geometry and follow a highly irregular pattern of the control processor and carries out the pre-processing zonal decomposition.

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7 Results The first test case considered here is that of scattering of a wideband pulse from a perfectly conducting sphere. The grid used for this case was an 0-0 type grid and consisted of 37x46~73 grid points. The sphere is of radius 0.1 meters. The incident pulse is represented Ily the derivative of the Gaussian and can be explicitly writ,ten as

(20) a(7-0At)Z f ( t ) = -2a ( T - Oat) e -

where T is the time delay parameter

r = t - ( F . i ) / c

2 (&) and 0 is a user defined parameter that dictates the

width of the pulse. The value of p used for the FVTD run in this case was 756. It should be mentioned at this point that the FVTD results are being comparcd to the results obtained with the PSU-FDTD code that ernploys the Yee algorithm. An equivalent of 64 was used in the FDTD run. (The FDTD run was made with a cell size of 0.01m or ten cells per radii) The difference in the f l value can be attributed to the difference in the time step, because the value of flat has to be the same for both codes. Since the value of 0 defines the spectral band width it has to be chosen judiciously so as not to create any unnecessary noise. Figure 8 is a compari- son of the backscatter cross-section against frequency of the FVTD and FDTD algorithms and the exact solu- tion. The comparison between the exact solution and the FVTD solution is exceptionally good especially at low frequencies and is seen to be an improvement over the FDTD solution.

The second test case investigated was that of the metallic ogive. The ogive has a half angle of 22.62 de- grees and an aspect ratio of 5:l. The length of the ogive is 10 inches. An 0-H type grid was used in this case (Fig- ure 9). The grid consisted of 200 x 31 x 32 cells and was clustered near the leading and trailing edges and close to the surface of the scatterer. A wideband Gaussian pulse is incident normal to the leading edge. The direction of propagation of the incident pulse is parallel to the major axis of the ogive. In Figure 10 the co-polarized RCS is plotted versus frequency for the FVTD and FDTD al- gorithms. The FDTD results were obtained in this case with 4 million grid points.

A third test case was run with a perfectly clectri- cally conducting trapezoidal wing. The wing used in this case is the Lockheed Wing-C [21] with camber and

twist. The purpose of this test case was primarily to test the robustness of the dual grid based algorithm in working with different types of grids. A C-H type grid was used in this case. The grid pertaining to a single zone of the wing is depicted in figure 11. Ghost points pertaining to the magnetic field were introduced to solve for the electric field points that lie on the split line he- tween the trailing edge and the outer boundary. The grid consisted of a 109 x 42 x 146 grid points. Domain de- composition was performed along the largest dimension. A wideband Gaussian pulse was incident on the leading edge of the wing. The incident field is linearly polarized along the z-direction and the direction of propagation is one of forward incidence to the wing. Figure 12 shows the scattered field along various cross-sections of a part of the wing. Electric field continuity is maintained along the split line, thereby showing that the dual grid did not create any unnecessary problems in treating grid config- urations of this type. Figure 13 depicts the intensity of theH, field component of the magnetic field dong the various cross-sections of the dual grid of a section of the wing.

\ 4 d

Since the wing, fuselage and engine are important parts of any aircraft configuration the last test case per- tains to an aircraft engine. In this case an engine inlet was used as the scatterer. The center body, and the engine cowl are treated as perfectly conducting. The fan iction was also treated as a closed PEC boundary. This ,rems to be a reasonable approximation when using wavelengths that are large compared to the inner annu- lus height. Figure 14 depicts a planar slice of the grid that was used in this case. It consists of 242 x 40 x 65 grid points. As is evident outside the cowl this grid is rather coarse. Hence, a low frequency sinusoidal exci- tation was used in this case. The incident wave travels along the axial direction of the engine and is linearly polarized along the transverse direction. Although the geometry is axisymmetric the scattering problem is not. Figure 16 is a Fore Looking Aft view of the scattering patterns on equi-angular planes dong the rotational di- rection at three different axial locations of the grid that are marked out in Figure 15. As is illustrated in Figure 16 the intensity of the scattered fields is greater in planes that have a component of the incident polarisation tan- gential to them.

Table 1 is a comparison of run times for the algorithm on a 32 node CM-5 partition with a 4 node and 16 node partition for the SP-2. As is evident, using the message passing paradigm on the SP-2 gives significantly faster run times.

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8 Conclusions and Future Work A finite volumc code has been developcd based on

[4] Shankar, V., “A Gigaflop Performance Algorithm for Solving Maxwell’s Equations of Electromagnet- ics”, AIAA Paper 91-1578, Honolulu, June 1991.

W’

a dual grid approach utilizing Message Passing Lan- guage(hlPL) for the IBM SP-2. The zonal bascd mes- [5] Noark, R.W. and Anderson, D.A., “Time Domain sage passing approach presented here has bccn found to be more efficient than the data parallel approach. This is due mostlv to Drocesses like the near to far zone trans-

Solutions of Maxwell’s Equations Using a Finite Volume Formulation”, AIAA Paper 92-0451, Reno, Nevada, Jan 1992. ” .

formation that are inherently serial in nature and can he better exploited using message passing. The algorithm and the code can be used to solve scattering problcms for bodies of arbitrary shapes. The results obtaincd are in

[6] Shang, J.S. and Gaitondc D., “Scattered Electro- magnetic Field of a Reentry Vehicle”, AIAA Paper 94-0231, Reno, Nevada, Jan 1994.

good agreement with the exact solution and conventional FD-TD methods. In some cases, like the metallic ogive reasonable results were obtained with 20 times lcss grid points over conventional FD-TD methods. Scattering from the wing and the engine show that the algorithm is capable of cffcctively handling complex configurations.

Further testing is underway for scattcring from the wing at arbitrary angles of incidcnce to study in dctail the cffect of edge diffraction from the wingtip. Radar Cross Section (RCS) calculations are also being carried out for the engine inlet and for full aircraft configurations on the SP-2.

v‘ 9 Acknowledgements

[7] Weinberg, Z. and Long L.N., “A Massively Paral- lel Solution of the Three Dimensional Navier-Stokes Equations on Unstructurcd Adaptive Grids”, AIAA Paper 94-0760, Reno, Nevada, Jan 1994.

[8] Ozyoruk, Y . , and Long L.N. .‘A Navier-Stokes Kirchhoff Method For Noise Radiation From Ducted Fans”, AIAA Papcr 94-0462, Reno, Nevada, Jan 1904.

[9] Chyczewski, T.S., and Long, LA”, “A Higher Or- der Accurate Parallel Algorithm for Aeroacous- tic Applications”, AIAA Paper 94-2265, Colorado Springs, Colorado, June 1994.

[lo] Liu, Z.M., Mohan, A.S., Aubrey, T.A., and Belcher, W.R., “Parallelized FDTD for Antenna Radia- tion Pattern Calculations”, 11th Annual Review of Progress in Applied Computational Electromagnet- ics, Monterey, CA, March 1995.

The work done here mas partially supported by NASA undcr rrrant NAG2-867. The authors would like to thank

Y

thc Maui High Performance Computing Center and the [111 “,, and Long, L,N,, FVTD Algorithm Ccntcr for Academic Computing at Penn State for the for Equations on Massively Parallel Ma- computing timc made available on the IBM SP-2. The chines”, 11th Annual Rcview of Progress in Ap-

plied Computational Electromagnetics, Monterey, authors would also like to thank Mr Joe Schuster and

CA, March 1995. Mr Yusuf Ozyoruk for their valuable insights.

References [l] Yce, K.S., “Numerical Solution of Initial Boundary

Value Problems Involving hlaxwell’s Equations in Isotropic Media”, IEEE Transactions on Antennas and Propagataon Vol. AP-14, No. 3, May 1966

(21 Coorjian, P.M., “Algorithm Development for Maxwell’s Equations for Computational Electro- magnetism”, AIAA Paper 90-0251, Rcno, Nevada, Jan 1990.

[3] Madsen, N.K., and Ziolkowski, R.W., “A Three- Dimensional Modified Finite Volume Techniquc for Maxwell’s Equations”, Electromagnetics Vol 10, pp 147-161, 1990.

W

[12] Shang, J.S., Calahan, D.A., and Vikstorm, B., “Performance of a Finite Volume CEM Code on Multicomputers”, AIAA Paper 94-0236, Reno Nevada, Jan 1994.

[13] Rowell, C., Shankar, V., Hall, W.F., and Moham- madian, A., “Advances in Time-Domain CEM us- ing Massively Parallel Architectures”, 11th Annual Review of Progress in Applied Computational Elec- tromagnetics, Monterey, CA, March 1995.

[I41 Nguyen, B.T., Hutchinson, S.A., “The Upwind Leapfrog Algorithm Scheme for 3-D Electromag- netic Scattering and its Implementation on Two Massively Parallel Computers”, 10th Annual Re- view of Progress in Applied Computational Elec- tromagnctics, Montercy, CA, March 1994.

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1151 Kordulla, W., and Vinokur, M., "Efficient Compu- tation of Volume in Flow Predictions", AIAA Jonr- nal, Vol. 21, No. 6, pp 917-918, June 1983.

[16] Agarwal, R.K., and Deese, J.E., "Transonic Wing- Body Calculations Using Euler Equations", AIAA Paper 83-0501, Reno, Nevada, Jan 1983.

[17] Jameson, A., Schmidt, W., and Turkel, E., "Sumer- ical Solutions of the Euler Equations by Finite Vol- ume Methods Using Runge-Kutta Timc-Stepping Schemes", AIAA Paper 81-1259, Palo Alto, Cali- fornia, June 1981.

[l8] Luebbers, R.J., Kunz, K.S., Schneider, 11. and Hunsherger, F., "A Finite-Difference Time-Domain Near Zone to Far Zone Transformation', IEEE Transactions on Antennas and Propagation. Vol39, No. 4, April 1991.

[lo] Mur, G., "Absorbing Boundary Conditions for Finite-Difference Appro ximation of the Time- Domain Electromagnetic Field Equations", IEEE

Y.F. "A Transmitting boundary for Transient Wave Analysis", Scietia Sinica, Vol. 28, No. 10: pp 1063- 1076, Oct 1984.

E

. Machine No. Of nodes I.1- sec/gridPt/iter . CM-5 32 nodes 17.2 SP-2 4 nodes 12.76 SP-2 16 nodes 3.99

H / Y

E E

Figure 1: Combined E and H field cells

Table 1 : Parallel Performance Transactions in Electromagnetic Compatibility Vol. EMC-23, pp 1073-1077, Nov 1981.

R

w E(i, j, k+"

E(i, j+l, k)

Figure 2: Points contributing to evaluation of flux.

PEC Surface

'ncident

11 /

flux for the shaded surface is = - S x d -'

Figure 3: Flux Evaluation on PEC Surface.

PEC Surface PEC Surface

For each of the shade !l surfaces only points marked o contribute to calculation of the Gauss

+

PEC Surface

Processor Domain I Processor Domain 1+1

' \ H-fieid grid Y E-field grid Interior points on PD i

are boundary ioints for PD i+l

Figure 5: Processor Domain Decomposition

I Read grld lrom Proc 0 Read grld lrom Pia: 0 F

Figure 6: Message Passing Algorithm.

u Figure 4: PEC Surface Boundary implementation

Figure 7: Grid for the Sphere.

Figure 9: Grid for the Ogive

. . . . . . . . . . . . .......................................

. . . . . . . . . .

iusqa) s a

Figure 8: RCS for PEC Sphere.

m

m 0

U w P

u 0 I

._I

--T7-

................................

. . . . . . . . . . . .

.. ..i .........

.. , . ,..... ..

. . . . . . . . .

. . . . . . . . . - , . . . . . . .

.......................

. . . . . . . . . . .

i j n o L o 0 L o o L n o j n o N O I I - , - , . Y I j n Y I Y I C I , , , , , , , , I

iwsqal sari

Figure 10: RCS for PEC Ogive.

10

Figure 11: Grid for the wing

Figure 12: Scattered E, component for wing.

11

Figure 13: Scattered H, component for wing.

-

Figure 14: Grid for cngine inlet.

12

3 t

Figure 15: zones for engine inlet

Figure 16 Scattered E, Fields for Engine Inlet.

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