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Naval Air Warfare Center Weapons Division

FOREWORD

This report details the implementation of applying the bi-conjugant gradient algorithmto solve a volume integral equation. The integral equation is a solution to Maxwell 'sequations and involves scattering of a plane wave from a metallic nano-cyclinder. Thesolution algorithm is outlined, a numerical example is given, and a FORTRAN codelisting is provided. This work was done during a 2-month period at the Naval AirWarfare Center Weapons Division (NAWCWD), China Lake, California, during Juneand Ju ly 2004. The ONR-ASEE Fellowship Program funded it.

Th is report was reviewed for technical accuracy by Pamela L. Overfelt of the Sensorand Signal Sciences Div ision, Research Department, NAWCWD Code 4T4100D.

Approved byC. F. MARKARIAN,HeadResearch Department2 August 2004

Released for publication byKAREN HIGGINSDirector for Research and Engineering

NA WCWD Technical Publication 8590

Under authority ofD. J. VENLET

RADM, U.S. NavyCommander

Published by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Technical Information DivisionCollation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cover, 11 leavesFirst printing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 copies

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September 2004 June-July 20044. TITlE AND SUBTITlE 5. FUNDING NUMBERS

Bi-Conjugate Gradient Algorithm for Solution of Integral EquationsArising in Electromagnetic Scattering Problems (U)6 AlffilOR(S)

Surendra Singh, Klaus Halterman, and J. Merle Elson7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION

REPORT NUMBER

Naval Air Warfare Center Weapons Division NAWCWD TP 8590China Lake, CA 93555-6100

9. SPONSORINGlM ONITORING AGENCY NAM E(S) AND ADDRESS(ES) 10. SPONSORINGlMONITORINGAGENCY REPORT NUMBER

Office ofNaval Research1900 N. Quincy Ave.Arlington , VA 2220111. SUPPLEMENTARY NOTES

12a DISTRIBUTIO N/AVAILABILITY STATE MENT 12b. DISTRIBUTION CODE

A Statement; public release; distribution unlimited.13. ABSTRACT {Ma Ximum 200 words)

(U) The hi-conjugate gradient (bi-CG) algorithm is applied to numerically solve linear equation systemsresulting from integral equations arising in electromagnetic scattering problems. The basic advantage ofusing this algorithm over traditional methods , such as matrix inversion, is that the algorithm is iterative innature . The iterative nature allows the user to control the residual error in the final solution. Also , thealgorithm can be implemented without storing the coefficient matrix, thus providing huge saving in storagerequirements. It was realized that the existing code that utilized matrix inversion to solve the linear equationsystem was limited to a coarse discretization of the geometry. This code could not handle very fine geometrydiscretization due to storage limitations. With the implementation of the bi-CG algorithm, this limitation wasovercome . The present code can very easily handle very fme discretizations, thereby vastly improving theutility of the computer code. This report outlines the bi-CG algorithm and provides its implementation tosolve an electromagnetic scattering problem of a nanowire illuminated by a plane wave. The report alsoincludes a complete FORTRAN listing of the code.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Iterative Methods , Integral Equations, Electromagnetic Scattering, Bi- 21Conjugate Gradient Algorithm 16. PRICE CODE

17. SECURITY CLASSIFICATION l B. SECURITY CLASSIFICATION 19. SECURITY CLASSIF ICATION 20. LIMITATION OF ABSTRACTOF REPORT OF ABSTRACT OF THIS PAGE

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED SAR

NSN 7540-01-280- 5500 Standard Form 298 (Rev. 2-89)Prescnbed by ANSI Std. 239-18

29-102

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SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

Standard Form 298 Back (Rev. 2-89) SECURITY CLASS IFICATION OF THIS PAGE

UNCLASSIFIED

NAWCWD TP 8590

CONTENTS

Introduction 3

Bi-Conjugate Gradient Method (hi-CG) 4

Application ofBi-CG Method To Solve an Intregal Equation 6

References 9

Appendix: FORTRAN Code of the Bi-CG Algorithm and Associated FunctionSubprograms. . . .. .. . . . . .. ..... . ... . .. .. .. .. .. . . . ... .. .. . .. . ..... . . .. . .. . . . . .. . .. . .. . . . . . . . .. .. 10

---~-----~-~---------~~~------_._----------- --

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INTRODUCTION

The work described in this report was performed in collaboration with ProfessorSurendra Singh, who came to the Naval Air Warfare Center Weapons Division(NAWCWD), China Lake, California, during June and July 2004. Professor Singh, afaculty member in Electrical Engineering Department at The University of Tulsa, Tulsa,Oklahoma, was visiting as an Office of Naval Research-ASEE Summer Faculty Fellow.He worked in the Optics, RF and Material Physics Branch, Sensor and Signal SciencesDivision, Research Department. This work involved learning and implementing the bi­conjugate gradient algorithm in order to numerically solve linear equation systemsresulting from integral equations arising in electromagnetic scattering problems.

The hi-conjugate gradient (bi-CG) algorithm (Reference 1) is a well-known andestablished iterative technique to solve a system of linear equations of the form A x =b.In this equation the square matrix A and the source or excitation vector b are bothcomplex. The algorithm converges most rapidly if the matrix A is square, symmetric,and positive-definite, This does not imply that the algorithm is not suitable if the matrixis not symmetric. We have applied the algorithm to non-symmetric matrices with goodresults . Details of the bi-CG algorithm are provided in the next section. A variant of thisalgorithm, the bi-CG stabilized (bi-CGSTAB) algorithm (Reference 2), was alsoimplemented. But in the numerical experimentation, we found that the stabilized versiondid not provide any advantage over the earlier version. A very good source for a basicunderstanding of the conjugate gradient method is provided in Reference 3. In the Bi­Conjugate Gradient Method section, we provide the solution of an integral equationknown as the Lippman-Schwinger equation. The 2-dimensional physical probleminvolves a plane wave incident on a cylinder. Note that the original computer code toobtain the numerical solution of the integral equation was developed by Klaus Haltennanand J. Merle Elson at China Lake . This code utilized matrix inversion to solve the systemof equations. The code was modified to accommodate the bi-CG algorithm. A listing ofthe FORTRAN code of the bi-CG algorithm and associated function subprograms isprovided in the Appendix.

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BI-CONJUGATE GRADIENT METHOD (bi-CG)

The bi-Cfl method is an iterative technique used to solve a system of linear equationsof the form A x = b. The method is most suitable in situations where matrix A is so largethat storing the matrix may pose a significant storage problem. This means that themethod can be implemented without storing the entire A matrix. This is accomplished byjust calculating a single row or column of the matrix at a time. This provides tremendoussavings in storage requirements and allows the user to solve a very large system ofequations. Note that the savings in storage requirements comes at the cost of increasedcomputation time because each iteration of the algorithm requires computing theelements of matrix A twice. In order to increase the computational efficiency inevaluating the matrix elements, we found that the matrix elements involve thecomputation of two types of Hankel functions. The two Hankel function values arecomputed once and stored in two arrays. The arrays are then utilized in computing thematrix elements. This technique provided considerable saving in computation time. It isobserved that the bi-CG algorithm converges a little slower when the matrix A is dense,and that the convergence is faster when the matrix A is sparse. The pseudo-code for thebi-Cfl method is given below:

Initialize the following variables: conv = 10-4, rerr = 100, P=0, and the followingvectors: r = b, rn = 0, x = 0, p = 0, pn = 0, t = 0.

rn = r* (* denotes complex conjugate)vall = b*b

do i = I,nitmif rerr>conv

va12 = r*rrerr = abs(val2/vall)va13 = rn*rif (i* I) P= va13/vaI5p =r+ P ppn =rn + P pnt = A P (This operation can be performed by computing one

row ofA at a time, thus avoiding the storage of matrix A.)

val4 = pn*xa = va13/vaI4x =x+ap

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r = r-att =transpose(A*) pn (This operation can be performed by

computing one column of the matrix A at a time, thus avoiding the storage of the matrixA.)

rn=rn-atval5 =va13

end ifend do

SUBROUTINE beg (b,nunks,nitm,eonv,ci,nit,rerr)

This subroutine iteratively solves a system of linear equations of the form [A][ei]=[b]using the bi-CO method. The subroutine does not require the storage of the matrix A.Rather it uses two function programs, arow and acol which provide a specific row orcolumn of the matrix, respectively. Note that vector x is replaced by ci.

Input Variables:

nunkns (integer) - Total number of unknowns

b (complex vector) - Righthand side or excitation vector oflength nunkns

nitm (integer) - Maximum number of iterations for bi-CO (typically set it equal tonunkns)

cony (real) - Convergence factor for bi-CO (typically 10-4 to IO-{;)

Output Variables:

ci (complex) - Solution vector oflength nunkns

icount (integer) - Number of iterations needed for bi-CO method to converge

rerr (real) - Residual error. The bi-CO algorithm will stop when the residual error (rerr)becomes less than the pre-specified convergence factor (conv).

FUNCTION arow(i, vee)

This function provides the ith row of matrix A and returns a vector, vec, of lengthnunkns.

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Input: i (integer) - ith row number of matrix AOutput: vec (complex vector) - ith row ofmatrix A vector oflength, nunkns.

FUNCTION acol (j, vec)

This function provides the jth column of matrix, A, and returns a vector, vec, oflength nunkns.

Input:j (integer) - j th row number of matrix AOutput: vec (complex vector) - jth column of matrix A vector of length, nunkns.

COMPUTATION OF HANKEL FUNCTIONS HO AND HI

The tensor Green's function components require the evaluation of Hankel functionsHo andH.. Because the argument of the Hankel functions depends on the relativedistance between the source (x ',y ') and observation (x,y) point, it was determined thatthere are only a limited number of observation and source point combinations that arerepeated over and over again in the computation of the Green's function. The Hankelfunctions are computed for different combinations of source and observation points andstored in two arrays. Then the stored values are utilized in the computation of the tensorGreen's function, thereby reducing computation time by avoiding repeated computations .

APPLICATION OF bi-CG METHOD TO SOLVE ANINTERGAL EQUATION

Here we provide an example of solving a system of linear equations using the bi-CGmethod. The system of equations is obtained as part of the numerical solution of anintegral equation. The first attempt at solving the equations was done by using thetraditional approach of matrix inversion. However this required storing the matrix A Asthe dimensionality of the problem increased, thereby increasing the number of unknownsas well as the size of matrix A, it was clear that the resulting problem could not be solvedbecause of the limited storage capacity of the personal computer: hence the move to bi­CG method.

Consider a scattering system described by a dielectric function,E(r), embedded in an

infinite homogeneous background material, EB' When the system is illuminated by an

incident field, EO (r), the total electric field, E(r), is given by the following integralequation (Reference 4):

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E(r) = EO(r) + f dr lG B(r,r') kg tic(r')E(r')v

where G B(r ,r') is the Green's tensor associated with the infinite background CB'

tiC(r) = C(r) - cB and the integration is over the entire scatterer volume, V. Toimplement a numerical solution to the above integral equation, we define a grid with Nmeshes. Each mesh, i, is centered at position, r., and has a volume, V;, i = 1,2,...N (for a

2-dimensional systems, V; will be replaced by the area of the mesh, i). Representing the

discretized electric field, E; = E(r;) , the dielectric constant, tic; = tic(r;) , and the

Green's function, GB(i,j) = G B(r;,rj ) , the integral equations can be written as a system

of linear equations:

The Green's tensor GB(r,r') is given by

G B(r,r') = l[G~ ]

[G~,]

[G~]

For a complete description of the Green's tensor components, refer to Reference 4.For a geometry with N meshes , N= nx*ny, where nx and ny represent the number ofdivisions of the scatterer in x and y directions , respectively, the total number of unknownsis 3*N to account for the three components of the electric field. Each of the sub-matricesin the tensor Green 's function is now of the order of NxN. Now we provide a 2­dimensional scattering problem that uses the above formulation to compute the totalelectric field.

UME RICAL EXAMPLE

We provide a numerical example (Reference 5) utilizing the formulation presented inthis Section. We consider a metal wire of radius = 10 nm with a dielectric constant,E = (-1.07,0.29). The wavelength of the incident field is A=388 nm. The geometry is

discretized in a grid of 250x250 (nx=ny =250), resulting in a total of 125,000 (2*nx*ny )unknowns. The number of unknowns in this example is 2*nx*ny, as only twocomponents of the electric field are needed due to the polarization of the incident field .(Note that if we needed to store the matrix A of dimension (l25,000xI25,000), thestorage space needed would be very large. This provides the motivation for using the bi-

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NAWCWDTP 8590

CG iterative method , in which case we do not store this matrix) . Figure 1 show theelectric field amplitude relative to the incident field around the wire as it is illuminatedwith a plane wave.

Contour Plot of Relative Field Amplitude

403530252015105

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+0#roC 20

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>-10

X-Coordinate (nm)

FIGURE I. Contour Plot of Relative Field Amplitude.

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REFERENCES

1. T. K. Sarkar and S. M. Rao. "The Application of Conjugate Gradient Method for theSolution of Electromagnetic Scattering From Arbitrarily Oriented Wire Antennas,"IEEE Trans. Antennas Propag., Vol. 32, No. 4 (April 1994), pp. 398-403.

2. H. A. Van Der Vorst. "Bi-CGSTAB: A Fast and Smoothly Converging Variant ofBi-CG for the Solution of Non-symmetric Linear Systems," SIAM J. Sci. andStatistical Computing, Vol. 13, No. 2 (March 1992), pp. 631-44.

3. Carnegie Mellon University. An Introduction to the Conjugate Gradient MethodWithout the Agonizing Pain, by J. R. Shewchuck. Pittsburgh, Pennsylvania, CMUSchool of Computer Science (http://www.cs.berkley.edu/-jrs/).

4. O. J. F. Martin and N. B. Piller. "Electromagnetic Scattering in PolarizableBackgrounds," Phys. Rev. E, Vol. 58, No. 3 (September 1998), pp. 3909-15.

5. J. P. Kottmann and O. J. F. Martin. "Plasmon Resonances of Silver Nanowires Witha Nonregular Cross Section," Phys. Rev. B, Vol. 64 (8 November 2001) 235402­1:235402-10.

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AppendixFORTRAN CODE OF THE BI-CG ALGORITHM AND ASSOCIATED

FUNCTION SUBPROGRAMS

Here is the listing of the computer code with subroutine beg and functionsubprograms arow and aeol:

bi-conjugant gradient (works)solution to integral equation for e-field

program bcg_methodimplicit nonereal,parameter :: systern_size=40 !total grid size i n nrnreal, parameter :: lambdaO=338 !incident wavelength i n nrnreal, parameter :: L=systern_size /lambdaO !total system size i n

wavelengthsinte g e r, parameter . . nx=60, ny=nx , nxy = nx*ny ! t o t a l number of grid

pointsinteger, parameter .. maxi=3*nxycomplex, parameter .. Im=(O., 1.),zero=(O., O.),one=(l., 0.)complex , dimension(l:3*nxy) .. EO,etotalinteger, dime nsion(l:3*nxy) :: i pivcomplex, d imens i o n ( l: 2* nxy ) : : HO, H1integer , d imension (l:nxy ) :: ix,jycomplex , dime nsion(l: nxy) : : epsreal:: a rg , r_eff,theta,psi, krh o , phi, pi , kx,ky,x,y, wc, wcda,dx,dyreal :: xd,yd, xp, ypcomplex : : mfac t or ,Mx,My ,Mz , k z ,eb , z l ,rnxx, myy , mz zinteger :: i,j,k,icount,ijreal::conv,rerr,rho1,rho2,rho3,rhoINTEGER : : clock_start,c1ock_end,c1ock_rateREAL :: ela p sed_time,fr eal :: r1

CALL SYSTEM_CLOCK (COUNT_RATE=clock_rate) ! Find the rateCALL SYSTEM_CLOCK(COUNT=clock_start) ! Start timing

conv= .OOOl

EO=O.eb=onedx=L/nxdy=dx!dy=.025 !grid size p e r wa ve leng t hpi = acos ( - 1 . )wc = 2. *pi

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wc d a = wc ** 2 *dx*dytheta= 90.

phi = 90 .I p s i =O.kx=-cos(theta*pi /180 .)ky=- s in( theta*pi /180.)kz=O .k rho =sqrt(eb) !krho /kOr_ef f = sqrt(dx*dy/pi)rnf a c tor=wc*r_eff*Hl_fun(wc*r_eff*krho) /krho+2* Irn/(pi*krho**2 )Mx = I rn*p i/ 4* (2 .-kr h o **2 ) *rnf a c t o rMy=MxMz = Irn*pi /2 *(1 .-kz**2)*rnfactor

k = 0do i = 1 , nx

do j = 1, nyk = k + 1 !k goes up to ny*nxx =i*dxy = j*dyi x (k ) = ijy(k) = ja rg=kx*x+ky*yz l =exp(wc*Irn*arg)

EO(k) =z1*(cos(phi*pi /180.)*sin(theta*pi /180 .)*cos(psi*pi /180.) -&

! sin(phi*pi/180.)*sin(ps i*pi /180.»! EO(k+nxy) =

z1 *( s i n (phi *p i/1 8 0 . ) *s i n ( t h e t a *p i / 18 0 . ) *c os (p s i *p i/ 1 8 0 . ) +&! cos(phi*pi/180.)*sin(psi*pi /180.»!EO(k+2*nxy) = z1*cos(theta*pi /180.)* cos(ps i*pi /180.)

EO (k+2*nxy) = zlend do

end do

eps = zeror1=10./lambd a O !radius in wavelengths

choose sites of delta_epsilon over 20 gridk = 0do i = 1,nx

do j = 1,nyk = k + 1

eps(k) =zero! eps(k) = one*l.Srho1=sqrt(real(i -nx/2)**2+real(j -ny/2) **2)i f (rho1<=int(r1/dx» eps(k) = (-1.07,0.29) - one

r h o 2= s q r t (r e a l (i-nx/ 2 ) **2+r e a l ( j - ny/ 2 ) **2 )if (rho2< =3 .) eps(k) = (-100.,0.) - oner h o 3=s q r t (r e a l (i - 3 * (nx / 4 » **2 +r e a l ( j - ny/ 2 ) **2 )if (rho3 <=3.) eps(k) = (-100.,0.) - one

i f (i >= nx/2-S .and. i <= nx/2+S .and. &j >= ny/2 -S .and. j <= ny/2+S) eps(k) (-100.,0.) - one

! write(4S,44S) ix(k),jy(k) ,real(eps(k»end do

end do!clos e (4 S )

i =1do j=2, nxy

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x = ix(i)*dx;y=jy(i)*dyxp = ix(j)*dx;yp = jy(j)*dyxd=x-xp;yd=y-yprho = sqrt(xd*xd+yd*yd)arg = wc*krho*rhoij=(ix(i)-ix(j))**2 + (jy(i)-jy(j))**2

HO (ij)=HO_fun(arg)H1(ij)=H1_fun(arg)end do

mxx=Mx-O.5myy=My-O.5mzz=Mz!*****************************************************************************

call bcg(EO,3*nxy,maxi,conv,etotal,icount,rerr)print *, 'number of iterations=',icount-1print *, 'the residual error =',rerr

! print *,testx!***********************************************************************

******

do k = 1, nxy ! Exx = ix(k)*dx ; y = jy(k)*dywrite(97,*) x*lambdaO, y*lambdaO, abs(etotal(k))end do

do k = nxy+1, 2*nxy !Eyx = ix(k-nxy)*dx ; y = jy(k-nxy)*dywrite(98,*) x*lambdaO, y*lambdaO, abs(etotal(k))end do

do k = 2*nxy+l, 3*nxy !Ezx = ix(k-2*nxy)*dx ; y = jy(k-2*nxy)*dywrite(99,*) x*lambdaO, y*lambdaO, abs(etotal(k))end do

do k=1,nxyx = ix(k)*dx ; y = jy(k)*dy

arg=etotal(k)*conjg(etotal(k))+etotal(k+nxy)*conjg(etotal(k+nxy))+etotal(k+2*nxy)*conjg(etotal(k+2*nxy))

write(96,*) x*lambdaO, y*lambdaO, argend do

!**********************************************************************.******

444 format(F6.3,lX,F6.3,1X,F6.3)

CALL SYSTEM_CLOCK (COUNT=clock_end) ! Stop timing! Calculate the elapsed time in seconds:elapsed_time=REAL((clock_end-clock_start))/clock_rateprint *, 'elapsed time =',elapsed_time/60., 'minutes'

contains

subroutine bcg(b,nunkns,nitm,conv,ci,nit,rerr)implicit nonecomplex, intent(in), dimension(1:nunkns) :: b

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complex, intent(out), dimension(l:nunkns) :: ciinteger, intent(in) nunkns,nitminteger, intent(out) nitreal, intent(in) convreal, intent(out) rerrcomplex,dimension(l:nunkns) :: x,p,pn,r,rncomplex :: beta,alpha,vall,val2,val3,val4,val5,sumxinteger ::i

x=O;p=O;pn=O;ci=O;icount=O;beta=Orerr=lO0r=brn=conjg(r)vall=dot_.product(conjg(b),b)

do nit=1,nitm

if (rerr>conv) thenval2=dot-product(conjg(r),r)rerr=abs(val2/vall)val3=dotproduct(conjg(rn),r)if (nit/=l) beta=val3/val5p=r+beta*ppn=rn+beta*pn

forall (i=l:nunkns) x(i)=dotproduct(conjg(arow(i)),p)

val4=dot-product(conjg(pn),x)alpha=val3/val4ci=ci+alpha*pr=r-alpha*xforall (i=l:nunkns) x(i)=dotproduct(conjg(acol(i)),pn)

rn=rn-alpha*xval5=val3icount=icount+l

print *,'iteration#',nit-iprint *,'residual error',rerrelseexitend if

end doend subroutine bcg

function arow(i)implicit noneinteger, intent(in) :: icomplex, dimension(l:3*nxy) arowinteger :: j,n

n=nxy

do j=l,3*n

if (i<=n) then

if (j<=n) thenif (i==j) thenarow(j)=l.-mxx*eps(j)

elseif (eps(j)==zero) then

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arow(j) =0.else

arow(j)=-gxx(i,j)*eps(j)end if

end ifelse if (j>n .and. j<=2*n) then

if (eps(j-n)==zero) thenarow(j)=O.elsearow(j)=-gxy(i,j-n)*eps(j-n)end if

else if (j>2*n) thenif (eps(j-2*n)==zero) then

arow(j)=O.elsearow(j)=-gxz(i,j-2*n)*eps(j-2*n)end if

end if

end if

if (i>n .and. i<=2*n) then

if (j<=n) thenif (eps(j)==zero) then

arow(j)=O.elsearow(j)=-gxy(i-n,j)*eps(j)

end ifelse if (j>n .and. j<= 2*n) thenif (i==j) then

arow(j)=l.-myy*eps(j-n)else

if (eps(j-n)==zero) thenarow(j)=O.elsearow(j)=-gyy(i-n,j-n)*eps(j-n)end if

end ifelse if (j>2*n) then

if (eps(j-2*n)==zero) thenarow(j)=O.else

arow(j)=-gyz(i-n ,j-2*n)*eps(j-2*n)end if

end ifend if

if (i>2*n) then

if (j<=n) thenif (eps(j)==zero) then

arow(j)=O.else

arow(j)=-gxz(i-2*n,j)*eps(j)end if

else if (j>n .and~ j<= 2*n) thenif (eps(j-n)==zero) then

arow(j)=O.

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elsearow(j)=-gyz(i-2*n,j-n)*eps(j-n)end if

else if (j>2*n) thenif (i==j) then

arow(j)=1.-mzz*eps(j-2*n)else

if (eps(j-2*n) ==zero) thenarow(j)=O.else

arow(j)=-gzz(i-2*n,j-2*n)*eps(j-2*n)end if

end ifend if

end if

end doend function arow

function acol(j)implicit noneinteger, intent (in) .. jcomplex, dimension(1:3*nxy) .. acolinteger :: n , i

n=nxydo i=1,3*n

if (i<=n) then

if (j<=n) thenif (i==j) thenacol(i)=l.-mxx*eps(j)

elseif (eps(j)==zero) then

acol(i)=O.else

acol(i)=-gxx(i,j)*eps(j)end if

end ifelse if (j>n .and . j<=2*n) then

if (eps(j-n)==zero) thenacol(i)=O.else

acol(i)=-gxy(i,j-n)*eps(j-n)end if

else if (j>2*n) thenif (eps(j-2*n)==zero) then

acol(i)=O.else

acol(i)=-gxz(i,j-2*n)*eps(j-2*n)end if

end ifend if

if (i>n .and. i<=2*n) then

if (j<=n) thenif (eps(j)==zero) then

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acol(i)=O.else

acol(i) =-gxy(i-n,j)*eps( j)end if

else if (j>n .and. j<= 2*n) thenif (i==j) then

acol(i) =l.-myy*eps(j-n)else

if (eps(j-n)==zero) thenacol(i)=O.else

acol(i)=-gyy(i-n,j-n)*eps(j-n)end if

end ifelse if (j>2*n) then

if (eps(j-2*n)==zero) thenacol(i)=O.else

acol(i)=-gyz(i-n,j-2*n)*eps(j-2*n)end if

end ifend if

if (i>2*n) then

if (j<=n) thenif (eps(j)==zero) then

acol(i)=O .else

acol(i) =-gxz(i-2*n,j)*eps(j)end if

else if (j>n .and. j<= 2*n) thenif (eps(j-n)==zero) then

acol(i)=O.else

acol(i)=-gyz(i-2*n,j-n)*eps(j-n)end if

else if (j>2*n) thenif (i==j) then

acol(i)=1.-mzz*eps(j-2*n)elseif (eps(j-2*n)==zero) then

acol(i)=O.else

acol(i)=-gzz(i-2*n,j-2*n)*eps(j-2*n)end if

end ifend if

end if

end doend function acol

complex function gxx(i,j)implicit none

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integer, intent (in) :: i,jinteger:: i j,difl,dif2real :: rho,x,y,xp,yp,rx,ry,c2theta ,xd,yd

x = i x (i ) *dx; y = jy(i)*dyxp = i x ( j ) *dx ; yp = jy(j)*dyxd=x-xp;yd=y -yprho = sqrt(xd*xd+yd*yd)d ifl= i x ( i ) - i x (j ) ; d i f 2= j y ( i )-jy(j )ij =difl*difl+dif2*dif2rx = xd /rhory = yd /rhoc 2 t h eta=r x *r x- ry*rygxx=wcda*Im/4.*((1 . -

krho*rx*krho*rx)*HO(ij)+krho*c2theta*Hl(ij) /(wc*rho))

end function gxx

complex function gxy(i,j)implicit noneinteger, intent (in) :: i,jinteger : : ij,difl,dif2real :: rho,x,y,xp,yp,rx,ry,s2theta,xd,ydif(i /= j) thenx = ix(i)*dx;y=jy(i)*dyxp = ix(j)*dx;yp=jy(j)*dyxd=x-xp;yd=y-yprho = sqrt(xd*xd+yd*yd)arg = wc *krho*rhod ifl= i x ( i ) - i x ( j ) ; d i f 2= j y ( i ) - j y ( j )i j=d i f l *d i f l +d i f 2 *d i f 2rx = xd/rhory = yd/rhos2theta=2*ry*rx

gxy=wcda*Im/8.*krho*krho*s2theta*(2./arg*Hl(ij) -HO(ij))elsegxy = zeroend if

end function gxy

complex function gxz(i,j)implicit none

integer, intent (in) : : i,jinteger :: ij,difl,dif2real :: rho,x,y,xp,yp,rx,xd,ydif(i /= j) thenx =ix(i)*dx ; y =jy(i)*dyxp=ix(j)*dx ; yp=jy(j)*dyxd=x-xp;yd=y-yprho = sqrt(xd*xd+yd*yd)arg = wc*krho*rhodifl= ix(i)-ix(j);dif2=jy(i)-jy(j)i j =difl*difl+dif2*dif2rx = xd/rhogxz=wcda*1./4 . *krho*kz*rx*Hl (ij)elsegxz = zeroend if

end f un c t i on gxz

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c omple x f unction gyy(i, j )i mp l i c i t nonei n t e g e r , intent (in ) :: i, ji n t e g e r :: ij , d i f l, d i f 2real :: rho ,x ,y,xp ,yp ,rx,ry,c2theta,xd,ydx =i x( i ) *dx iy=jy( i )*dyxp=ix( j)*dx iYP=jy(j)*dyXd=X-XPiyd=y-yprho = sqrt(xd*xd+yd*yd)arg = wc*krho*rhodifl= ix (i)-ix(j);dif2=jy( i) - jy( j )ij =difl*difl+dif2*dif2rx = xd/rhory = yd/rhoc2theta=rx*rx-ry*rygyy=wcda*Im/4 .*((1 .-krho*ry*krho*ry)*HO(ij) -

krho*c2theta*Hl(ij)/(wc*rho))

end function gyy

complex function gyz(i,j)implicit noneinteger , intent (in) :: i, jinteger :: ij ,difl,dif2real:: rho,x,y,xp,yp,ry,xd,ydif(i 1= j) thenx =ix (i)*dx ;y = jy( i )*dyxp=ix(j)*dx ;yp= jy( j)*dyxd=x-xp;yd=y-yprho = sqrt(xd*xd+yd*yd)arg=wc*krho*rhodifl = ix (i ) -ix(j)idif2=jy (i ) - jy( j )ij =difl*difl+dif2*dif2ry=yd/rhogyz=wcda*l. 14 . *krho*kz*ry*Hl (ij)elsegyz = zeroend if

end function gyz

complex f unction gzz (i ,j)implicit noneinteger, intent (in) :: i,jinteger :: ij,difl,dif2real : : rho, x,y,xp,yp,xd,ydx = ix(i)*dx ; y = jy(i)*dyxp = ix(j)*dx i yp = jy(j)*dyxd=x-xp;yd=y-yprho = sqrt(xd*xd+yd*yd)arg = wc*krho*rhodifl= ix(i)-ix(j);dif2=jy(i) - jy( j)i j=difl*difl+dif2*dif2gzz= wcda* Im/4.*(1.-kz*kz)*HO(ij)

end function gzz

FUNCTION HO_fun(z) RESULT (HOou t )IMPLICIT NONEreal, INTENT (IN) :: zreal:: x,J_O,Y_O,f_O,theta,x2,x3,x4,x5,x6,x8,xlO,x12,sqz

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complex: : HOout

if ( z <= 3.) thenx= z/ 3 .x2 =x*x;x4 =x2*x2;x6=x4*x2;x8=x6*x2;xlO=x8*x2;xl2 =xlO*x2J_O=1 .-2 . 2 4 9 9 9 97 *x 2 +1 . 2 65 62 0 8 *x 4- . 3 l 63 8 6 6 *x 6+ . 044 44 7 9 *x 8 ­. 0 03 9 4 4 4 *x l O+ . 00 02 l 0 0 *x 1 2Y_O=(2 /pi)*log(z/2.)*J_O+.3674669l+.60559366*x2­.74350384*x4+ .25300ll7*x6-.04261214*x8+.00427916*xlO -.00024846*x12e l s ex =3. / zx 2=x*x;x3 =x2*x ;x4=x3*x;x5=x4*x;x6=x5*x;sqz=sqrt(z)f_O=. 7 97 8 84 5 6- . 0 0 00 0 07 7 *x- . 0 05 52 7 4 0*x 2 - . 0 0 0 0 9 5l 2 *x 3 +. 0 0 1 3 7 2 3 7 *x 4 &

- . 0 0 07 2 8 0 5 *x 5 + .OOOl4476*x6

t h e t a= z - . 7 8 53 9 8 l 6 - . 04 1 6 63 97 *x - . 0 0 0 03 9 54 *x 2 + . 0 02 62 57 3 *x 3 - . 0 0 0 54 l 2 5 *x 4 && - . 0 002 9 3 3 3 *x 5+ . OOOl 3 5 5 8 *x 6

J_O =f_O*cos(theta)/sqzY_O=f_O*sin(theta)/sqzend if

HOout=cmplx(J_O,Y_O)END FUNCTION HO_fun

FUNCTION Hl_fun(z) RESULT (Hlout)IMPLICIT NONEr e a l , INTENT (IN) :: zreal:: x,J_l,Y_l,f_l,thetal,x2,x3,x4.x5,x6.x8,xlO,xl2 ,sqzc omp l e x : : Hlout

if (z <= 3.) thenx =z /3.x2 =x*x;x4 =x2*x2;x6=x4*x2;x8=x6*x2;xlO=x8*x2;xl2=xlO*x2J _l=.5*z - .56249985*z*x2+.21093573*z*x4-.03954289*z*x6+.00443319*z*x8­.0003l761*z*xlO+.OOOOl109*z*xl2

Y_l=(2 /pi)*log(z /2 .)*J_l-.6366l98/z+.22l209l*x2/z+2.1682709*x4 /z ­l . 31 64 8 27 *x 6 / z + . 31 2 3 9 51 *x 8 / z - . 04 0 0 97 6*x l O/ z + . 0 02 7 87 3 *x 1 2 / zelsex =3 . /zx2 =x*x;x3 =x2*x;x4=x3*x;x5=x4*x;x6=x5*Xisqz=sqrt(z)f_l= . 7 97 884 5 6+ . 0 0 0 0 01 5 6 *x + . Ol 65 9 6 67 *x 2 + . 0 0 01 7 1 0 5 *x 3 ­.0024951l*x4+.00ll3653*x5-.00020033*x6

thetal=z-2.356l9449+.12499612*x+.00005650*x2­.00637879*x3+.00074348*x4+.00079824*x5-.00029l66*x6

J _l=f_l*cos(thetal)/sqzY_l=f_l*sin(thetal)/sqzend if

Hlout=cmplx(J_l.Y_l)END FUNCTION Hl_fun

end program bcg_method

19

INITIAL DISTRIBUTION

1 Naval War College, Newport (IE22, President)1 Headquarters , 497 INOT, Falls Church (lOG Representative)2 Defense Technical Information Center, Fort Belvoir1 Center for Naval Analyses, Alexandria, VA (Technical Library)8 Surenda Singh, Department of Electrical Engineering, The University ofTulsa, 600

S. College Avenue, Tulsa OK 74104

ON-SITE DISTRIBUTION

4 Code 4BLOOOD (3 plus Archives copy)1 Code 4TOOOOD, Markarian12 Code 4T411OD

Chesnut (1)Elson (5)Haltennan (5)Overfelt (I)