Issued as Progress report, Report, and Final report, Project no. G-37-641Library lacks diskett

Meyer, Gunter H.

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Scholarly Materials And Research @ Georgia Tech

fa 17:33:11 	 OCA PAD INITIATION - PROJECT HEADER INFORMATION 	 08/17/88 Active Project #: G-37-641 	 Cost share #: Center # : R6563-0A0 Center shr Contract#: Z81156 	Mod #: Prime #: Subprojects ? : N Main project #: Rev #: 0 OCA file #: Work type : RES Document : PO Contract entity: GTRC Project unit: 	MATH 	Unit code: 02.010.144 Project director(s): MEYER G H 	MATH Sponsor/division names: MCDONNELL DOUGLAS CORP 	/ ST LOUIS, MO Sponsor/division codes: 212 	/ 045 Award period: 	880801 	to 	881231 (performance) 	881231 (reports) Total to date 29,827.00 29,827.00 0.00 Sponsor amount 	 New this change Contract value 	29,827.00 Funded 	 29,827.00 Cost sharing amount ntracting plan apply ?: N SIS OF EM SCATTERING FROM CURVED SURFACES PROJECT ADMINISTRATION DATA OCA contact: E. Faith Gleason 	894-4820 Sponsor technical contact 	Sponsor issuing office RON PEARLMAN, DEPT. 313, BLDG. 61 (314)777-3407 MCDONNELL AIRCRAFT COMPANY PO BOX 516 ST LOUIS, MO 63166 Security class (U,C,S,TS) : U Defense priority rating 	: Equipment title vests with: 	Sponsor Administrative comments -INITIATION. J. PENNY, DEPT 760, BLDG 33, LEV. 5N (314)233-9660 MCDONNELL DOUGLAS CORPORATION MCDONNELL AIRCRAFT COMPANY PO BOX 516, ST LOUIS, MO 63166 ONR resident rep. is ACO (Y/N): N supplemental sheet GIT GEORGIA INSTITUTE OF TECHNOLOGY OFFICE OF CONTRACT ADMINISTRATION NOTICE OF PROJECT CLOSEOUT ■ Project No. G-37-641    Closeout Notice Date 07/08/91 Center No. R6563-0A0 	 School/Lab MATH 	  Project Director MEYER G H       Sponsor MCDONNELL DOUGLAS CORP/ST LOUIS, MO 	  Contract/Grant No. Z81156 	  Contract Entity GTRC Prime Contract No. 	  Title ANALYSIS OF EM SCATTERING FROM CURVED SURFACES 	  Effective Completion Date 881231 (Performance) 881231 (Reports) Date Closeout Actions Required: 	 Y/N Submitted Final Invoice or Copy of Final Invoice 	 Y 	900713 Final Report of Inventions and/or Subcontracts 	 N Government Property Inventory 8 Related Certificate 	N Classified Material Certificate 	 N Release and Assignment 	 N Other 	 N Comments 	  Subproject Under Main Project No. 	  Continues Project No. 	  Distribution Required: Project Director 	 Y Administrative Network Representative 	 Y GTRI Accounting/Grants and Contracts Y Procurement/Supply Services 	 Y Research Property Managment Y Research Security Services 	 N Reports Coordinator (OCA) Y GTRC 	 Y Project File 	 Y Other 	 N N To: 	Ms. F. Gleason From: G. Meyer. Mathematics November 14, 1988 Enclosed please find a copy of my reports which I have sent to Ron Pearlman at McDonnell Douglas Aircraft Company. They are for your records. In a telephone conversation on Thursday last week, Ron has asked that Georgia Tech bill McAir for the balance of the contract at its earliest  convenience to avoid getting caught in some accounting backlog at McAir. Thanks. G-37 6'11 Progress Report on WS-NAPD-2819 Analysis of Scattering from Curved Surfaces Gunter H. Meyer Georgia Tech September 30, 1988 The Fock theory as presented in [1 , 21 is designed to provide the electromagnetic fields near the vertex of a conducting scatterer S of the form z-- 	(alt + 2bxy + cy l"). In this theory it is assumed that the scatterer is a paraboloid (ac-b > 0) and that the incident wave is a time harmonic plane wave traveling along the x-axis. The total field satisfies the Helmholz equation (Vx x 4 	F / t z 	4 kl.tr .1 0 with appropriate boundary conditions on S. The key assumption is made that =*ttiti)z)e , 	 X `r where xx and 7,3 ■3 are small compared to ICrx and Vivi. . A t  formalism is then presented which allows an approximate solution kr n terms of Airy functions. The work carried out so far is based on the observation that after the order of magnitude approximations the resulting equations, including the reduced Leontovich boundary condition, contain the y-coordinate only as a parameter. Specifically, it is shown that the Fock transformations remain valid essentially without change if the scatterer S is of the form z = - 7_  (a(y)xt + 2b(y)x + c(y)) where a(y)c(y)-bi (y) > 0. In contrast to the old theory the cross section through S at constant z no longer need to be elliptic. For example, - y 6 now is permissble. In general, the new formulation allows somewhat more freedom in approximating a given scatterer S with a shape with parabolic cross sections in x only. For this more general setting the fields near the vertex are again expressible in terms of Airy functions. Given an arbitrary scatterer and a point x o near the shadow boundary the question now arises of where to place the origin for the Fock theory in order to find the fields near x o . Ray theory suggests that it be determined by the geodesic from x e to the shadow boundary which has the direction of the incident wave there. A computer program based on a piecewise polynominal representation of the geodesic is currently in the debugging stage. Concurrent work carried out in collaboration with the EM signatures group of MDRLsuggests that the whole Fock theory can be based on a local near-geodesic coordinate system. If successful, this approach would free the theory from any specific representation of the scatterer but would place a premium on geodesic calculations. This serves as an added incentive for the code development. 1.V. A. Fock, Electromagnetic Diffration and Propagation Problems, Pergamon Press, Oxford, 1965. 2. G. T. Ruck, et al., RCS Handbook Vol. 1, Plenum Press, New York, 1970. (3eorgia institute of Technology Schooi of Mathematics At!anta, Georgia 30332 .5 	(404 894-2700 DE 5 VitiNG T. I 	 DDAY • - November 14, 1988 Dr. Ron Pearlman Dept. 313, Building 61 McDonnell Douglas Aircraft Company P. 0. Box 516 St. Louis, Missouri 63166 Dear Ron: Enclosed please find a listing and a disk for a Fortran program to compute geodesics. The program was written and checked out on the Cyber 855 of Georgia Tech. It was transferred with the CDC communications package Connect to my Macintosh II and copied to the enclosed floppy disk. The listing was taken from the disk and printed with Microsoft Word. Please let me know if this program can be easily transferred to the VAX. If not I shall try to get a VAX tape made here at Georgia Tech. I shall attach a brief ,lescription of the program and a representative calculation. Please let me know if the present program meets your needs or whether modifications are required. Sincerely yours, Gunter H. Meyer GHM:cw An Equal Education and Employment Opportunity Institution 	 A Unit of the University System of Georgia Program GEODESY is a Fortran 77 program to compute a geodesic curve (x(t),y(t),z(t)) on a surface z = f(x,y) One end of the geodesic is a specified target point (x i .y i ,z i ). The other end is either a specified point (x0 ,y 0 ,z 0 ) on the surface z = f(z,y), or to be computed such that the geodesic ends on a prescribed curve x = sb(y) on the surface with a specified direction (k 1 ,k 2 ,k 3 ). (The idea is that the curve x = sb(y) is the shadow boundary on the surface for an incoming plane wave traveling in the direction (k 1 ,k 2 ,k 3 ). The geodesic then is the ray path to the target point.) The program requires the following modifications before use. The user must provide in SUBROUTINE SURFACE the function f(x,y) and its first and second partial derivatives. The user must provide in FUNCTION SB the equation x = sb(y) for the shadow boundary. It is obtained by solving ((—df/dx, —df/dy,1),(k 1 ,k 2 ,k 3 )) = 0 ((x,y) denotes the usual dot product). On execution the program asks for the following input: 1. Number of nodes for the approximation of the geodesic. 3 to 5 nodes suffice for a short geodesic. Up to 20 nodes are allowed. The execution time increases drastically with the number of nodes. 2. Base coordinate (x 1 ,y 1 ) of the target point (x1,y1,z1). 3. Option which determines whether (1) the geodesic from point to point or (2) from curve to point is to be computed. 4. Base coordinates (x o ,y 0 ) of the starting point (x 0 ,y 0 ,z 0 ) in the case of option 1. 5. Direction of the incident wave in case of option 2. 6. Guess for the coordinate y o of the endpoint on the shadow boundary in case of option 2. Output: 1. After completion of each secant step the program prints out the number of conjugate gradient iterations and the length of the computed curve. These lengths must converge if the program is successful. 2. After convergence the program prints the endpoints of the geodesic, its direction at the endpoint (x 0 ,y 0 ,z 0 ) and the length of the geodesic. 3. If the last step of the conjugate gradient calculation terminates with an error message (e.g. IER = 129) the final result is of doubtful value. Restrictions: 1. The program uses the conjugate gradient subroutine ZXCGR of the IN1$L Program Library which must be attached before execution. 2. The function f(x,y) and its derivatives must be defined for all x and y in order for the conjugate gradient method to work. This limits use of the program for surfaces like z = ,11 — x 2 — y 2 . 3. The program will likely experience difficulties if the geodesic is not uniquely defined. Expianaiun for the sample output: The program is provided is set up for the surface z-ci 5' Ax 1. where B=1.C-3. The plot shos the surface over the rectangle ,  i I Run 1: 	5 nodes are requested. The incident vector is 11,1,-1 The prgracn did not converge within the error tolerance of I e-5 in the allowable 20 secant iterations On the other hand. the length of the geodesic remains nicely stationary. Run 2 	10 nodes are requested and the final point from run 1 is given is initial guess. We have convergence after 9 secant iterations Run 3: 	20 nodes are requested The final point From run 2 is entered as the endpoint under option 1. The answers are the same as in run 2. - _ 	_ : - - - _ Final Report on WS—NAPD-2819 Analysis of Scattering from Curved Surfaces Gunter H. Meyer Georgia Institute of Technology December 15, 1988 The Fock theory as presented in [1, 2] is designed to provide the electromagnetic fields near the vertex of a conducting scatterer S of the form 1 Z2(ax` 2bxy + cy2 ). In this theory it is assumed that the scatterer is a paraboloid (ac — b2 > 0) and that the incident wave is a time harmonic plane wave traveling along the x—axis. The total field satisfies the Helmholtz equation Oxx Zkyy Ozz k2 0 = 0 with appropriate boundary conditions on S. The key assumption is made that = 0*(x,y,z)e 2k x where Dix 	Y and 	are small compared to 1c0 x* and 0z* z . A formalism is then presented which allows an approximate solution b* in terms of Airy functions. The work is based on the observation that after standard order of magnitude approx-imations the resulting equations, including the reduced Leontovich boundary condition, contain the y—coordinate only as a parameter. Specifically, it is shown that the Fock transformations remain valid essentially without change if the scatterer S is of the form z = (a(y)x 2 + 2b(y)x + c(y)) 1 where a(y)c(y) — b 2 (y} > 0. In contrast to the old theory the cross section through S at constant z no longer need to be elliptic. For example, z = x2 — y4 now is permissble. In general, the new formulation allows somewhat more freedom in approximating a given scatterer S with a shape with parabolic cross sections in x only. For this more general setting the fields near the vertex are again expressible in terms of Airy functions. Given an arbitrary scatterer and a point x o near the shadow boundary the question now arises of where to place the origin for the Fock theory in order to find the fields near x o . Ray theory suggests that it be determined by the geodesic from xo to the shadow boundary which has the direction of the incident wave there. A computer program for its calculation which is based on a piecewise polynomial representation of the geodesic has been submitted earlier. Concurrent work carried out in collaboration with the EM signatures group of MDRL suggests that the whole Fock theory can be based on a local near—geodesic coordinate system. If successful, this approach would free the theory from any specific representation of the scatterer but would place a premium on geodesic calculations. This serves as an added incentive for the code development. 1. V. A. Fock, Electromagnetic Diffration and Propagation Problems, Pergamon Press, Oxford, 1965. 2. G. T. Ruck, et al., RCS Handbook, Vol. 1, Plenum Press, New York, 1970. 2 

Analysis of EM scattering from curved surfaces

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Analysis of EM scattering from curved surfaces

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