Electromagnetics Laboratory Report No. 76-6

A METHOD FOR COMBINING INTEGRAL EQUATION AND ASYMPTOTIC TECHNIQUES FO,, SOLVING ELECTROMAGNETIC SCATTERING PROBLEMS

Technical Report

W. L. Ko R. Mittra

r-tay 1976

U.S. Army Research Office Grant No. DAHC04-74-G-0113

Office of Naval Research Grant No. N00014-75-C-0293

Electromagnetics Laboratory Department of Electrical Engineering ~

Engineering Experiment Station University of Illinois at Urbana-Champaign

Urbana, Illinois 61801

Approved for public release; distribution unlimited.

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THE Fl NDI NGS IN THIS REPORT ARE NOT TO BE CONSTRUED AS AN OFFICIAL DEPARTMENT OF THE ARMY POSITION, UNLESS SO DESIGNATED BY OTHER AUTHOR I ZED DOCUMENTS.

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1 Radar Scattering

i Fourier Transform Technique

1 Asymptotic Methods

1 Integral Equations

1 Numerical Solution

1

1 1 1 1 I

(continued)

13. Diffractions by a strip, a thin plate, a rectangular cylinder, and a circular

cylinder are presented as illustrative examples that demonstrate the usefulness of the approach for handling a variety of electromagnetic scattering problems in the resonance region and above. Some concluding remarks and comparison with other methods are also Included.

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ILU-ENG-76-2549

A METHOD FOR COMBINING INTEGRAL EQUATION AND ASYMPTOTIC

TECHNIQUES FOR SOLVING ELECTROMAGNETIC SCATTERING PROBLEMS

Technical Report

W. L. Ko R. Mlttra

May 1976

U.S. Army Research Office Grant No. DAHC04-74-G-0113

Office of Naval Research Grant No. N00014-75-C-0293

Electromagnetics Laboratory Department of Electrical Engineering

Engineering Experiment Station University of Illinois at Urbana-Champaign

Urbana, Illinois 61801

Approved for public release; distribution unlimited.

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ACKNOWLEDGMENT

The work reported In this paper was supported In part by the

U. S. Army Research Office under Grant DAHC04-74-0-0113 and In part

by the Office of Naval Research under Grant NOO014-75-C-0293.

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PREFACE

A little over a century ago, Maxwell put together the fundamental

laws that govern all the electromagnetic phenomeu In an amazingly

elegant form, i.e., the set of equations named after him. Ever since

then, the major contributions in the field of electromagnetics have been

solving boundary value problems, or seeking solutions which satisfy

the Maxwell's equations and match the boundary conditions (including

radiating conditions as special cases)in specific environments under

consideration.

The dyadic Green's functions technique, an elegant and powerful

method for solving boundary-value problems, was first formulated by

Schwinger in the early 1940's. However, solutions obtained by that

technique are extremely complicated and are not always best suited for

numerical computations.

In the late 1950's, Keller conceived the concept of a diffraction-

coefficient approach to the high-frequency scattering, and was able

to obtain approximate solutions for far-field patterns in a very simple

and straightforward manner. Although his method fails at certain aspect

angles in space, it has been applied to solve many practical problems.

In the 1960's, the moment method became popular due to the avail-

ability of the large scale computer systems. However, for electrically

large scatterers, the moment method is limited by the storage of a

computer.

In the early 1970's, a trend of combining the Keller's geometrical

theory of diffraction with the moment method to solve high-frequency

scattering problems was initiated. However, a general approach

for combining the two techniques has yet to be developed.

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It Is the purpose of this study to Introduce a method of combining

the integral equation and asymptotic techniques for solving electro-

magnetic scattering problems. The method is conceptually simple and

computationally efficient. Comparisons with contemporary combinational

approaches show that the method, when fully developed, appears to be

highly promising in solving practical problems. Of course, this thesis

is merely a start, and there is much left to be studied and investigated.

It is the author's hope that future research activities along the same

lines as described herein will result in not only a feasible but also

a practical way of handling real-world high-frequency scattering problems.

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ABSTRACT

This paper Introduces a new approach for combining the Integral

equation and high frequency asymptotic techniques, e.g., the geometrical

theory of diffraction. The method takes advantage of the fact that the

Fourier transform of the unknown surface current distribution is pro-

portional to the scattered far field. A number of asymptotic methods

are currently available that provide good approximation to this far

field in a convenient analytic form which is useful for deriving an

initial estimate of the Fourier transform of the current distribution.

An iterative scheme is developed for systematically improving the

initial form of the high frequency asymptotic solution by manipulating

the Integral equation in the Fourier transform domain.

A synthetic-aperture-distribution scheme is also developed in which

the approximate scattered far-field pattern obtained by asymptotic tech-

niques is improved by systematically correcting the scattered field

distribution on an aperture erected in juxtaposition with the obstacle.

The Introduction of such a planar aperture not only provides an additional

degree of freedom in performing improving operations, but also renders

the scheme to handle n-dimensional geometries by (n - 1)-dimensional

fast Fourier transform (FFT), where n - 2,3, and circumvents the un-

wieldy three-dimensional FFT, making it a conceptually simple and com-

putationally efficient method.

ix

A salient feature of the method Is that It provides convenient

validity checks of the solutions for the surface current distribution

and the scattered far-field pattern by verifying that the scattered

field obtained Indeed satisfies the boundary conditions at the surface

of the scatterer. Another Important feature of the method Is that It

yields both the Induced surface current density and the far field.

Diffractions by a strip, a thin plate, a rectangular cylinder, and

a circular cylinder are presented as illustrative examples that demon-

strate the usefulness of the approach for handling a variety of electro-

magnetic scattering problems In the resonance region and above. Some

concluding remarks and comparison with other methods are also Included.

TABLE OF CONTENTS

xi

Page

1. INTRODUCTION 1

2. FORMULATION OF AN ITERATION METHOD 5

2.1 Derivation of the Method 5 2.2 Recipe for Applying the Method 8

3. DIFFRACTION BY AN INFINITE STRIP 11

3.1 Introduction 11 3.2 Geometry of the Strip Problem 11 3.3 Iteration Method Applied to the Strip Problem 12 3.A Numerical Results and Discussions 16

3.4.1 Normal Incidence 16 3.4.2 Near grazing Incidence 18

3.5 Summary 25

4. DIFFRACTION BY A FINITE THIN PLATE 30

4.1 Introduction 30 4.2 Iteration Method Applied to the Plate Problem 31 4.3 Numerical Results and Discussions 38

5. A NEW LOOK AT THE SCATTERING OF A PLANE WAVE BY A RECTANGULAR CYLINDER 47

5.1 Introduction 47 5.2 Formulation 49

5.2.1 Pole singularities In the diffraction co- efficients 55

5.2.2 Pole singularities In the physical optics currents 59

5.2.3 Discontinuities In the far-field pattern at * - 0,

j, it, and y 72

5.2.4 Physical Interpretation of the existence of the discontinuities In the far-field pattern at 4) 0, ir j 3Tr ,, y, ir, and = 76

5.3 Improved Far-Field Pattern and Comparison with Results Obtained by Other Approaches 90

5.4 Accuracy Check 95

5.4.1 Method of computation 96 5.4.2 Results and comments 99

5.5 Summary 102

xil

Page

6. A SYNTHETIC-APERTURE-DISTRIBUTION APPROACH TO THE HIGH- FREQUENCY ELECTROMAGNETIC SCATTERING OF OBSTACLES WITH CONVEXLY CURVED SURFACE 104

6.1 Introduction. 104 6.2 Exact Solution Ill 6.3 Geometrical Optics Solution 118

6.3.1 Fields of a ray 118 6.3.2 Reflection at a curved surface boundary separating

two different media 126 6.3.3 Scattering of a plane wave by a circular

cylinder 129

6.4 Synthesizing the Approximate Aperture Field Distribution. . 132

6.4.1 Method I 135 6.4.2 Method II 138 6.4.3 Method III 141 6.4.4 Method IV 147 6.4.5 Method V 153

6.5 Computation of Induced Surface Current 169

6.5.1 Method of computation 170 6.5.2 Results and comments 173

6.6 Accuracy Check 1