Optimization methods in electromagnetic radiation

Springer Monographs in Mathematics Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo Thomas S Angell Andreas Kirsch Optimization Methods in Electromagnetic Radiation With 78 Illustrations Springer Thomas S Angell Department of Mathematical Sciences University of Delaware Newark, DE 19716 USA angell@math.udel.edu Andreas Kirsch Mathematics Institute II University of Karlsruhe Englerstr D-76128 Karlsruhe Germany kirsch@math.uni-karlsruhe.de Mathematics Subject Classification (2000): 78M50, 65K1O, 93B99, 47N70, 35Q60, 35105 Library of Congress Cataloging-in-Publication Data Angell, Thomas S Optimization methods in electromagnetic radiation / Thomas S Angell, Andreas Kirsch p cm - (Springer monographs in mathematics) Includes bibliographical references and index Maxwell equations-Numerical solutions Mathematical optimization Antennas (Electronics)-Design and construction I Kirsch, Andreas, 1953- II Title III Series QC670.A54 2003 2003065726 530 14'I-dc22 ISBN 978-1-4419-1914-4 ISBN 978-0-387-21827-4 (eBook) DOl 10.1007/978-0-387-21827-4 © 2004 Springer-Verlag New York, Inc Softcover reprint of the hardcover 1st edition 2004 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights 87 54 SPIN 10951989 Typesetting: Pages created by the authors in LaTeX2e using Springer's SVMono.c1s macro www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH Contents Preface IX Arrays of Point and Line Sources, and Optimization 1.1 The Problem of Antenna Optimization 1.2 Arrays of Point Sources 1.2.1 The Linear Array 1.2.2 Circular Arrays 1.3 Maximization of Directivity and Super-gain 1.3.1 Directivity and Other Measures of Performance 1.3.2 Maximization of Directivity 1.4 Dolph-Tschebysheff Arrays 1.4.1 Tschebysheff Polynomials 1.4.2 The Dolph Problem 1.5 Line Sources 1.5.1 The Linear Line Source 1.5.2 The Circular Line Source 1.5.3 Numerical Quadrature 1.6 Conclusion 10 15 15 19 21 22 24 26 30 36 43 47 Discussion of Maxwell's Equations 2.1 Introduction 2.2 Geometry of the Radiating Structure 2.3 Maxwell's Equations in Integral Form 2.4 The Constitutive Relations 2.5 Maxwell's Equations in Differential Form 2.6 Energy Flow and the Poynting Vector 2.7 Time Harmonic Fields 2.8 Vector Potentials 2.9 Radiation Condition, Far Field Pattern 2.10 Radiating Dipoles and Line Sources 2.11 Boundary Conditions on Interfaces 49 49 49 50 51 52 55 56 58 60 63 68 VI Contents 2.12 Hertz Potentials and Classes of Solutions 70 2.13 Radiation Problems in Two Dimensions 73 Optimization Theory for Antennas 77 3.1 Introductory Remarks 77 3.2 The General Optimization Problem 80 3.2.1 Existence and Uniqueness 81 3.2.2 The Modeling of Constraints 84 3.2.3 Extreme Points and Optimal Solutions 88 3.2.4 The Lagrange Multiplier Rule 93 3.2.5 Methods of Finite Dimensional Approximation 96 3.3 Far Field Patterns and Far Field Operators 101 3.4 Measures of Antenna Performance 103 •••••••••••••• The 4.1 4.2 4.3 4.4 4.5 Boundary Value Problems for the Two-Dimensional Helmholtz Equation 145 5.1 Introduction and Formulation of the Problems 145 5.2 Rellich's Lemma and Uniqueness 148 5.3 Existence by the Boundary Integral Equation Method 151 5.4 L -Boundary Data 157 5.5 Numerical Methods 163 5.5.1 Nystrom's Method for Periodic Weakly Singular Kernels 164 5.5.2 Complete Families of Solutions 168 5.5.3 Finite Element Methods for Absorbing Boundary Conditions 174 5.5.4 Hybrid Methods 181 Boundary Value Problems for Maxwell's Equations 185 6.1 Introduction and Formulation of the Problem 185 6.2 Uniqueness and Existence 188 6.3 L -Boundary Data 193 Synthesis Problem 113 Introductory Remarks 113 Remarks on Ill-Posed Problems 115 Regularization by Constraints 121 The Tikhonov Regularization 127 The Synthesis Problem for the Finite Linear Line Source 133 4.5.1 Basic Equations 134 4.5.2 The Nystrom Method 135 4.5.3 Numerical Solution of the Normal Equations 137 4.5.4 Applications of the Regularization Techniques 138 Contents VII Some Particular Optimization Problems 195 7.1 General Assumptions 195 7.2 Maximization of Power 197 7.2.1 Input Power Constraints 198 7.2.2 Pointwise Constraints on Inputs 202 7.2.3 Numerical Simulations 204 7.3 The Null-Placement Problem 211 7.3.1 Maximization of Power with Prescribed Nulls 213 7.3.2 A Particular Example - The Line Source 216 7.3.3 Pointwise Constraints 219 7.3.4 Minimization of Pattern Perturbation 221 7.4 The Optimization of Signal-to-Noise Ratio and Directivity 226 7.4.1 The Existence of Optimal Solutions 227 7.4.2 Necessary Conditions 228 7.4.3 The Finite Dimensional Problems 231 Conflicting Objectives: The Vector Optimization Problem 239 8.1 Introduction 239 8.2 General Multi-criteria Optimization Problems 240 8.2.1 Minimal Elements and Pareto Points 241 8.2.2 The Lagrange Multiplier Rule 247 8.2.3 Scalarization 249 8.3 The Multi-criteria Dolph Problem for Arrays 250 8.3.1 The Weak Dolph Problem 251 8.3.2 Two Multi-criteria Versions 253 8.4 Null Placement Problems and Super-gain 262 8.4.1 Minimal Pattern Deviation 264 8.4.2 Power and Super-gain 270 8.5 The Signal-to-noise Ratio Problem 278 8.5.1 Formulation of the Problem and Existence of Pareto Points 278 8.5.2 The Lagrange Multiplier Rule 280 8.5.3 An Example 282 A Appendix 285 A.1 Introduction 285 A.2 Basic Notions and Examples 286 A.3 The Lebesgue Integral and Function Spaces 292 A.3.1 The Lebesgue Integral 292 A.3.2 Sobolev Spaces 295 A.4 Orthonormal Systems 298 A.5 Linear Bounded and Compact Operators 300 A.6 The Hahn-Banach Theorem 307 A.7 The Frechet Derivative 310 A.8 Weak Convergence 312 VIn Contents A.9 Partial Orderings 315 References 319 Index 327 Preface The subject of antenna design, primarily a discipline within electrical engineering, is devoted to the manipulation of structural elements of and/or the electrical currents present on a physical object capable of supporting such a current Almost as soon as one begins to look at the subject, it becomes clear that there are interesting mathematical problems which need to be addressed, in the first instance, simply for the accurate modelling of the electromagnetic fields produced by an antenna The description of the electromagnetic fields depends on the physical structure and the background environment in which the device is to operate It is the coincidence of a class of practical engineering applications and the application of some interesting mathematical optimization techniques that is the motivation for the present book For this reason, we have thought it worthwhile to collect some of the problems that have inspired our research in applied mathematics, and to present them in such a way that they may appeal to two different audiences: mathematicians who are experts in the theory of mathematical optimization and who are interested in a less familiar and important area of application, and engineers who, confronted with problems of increasing sophistication, are interested in seeing a systematic mathematical approach to problems of interest to them We hope that we have found the right balance to be of interest to both audiences It is a difficult task Our ability to produce these devices at all, most designed for a particular purpose, leads quite soon to a desire to optimize the design in various ways The mathematical problems associated with attempts to optimize performance can become quite sophisticated even for simple physical structures For example, the goal of choosing antenna feedings, or surface currents, which produce an antenna pattern that matches a desired pattern (the so-called synthesis problem) leads to mathematical problems which are ill-posed in the sense of Hadamard The fact that this important problem is not well-posed causes very concrete difficulties for the design engineer Moreover, most practitioners know quite well that in any given design problem one is confronted with not only a single measure of antenna perfor- X Preface mance, but with several, often conflicting, measures in terms of which the designer would like to optimize performance From the mathematical point of view, such problems lead to the question of multi-criteria optimization whose techniques are not as well known as those associated with the optimization of a single cost functional Sooner or later, the question of the efficacy of mathematical analysis, in particular of the optimization problems that we treat in this book, must be addressed It is our point of view that the results of this analysis is normative; that the analysis leads to a description of the theoretically optimal behavior against which the radiative properties of a particular realized design may be measured and in terms of which decisions can be made as to whether that realization is adequate or not From the mathematical side, the theory of mathematical optimization, a field whose antecedents pre-date the differential and integral calculus itself, has historically been inspired by practical applications beginning with the apocryphal isoperimetric problem of Dido, continuing with Newton's problem of finding the surface of revolution of minimal drag, and in our days with problems of mathematical programming and of optimal control And, while the theory of optimization in finite dimensional settings is part of the usual set of mathematical tools available to every engineer, that part of the theory set in infinite dimensional vector spaces, most particularly, those optimization problems whose state equations are partial differential equations, is perhaps not so familiar For each of these audiences it may be helpful to cite two recent books in order to place the present one amongst them It is our view that our monograph fits somewhere between that of Balanis [16] and the recent book of Cessenat [23], our text being more mathematically rigorous than the former and less mathematically intensive than the latter On the other hand, while our particular collection of examples is not as wide-ranging as in [16], it is significantly more extensive than in [23] We also mention the book of Stutzman and Thiele [132] which specifically treats antenna design problems exclusively, but not in the same systematic way as we here Moreover, to our knowledge the material in our final chapter does not appear outside of the research literature The recent publications of the IEEE, [35] and [84], while not devoted to the problems of antenna design, are written at a level similar to that found in our book While this list of previously published books does not pretend to be complete, we should finally mention the classic work of D.S Jones [59] Part of that text discusses antenna problems, including the synthesis problem The author discusses the approach to the description of radiated fields for wire antennas, and dielectric cylinders, and the integral equation approach to more arbitrarily shaped structures, with an emphasis on methods for the computation of the fields But while Jones does formulate some of the optimization problems we consider, his treatment is somewhat brief 316 A Appendix To be more precise, we start with the definition of a general relation on an arbitrary set Definition A.59 Let S be any set and P a relation defined on S i.e., P is a subset of the set of all ordered pairs of elements (Sl' S2) E S x S Then we say that Sl precedes S2 if the ordered pair (Sl' S2) E P We will use the notation Sl -< S2 if (Sl' S2) E P Given such a relation (and we have introduced the word "precedes" advisedly) it may have certain properties In particular, it may have the properties that make it suitable for us to use as a (partial) ordering Definition A.60 Given a set S and a relation -< defined as above, we say that -< defines a partial ordering of the set S provided: (reflexivity), (i) for all s E S, s -< s (anti(ii) if 81, S2 E S and both Sl -< S2 and S2 -< Sl then Sl = S2 symmetry), (transitivity) (iii) if Sl, S2, S3 E S and if Sl -< S2 and S2 -< S3, then Sl -< S3 Otherwise said, a relation -< is a partial ordering on S if it is a reflexive, anti-symmetric, and transitive relation defined on S We should point out that the usual ordering on lR is a partial ordering of that set It is in fact more than that since any two elements are related, one being less than the other In this case we have a total- or well-ordering The difference is that, with a partial ordering, not every two elements are necessarily comparable The notion of a partial ordering is independent of the nature of the set S However, in our applications we work in a vector space, Z, and we want to find a systematic way to impose a partial ordering on it We first introduce the notion of a cone This definition is dependent only on the algebraic structure of the vector space Definition A.61 Let Z be a real linear space, and let A c Z be non-empty Then A is called a cone with vertex E Z provided that for every z E A, and \ > we have \ z E A We will find that it is useful to use the standard notation -A := {-z : z E A} It is easily seen that the set {O} C Z satisfies the definition of a cone On the other hand, the entire vector space Z is also a cone and that instance is trivial as well We are obviously interested in less trivial cases, for example the set A C lR defined by (A.24) A := R:::o, which is clearly a non-trivial cone in R Indeed, it generates the usual ordering in lR as will become clear below More important for our work is the generalization of (A.24) given by A.9 Partial Orderings A := lR~o := {x E lRn : Xj ~ 0, j = 1,2, ,n} 317 (A.25) which we will refer to as the usual order cone in lRn The cone A is just the first quadrant in the case that n = We isolate two properties that a cone may have since each one, separately, relates to one of the defining properties of a partial order Definition A.62 A cone A c Z with vertex is called non-trivial provided A -=I- {O} and A -=I- Z The cone is called line-free provided E A and An (-A) = {O} It is easy to see that the cone described above in (A.25), namely A := {x E lRn : Xj ::; 0, j = 1,2, ,n} is non-trivial and is line-free Indeed, one need only notice that -A is just {x E lRn : Xj ::; 0, j = 1,2, ,n} Example A.63 The set A := lR>o is also a cone, is still convex, but does not contain the origin In lR the set is an example of a cone that is not convex Moreover, it fails to be line-free since it contains a line, namely the line Xl = o The point of introducing these definitions is to show that a cone, with these properties, can be used to define a partial ordering in Z Indeed, given a cone A with vertex 0, we may define a binary relation -< by x -< y provided y - x EA (A.26) With this definition -< we can easily check that this binary relation is a partial ordering of the vector space Z provided A is convex, contains the origin, and is line free (a) If E A, then -< is reflexive This follows from the observation that for any x E Z, x - x = E A which implies that x -< x (b) If A is convex then -< is transitive, for if x, y, z E Z, and if x -< y and y -< z then y - x E A and z - YEA Since A is convex, 2(y-x) + 2(z-y) E A, and so ~(z - x) E A from which it follows that z - x E A Hence x -< z (c) If A is line-free, then -< is antisymmetric Indeed, if x -< y and y -< x then y - x E An (-A) = {O} so that x = y To summarize, these three observations show that the following theorem is true 318 A Appendix Theorem A.64 If Z is a linear space and A c Z is a line-free, convex cone with E A, then the binary relation -< defined by x -< Y if and only if y - x E A , defines a partial order on the vector space Z There is also the partial converse of this theorem If -< is a partial order on X which respects the operations Le x -< y =? x + z -< y + z and AX -< AY for all x, y, z E X and A > then A := {x EX: -< x} is a line-free, convex cone, and contains O This is easily proven by arguments similar to those above This leads to the following standard terminology: Definition A.65 A pair {Z, -