Springer optimization methods in electromagnetic radiation (2004) ling lotb

Springer Monographs in Mathematics Thomas S Angell Andreas Kirsch Optimization Methods in Electromagnetic Radiation With 78 Illustrations 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 D-76128 Karlsruhe Englerstr Germany kirsch@math.uni-karlsruhe.de Mathematics Subject Classification (2000): 78M50, 65K10, 93B99, 47N70, 35Q60, 35J05 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 ISBN 0-387-20450-4 (alk paper) Maxwell equations—Numerical solutions Mathematical optimization Antennas (Electronics)—Design and construction I Kirsch, Andreas, 1953– II Title III Series QC670.A54 2003 530.14′1—dc22 2003065726 ISBN 0-387-20450-4 Printed on acid-free paper  2004 Springer-Verlag New York, Inc 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 Printed in the United States of America SPIN 10951989 Typesetting: Pages created by the authors in LaTeX2e using Springer’s SVMono.cls 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 1 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 Synthesis Problem 113 4.1 Introductory Remarks 113 4.2 Remarks on Ill-Posed Problems 115 4.3 Regularization by Constraints 121 4.4 The Tikhonov Regularization 127 4.5 The Synthesis Problem for the Finite Linear Line Source 133 4.5.1 Basic Equations 134 4.5.2 The Nystrăom Method 135 4.5.3 Numerical Solution of the Normal Equations 137 4.5.4 Applications of the Regularization Techniques 138 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 L2 −Boundary Data 157 5.5 Numerical Methods 163 5.5.1 Nystră oms 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 L2 −Boundary Data 193 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 Fr´echet Derivative 310 A.8 Weak Convergence 312 VIII 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- A.9 Partial Orderings Λ := Rn≥0 := x ∈ Rn : xj ≥ , j = 1, 2, , n 317 (A.25) which we will refer to as the usual order cone in Rn The cone Λ 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 Λ ⊂ Z with vertex is called non-trivial provided Λ = {0} and Λ = Z The cone is called line-free provided ∈ Λ and Λ ∩ (−Λ) = {0} It is easy to see that the cone described above in (A.25), namely Λ := x ∈ Rn : xj ≤ , j = 1, 2, , n is non-trivial and is line-free Indeed, one need only notice that −Λ is just x ∈ Rn : xj ≤ , j = 1, 2, , n Example A.63 The set Λˆ := R>0 is also a cone, is still convex, but does not contain the origin In R2 the set x ∈ R2 : x1 ≥ ∪ x ∈ R2 : x1 ≤ 0, x2 ≥ −x1 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 x1 = 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 Λ with vertex 0, we may define a binary relation ≺ by x≺y provided y − x ∈ Λ (A.26) With this definition ≺ we can easily check that this binary relation is a partial ordering of the vector space Z provided Λ is convex, contains the origin, and is line free (a) If ∈ Λ, then ≺ is reflexive This follows from the observation that for any x ∈ Z, x − x = ∈ Λ which implies that x ≺ x (b) If Λ is convex then ≺ is transitive, for if x, y, z ∈ Z, and if x ≺ y and y ≺ z then y − x ∈ Λ and z − y ∈ Λ Since Λ is convex, 1 (y − x) + (z − y) ∈ Λ , 2 and so 12 (z − x) ∈ Λ from which it follows that z − x ∈ Λ Hence x ≺ z (c) If Λ is line-free, then ≺ is antisymmetric Indeed, if x ≺ y and y ≺ x then y − x ∈ Λ ∩ (−Λ) = {0} 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 Λ ⊂ Z is a line-free, convex cone with ∈ Λ, then the binary relation ≺ defined by x≺y if and only if y − x ∈ Λ, 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 i.e x ≺ y ⇒ x + z ≺ y + z and λx ≺ λy for all x, y, z ∈ X and λ > then Λ := {x ∈ X : ≺ x} is a line-free, convex cone, and contains This is easily proven by arguments similar to those above This leads to the following standard terminology: Definition A.65 A pair {Z, ≺} where Z is a real linear space and ≺ is a partial order defined on Z is called an ordered vector space The cone which induces the partial order is called the order cone We write