Conformal Mapping: Methods and Applications

ROLAND SCHINZINGER

Electrical Engineering Department, University of California, Irvine, CA 92717, U.S.A.

and

PATRICK) A.A. LAURA Universidad Nacional del Sur, 8000 Bahia Bianca, Argentina and Institute of Applied Mechanics (CONICET)

ELSEVIER Amsterdam - Oxford - New York - Tokyo 1991

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CONTENTS

Preface i

1 Introduction and Overview 1

1.1 Structure of the Book 3

1.2 Modern Applications of Conformal Mapping 7 Electromagnetics 8, Vibrating Membranes & Acoustics 8, Transverse Vibrations & Bückling of Plates 9, Elasticity 10, Heat Transfer 11, Fluid Flow 12, Other Areas 12

1.3 Growth in Scope of Applications 14

2 Basic Mathematical Concepts

2.1 Transformation of Coordinates 16

2.2 Transformation by Means of Complex Functions 21

2.3 Analytic Functions 23

2.4 Conformality and Uniqueness 25

3 A Selection of Mapping Functions 33

3.1 Elementary Transformations 34 3.1.1 Linear Transformation 34 3.1.2 Power Transformation 3 6 3.1.3 Inverse or Reciprocal Transformation 40 3.1.4 Logarithmic and Exponential Transformations 43 3.1.5 Hyperbolic Transformation 45 3.1.6 Interpretation of Elementary Transformation 47

3.2 Composite Transformations 48 3.2.1 Bilinear Transformation 49 3.2.2 Joukowski Transformation 53 3.2.3 Trigonometrie and Hyperbolic Transformations 57 3.2.4 Maxwell's Transformation 58

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3.3 Schwarz-Christoffel Transformation for Polygons 60 3.3.1 The Basic Method 60 3.3.2 Trigons 68 3.3.3 Maxwell's Transformation Revisited 71 3.3.4 Joukowski's Transformation Revisited 76 3.3.5 Mapping the Interior of a Rectangle 77

The Rectangle 77, The Square 81, Weierstrass Formulation for Rectangle and Triangle 82, Application 83, The Inverse and Further Notes 84

3.3.6 Mapping the Regions Exterior to Rectangles and 84 Between Rectangles

3.3.7 Polygons with Rounded Corners 88 3.4 Exploiting Symmetry 93

4 Numerical Methods 95

4.1 Methods of Approximation 95

4.2 Series Approximations 98 4.2.1 Mapping the Interior of the Unit Circle 99 4.2.2 Method of Simultaneous Equations (Kantorovich) 100 4.2.3 Mapping Exterior Regions 104 4.2.4 Successive Approximations (Fornberg) 104

4.3 Variational Methods 105 4.3.1 Carrying out the Minimization 108 4.3.2 Orthogonalization 111 4.3.3 Minimizing the Perimeter, Orthogonal Polynomials 113 4.3.4 Example of Minimum Perimeter Method 116 4.3.5 Minimizing the Area, Comparison of Methods 119 4.3.6 Additional Extremum Principles 119 4.3.7 Additional Approximation Methods 120

4.4 Integral Equation Methods 120 4.4.1 Method of Lichtenstein and Gershgorin 122 4.4.2 Method of Theodorsen and Garrick 128 4.4.3 Method ofSymm 133 4.4.4 Comments on Integral Equation Methods 135

4.5 Numerical Determination of Schwarz-Christoffel 135 Transformation

4.5.1 The Parameter Problem 137 4.5.2 Averting Singularities at Infinity 138 4.5.3 Method ofTrefethen 139

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4.5.4 MethodofFoster and Anderson 141

4.6 Doubly Connected Regions 145 4.6.1 General Observations 146 4.6.2 Symmetry and Circles 148 4.6.3 Symmetrie Region with an Inner Circle 150 4.6.4 Symmetrie Region with an Outer Circle 152 4.6.5 Other Configurations; Accuracy 153

Mathematical Models 155

5.1 Potential Fields and the Laplace Equation 155 5.2 The Laplace Equation under Conformal Mapping 158 5.3 Steady Current Flow, Electrostatics, Magnetostatics 161 5.4 Temperature Field 165 5.5 Fluid Flow Field 166 5.6 The Pois son Equation 167 5.7 The Two-Dimensional Wave Equation 169 5.8 The Two-Dimensional Diffusion Equation 170 5.9 Bending, Bückling, and Vibrations of Plates 171 5.10 Boundary Conditions 173

5.10.1 Sources and Sinks 173 5.10.2 Lines of Symmetry 176 5.10.3 Dirichlet and Neumann Conditions 179

5.11 Extracting the Results 181

Nonplanar Fields and Nonuniform Media 183

6.1 Nonplanar Fields 183 6.1.1 Description of Difficulties 183 6.1.2 Potential Fields with Rotational Symmetry 188 6.1.3 Symmetrie Field in a Wedge-Shaped Region 193 6.1.4 General Coordinate Transformations in Three 198

Dimensions 6.1.5 Cases with Rotational Symmetry Revisited 201 6.1.6 Transformation in Two Planes 205

6.2 Nonuniform Media 209

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6.2.1 Field at Boundary Separating Different Media 209 6.2.2 Anisotropie Media 214

6.3 Quasiconformal Mappings 220

7 Static Fields in Electricity and Magnetism 222

7.1 Area-Networks and Hall Generators 223 7.1.1 Two-Terminal Devices 223 7.1.2 Three-Terminal Devices 227 7.1.3 Resistivity Measurement with Four Electrodes 232 7.1.4 Hall Effect Transducers 235 7.1.5 Nonuniform Resistivities 237

7.2 Electric Fields in Dielectrics 238 7.2.1 Electrode Shapes for Uniform Field Strength 239 7.2.2 Deflection Head for an InkJet Printer 241

7.3 Magnetic Fields of Stationary Structures 244 7.3.1 Transformers 244 7.3.2 Magnetic Recording Heads 251

Perpendicular Field 251, Longitudinal Field 254 7.4 Magnetic Field of Rotating Machines 259

7.4.1 Field Analysis in Salient Pole Machines 259 7.4.2 Other Applications 265

8 Transmission Lines and Waveguides 266

8.1 The Basic Equations 267 8.1.1 Max well's Equations under Conformal Transformation 269 8.1.2 The Wave Equations 270

8.2 Transmission Lines 273 8.2.1 Two Basic Geometries ^ 213 8.2.2 Wires within Parallel Plate Boundaries 275 8.2.3 Wire Inside a Square Revisited 279 8.2.4 Other Shapes 283 8.2.5 Transmission With Lossy Conductors 284 8.2.6 Pipe-Enclosed Cable 288 8.2.7 Single Wire Above Lossy Ground 289

8.3 Strip Lines 290 8.3.1 Homogeneous (Single Dielectric) Strip Lines 290

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8.3.2 Inhomogeneities and Microstrip Lines 301 8.3.3 Anisotropie Striplines 308

8.4 Waveguides 309 8.4.1 Waveguide Characteristics 310 8.4.2 Conformal Mapping of the Transverse Section 313 8.4.3 Nonuniform Waveguides 315

8.5 Other Applications in Electromagnetics and 318 Electrooptics

9 Vibrating Membranes and Acoustics 320

9.1 Membrane Vibrations 320 9.1.1 General Considerations 320 9.1.2 Upper Bound on Fundamental Frequency 322 9.1.3 Analysis of Free Vibrations 324 9.1.4 Applications 326 9.1.5 Other Solutions of Heimholte Equation 328 9.1.6 Eigenvalues in Some Doubly Connected Regions 332 9.1.7 Optimization of Eigenvalues 332 9.1.8 Transverse Vibrations of Composite Membranes 337

9.2 Acoustic Wave Guides 339

10 Transverse Vibrations and Bückling of Plates 341

10.1 Approximate Determination of Fundamental 342 Frequency

10.2 Examples 344 10.3 Plates of Arbitrary Shape Subjected to In-Plane 348

Stress 10.4 Higher Frequencies of Vibration, Regulär Polygons 350 10.5 Plates Carrying Concentrated Masses 355

10.5.1 Simply Supported Plates 356 10.5.2 Clamped Plates 360

10.6 Stepped Thickness over a Concentric Circular 366 Region

10.7 Bückling under In-Plane Compression 371 10.8 Optimization of the Calculated Eigenvalues 373

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11 Stresses and Strains in an Elastic Medium 375

11.1 Stresses in a Plane 375 11.2 Torsion and Crack Problems 385

12 Steady State Heat Conduction in Doubly Connected 389 Regions

12.1 Shape Factors for Circular Bars with Polygonal 389 Perforations

12.2 Analysis of Certain Composite Configurations 393 12.3 Temperatures in Composite Rod with Heat 399

Generation in a Circular Core 12.4 Heat Transfer in Internally Cooled Fuel Elements 407

12.4.1 ACase Study 408

12.5 A Sampling of Other Problems 422 12.5.1 Mixed Boundary Value Problems 422 12.5.2 BuriedCables 423 12.5.3 Porous Region with Curved Boundary 424 12.5.4 Heat Flow in Conjunction with Other Fields 425

13 Transient Heat Transfer in Isotropie and Anisotropie 426 Media

13.1 Isotropie Configurations 426 13.2 Orthotropic Configurations 434

13.2.1 Numerical Results 438 13.2.2 Effect of Changing the Orientation of the Axes 443

of Orthotropy 13.3 Orthotropic Plates with Complicated Initial 447

Conditions 13.4 Long Rods with Complicated Initial Conditions, 455

Adiabatic Boundaries 13.5 Combined Fluid Flow and Heat Transfer 458 13.6 Optimized Rayleigh-Ritz Approach 460

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14 Fluid Flow 464

14.1 Potential Flow 464

14.2 Simple Cases of Two-Dimensional Potential Flow 465 14.2.1 Flow Around Circular Cylinder 467 14.2.2 Presence of Circulation 468

14.3 AirFoils 470

14.4 Ship Hulls 473

14.5 Free Streamline Flow: The Hodograph 474

14.6 Unsteady Flow and Waves 477

14.7 Porous Media, Diffusion, Flow Normal to 479 Boundaries

14.7.1 Seepage Underneath a Dam 479 14.7.2 Discharge Wells 481 14.7.3 Slotted Well Casing 483 14.7.4 Measurement of Diffusion Constant 484 14.7.5 Vortices and Flow Normal to Boundaries 485

14.8 Hydrostatic Bearing 486

14.9 Other Applications

15 Conformal Mapping and Other Methods 491

15.1 Methods of Solution 492

15.2 Analytical Methods 493 15.2.1 Green's Function 493 15.2.2 Conformal Mapping 495

15.3 Numerical Methods 496 15.3.1 Interior Methods 497

Finite Differences 497, Finite Elements 497 15.3.2 Boundary Methods 499

Source Simulation 499, Matching or Collocation 500, Boundary Elements 500, Moments 500, Galerkin 500

15.3.3 Monte Carlo Method 501

15.4 Analogs and Graphical Methods 501 15.4.1 Conductive Sheet Analog 502 15.4.2 Graphical Field Plotting 504

15.5 Comparison and Combination of Methods 505

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15.5.1 Conformal Mapping Coupled With Other Methods 505 Emphasis/Deemphasis of Regions 505, Infinite Boundaries 506, Boundary Simplification 507, Boundary Fitted Coordinates 507, Mesh Generation 508, Anisotropie Media 508, Inverse Problem 509

15.5.2 Comparison of Numerical and Analog Methods 509

15.6 Concluding Remarks 511

Appendices

A-1 List of Symbols 513

A-2 Index of Transformations 516

A-3 Selected Bibliography 522

A-4 Name and Author Index 568

A-5 Subject Index 575