Conformal Mapping Methods and Applications

by Schinzinger, Roland; Laura, Patricio A. A.

• ISBN13: 9780486432366
• ISBN10: 048643236X
• eBook ISBN(s): 9780486150741
• Format: Paperback
• Copyright: 2003-11-20
• Publisher: Dover Publications
• More Book Details

Summary

Beginning with a brief survey of some basic mathematical concepts, this graduate-level text proceeds to discussions of a selection of mapping functions, numerical methods and mathematical models, nonplanar fields and nonuniform media, static fields in electricity and magnetism, and transmission lines and waveguides; and much more. Revision of 1991 edition. Includes 247 figures and 38 tables. Appendices.

Author Biography

Roland Schinzinger: Department of Electrical Engineering and Computer Science University of California, Irvine Patricio A. A. Laura: Universidad Nacional del Sur, Argentina and Institute of Applied Mechanics (CONICET)

Table of Contents

Preface	vii
Errata	xxi
Introduction and Overview	1	(32)
Structure of the Book	3	(4)
Modern Applications of Conformal Mapping	7	(7)
Electromagnetics	8	(1)
Vibrating Membranes & Acoustics	8	(1)
Transverse Vibrations & Buckling of Plates	9	(1)
Elasticity	10	(1)
Heat Transfer	11	(1)
Fluid Flow	12	(1)
Other Areas	12	(2)
Growth in Scope of Applications	14	(2)
Basic Mathematical Concepts
Transformation of Coordinates	16	(5)
Transformation by Means of Complex Functions	21	(2)
Analytic Functions	23	(2)
Conformality and Uniqueness	25	(8)
A Selection of Mapping Functions	33	(62)
Elementary Transformations	34	(14)
Linear Transformation	34	(2)
Power Transformation	36	(4)
Inverse or Reciprocal Transformation	40	(3)
Logarithmic and Exponential Transformations	43	(2)
Hyperbolic Transformation	45	(2)
Interpretation of Elementary Transformation	47	(1)
Composite Transformations	48	(12)
Bilinear Transformation	49	(4)
Joukowski Transformation	53	(4)
Trigonometric and Hyperbolic Transformations	57	(1)
Maxwell's Transformation	58	(2)
Schwarz-Christoffel Transformation for Polygons	60	(33)
The Basic Method	60	(8)
Trigons	68	(3)
Maxwell's Transformation Revisited	71	(5)
Joukowski's Transformation Revisited	76	(1)
Mapping the Interior of a Rectangle	77	(1)
The Rectangle	77	(4)
The Square	81	(1)
Weierstrass Formulation for Rectangle and Triangle	82	(1)
Application	83	(1)
The Inverse and Further Notes	84	(1)
Mapping the Regions Exterior to Rectangles and Between Rectangles	84	(4)
Polygons with Rounded Corners	88	(5)
Exploiting Symmetry	93	(2)
Numerical Methods	95	(60)
Methods of Approximation	95	(3)
Series Approximations	98	(7)
Mapping the Interior of the Unit Circle	99	(1)
Method of Simultaneous Equations (Kantorovich)	100	(4)
Mapping Exterior Regions	104	(1)
Successive Approximations (Fornberg)	104	(1)
Variational Methods	105	(15)
Carrying out the Minimization	108	(3)
Orthogonalization	111	(2)
Minimizing the Perimeter, Orthogonal Polynomials	113	(3)
Example of Minimum Perimeter Method	116	(3)
Minimizing the Area, Comparison of Methods	119	(1)
Additional Extremum Principles	119	(1)
Additional Approximation Methods	120	(1)
Integral Equation Methods	120	(15)
Method of Lichtenstein and Gershgorin	122	(6)
Method of Theodorsen and Garrick	128	(5)
Method of Symm	133	(2)
Comments on Integral Equation Methods	135	(1)
Numerical Determination of Schwarz-Christoffel Transformation	135	(10)
The Parameter Problem	137	(1)
Averting Singularities at Infinity	138	(1)
Method of Trefethen	139	(2)
Method of Foster and Anderson	141	(4)
Doubly Connected Regions	145	(10)
General Observations	146	(2)
Symmetry and Circles	148	(2)
Symmetric Region with an Inner Circle	150	(2)
Symmetric Region with an Outer Circle	152	(1)
Other Configurations; Accuracy	153	(2)
Mathematical Models	155	(28)
Potential Fields and the Laplace Equation	155	(3)
The Laplace Equation under Conformal Mapping	158	(3)
Steady Current Flow, Electrostatics, Magnetostatics	161	(4)
Temperature Field	165	(1)
Fluid Flow Field	166	(1)
The Poisson Equation	167	(2)
The Two-Dimensional Wave Equation	169	(1)
The Two-Dimensional Diffusion Equation	170	(1)
Bending, Buckling, and Vibrations of Plates	171	(2)
Boundary Conditions	173	(8)
Sources and Sinks	173	(3)
Lines of Symmetry	176	(3)
Dirichlet and Neumann Conditions	179	(2)
Extracting the Results	181	(2)
Nonplanar Fields and Nonuniform Media	183	(39)
Nonplanar Fields	183	(26)
Description of Difficulties	183	(5)
Potential Fields with Rotational Symmetry	188	(5)
Symmetric Field in a Wedge-Shaped Region	193	(5)
General Coordinate Transformations in Three Dimensions	198	(3)
Cases with Rotational Symmetry Revisited	201	(4)
Transformation in Two Planes	205	(4)
Nonuniform Media	209	(11)
Field at Boundary Separating Different Media	209	(5)
Anisotropic Media	214	(6)
Quasiconformal Mappings	220	(2)
Static Fields in Electricity and Magnetism	222	(44)
Area-Networks and Hall Generators	223	(15)
Two-Terminal Devices	223	(4)
Three-Terminal Devices	227	(5)
Resistivity Measurement with Four Electrodes	232	(3)
Hall Effect Transducers	235	(2)
Nonuniform Resistivities	237	(1)
Electric Fields in Dielectrics	238	(6)
Electrode Shapes for Uniform Field Strength	239	(2)
Deflection Head for an Ink-Jet Printer	241	(3)
Magnetic Fields of Stationary Structures	244	(15)
Transformers	244	(7)
Magnetic Recording Heads	251	(1)
Perpendicular Field	251	(3)
Longitudinal Field	254	(5)
Magnetic Field of Rotating Machines	259	(7)
Field Analysis in Salient Pole Machines	259	(6)
Other Applications	265	(1)
Transmission Lines and Waveguides	266	(54)
The Basic Equations	267	(6)
Maxwell's Equations under Conformal Transformation	269	(1)
The Wave Equations	270	(3)
Transmission Lines	273	(17)
Two Basic Geometries	273	(2)
Wires within Parallel Plate Boundaries	275	(4)
Wire Inside a Square Revisited	279	(4)
Other Shapes	283	(1)
Transmission With Lossy Conductors	284	(4)
Pipe-Enclosed Cable	288	(1)
Single Wire Above Lossy Ground	289	(1)
Strip Lines	290	(19)
Homogeneous (Single Dielectric) Strip Lines	290	(11)
Inhomogeneities and Microstrip Lines	301	(7)
Anisotropic Striplines	308	(1)
Waveguides	309	(9)
Waveguide Characteristics	310	(3)
Conformal Mapping of the Transverse Section	313	(2)
Nonuniform Waveguides	315	(3)
Other Applications in Electromagnetics and Electrooptics	318	(2)
Vibrating Membranes and Acoustics	320	(21)
Membrane Vibrations	320	(19)
General Considerations	320	(2)
Upper Bound on Fundamental Frequency	322	(2)
Analysis of Free Vibrations	324	(2)
Applications	326	(2)
Other Solutions of Helmholtz Equation	328	(4)
Eigenvalues in Some Doubly Connected Regions	332	(1)
Optimization of Eigenvalues	332	(5)
Transverse Vibrations of Composite Membranes	337	(2)
Acoustic Wave Guides	339	(2)
Transverse Vibrations and Buckling of Plates	341	(34)
Approximate Determination of Fundamental Frequency	342	(2)
Examples	344	(4)
Plates of Arbitrary Shape Subjected to In-Plane Stress	348	(2)
Higher Frequencies of Vibration, Regular Polygons	350	(5)
Plates Carrying Concentrated Masses	355	(11)
Simply Supported Plates	356	(4)
Clamped Plates	360	(6)
Stepped Thickness over a Concentric Circular Region	366	(5)
Buckling under In-Plane Compression	371	(2)
Optimization of the Calculated Eigenvalues	373	(2)
Stresses and Strains in an Elastic Medium	375	(14)
Stresses in a Plane	375	(10)
Torsion and Crack Problems	385	(4)
Steady State Heat Conduction in Doubly Connected Regions	389	(37)
Shape Factors for Circular Bars with Polygonal Perforations	389	(4)
Analysis of Certain Composite Configurations	393	(6)
Temperatures in Composite Rod with Heat Generation in a Circular Core	399	(8)
Heat Transfer in Internally Cooled Fuel Elements	407	(15)
A Case Study	408	(14)
A Sampling of Other Problems	422	(4)
Mixed Boundary Value Problems	422	(1)
Buried Cables	423	(1)
Porous Region with Curved Boundary	424	(1)
Heat Flow in Conjunction with Other Fields	425	(1)
Transient Heat Transfer in Isotropic and Anisotropic Media	426	(38)
Isotropic Configurations	426	(8)
Orthotropic Configurations	434	(13)
Numerical Results	438	(5)
Effect of Changing the Orientation of the Axes of Orthotropy	443	(4)
Orthotropic Plates with Complicated Initial Conditions	447	(8)
Long Rods with Complicated Initial Conditions, Adiabatic Boundaries	455	(3)
Combined Fluid Flow and Heat Transfer	458	(2)
Optimized Rayleigh-Ritz Approach	460	(4)
Fluid Flow	464	(27)
Potential Flow	464	(1)
Simple Cases of Two-Dimensional Potential Flow	465	(5)
Flow Around Circular Cylinder	467	(1)
Presence of Circulation	468	(2)
Air Foils	470	(3)
Ship Hulls	473	(1)
Free Streamline Flow: The Hodograph	474	(3)
Unsteady Flow and Waves	477	(2)
Porous Media, Diffusion, Flow Normal to Boundaries	479	(7)
Seepage Underneath a Dam	479	(2)
Discharge Wells	481	(2)
Slotted Well Casing	483	(1)
Measurement of Diffusion Constant	484	(1)
Vortices and Flow Normal to Boundaries	485	(1)
Hydrostatic Bearing	486	(5)
Other Applications
Conformal Mapping and Other Methods	491	(86)
Methods of Solution	492	(1)
Analytical Methods	493	(3)
Green's Function	493	(2)
Conformal Mapping	495	(1)
Numerical Methods	496	(5)
Interior Methods	497	(1)
Finite Differences	497	(1)
Finite Elements	497	(2)
Boundary Methods	499	(1)
Source Simulation	499	(1)
Matching or Collocation	500	(1)
Boundary Elements	500	(1)
Moments	500	(1)
Galerkin	500	(1)
Monte Carlo Method	501	(1)
Analogs and Graphical Methods	501	(4)
Conductive Sheet Analog	502	(2)
Graphical Field Plotting	504	(1)
Comparison and Combination of Methods	505	(6)
Conformal Mapping Coupled With Other Methods	505	(1)
Emphasis/Deemphasis of Regions	505	(1)
Infinite Boundaries	506	(1)
Boundary Simplification	507	(1)
Boundary Fitted Coordinates	507	(1)
Mesh Generation	508	(1)
Anisotropic Media	508	(1)
Inverse Problem	509	(1)
Comparison of Numerical and Analog Methods	509	(2)
Concluding Remarks	511	(2)
Appendices
A-1 List of Symbols	513	(3)
A-2 Index of Transformations	516	(6)
A-3 Selected Bibliography	522	(48)
Additional References	568	(2)
A-4 Name and Author Index	570	(7)
A-5 Subject Index	577