Conformal Mapping

Methods and Applications

Roland Schinzinger, Patricio A. A. Laura

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486150741
• Topic: Mathematics
• Subtopic: Mathematics Reference
• Index: Mathematics

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. Other topics include vibrating membranes and acoustics, transverse vibrations and buckling of plates, stresses and strains in an elastic medium, steady state heat conduction in doubly connected regions, transient heat transfer in isotropic and anisotropic media, and fluid flow. Revision of 1991 ed. 247 figures. 38 tables. Appendices.

Chapter 1

INTRODUCTION AND OVERVIEW

This book presents examples of successful applications of conformal transformations by the authors and others in various fields of engineering and applied science. Our intention is to interest the community of engineers, physicists, and applied mathematicians in this field which we find so useful and fascinating.

We assume that many readers would like a brief review--or perhaps even a first introductory tour--to the classical methods of conformal mapping. Such basic material will also acquaint the reader with the notation adopted by the authors. (An abbreviated list of symbols is included in Appendix A-1.)

Accordingly, we begin the book and most of the chapters on applications with field problems which satisfy Laplace’s equation in two dimensions for a uniform medium. Thereafter we examine more complicated equations and media. Initially, then, we would consider a problem “solved” when we have identified equipotential lines and lines of force or flow in sufficient detail to interpret important field phenomena locally or globally.

The results will be sets of orthogonal lines u=constant and v=constant in, say, an x – y plane where u(x,y) and v(x,y) satisfy Laplace’s equation. The equipotential and flux lines can be interpreted as coordinates u and v of another plane to which the original field has been transformed in such a manner that the field lines eventually form a rectangular or polar grid depending on the choice of coordinate system.

An interpretation in terms of another coordinate system is easy once a solution has been found. But the trick is to find the solution in the first place. The task is facilitated by designating both the original x – y plane (in which the physical problem is presented) and the u – v plane (of the transformed model) as complex planes: z = x + i y and w = u + i v. The transformation or mapping function is then a complex function of a complex variable, w = f(z), as indicated in Fig. 1.1. Such functions, if fairly well behaved (i.e. analytic), will map a region in one complex plane onto another complex plane in a way which preserves the magnitude and direction of angles between intersecting arcs. Thus Laplace’s equation for potential fields is satisfied in the transformed plane as well. The partial differential equations describing wave propagation, diffusion, and vibration, along with boundary conditions, can also be transformed.

Figure 1.1 The physical plane z and the model plane w.

The z-plane is also known as the “original” or “problem” plane and the w-plane as the “computational” or “standard” plane (the latter when the transformed image is the unit disk or a square).

Let us return for a moment to Figure 1.1. There we have labeled the transformations as w = f(z) and z = f^-1(w), implying “forward” and “inverse” directions. We will also use the notation w=w(z), z=z(w), or z → w and w → z. Whether we employ a mapping in one direction or the other will depend on the task at hand. If an existing physical configuration is to be analyzed, its transformation into the model plane will be carried out by w = w(z). On the other hand, an ideal model which satisfies certain performance criteria may be first fashioned in the w-plane. Thereafter it can be brought to realization in the z-plane by a transformation z = z(w). We may compare this inverse transformation w → z as an act of synthesis or engineering design in the idealized sense. Often engineers must resort to repeated analysis imbedded in a trial and error process to carry out their design tasks. In conformal mapping a similar situation frequently occurs, but in an opposite sense. While a forward mapping is usually desired, it may be easier to carry out trial mappings starting from standard model images (such as unit circles and squares) and to make adjustments to fit the given configurations in the problem plane. For this reason the transformations presented by us may appear as w(z), z(w), or (where necessary) in both forms.

1.1 STRUCTURE OF THE BOOK

The book is organized as follows. Chapters 2 and 3 are of an introductory nature. The reader who is familiar with analytic function theory and the classical approaches to conformal mapping can bypass these chapters and move right on to the modern work beginning in Chapter 4. Those acquainted with the latest numerical techniques will want to start with Chapter 5 (physical models) or Chapter 6. Chapter 2 describes the classical methods of conformal mapping applied to two-dimensional fields in uniform media. It also serves as a review of the basic properties of analytic functions--those functions of a complex variable which can transform orthogonal grids in one plane to orthogonal grids in another plane.

Figure 1.2 The bilinear function

It is widely used for mapping adjacent or included circles into pairs of concentric circles. Application example: heat flow through the thermal insulation separating two pipes, one inside of and parallel to the other.

Representative examples of basic but useful mapping functions as exemplified in Figures 1.2 and 1.3 are given in Chapter 3. We limit ourselves there to fields in singly connected regions as in Figure 1.3 and to easily handled doubly connected regions as in Figure 1.2.

Figure 1.3 A composite mapping function

This function can be derived by applying the Schwarz-Christoffel transformation. Shown is the fringing field in the endzone of an electric capacitor.

In Chapter 4 we present the first departure from the idealized cases under review in Chapter 3. The boundaries, in particular, will be difficult to describe mathematically, or they will be composed of portions which are readily transformed individually but not jointly. See Figure 1.4 for such a deceptively simple looking example. In these cases numerical approximation methods will be required. Accordingly that chapter will contain a review and discussion of recent developments in the numerical generation of mapping functions.

Chapter 5 presents an overview of the functional relationships which govern potential fields and related physical phenomena in the various areas of application to be covered in the following chapters. In many cases there are analogies among fields of different types. It is then not necessary to transform boundary configurations of a given sort again and again. The reader will be able to use the results of many transformations in, say, heat flow problems to solve a problem with like boundaries in electrostatics.

Figure 1.4 Numerical result for a doubly connected region

The square and circle are easily transformed separately but not jointly. A suitable transformation is found by numerical means. Example: heat flow in a graphite brick of a graphite-moderated nuclear reactor.

Figure 1.5 Vector components in more than two dimensions.

The curved waveguide supports an electromagnetic wave traveling in the direction of Poynting vector P. The electric field vector

_zhas x and y components, the magnetic field vector

a ζ component. Conformal mapping is possible if the permeability in the w-plane is modified by the metric coefficient m as indicated.

In Chapter 6 we examine methods to overcome some of the problems presented by field components in a third dimension and by nonuniformity of the medium. An example of this is shown in Figure 1.5 where the curved boundaries of the waveguide cause the electric field to have components in two directions (in the plane of the paper). Combined with the magnetic field vector in the third direction (normal to the plane of the paper), this leads to a departure from propagation in a plane wave mode. Conformal mapping may nevertheless be carried out if the medium is modified by paying heed to the metric coefficients which usually disappear through cancellation in the two dimensional case.

Another example describes the reverse of the above process: Starting out with an anisotropic medium, we render it isotropic in the model plane by means of the metric coefficient. The result is a modification of the boundary. See Figure 1.6.

The main body of the book consists of the chapters on applications. These are arranged as follows:

• 7 Static fields in electricity and magnetism
• 8 Transmission lines and wave guides
• 9 Vibrating membranes and acoustics
• 10 Transverse vibrations and buckling of plates
• 11 Stresses and strains in an elastic medium
• 12 Steady state heat conduction
• 13 Transient heat conduction
• 14 Fluid flow

It should be stressed here that many examples depart from the simple functional relationships expressed by Laplace’s equation.

Chapter 15 concludes our presentation with a discussion of the use of conformal mapping in conjunction with other methods of solving boundary value problems.

There are several useful appendices which should be consulted while reading the book: A-1 gives a list of symbols which recur regularly, A-2 is an index of transformations, and A-3 is a selected bibliography and list of references. A-4 is the name/author index and A-5 is the subject index.

Our style of writing may appear to be inconsistent: in one section we skim the surface, in another we appear almost bogged down with minutiae. The reason is our desire to give the reader a feeling now and then of how a particular methodology came about, while at other times we wish to stress the breadth of present or possible applications.

Figure 1.6 A nonuniform medium.

A parallel plate capacitor with an anisotropic dielectric is transformed into an isotropic capacitor of different thickness.

We now turn to such an overview. It is an illustration of the nonclassical uses of conformal mapping by means of examples of the type presented in Chapters 7 through 14. In several chapters the modern approaches are preceded by discussion of some classical problems to highlight the basic physical concepts. The chapters may also include examples of how new technologies benefit from conformal mapping, be it of the classical or nonclassical type.

1.2 MODERN APPLICATIONS OF CONFORMAL MAPPING

Classical applications of conformal mapping to many stationary problems of mathematical physics go back over a century and continue to the present. As was indicated, these applications usually deal with solutions of Laplace’s equation which remains invariant if the original plane is subjected to a conformal transformation. Consequently, complicated configurations can be transformed into more convenient ones without modification of the governing partial differential equation.

Applications of conformal mapping techniques in certain areas, such as the mathematical theory of elasticity, are considerably more complex. In other cases the physics may not be the problem, but the shape of the boundary is. In both respects new ground has been broken. A brief sampling of such nonclassical approaches follows. Specific examples, detailed solutions, and additional references are provided in Chapters 7 through 14.

Electromagnetics

As integrated circuits are pushed to new limits of performance, more exacting information is required on the characteristics of microcircuit components. Promising numerical approaches have been applied by Trefethen [1981, 1984] and others to planar semiconductor segments such as resistors and Hall effect transducers, while the more classical approaches continue to be of importance in the design of microstrip lines and antennas. See Chapter 7.

In waveguide analysis the seminal contributions of Meinke [1949-1,2; 1963], which involve trading boundary shapes (or the effects of a third dimension) for uniformity of the medium, are being followed by investigators interested in propagation along lossy transmission lines on the one hand [Schinzinger &am...