Conformal mapping : methods and applications
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Bibliographic Information
Conformal mapping : methods and applications
Elsevier , Distributors for the U.S. and Canada, Elsevier Science Pub. Co., 1991
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Includes bibliographical references (p. 522-567) and indexes
Description and Table of Contents
Description
Conformal mapping is a powerful method of analysis with many successful applications in modern technology. The aim of this book is to enlighten readers on the advantages of conformal mapping by illustrating its wide applicability and describing the new mathematical techniques available. Conformal mapping uses functions of complex variables to transform complicated boundaries into simpler, more readily analyzed configurations. It has usually been assumed to be restricted to planar fields satisfying Laplace's equation, fields in uniform media, and regions which are not awkwardly connected in multiple ways. This book shows how these restrictions can be lifted in many cases of practical significance by analytical and numerical techniques. Modern Applications involve not only new or revised algorithms; equally significant is the use of classical methods in new technologies. It is particularly in the latter category that many problems are found which are amenable to elegant, insightful solutions by conformal mapping without resorting to routine, lengthy numerical procedures.
Table of Contents
1. Introduction and Overview. Structure of the book. Modern applications of conformal mapping. Growth in scope of applications. 2. Basic Mathematical Concepts. Transformation of coordinates. Transformation by means of complex functions. Analytic functions. Conformality and uniqueness. 3. A Selection of Mapping Functions. Elementary transformations. Composite transformations. Schwarz-Christoffel transformation for polygons. Exploiting symmetry. 4. Numerical Methods. Methods of approximation. Series approximations. Variational methods. Integral equation methods. Numerical determination of Schwarz-Christoffel Transformation. Doubly connected regions. 5. Mathematical Models. Potential fields and the Laplace equation. The Laplace equation under conformal mapping. Steady current flow, electrostatics, magnetostatics. Temperature field. Fluid flow field. The poisson equation. The two-dimensional wave equation. The two-dimensional diffusion equation. Bending, buckling, and vibrations of plates. Boundary conditions. Extracting the results. 6. Nonplanar Fields and Nonuniform Media. Nonplanar fields. Nonuniform media. Quasiconformal mappings. 7. Static Fields in Electricity and Magnetism. Area-networks and hall generators. Electric fields in dielectrics. Magnetic fields of stationary structures. Magnetic field of rotating machines. 8. Transmission Lines and Waveguides. The basic equations. Transmission lines. Strip lines. Waveguides. Other applications in electromagnetics and electrooptics. 9. Vibrating Membranes and Acoustics.. Membrane vibrations. Acoustic wave guides. 10. Transverse Vibrations and Buckling of Plates. Approximate determination of fundamental frequency. Examples. Plates of arbitrary shape subjected to in-plane stress. Higher frequencies of vibration, regular polygons. Plates carrying concentrated masses. Stepped thickness over a concentric circular region. Buckling under in-plane compression. Optimization of the calculated eigenvalues. 11. Stresses and Strains in an Elastic Medium. Stresses in a plane. Torsion and crack problems. 12. Steady State Heat Conduction in Doubly Connected Regions. Shape factors for circular bars with polygonal perforations. Analysis of certain composite configurations. Temperatures in composite rod with heat generation in a circular core. Heat transfer in internally cooled fuel elements. A sampling of other problems. 13. Transient Heat Transfer in Isotropic and Anisotropic Media. Isotropic configurations. Orthotropic configurations. Orthotropic plates with complicated initial conditions. Long rods with complicated initial conditiions, adiabatic boundaries. Combined fluid flow and heat transfer. Optimized Rayleigh-Ritz approach. 14. Fluid Flow. Potential flow. Simple cases of two-dimensional potential flow. Air foils. Ship hulls. Free streamline flow: The Hodograph. Unsteady flow and waves.
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