Iterative Methods for Calculating Static Fields and Wave Scattering by Small Bodies

by Alexander G. Ramm

Overview

Iterative methods for calculating static fields are presented in this book. Static field boundary value problems are reduced to the boundary integral equations and these equations are solved by means of iterative processes. This is done for interior and exterior problems and for var­ ious boundary conditions. Most problems treated are three-dimensional, because for two-dimensional problems the specific and often powerful tool of conformal mapping is available. The iterative methods have some ad­ vantages over grid methods and, to a certain extent, variational methods: (1) they give analytic approximate formulas for the field and for some functionals of the field of practical importance (such as capacitance and polarizability tensor), (2) the formulas for the functionals can be used in a computer program for calculating these functionals for bodies of arbitrary shape, (3) iterative methods are convenient for computers. From a practical point of view the above methods reduce to the cal­ culation of multiple integrals. Of special interest is the case of inte­ grands with weak singularities. Some of the central results of the book are some analytic approximate formulas for scattering matrices for small bodies of arbitrary shape. These formulas answer many practical questions such as how does the scattering depend on the shape of the body or on the boundary conditions, how does one calculate the effective field in a medium consisting of many small particles, and many other questions.

Product Details

ISBN-13:	9780387906829
Publisher:	Springer New York
Publication date:	05/11/1982
Edition description:	Softcover reprint of the original 1st ed. 1982
Pages:	123
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

1. Basic Problems.- §1. Statement of the Electrostatic Problems for Perfect Conductors.- §2. Statement of the Basic Problem for Dielectric Bodies.- §3. Reduction of the Basic Problems to Fredholm’s Integral Equations of the Second Kind.- §4. Reduction of the Static Problems to Fredholm’s Integral Equations of the First Kind.- 2. Iterative Processes6 for Solving Fredholm’s Integral Equations for the Static Problems.- §1. An Iterative Process for Solving the Problem of Equilibrium Charge Distribution and Charge Distribution on a Conductor Placed in an Exterior Static Field.- §2. An Iterative Process for Solving the Problem of Dielectric Bodies in an Exterior Static Field.- §3. A Stable Iterative Process for Finding the Equilibrium Charge Distribution.- §4. An Iterative Process for Calculating the Equilibrium Charge Distribution on the Surface of a Screen.- 3. Calculating Electric Capacitance.- §1. Capacitance of Solid Conductors and Screens.- §2. Variational Principles and Two-Sided Estimates of Capacitance.- §3. Capacitance of Conductors in an Anisotropic and Nonhomogeneous Medium.- §4. Physical Analogues of Capacitance.- §5. Calculating the Potential Coefficients.- 4. Numerical Examples.- §1. Introduction.- §2. Capacitance of a Circular Cylinder.- §3. Capacitance of a Parallelepiped of Arbitrary Shape.- §4. Interaction Between Conductors.- 5. Calculating the Polarizability Tensor.- §1. Calculating the Polarizability Tensor of a Solid body.- §2. The Polarizability Tensor of a Thin Metallic Screen.- §3. The Polarizability Tensors of a Flaky-Homogeneous Body or a System of Bodies.- §4. Variational Principles for Polarizability.- 6. Iterative Methods of Solving some Integral Equations Basic in the Theory of Static Fields: Mathematical Results.- §1. Iterative Methods of Solving the Fredholm Equations of the Second Kind at a Characteristic Value.- §2. Iterative Processes for Solving Some Operator Equations.- §3. Iterative Processes for Solving the Exterior and Interior Boundary Value Problems.- §4. An Iterative Process for Solving the Fredholm Integral Equations of the First Kind with Pointwise Positive Kernel.- 7. Wave Scattering by Small Bodies.- §1. Introduction.- §2. Scalar Wave Scattering: The Single-Body Problem.- §3. Scalar Wave Scattering: The Many-Body Problem.- §4. Electromagnetic Wave Scattering.- §5. Radiation from Small Apertures and the Skin Effect for Thin Wires.- §6. The Inverse Problem of Radiation Theory.- Problems.- Bibliographical Notes.- List of Symbols.