Field Solutions on Computers

Stanley Humphries Jr.

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9781000144413
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Engineering General
• Index: Technology & Engineering

Field Solutions on Computers covers a broad range of practical applications involving electric and magnetic fields. The text emphasizes finite-element techniques to solve real-world problems in research and industry. After introducing numerical methods with a thorough treatment of electrostatics, the book moves in a structured sequence to advanced topics. These include magnetostatics with non-linear materials, permanent magnet devices, RF heating, eddy current analysis, electromagnetic pulses, microwave structures, and wave scattering. The mathematical derivations are supplemented with chapter exercises and comprehensive reviews of the underlying physics. The book also covers essential supporting techniques such as mesh generation, interpolation, sparse matrix inversions, and advanced plotting routines.

1

Introduction

Delightful task! to rear the tender Thought,

To teach the young Idea how to shoot,

To pour the fresh Instruction o’er the Mind,

To breathe the enlivening Spirit, and to fix

The generous Purpose in the glowing breast.

James Thompson

1.1 Overview

Numerical techniques often give straightforward answers to problems that are difficult or impossible to solve with analytic methods. This book is a toolbox of practical numerical methods for scientists and engineers who deal with electric and magnetic fields. It is designed for self-study or as a one-term course for advanced undergraduates. The emphasis is on representing physical problems by numerical simulations and understanding the results. We shall concentrate on finite-element techniques that can handle complex geometries and materials. The methods apply to a broad range of systems, including biological media, plasma processing reactors, accelerators, solid-state devices, and rotating electromagnetic equipment.

The material can help you write your own programs, either for direct field solutions or analyses of results from other programs. It can also help you apply packaged software more effectively. Because hands-on experience is an essential to understand the methods, educational software keyed to the text is available. The two-dimensional programs handle the full spectrum of physical systems that we shall study.

The goal of this book is to act as a bridge between introductory texts on electromagnetism and the growing list of advanced references on numerical field techniques. Standard texts emphasize analytic techniques and usually include exercises that are solvable on hand calculators. Despite the growing importance of computers in electromagnetic design, limits on time and hardware preclude detailed coverage of numerical techniques in introductory courses. This book addresses the problem partly by coordinating the material with integrated software for personal computers. Furthermore, electromagnetic theory is cast in a form aligned to numerical techniques. This approach has an added advantage — it can help enhance your intuitive grasp of field theory. Computer solutions are concrete representations of the abstract concepts of vector calculus. The viewpoint of fields as the interactions of simple elements is particularly valuable to students familiar with circuit theory. Another feature of the book is an emphasis on interdisciplinary applications. We shall touch on related areas of physics and engineering, including gas dynamics, thermal transport, and charged-particle optics. One motivation is to demonstrate the versatility of the numerical methods, which extend to a remarkable spectrum of applications. A second goal is to encourage a broad viewpoint that can be helpful working with electromagnetic devices. For example, in the design of a magnet we can apply similar techniques to calculate field strength, magnetic forces, strain components and cooling requirements.

The literature on numerical techniques for electromagnetism has grown considerably in the past decade. With few exceptions, available books are comprehensive reviews of advanced work aimed at experienced readers. A consequence is that it is often difficult to find practical guidelines for applying the methods. In contrast, this book has the modest goals of summarizing underlying physics and describing methods that are easy to use. To achieve a manageable length, many topics are not covered, including moment methods and high order finite-elements. Nonetheless, the book can serve as an introduction to the literature when your application demands more advanced techniques. The criterion for choosing topics was that solutions could be accomplished on standard personal computers. For example, although we shall study conformal triangular meshes in two dimensions, the discussion of three-dimensional solutions is limited to regular box elements.

1.2 The Load Ahead

The material roughly follows the order of an introductory electromagnetism course, starting with electrostatics and progressing to electromagnetic waves. The topics are arranged in a linear progression — early chapters provide a groundwork for more involved treatments that follow. Numerical techniques are introduced as needed for the increasingly challenging solutions. The book includes sections on good design practices, such as choosing run parameters, applying dimensionless variables, making effective benchmark tests, and interpreting results. There are also sections on auxiliary numerical techniques, such as interpolation and matrix inversion.

We begin with boundary value problems where the goal is to find static solutions in space that are consistent with conditions on surrounding boundaries. We then proceed to initial value problems, in which we start from a given state of a system and follow its evolution in space and time. The first eight chapters emphasize electrostatic solutions. Electrostatics has a strong intuitive appeal, and the derivations create a foundation of theory for applications that follow. Chapter 2 plunges immediately into finite-element solutions on arbitrary triangular meshes. This method gives two-dimensional solutions with high accuracy because a set of triangles can conform to curved and slanted boundaries of electrodes and dielectrics. To begin, the chapter reviews the differential and integral equations of electrostatics with dielectrics. The first task in a numerical solution is to convert the continuous equations into a set of difference equations suitable for digital computers. The approach in Chapter 2 is to apply Gauss’s law over a volume defined by triangular elements surrounding a mesh vertex. In contrast, Chapter 3 derives the finite-element equations from the principle of minimum field energy. Although the approach is less intuitive, the formulation is more easily extended to higher-order field approximations and three-dimensional solutions.

Chapter 4 introduces electrostatic solutions on regular meshes with elements shaped like boxes. We review finite-difference methods. Here, the idea is to generate difference equations by direct conversion of differential equations like the electrostatic Poisson equation. The conversion is straightforward when the solution space is divided into box volumes with rectangular sides parallel to the coordinate axes. This type of mesh limits the solution accuracy but minimizes the amount of information that must be stored in memory. Finite-difference expressions play an important role in the time-dependent solutions of Chapters 12, 13, and 14.

Working from a set of difference equations, the following three chapters concentrate on techniques for solution and analysis. Chapter 5 covers the preliminary task of mesh generation. The term mesh refers to the way that a solution area or volume is divided. Figure 1.1 shows an example of a two-dimensional conformal mesh. The triangles constitute elements, the fundamental spatial unit. Material properties, like the dielectric constant, are assumed uniform over each element volume. Vertices are the intersections of element boundary edges. Solutions of the difference equations yield values for the electrostatic potential at the vertices. Values at intervening points and the field components can be estimated by interpolation. The chapter describes techniques to set up regular rectangular meshes in three dimensions. It also covers the more challenging task of defining a set of irregular triangular elements for two-dimensional solutions. Here, the triangle sides closely follow the boundaries of physical objects. Another task is to set up a mesh indexing system so that we can determine the elements surrounding a vertex, the neighboring vertices and the material identity of elements. The result of mesh generation and difference conversion is a large set of coupled linear equations. The final section of Chapter 5 covers solution of equation sets by relaxation methods. These methods are easy to program and run rapidly, but they may fail with difficult geometries or complex materials. As an alternative, Chapter 6 reviews direct inversion of linear equation sets by matrix methods. Initial sections cover familiar techniques like Gauss-Jordan inversion that apply to moderate equation sets. The chapter concludes with a discussion of block matrix methods to solve the large equation sets associated with field problems.

FIGURE 1.1
Conformal two-dimensional mesh. The solution area is divided into triangular elements to represent different materials. A numerical solution gives field quantities at the vertices.

Chapter 7 covers the important topic of what to do with electrostatic potential values after you find them. The first sections deal with the calculation of gradients to determine electric fields. The least-squares-fit method is useful for arbitrary meshes because the number of available data points may vary. The remaining sections cover techniques for graphical display of data, including plots of boundaries, meshes, and element properties. The chapter also describes finding equipotential contours, making field line plots, and displaying potential as a three-dimensional wireframe elevation.

Chapter 8 addresses field solution techniques for materials with complex properties. The topic is an important preliminary for the magnetic solutions of the following chapter. Nonlinear materials have local properties that depend on the field quantities. Because the fields are not known in advance, we must employ a cyclic process to derive solutions. This usually involves an initial field approximation, calculation of the local material properties, correction of the fields, and so forth. The chapter addresses the convergence of cyclic calculations and interpolation methods to extract nonlinear material parameters from numerical tables. The response of anisotropic materials (such as birefringent crystals or permanent magnets) to fields depends on orientation. The final section reviews finite-element equations for these materials.

We proceed to magnetostatics in Chapter 9. The first section reviews Ampere’s law in differential and integral form. The relationships lead, respectively, to finite-difference and finite-element equations. The chapter concentrates on two-dimensional solutions in planar and cylindrical geometries. In this limit, the vector potential plays a role analogous to the electrostatic potential. We can directly apply the solution methods developed in previous chapters. The chapter also discusses the properties of permanent magnets, materials with both nonlinear and anisotropic properties. The final section covers cyclic methods to find self-consistent operating points in permanent magnet devices.

Chapter 10 reviews several applications for static field solutions. Initial sections deal with volume and surface integrals of electric and magnetic field quantities over the regions of a triangular mesh. The integrals yield useful quantities like field energy and forces on structures. For electrostatic solutions we can find induced surface charge to determine self and mutual capacitance. Integrals over magnetic solutions yield information on inductance, force, and torque. The remaining sections review three applications. Sections 6 and 7 cover charged-particle devices. The first section addresses relativistic equations of motion and time-centered solutions for ordinary differential equations. The next section covers advanced techniques like self-consistent space-charge forces and field-limited flow in electron guns. Section 8 reviews advanced boundary conditions in finite-element solutions with application to the design of Hall effect sensors.

Chapter 11 initiates our study of electric and magnetic fields that change in time. We first address frequency-domain solutions where steady-state fields vary harmonically. The assumption is that the frequency is low enough so that the effects of radiation are small. The approximation allows neglect of inductively generated electric fields in electrical problems and displacement currents in magnetic problems. In this limit, electric and magnetic field solutions are separable. The governing equations are similar to those for static fields and we can adapt methods of previous chapters. The models of Chapter 11 apply to a variety of applications, including eddy currents in alternating current (AC) transformers, radio frequency (RF) electric fields in biological media, and inductive coupling in microcircuits. Chapter 12 introduces the topic of initial-value solutions through a detailed study of the diffusion equation. This relationship is critical to almost all areas of physics and engineering. We use thermal transport as a framework to develop time-dependent finite-element equations. These equations are applied to simulations of pulsed magnetic fields in the presence of conducting materials. The chapter also addresses issues of numerical stability for initial value problems.

The final two chapters remove limits on frequency to model radiation effects with coupled electric and magnetic fields. The numerical techniques incorporate the full set of Maxwell’s equations. Applications include microwave devices, RF shielding, and communications. Electromagnetic solutions encompass a much broader range of possibilities than the static and quasi-static results of previous chapters. Often, solutions are easy to generate but challenging to understand. We shall take a step-by-step approach, starting from an extensive discussion of one-dimensional solutions in Chapter 13. The advantage is that we can address the basic physics issues without worrying about details of mesh generation and complex interacting waves. With confidence in the validity of the methods, we proceed in Chapter 14 to two-and three-dimensional solutions.

The first section ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Preface
• Author
• Table of Contents
• Chapter 1 Introduction
• Chapter 2 Finite-Element Electrostatic Solutions
• Chapter 3 Minimum-Energy Principles in Electrostatics
• Chapter 4 Finite-Difference Solutions and Regular Meshes
• Chapter 5 Techniques for Numerical Field Solutions
• Chapter 6 Matrix Methods for Field Solutions
• Chapter 7 Analyzing Numerical Solutions
• Chapter 8 Nonlinear and Anisotropic Materials
• Chapter 9 Finite-Element Magnetostatic Solutions
• Chapter 10 Static Field Analysis and Applications
• Chapter 11 Low-Frequency Electric and Magnetic Fields
• Chapter 12 Thermal Transport and Magnetic Field Diffusion
• Chapter 13 Electromagnetic Fields in One Dimension
• Chapter 14 Two- and Three-Dimensional Electromagnetic Simulations
• Bibliography
• Index