The Finite Element Method in Charged Particle Optics

281 Pages·1999·7.023 MB·English

by  Anjam Khursheed (auth.)

THE FINITE
ELEMENT METHOD
IN CHARGED
PARTICLE OPTICS
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
THEFINITE
ELEMENT METHOD
INCHARGED
PARTICLE OPTICS
by
Anjam Khursheed
National University of Signapore, Singapore
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
Khursheed, Anjam.
The finite element method in charged partide optics / by Anjam
Khursheed.
p. cm. --(The Kluwer international series in engineering and
computer science; 519)
Indudes bibliographical references and index.
ISBN 978-1-4613-7369-8 ISBN 978-1-4615-5201-7 (eBook)
DOI 10.1007/978-1-4615-5201-7
1. Electron optics. 2. Finite element method. 3. Partide beams.
4. Electron beams. 1. Title. II. Series: Kluwer international
series in engineering and computer science ; SECS 519.
QC793.5.E62K48 1999
537.5'6--dc21 99-40720
CIP
Copyright © 1999 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint of the hardcover 1 st edition 1999
AII rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, mechanical, photo-copying, recording,
or otherwise, without the prior written permission of the publisher, Springer Science
+Business Media, LLC.
Printed on acid-free pap er.
CONTENTS
PREFACE ix
ACKNOWLEDGEMENTS xiii
1. FIELD THEORY 1
1. Electrostatics 1
1.1 The Conservative Property 1
1.2 Gauss's Law and Poisson's Equation 2
1.3 Orthogonal Coordinate Systems 5
1.4 Boundary Conditions 8
1.5 Electrostatic Energy 9
2. Magnetostatics 11
2.1 Basic Laws 11
2.2 Boundary Conditions 12
2.3 The Vector Potential 12
2.4 The Vector Potential in two dimensions 13
2.5 The Magnetic Scalar Potential 17
2.6 The Reduced Scalar Potential 18
2.7.The Two Scalar Potential approach 20
2.8 The Vector Potential in three dimensions 22
2.9 Magnetic Energy 23
2. FIELD SOLUTIONS FOR CHARGED PARTICLE OPTICS 25
1. The Equations of motion 25
2. The Paraxial Equation of Motion 29
3. On-axis Lens Aberrations 32
4. Electrostatic and Magnetic Deflection Fields 37
4.1 The Deflector Layout 37
4.2 Harmonic expansions of the scalar potential 39
4.3 Source term for Toroidal Coils 40
4.4 Source term for Saddle Coils 42
4.5 Boundary conditions on Multipole Electrodes 43
vi CONTENTS
3. THE FINITE DIFFERENCE METHOD 45
1. Local finite 5pt difference equations 46
2. The Matrix Equation 47
3. Truncation errors 49
4. Asymmetric stars 50
5. Material Interfaces 53
6. The nine pointed star in rectilinear coordinates 55
7. Axisymmetric cylindrical coordinates 56
4. FINITE ELEMENT CONCEPTS 61
1. Finite Elements in one dimension 62
1.1 The weighted residual approach 62
1.2 The variational approach 66
2. The Variational method in two dimensions 70
2.1 Square elements 70
2.2 Rectangular elements 72
2.3 Right-angle triangle elements 73
3. First-order shape functions 75
3.1 Shape functions in one dimension 76
3.2 Shape functions in two dimensions 77
3.3 Example of right-angle triangle 79
3.4 Quadrilateral element shape functions 82
4. The Galerkin Method 84
5. Nodal equations and Matrix Assembly 87
6. Axisymmetric Cylindrical Coordinates 92
6.1 The general nodal equation 92
6.2 First-order elements 92
6.3 On-axis nodes 94
6.4 Near-axis r2 correction 95
7. Edge elements 96
5. HIGH-ORDER ELEMENTS 99
1. Triangle elements 99
2. Quadrilateral elements 105
3. The Serendipity family of elements 106
CONlENTS VB
6. ELEMENTS IN THREE DIMENSIONS 111
1. Element shape functions 112
1.1 The Brick element 112
1.2 The Tetrahedral element 113
1.3 Prism elements 113
2. Generating tetrahedral elements to fit curved
boundary surfaces 114
7. FEM formulation in Magnetostatics 125
1. Magnetic vector potential 125
1.1 Two dimensional field distributions 125
1.2 Three dimensional field distributions 127
2. The magnetic scalar potential in three dimensions 131
2.1 The weighted residual approach 131
2.2 The Reduced scalar potential formulation 132
2.3 The two scalar potential formulation 133
3. Saturation Effects 135
8. ELECTRIC LENSES 141
1. Accuracy issues 141
1.1 The two-tube lens example 141
1.2 First-order and second-order elements 142
1.3 Cubic elements 152
2. Direct ray tracing using off-axis mesh node potentials 154
2.1 Direct tracing vs perturbation methods 154
2.2 Trajectory integration errors 156
2.3 Field interpolation errors 157
2.4 Truncation errors 159
9. MAGNETIC LENSES 161
1. Accuracy issues 162
1.1 The First-order approximation 162
1.2 The Solenoid example 164
1.3 Axial field errors for first and second-order elements 165
1.4 Cubic elements 173
2. Magnetic axial field continuity tests 176
2.1 First-order elements on a trial region block mesh 176
2.2 Mesh refinement 183
2.3 Direct ray tracing from axial field distributions 185
3. Magnetic field computations in three dimensions 187
viii CONTENTS
10. DEFLECTION FIELDS 191
1. Finite elementformulation 191
1.1 Energy functional 191
1.2 Element nodal equations 193
2. Accuracy tests 195
2.1 The analytical solution for magnetic deflectors in free space 195
2.2 The analytical solution for a magnetic core deflection test
Example 197
2.3 FEM results for magnetic deflector test examples 199
2.4 Electrostatic deflection 204
11. MESH RELATED ISSUES 209
1. Structured vs unstructured 209
2. The Boundary-fitted coordinate method 217
3. Mesh refinement for electron gun simulation 224
4. High-order interpolation 230
4.1 CI triangle interpolant 231
4.2 Normalised Hermite-Cubic interpolation 232
4.3 Laplace polynomial interpolation 234
5. Flux line refinement for three dimensional electrostatic
problems 238
6. Accuracy tests 239
6.1 High-order interpolation in two dimensions 239
6.2 The Ampere Circuital mesh test 248
6.3 High-order interpolation in three dimensions 253
Appendix 1: Element Integration formulas 257
1. Gaussian Quadrature 257
2. Triangle elements 258
Appendix 2: Second-order 9 node rectangle element pictorial stars 259
Appendix 3: Green's Integration formulas 265
Appendix 4: Near-axis analytical solution for the solenoid test example 267
Appendix 5: Deflection fields for a conical saddle yoke in free space 269
INDEX 273
PREFACE
In the span of only a few decades, the finite element method has become an
important numerical technique for solving problems in the subject of charged
particle optics. The situation has now developed up to the point where finite element
simulation software is sold commercially and routinely used in industry.
The introduction of the finite element method in charged particle optics came by
way of a PHD thesis written by Eric Munro at the University of Cambridge,
England, in 1971 [1], shortly after the first papers appeared on its use to solve
Electrical Engineering problems in the late sixties. Although many papers on the use
of the finite element method in charged particle optics have been published since
Munro's pioneering work, its development in this area has not as yet appeared in
any textbook. This fact must be understood within a broader context. The first
textbook on the finite element method in Electrical Engineering was published in
1983 [2]. At present, there are only a handful of other books that describe it in
relation to Electrical Engineering topics [3], let alone charged particle optics. This is
but a tiny fraction of the books dedicated to the finite element method in other
subjects such as Civil Engineering. The motivation to write this book comes from
the need to redress this imbalance.
There is also another important reason for writing this book. The development of the
finite element method in charged particle optics is relatively unknown amongst the
wider community of finite element researchers. This is unfortunate, since the finite
element method in charged particle optics has inspired unique developments of its
own, and some of these may well be beneficial to other subjects.
This book aims at providing an introduction of the finite element method in charged
particle optics to the general reader. It goes beyond merely summarising previous
work, and attempts to place the finite element method within a wider framework.
Examples of mesh generators and extrapolation methods for the finite element
method developed in fluid flow and structural analysis, not hitherto used in charged
particle optics will be presented. Detailed test examples are given and it is hoped
that they may serve as bench mark tests and be a way of comparing the finite
element method to other field solving methods. The book reports on some high
order interpolation techniques and mesh generation methods developed within
charged particle optics which may be of interest to other finite element researchers.
x PREFACE
No attempt here is made to describe the growing number of programs being
developed for charged particle optics applications. This is being done elsewhere [4].
The subject of charged particle optics covers a wide area of research that ranges
from the design of particle accelerators, scanning electron microscopes through to
the simulation of television tubes. It is important to state from the outset that the
kind of finite element developments reported here only relate to the design of
instruments like electron microscopes. This naturally means that subjects such as
calculating high frequency (RF) field distributions are not described. This book is
primarily concerned with the deflection, guidance and focussing of charged particle
beams by static electric or magnetic fields.
The material covered in this book is intended to serve as graduate text, suitable for
postgraduate students and researchers working in the subject of charged particle
optics and other finite element related applications.
An introduction to the kinds of field problems that occur in charged particle beam
systems is presented in Chapters 1 and 2. These chapters are primarily intended for
the non-specialist in charged particle optics.
Chapter 3 summarises some aspects of the finite difference method, which are
closely related to the finite element method. Chapter 4 outlines finite element theory
and procedure. Starting with simple examples in one dimension, it shows how
partial differential equations can be cast into integral form and made suitable for
finite element representation. An introduction to the local polynomial function
expansions used for each element, known as shape functions is given. The detailed
formulation of local and global matrices is also presented with simple examples.
Chapter 5 covers the important subject of higher-order elements, which can be an
effective way of improving finite element accuracy, while Chapter 6 describes the
finite element method in three dimensions. Different ways to formulate scalar and
vector problems for magnetic fields are presented in Chapter 7.
Some test field distributions are examined to assess the accuracy of finite element
solutions in Chapters 8 and 9. They show how high-order elements and
extrapolation methods can significantly reduce truncation errors.
Chapter 10 deals with deflection fields, while Chapter 11 describes various
developments in mesh related issues. This latter chapter demonstrates how mesh
generation and refinement is an integral part to finite element accuracy.
ANJAM KHURSHEED
Singapore