The Finite Element Method in Charged Particle Optics

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.

Klappentext

This multidisciplinary book is intended to serve as a reference for postgraduate students and researchers working in the fields of charged particle optics or other finite-element-related applications. It is also suitable for use as a graduate text. For the non-specialist in charged particle optics, the opening chapters provide an introduction to the kinds of field problems that occur in charged particle beam systems. br/ A new and comprehensive approach to the subject is taken. The finite element method is placed within a wider framework than strictly charged particle optics. Concepts developed in fluid flow and structural analysis, not hitherto used in charged particle optics, are presented. Benchmark test results provide a way of comparing the finite element method to other field-solving methods. The book also reports on some high-order interpolation techniques and mesh generation methods that will be of interest to other finite element researchers. br/ Additional coverage includes: ullifield theory and field solutions for charged particle optics; /li liaspects of the finite difference method related to the finite element method; /li lifinite element theory and procedure, including detailed formulation of local and global matrices; /li lihigher-order elements, which can be an effective way of improving finite element accuracy; /li lithe finite element method in three dimensions; /li liways to formulate scalar and vector problems for magnetic fields; and /li lisignificant reduction of truncation errors using higher-order elements and extrapolation methods./li /ul

Inhalt

1. Field Theory.- 1. Electrostatics.- 2. Magnetostatics.- 2. Field Solutions for Charged Particle Optics.- 1. The Equations of motion.- 2. The Paraxial Equation of Motion.- 3. On-axis Lens Aberrations.- 4. Electrostatic and Magnetic Deflection Fields.- 3. The Finite Difference Method.- 1. Local finite 5pt difference equations.- 2. The Matrix Equation.- 3. Truncation errors.- 4. Asymmetric stars.- 5. Material Interfaces.- 6. The nine pointed star in rectilinear coordinates.- 7. Axisymmetric cylindrical coordinates.- 4. Finite Element Concepts.- 1. Finite Elements in one dimension.- 2. The Variational method in two dimensions.- 3. First-order shape functions.- 4. The Galerkin Method.- 5. Nodal equations and Matrix Assembly.- 6. Axisymmetric Cylindrical Coordinates.- 7. Edge elements.- 5. High-Order Elements.- 1. Triangle elements.- 2. Quadrilateral elements.- 3. The Serendipity family of elements.- 6. Elements in Three Dimensions.- 1. Element shape functions.- 2. Generating tetrahedral elements to fit curved boundary surfaces.- 7. FEM formulation in Magnetostatics.- 1. Magnetic vector potential.- 2. The magnetic scalar potential in three dimensions.- 3. Saturation Effects.- 8. Electric Lenses.- 1. Accuracy issues.- 2. Direct ray tracing using off-axis mesh node potentials.- 9. Magnetic Lenses.- 1. Accuracy issues.- 2. Magnetic axial field continuity tests.- 3. Magnetic field computations in three dimensions.- 10. Deflection Fields.- 1. Finite element formulation.- 2. Accuracy tests.- 11. Mesh Related Issues.- 1. Structured vs unstructured.- 2. The Boundary-fitted coordinate method.- 3. Mesh refinement for electron gun simulation.- 4. High-order interpolation.- 5. Flux line refinement for three dimensional electrostatic problems.- 6. Accuracy tests.- Appendix 1: Element Integration formulas.- 1. Gaussian Quadrature.- 2. Triangle elements.- Appendix 2: Second-order 9 node rectangle element pictorial stars.- Appendix 3: Green's Integration formulas.- Appendix 4: Near-axis analytical solution for the solenoid test example.- Appendix 5: Deflection fields for a conical saddle yoke in free space.