Field Precision LLC Electric and Magnetic Field Calculations with Finite-element Methods User Manual (EN)

Electric and magnetic field calculations with finite-element methods Stanley Humphries, Ph.D. President, Field Precision LLC Professor Emeritus, University of New Mexico Published by Field Precision LLC E mail: techinfo@fieldp.com Internet: https://www.fieldp.com Copyright by Field Precision LLC All rights reserved. This electronic book may be distributed freely if (and only if) the file is distributed in its entirety. Sections of the file, text excerpts and figures may not be reproduced, distributed or posted for download on Internet sites without permission of the publisher. 2

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Contents 1 Introduction 4 2 First 2D electrostatic solution 6 3 Electrostatic application: building the mesh 14 4 Electrostatic application: calculating and analyzing fields 21 5 Electrostatic application: meshing and accuracy 25 6 Magnetostatic solution: simple coil with boundaries 33 7 Magnetostatic solution: boundary effects and automatic operation 40 8 Magnetostatic solution: the role of steel 45 9 Magnetostatic solution: when steel gets complicated 51 10 Magnetostatic solution: permanent magnets 59 11 3D electrostatic example: STL input 65 12 3D electrostatic example: mesh generation and solution 69 13 3D electrostatic application: getting started 73 14 3D electrostatic application:

extrusions 79 15 3D electrostatic application: mutual capacitance 86 16 3D magnetic fields: defining coil currents 93 17 3D magnetic fields: free-space calculations 100 18 3D magnetic fields: iron and permanent magnets 108 3

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Introduction

1 Introduction Field Precision finite-element programs cover a broad spectrum of physics and engineering applications, including charged particle accelerators and X-ray imaging. The core underlying most of our software packages is the calculation of electric and magnetic fields over three- dimensional volumes. To use our electric and magnetic fields software effectively, researchers should have a background in electromagnetism and should be able to make informed decisions about solution strategies. First-time users of finite-element software may feel intimidated by these requirements. My motivation in writing this book is to share my experience in field calculations.

I hope to build users’ knowledge and experience in steps so they can apply finite- element programs confidently. In the end, readers will be able to solve real-world problems with the following programs: EStat (2D electrostatics) HiPhi (3D electrostatics) PerMag (2D magnetostatics) Magnum (3D magnetostatics) To begin, it’s important to recognize the difference between 2D and 3D programs. All finite- element programs solve fields in three-dimensions, but often systems have geometric symmetries that can be utilized to reduce the amount of work. The term 2D applies to the following cases: Cylindrical systems with variations in r and z but no variation in θ (azimuth).

Planar systems with variations in x and y and a long length in z . Which brings us to the first directive of finite-element calculations: never use a 3D code for a calculation that could be handled by a 2D code. The 3D calculation would increase the complexity and run time with no payback in accuracy. We need to clarify the meaning of static in electrostatics and magnetostatics. The implica- tion is that the fields are constant or vary slowly in time. The criterion of a slow variation is that the systems do not emit electromagnetic radiation. Examples of electrostatic applications are power lines, insulator design, paint coating, ink-jet printing and biological sorting.

Magne- tostatic applications include MRI magnets, particle separation and permanent magnet devices. A following coarse will cover simulations of electromagnetic radiation ( e.g. , microwave devices). Secondly, it’s important to have a clear understanding of the purpose of computer calcula- tions of electric and magnetic fields. Numerical methods should be used when it is not possible to generate accurate results with analytic methods. Numerical solutions are necessary in the following circumstances: 4

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Introduction

Figure 1: Screenshot of the MagView postprocessor for 3D magnetostatics. The system has a complex asymmetric geometry. The solution volume contains many objects with different material properties. Materials have complex properties ( e.g. , saturation of iron in magnetic circuits) In an ideal case, a user makes analytic estimates of field values and then applies numerical methods to improve the accuracy. The initial analysis gives an understanding of the physics involved and the anticipated scales of quantities essential information for effective solution setups. The worst case is when a user treats a program as an omniscient black box.

No matter what software manufacturers may claim, using a field program without understanding fields is at best a gamble. Sometimes you may get lucky, but most of the time considerable effort is wasted generating meaningless results. In summary, I would like to help you become an informed software user. I suggest you start by downloading a free textbook that will help you brush up on electric and magnetic field theory. The book also gives a detailed description of the FEM techniques I will discuss: S. Humphries, Finite-element Methods for Electromagnetics (CRC Press, Boca Raton, 1997) (available for free download at http://www.fieldp.com/femethods.html ). 5

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