NOTE TO USERS

Page(s) not included in the original manuscript are unavailable from the author or university. The

manuscript was microfilmed as received.

128 AND 129

This reproduction is the best copy available.

UMI

Accuracy Criteria

and Fiaite Element Study of a

Highly Sahvated Magnetic Device with a Large Air-Gap

by

Masoud S harifi

A t hesis submitted in confonni ty with the requirernents

for the.degree of Doctor of Philosophy

Graduate Department of Elecvical and Compter Engineering

U~versity of Toronto

Cs Copyright by Masoud Sharifi 2000

National Library Biblbthèy n a t i o ~ k du Cana a

Acquisitii and Acquisitions et Bibliographie Sewicûs rwwices bibliqraphiques

The author has granted a non- exclusive licence aliowing the National Library of Canada to reproduce, loan, distn'bute or seii copies of this thesis in microforni, paper or electroaic formats.

nie author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or otherwise reproduced without the autbor's permission.

L'auteur a accordé une licence non exclusive pkettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/film, de reproduction sur papier ou sur format électronique.

L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

Canada

To Mehri, Mohammad-Hassan, Mohsen and MeMi:

my roofs

To Mansureh:

rny ground

To Yasamin:

my ski

Accuracy Criteria and F i t e Elemnt Study of a

Highiy Saturated Magnetic Device with a Large Air-Gap

Department of Electrical and Computer Engineering University of Toronto

Abstract

The F i t e Element Method hm ken frequently used to obtain a nüable virtual prototype

of a device. The accuracy of the device mode1 gains considerable importance when

measured data are not available for cornparison. This thesis proposes to use Ampere's

circuital law to obtain a global accuracy criterion, Arnpere's law enor, which rneasures

the accuracy of the FE mode1 and solutions.

The C-shaped magnet of an open-concept Magnetic Resonance lmaging system is

used as the focus for this thesis. Being threeilimensional, highly saturated, and havhg a

large air-gap, the open-concept MRI magnet represents an interesthg modeling

change. The objective of the thesis is thus to use this device as a vehicle to explore

accuracy issues when modeüng saturated. 3D. non-symmetric rnagnetostatic devices.

A stable 3D modeling approach is developed tbat encloses the MRI rnagnet in a

spherical FE volum, the exterior surface of which cm reprisent the true infinite

boundary condition. A FE made1 of the MN magnet is thus developed and studied. It is

concluded thnt the average and standard deviation of Ampere's law errors provide a vaiid

globd accuracy masure for thip class of FE solutions.

A selection study is also necessary to obtain the most accurate FE mode1 of a device.

This thesis descriis (a) a sekction strategy and @) selection cnteria A seiection strategy

incorporates two rnethods that simplify the search for better FE models of the device. . ,

'Both methods have proven to k practical and constructive. The selection criteria include

Ampere's law error, the energy content of the model, the energy content in specific

regiow of tk model. and the average of the magnetic fieM data at specific regions of the

magnet. These selection criteria have proven to be feasible for selectuig the best FE

model of the device.

An alternative design of the MRI magnet is introduced. The performance of the

alternative design is shown to be supenor to the original design of the MRI magnet.

In addition, the thesis introduces two two-dimcasional rnodeling approaches to

simpw the FE study of unsymrnetrical3D devices. One of the 2D modehg approaches

is shown to be applicable for calculating the magnet data.

Acknowledgernents

1 would iike to express my smcere gratitude and appreciation to my supervisor, Professor

J.D. Lavers, for his invaluable guidance, patience, advice and encouragement t hroughou t

the course of my P1i.D. program and preparation of dus thesis.

1 would also like to thank Professor A. Konrad, Professor F.P. Dawson and Dr. H.

K a d e r for their advice and commnts during my fuial oral Ph.D. examination.

I thank my hiends in the power Group, in particular, Dr. ICV. Narnjoshi and Dr. M.

Graovac for theu ûui',ful comments and discussions.

The hancial support provided by the University of Toronto Open Fellowship Program is

gratefully acknow ledged.

Special appreciation goes to my wife, Mansureh, and to my daughter, Yasamin, for their

understanding, patience and continuous support during my education.

Table of Contents

1 . 1 FIN~IEELEMENTMETHOD ................... .......- ................................................ 2

.......... ... ...............................*..........,**.....**...*....... 1.1.1 Accurocy Criteria ..... ... 2

.......................................................... .................... 1.1.2 A Seleetion Approach .. 3

............ .................*.......*...............*... 1.2 MAGN~CRESONANCE~MAGING .............. 4

1.2. I A ir-gap and Magnetic Mate rial of the M N Magner .................................... 6

1.2.2 Coi1 of the M N Magnet ............. .. ................................................. 7

............................................................................. 1.2.3 Size of the M N Magnet ... 7

1.3 THESIS OBEcî"M% ............................................................................................... 8

I . 4 THESIS O- ......... ........ ..... ........*........ *....................*......*.......................... 9

ACCURACY AND SELECTION OF A FINITE ELEMENT MODEL omaaoeooaoo~o~..aaemaooooll

2.1 THE FE MODELOFTHE URI MAGNET ............................................................ 13

........... 2.1.1 TheMMMugnetSpccifiation ........................ ......C.................. 13

2.1.2 The Finite Element Method ............ ........,,, ..................................... 1 4

2.1.3 Boundary Value Problem Definition ............ ..... ...............................W. I 5 .. 2.1.4 Donuiin Definitaon ................................................................................. 1 7

2.2.5 Domarin Discretitation .......................................................................... 1 9

2.2 VERIFKATION OFTHE ~ U L T S .................... Ct ........................cc...................... 23 2.2.1 Magnetic Field ................. ... ............... 2.2.2 Gauss1sL4w .................... ............. ... ..... ,... 26

2.2.3 Amperetsk»v ............? .............. ................................................................... 26

....................................................... 2.2.4 Stored Energy ................... ......... 27

......................................... .... ................. 2.3 S ~ O N APPROACH ....... ...... 28

.............................................................................................. 2.3.1 Initial Model 28

.................................................................................... 2.3.2 Selection Crirericr 31

..................... 2.4 MODIFICATION OFTHE FINIE E m MODU ,.,, .................. 1

............................ ................... . 2.4.1 Boundary Value Problem Definition .... ... 3 2

.............................................................. 2.4.2 Definition of the Problem Domain 34

2.4.3 DomainDiscretiztation ................... ... .................................................. 36

2.4.4 Element Type ..................... .............. ...................................................... 38

2.4.5 Material Characteristic ...... ..... ...... ... . .- . 3 9

2.4.6 CurrentExcitation .............. ................... .............................................. 40

............................... ............................... 2.5 SUMMARY .. 40

............................................................... 3.1 ~ENT~FICATION OF AN I N ~ A L MODEL 47

...................................................................... 3.1.1 Free-Spuce Material Methud 47

................................................ 3.1.2 Reduced Excitation Method: Lwding Steps 49

.................. ........*...*.............................. 3.1.3 TheSturting,orlnitial. Moàel .. 50

............................................. ............. . 3.2 SELECIION OF AN OPT~MAL MODEL .. .., 5 1

............. 3.2.1 AnEquivalentLineurModcl .......... .................................. 5 2

.................................................. 3.2.2 Puth Dependency cf Ampere 's Law Enor 54

3.2.3 Magnetic Field us Selection Criteria ..................................................... 55

3.2.4 MagneticEncrgyasSeleciionCrircria ...................................................... 56 ........................................ ............... 3.2.5 The SeIected Optimum Mode1 ... .... 58

........................... .................. 3.3 THE OPTIMUM MODEL OF THE MRI MAGNET ... 63

...................................... ................... 3.3.1 Magnetic Field Distribution ... ........ 63

................................................................................. 3.3.2 Coif Modcling Effect 66

3.3.3 Coi1 Cunent Lcvel Analysis ........................ .... ... ..... .. .. ............. 3.3.4 Perfomunce Swnmary of the Optimum Mode1 of the MM Magnet 70

..................... ... ...........*.... 3.4 DESIGN ALTERNATIVE: COL AROUND THE POLE , .., 79

................................................ 3.4.1 FE Analysis of the Alternative MM Magnet 79

..........*.. ..................................... 3.4.2 Magnetic Field Groghical Results ... 8 2

....................................... ...................... 3.4.3 Coi1 Current Level Analysis .... 84

..................................... ................... 3 . 5 COMPARISON OF MRI MAGNET DESIGNS ,.. 85

....................................... 3.6 CONCLUSION ..................................... .... ... .. .. ... 86

.................... STUDY OF THE MRI MAGNET WITH 2D FEM eoooooeoooo~~eoeo~ooo~~oooo~oooooeeo~96

..................................... 4.1 I ~ O D U C T I O N 96

...................................... 4.1.1 Axisynmetn'c A pproximtion 4 the M M Magnet -97

................. 4.2 A PARTIAL 2D MODEL OF THE COIL AND THE CORE ...................... ... -98

........................................ ............ 4.2.1 Problem Definition und Results ........ 99

.............................................. 4.2.2 Limiting Effect of the Coi1 Excitation Level 102

4.3 AN ACCEPTABLE PARTIAL 2D MODEL OF THE COL AND THE POLE .................. 105

4.3.1 Problem Domain and Results ............... ................. ................................ 105

.................................................... 4.3.2 TheEffecroftheCoilExcitationLevel 1 0 6

............................................... 4.4 A Nom Comsm 2D M O D ~ G METHOD 109

4.4.2 ProblemDomain ................... ..........,. ............. .. 110

4.4.2 ACompositeZDModcloftheO-MRIMugnet .......................................... I I I

........................................... 4.4.3 A Composite 2D Mode1 of the A-MM Mugnet if6

4.5 CONCLUSIONS .................. ..*.......................... .............................................. 120

APPENDIX B oooo.ooooaaaroooaoo~aoooo.oaooooaa~ooa~o*oooooooeoaoo.o~oooo~~oooaaoaaooooo~oeooooooooooo~eoaommooaoeeoaoo~oaooo*of 3O

CZ006 CHARACTERISTIC CURVES a a a * o e o o o o o . ~ a e o a o o o a a o o a o a o ~ o o o o ~ e o a a a o o o o o o o o o o o o a a o o o a o a e o a 1 3 O

C.1 BOUNDARYVALUEPROBIEMD~ON ............ .............. .,. ................ 1 3 3

Ci . I M ~ l f ' s Equations .......... .. ..................................................... 1 3 4

C.I.2 Magnetic Vector Poruitial ............... ......................................................... 135

................................................................................ C.i.3 Problem Description 136

.......................................................................... C.2 VAR~A~ONALFORMU~A~ON 136

C.3 DOM DEFIN~~ION ............................................................. , ... 1 3 9

.......... C.4 Donuau DISCREI~ZA~ON ....................................... f 39

CS INTERPOLAT~ON FUNCI~ONS S ~ O N ...........................................*............. 140

.............................................. C S 1 Linear Isopurametric Hexahedral Element 1 4 1

............ .............*.**.......*.........*...*...*......*.................... C.5.2 Infinite Efement .. 143

C.6 SYSTEM FORM~LA~ON ................................................................................... 146

C . 6.1 Variational Implementation .... .. .. .. ................................................. 146

.................................................................. C.6.2 Gaussicrn Element Integrution 149

C.6.3 Global Derivative of Interpolation Functionr ....................... .... .......... 149

C.6.4 Element Matrix Calculation ..................................... ... ....................... 1 SI

C . 6.5 toad Vector Culcuiation .................... ... .. ... ............................................. 152

C.6.6 Sy~ofEquutionsAssembly ....................... ........................................... 153

............................................... .................. C.7 SYSTEM SOL- ........................ 154

................................ C . 7.1 Newton-Raphson Method ........................................ 155

C.8 EVALUATIONOFTHEFINALRESULTS ............................................................... 157

C.8.1 MagneticField ................................. ............................................... 1 5 8

C.8.2 Magnetic Flux ......................................................................................... 159

C . 8.3 Magnetomutive Force ...................... ... .. ... .................................... 1 6 0

C.8.4 StoredEncrgy ............................... .... ................................................ 160

C.9 CONCLUSION .. .....*.+~...*..r~~*.**~****...v.*r...... ............................................ 161

Chupter 1

Introduction

TypicaUy, most engineering applications requin an intensive performance evaluation of a

device or a system during the design and development stages. Often, it is not possible to

buüd an actual full-scale prototype of the device or system Thus, one devebps as realistic a ,#

simulation model as is possible with F i t e Ekmtnts, Boundary Elemnts, F i t e DEerences,

Integral Equations. or other such mcthods.

Mathematical rnodels of physical devices cornmoniy are posed in terms of boundary

value problems. The F i t e Ekment Method sirnply employs a numerical approach in

achieving an approximate solution to such problems. Although the FEM has k e n applied to

and has been calibrated for many problems. each application is unique and each requires its

own validation. The output of any FE model is only as good as the model itseif. Caliiration

problem arise when masured data are not avaüable, as will k the case whenever the device

in question is large and the cost of t s manuftxture is high. Therefore, a centrai modehg

issue is how best to validate the FE model under such circwnstances.

The accuracy of the FE approximate solutions to highiy saturated, nonlinear,

mapetostatic field problems will provide a central focus for this thesis. Just as the FEM

employs the laws of physics to mathernatically represent a physical device, the accuracy of

the approximate solutions for that device will be verified in accordance with those laws.

A large air-gap C-core electromagnet serves as a focus for the FE modehg issues that

are discussed in this thesis. This partkular device is also of considerabk practical importance

since it is the key component in a ncently developed open-concept Magnetic Resonance

Imaging (MRI) system The Cson magnet for this application represents a challenging

Chapta 1. introduction 2

problem fiom a modeiing viewpoint. Fust, the magnetic material of the device, in certain

ngions, is âriven weli into saturation. Therefore, there is a considerable leakage rnagnetic

field. Additionally, the device is not entirely symmetrical. It has two s h t r y &nes at Y=û

and Z=O planes of a Cartesian coordinate system; however. such symmetry does not permit a

2D study of the device either in 2D planiu or in 2D axisymmetric fashions. Therefore, The

MM magnet problem must be modeled in th-dimensions (3D). The thesis therefore

considers the development of a Wtual prototype for this device as a means of exploring

accuracy issues when modeling highly saturated rnagnetostatic devices.

Finite Element Method

When applying the F i t e Elernent Method, there are several issues that have not received

adequate attention in the past. First, there is the question of uccuracy, particularly in ternis of

quantities that an derîved fiom the basic solutions. Second, there is the question of

identification and selection, which mans the achievernent of the bea possible models and

solutions for a given probkm. The accurucy and the selection issues are discussed in the

fo 110 wing subsections.

1.1.1 Accuracy Criteria

The goal of rnost FE studies is to obtain the rnost accurate model possible for a given

physical device. Having d e f d a problem, and then having modeled and solved it ushg a

proper FE formulation, one must asscss the quaüty of the nsults thus obtained. This leads to

an evahation of rnodel andlor solution error(s) and an assesment of rnodel accuracy. 8

Fmaüy, dinerent models can ôe compared in order to identQ the k s t possible model for the

given probkm

An assesment of the solution quaiity can bt considered in t e m of either local or global

criteria. Local accuncy criteria have often k c n discussed in the üterature within the context

of adaptive rneshing [l-31. It should k noted that a mode1 that compks with local accuracy

criteria is not neccssariiy the best possibie model m terms of o v e d solution quality.

Global accuracy criteria masure the vaiidity and reliabiüty of a rnodel in terrns of its

solution as a whole. The integral f o m of Maxwell's equations, for example, are essentially

global criteria and can be used to* confïrm and evaluate FE solutiork. It is suhrising that

global accuracy criteria have not ken widely discussed in the literatun. Recently, however,

Bossavit has questioned the approhte nahm of the FE solutions (41. He concludes that the

FE solution of magnetostatic problem often 'Tails to have a general validity." The reason is

that the FE approximate solution satisfks only one of Maxwell's equations - in theu integral

f o m .

In another recent study, Ampere's circuital law is used to evaluate a posteriori error in

the 3D FE solutions of a nctangular coi1 that surrounds a linear magnetic core of

20M00xlûû mm volume size [5 ] . Two FE models of the problem are introduced: the coarse

mesh model (with 8640 tetrahedral elemnts,) and the fine mesh model (with 43560

tetrahedral elemnts.) Five integral paths are chosen in the air region of the problern, t h e of

them linking the curnnt source. It is shown that the fine mesh model yields a lower value of

the posteriori error. It is also claimed that the posteriori enon of the fme mesh mode1 "show

no dependency with path location," - within a 4% discnpancy in errors dong five paths.

In this thesis. it is proposed that Ampen's circuital law provides a convenient and

reliabk quality rneasure when judging FE solutions to nonünear, 3D magnetostatic problem.

It wiii be shown that this measure can indicate when a particular model is produchg accurate

results. However, it should be noted that such a masure does not necessarily show when the

mst accurate FE results are obtained. Thus. a second aspect of the problem is to propose

mans, conditions and restrictions whereby the most (possible) accurate FE mode1 can be

developed.

Powemil software packages impkment the FEM and provide the necessary numerical tools

to obtain requind data such as magnetic field distnions, stored magnetic energy, magnetic

flux, etc. The analyst must then examine the solutions, tirst, to verify the accuracy of the

solutions, and second, to fhd ways to improve the quality of the model and its results. The

Chaptcr 1. Introduction 4

accuracy of the solutions should be detennined from the accuracy criteria. Therefore, the

hem of the selection issue is how to improve the quality of the solutions. how to measun the

irnprovement of the solutions' and how to distinguish the best model'for a given

pro b1em

This thesis proposes:

a) to develop convenient rnethods to obtain an initial mudel for a given problem; Le. a

model that yields a solution of acceptable accuracy such that the selection process can k

initiated,

b) to define criteria to assess the quality of the solutions yielded by the initial as weil as the

subsequent models,

C) to identify simple mthods that improve the efficiency of the selecion process. and

d) to defhe criteria to fmd the most accuraie model for a problem.

1.2 Magnetic Resonance Imaging

Nuclear Magnetic Resonance Imaging (MRI) was discovered in the 1950's and has been

used for irnaging human body parts since 1970 [6-71. The major advantages of MRI are the

non-invasive operation of the device, the high resolution of its images, and its capability to

show both soft and hard tissues. Better and faster cornputhg facilities have substantially

decreased the MR-imaging t h e . Recently, high-speed MRI units have been used to achieve

real time monitoring of human tissue. A new developmnt in MR-imaging has been the

introduction of smail, open-concept MN units that are partîcularly attractive for use with

surgical procedures that avoid major cuts or openings of the hurnan body. The tissue or body

part in question can be imaged in na1 time and the image then used to guide surgical

procedures. In order to achieve these ends, the open concept MRI units represent a major

departure from conventional MRI designs.

In a typical air-con MRI unit 181, the human body or the imaged tissue is located in the

central voiumc of a large abcore magnet. Superconducting cob are used to generate a

uniform, static magnetâ field withui the volume containing the tissue to be Maged. Typical

levels of mch a magnetic field may be up to 3 T. Coüs and devices that are

Chapter 1. Introduction 5

necessary to operate the MRI system surround the imaging volume. Consequently, the inner

volume of the rnagnet, and thus the imaged tissue, are not accessible to the extent that would

be required in a surgical procedure.

A recent project at the University of Toronto bas focused on the development and testing

of an open concept system t hat has ken t e m d Image-Guided Minimally Invasive Therap y

(IGMIT). The main component in this process is an openîoncept C-shaped magnet. The C-

shaped MRI rnagnet that is proposed for this system is shown h Figure 1.1, while hirther

detaik are provided in Appendix A. Open-concept MRI units are economicaiiy very

promising due to the reduced size, cost, and power consumption of the MRI unit. Thus, there

is the possibility that specidy designed units for particular parts of the human body can be

designed.

Figure 1.1 : The half symmtry of the MRI magnet where the coi1 is around the core.

The main magnetic component of the open-concept MRI unit is its C-shaped magnet. For

the purpose of the prototype IGMK unit, the superconducting magnet consisted of a

Chapta 1. Introduction 6

cyündrical core. two cyündricai poles, and two slab yokes connecting the core and poles. The

h g e d tissue was to be placed within the large air-gap opening of the magnet; i.e. the the-

space volume between the poles. Thus, t h e sides of the imaged tissk would be accessible

for medical treatmnts. The magnet is energized with a superconducting coü using 120 kA-t

(12(b<lo3 Ampere-turns) of direct cumnt. With this design. the operational magnetic flux

density in the air-gap of the MRI mgnet is reported at 0.27 T [Appendix A].

Throughout this thesis, the term 'WU magnet" will be used to refer to such a magnet.

The magnet material used for core. yokes. and poles is Cl006 steel. The B-H characteristic

for this material can k found in Appendix B, Figure B. 1.

The MRI magnet used in the IOMIT system provides a focus for the research that is

reported in this thesis. The primary objective of the thesis is to examine the factors that

govem the accuracy of F i t e Element models that are used to represent 3D. highly saturated

mapetostatic devices. The key issues are the accuracy of a given model and the modification

of the model (i.e. the solution grid) such that the most accurate solution possible cari be

identified. Once an optimum mode1 of the core and the surroundhg air has been obtained,

alternative coi1 configurations can be examined.

In the foiiowing subsections, the principal fatures of the C-core MRI magnet wiiï k

1.2.1 Air-gap and Magnetic Material of the MRI Magnet

In most rnagnetic devices. the air-gap constitutes a smaii fiaction of the magnet volume. For

example, power electronic transformers and inductors typicaiiy have a srnall air-gap in order

to lincarise the magnetic circuit. Similarly, electric machines also employ a srnall air-gap to

facihate the rotaiional function of the machine. For devices such as these, the magnetic field

dktriition within the core is of interest during the design stage.

When the air-gap volum of a magnetic device is increased. more cumnt must be

suppüed to the circuit of the device in order to keep the samc level of rnagnetic field in the

air-gap. Comequently, the magnetic material stacts to saturate and stores more magnetic

Chapta 1. Introduction 7

energy. This reduces the hear effcct of the air-gap to a point where the nonlinear rnagnetic

material must be included in any mdeL

The air-gap volume of the M'RI -mgnet accounts for 9% of the rnabet voluk. However,

hinctiondy, it must be large enough to hold the imaged tissue and any accessory equipment.

A very high kvel of magnetic flux density is technicaiiy preferred at the air-gap volume.

This, in tum, requires a high excitation current in the coi]. Consequently, the rnagnetic flux

density in the core is increased to high saturation levels and the con stores a considerable

amount of energy. In addition. rnagnetic leakage flux hcnases to such kvels that the fm-

space volume around the magnet aiso stores a signdicant amount of energy and cannot be

neglected. For these Rasons, together with the inherently 3D nature of the geometry, the

large air-gap MRI magnet represents a challenging modeling problem.

1.2.2 Coii of the MRI Magnet

As is shown in Figure 1.1, the C-shaped MRI rnagnetic used in the IGMIT system

incorporates a long, thin superconducting coil to establish the required magnetic field. This

coil encompasses, and is closely coupkd to. the core section of the magnet (in the upper left-

hand portion of the figure.) It is isolated from the magnet by a cryo-cooler shell. As noted,

the coil excitation level is very high in order to maximize the magnetic flux density in the

magnet air-gap. Consequently, the magnetic field level in the vicinity of the coil is very high

and changes rapidly. Modeling the coil within the large rnagnet is thus difficult and plays an

important rok in the accuracy of the simulation results. Eudimg a convenient method to

reprisent the effect of the long, thin coil would reduce the modeling dificulties. The search

for such a mthod is a consideration in this thesis.

1.2.3 Size of the MRI Magnet

The magnet is exccptiondy large. Its fiame size is 1.30~1.23~0.50 m and its volume is

0.615 m3. nK ûee-space region that surrounds the magnet stores a considerable amount of

energy due to t k kakage of the magnetic field. Thus. the modeling of the infinite space

Chapia 1. introduction 8

exterior ngion must be done carefuUy and wül play an important mle in determinhg the

accuracy of the results. Due to the extenor region, the overaü sizc of the resulting FE model

will be very large. This in turn &ses a constraint on the mode1 since the number of

avaüable nodes and elemcnts wiU k limited by the FE simulation package king used. The

tradesff between the üMtad number of nodes and elements, on the one hand, and optimal

accuracy on the other hand, is a central issue in any modeling exercise.

1.3 Thesis Objective

Two aspects of the F i t e Elernent Method provide the focus for this thesis: the global

accuracy of the solutions obtained, and selection strategies toward a final and best solution of

a given problem These wül be applied and examined in relation to the C-shaped magnet of

the open-concept MRI magnet. The thesis objectives are divided among the foiiowing

topics:

1. F i t e Element Modehg:

To rnodel the MRi magmt with cifiennt gridding strategies, and

To develop a stable rnodeling practice suitable for the MRI magnet.

2. Global accuracy criteria:

To develop a global accuracy cntenon using Ampere's circuital law as a basis, and

To obtain the limits and conditions under which it can k applied.

3. Sekction approach:

To develop convenient rnethods to obtain an initial model of a problem, where the

initial model is defined as yielding nasonable and acceptable accuracy,

To identw a simple mthod to pide the improvement of the model such that overall

accuracy is increased,

To verify that the de- accuracy critena can k used to assess the quality of the

solutions,

To define selection criteria that can be used to select the most accurate model of the

probkm and

To study selection criterion baseà on the energy content of the FE model.

Chapta 1. ïn~oduction 9

4. The MRI rnagnet study :

To analyze the MRI magnet perfocmance under different operating conditions, and

To introduce and analyze an-altemative design for the MRI makit.

5. Equivaknt 2D models:

To develop suitable 2D approaches to simplify the study of the MRI magnet and

sirnilar devices, and

To vaiidate the use of the 2D equivaknt approaches and exarnin their limitations.

1.4 Thesis Outline

This thesis is composed of five chapters. The tint chapter provides the introduction, the

objectives and the outline of the thesis.

Chapter 2 discusses the key issues that arise when the Finite Element Method is appiied

to 3D problerns. These include the accuracy of the FE model, and the notion that a selection

strategy is required to guide the modification of the model. First. the FE model of the focus

problem used in this thesis. namely the MRI magnet, is completely defmed and descrikd.

This description includes: (a) the specitications of the MRI magnet. (b) an outline of the

Fmite Element modeling basic steps, (c) the different approaches that are possible when

using the FEM to mode1 a problem, and (d) the domain definition and discretization of the

MRI magnet problem Second, the applications of Maxwell's equations are discussed in

general terms. and Ampere's circuital law is used to define a global accuracy criterion. Third,

the selection strategy that is used in this thesis is completeiy htroduced. Founh and last, the

modifications of the Finite Element moâeüng steps are discussed. Such modifkations are

necessary either to achieve an acceptable solution or to improve the accuracy of the solution.

Chapter 3 examines the thesis proposais: the global accuracy criterion and the selection

approach. This study focuses on the MRI magnet problem whert the magnetic tlux density in

the center of the air-gap is measurcd and known. Fit. an initial model of the probkm is

obtained and veritied against the known solutions in the air-gap of the rnagmt. Second. a

novel and simple equivaknt nrthod that improvcs the efficiency of the selection process is

introduced and employed. niird. the validity of Ampere's law accuracy criterion is studied

Chaptcr 1. Introduction 10

and it is shown that such a masure is not completely flawless and should be used canfully.

Fourth, selection criteria based on the energy content and the magnetic field data in a region

of the magnet are investigated foi the M M magnet problem. Fifkh, ' the most accurate FE

model of the MRI rnagnet is selected. Its solution is given graphicaîiy and numerically. The

magnet is ais0 analyzed for difftzent operational conditions. Sixth, an alternative design of

the MRI magnet is introduced and its FE model is studied. Fiialiy, the original and

alternative designs of the M'RI magnet are compared and conclusions are drawn.

Chapter 4 studies diffennt equivalent two-dimensional (2D) models of the MRI magnet

with the caveat that the sirnpwed rnodels should not simplify the basic 3D problem out of

existence. Two approximate approaches are introduced. In the first approach, a magnetically

isolated region of the magnet is separately modeled and studied. The core and the pole of the

magnet are modeled and studied by this approach. In the second approach, the geometry of

the magnet is simplified while preserving its cntical features. The nsultant composite

problem has a cylindrical symmtry and is modtled in 2D. The FE solutions in the regions

of interest are then sought. The original and alternative designs of the MM magnet are

studied by this approach and the solutions are shown satisfactory.

Chapter 5 summafizes the conclusions and conuiiutions of the thesis. FoUowing this are

the References and Appendices.

Appendix A describes the Image-Guided Minirnally Invasive Therapy (IGMIT) project.

Appendix B shows the B-H and V - B ~ characteristic curves of the material used in the MRI

magnet. Appendix C details the 3D linear and non-linear magnetostatic Finite EIement

Method based on the magnetic vector potentials. The boundary value problems and the FE

approach for solving them are d e s c n i in a detded series of steps that lead to the solution

of the problcm. Worbg effectively with the FEM - which is the focus of the thesis

objectives - requks the compiete understanding of the method. Appendix C tries to cover

such a background cornpnhcnsively. Appendix D su- the necessary steps in the

appiication of the axisymmetric Fmite Element Method.

Chapter 2

Accuracy and Selection of a Finite Element Mode1

Over the p u t 30 years. the F i t e Elemnt Method has ken widely used to solve a broad

class of problems in most fields of engineering. Today, the FEM is the solution rnethod of

choice in many of the appücation areas. This has ken accompanied by the wide avaüabiiity

of commercial software packages that impiement the method.

Whenever any simulation software is used to rnodel a physical device or process, one

should never lose sight of the fact that the results obtained are only as good as the mode1 that

is used. This is particularly true when the device king modeled is non-linear and 3-

dimensional. The central issues involved in any FE modeling exercise include the following:

a) Generaliy, the FE software packages have an upper bound on the number of elements or

nodes that can bc used in the rnodel. nius, there is an important trade-off between the

accuracy desùed and the resources available.

b) Calibration and mcasured data are often not availabk. This is particuIarly true when the

device king modeled is large or expensive. Indeeci, the device may not even exist in

prototype form until a significant arnount of modehg has k e n cornpkted. The question

is then one of judggig the quality of the results that have ken predicted by a given

model. This cm k t e m d the qirolity problem; the issue is that of choosing one or more

criteria by which the general quaüty or accuracy of a particular solution cm be verified.

C) A related issue is how to deploy the resources available within a given FEM package in

order to obtain the rnost accurate solution to a given problem This can be t e m d the

Chaptcr 2. Accuracy and Seltction of a Finite Elcment Mode1 12

selection problem; the FE model must be rnodified in order to mlliimize one or more

error masures as defmed by the chosen quality factor(s).

The quulity and selection problem an discussed in this chapter and, in particular,

masures that can effectively k used to judge the quaiity of a FE solution are descrikd. In

order to focus the discussion throughout the thesis, the particular problem of developing an

accurate mode1 for an open-concept MRI magne< is considered. ANSYS, a widely used and

commercially availabk FE software package, provides the modeling and solution platform

for the thesis.

The FE mode1 of the MRI rnagnet is detaikd in Section 2.1. The MRI magnet

specitications are given fnt. Next. a brief introduction to the F i t e Ekment Method is given

which provides the necessary background to the FEM appücation. The boundary value

problem, and the problem domain of the MN magnet are then defmd. Dinerent modehg

practices, and the final discretized FE model of the magnet are fmdy discussed and

presented.

The quality of the FE solutions is discussed in Section 2.2. The Finite Elemnt Method

employs Maxwell's equations to rnathematicaliy mode1 a device. The integral f o m of these

equations are used to assess the FE approximate solutions and to estirnate the general

accuracy and qudty of the FE results. Different criteria have been defmed to assess and

ver@ the accuracy of the results. These criteria are discussed generally and also in regard to

the formulation of the FEM that is used in this thesis.

The selection strategy that is used in this thesis is defmd in Section 2.3. The selection

strategy defines the mans and approaches for masuring and improving the accuracy of the

solutions. The criteria for the most accurate mode1 are also &fincd in this section.

Diffèrent moâif'ications of the FE rnodei are discussed in Section 2.4. These

modifications are introduced as a rcsult of the model selection process; Le. the modifications

of the model irnprove the accuracy of the soIution.

Chanta 2. Accuracv and Sclcction of a Finite Elemcnt Mode1 13

2.1 The FE Mode1 of the MRI Magnet

The main objectives of this section ike (a) to discuss the approach use; in this the& to mode1

the MRI rnagnet, and (b) to introduce the FE model of the rnagnet. Therefore, the MRI

magmt specifications and s o m general t e m are given fkst in Subsection 2.1.1. The Finite

Ekmnt Method and the related modehg issues are then discussed in Subsection 2.1.2,

leading to Subsection 2.1.3 where the boundary value probkm for the M M magnet is

descriid. The problem domain is defined within Subsection 2.1.4. Diiferent discretization

mthods are discussed next. and fmally, the best modeling practice for the MRI rnagnet is

presented.

2.1.1 The MRI Magnet Specification

The MRI rnagnet includes one cyündrical core, two cyündrical poles. two yokes, and one

coil, d of which were shown in Figure 1.1. The central volume of the air-gap is the main

region of interest for MR imaging purposes. There are two syrnmetry planes in the magnet.

The fmt is the z = O plane which cuts the core at its horizontal mid-plane: the tangentid

component of the magnetic flux density must vanish across this plane. The second plane of

symmetry is the y = O plane. This plane divides the rnagnet in haif venically and is subject to

the boundary condition that the normal components of the magnetic fiux density must vanish

across it. By exploithg these symxmtry planes, a quarter symmetry model of the magnet is

used, as is illustrated in Figure 2.1, page 41 - at the end of this chapter. The coil is shown as a

white cyündricai sheD in the figure.

The y = O symmetry plane is an important cross-section since it shows al1 the basic

features and components that comprise the magnet. This cross-section is shown in Figure 2.2

together with the various paths that will k used to calculate Ampere's Iaw error.

The MRI magnct specifkations are given in Table 2.1. The magnetic material used in the

rnagnet is Cl006 steel and its B-H c u m characteristic is shown in Figure B. 1. Appendix B.

Chanter 2. Accuracy and Sclection of a h i t e Elemmt Mode1 14

Table 2.1: The MRI Magnet specifcations that are used in this thesis.

Component:

Co re

Yoke

Pole

Air-gap

Coi1

Coi1

Material 1 Data

Magnetic Material Radius = 250 mm Height = 300 mm

Magnetic Materid C.L distance = 800 mm Height = 3 15 mm

Magnetic Material Radius = 250 mm Height = 150 mm

Free-space Material Radius = 250 mm Height = 15Omm

Super-conductor Radius = 275 mm Thickness = 25 mm

Super-conductor Length = 265 mm 95 Current = 60 kA-t.

2.1.2 The Finite Element Method

The Fite Element Method is a numerical technique for obtalliing approximate solutions to

boundary value problem. The 3-D iinear and non-iinear magnetostatic Fiite Element

Method - which are used in this thesis - are detaikd În Appendix C. ANSYS, a widely used

and commerciaiîy available FE software package, provides the modeling and solution

platfonn for the thesis.

The Fhite Element analysis of a bundary value problern incorporates the foilowing

main steps:

Dennition of the boundary value problem.

Formulation of the probkm in tenm of the Ritz (variational) or Galerkin approaches.

Dennition of the problem dornain and its boundaries.

Discretization of the problem domain by elements and nodes.

Sekction of the Înterpolation hinctions for each category of element.

Formulation of the system of the equations in linear or non -ünear terms.

Solution of the system of equations.

Evaluation of the flnd results.

Vedïcation of the final results for their accuracy.

Modifications of the steps in the FE approach, if necessary, in order to improve the

solution accuracy.

Chanter 2. Accuracv and Selcction ofa FuUte Elcnitnt Mode1 15

Idormation as to how these steps affect each other, even at the stages very close to the final

solution, is not widely available. The entire model, as well as all the steps, rnay be modified

at any stage due to a failure of the &M to achieve an acceptable soiuti&.

Wheiwer a FEM software is used to model a device, one has control over some steps of

the FEM appiication. These rnodeling selections and controls are:

a The definition of the boundary value problem in step (a),

Thedefinitionoftheproblemdomain instep (c), and

O The discretuation of the problem dornain in steps (d) and (e).

These modeling controls are discussed in the foiiowing subsections while the software

impkmentation of the other steps - steps (b), (0, (g), and (h) - are detailed in Appendix C.

Verüration of the remlts. step (i). is discussed in Section 2.2, and modification of the FEM

approach, step 0). is given in Section 2.4.

2.1.3 Boundary Vaiue Problem Definition

Mathematicai models of distnited parameter physical systems are typicaiiy formulated in

tenns of boundary value problems. Any boundary value probkm can k defincd in terms oE

a) The conditions that mus< be satisfied on the boundary f that enclose the problem dornain

D, and

b) A goveming differential equation in the domain D. This equation can be written as:

= f * (2.1)

where L is a differential operator, u is the unknown quantity, and f is the excitation or forcing

fiinc tion.

In elcctromagnctics. the dinerential operator L, the unknown quantity u, and the forcing

firnction f are aU obtained h m Maxweii's equations. In the case of magnetostatic field

problems, the difkrential f o m of Maxweli's equations are

VXH = J (Ampcre's law), (2-2)

Chabter 2. Accuracv and Selection of a Finite Elemart Modcl 16

VsB = O (Gauss's law ), (2-3)

where B is the magnetic field intensity vector. B is the rnagnetic flux density vector, and J

is the electric cumnt density vector. Using p for the permeability and v for the reluctivity of

the medium, the constitutive relation is written as

B = pH or H = vB.

Mannetk Vector Potential:

A FE mode1 of a static magnetic field is traditionaiîy defined in t e m of magnetic potentials.

The problem can be defined either in tenir9 of the magnetic scalar potential [9-101, or by

using the magnetic vector potential [Il.-131, or by a combination of the scalar and vector

magnetic potential [14]. The rnagnetic vector potentid approach is used in this thesis and is

introduced in the following subsections. The reasons for not using one of the other two

approaches (a scalar potential; a combination of scalar and vector potentials) are given in

Section 2.4.

Maxwell's Equations govem magnetostatic field problem. Since the magnetic flux

density B is a divergence free field. it can k represented in terms of a magnetic vector

potential A as

B = VXA. (2-5)

By substituthg (2.5) in (2.2). it foilows with the aid of (2.4) that a second order differential

equation

Vx(vVxA) = J . (2-6)

is obtained for the magnetic vector potential A. This equation only partially represents the

magnetostatic hundary value problem over the domain D since any vector field can only be

compktely defined by imposing conditions on its curl and its divergence. A gauge condition

must therefore be Biiposed on the divergence of A in order that A can be uniquely detined.

The commonly used gauge condition for magnetostatic probiems is the Coulomb gauge,

which is dehed as

V-A = O (Coulomb gauge). (2.7)

Chapia 2, Accuracy and Sclection of a Finite Elcmcnt Mode1 17

In this thesis, the boundary value problern is descnbcd by using a v-tor potential A that

satisfis the follo wing vector differential equation

Vx(vVxA) - V(vV*A) = J , (2.8)

which is obtained by enforcing the Coulomb gauge on to (2.6) and comprehensively

explained by Biro and Preû in 1131. The boundary conditions at the outer boundary Tof the

probkm domain D are of two kinds

A = P Dirichlet condition on r', (2.9)

ia<(VxA) = O Neumann condition on &, (2.10)

These conditions are applicable at planes of symmetry and on r = TI v r2 . The continuity

conditions apptied at the interface ktween two media of dwerent reluctivity are

v'itxVxA' = v'i^utVxA' Continuity of tangential H, (2.1 1)

kVxA' = &VxA' Continuity of n o r d B , (2.12)

&A' = iixA' Continuity of A, (2.13)

where superscripts + or - refer to different sides of the discon tinuity interface Sd.

2.1.4 Domain Definition

The solution dornain in ms t electromagnetic problems contains the main physical device

and the surrounding fiee-space region. The surrounding region should be defined properly,

but this definition wili depend on the type and nature of the problem and the objective of the

analysis. It is impossible to direftly mode1 the whok infinite space around the device, and

therefore, the exterior domain should be restricted and should consist of a reasonable volume

sumunding of the device. Several rnethods exist to compensate for those portions of the

infinite space region that have been tnincated.

The so-calkd far f ~ l d or infinite elements an often ernployed to take the intinite extenor

fi.ce-space region into account. By employing them, the entire problem domain is dividd

into an intcrior region, an exterior ngion, and a transitional region between the interior and

exterior regions. Within the interior and transitional regions, regular elemnts of finite size

Chapta 2. Accuracy and Sclcction of a Finite Elcmcnt Modcl 18

are used. For the exterior region, ekrnents that effectively have an infinte volume are

employed. nie proper geometry and location of these elements cm greatly affect the

accuracy of the results. It is knoA that the infinite elements performkst when-the exttrior

ngion is symmetrical with respect to the geometric origh of the domain. This symmetry

leads to the use of a circular exterior boundary in 2-D problems, and a cylindrical or

spherical exterior boundary in 3-D problems.

The interior region contains the device, its components, the regions of interest and the

immdiate surroundings. A free-space c yhdrical or spherical volume envelo ps t his interior

region and provides the transitional ngion ktween the interior region and the exterior

region, whose surface is the infinite side of the M t e element.

The volume of the enveloping cyünder or sphere should be large enough to permit the

proper accommodation and performance of the normal and infihite elements. On the other

hand, it should be small enough that the rnodel, as a whole, employs a reasonable and limited

number of nodes or degrees of fmdorn

When a cylindrical volume is used, the Wite elemnts can be used either on the curved

surface or on the flat surf'ace(s) of the cylinder (which can k seen in Figure 2.3.5.) The 3-D

W t e elements can have ody one face toward infjnity. Unfortunately, the corner region of

the cylinder has two faces toward W t y and cannot be rnodeled with infinite elements.

Therefore, the boundary conditions on one surface of the cyiinder should be set properly - whüe the W t e elements are employed on the other surface of the cylinder. On the other

hand, when a spherical envelope is used these problems are not encountered (as can be seen

in Figure 2.4.1 .) However, the spherical envelope pnsents pater difficulties at the meshing

stage, as compared to the cylindrical approach. The unsatisfactory results of the cylindrical

envelope approach wiii be given in Subsection 2.4.2. In this thesis, the spherical envelope

approach is prirnarily used to mate the transitional and extemal ngions of the problem

domain.

In the study of the MRI magnet, the problem domin is divided into interior, transitional, and

exterior ngions. The interior region contains the device, the fric-space between different

Chapta 2. Accuracy and Sclcction of ri Fuiitc Elcmcirt Modtl 19

parts of the device, and the region of interest. In the case of the MRI rnagnet, the region of

the interest is p M y the air-gap between two pole pieces. The device consists of the

excitation CO& the core. the y okes, and the poie pieces.

A frecspace spherical volume envelops the interior ngion and provides a transitional

region between the interior region and the exterior ngion. Then. the exterior of such a

cylinder or sphere will face infinity and accommodate the infinite side of the infinite element.

During the modeling of the MRI magnet, different radii and element sues were used to

define the envelope spherical volume, with the interior region king kept constant. After

several successhil solutions, a set of parameten was established for the envelope volume.

These parameters are kept as long as a change in them does not improve the accuracy of the

resu lt. The fmal parameten are presented in the foilowing section.

2.1.5 Domain Discretization

The discretkation of the problem domain is the most important step when developing a FE

model for a given device. The manner in which the domain is discretized wili affect the total

element and node numkrs, and the computer storage requirements. It aiso defines the

computation thne, the software requiremnts, the convergence of the analysis to a solution,

and the accuracy of the numerical nsults. The efficiency or even propriety of the

dûcretization cannot be completely judged till the nsults are obtained and verified.

The problem domain is typicaüy âiiretized using one or more of the foiiowing

techniques:

Semitommtcr-aided discret uation or extrusion in w hich the analyst decides on ho w and

where to use a sofiwarc mhing capabiiities. One of the most effective rnethods of m a h g a

3 D FE mode1 is by extniding a proper 2-D mdel dong a pcedefined path. F i t , the analyst

makes a 2-D projection model, which includes mas meshed with 2-D elements. Second,

software extrudes the 2 0 model (or a spceifïc part) dong a path to mate the 3-D volume.

As a nsuh, a circdar do&, a t rbp la r elcmnt or a quadrüateral ekrncnt cm k extmded

to a cyüadricd volumt, a prisrn ekment, or a brick elemtnt. respectively. The path should

provide enough information to comctly büd the model. The extrusion divisions define the

Chaptcr 2. Accuracy and Sclcctian of a Finite Elment Mode1 20

tineness of the elements in the direction of the extrusion path. The extrusion method is

mainly used in this thesis.

Computcr-aided discretization or kee-meshing in which the analyst ' defines the geometry,

sets some control and meshing parameters, and the software creates the ekments and the

correspondhg nodes. To obtain a given quality of discretization, more parameters should be

set. and the mshing control shouM k adjusted fkom the software to the analyst. Software

programs, or at lest ANSYS 5.3, fail to discretize a slightly complicated 3-D domain

containhg many volumes rhat have curved surfaces - such as the MRX rnagnet problem

dornain which include the thin cylindncai coil. the cyhdrical con connected to the yoke

slab, etc. This fdure is mainly fiom the loss of the node connectivity ktween adjacent

elements of different volumes. The fkee-meshing method is used selectively in this thesis.

Ada~tive discretization in which the anaiyst only defines the problem domain. A FEM

software is then employed to discretize and solve the problem. DifTerent adaptive

discretization schems have ken developed and used for 2-D problems. This method is not

as widely used in 3-D probkms; however, new developments have been reported (15-161.

Tetrahedral elemnts are used exclusively in adaptive discretization. Such elemnts are

avoided in this thesis due to nasons that wül k discussed in Subsection 2.4.4. The adaptive

meshing is not used in this thesis.

Non-ada~tive dûcretization and modification in which the meshing of a domain is performed

according to the geometry of the domain and some meshing guidelines. These controllhg

pideihes are rnainiy the anaiyst's specified division number of the geomtrical entities, the

aspect ratio between diffierent sides of an ekment, the span angle ktween adjacent iines or

areas of the element. and the visual quaüty of the element. The discretized problern is solved

by the FEM and the accuracy of the result is then verified. If the quaüty of the results is not

satisfactory the mshing parameters are adjusted, and the process starts h m the kgllining.

The principles of this approach is used in this thesis.

n Mscretizrition of the MRI Mamet:

A convcnknt, overall problem domain for the MRI mapet is an overall spherical volume

which, by definition, inchides the previously noted interior, transitional, and exterior regions.

Chaptu 2. Accuracy and Sclection of a Finite Elcmcnt Modcl 21

The interior region is cylindrical in shape and contains the entire magnet together with a

portion of the surrounding free-spact (shown in Figure 2.3.5.) The transitional region is the

f ~ s t layer around the cyiindrical interior mgion and accommodates &mial ele&nts (shown

in figure 2.4.1.) The exterior region envelops the transitional region and provides a volume

to ward the infinitely extending free-space exterior, which accommodate the Uifinite elements

(show in Figure 2.4.4.) A conirolled combination of the extrusion and free-meshing

techniques are used to non-adaptively discretize the domain of the MRI magnet problem.

The problem domain data includes two types or sets of parameters. The fust set of

pararneters includes the M N magnet specifications that were given in Table 2.1 of

Subsection 2.1.1. These are, in essence, the dmiensions of the rnagmt and the materiai

parameters. The second set of pararneters includes the geometrical information of the

surrounding he-space. These data wiil be given below in order to complete the defmition of

the probkm domain.

The discretization pararneters of the most accumte model of the MRI magnet include

the element size setting, the number of nodes per line, and other pararneters. The

discretization method used in this thesis has rnany steps, and each step includes rnany

dimetization pararneters. A detded report of these discretization parameters is not

particularly useful. Thenfore, only the important discretization parameters of the final FE

model are provided in this subsection.

Discretization guideks have ken developed during the modeling of the MN magnet.

These pidelines, with carehil modification, are also applicable to other simiiar problems.

These pideîines and the rneshing paramtcrs of the MRI magnet are summarlled in the

followhg four steps.

The starting pomt for the model development is the H cùcular 2-D projection area shown in

Figure 2.3.1. This area contains aii the necessacy details that will be nquired in the fuial 3-D

mode1 of the MRI magnet. It wiil be used, together with an extrusion approach. in developing

that model. The radius of this circular area is Ri=900 mm and it contains a totai of 85 areas.

Controlied free-meshing was used to dimethe the areas with 1620 ekments and 16 11 nodes.

Chaptcr 2. Accuracy and Sclection of a Finite Elcment Modcl 22

Quaddateral ehments are mainly used in this 2-D cross-section. This choice anticipates

preferred hexahedral elements that will result during the extrusion stage. (Finer details of this

2-D grid are shown in Figure 2.5.5.)

2. Interior d o n : p

The 2-D projection arca of Figure 2.3.1 is then extmded dong the z-axis - with proper setting

for the element height - to create the magnet mode]. Figure 2.3.2 represents a partial model

of the core region, as obtained during the extrusion stage. It shows a portion of the central

core, the fke-space volume ktween the core and mil, and the coil.

Figure 2.3.3 shows the continuation of the extrusion that yieids the core (the dark

cylindrical region), a portion of the yoke (the gray quarter cylindrical region attached to the

core), and the tkee-space above the yoke. The coil volume and the free-space between the

core and coil are also shown. Continuing with the development of the rnodel, Figure 2.3.4

shows the model after completion of the con, the pole and the air-gq (both on the right hand

side of the rnodel.) Note the creation of the yoke and the fm-space beneath the yoke by the

extrusion and setting Mennt materids to the extmded volumes.

The complete extrusion of the 2-D projection model yields the interior region of the M M

magnet model. This cyhdtical volume is shown in Figure 2.3.5. The height of this cyünder

is Hi=700 mm Hexahedral elements are rnainly used and prism ekments are only employed

to accommodate the geometry where necessary. The extrusion is done in seven steps and

yields &5x7=595 volumes, 1620x19=30780 elements, and 161 1~20r32220 nodes.

3. Transîtional Redon:

In the final modcl, mtinite elements wül be used on a spherical bounding surface to properly

represent the far field behavior of the magnetic vector potential, and thus the magnetic flux

density. It is therefon necessary to properly npresent the transition from the previously

d e r n i d (cyündrical) interior region to the outer (spherical) exterior bounding surface.

A 51 sphen with a radius of R,-=1600 mm envelops the inner cylindrical volume. The

b e r cylindrical volume is then excluded from the sphere to create the txansitional volume.

ControlLd mesbing was found to be unsuitable for use with this complementary volum.

Chaptcr 2. Accuracy and Selection of a F i t e Elcmcnt Modcl 23

Therefon. it is divided into the thm sub-volumcs that are shown in Figure 2.4.1 - one face

of each v o b m is marked with +X. -X, and +Z volumes. Note the cutting angles of the

cornplementary sphere, which are in the direction of the future radial extrusion.

The two volumes on either side of the y-z plane are discretized using a proper extrusion

that yields %(4x 1 8x l9)=27 36 hexahedral elements. These volums are sho wn in Figure

2.4.2.

In order to discretize the remaining sub-volume (the +Z volume that is absent from

Figure 2.4.2). its surface is fkst rneshed with quacidateral elements. Then, free meshing is

used to mesh the volume with 26340 tetrahedral elements. The final form of the discretized

transitional region is shown in Figure 2.4.3.

4. Exterior Reaion:

In order to complete the transition to the spherical fat= field surface, a U sphere sheU having

an outer radius of R, =1600+1000 mm is used to envelop the transitional volume. This

sheil is tenned the exterior volume and is divided into four sub-volumes, which are shown in

Figure 2.4.4. These volumes are discntized with radial extrusion, which yklds 1332

hexahedral innnite elements. The h a 1 mode1 is shown in Figure 2.4.5, where a cut-out

section bas been included to show the transition to the exterior volumes.

For the most accurate moàel of the MRI magnet, the probkm domain is a U sphere

having a radius of R, =26ûû mm A total 61 188 ekments and 41 193 nodes were used to

discretize the probiem domain, resulting in a total of 117,345 degrees of Freedorn. Figure

2.4.6 shows details of the interior region of the FE rnodel. two symrnetry planes of the rnodel.

and details of the exterior grid.

2.2 Verifi~cation of the Results

The F i t e Elernent Method provides an approximate solution to a given boundary value

probkm Since the method itself is an approximation and the solution is numerical, there are

inherent emrs not only in the numerical solution for the potentials. but ais0 in data derived

Chaptcr 2. Accuracy and Sclection of a Finite Elcment Mode1 24

from that solution. The main objective of this section is to introduce criteria that can k used

to evaluate the accu- of the FE solution.

Different criteria are used to askss the quality and acceptabiiity of the FE rèsults. These

criteria can be divided into (a) local quaüty criteria, and (b) global quaüty criteria. For the

purpose of local verification, a specüic location or a region within the problem domain is

investigated. On the other hand, global cnteria may involve the entire problem domain, or a

large number of subregions. Either dincrential or integral forms of Maxwell's equations can

be used as a basis for estabüshing local or global accuracy criteria.

In Subsection 2.2.1, the errors in the derived magnetic field data are exaqined. It is

argued that the error mesures based on the magnetic field discrepancy cannot provide global

accuracy criteria. Subsection 2.2.2 and 2.2.3 investigate the agreement between the FE

approximate solutions and Maxwell's equations. It is shown that the FE approximate

solutions always satisfy Gauss's law when magnetic vector potentials are used to define the

FE problem A globd accuracy masure is then dehed based on Arnpere's law. Subsection

2.2.4 introduces the stored rnagnetic energy as a selection criterion. In each section. the

arguments are first discussed in general terms and then are prescnted more specifically in the

context of the vector potential FE formulation used in this thesis.

2.2.1 Magnetic Field

The rnost important derived-data of the FE solutions are the rnagnetic fields B and A. A

convenient location to observe and estimate the pst-processing error is at the planes of

symmetry, on which the field is either tangentid or normal, and where one or two

component(s) of the magnetic field must vanish. However, the FE solution shows that the

out-of-normal or out-of-tangentid magnetic fields are not equal to zero. These rnagnetic field

data include the differentiation or pst-processing errors. However, such error cannot be used

as either a globai or a local accuracy criterion.

The megnetic field error can be obsemd by the foUowing approaches:

a) Us& the masund data: The rneasured magnetic field data cm be used to assess the

quality of the iesults if they are available. This is the strongest accuracy check on the FE

Chapter 2. Accuracy and Sclcction of a Finite Elemcnt Modcl 25

results. However, this mcthod shouM be applied with caution where the results are verified at

a specifc location. It is always possiùle to select or modify the discretized location, node or

elemcnt so as to minimize the diffeknce between the appmximate and the masured data. In

this thesis, the average of the magnetic hu density in the air-gap center of the MRI magnet

is calculated fiam the FE solution and compared with the measured value. Proximity of two

values indicates that the solution of a particular FE mode1 is generaily correct. The most

accurate mode1 of the problern should then be selected from different correct FE models of

the problem This is the subject of the selection process that U given in the next section.

b) Usinn contour ~lots: Contour plots of the magnetic field are frequently used to hvestigate

visuaiiy the results. Contour plots can provide a designer with invaluable information, but

cannot be used as a local or global accuracy measure for the FE solution.

cl Usinn a discontinuitv error criterion:

The magnetic field data are not continuous at the comrnon node between adjacent elements

due to the formulation of the nodal FEM based on the magnetic vector potentials. The

elernent nodal field data are usualiy averaged to obtain more acceptable and continuous

looking results. The discontuiuity enore,, of the B for th element is then defined as

where Bi is the nodal (averaged) B at node j, BQ is the B of the elemnt i at node j, and n is

the number of nodes in elemnt i.

The application of the discontinuity error to 2 0 and 3-D problems has produced some

guideiines such as avoiding n m w , fat, thin ekrnents, and keeping the aspect ratio of the

elements in a reasonable range. However, the local smoothness of the results does not

exclusively parantee the accuracy of the FE solutions. Therefore, a global accuracy criterion

carmot be de- based on discontinuity error (2.14). The continuity of the magnetic field,

which is the result of MaxweWs equations, is discussed separately in the following.

Chaptcr 2. Accwacy and Selection of a Finitc Elcmcirt Model 26

2.2.2 Gauss's Law

The diffetentid form of Gauss's law (2.3) States th& the divergence of the magnetic flux

density B is always zero. However, when the rnagmtic vector potential is used to formulate

the nodal FEM, the divergence of the FE solutions of B is obtained as

VgB, =V.(VxA,)=O. (2.1 5 )

Therefore, by mrit of vector identities, (a) the MVP based FE solutions always satisfy

rnagnetic Gauss's law, (b) the B solutions are always solenoidal, and (c) the total rnagnetic

flux is always preserved.

The continuity of the normal magnetic flux density between different media is also

concluded from Gauss's law. From the relation B=VxA together with the continuity of the

vector magnetic potentials A, it is concluded that the normal B of the FE solutions is always

continuous bet ween difTerent media,

In conclusion, when magnetic vector potentials are used to formulate the FEM, the

approximate FE solutions always satisfy the magnetic fom of Gauss's law. and thus. a global

accuracy criterion cannot be dehned based on Gauss's law.

2.2.3 Ampere's Law

The continuity of tangential H between different media is the direct result of Ampere's law

where the current is zero at the interface between two media or elements, and the

magnetostatic field is studied. The compkte satisfaction of this condition, in addition to the

natucal satisfaction of Gauss's law - continuity of the normal B & tangential H - yields the

exact solution of a boundary value probien The exact solution to even a simpk problem is

not practically achievable. Within an approximate FE solution, the tangential H is not

continuous between daennt elemcnts and mdia. However, the local discrepancy of the

tangential magnetic fîeld may be used in refining of the elements.

Cornpliance of the FE solution with Maxwell's equations is related to the weak or panid

satisfaction of Maxwell's equations by the FE apptoxhte solutions. Regardlas of the FE

Chapta 2. Accuracy and Selcction of a Finitc Elcment Modcl 27

formulation used, The FE solutions satisfy one of Maxweil's equations cornpletely and

satisfy the other one approxirnately. In the magnetic scalar potential formulation of the FEM,

Arnpere's law is completely satisfiid whüe Gauss's law is partially satisfkd. In the magnetic

vector potential formulation of the FEM, Gauss's law is completely satisfied while Ampere's

law is not completely satisfied. It is therefore proposed that the partial satisfaction of

Ampere's law can effectively be used to defue a global accuracy rneasure for the FE

solutions that are considered in thi s thesis.

The magnetornotive force drop along a contour line is used to defme a global accuracy

criterion for the approximate solutions of the FEM. Arnpen's circuital law States that the

total mmf along a closed path must equd the total current passing through the surface that the

path bounds. Thus, a convenient error measure is

This emr includes the errors originating fiom FEM application, from the difîerentiation in

pst-processing of potentials A to obtain B, from material curve fitting or B-H curve

interpolation to obtain B, anci fiom numerical integration as in (2.16).

It should be noted that the emxl is path dependent, and that the path may have k e n

selected as a way to incnase or decrease this error. As a result, a single path cdculation

should be avoided. To avoid this rnisjudgment, different paths, which enclose the same

cumnt, should be chosen, and the relevant errors should k studied statistically. This data

can then be used to compare différent models of the same boundary vaiue problem Figure

2.2, page 41. as was noted previousiy, shows the different paths that were used in this thesis

for the study of the MRI magnet.

The FE formulation is generally bascd on the ngnimization of an energy related hnctional.

This mans that the FE solution for each probiem has the minimum energy of the respective

discretized model. Monover, different FE models of the same boundary value problem

would introduce diffcnnt levels of stockd energy. Physks states thai any naturai system is

Chanta 2, Accufacv and Selcction of a Finite Elcment Mode1 28

stable o d y at the minimum level of its potcntid energy content. This axiom cm ôe used to

compare dinerent models and solutions of a boundary value probkm. The FE model which

has the lowest emrgy content between different models of a problem is expe&d to k the

k s t appioximate model of that problem This approach is used as one of the selection criteria

in this thesis.

2.3 Selection Approach

The Fii te Element Method provides an approximate solution to a boundary value probien

and as any numerical method, the results are as good as the model. Accuracy criteria are

therefore used to assess the quality of the resuhs. The rnodeling approach may then be

moditied to obtain a better (more accurate) model, in which the availabk resources (Le.

elemcnts) are more efficiently employed.

The basic issues in tnodeling a complex non -1inear pro blem are:

1) How can one efficiently and conveniently get a correct starting initial model.

2) How can one retine and modify the initiai mode1 toward a more accurate model,

3) How much more accuracy improvement is achievable, and

4) What are the selection criteria to measure the accuracy and to select the most accurate

model?

The foiiowing subsections discuss the approach used in this thesis (a) to obtain the initial

rnodel. and (b) to select the most accurate model of the problem.

2.3.1 Initiai Mode1

The initial rnodel of a problem is a correct model: the problem is defined. discretized and

solved properly. and thus, the accuracy of the FE solution is acceptable. The initial mode1 is

the starting mode1 in t k search for the most accurate mode1 of the problem Achieving the

initial mode1 and solution of a pmblem:

a) is the most time consuming part of solving a boundary vaiue probem.

Chapter 2. Accwacy and Sclcction of a Finite Elcmcnt Mode1 29

b) may requin many modifications in the FE application, e.g. defhition of the problem,

dennition of the probkm dornain, and meshing of the problem dornain. and

c) is the first and biggcst milestone-of the FEM toward the final solution of the p&blern.

This thesis employs the foUowUIg three steps to simplify achieving the initial FE model of

the MRI magnet.

Free-S~ace Matenal Method

In the Free-Space Material Method, the magnetic material of the discretized mode1 is

replaced by free-space, and the total excitation is appiied to the model. The MRI magnet

problem thus changes to another problem: a coil located in free-space. The FE solution is

used to calculate the total stored magnetic energy, the inductance of the excitation CO& and

Arnpere's law enor (2.16). These are compared with the c o n h d and pubiished solutions

of the new problem The FE modeiing continues tül a close agreement is reached between the

FE solutions and published solution. Agreement of two results confimis, to sorne extent, that

a) The boundary value probkm is defined and formulated properly, and

b) The cunent source (coil) and boundary conditions are applied correctiy.

This new model of the problem is ready for more examination by the following method.

Reduced Excitation Method

In this approach, the coü of the he magnet is energized with substantially lower cumnt.

Therefore, the rnagnetic material is not over-saturated. and a mild non-iinearity exists in the

problem and system of equations. The emf (2.16) error and the field at material discontinuity

surfaces are studied for hiture grid refmement. The solutions confirm, to sorne extent, that

a) The cumnt source (coil) and boundary conditions are applied c o d y .

b) The proper interpolation functions or ekrnent types are selctcd, and

c) The material characteristic of the non-ünear material is properly d e h d .

The problem rnodel is modified tül a successful solution is achieved. This mode1 is used to

obtain the initial mode1 of the given problem.

Cha~tcr 2. Accuracv and Sclection of a Finite Elcmcnt Mode1 30

In continuation of the above method, the current excitation of the FE model is increased in a

step-by-step rnanner. At each level of the excitation, the FE model is modified to maintain a

nasonable Ampere's law ercor e-f within the solution. The fuial modifkd model is then

hilly excited with the non-hear magnetic material in place. If this rnodel is proved as a

correct model with an acceptable accuracy, then it is the initial model of the problem The

correctness of this FE rnodel (or any given problem) is discussed under three categorics:

(a) The measured data of the ~robkm are availabb. Agreements of the FE solutions and

masured data c o n h the correctness of the model. This is the strongest verifkation

argument. The MRI magnet measured data an avaiîable in the center of the air-gap. The FE

solutions of the FE model are calculated and compared with such data. A close agreement of

two data verüies the valjdity of the rnodel, and thus, such model is introduced as the initial

FE model of the problem For the purpose of this thesis, the initiai model is then used to

investigate the application of the proposed gened accuracy criteria

Jb) The measured data are available for a similar ~roblern of eaual or meater difficultv. The

FE approach, which is used for the FE model king investigated. can be applied CO a simüar

probkm Agreement of the FE solutions and the measured data would c o n f i the validity of

the FE model. The similarity of the two problems should have ken estabüshed kforehand.

The thesis wiii întroduce an alternative design to the MRI magnet where the rneasured data

are not avaiiabk. The FE model of such a design would be acceptable because its application

to a similar and more difficult problem - the original MRI magnet - yields solutions close to

the masureci &ta of the sKnilar problem.

{cl The masured data of the ~roblem are not availabk. A globai accuracy criterion should

be used to assess the accuracy and comctness of the FE model. Such an accuracy critenon

masures the approximation of Maxwell's equations in the FE model of the problem The

thesis tries to prove the applicability of Arnpere's law as a global accuracy criterion.

The corrcct initial mode1 serves as the foundation in the search for the most accurate FE

mode1 (optimum model) of the problem

Chapta 2. Accwacy and Selcetion of a Finite Elunent Modcl 3 1

2.33 Selection Criteria

Achieving the initial mode1 c o n f i that the basic approach to the problem is correct. and

that the fmal result would be obtained by fie-tuning of the Finite Element Method steps. The

main possible modification fields of the mthod are (a) the modification of the problern

domain defmition, and (b) the modifcation of the probiem domain discretization. Beginning

with the initial model. a mode1 parameter is changed. and the sensitivity of the solution to

such a change is assessed. Improvements in the quality of the results yield a better model,

which may be introduced as the new solution. The final FE model of the problern is obtained

when the model modifications do not change

(a) The average of Ampere's law error (2.16) dong differen t paths, and

(b) The stored magnetic energy in the problem domain.

The applicability of these measures wül be proved in the next chapter.

In conclusion. the most accurate mode1 of a boundary value problem, solved with the

Finite Element Method, should:

a) Comply with the known magnetic field data in ngions of interest, if such are available,

b) Confom to the integral form of Arnpen's law as close as possible, and

c) Contain the minimum amount of stored magnetic energy compared to otha models.

2.4 Modification of the Finite Element Mode1

A key issue in any Fuite Ekment modehg exercise is to ensure that the model yields the

best possible results. It is therefore necessacy to subject the mode1 to a sequence of

modifications in an effort to improve the prtdrted results. It should be noted, however, that

it is not possible to produce an exact diagnostic niap to guide this process. Nevertheless, the

modifiation may involve one or some of the foliowing steps:

1) The definition of the boundary value problem.

2) The definition of the problem domain.

3) The discretization of thc probkm doniain.

4) The use of elemcnt type.

Cha~tcr 2, Accuracy and Selcction of a Finitc Elemcnt Mode1 32

5) The definition of the matenal characteristic.

6) The application of the cumnt excitation or load.

Ther steps are fim discussed inVgeneraI tems and then for paai&lar case of the MRI

2.4.1 Boundary Value Problem Definition

A boundary value problern is a mathematical mode], of the form Lu=J that represents a given

physical system The selection of the unknown quantity u, together with the appropriate

boundary conditions, dehes the probkm Therefore, the variable u should be selected

carefuiiy kcause some available choices wiii not yield solutions of acceptable accuracy. In

magnetostatic problems, for example, the selection of the unkno wn quant ity is mainly

aected by the level of magnetic saturation, the connection of the magnetic circuit parts, the

geornetry and topology of the pro blem, and the excitation method.

The magnetostatic nodal FEM can be defmed either in ternis of (a) the Magnetic Vector

Potential (MW), or (b) a Magmtic Scalar Potential (MSP), or fmally (c) a combination of

the MVP and a MSP. The MVP based FEM is used in this thesis. Other formulations have

ken tried unsuccessfuiiy. The foilowing subsections descrii the MSP and the MSP+MVP

approaches and their shortcoming for the MRI rnagmt problem

Masnetic Scalor PotentCak

in a magnetic scalar potential (MSP) formulation of the FEM, the total magnetic field

intensity H in the problem domain D can be decomposed into two ternis [IO]

H = H,-V@E. (2.17)

Then, the analytical or numerical solution of H should satisQ Gauss's law in domain D

v-B=v-~(E, -v+~) = O. (2.18)

The magaetk field intensity 8, is a generalized or a guess magnetic fdd, which satisfies

Ampere's Law and should k deiïncd according to the problem type. Then? the remaining part

of the E, can be deciveci as the gradient of the generaiized scaiar magnetic potential &. This

Chapta 2. Accuracv and Scloction of a Finite Elemcnt Mode1 3 3

general definition then allows for three particular choices of scalar potential: the Reduced

Scaiar Potential (RSP), the Diffennce Scalar Potential (DS P), and the Generalized Scalar

Potential (GSP). The choice of which scalar potential to use depends only on the problem

type. The main advantage of the MSP formulation of the FEM is the reduction of the degrees

of fieedorns to be solved.

In order to test the accuracy that cm be obtained when a scalar potential is used as a bais

of formulating a model of the MRI magnet, the DSP-based FEM was used on the ANSYS

platform. For a model with less than 1 6 0 elements, the DSP approach yielded the average

B at the center plane of the air-gap of 0.384 T, which is 42% mon than the rneasured value

of 0.27 T. This test illustrates that the Mproper choice of potential may yield results that are

signifîcantly in error. Meanwhiie, the MVP based FEM cannot converge to a solution when

the same mode1 is used.

The DSP-based FEM was also tried for the fuial optimum model of the MRI magnet (this

model was obtained by the MVP-based FEM and presented in Subsection 2. L S.) The results

are summarized in Table 2.2. These remlts clearly show that for a weli-developed and

optimal model of the MRI magnet, the DSP based formulation of the FEM fails to achieve an

acceptable solution. The insufficiency of the DSP-FEM may weli be due to (a) the

complicated geometry of the MRI magnet. (b) the over saturation of the magnet material, and

(c) the weakness of the DSP based ekments due to reduced degrees of freedom.

Table 2.2: The results from DSP and M W based FE models of the MRI magnet.

A cornbined vector and scalar potential foda t ion has ken proposed 1141 to overcom the

shortcoming of the scalar potential formulation. In the combined potential approach, the

probkm domain is divided into two regions. One region contains the current source and the

DSP FEM

MVP(A) FEM

Discrepancy 96

0.3 148 0.2889 2.1873 593 377 4565

0.2903 0.2658 2.1913 512 375 4464

8.4% 8.6% -0.2% 16% 0.1% 2.2%

Chanta 2. Accuracv and Sclection of a Cruùtt Elcment Maiel 34

surrounding free-space and hem, the reduced scalar potential (RSP) fomailation is used. In

the second region, containhg the magnetic material together with the space in between. the

magnetic vector potential (MW) is used.

The çombined RSP+MVP based FEM, on the ANSYS platform. has also ken used to

model the MRI rnagnet for teshg purposes. The MVP based ekmnts were used to model

the magnetic material of the rnagnet. whereas the RSP based elemcnts w e n used to model

the coil of the magnet together with the surrounding fiee-space region. Interface elements

were then employed to connect two segments of the rnodels. For this test. the non-linear

iterative solver of the FEM could not converge to a solution for a cumnt excitation higher

than 30 kA-t, which is ody half of the rated operational current level of the magnet. The

results for the 30 kA-t test problem are shown in Table 2.3. The close agreement between the

two sets of results seems to confinn that the combined potential formulation of the FEM can

be used with confidence when the magnetic material is not heavily saturated. However, as

noted above, the combined potential formulation fds to converge when the material is

pushed deeply into saturation.

Table 2.3: The results from RSP+MVP and MVP based FE rnodels of the MRI magnet.

1 Cumnt = 30 ICA-t 1 Bgap ,,, T BCore ,,, T Arnpere 's Law error ]

2.43 Definition of the Problem Domain

RSP+A FEM

MVP(A) FEM

The domain of a problem contains the main physical device together with the surrounding

regions. The h t step in d e m g the problem domain is to decide how to best represent the

dimensionality of the device king studied. In practice, there are only three choices: a planar

2-D approximation, an axisynmietric 2-D apprioxllnation, or a hiIl 3-D representation of the

device. The anaîyst must decide which approach represents the problem more ~ccuratcly.

This choice must then be validated.

0.202 1 S6 10%

0.203 1.67 4%

Chapta 2 Accuracy and Selection of a Finitc Elcment Malt1 35

In the case of the MRI rnagnet, the choice of a planar 2-D FE approximation, using r

magnetic vcctor potential formulation. certainly yielded a solution. The results satisfied both

Ampere's law and Gauss's law, i d the contour plots of the magne& field apkared to be

quite satisfactory. However. for an excitation cumnt of 60 kA-t. the average magnetic flux

densities at the center plane of the air-gap and core were computed to be 0.495 T and 1.35 T,

respectively. When a 3-D FE mode1 (whose results wüi be presented in the next chapter) was

used, the corresponding values were found to be 0.268 T and 2.19 T, respectively. The

comsponding errors of 85% and 38% are due to an incorrect representation of the problem

domain. In other words, a 2-D planar geometry shply cannot represent the magnetic flux

density in a core that has a f ~ t e cross section, particularly when there is a significant

kakage flux due to saturation effects.

The second step in defining the probkm domah is to decide how to best represent the

fne-space region surrounding the device. The surrounding region should be defmed properly

while the exterior surfaces of this region rnake most of the FE mode1 boundary. Having a

good representation of the surrounding region becornes particularly important when that

region stores a s i m a n t proportion of the total magnetic energy in the system.

In the case of the MRI rnagnet, the solution domain can be divided into interior and

exterior regions. As noted previously, the interior region contains the magnet con and the

coü whereas the exterior region represents free-space and. ultimately, the far fields. The

exterior surrounding region can k a cyhdrical or spherical volume, either of which will

envelop the interior region. Table 2.4 shows the results from two models of the MRI magnet

where both cyiindrical and sphericai exterior envelopes were used. Two designs of the MRI

magmt were used, one where the coil was conventionaliy positioned around the core. and the

othet where the coil was positioned around each pole piece.

Solving the cylindrical mode1 of the magnet where the coii is around the core, the FE

Newton-Raphson mthod fded to converge to a solution. Initially, 15 iterations were used.

Additiondy, the convergence could not be achieved with increasing the numkr of

iterations, where a total of 30 iterations were used. Where a cyiindrical sunounding ngion

was used for the exterior region of the rnagnet where the coil is around the pole, the solution

Chabta 2. Accurac~ and Selection of a Finite Element Modcl 36

accuracy was not acceptable. The spherical model. which was descnhed previously in

Subsection 2.1 5, was chosen as being the best npresentation of the exterior region.

On the buis of these tests, it was thenfore concluded that the solution accuracy was

severely compromised when a cylindricai exterior region was used to represent the effect of

the far fields. This accuracy problem seem to be a function of the magnet operating

characteristics: the device is highly saturated, the air-gap is large, and a considerable amount

of rnagnetic energy is stored in the surrounding region. Consequently, the spherical

surrounding region was used for the models that were descnid in this and next chapter. The

main parameten of the sphericai surrounding region are its inner and outer radii. Different

radii were used to defîne Merent models of the problem, and the fmal radius - which was

given in Subsection 2.1.5 - was obtained from the most accurate FE model.

Table 2.4: The results from different models of the MRI rnagnet.

Cyündrical mode1 2.46 0.277 14% Not converged 30

Sphericd mode1 2. 19 0.268 4% Converged 16

Discrepancy % +12% +3.4% b

Coi1 around Pole ( B~~~~~ ,, T B,,, T emf Newton-Raphson

Cyündrical mode1 0.90 1 0.397 6.5% Converged 12

Spherical mode1 0.835 0.4 19 2.5% Converged 12

Discrepancy 8 +8% -5%

One of the purposes of this thesis is to snidy the effects that modifkations of the solution grid

have on the solution accuracy. For the purpost of this thesis, a controllcd combination of the

extrusion and fiee-meshing techniques was used to develop various grids for the problem.

The modification of each soiution grid involves many parameters. and because each

parameter is intimately nlatcd to severai other parameters, it is very difficult to separate the

effect Lat each individual parameter has on the overall solution accuracy.

Chapta 2. Accuracy and Sclection of a Finite Elcmcnt Modtl 37

model. The problem domain was then discretized and solved with the mM. The solution for

a given grid was examined and a small subset of the disaetbation parameters was changed.

The solution results for the new k d e l were then studied to unde&tand the effect of the

parameter changes. This step-by-step approach yielded a sequence of darerent models and

solutions, which can k judged by the accuracy criteria The knowledge thus acquired was

then used to define a new set of parameten and to create the next version or generation of the

magnet model. The process was continued until a model obtained that yielded a solution of

acceptable accuracy . Six different models from sequential versions of the MFü magnet model were selected in

order to briefly describe the process of grid retinemnt. These models are shown in Figures

2.5.1 to 2.5.6, inclusive. In order to simplify matters, a 2-D projection of each model is used.

As noted in the figures, the extenor region of each model is a sphencal envelope, and infilnite

ekments an used to npnsent far field effects. In each case, the current excitation is 60 kA-t.

The non-bar FEM was found to converge for all models. A Mted number of parameters

and their effects are discussed in the following:

(1) Fimire 2.5.1 shows a rectangular 2 0 projcction rnodel where 270 18 ekments have been

used. Ampere's law e m r e ,,,,,,f, as specifïed by (2.16), was found to be 13% for this model.

The mid-path of the magnet. as defined in Figure 2.2, was used in order to evaluate e ~ f .

This mode1 has an unacceptably large Ampere's law error and thus provides a starting point

for the grid refmement process.

(2) F&re 2.5.2 shows a circular 2-D projcction model where 47833 elements have been

used. where a rectangular model was used previously. Note also the extra layer around the

coi1 and the triangular ekmtnts in the outer area The outer W circular line is divided into n ..t

=10 divisions. The error e ,f of this mode1 was calculated to be 1 1%.

(3) Fimire 2.5.3 shows a 2-D projection mode! where 31892 elemnts (half of the previous

model) have k e n used In this case, quadrilateral elements have been used in the outer area,

and the core region has been sub-divided into diaerent areas. The outer !A circular h e

divisions have bcen increased to n ,r =12. As a conscquence of these modifications. the error

has been reduced to e ,,,,,!g= 8%.

Chapta 2. Accuracv and Sdcction of a Finite Elcmcnt Modcl 38

(4) Bmre 2.5.4 shows a 2-D projection model where 54941 ekmnts have ken used. In this

model, triangular elernents have been used in the outer area, and there is a better

representation of the yoke pojection area Fmally, the outer !A circ& line dihsions have

been fbrther increased to n ,.[ =14. As a consequence of these modifications, the error has

been hirther nduced to e -f= 6%.

(5) Fimire 2.5.5 shows a 2-D projection model where 61 188 ekments have been used. The 3-

D model for this particular case wns previously shown in Figures 2.3 and 2.4. Note the

quadrilateral ekments in the outer area. together with the division of the core area and its fiee

meshing. The outer % circular iine divisions have ken increased again. this time to n = 18,

and the error has continued to decrease. now to the level e mf= 3.8%.

6) Fimire 2.5.6 shows a 2-D projection model where 63650 ekments have been used. The

discretization of the core area has been refmed and the outer '/r circular line divisions has

been kept constant at n 01 =18. In this final case. Ampere's law error dong the mid-pth of

the rnagnet was found to be e mf= 1.5%.

It is interesting that the better FE models cf the MM rnagnet include the better

representation and discretization of the surrounding fm-space. The complete advantage of

hexahedral e k m t s (made from quacirilateral 2-D elements) over the prism and tetrahedral

ekments (made nom triangular 2-D elernents) can also be concluded.

2.4.4 Element Type

The accuracy of the FE results is a&cted by the order and type of the elemnts used in the

discntization of the probkm domain. Increasing the order of the elernent type - either

unifody over the problem domain or selectively at specinc regions - is not used in this

thesis. The type of elemnt (tetrahedral or hexahedrai) that is used in developing the FE

mode1 has a signincant inmience on the accuracy of the magnetic k l d data Afier the FE

problem is solved, the magnetic field data cm k derived ushg the differential relation

between the magnetic vector potential and the magnetic flux density.

nie first order tetrahedral elemnt has a potentiai shape function of the form

'ax+by+cz+d'. Thus. the extracted components of the magnetic field - e.g. the curl of the

C h a ~ t u 2. Accuracy and Selcction of a Finite Elcmcnt Mode1 39

magnetic vector potential - are constant over each element. Consequently, when developing a

FE grid using tetrahedral elements, each element must k sufficiently small to just@ the

Unplicit assumption that the magnetic field is constant across that eleknt. ~ h i s is the main

drawback of using the tetrahedral elements, especially where the domain is srnall and the

field is changing rapidly. In the modeiing of the MRI magnet, the tetrahedral elements are

not employed in the interior region of the FE model. They are only used in one of the fiee-

space transitional volumes to accommodate the geomtry - which was show Figure 2.4.3.

In contrast, the fùst order hexahedral or brick element has a potential shape hinction of

fom 'axyt+bxy+cyz+da+ex+fy+gz+h' - eight nodes and thus eight coefficients. Thus the

nsulting magnetic field wiii have a linear variation within each element. Hexahedrai

elements are also easy to visualize and convenient to mate by extruding mthods. They are

p M y used in this thesis. The prism elements are a special form of the brick ekments and

are generdy avoided in this thesis. They are only employed to accommodate the geometry.

2.4.5 Material Chaiacteristic

A FE nonlinear solver program may corne very cbse to the solution of a problexn, iterate

around that solution, and ultirnately fail to converge. Further investigation of the non-

converged solution may even show partly satisfactory results. This rnay be due to the least

suspicious factor of the entire Fie Element Method: a poor definition of the non-linear B-H

characteristic. The basic issue here is that the local reluctivity of the material is required in

order to cakulate-stiffness matrix [q for each element. In addition to the local reluctivity,

the slope of the material reluctivity with respect to the square of rnagnetic flux density is also

rquired in this case to calculate each element's Hessian matrix [M. Therefore, to insure the

uniqueness of the FE result, the reluctivity should be (a) single valued, and (b) monotonic.

The visual checkhg of either the B-H or the V - B ~ characteristic curve rnay be satisfactory

to confirm the single-valwd relation, but it is not enough to confimi the monotonie change of

the nluctivity The v-B* characteristic is usually ârawn and used to c o n h the monotonic

change of the reluctivity v with respect to B'.

Chapta 2. Accuracy and Sclcction of a Finite Elcment Mode1 40

For the purpose of this study, Cl006 steel is used for the MRI magnet core. The v-B?

characteristic for this material is shown in Figure 8-2 of Appendix B.

2.4.6 Current Excitation

in the case of the MRI rnagnet application, the FEM solver may approach but not actually

converge to a fuiai solution. Another factor (other than material characteristic) that has a

strong impact on this process is the method that is used to apply the current or load to the

relevant portions of the problem domain. Usuaily. the total excitation cumnt is not appiied in

one step when nonhear materials exist in the problem domain. Rather, the current or load

should be applied in a step-by-step manner, with the solver king permitted to run for several

iterations befon applying extra loads. The immediate advantage of such a loading approach

is that the elementd non-linear material data can be obtained and updated in each step. This

in tum provides good initial material data for further loading of the problem domain.

In a highly saturated problem nich as the MFü mgnet, an insufficient number of loading

steps wiii result in the loss of material data fiom the no-load to the hiii-load state. As a

result, during the solver iterations in the Mi-load state. the convergence criteria cannot be

satisfied. Although this problem is not easily detectable. it is easiiy resolved by increasing the

number of steps fiom no-load conditions to the full-load excitation of the FE model.

Chapter 2 has detailed the application of the Fite Element Method, in general tenns, and to

the MEü rmgnet problem. The discussion has focussed on the question of how best to

evaluate the accuracy of the solutions, and on the question of how to modidy the FE grid such

that the best solution can be obtained. The application of the FE modeling to the MRI magmt

problem was used to focus the dimission.

Chapter 3 wül employ approaches diPcussed m this chapter for study of the MRI magnet

probkm It wül ver@ Ampere's law error as a diable global accuracy measure. and

furthcrmon, it wül examine the selection criteria that were proposai in this chapter.

Chapta 2. Accuracy and Setecticm of a Finite Element Mode1 41

Figure 2.1 : The If4 mode1 of the MRI magnet where the coi1 is around the core.

Figure 2.2: The y = O siice of the MRI magnet showing different paths which are used to calculate Ampere's law e m .

Chaptcc 2. Accuracy and Selection of a Fite Elment Mode1 43

Figure 2.3: The steps in defhing and discretking the MRImagnei dunain: 1 : 2-D projection d e l , Ri=0.900 m. 2: Extrusion of the core, fke-space, con. 3: Extrusion of the COR to yoke, air. 4: Extrusion of air-gap and yoke. 5: Intenor cyiindricai region or volume, 6: The MRI mgnet inside the cylindncal

Hi=0.700 m. volume.

Chapter 2. Accuracy and Sclection of a Finite Elemmt Mode1 43

Figure 2.4: The steps in d e h g and discreiizing the MRI magnet domain: 1 : Transitional envelope region Rmm= 1.6 m. 2: Hexahedral ele- for side volums. 3: Tetrahedral elements for fiont volume, 4: Exterior envekpe volumc(s), L ' 2 . 6 m. 5: Hexahedral elements for al1 volumes. 6: The finai mode1 of the magnet wPhin two

symmtry plane limits.

Chapter 2. Accuracy and Selection of a Fiite Elemait Modcl 44

Figure 2.5: Six 2-D projection models h m dWerent generations of the MRI rnagnet model: 1: RectanguQr interior region, 13% 2: n c-l 40, e m m p 1 1 % . 3: n,r=12,emmf=8%. 4: n '14, emmf ~ 6 % . 5: nCp18,emmf=3.8%. 6: 1 ~ ~ - ~ 4 8 , e , , , , ~ ~ 1 . 5 % .

Chapter 3

Study of the MRI Magnet with 3D FEM

The Finite Ekmnt Method and various aspects of its application to the MRX magnet probkm

were detded in Chapter 2. It was proposed that the integral fonn of Arnpere's iaw could

provide a usehl quantitative masure when assessing the quality of an individual FE solution

for the MRI magnet problem This accuracy masure could thus guide the process of

identifjhg the best possible solution to the problem.

The present chapter undenakes a systernatic study in which the use of Arnpere's law as

an error rneasure is examimd for a specific magnet geomtry. Its validity as a global e m r

measure is examined and the conditions for its successfùl application are identified. It is

show that the use of Ampere's circuital law as an error measure allows one to ntiably

identify a F i t e Ekment model, arnong several, that offen optimal accuracy. This can k

termed the modcl selection process. Similady, the strategy used in this process cm be t e m d

the model selection strategy. The applicabüity of the selection strategy is verikd in the

search for the most accurate model of the MRI magnet. In addition to using Ampcre's

circuital law as an error measun, it is also demonstrated that the magmtic field data and the

enetgy content of the FE model can k niiably u r d as selection cntena.

In order to start the proccss of selecting an optimal model, it is first necessary to identm

a starting or initial rnodel of the magnet. The initial model is simply a model that provides

reasonabk results. The grid for this model is rnoditied in the search to improve the quality of

the solution. Means of efficiently identifjhg the initial model are discussed in Section 3.1.

The process of rkcting the best FE rnodel of the MRï magnet, h m among many such

modets. is detaüed in Section 3.2. A basic isme that is f n d m this process is the need to

Chaptcr 3. Study of the MRI Magnct witb 3D FEM 46

potentidy formulate and solve many nonhear models of the magnet. This can k

prohibitively expensive, certainly in tenns of manpower. To avoid this difficulty, an

equivalent ünear rnodel of the maiet is used that substantiaiiy improves the efficiency of the

selection process. It is show that the accuracy of any given nonlimar FE mode1 is very

closely matched by an equivalent lïnear model that uses an appropriately chosen constant

permability. In a second study, the path dependency of Ampere's law error is investigated.

It is concluded that the average and standard deviation of Ampere's law enors dong different

paths provide a diable global accuracy criterion. Thirdly, in this section, the use of magnetic

field data and energy content are examined as alternative selection criteria. The agreement

between two selection criteria, (a) minimum of Ampere's Iaw error, and (b) minimum of

energy stored in the model, is then confmned.

Having rlected a made1 that provided the most accurate solution for MRI magnet in

Section 3.2, the rnodel is then used in Section 3.3 to examine the magnet performance in

detaii. The contour plots of the magnetic flux density are shown for different regions of the

rnagmt. The performance of the magnet is also analyzed for dflerent levels of the coil

current. The effects of the coü mode1 are then studied. It is shown that an equivaknt currcnt

sheet coil can effectively simulate the coi1 of the MRI magnet.

In Section 3.3, it is shown that the existing design of the MRI magnet requins an

extraordinarüy high excitation in order to produce a relatively modest air-gap magnetic flux

density. Therefore, an alternative to the original design is introduced and analyzed in Section

3.4. The rnagnet geomtry is preserwd but the coil is split and thcn positioned around each

pole piece, as opposed to king around the core in the original design. In the case of thû

modified design, no experimcntal data was availabk for model caiiimtion purposes.

Therefore, the pceviousîy d e s c n i selection process was repeated in order to idenUfy a final

mode1 that is kiieved to yield accurate mults. This rnodel was then used to undenake a

detaiied analysis of the proposed design. The original and the modified design are compared,

particularly in terms of magnetic flux density magnitude and the emrgy content

Chapter 3. Study of the MRI Magna with 3D FEU 47

3.1 Identification of an Initùzl Mode1

The identification of a good star&, or initiai. rnodel for the proMe& at hand k one of the

most important destones in the FEM process. The starting, or initial, model is sirnply a

model that yields a solution having acceptable, but not necessanly optimal, accuracy. This

model confirms that (a) the problem and the problem domain are correctly defined. and (b)

the problem domain is properly diretized. The initial model thus serves as the foundation in

the selection search of the final and most accurate model of the problem The development of

the initial model for the MRI rnagnet can k divided into two steps, namely, to initiaily

model the coü alone, and then to iylude the nonlincar core. but with a reduced coil

excitation. These approaches were initidy discussed in Subsections 2.2.1 to 2.2.3. Theù

applications to the MN magnet problem, together with the results thus obtained, are given in

the following su bsections.

3.1.1 Free-Space Material Method

Using good engineering judgment, a discnte model of the MRI magnet probkm, including

the coil, the con and the surroundYig k-space ngion is developed. However, the magnetic

materiai of the discntized problem domain is replaced by fke-space, but the coii excitation

and other features of the FE model remani unchanged. Thus, the problem is that of a coil

Iocated in fkee-space, and the FE solution is used to obtain the total energy of the system.

This can k compared to values that are found in the üteratun for an empty solenoidal coil

[17]. If good agreemnt is obtaimd between the published and the FE resuls, the model wül

k furthet tested by nsetting the pioperty parameters of the con region to those of a

nonlincar magne tic material.

As was noted above, the energy stored in an empty solenoidal coii can k easily

computed using known expressions. Fust, the inductance of a cyiindrical current sheet of N

tums is @en by

(3- 1)

Cha~ter 3. Studv of thc MRI Mamet with 3D FEM 48

where the correction factor K is Nagaoka's constant [17]. This constant is tabulated in the

literature [17] and accounts for the coii end effects. The stond energy of the coii carrying a

current I is then given by

Substituthg NI with the total excitation of 120 kA-t and the area Acdr with tc~-'caiI yields the

energy

Severai difkrent models of the MRI magnet have been solved with this approach of

replacing the c o n cegion by fne-space. The results for five such models are summarized in

Table 3.1. This table compare the energy predicted by the FE models to the values given by

(3.3). The tabie also includes values for e -J , which represents the mid-path Ampere's

kw enor, as was defmed by (2.3). Arnpen's law errors were obtained by performing a fùli

nonlimar solution for the model in question and have ken included for the purpose of

cornparison. It should be noted that Ampere's law error for the iinear models (Le. the models

with the material properties of the core region set to those of fke-space) are less than 0.5%

and thus are not rnentioned.

Table 3.1 : Five 3D models of the problem domain, solved by FEM.

Mode1 1 Mode1 2 Mode1 4 Mode1 5

The first and second modcis present an acceptable level of e W . , fiom correspondhg

nonlinear analyses (a single digit enor); however, the energy diffennces between EGniver and

EW are not acceptabk, and the mdels are thus rejected. Model #l has a grid structure

sllnüar to the one that was show in Figure 2.5.1. The rectangular 2D projection mode1 is the

main weakness of this moâel. A circular 2D projection model and more ekmnts wen used

Chapta 3. Study of the MRI Magnct with 3D FEM 49

in the grid of Model #2. Model #3 was obtained by (a) better representation of the exterior

frre-space and (b) by refinhg the elemnts in the magnet core where the rnagnetic flux

density is between 1 and 1.7 T - the h e e point of the magnetic material B-H curve.

A mode1 with energy discrepmcy of less than 1% is considered acceptable. Therefore.

the third model is rlected as the candidate model for further testing. The 2D projection

modeis of the 3d, 4', and 5" modcls were given in Figures 2.5.3, 2.5.4 and 2.5.6.

respectively. The fourth and tifth rnodels are oniy given here to show the validity of the free-

space approach, and Section 3.2 will hirther inspect their perf<nmance.

3.1.2 Reduced Excitation Method: Loading Steps

The next step in the process of determinhg a good staning, or initiai, model is to reset the

material properties in the core region to those of a nonîinear magnetic material. Since the

problem is now nonhear, it is necessary to ensure that the Newton-Raphson iteration for a

given probkm (Le. model) converges. It is known that the Newton-Raphson mthod

converges easiiy to a solution when the initial values of the degrees of fieedom are close to

the ha1 solutions. However, when the hiil magmt excitation cumnt is applied in one step,

the initial and final values of the solution variables can k sufficiently separated that the

Newton-Raphson cannot converge to a solution. In order to avoid this problem, the cumnt

excitation is applied to the nonlinear mdel of the MRI magnet in relatively fine steps. The

current increments are kept smali enough to ensure the convergence of the Newton-Raphson

within the first iteration of each step. Should the method fail to converge in the second

iteration at the full loaâ condition, the cumnt loading rheme has to be modified to provide

smaller cumnt increment. For the purpose of this study, a total of 15 loading steps were used

in order to estabüsh the hiil Ioad excitation of 60 ICA-t when energizing the nonümar magnet

model-

Chapta 3. S tudy of the MRï Magna with 3D FEM 50

3.1.3 The Starthg, or Iniliol, Model

Of the t h e models that were exakined in Table 3. i, Model #3 was chosen as ' pviding a

good bais for further refmemnt. Several factors were considered when making this choice.

In particular:

@ In the fusf stage of the development, when the core region materid properties were set to

those of fne-space, Model #3 provided an acceptable estimate of the stored magnetic energy.

When the rnagnetic core properties were nhtroduced. the solution of Model #3 had a

nasonable Arnpere's law emr.

a There was good agreement with masured data. The magnetic flux density at the

perimeter of the air-gap region of interest is calculated as B .,, (2 = O, r = R ,,,,,+- 2) = 0.273 T. The average rnagnetic flux density at the air-gap center plane (z = O, r 6 R air-g,) is

computed as B ai~gap = 0.26 T . The masured value of the same quantity is reported

[Appendix A] as B M ~ ~ ~ ~ ~ < u ~ . ~ ~ ~ = 0.27 T m the air-gap region of interest at (z = 0, r 5 R .ir-,,+

2). The shjrnming coüs and shaping of the pole faces are conventiondy used to homogenize

the magnetic field within the required Limits in the volume of interest - which are out of the

rope of this thesis. Thenfore, it is not possible to numerically achieve the exact

homogeneous measured magnetic field. However, the numerical result should be close to the

measured data in order to suggest the possbility of the shimming and calibration procedure.

It is possible to defuie the volume of the interest in the FE model of the problem in order

to s h p w the cornparison of the solutions with the published data The initial models of the

MRI magnet had such geomctric identification in the air-gap of the magnet. However, such

extra modehg requhmnts have nstricted the mode1 discretization and reduced the

efficiency of the FEM. Thenfore, the volum of interest has not k e n identifred in pnsented

models of the MRI magnet.

Having systematicdy chosen a starting, or initial, model. the next step in the process is

to refine that mode1 m ordcr to improve the quality of the solution. This refinement process

yields several models of the probkm. The process of selecting the most accurate model of the

magnet, among many models, wül be discussed in the next section.

Chapttr 3. Smdy of the MRI Magna with 3D FEM 5 1

3.2 Selection of an Optimal Mode1

Once an initial FE model has beenidentihd. modifications are introdked in order to obtain

the most accurate rnodel of the problem The two main areas where modifications can be

introduced are (a) the problem dornain definition, and (b) the probkm dornain discretization.

The rnoditication process includes (1) changing a parameter or a set of paramters, (2)

solving the model, (3) investigating the quality of the results, and (4) using the results to

guide the introduction of hirther modifications. The search for the k s t model obviously ends

when no further improvemnt is achieved by rnodel modification.

The two key issues in this process are the accuracy criteria by which individual solutions

are judged, and the selection criteria by which the best solution among a group of solutions

cm be identified. In addition to specificaliy iden twg the k s t possible solution, this

section also studies, appües, and vetifes a global accuracy criterion: Arnpere's law error,

and a global selection criterion: the energy content of the model. As opposed to local or

semi-global criteria, a global selection criterion indicates a ktter mode1 of a problem with a

unique, unambiguous and predetermined masure; e.g. the rnost accurate model has the

minimum of the total energy stored in the system This section also investigates the use of

semi-global selection criteria; e.g. the regional average of magnetic k l d data, and the

magnetic energy s tond in a region of the model.

Subsection 3.2.1 introduces a novel rnethod. based on an equivaknt linear model, that

can conveniently and efficiently be used to obtain hproved models for a given problem. A

set of models is thus generated using this approach. A final subset of mo&ls is then sekcted

for hirther study. In Subsections 3.2.2 to 3.2.5, this subset is used to ülustrate the application

of different accuracy critena and to select the best mode1 representing the problern.

The use of Ampm's law as an error rneasure is the subject of Subsection 3.2.2. The

path dependency of this quality rneasure is examined in detail and it is show that an accurate

mode1 can be identified,

The possibility of using the magnetic field data as a model selection criterion is discussed

in Subsection 3.2.3. Düferent locations of the MRI magnet are selected, and the average

Chapta 3. Study of the MM Magnct with 3D FEM 52

magnetic flux densities for different models are examined. It is shown that the application of

the average of magnetic field data as selection criteria are limited and should k used in

combinat ion with other selection c&eria

Global and semi-global quaüty measures based on the energy content of different parts of

the mode1 ais0 provide a mcans of comparing the quality of a solution. Their applications are

compared to Ampere's law emr approach in Subsection 3.2.4. The selection of the optimum

model of the MRI magnet is then concluded.

3.2.1 An Equivalent Linear Mode1

The process of identifyuig a good, accurate FE mode1 for a noniinear 3D problem such as

the one king considered in this thesis, is very t h e consurring in ternis of both human and

computer resources'. The difficulty arises fiom two characteristics of the MRI magnet

pro blem:

a) the three dirnensionality of the model. which cannot k simplified, and

b) the non-ünearity and high saturation of the magnetic material in the rnagnet core.

It is therefore important to identify efficient means of developing and testing noniinear FE

models. The objective in this subsection is to introduce and develop one such method.

A scheme is proposed to approximate the nonlinear material of the magnet using a

properly selected iinear material. It is postulated that the solutions of the hear and noniinear

models may follow the same quality patterns and may yield similar error masures.

Therefore, the accuracy of a given noniinear model can be estimated by fust solving the

equivaknt ünear probkm, naturdy on the same grid. This scheme to accelerate the selection

study of the MRI magnet is descriid in the paragraphs that foilow.

The first probkm that must k nsolved is to identify a suitable equivaknt permabüity.

In order to do so, four different models (i.e. solution p*&) of the MRI magnet were chosen.

Fun nonlincar solutions were available for each of the solution grids. The four models were

' The CPü time on an Ultra Sun Microsystcms computa was 8 h o m for solving a nonlinear model of the MRI magnet whm (a) 61188 elemcnts, (b) 117345 &grtes of frcodom, and (c) 16 itcrations wcrt used. The data arc ftm tbc most accurate modcl of the MRI magrnt that has k e n introduccd in Saction 2.1 and Section 3.3.

Chapta 3. Study of the MRI Mamer wiîh 3D FEM 53

chosen such that the solutions covered a reasonable range of Ampere's law error estimates.

Mon specificaUy, the e m r estimmtos ranged from a high of 8% to alow of 4%.

The next step in the process w& to re-solve each of the four models for constant relative

permeability values that ranged from 50 to 2000. Ampere's law error estimates for al1 such

solutions are shown in Figure 3.1 (page 59 at the end of this section) as a function of relative

permeability. Clearly, the trend shown in Figure 3.1 is for each model to converge to a

constant Ampen's law error as the relative permeabîüty increases.

A more interesting feature becomes apparent when the error for each of the nonlinear

rnodels is plotted on the same set of curves. The four horizontal lines that are shown in the

figure indicate these error kvels. It will be noted that linear models having a relative

pemabiiity in the range 100 to 175 yield error estirnates that are comparable to the fùlly

nonlimar cases. It is thenfore postulated that iinear models of the MN magnet, with relative

permeability values in this range, can be used for the purpose of model refmement when

seeking to improve overall solution accuracy. The advantage of this approach is clearly that

the effect of a particular rehemnt to the model is king assessed using a one-step iinear

solution, For the purposes of this thesis, a constant relative permeabüity of 100 was chosen

for use in the ecpivalent ünear models.

In order to illustrate that the equivalent linear models do, in fact, provide a very good

estimate of Ampere's law error, and thus provide a simple means of reducing the

computational effort when refitning a solution grid, a study was undertaken. A sequence of

equivalent linear models, each representing a particular refuiement of the solution @d, were

solved and Ampere's law erroa were computed. The full nonlinear solutions were also

cornputed for a subset of these models and, again, Arnpen's law errors were determined. In

all cases, Ampere's law e m r was computed dong the mid-path of the rnagnet.

Figure 3.2 shows the resulting vaiues of the error estimate for each of the 43 models in

the sequence. The results for the linear models are shown as white ban, while the black bars

arc the error estimates from the fidl nonlinear solutions. There is a very close correspondence

between the iinear and nonliaear mdels. Thus, it cm be concluded that the equivaluit linear

approach can indeed be used with advantage when refining the solution grid for problems

such as the MRI magnet.

Cha~ta 3. Studv of the MRI Maenct with 3D FEM 54

3.2.2 Path Dependency of Ampen9s Law Error

The integral fom of Arnpen's law: - emqf . f =f ,H*dl -z#ncfbserf (3.4)

has been proposed as a global accuracy criterion When this criterion was used in Section 3.1,

it was evaluated dong the mid-path of the magnet. Note that the rnid-path was previously

defined in Figure 2.2. In using this definition, it was presumed that the rnid-path error

provided a good estimate of the global accuracy of the model. However. this assumption was

not verified in detail. Thenfore. the purpose of the present subsection is to examine the path

dependency of Ampere's law error and to determine whether the mid-path assumption is

valid.

From aii the models that were developed for the M'RI magnet study, a subset of 20

models was selected for discussion in this and the foilowing subsections. The first model of

this set is the initial model of the rnagnet that was identifiecl in Subsection 3.1.3 of this thesis.

For each model, Arnpere's law error was calculated dong different paths that were shown in

Figure 2.2. The models were then sorted with respect to the average of the errors. The mici-

path error, the maximum enor considering aii paths, the minimum error also considering ail

paths, and the average error were plotted for each rnodel, and the results are shown in Figure

3.3-a. In this figure, the mid-path, maximum and minimum erron are denoted by symbols

whüe the average enor is denoted by a bar. The standard deviations of the mode1 erroa were

also deterrnined and the results are shown in Figure 3.3-b. Note that the 2D projection

models of Models 1. 10, 15, 20 were previously given in Figures 2.5.3, 2.5.4, 2-55. and

2.5.6, respectively.

On the basû of the results shown in Figure 3.3. it can be concluded that:

a) The mid-path Amperc's law error is a good accuracy measun and can be used to judge

and compare di&rent modeis of the rnagnet. In ali but two cases (Models 13 and 20), the

mid-path error is very close to the average of errors taken over ali paths. However, the

cesulis for Modeis 13 and 20 confïrm that modtl modification can minim&e the mid-path

Chapta 3. Study of the MRI Magnet with 3D FEM 55

Arnpen's law error. Therefore. the mid-path error should k used judiciously and in

conjunction with other accuracy masures.

The average of Ampere's law-errors along dinerent paths is a better global accuracy

masure. However, the results for Models 13 and 20 show that a rnodel with a low

average error may have a wide range of error. Therefon, this masure does not provide

the best criterion for selection of the most accurate model, and should not be used as the

sole quality measure. Models 10 to 20 have the same average error of 4%.

The standard deviation of Ampere's law errors along different paths can be used to

overcome the uncertainty of the best rnodel selection. Figure 3.3-b shows the lowest

standard deviation of an 0.5 for Model #l 5, which has an average error of 4%.

On the bais of the forgoing discussion, it can be concluded that. Model #15 is the fvst

candidate for the most accurate rnodel.

3.23 Magnetic Field as Selection Criteria

The rnagnetic field solution was used in Subsection 3.1.3 as an occuracy criterion in order to

verify the comctness of the initial model. The appücation of the magnetic field solution as a

cnterion in selecting the best model among a set of models is examined in this subsection.

The main problem with using the magnetic field as a selection cnterion is that it is a local

quaüty of the mdeL Using local quality masures to globaiiy compare difTerent models of a

probkm is unproductive and even misleading. Rather than use point values of the rnagnetic

field, it is suggested that the average of the magnetic field be taken over a region in order to

obtain a semi-global masure of solution quality.

The fist magnetic field selection criterion is the average magnetic flux density dong the

air-gap centerüne. Its normaiized value for the seiected set of 20 models is shown in Figure

3.4-a This Line integral selection criterion changes less than 2% nom the f ~ s t to the IO&

model; however, it osciuates within a 0.5% range afterward. Therefore, the line integral or

average of the magnetic field solution is not very useful as a selection criterion.

The second sebaion criterion to te examined is the average magnetic flux density at the

con center plane at z = O, a s u r f a integral quaüty measure. It provides the magnet

Chapts 3. Study of the MRI Magna wiih 3D FEM 56

saturation information and it is show in Figure 3.4-b. The model modification from the

initial model to the 10' model increases such a m u r e by kss than 1%. Models 10 to 20

introduce an almost identical average magnetic flux density of 2.19 T across the core.

The ihird rnagnetic field selection criterion is the average rnagnetic flux density across

the air-gap center plane. The data for this criterion are shown in Figure 3.4-c. This surface

integral selection criterion changes around 2% from the fust to the 1 0 ~ model. Models 10 to

20 introduce an almost identicai average rnagnetic flux density of 0.266 T across the air-gap.

This is very close to the measured value of 0.27 T in the air-gap.

These tests indicate that the averages of magnetic flux density across the core and air-gap

are consistent ktween Models 10 to 20. Therefon. (a) the optimum model is one of these

models, and (b) the average magnetic field cm k kneficiaily used as a selection criterion in

conjunction with O t her global accuracy measures.

The magnetic flux density across the air-gap is reported as 0.27 T and computed as 0.266

T from models 10 to 20. Cornparison of Figure 3.4-c (the average B across the air-gap; a

semi-global sekction criterion) and Figure 3.4-d (the average of Arnpere's Iaw error in

different paths; a global accuracy criterion) c o n f m that Arnpere's law error provides a

mon nliabk predictor of the solution's global accuracy. Thus, Mode1 #15 is still the best

candidate for the most accurate model.

3.24 Magnetic Ewrgy as Selection Criterla

The total rnagnetic energy stored in a mode1 is one c o m n global selection criterion that

bas been used for electrornagnetic problems. The selection direction is also kno wn to ward the

minimum energy; the model with the lowest stond energy is the k s t model among many

models. However, it should be noted that the stored energy does not provide a direct measure

of the solution accuracy. The stored energy in a group of elements can also be used as a

semi-globo1 selection criterion, but the selection direction is not clear; the model with the

lower energy in a group of elenvnts is not necessarily the better mode1 for the probkm as a

whole. This subation mvestigatcs the use of magnetic energy as selection criteria (both

global and semi-globai) on the MRI magnet probiem

Chaptu 3. Study of the M . Mamet with 3D FEM 57

Figure 3.5 shows the energy content of the MRI magnet, in addition to the average of

Ampere's law e m r and the magnetic field at the air-gap. The same set of 20 models is

chosen for this study. The energy values are fiom '/. mode1 of the MRI mapet: The values

are normaiized with respect to the comsponding values of the initial mode].

Figure 3.5-a shows the magnetic energy stond in the magnetic components of the

rnagnet, namely the con, the yoke, and the pole. This energy increases by 10% fiom the fvst

model to the 10' model. From Model 10 to 19, the magnetic material energy stays at an

almost constant value of 375 J. Figure 3.5-b shows the magnetic energy stored in the au-gap

of the MRI magnet. It increases by 2% fiom the first mode1 to the 10' model. From Model

10 to 19. the air-gap energy remains at 5 12 J kvel.

These cases indicate that better models of the magnet do not necessarüy have a lower

ngional magnetic energy. In other words, minimizing a regional magnetic energy does not

produce the better and best model of a problem It is concluded that the direction of a serni-

global selection criterion (magnetic energy stored in a region of the FE model) is not

predetermined (toward the minimum or maximum) and better models may have lower or

higher values of a cntcnon. The direction of these criteria should be deterrnined based on

other global accuracy or global selection criteria for a given problem

Figure 3.5-c shows the total magnetic energy stored in the MRI magnet model. This

hcludes the energy stored in the magnetic material, the air-gap. and the surrounding fne-

space volume. The total energy decreases continuously fiom 4621 1 for the tirst model to

4477 J for the 10' model. It stays at a minimum value of 4463 J afterwards. The minimum of

the total stored energy in a model is associated with the most accurate model of a problem

Thus. any of Models 10 to 20 can be the optimum model of the MRI magnet.

Note the irregular solutions of Model U20. The total energy is similar to the other models,

but the energy in the air-gap and magnetic components khave erraticaily. The Ampeds law

errors almg M e m i t paths were also found to k erratic. Thus, the selection process focuses

on Models 10 to 19. This also confirms that more than one criterion should be used to select

the ma wurate model of a probkm.

The pattern and behavior of the energy content of the mode1 (Figure 3.5-c), the average

Amperc's law error (Figure 3.5-d), and the average magnetic fîux density across the air-gap

Chapta 3. Study of the MRI Magna wiOi 3D FEM 58

(Figure 3.5-e), are very simiiar. This c o n f i i the validity of using the average of Ampen's

law error as a global accunicy criferion, and the energy content of the model as a global

seLecrion crire~on for the FE soluti&s of the problem Monover, the kgnetic e k g y stond

in a region of the FE model is also concluded as a useful and practical selection criterion. AU

of these measures indicate that any of Models IO to 19 can be the optimum model of the

magnet. Thus, the final selection would be based on the statistical masures of Ampere's law

error.

3.2.5 The Selected Optimum Model

This section started with an initial correct, but not optimum, FE model of the MFü magnet.

Better models were efficiently sought using a novel equivalent linear rnodeling approach.

This study then introduced 20 different FE models of the problem fiom which the most

accurate or the optimum model was to be sebcted. A considerable improvement was shown

from Model#l to Model #IO. In the cases of Models 10 to 19, however, modifications to the

magnet model did not enhance the solutions. Difkrent quantities of interest - such as the

average magnetic field actoss the core and air-gap of the rnagnet, and the stored magnetic

energy in the rnagnetic part, air-gap and whok rnodel of the magnet - were found to be very

similar for this set of rnodels. Even the average of the Ampere's law errors dong different

paths was almost stable at a 4% level for these models. It is thus concluded that further rnodel

modification cannot signifcantly improve the quality of the solutions either meaningfidiy or

within a nasonable resource costs. Therefore. the optimum model of the MRI magnet is any

of Models 10 to 19.

Whüc al1 the global and semi-global selection criteria and global accuracy criteria point

to the same range of Models 10 to 19, the final model sekction is based on the standard

deviation of Ampere's law errors dong daferent paths for each model. This is a statistical

measure of a global accuracy criterion and thus a soüd selection criterion. It c m dEerentiate

between mdels that are very smiüar in ternis of other accuracy and selection criteria. Model

t15 was thus sclected as the most accurate or the optimum FE mode1 of the M . magnet.

Chaptcr 3. Study of the MRI Magnct with 3D FEM 59

Figure 3.1 : Ampere's law error for four models of the MRI magnet. The hem analyses utüize dEerent values of p, for the magnet material.

1 1 1 1

9 1 I I 1

1 - HooHow bars: Linear analyses 1

-+ The initial model with 8% emr +. A 3-0 mode1 with 6% error + A 3-0 mode1 witn 23% enor

- + A 3-0 mode1 wiai 1.5% enor * Enor from non-linear sîudies of the mode1

. . . . . . . . . . . . . . . . . . .: . . . . . . . . . . . . . . . . . . . . . .:. . . . . . . . . . . . . . . . .I - solid ban: s on-~near analyses

I

6

20 30 3-0 Model No.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . .-

. . . . . . . . . . . . . . . . . . . . . . . * . . . . . . . . . . . . . . . . . . . . . . . . -

. . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. ........ " . . . . . . . . . . . . . . . . . * . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . . . . . . . . - ,-.. . . . . . . . . .:.. .:... . ; . . . . . . . . . . . . . . ......:.......*. . . . . . . . . . . . . . .

m iiI

. . . . . . . . . . . . . . . . . . . * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _ _ _ -

. . . . . . . . . , . . . . . . . . , . . . ................... " . . . . . . ' . . . . . . . . . . . .-

1 l t 1 I I

O 200 400 600 800 1000 1200 1400 1600 1800 2000

ril

Figure 3.2: Amperc's law e m r for di&rcnt modeis of the MRI magnet. The linear analyses utiiize k=100 for the mgnet material.

Chapter 3. Study of the URI Magna wviîh 3D FEM 60

L 8 I

-. Average of the enors + Mid-pathemr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :'. .: v Minimum error

A Maximumenor

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;.. .:. ..;.

. . * ?

1

Figure 3.3: Ampere's k w error analysis for 3D rnodels of the MRI magnet: a) Error along different p t h s is show for each model. b) The standard deviations of Ampere's law errors along different paths.

Chapta 3. Study of the MRI Magna with 3D FEM 6 1

104 I I 1

œ - B acms the alr-gap

2

102- 1 I I 1

3 'iC..n... . . . . . . . ........'.......... . . . . . . . . . . . . : . , . . . . . . . . . . . . . . . . . . . ' 1 1 - Average of AL enon 1

Y

3

Egure 3.4: The a v m p magnetic flux density at diffennt locations of the MRI magnet b) Along the air-gap centerline (at y=û) normalized w.r.t. 0.28877 T. b) Across the core center plane (at z=û) normalized w.r. t. 2.1699 T, C) AC~OSS the air-gap center plane (at &) normalized w.r.t. 0.2602 T, d) The average Ampere's law error for dûfennt models (repeat from Figure 3.3).

The resulu are show for dürerent models and are normalized with respect to the initial mode1 value. The relevant data for Mode1 #t 5 are: L,,,,e,z=o = 2.1913 T, Bacmm aï,,. z d = 0.2658 TI Bals mr.80P* ,?=O = 0.2903 Tl.

[ - B across the cor8 1

Chaptu 3. Snidy of the MRI Magna with 3D FEM 62

Figure 3.5: The energy contents in diffennt parts of the MRI magnet: a) The magnetic part energy Norrna1izedw.r.t. 338J, b) The air-gap energy Nonnalized w.r.t. 500 J, C) The whole mode1 energy Normalized w.r.t. 462 1 JI d) The average Ampett's law error for âif'ferent models (repeat from Figure 3.3). e) The average magnetic flux densi ty across the air-gap (repeat from Figure 3.4).

The results are shown for di&nnt modeis and are normaliped with respect to the initial mdel value. The relevant data for Mode1 #15 are: r Empkw0375 J, Eaipgw=5 12 J, E,O#M JI-

104 1 ï I 1

x 1 - B acmss the air-gap 1

Chabta 3. Studv of the MRI Mannet with 3D FEM 63

3.3 The Optimum Mdel of the MRI Magnet

The selection study concluded that Model #15 was the most accurate of the various models

that were developed for the MRI magnet. The magnet specincations were given in Section

2.1, and the basic FE mode1 was shown in Figures 2.1 to 2.4. The numericd quantities of

interest were summacized in the previous section. Next, the performance analysis of the MM

magnet is presented in this section. The graphical results obtained using Model #15 are

shown in Subsection 3.3.1, and then, two mthods to model the magnet coil are studied in

Subsection 3.3.2. It û shown that an equivalent current sheet coil can simulate the coil

closely. The magnet is also analyzed for different values of the coil current and the results are

given in Subsection 3.3.3.

3.3.1 Magnetic Field Distribution

Results have been presented in previous sections for (a) the average rnagnetic flux density at

duennt locations of the MRI magmt, and (b) the energy content of different parts of the

MRI magnet. Selected magnetic field distributions for the optimum FE mode1 of the MRI

magnet are presented in this subsection. The reai chailenge in presenting these results

graphically is how to best capture the 3D nature of the model. For example, 3D arrow plots

of the rnagnetic field are too packed and hide rather than show information in a usefbl sense.

Therefore, this type of plot has not ken used.

More usefuily, contour plots of the magnetic flux density magnitude can be used to

graphicaiiy show distributions. However, the connectivity and clarity of the resuhs are

reduced when many 2D süces of the rnodel are uscd. On the other hand, complete isotropie

3D plots do not show the data properly and should not be used. Therefore, the approach used

in this subsection is to graph the data on two Unportant slices of the magnet parts (the con,

the yoke, the pole and the air-gap) but within a 3D frame. These slices are selected at the y0

and -onstant planes of the magnet parts. As a result, the rnagnetic field on the extemai

Chaptcr 3. Study of the MRI Magna with 3D FEM 64

surface of these parts is not presented. To compensate for such a shortcoming, Figure 3.6

shows the magnetic flux dcnsity. B. on the extemal surface of the magnet. * .

Figure 3.6 - page 71 at the end of this section - sets the coordinate and the view direction for

the subsequent figures. It shows the magnetic f ~ l d distribution on the extemal surface of the

rnagnet. The core region (in the upper left-hand portion of the figure) is highly saturated

while the pole (in the upper right-hand) remains unsaturated. The yoke experiences a rniid

saturation close to the core, but is not magnetically saturated close to the pole. The rotational

symmetry of the field at the core and pole exteriors. together with the signiticant changes of

the magnetic flux density in Merent parts of the rnagnet, are noteworthy features of the

distribution.

It should also be noted that the results are not the most exact ones at the externai surface

of the magnet. Errors are caused by the material discontuiuity. the lack of boundary condition

enforcemnt and the resultant discontinuity of the tangentid H ktween different media, and

the post-processing of the magnetic vector potentiai data.

The results at two important 2D slices of the con are shown in Figure 3.7. The sernicircle

wue fiame is the connection area of the cote to the yoke. The z = O plane of the core (the

upper right hand sernicircular area) has a nomial magnetic field boundary condition whik the

y = O plane has a tangenthl magnetic field boundary condition. The t h e degrees of fieedom

at each node are thus reduced to two degrees and one degree, correspondingly. Therefore, the

results at these slices are considered to be the most accurate ones. The con is highly

saturated and the magnetic field is almost uniform at a z = c plane. Both are due to the

presence of the coi1 around, dong. and close to the core. The rnagnetic field has rotationai

symmetry with respect to the core axis at (x = -400 mm, y = O).

Chrtpta 3. Study of the M N Magnct with 3D FEM 65

The magnetic nkd distnlution for the lower plane (connection regions to the core and the

pole at z = 300 mm) and syrmnetry plane of the yoke is shown in Figure 3.8. The wüe frame

(at z = 615 mm) shows the airside of the yoke. The symmetry plane y = O has a tangential

boundary condition and thus shows less disruptive or enoneous data. The worst elemnts of

the magnet mode1 are located at (a) the inner connection region of the yoke and the core and

(b) the inner connection region of the yoke and the pole. The rapid changes of the field at

these locations are shown in Figure 3.8. Refining the size of these elements has not improved

the quality of the solutions. The cornparison of Figure 3.6 and Figure 3.8 reveais that the

yoke is a ieaky magnetic flux pipeline corn the highly saturated core to the unsaturated pole.

Pole: - Figure 3.9 shows the magnetic fhix density distribution at the y = O plane and the z = 150 mm

plane of the pole - which is the upper right hand semicircular area Note that the y = O plane

(the rectangular area of the figure) has an imposed tangential boundary condition. This

figure, together with Figure 3.6, shows that the exterior of the pole cylinder is moderately

saturated and has a semi-rotational symmetry. The y = O symrnetry plane shows the rapid

reduction of the rnagnetic flux density fiom the yoke side to the air-gap side of the pole. The

pole lower plane at z = 150 mm has a low rnagnetic flux density level which shows a semi-

rotationai symmetry with respect to the pole axis. This is due to the presence of the hre-

spce on the air-gap side of the pole face.

The magnetic flux density of the ait-gap is show in Figure 3.10. The z = O plane represents

the mid-plane of the air-gap and therefon is a symmtry plane with a nomial magnetic field

boundary condition. This plane is shown as the vertical sernicircle in the figure. Sunilarly,

the horizontal plane in the figure is taken at y = O and is also a symmetry plane, but with a

tangential magnetic field boundary condition. The z and y = O planes in this figure are &O

two siices of the MRI magnet's volume of interest within which there shoukl be a high

Chanter 3. Studv of the MRI Mannet with 3D FEM 66

degree of field uniformity. Recall that this region of field unifodty was specified to be a

concentric cylinder within the air-gap but with half of its radius and length. Its borderlines

are shown in Figure 3.10. It will b; noted in the figure that the magnetic flux density has a

general rotational symmetry with respect to the air-gap axial axis. This symmetry is more

pronounced in the volume of intenst and off-centered outside this volume. The numerical

results at these two ngions, the air-gap and the region of interest, are:

a) The rnagnetic flux density is 0.270 T dong the perimeter of the volume of interest and

reaches a maximum of 0.288 T in the volume center.

b) The magnetic flux density is between 0.225 T and 0.245 T at the periphery of the air-gap

volume. The average magnetic flux density across the air-gap, at z = O plane, is 0.2658.

It should be noted that the above-noted figures for field uniformity relate only to the FE

model. A very uniform magnetic field is practicaüy produced, once the equiprnent has ken

installed. by introducing shunming coils and other devices.

3.33 Cou Modeling Effect

The coü representation that is used in the FE model of the MRI magnet should closely

approximate the actud MRI coü. The model used for the coi1 affects the accuracy of the

solutions, especially a global accuracy mesure such as Ampere's law error. The FE

rnodeiing and representation of the coi1 is the focus of this subsection. There are two

methods for rnoàehg the coil: (a) to model its volumetric geometry, or (b) to simulate the

coil by an equivaknt cunent sheet. It is shown that a current sheet effectively simulates the

coil of the MRI magnet. This is very benekial where the exact data of the coi1 are not

avaiiable (e.g. in initial stages of designing a device) and an equivalent coil can k used to

simulate a real coil and energize the magnet model.

In the fmt mcthod, the col geomctry is modeled accurately. The main diffculty is the

thinness of the coü with respect to the coil length and the magnet dimensions. The modehg

of such a coii is very daficult and tim consuming. However. the cumnt loading is easy

where the cumnt density is calculated once and appiied to the coü elemcnts. The

Chaptcr 3. Study of the MM Magna wittr 3D FDI 67

interpolation functions of each ekmnt are then used to obtain the proper cumnt segment

value of each node. This method is caiied the elemental loading of the FE model.

In the second approach. whem the coil is n m w enough. a cu&nt sheet s'imulates the

coiL Such a coil is very easy to model. The properly placed nodes -at the mid-radius of the

coi1 and along the coil - are used to rnodel the coi]. The values of the cumnt segments are

calculated and assigned to these nodes dinctly. In this respect. the coil representation

cequires more h u m resources before the solution stage. This method is caiied the nodal

loading of the FE model.

AU the magnet modeis used in this thesis accommodate both types of coil representation.

To c o n h the accuracy and performance of the magnet models regardkss of the coii model.

four models of the rnagnet were selected. These rnodeis are - fiom the pnvious set of 20

models - Model #l as the initiai model, Model #10 as the entrance model to the optimum

model neighborhood, Model #15 as the optimum mode], and Model #20 as the erratic mode1

of the set.

These models were solved with the volumetric coil and with the current sheet coii.

Different quantities of interest are shown in Figure 3.1 1 (page 76) for both approaches. The

results for the volumetric coil models are shown as white bars, whiie black bars are used to

show the resuhs of the current sheet coil modeh. The results are &O grouped for each

quality of interest; namely (a) the average of magnetic flux density across the core (group

ni). across the air-gap (group a), along the air-gap (group #3), (b) the stond magnetic

energy in the air-gap (group W), in the magnetic part (group #5). in the whole model (group

#6), and (c) the average of Ampere's law error along different paths (group #7). The results

are aiso normalized with respect to Model #15 results.

The quaüty masures of both coil models agree closely except for the air-gap energy of

the erratic Model 1120. Thmfore, it is concluded that acceptable solutions can be obtained

with a proper equivaient cummt sheet. This conclusion wiil also k used when an alternative

to the MM rnagnet design is examuicd for the case where the coil data are not availabk.

C b t a 3. Study of the MRI Magnct with 3D FEM 68

3.3.3 COU Current Level Adysis

The main objective of this subsection is to study the magnet performance for dinerent values

of the coil current. The results provide engineering insights for simpürying the magnet mode1

in later studies and for the design of similar devices. This study also explains many questions

regarding the mdeling of the magnet. The cumnt load of the magnet coii is varied from 10

kA-t to 60 kA-t and quantities of interest are obtained. These quantities are discussed in the

following subsections.

Mametic Fieid:

The values of rnagmtic flux density, averaged across the core as well as across and dong the

air-gap, are shown in Figure 3.12 as a fùnction of the coi1 excitation. The results are ail

nonnallled to 100% for the 60 kA-t excitation. It wili be noted that the three graphs are

a h s t identical.

It can k concluded that for this device, one weii-defined normalized quality measure can

effectively represent other quaüty masures, regardes of the coil excitation level. The effect

of saturation û clear for excitation leveis higher than 30 kA-t. The averaged field is hcnased

by only 10% after raising the current level fiom 40 to 60 kA-t. This brings up the question of

the magnet design and its operating efkiency.

Maimetic Enem:

Figure 3.13 shows three normaiized magnetic energy h the magnetic parts (including the

core, the yoke and the pole), in the air-gap, and in the total volume of the rnodel. The results

are normalized to 100% for the 60 kA-t excitation.

For excitation ievels las than 30 kA-t. the rnagnet is not saturated and most of the coil

energy is stored in the air-gap magnetic field. At higher excitation Ievels, mon energy is

stored in the magnetic portion of the device in almost a ünear fashion. The slope of the total

energy graphs changes around 30 kA-t, which indicates that saturation has begun to set in.

Chapta 3. Study of the MRi Magnct with 3D FEM 69

E n e m Ratio:

To clarify the energy content and its distributon in the magnet, an energy ratio is defined for

a volume of interest as follows:

E.R.,= Volume Energy + Total Energy . (3.5)

Figure 3.11 shows this ratio for the magnetic part of the rnagnet, for the air-gap of the

rnagnet, and for the total volume of the model excluding the magnetic part and the air-gap.

The results are normalized to 100% for total energy of the model at each cumnt level.

As the coi1 cumnt is increased, the magnetic part becomes more saturated and stores

more of the added energy. Meanwhile, the air-gap share of the energy is reduced, but not

substantially. Thus, the increased energy share of the magnetic part cornes from the energy

stored in the external volum. For cumnt levels higher than 30 kA-t, the E.R., changes

h o s t Iinearly versus the currem excitation.

It is noteworthy that around 80% of the magnetic energy is stored in the k-space region

surrounding the magnet and the air-gap, regardless of the coi1 cumnt level. This jusrifies the

precise and carefùl approach chosen in this thesis to mode1 the extemal the-space region.

This also explains why the FE model and its solutions wen mon accurate when a better

modeling of the exterior free-space was employed.

Mannetomotive Force:

The magnetomotive force drops almg the magnetic part and the air-gap of the magnet are

shom in Figure 3.15. These results show the contributions of the rnagnetic portion of the

magnet, as weil as the contn ïon of the air-gap, to Ampere's law error. At the hiil

excitation. 38% and 58% of the 60 kA-t are dropped dong the rnagnetic part and air-gap of

the magnet, respactively. Thus 4% of the excitation is not accounted for and is reported as

the enor. The graphical similarity of the energy ratio in figure 3.14 and the rnagnetomotive

force in Figure 3.15 arc noteworthy.

Chapta 3. Study of the MRI Magnet wiih 3D FEM 70

3.3.4 Performance Summry of the Optimum Mode1 of the MRI Magnet

For the original model of the MRI rnagnet. when the c d is placed Gound the hre, and at

the operational current excitation level of 60 kA-t, the device performance sumrnary and the

general qualities of interest are given as follows:

The average of Arnpere's iaw error dong different paths is 4%.

The standard deviation of Arnpere's law enor dong different paths is 0.5,

The magnetic energy stored in the model as 4464 J,

The magnetic energy stored in the air -gap of the mode1 as 5 12 J,

The magnetic energy stored in the magnetic part of the model as 375 1.

The average magnetic flux density across the core as 2.19 12 T,

The average magnetic flux density across the air-gap as 0.2658 T.

It should be noted that:

a) Only a srnall arnount, 11.5%, of the total magnetic energy is stored in the air-gap volume.

b) Only a small amount, 8.496, of the total magnetic energy is stored in the rnagnetic part of

the rnagnet.

C) Most of the total magnctic energy, 80.1 %, is storeci in the volume exterior to the magnet.

d) Only 12 % of the magnetic flux at the core center plane teaches the air-gap center plane.

Therefore, the original design of the MRI magnet, where the coi1 is placed amund the con, is

not satisfactory. A better alternative design of the MRI magnet is given in the next section.

Chapta 3, Sndy of the MW Magnet with 3D FEU 72

AlasYs 5 -3 JUL 0 1999 14 :44 :30 PLOT 110. 5

Figure 3.7: The contour plots of the magnetic flux density magnitude in the MRI mgnet core at z = O plane (acfoss the core cyünder and in the upper right hand portion of the figure) and at y = O symmetry piane (dong the core cyünder.) The c d 19 muod the CO-. . The semicircle wire h (in the lower kfi hand portion of the figure) is the connection ana of the core to the yoke.

Cba~tcr 3, Studv of the MM Mamet with 3D FEM 74

AIISYS 5 . 3 JUL 8 1999 11:36:34 PLOT NO. 3

Figure 3.9: The contour plots of the mgnetic flux density mgnitude in the Miü mgnet pde ai z = 150 rmn plane (across the pole and adjacent to the air-gap) and ai y = O symnetry plane (the rectanguk area along the pole cyünder.) The COU is amoiid the core. The semicircular borderline or wire frame is the connection area of the pole to the yoke.

Chaptcr 3. Study of the MRI Magnct with 3D FEM 75

Egure 3.10: The contour plots of the magnetic flux density magnitude in the MRI magnet a b p p ai = O plane (across the air-gap) and at y = O symmetry plane (dong the air-gap.) The mil is amund the corn. The borderlines of the region of interest, that is a concentric cylinder within the air-gap but with hnlf of its radius and length, are shown Uiside the air-gap. The semicirculat boderline or wire fhme (in bottom left hand portion of the figure) is the connection area of the air-gap to the pole.

Cha~rer 3. Study of the MRI Magner with 3D FEM 76

Hollow bar, nsuiis from volumetric mil Solid bar: resutts fmm cunent sheet mil In each gmup (left to rigth): Modd no* ',10*15* 20

a-.

( Average magnetic flux density ) ( Stored magnetic energy )

Figure 3.1 1 : The average magnetic flux density across the core Grou across the air-gap Gr ou dong the air-gap Grou

The stored magnetic energy in the air-gap Group #4, in the magnetic part Gmup US, in the whole mode1 Group #6.

The average of Arnpen's law error Along diffemnt paths Group #7.

The nodal (equivaknt cumnt sheet coii) and ekmental (volumetric coii) cumnt excitations are used. The data are normalized with respect to the Mode1 11 5 results.

Chapta 3. Study of the MM Mannet wiih 3D FEM 77

Figure 3.12: The average magnetic flux density at dwerent locations of the magnet versus the coil excitation levels of 10 to 60 kA-t. The results are norrnaiized with respect to the final results of the 60 kA-t excitation. The norrnalization values are given on each graph.

Figure 3.13: The stored magnetic energy in different parts of the magnet excitation kvels of 10 to 60 kA-t. The results are normalized with respect to of the 60 Wt excitation- The nomakation values are given on each graph.

versus the coil the fmal results

Chapta 3. Study of the MRI Magnet with 3D FEM 78

Figure 3.14: The energy ratio of the magnetic part of the magnet, the air-gap o f the magnet, and the remairiing volume out of the rnagnet venus the coil excitation levels of 10 to 60 kA-t. The results are nonnaüzed with respect to the total energy of the model at each load step.

Figure 3.15: The average of the magnetomotive force drop dong the magnetic part, the average of the magnetomotive force drop dong the air-gap, and Arnpere's law error versus the coü excitation kvel of 10 to 60 kA-t. The results are nomialized with respect to the cumnt kvel at each load step.

Chabta 3. Studv of the MRI Mannci with 3D FEU 79

3.4 Design Alternative: C d around the Pole . .

Amongst diffemnt components and parameters of the original MRI (O-MRI) mgnet. the

location of the coi1 offers one attractive possibüity to improve its performance. The

puformance of the mgnet, as orighdy designed, was summanZed in Subsection 3.3.4. The

study of an alternative design of the MRI (A-h4RI) magnet and its performance is the main -. focus of this section. The FE model and solution of this rnagnet is given in Subsection 3.4.1.

The contour plots of the magnetic flux density are presented in Subsection 3.4.2. The A-MRI

magnet is also analyzed for different levels of the coi1 current and the results are given in

Subsection 3.4.3.

3.4.1 FE Analysis of the Alternative MRI Magnet

The anaiysis of the alternative design of the MRI (A-MRI) magnet is a typical engineering

problem: the analysis of a physical device kfore its manufacture. Questions iike how best to

defm the boundary value problem should it be dehed in ternis of MVP or MSP? What is

the domain definition and discretization? Are the solutions of the FE rnodel correct, and by

wbat masure? 1s the developed FE model, an optimum model? Can its accuracy k

enhanced, and if so by how much, and at what cost? In other words, di the steps taken for the

study of the O-MRI magnet canlshould be repeaied.

Previous studies of this thesis have pmvided the advantage of having the answers to most

of these questions. The cumnt A-MRI rnagnet is a simüar probkm to the previous O-MRI

mgnet, except it is less difFicult due to a low level of magnetic saturation in the rnagnet.

Therefore, the FE study of the A-MRI mgnet would k brief. It focuses only on some

criticai issues such as accuracy and the &velopment and identification of an optimum rnodel.

The steps that were taken to study this device are summarized in this subscdion.

Cha~ter 3. Studv of the MRI Mamct with 3D FEM %O

Sirecifications and Model:

The geomtry of the AMRI magnet is similar to the O-MRI magnet. the latter having ken

given in Section 2.1. The s a m rnagnetic material, core, yokes, and pies are used for the

altemate design The only modification is that the coil is split and positioned around the

poles, as opposed to being around the con. The total current of the two coils in the altemate

design is equal to the O-MM magnet current of 120 kA-t. Due to symmetry, only one quater

of the magnet, and thus only one of the coils, needs to be modeled. The current excitation of

each coii is 60 kA-t. The coii height is 115 mm, while the pole height is 150 mm. The air-gap

side of the coil is flush with the pok face at ~ 1 5 0 mm.

As before. the boundary value problem was defmed in t e m of the magnetic vectot

po tential. The pro blem domain was then defmed and discretized. Next , the New ton-Raphson

method was used to solve the nonünear system of equations. The rnagnetic field data were

then obtained and investigated.

It should be noted that all the FE models of the MRI magnet were developed with the

provision of accommodating a volumetric coü or a cunent sheet coil being placed either

around the core or around the pole. To differentiate ktween models of two magnets, the

altemate MRI magnet rnodels are designated with a suffix "A." For exiunpk, Model #lSA is

the FE model of the A-MRI rnagnet whose problem defmition and formulation, dornain

dimension, dornain grid. and material all are the same as Model # 15 (shown in Figures 2.1 to

2.5), but with a cunent sheet coil king around the pole, and of course. different solutions.

Initial Model:

Model #lA is the fust candidate for the inirial model. Its problem domain and dimetization

are identical to the rnodel that was detailed in Section 2.1. Its 2D projection rnodel is also

identical to the one shown in Figure 2.5.3. Ampere's law errors were cdculated almg the

same p a h of the magnet that were shown in Figure 2.2. The average of errors was obtained

as 4.78, which is within a reasonable Limit. Therefore, Mode1 #1A is introduced as the initial

rnodel of the A-M'RI magnet.

Chapta 3. Study of the MRI Magnct with 3D FEM 81

Sektion Studv:

The initial model was modified systematicaly in order to identw the most accurate model of

the A-MRI magnet. The chosen selection criterio were (a) the magnetic flux density, (b) the

energy contents, and most importantly (c) the Arnpere's Iaw enon of the rnodels.

From the set of 20 models, pnsented in Section 3.2, only four models were selected:

1 ) Model # 1 A, as the initial model of the OMRI model,

2) Model X I OA, as the enirana model to the optimum mode1 neigh borhood,

3) Model #lSA. as the optimum rnodel candidate, and

4) Model #20A, as the erratic model of the MRI magnet.

These FE models were solved and the results are shown in Figure 3.16 - page 88 at the end of

this section . The quantities of interest are normalUed with respect to the solutions of Model

#l SA of the AMRI magnet, which are also given on the graphs.

The O~timum Model:

Arnong different models. Model t15A is the optimum model of the A-MRI magnet because it

provides:

a) The minimum average of Ampere's law errors dong different paths at 2.546,

b) The minimum standard deviation of Ampem's law errors dong different paths at 0.27,

C) The minimum rnagnetic energy in the wwhok mode1 of the magnet at 2990 1,

d) The maximum magnetic flux density across the air-gap center plane at 0.419 T,

e) The maximum magnetic flux density across the core center plane at 0.835 T,

f) The maximum rnagnetic energy stored in the air -gap of the rnagnet at 1477 J, and

g) The maximum magnetic energy stored in the magnetic part of the magnet at 22 J.

This of course is not surprishg since Model #15 has closely ken identified as king the

optimum rnodel for the original MRI rnagnet. AU that has changed in Model #lSA is the

location of a cumnt sheet approximation of the c d .

Chaptcr 3. Study of the MRI Magnct with 3D FEM 82

3.4.2 Magnetic Field Grapbicai Results

Contour plots of the magnetic fl& density arc used to show the r&lts. The secpence of

graphs, as before, starts with the general extemal overview of the rnagmt, as shown in Figure

3.17. The magnetic field at selected slices of the magnet core, yoke. pole, and air-gap are

then pnsented.

Mamet:

The distribution of the magnetic flux density magnitude over the extemal surface of the core.

yoke and pole is shown in Figure 3.17. This figure also defines the coordinates and the view

direction of the subsequent figures. As can be seen in the figure, the pole is only saturated in

a relatively small region at the pok face edge. The magnetic field over the surface of the

yoke is lower than the saturation kvel of 1.5 T, and its value dingnishes at the corner

sections of the yoke. The magnetic flux density at the inner side of the core (toward the air-

gap) is around 1 T, while its value is reduced to 0.5 T at the outer side of the core (far €rom

the air-gap). The following observations are noteworthy:

The magnetic field at the exteriors of the pole is rotationally symmetric,

0 The magnetic flux density changes smoothly between different parts of the magnet, and

0 The magnetic flux density kvel is between 1 to 1.2 T in most of the magnet volume.

The nsults at two important 2D slices of the core are shown in Figure 3.18. The normal and

tangentid magnetic field boundary conditions are set at the z = O and y = O planes of

symrnetry, respectively. The results at these slices are thus the most accurate ones. The core

magnetic flux density is slowly changing fiom 1 T to 0.7 T, and its average across the core at

the t = O plane is 0.8350 T. The coic magnetic flux density lacks a rotational symmetry.

It is concluded that the core only closes the magnetic circuit of the magnet and that it is

not saturated. This cm be used to enhance the future designs of the rnagnet in two ways: h t ,

the radius of the core can be reduced in order to decrease the core volume, and second, a less

expensive magnetic material can k used in order to d u c e the cost of the core.

Chaptu 3. Study of the MRI Magnct with 3D FEM 83

The resuhs for the yoke lower plane at z = 300 mm and the yoke symmetry plane, at y = O are

shown in Figure 3.19. In the major volum of the yoke - fiom the centeriine of the con to the

centerhe of the pok - the magnetic flux density inside the yoke is approximately 1.2 T. On

the other hand. in the top corners of the yoke, the magnetic flux density is ktween O and 0.4

T. A rapid change of the magnetic field exists only at the inner connecting regions of the

yoke with the core and the pole. The following observation can be concluded:

The magnetic flux density in the active volume of the yoke is around 1.2 T,

r The low magnetic field regions are in the outer sides and corners of the yoke, and

r The yoke acts as a gwd magnetic flux pipeline between the core and the pole.

Pole: - The results for the pole face at z = 150 mm and the pole symmetry plane at y = O are shown

in Figure 3.20. The magnetic flux density in most of the pole volume is between 1 T and 1.4

T. It is ktween 0.55 T and 2 T at the pole face. However, it changes only between 0.55 T

and 0.77 T in the center region of pole face. The average magnetic flux density at the pole

face is 0.62 T.

The rapid change of the magnetic field is due to the proximity of the coii and the air-gap

to the pole fxe. The magnetic flux density at the pole face has a ckar rotational symmetry,

which is mainly due to the location of the coi1 around the pole.

Air-GID:

The magnetic flux density in the air-gap is shown in Figure 3.2 1. The z = O and the y = O

planes are the normai and tangentid magnetic field planes. The z, y = O planes include two

slices of the MRI magnet volume of interest; i.e. the volum within which maximum

uniformity is desirrd for the flux density distribution. These are also shown in Figure 3.21 by

the borderlines of the inner cyhder. The cesults at the air-gap regions are:

Chapta 3. Study of the MRI Magna witb 3D FEM 84

a) The magnetic flux density is 0.45 T around the volume of interest and reaches a

maximum of 0.48 T in the volume center. It is suggested that the 0.03 T deviation could

k reduced by introducing sorne-masure of pole shaping.

b) The magnetic flux density's approximate value is 0.33 T at the periphery of the air-gap

volume. The average magnetic flux density across the air-gap at z = O plane is 0.4 193 T.

The magnetic flux density has a pmnounced rotational symmetry with respect to the air-gap

axial axis. especiaiiy in the air-gap volume of interest.

3.43 Coil Curent Level Analysis

The main purpose of this subsection is to study the alternative mode1 of the M N magnet for

different b e l s of the cumnt excitation. The current level of the coil is changed from 20 kA-

t to 120 kA-t and the quantities of interest are calculated. The operational current kvel of the

coil is 60 kA-t. The quantities of interest are discussed in the foliowhg paragraphs.

The magnetic field withm the core was found to be lower than the saturation level, even for

the high excitation of 120 kA-t. The pole and yoke connection area was thus selected to show

the highest magnetic flux densities in the rnagnet. Figure 3.22 shows the nomialized average

magnetic fiux densities at the pole and yoke connection area, across the air-gap, and dong

the air-gap. The magnetic fiux density across the yoke-pole area changes linearly with

respect to the current levels that are lower than 80 kA-t. However, for excitation levels higher

than 80 IrA-t, the magnetic flux density in this ngion exceeds the saturation level of 1.5 T.

The magnetic field at the air-gap of the magmt changes almost linearly with respect to the

current excitation.

The normaiked stored magnetic energy in Merent parts of the magnet is shown in Figure - 3.23. The nsults are normaüzed to 100% for the 120 kA-t excitation. The graph of the total

encrgy shows that a rdd saturation dominates the magnet at cumnt levels higher than 80

C h a ~ t a 3. Snidv of the MRI Mamet with 3D F E . 85

kA-t. The s a m holds tnic for the air-gap energy content because it holds half of the magnet

energy. The magnetic part of the mgnet begins to store energy at the outset of the saturation

approximately at 80 kA-t.

Enem Ratio:

The energy ratios of the magnetic part, the air-gap, and the free-space exterior to the rnagnet

are shown in Figure 3.24. The magnetic part of the magnet stores only 1% of the total energy

while the air-gap stores around 50% of the total energy - regardless of the cunent level. At

the operational current level of 60 kA-t, 0.7%,49.3%, and 49.9% of the total 2990 J is stored

in the magnetic part, the air-gap, and the exterior free-space region, respectively.

Mametomotive Force:

The rnagnetomotive force drops and the mmf error are shown in Figure 3.25. Most of the

mm. drop occun dong the air-gap. For cumnt levels higher than 80 kA-t, the saturation

increases the mmf drop dong the magnetic part of the magnet, which consequently decreases

Ampere's law error.

The surnmary of the O-MRI magnet performance data wül be given in the next section.

3.5 Cornparison of MRI Magnet Designs

Akhough the focus of the thesis has k e n to develope global accurucy criteria and selection

criteria, the comparative study of two MRI magnet designs offer engineering insights that

kad toward a better design of the magnet. Table 3.2 compares the nsults fiom two designs of

the MRI magnet. The last column is the ratio of the alternative (A-MRI) design results to the

original (O-MRI) mode1 results. The cumnt excitation level for both models is 60 ICA-t for

the % symmetry model, or equivaltntly 120 M-t for the whole magnet.

The stored magnetic energy in the AMRI magnet is 67% of the O-MM magnet storrd

encrgy. Such a ceduction in total energy is accompanitd (a) by an hcrease in the air-gap

energy by 188%. and (b) by a substantial decrease of the energy stored in the magnetic part.

Tht encrgy ratio of the A-MRI air-gap is three times more than the original design's value.

Chanta 3. Studv of the MRX Mamet with 3D FEM 86

Therefon, the AMRI magnet is far better than original design in ternis of the rnagnet

energy. Most importaatly, the better energy distribution and efficiency of the alternative

design is accompanied by a signiticant 58% increase in the magnetic hx densit; across the

air-gap. and moreover, such increase is achieved without significantly sacrifcing the field

uniforrnity. It is conciuded that the alternative design performance is far better than the

original model of the MRI magnet.

Table 3.2: Cornparison of the O-MRI and A-MRi rnagnets. Quantities of interest [ AMRI 1 O-MRI 1 A/O

The magnetic energy stored in the whole model:

The magnetic energy stored in the air-gap:

The rnagnetic energy stored in the magnetic part:

The magnetic energy stored out extenor to the magne t:

The energy ratio, E.R., of the air-gap:

The energy ratio of the region exterior to the magnet:

The energy ratio of the magnetic part:

The average magnetic flux density, B, across the air-gap:

The average magnetic flux density across the core:

The rnagnetic flux of the air-gap + the con magnetic flux:

The average of Ampere's law enors, e , ~ :

The standard deviation of Ampere's law errors:

3.6 Conclusion

The accunicy and selection issues of the basic FE model were generaily discussed in Chapter

2. The MRI magnet probkm was then solved in ths chapter. An alternative design for the

MRI mgnet was also suggested. The accuracy of the A-MRI model and the optimum model

were studied. The performances of both designs of the MM magnet were then compared, and

the AMRI design was concluded to be mpiot to the original design of the M'RI magnet.

nie m4Ui conclusions of this chapter are the folowing:

C h a ~ t a 3. Studv of the MRI Marna with 3D FEM 87

Ampere's law error masure is a valid global accuracy m u r e that should k used

carefhlly. The path dependency of such a measure causes a few lirmtations in its

application as a globd accuracy criterion.

Statistical indicaton of Ampere's law error dong diffemt paths. its average and standard

deviation, are reiiable global accuracy measures.

The stond magnetic energy in the FE model is a valid selection criterion. The magnetic

energy in the FE model parts, such as the magnetic part, and the air-gap of the MRI

magnet, can also be used as selection criteria

The average of the magnetic flux density at different locations of the magnet is a good

selection critenon. When such an average is taken across the core or air-gap, it shows a

trend simüar to the average of Ampere's law error measure.

The fkee-space material rnethod and the reduced excitation rnethod were very effective in

obtaining the initial model of the MN rnagnet.

The equivaient hear modeling method - introduced in Subsection 3.2.1 - substantially

improved the emciency of the selection search for better models of the M N magnet. It

was shown that the accuracy of the noniinear mode1 could be estimated fiom a ünear

mode1 of the problem

C h a ~ t a 3. Studv of the MRI Maenet with 3D FEM 88

In each group (left to rigth): Model no. 1 A, 104 ISA, 20A ..............................................................................

( Average inagneîic flux dense ) ( Stored magnetic energy )

Figure 3.16: The average magnetic flux density across the core Group #1, across the air-gap Group lf2. dong the air-gap Group #3.

The stored magnetic energy in the air-gap Group M. in the magnetic part Group US, in the whole mode1 Group 116.

The average of Ampere's law emr Along different paths Group #7.

Modeb lA, IOA, HA, 20A are used. The nodal current excitation of 60 kA-t is used. The data are nonnalized with respect to the Model #15A results: B - 4 0.8350 T Bmw+p 0.4193 T Bdag 0.4840 T Energy&e 1477 J Energy-=,= 1477 J EnergyId= 2990 1.

Figure 3.17: The contour plots of the magnetic flux density magnitude in the A-MRI magnet, where the coi1 is amund the pde.

Chapter 3. Stidy of the MRI Magnet with 3D FEM 90

Figure 3.18: The contour plots of the magnetic flux density magnitude in the AMR[ magnet cote at z = O plane (across the core cyünder) and at y = O symmetry plane (dong the core cyünder.) The COU LF amund the pok.

Chapicr 3. Study of the MRI Mamet with 3D FEM 91

AklSYS 5.3 JUL 16 1999 14:14:19 P m NO. 3

Figure 3.19: The contour plots of the mgnetic flux deirsity magnitude in the A-MRI mgnet yoke at z = 300 mm plane (iower plane of the yoke slab) and at y = O synunetry plane. The coü is amund the pie.

Chapter 3. Snwly of the MRI Magnet with 3D FEM 92

MISYS 5 . 3 JUL 16 1999 34:14:54 PLOT !IO. 4 9 .4 9 .45 9 - 5 - .55 9 -6

.O5 9 .7 œ .a - . B 5 œ .9 = .95

1 - 1.05 œ 1.1 œ 1.2

1.25 = 1.3 - 1.35 œ 1.4

1.45 1.5

0 1.6 0 1.65

1.7 m 1.75 œ 1 .0

1.85 - 1.9 œ 2

Figure 3.20: The contour plots of the magnetic flux density magnitude in the A-MRI magnet poie at z = 150 mm plane (across the pole and adjacent to the air gap) and at y = O symmetry plane (dong the pole cylinder.) The cou is acouad the pok.

Chapia 3. Study of the MRI Magna wiih 3D FEM 93

ATISYS 5 - 3 JUL 16 1999 14 :21 : O 9 P K n t10. 5

.31

. 3 2

. 3 3

.34

. 3 5

. 3 5

.37

.38 œ .39 œ .4 œ - 4 1

.42 9 .43

*-14 . 4 5 .46 .47 . 4 8

œ . 4 9 m .5

Figure 3.21: The contour plots of the magnetic flux density magnitude in the A-MRI mgnet abgap at z = O plane (across the air-gap) and at y = O symmetry p h e (along the air-gap.) The coi1 is amund the pok.

Chauta 3. Studv of the MRI Mamct with 3D FEM 94

Figure 3.22: The average magnetic flux density at diffcrent locations of the magnet versus the excitation level of 20 to 120 kA-t. The nsults are norniaüzed with respect to the fmal results of 120 kA-t excitation. The nomialization values are given on each graph.

figure 3.23: The stored magnetic energy in difKerent parts of the magnet venus the excitation kvel of 20 to 120 kA-t. The resuhs are nomializcd with respect to the final resulu of 120 lrA- t excitation. The normalization values are given on each graph.

Chaotcr 3. S tudy of th t M N Mama with 3D FEM 95

Figure 3.24: The energy ratio of the magnetic part of the magnet, the air-gap of the magnet. and the remaining volume out of the magnet versus the excitation level of 20 to 120 kA-t. The results are normalized with respect to the total energy of the mode1 at each load step.

Figure 3.25: The average of the magnetornotive force &op dong the magnetic part, the average of the magmto-motive force drop dong the air-gap, and Ampere's law error versus the excitation level of 20 to 120 LA-t. The results are nomiplized with respect to the current level at each load step.

Chapter 4

Study of the MRI Magnet with 2D FEM

4.1 Introduction

The 3D FE study of the MRI magmt was given in Chapter 3. However, the development of a

3D mode1 of even such a simple structure as the MRI rnagnet is very t h consuming, both in

tenns of human effort as weii as computationai resources. An attractive alternative is to use a

suitably designed 2D mode1 with the caveat that it does not simpl@ the basic problem out of

existence. Defined comctly, 2D models can be used cost effectively to explore design

alternatives prior to undenaking a full 3D modeling effort. The purpose of this chapter is to

compare 2D (approxmiate) and 3D (exact) rnodels of the MM magnet system with a view of

developing a method that simplüies the anaiysis without sacrificing the integrity of the

results thus obtained.

It is dways pnferable to undertale the analysis of an electromagnetic device using 2D

models, provided suitable planes of symmetry can be identified. The main objective of this

chapter is to show that even in the case of the highly saturated MRI magnet, enginee~g and

physical insight aid in idcntifjhg suitable 2D models. By direct cornparison with the results

of masuremnt and of an optimum 3D FE analysis, it is show that the 2D approximate

mMiels provide surprisingly good m l t s in rlected regions of the rnagnet systen

A physicai system with eithec longitudinal or rotational synnnetry can k entirely

modeled in 2D. In reality, most physicai systems have no such symmetry and cannot be

Chaota 4. Studv of the MRI Mamet witb 2D FEM 97

modeled entinly in a 2D space. However. there are two generic types of approximations to

reduce the dimensionality of a physical system and simpiiry its study:

a) If the portion of the physical sistem ihat is of intenst can be isol&ed frorn the system as

a whole and studied conveniently in 2D. one obtahs a panial2D model. This approach is

used and discussed in Sections 4.2 and 4.3.

b) If the geornetry of the physical system c m be modifkd such that some critical featuns

are preserved in either 2D translational or rotational symmetry. one obtains a composite

2 0 madel, This method is detailed in Section 4.4.

Both approaches are examined within the context of the 3D MRI magnet system The main

objective is to determine whether the 2D approximate models can yield usehil and accurate

results.

The remainder of this section discusses the axisymmtrical 2D approaches for study of

the MRI magnet. It also clarifies the objectives and structure of this chapter in more detail.

4.1.1 W y m m e t r i c Approximation of the MRI Magnet

In an auSymmetncal problem, a rotational or cyündrical symmetry exists in the problem that

simplifies its repnsentation. This method can &O be used to study some specific regions and

aspects of the MRI magnet problem. The steps in the axisyrmnetric FE modeling of the M M

magnet are summafized in Appendix D.

The focus of the thesis, the MRI mgnet, has been studied in 2D by employing thne

approximate models:

a) A panial2D rnodel where the coü is located around the core. The super-conductive coi1

and the encloscd core magnetic material were removed from the MRI magnet and studied

using an axisymmetric 2D FEM. This snidy is given in Section 4.2. It is shown that this

approach yiekis nasonable results when the corn is highiy saturated.

b) A punial 2D mode1 where the coils are p l a d amund the poks. The combination of the

pole piece, the coi& and the air-gap were separated h m the 3D problem domain and

aaaiyzed with the axjsymmetric 2D FEM. This study is detaüed in Section 4.3. It is

Chapta 4. Study of the MRI Magnet w i b 2D E;EM 98

confonncd that the air-gap magnetic field solutions can be conveniently approximated by

this approach.

c) A compmitc 2D mode1 where the coil is located around either the tore or thé poles. The

effect of the yoke on the core and air-gap magnetic field is assumd to be negligible.

Thus the 3D problem domah was modified, first, by preserving the core, poles. and coü

cylindncal configuration, and second, by attnbuting a cylindrical shape to the yoke. The

whole composite magnet was then modeled and studied with the axisymmetric 2D FEM.

The study and the relevant results are given in Section 4.4. This modehg practice has

proven to be the most practical one to estimate the magnetic field in the core and air-gap

of the magnet. The solutions are shown to be surprisingly close io the 3D solutions

ngardless of the coil location and excitation level.

The main objectives in the study of the approximate 2D FE models are (a) to detemine

whether they can yield usehl and accurate results. and (b) to obtain the extent of their

applicabüity. These issues are exploreci in this chapter. The quantities of interest are limited

to the magnetic fiux density in the core, the pole and the air-gap. The averages of the

magnetic flux density fiom 2D and 3D analyses are compared.

4.2 A Partial 2D Mode1 of the Coi1 and the Core

When the coü is concentric with the core of the MRI rnagnet, the partial modehg and study

of the core can k justifieci by two arguments:

a) The high level of the coil excitation causes a high magnetic saturation in the core. It is

assumd that the region of the magnet that is completely inside the coil is not affiited by

other componcnts of the magnet such as the yoke. pok and air-gap. In other words, it is

assumd that the CO= is rnagneticaly irolated h m the remainder of the magnet.

b) Figures 3.6 and 3.7 (nom 3D FE solutions of the MRI magnet) displayed the magmtic

flux demit. contour iines at the con. The core magnetic field was rotationaliy symmctric

with respect to the coce a n m l axis, and it was almost uniforni at any z = h plam - when

Cha~tcr 4. Studv of the MW Mamet with 2D FEM 99

h < L o i i . This suggests that the core magnetic field can be presented in an axisymmctrical

fas hion.

Therefore, it is suggcsted that the coil and the enclosed con magnetic material k separated

fiom the rest of the magnet and k modejed individually.

The new problem of the coü and the enclosed magnetic matMa1 has a cylindrical

symmetry, and tkrefore, can be modeled with axisymmetric 2D FEM. The problem dornain

of this partial 2D mode1 is shown in Figure 4.1 - page 103, at the end of this section. The

quantity of interest is the magnetic flux density at the con center plane (z=O). By comparing

with the hii 3D mode], it is possible to detennine the limits and conditions under which this

approximation is justified. The approximate rnodel c m be beneficiaiiy used to estimate B at

the con central plane as a function of the core geornetry.

The unckar part of this mode1 is the definition of the proper boundary conditions.

However, the field data are mainly sought in the center region of the core. which is

surrounded by the coü. Therefore, the boundary condition values should not affect the results

greatly. The normal, tangential and far-field boundary conditions are used and are shown in

Figure 4.1.

The 2D partial mode1 of the core is studied in the next subsection. The accuracy of ihis

approach is then investigated in Subsection 4.2.2.

4.2.1 Problem Definition and Resuits

In this subsection, the domain of the partial 2D mode1 is fint defmed, and then its dimensions

are obtaincd. The general guidelines for the specification of the dornain parameters are &O

devetoped and hold for the rest of this chapter. The solution of the partial 2D rnodel of the

con is finaiiy given.

The probkm domain is a rcctangular m a m the rot plane. wbich includes the core, the coü,

the air in ktween. and the surroundmg nCe-spacc. Figure 4.1 shows the sirnplified domain,

the dornain p ~ t c r s . and the boundary conditions. It should be noted that the kngth of the

Chapca 4. Stuày of the MRI Magnet with 2D FEM 100

core and the height of the problem domain are equal. The dornain width (RdoMi,,) and domain

height (ZdOnioin) wiU be obtained by using mthods that will be descriid next. The d e m g

parameters of the problem domai are given in Table 4. 1. The B H c u m of the core

rnagnetic matecial is given in Appendi B. The dimensions of the domain width and height

are obtained in the follo w ing paragraphs.

Table 4. 1: The problem domain parameters for partial model of the core and the coil.

FE Studv and Domain Dimensions:

Domain width, Rdomoin (to be obtained)

Core radius, &,, 250 mm

Coi1 width, Wwil 25 mm

Core - coi1 separation, Sec 25 mm

This study incorporates two issues: fust, determinhg the dornain of the problem, and second,

obtaining the proper discretization of the problem domain. These studies are related and

cannot be completely separated. The approach here is (a) to employ a very fuie mesh and

focus to obtaui the proper dimensions of the probkm do&, and (b) to use such problem

dornain and obtain the optimum discretization of the domain. This sequence yields the proper

domain dimensions and dismtization, which in tum yield the oplimm model of the probkm

in hand. The seleetion criterion during this study is the average of the magnetic flux density

at the core centet plane at z = 0.

To obtain the dimensions of the problem domain, a numrical vaiue was assigned to the

domain height, ZdOnioin, and the domain width was obtained fiom the following relation

domi in = a ~ h w i n (4. 1)

Domain height, Zdumin (to be obtained)

core height, zdOmPin (to be obtained)

Coü length. Lcoii 265 mm

Coi1 excitation, IteOu 60 kA-t.

where the coefficient a should be determined. The coil and core parameters were kept

constant. DüTerent models were then solved by axisymmetric FEM, and the average

magnetic flux density at the core ccnter plane were computed. These fieid data are shown in

Figure 4.2 as a function of aOIMUi and for different vaiues of coefficient a. It is concluded

that in order to approximate consistent solutiom, the domain paramters should satisfy the

followmg criteria:

Chaptcr 4. Study of the MRI Magna with 2D FEM 10 I

Different discretization Kh&S were applicd to the problem dornain. T& optimum

element sizes for difKercnt arcas were found to be:

O of the coil width, ( W , + 3 ) for mal1 areas such as the coil area, and

O coii width (W, ) for areas such as core and air, and with a proper spacing factor.

Further rehemnt of the elemnts did not change the average magnetic flux density at the

con center plane.

The Odimum Mode1 and Rtsults: ,

The problem domain of the optimum model was thus defined by:

Domain width ( Rdmn) = 2120 Domain height ( ) = 530 mm.

A total number of 4400 elemnts was used to discretize the probkm dornain. Figure 4.3

shows the contour plot of the magnetic flux density magnitude at the core of the magnet.

The folîowing observations are concluded from the results:

A high level of rnagnctic saturation exists in the core.

The average magnetic flux density at the core center plane is computed as 2.32 T versus

2.19 T that was computed from 3D analysis. This translates to a discrepancy of 6%.

Ampere's law emr ( e w 1 ) dong different paths is less than 5%.

The magnetomotive force &op ( mfi ) at the con and dong the line from (0,O) to ( -0.

z=LoiI), is around 37 kA-t. Thus the wnf drop dong the air-gap can be at most (60-37=)

23 kA-t, which would suggcst a magnetic flux density of around 0.20 T dong the air-gap.

The measured comsponding value is 0.27 T, and thus a 26% discrepancy would be

derived It shows that (1) the magnetic flux density of this model is ody acceptabk at tbe

zd'l plane of tbc core, and (2) even such a simple partial modtl yields betier nsults than

the 2D plansr mode1 of Subscction 4.1.1.

Chapter 4. Study of thc MW Magnet wiih 2D FEM 102

4.2.2 Limiting Effeet of the Coii Excitation Level

The objective here is to obtain the limits and the conditions to which the 2D approximation is

justified and the solutions of the partial 2D model of the core an accurate. Thus the magnetic

field of the core for differcnt levels of the coi1 excitation is examined. The optimum FE

mode1 was chosen and the coil excitation level was varied in the range of 10 to 60 kA-t. The

averages of magnetic fiux density at the core center plane were calculated. These data are

shown in Figure 4.4 as a function of the coil current dong with the resuhs of 3D analyses.

The error is also given in the same figure.

For the nominal excitation of the 60 kA-t, the results disagree within an acceptable

margin of 6%. As the excitation is reduced to 50 kA-<, the error is still acceptable and lower

than 10%. However, for lower leveis of the coil excitation. 2D and 3D studies diverge

considerably fiom each other. The error naches 200% for the case of 10 kA-t excitation

level. The results of the 3D analyses of the MRI magnet have been proven to be accurate.

Therefore, it can be concluded that the axisymmetric model fails to comctly evaluate the

core magnetic field for low levels of the coi1 excitation.

The solutions of the 3D mode1 of the MRI magnet, for a low cumnt level of 10 kA-t,

yielded a magnetic flux density of 0.7 T and 0.5 T at the inner side and the outer side of the

core, respectively. The results cleady did not have a rotationai symmetry and di&red in

value by more than 30%. For the same excitation level, the axisymmtric anaiysis computes a

smooth magnetic flux dcnsity of 1.7 T, which is more than double the actual m u n t of 0.58

T.

In conclusion. the 2D partial mode1 of the coii and the core yields acceptable result only

when the magnet corc is highly saturatai, and thus. the application of dus method is lirnited.

Chapta 4. Study of the MRI Mama with 2D FEM 103

Coil

flux N d Baunduy

Figure 4.1: Problem domain of the partial 2D (axisymmtricai) mode1 of the core and the coil.

Figure 4.2: The relation ktwecn the average magnetic flux density (B) across the cote center plane, and the dimensions of the pmbkm domain for the 2D partial mode1 of the core and the CO^. The kngth of tht c d is &&6S MA

Chapter 4. Study of the MRI Magnet with 2D FEM 104

Figure 4.3: The contour plots of the magnetic flux density magnitude at the core of the 0- MRI magnet. The 2D partial mode1 of the coi1 and the core (Figure 4.1) was used.

Figure 4.4: The average magnetic flux density (B) in the center plane of the core, for different levels of the coü excitation. The 2D partial mode1 of the coü and the core (Figure 4.1) was used.

Chanter 4. Studv of the MRI Maenet with 20 FEM 105

4.3 An Acceptable Partial 2D Mode1 of the Coi1 and the Pole - -

The partial modeling of the MRi magnet, where the cok are concentric with the poles, is the

subjtct of this section. The results of 3D analysis of the A-MN rnagnet (Section 3.4)

confirmeci that regardleu of the location of the coi1 and the excitation level of thecoil

a) the magnetic field is normal at the air-gap z=0 plane,

b) the poles are not saturated, and

C) the air-gap magnetic field has a cylindrical symmetry with rcspect to the air-gap ais.

These arguments suggest that the pole and air-gap can be separately studied. A new probkm

is defmed that includes only the pole, the air-gap, and a coil that is placed around the pole. In

this section, this simpiifïed version of the magnet is studied in 2D and the results are

compared with 3D results. The objectives of this study are:

1) to show the applicability of the 2D partial mdel, in providing useful solutions, and

2) to ob tain the extent and the condi dons to which the 2D approximation is justified.

The partial 2D mode1 can be used to estimate the air-gap magnetic flux density as a hinction

of the air-gap geometiy or the coil curent in initial stages of the magnet design.

4.3.1 Problem Domain and Resdts

The problem domain includes the pole, the air-gap, and the coil. Due to the symmtry of the

problem, only one hala of the domain is modeled in a cylindrical coordinate system The

problem domain is a nctanplar area in the rot plane. Figure 4.5 shows the simpKid

dornain and the boundary conditions. The domah parameters are given in Table 4.2. The

material and parameters were selected to simulate the samc magnet as in the 3D study given

in Section 3.4. Thc domain hcight, is a problem parameter. The domain width,

Rdo,, was sckctcd according to the optimizing relation of RdOIMiir L 4GdOmOiinl given in (4.2).

A total numbcr of 7500 ekmnts was used to discrctizc the probkm domin of this model.

Increasing the do& width and rcfining the dornain *d did not change the approximate

solutions. Thereforc, this model is the optimum partial 2D modcl of the problem

Chaptcr 4. Study of the MRI Magnet wiih 2D FEM 106

Figure 4.6 shows the contour plot of the rnagnetic flux density magnitude at the pole and the

air-gap. The foilo wing conclusions can be obtained from the results:

a) The pole is not rnagnetically saturated. and an almost uniform magnetic field exists in the

center region of the abgap.

b) The dûcontinuity of the rnagnetic field at the pole and air-gap interface is due to the

discontinuity of the tangentid component of the magnetic flux density. Br.

C) The nocmai component of the rnagnetic flux density. B,, is continuous across the

interface. It is the principal part of the magnetic field in the air-gap.

d) Ampere' s law error ( emg 1 ) dong different paths is las than 3%.

e) The magnetic flux density indicators at different locations are given in Table 4.3.

Table 4.2: The probkm domain parameters for partial model of the pole and the mil.

Tabk 4.3: The solutions fiom 3D and the partial 2D model of the pole and the mil.

Domain width, RdWi 1.2 m Pole radius. R,* 250

Coi1 width, Wc0u 25 mm

Pok - Coi1 separation. S, 25 mm

Domain height. ZdQnu 265 mm A.ir-GaP length. 4 i p d a P ' 1 50 mm

Coil fength, Lcoir 115 mm Coi1 excitation, Itail 60 ICA-t.

Therefore. the partial 2D mode1 of the pole and air-gap provides acceptabk rnagnetic field

data for the nominal currcnt excitation of 60 kA-t.

Bara ~ocariofi T

B oiong o i rgap

B am air-g4p

B umrrpte. o i ~ g a p

4.3.2 The Effect of the Coii Excitation Level

2D 3D Error 1

0.46 187 0.48396 4.5%

0.39476 0.4 1925 6%

0.64539 0.62279 3.5%

The objective of thip subsection is to determine the extent and the conditions to whkh the 2D

approximation is justiocd. Thus, the magnetic field of the abgap is studied for different

Chapter 4. Stisdy of the MRi Magnct with 2D EEM 107

levels of the coü excitation. The c u m t was changed h m a low kvel of 20 kA-t to a high

level of 120 k t . The average magnetic flux density across the air-gap, along the air-gap,

and acmss the pok and air-gap w&e computed. These are shown in Figure 4.7; dong with

those of the 3D analyses.

It is confimrd that the maximum discrepancy between 2D and 3D results are 6%. 5%

and 8% for the averages of magnetic flux density across the air-gap. along the air-gap, and

across the air-gap and pole interface uea, respectively. Moreover, a higher excitation level

reduces the error in the air -gap magnetic field.

Conclusion:

This study connmis that the partial 2D study of the pole and air-gap cm be justified. This is

rnainly due to the moderate rnagnetic field kvet in the pole, the proximity of the air-gap

region, and the cylindtical shag of the pole. The 2D axisymmetric study conveniently and

efficiently yielded satisfactory nsults in the air-gap. Such solutions can provide useful

engineering idormation about the deviœ in its initial stages of the design and study.

Figure 4.5: Robkrn domain of the partiai 2D (axisymmtricai) mode1 of the pole, the air- gap. and the coil.

Chapter 4. Study of the MRI Mrignet with 2D FEM 108

Figure 4.6: The contour plots of the rnagnetic flux density magnitude at the pole and air-gap (top) of the MRI rnagnet. The 2D partial mode1 of the coil and the pole was used.

Figure 4.7: The average of the magnetic flux density at the pole and air-gap. The 2D partial mode1 of the coil and the pole Figure 4.5) was used.

Chaptcr 4. Study of the MRI Magnct with 2D FEM 109

4.4 A Novel Composite 2D Modeling Method

The 2D partial model of the con. in Section 4.2. was shown to k satisfactory only when the

con was highly saturated. The partial 2D model of the pole and air-gap was examined in

Section 4.3, and the results were shown to be acceptabie. In those studies, the yoke was

excluded from the models. The main objective of this section is the study of a composite 2D

axisymmetric mode1 that includes all the components of the MRI magnet. It is shown that

such a novel modehg approach yields acceptabk solutions across the core and air-gap of the

MRI rnagnet.

Developmnt of the composite model is based on the following:

1) The hiil 3D analysis result shows that the yoke magnetic field is always between the

highly saturated levels of the con, and the unsahiratcd levels of the pole.

2) The cross-sectional ana of the yoke is 80% of the core cross-section.

3) The yoke is assumd as a rnagnetic transmission media that does not greatly affect the

magnetic field at the ngions of interest.

4) The yoke provides a magnetic path whereby magnetic mix closes on itseif. The

simpiified composite model should provide an altemate geometry for the same funaion.

5) Cylinders can approximate and simulate the actual yokes. The cyiindrical con, the

simulated yoke, the pok, and the air-gap aü have a cornmon axis in the composite model.

6) The material and radjus of the simulaicd cylindricai yoke are similar to the core and pole.

The 25% increase of the yoke cross-sectional area translates to a lower magnetic field in

the simulated yoke for the same transmitted rnagnetic flux.

7) The kngth of the simulated yoke is assurned to be equal to the man magnetic path of the

yoke in the 3D rnagnet.

8) At this point, ail the compomts of the composite probkm have a cyündrical symmetry

and can be conveniently modeled by a 2D axisymmeaic appmach.

Thecefore, a new probiem is defmed that includes the core, the simulated yoke, the pole.

md the air-gap. The problem domain of this composite MRI magnet is defimd in the next

subsection. Then, the O-MN magnct (where the coil is around the core) is modeleci and

solved. The nsults and conclusions are given in Subsection 4.4.2. Subsection 4.4.3 studies

the 2D composite rnodel for the AMRI magnet (when the coi1 is around the pole.) The

objectives of each study are:

O To determine the extent and conditions to which the 2D composite modeiing approach

provides useful, accurate and acceptable solution, and

O To calculate thne quantities of interest as the average of the magnetic flux density (1) at

the con center plane, at z = O of the 2D model, (2) at the air-gûp center plane, at =

ZdOMk of the 2D rnodel, and (3) dong the air-gap centerhe, at r = O of the 2D model.

These magmtic flux indicators are compared against the 3D counterparts in order to evaluate

the accuracy of the 2D composite moâeling approach.

4.4.1 Probiem Domain

The 2D problem domain is shown in Figure 4.8. Due to the symmetry, only half of the

domain is modeled in a cylurcirical coordinate system The material and the dornain

paramters, given in Table 4.4, were selected to sirnulate the same rnagnets as in the 3D

studies of O-MRI and A-MN rnagnets. The domain height, Lin, is the sum of the core, the

shulated yoke, the pk, and the air-gap lengths. The domain width, RdOMUi, was selected

according to the optimizing relation of Rdomh 2 4edoIMh in (4.2). A total number of 1 5 5 0

elements was used to discretize the probkm domain. Simüar solutions were obtained when

the domain grid was refïned and/or the domain width was increased. Therefon, this model is

the optimum composite rnodel of the MRI magnet.

Table 4.4: The problem domain parameten for the composite models of the MRI magnet.

Domain width, &,min 7000 mm

Core/Yoke/Pok tadius, REon 250 irni

Col length ot con, Leoit 265 mm

Coü length at pok, L i t 1 15 mm

Coi1 excitation, Itcoil 60 kA-t

Con - Coü separation, S, 25 mm

h ~ ' n a i n heighi, zdoiMin 1700 mm

Cote length, L0, 300 mm

Yoke kngth, 1100mm

Pok length, bk i 50 mm

Air-Gap length LifigaP 150 mm

Coii width, Wmil 25 mm,

Chapta 4, Study of the MRI Magnct with 2D FEM 11 1

4.4.2 A Composite 2D Mode1 of the O-MRI Magnet

The first objcctive of this study is to verify the accuracy of the composite mode& approach.

The rnagnetic flux density indicators were computed f'rom the FE solution of the composite

2D mdel of the O-M'RI magnet, where the coil was phced around the core and its excitation

ievel was 60 kA-t. For this nominal cumnt. the results agree within 2%. Therefore, the

accuracy of the composite 2D model Û acceptable. The magnetic flux density indicaton are

sumrnarized in Table 4.5.

Tabie 4.5: The results €tom 3D and composite 2D models of the A-MRI magnet.

Figure 4.9 shows the contour lines of the rnagnetic fiux density magnitude at the core of

the mode1 and from the 2D analysis. A high level of magnetic saturation enists in the core,

and the field slowly and smothiy changes in the z direction. Figure 4.10 shows similar data

from the 3D analysis of the OMRI magnet. The magnitude and the changing pattem of the

rnagnetic field in both graphs are very similar. It is concluded that the 2D composite model

efficiently reprcsents the core of the O-MN rnagnet.

The contour h e s of the magnetic flux density magnitude at the pole and the air-gap of

the 2D mode1 are shown in Figure 4.11. The regions and view directions were show in

Figure 4.8. The rnagnetic field is uniform in the center region of the air-gap. The pole is not

magnetically saturated. Figure 4.12 shows simüar data from the 3D study of the MRI

magnet. It is concluâed that the 2D composite model cm efficiently compute the magnitude

and pattem of the mgnetic flux density h the air-gap of the magnet. The 2D composite

mode1 also yields acceptable magnetic &Id data in the air -gap side of the pole.

The ncxt objcctive of the study is to find the extent and conditions under which the

composite 2D mdcling is valid. The 2D mode1 was solved for the coü excitation kvels of 10

to 60 kA-t. The magnetic flux M a t o r s werc calcuiated and compared with the results ftom

Bato iomiion T

ulo~g uir-gap

B atm air-8,

B ucrm mm

2 0 3D . Error

0.2886 0.2909 1%

0.2605 0.2658 2%

2.2307 2.1913 2%

Max. Error for current of 10 to 60 kA-t

1.5%

2.5%

4%

the 3D analyses of the O-MRI magnet. F i p 4.13 shows the average rnagnetic flux density

across the core. The error does not exceed 4%. Figure 4.15 shows the average magnetic flux

dcnsity across and dong the The discnpancy between 2D and 3D result~ is less than

2.5%.

Conclusion:

The composite 2D modeiing approach provided acceptable solutions when used to study the

OMRI magnet. The magnetic flux density at the air-gap and core were conveniently

obtained from the composite 2D model. The model also yielded reasonable results for

dinerent excitation levels of ihe coil. Thenfore. the 2D composite modeiing is concluded to

be an effective approach for approximating the OMRI magnet and simdar devices.

Air-Gap

n Coi1 ( m v e m u d ihc Pole: A-MRI)

1 Flux NomiPI Bwndary

Figure 4.8: Domain dennitions for composite 2D (axisymrrietric) models of the MRI magnet. The c d is placcd 6 t h around the core (GMRI) or amund the pole (A-MRI).

Chpiet 4. Sndy of the MRI Mamet with 2D FEM 113

Figure 4.9: Magnetic flux density contour line et the are of the OMRI magnet using a composite 2D model. The coil is around the con and its excitation level is 60 kA -t .

figure 4.10: Mpgnetic flux density contour liw at the CO= of the O-MRI magnet using a 3D model. The coil is around the c o ~ and its excitation level is 60 kA-t.

Figure 4.1 1 : Magnetic flux density contour h e at the pok and the air-gap (top) of the O- MRI mapnet using a composite 2D model. The coi1 excitation level is 60 kA-t.

Egure 4.12: Magnetic flux density contour line at the pole and the akgap (top) of the O- MRI mgnet using a 3D maiel. The coi1 excitation Ievel is 60 kA-t.

Chapta 4. Study of the MRI Magnet with 2D FEM 115

Figure 4.13: The average of the magnetic flux density across the core of the O-MRI magnet using 2D and 3D FE models. The coil is around the core.

Figure 4.14: nie average of the magnetic flux âensity actoss and dong the air-gap of the 0- MRI mgnet using 2D and 3D FE models. The coil is amund the corn.

Chapta 4. Study of the MRI Magnet with 2D FEM 116

4.4.3 A Composite 2D Mode1 of the A-MRI Magnet

The composite 2D modeling approach was appiied to the O-MRI rnagnet, and the solutions

were acceptable. The alternative design of the MR[ rnagnet (A-MRI). which was introduced

in Section 3.4. is a simiiar but less difilcuh device than the O-MRI magnet. Therefore, it is

suggested to use the composite 2D modehg approach and to study the A-MRI magnet. As

kfore, the main objective of this study is to detemine whether the composite 2D model can

yield usefbl and accurate results. T b other objectives are. first, to obtain the magnetic flux

density at the core and at the air-gap, and second, to fuid the extent and conditions to which

the 2D approximation is justifîed.

The composite 2D model of the AMRI magnet and the boundary conditions are shown

in Figure 4.8, page 112 - where the coil is located around the pole. The magnetic flux density

indicators of the FE solutions. at the coil excitation level of 60 kA-t, are summarized in Table

4.6. The solution of the composite model is found similar to the solution of the 3D analysis.

Table 4.6: The results h m 3D and composite 2D rnodels of the O-MRI magnet.

cxtcnt and conditions where the 2D composite mode1 is valid, the coil

excitation level was varied from 20 kA-t to 120 ICA-t. The study indicators were then

calculaied and cornparcd with resuhs from the 3D analyses of the A-MRI magnet. Figure

4.15 shows the average rnagnetic flux density across the corn, across the air-gap, and dong

the air-gap. The disctcpancy of the results does not exceed (a) 6% for the nominal current

excitation of 60 kA-t, and (b) 8% for the fidl range of the cumnt excitation.

Figure 4.16 shows the contour ünes of the magnetic flux density magnitude at the core of

the rnodel. This result from the 2D analysis shows th t the core is rot saturated. Figure 4.17

shows simüat data fiom the 3D analysis of the A-MRI magmt. Although the prof'îie of the

i

2D 3D Error

0.4596 0.4840 5%

0,3947 0.4193 6%

0.8243 0.8350 1.5%

Max. Error for curent of 20 to 120 kA-t

5%

6%

8%

Chaptu 4. SNdy of the MRI Magnet with 2D FEM 117

magnetic field from 2D model is di&rent fkom the 3D model, the averages of the magnetic

flux density nom the two models are very close.

The contour iines of the mgnetic fiux density magnitude at the pole and the air-gap of

the model are shown m Figwe 4.18. The magnetic field is unifonn in the center region of the

air-gap. The pok is not magnetically saturated. except at the corners. Figure 4.19 shows

similar data fkom the 3D study of the A-MRI magnet. The magnitude and the changing

pattern of the rnagnetic field in b t h graphs are similar. It can be concluded that the 2D

rnodel efficiently computes the magnitude and pattern of the magnetic flux density in the air-

gap and the pole of magnet.

Conclusion:

The magnetic flux density for the pole and the air-gap of the model were acceptable for

different excitation levels of the coü. The con magmtic field nsults were also acceptable for

coi1 excitation levels of kss than 80 kA-t. It is therefore concluded that the composite 2D

rnodeling approac h provides satisfac tory solutions for the A-MRI magnet design.

I I I I a 6u 10 t 00 120 - t K M l

Figure 4.15: The average mgnetic flux density across the core. dong the air-gap and across the air-gap. The A-MRI magnet is solved with 3D and 2D FEM, when the col is located around the pok.

Chaptcr 4. Stdy of the MRI Magne wiib 2D FEM 118

Figure 4.16: Magnetic flux density contour iine at the core of the A-MRI magnet using a composite 2D model. The coil is around the pole and its excitation kvel is 60 kA-t.

Figure 4.17: Mngnetic flux dcmity contour line at the eorc of the A-MRI magnet using a 3D model. The coil is around the pole and its excitation level is 60 kA-t.

Chaptcr 4. Study of the MRI Magnet with 2 0 FEM 119

Figure 4.18: Magnetic flux density contour ihe at the pole and the ai~gap (top) of the A- MRI magnet using a composite 2D model. The coi1 excitation level is 60 kA-t .

Fîgure 4.19: Magnetic flux density contour iine at the pole and the abgap (top) of the A- MRI magmt usïng a 3D model. The coi1 excitation level is 60 kA-t.

Chapta 4. Study of the MRI Magnet with 2D FEM 120

4.5 Conclusions

The primary objective of

which yield acceptable 2D

this chapter was to

modeis for magnetic

introduce and ernploy simplifying methods,

devices such as the MRI magnet. The partial

and composite 2D modeling appmaches were then inuoduced.

The 2D partial mode1 of the core only yielded an acceptable result when the core was

highly saturated. However, the 2D partial mode1 of the pole and air-gûp yielded reasonable

results in the air-gap. This was due CO the inclusion of the air-gap in the model and &O to the

moderate magnetic tield level in the magnetic poles.

Fiaiîy, the composite 2D model was shown to be the most practical 2D modeling

approach, giving acceptable results in the air-gap and the core of the rnagnet regardless of the

coil location and excitation level. The power of this model is that it can be used cost

effectively to explore design alternatives prior to undertaking a full 3D modeling effort.

The composite 2D rnodeling approach can k used for the study of other 3D physical

system that do not have a complete symmetry and cannot k modeled in 2D. The non-

symmetrical part of the system can be properly substituted and the resultant simulated system

can be studied.

Chapter 5

Conciusions

This thesis proposed to use Ampere's circuital law to obtain a globai occuracy measure for

asscssing the comctness of a FE mode1 and solution. The open-concept MRI magnet was

used as a basis for the study. This device includes a large air-gap, and a large and saturated

magnetic structure. A selection strategy and selection criteria were introduced to aid in the

study of the MRI rnagnet. A precise 3D modehg method, partly based on the use of a

constant permeabiüty mode1 of the saturated magnet. was developed and then used to obtain

the most accurate model of the MN magnet. This model was analyzed for Merent

operational conditions. An alternative design of the MRI magnet was then introduced and

studied. The thesis also pnsented 2D modeiing approsches to sirnpi8y the FE study of

magnetic devices that do not have a cornplete symmetry and cannot be modekd in 2D.

5.1 Contributions

The main contriiutions of the thesis are as follows:

a) The application of the Ampcre's law enor to evduate the accuracy of a 3D FE mode1 has

been extensively investigated. The conditions under which this global accurucy critenon

is reliable have bem obtained.

b) The applications of différent selection criteria have ken examined. Both global selection

criterion, e.g. the total energy stored in the model, and the semi-global selection

criterion, e-g. the energy stoced in a region of the mode1 or the integrai of the magnetic

Chaptcr 5: Conclusions 1 22

flux density over a region of the moôel, were shown to be practical in identifjmg the

optimum FE mode1 of a problem

C) A novel equivaient linear 3D modeiing approach'has been introciuced. Its appiication in

(a) evaluation of the model global accuracy. and (b) enhancing the efficiency of the

selection search and study, was shown. It was also shown that the global accuracy of the

nonlinear 3D model (Ampere's law error) of a problem could be estimated from the

equivalent linear 3D model of the probkm

d) A stable 3D modeiing approach has been developed that expands a 3D volume of a

device to a spherical volume and exterior with a limited number of elements. The main

advantage of this modeling approach is that it provides a representation of the infmite

boundary condition w hen infinite elements are used.

e) A 2D composite modehg approach has k e n introduced that simplifies the study of

magnetic devices that cannot be completely mdeled by conventionai 2D rnethods. The

2D composite model of the MRI magnet was shown to k a practical and reüable model.

giving acceptable solutions in the air-gap and the core of the MRI magnet.

f) Open concept MRI magnets, of the type that was considered in this thesis, conventionally

have the excitation coi1 located far from the air-gap. By sirnply moving the excitation

coil(s) to the pole region of the magnet, it was shown that the available air-gap flux

density, per unit of excitation, could k increased by almost 60% without signifcantly

compromising flux density uriiformity.

5.2 Conclusions

The foiiowing conclusions have ken drawn from the studies conducted in this thesis:

1) Arnpere's law enor has been show to k an acceptable globo1 accuracy meusure,

provided that its path dependency has been addressed. It was found that the error dong a

specific path could be muumized without a general improvement in the quality of remlts.

This limitation calkd for a statisticai exambation of the ercor dong different paths.

The average of Ampere9s law emrs dong di&rent paths has been shown to à a

better global accuracy meanire than the enor dong a singk path. However. such a

Chapta 5: Conclusions 1 23

rneasure could not properly address the wide range of errors along difkrent paths. The

combination of the standard deviation and the average of Arnpen's law emrs along

daerent paths have k e n show to be an appiicable global accuracy measun for

evaiuating the FE solution.

2) The total magnetic energy stond in the FE mode1 of the device was shown to be a

tmstworthy global selection criterion. The selection direction is predetemined and is

toward the minimum energy content of the model.

The magnetic energy stored in specific regions of the FE model was shown to k a

valid semi-global selection criterion. The selection direction of such a masure depends

on the chosen ngion and should be obtained for each problem and region.

3) The average of the magnetic field at.a specifk region of the FE mode1 was found to be a

vaüd semi-global selection criterion. When the region was selected within one material

neighborhood. the selection direction could be uniquely determined.

4) The equivalent hear 3D model of the MRI rnagnet - where the nonlimar material of the

magnet was replaced with an equivaknt linear matetiai - was concluded to be efficient in

the search for better models of the device.

5) The original design of the MRI magnet, where the coi1 is around the core, was found to

be inefficient. Less than 20% of the total energy was stored in the magnet and air-gap.

and only 12% of the cote magnetic flux reached the air-gap.

The alternative design of the MRI magnet, where the coils are placed around the

poles, was concluded to k mon efficient than the original design. It provided 58% more

magnetic flux across the air-gap whik using 33% less magnetic energy.

6) It was show that a curnnt sheet coü could effectively simulate the thin and long

superconducting coii of the MRI rnagnet.

7) The 2D composite modehg approach was found to be a useful and practical approach

for simplifying the study of 3D 'mgnetic devices that cannot be modeled by conventional

2D methods.

Chaptcr 5: Conclusions 1 24

5.3 Future Work

Further research studks, which cah be adhssed by appiication of mthods ar~d propos&

discussed in this thesis, are suggested. These are:

The MRX magnet can be studied with different materials for the core, yokes and poles.

These materials can be a permanent magnet, powder material. or magnetic materia of

different characteristics.

The design of the M N magnet can be modfied in more detaiis. The geometry of the air-

gap, poles, yokes and core can be ndesigned, and the device performance cm be studied.

The 2D partial and composite rnodehg approaches can be used to study smaller

magnets, and magnets with linear material.

The effect of the yoke length in the composite 2D mode1 of the MRI magnet can be

studied.

The rnodeiing approach used in this thesis, together with its principles, can k used for

the ac analysis of magnetic devices, such as inductors with large air-gaps that are used in

the power systems of moderate to large size industrial networks.

References

1. J.P. Webb, B. Forghani, "Adaptive Improvemnt of Fields using Hierarchal

Tetrahedral Fite Elements," IEEE Tmnsoctions on Magnetics, Vol.30, No. 5,

September 1994. pp.3511-35 14.

2. V.N. Panhasarathy, C.M. Graichen, A.F. Hathaway, "A cornparison of tetrahedron

quality masures." Finite Elements in Analysis and Design, Vol. 15. 1993, pp.255-

261.

3. Marcelo G. Vanti, Adroaldo Raizer, Joao P.A. Bastos, "A Magnetostatic 2D

Cornparision of Local Error Estimation in FEM," IEEE Transactions on Magnetics,

Vol.29. No. 2, March 1993, pp.1902-1905.

4. A. Bossavit, " How Weak is the 'Weak Solution" in Finite EIement Method?' IEEE

Transactions on Magnetics, Vol. 34, No. 5. September 1998, pp. 2429-2432.

5. K. Koibuchi and K. Sawa, "A New Posteriori Error Estimation Concept for Three-

dimensional Finite Element Solution," IEEE Transactions on Magnetics, Vol. 35, No.

3, May 1999, pp. 1446-1449.

6. P. Lauterbur, "Image Formation by Induced Local Interactions: Exampks Ernploying

NucIear Magnetic Resonance," Nature, Vol. 242, 1973. pp. 1% 19 1.

7. R.Damadian,'TumrDetectionby~"Science,Vol.171,1971.pp.1151.

8. Z.H. Cho, H.S. Kim, HB. Song and James Cuniming, 'Fourier Transfonn Nuclear

Magnetic Resonance Tornographic Imaging," IEEE Proceedings, VOL 70, No. 10,

Oct. 1982, pp.1152-1160.

OC. Zienkiewicz, J. Lyness, and J.R. Owen, 'Three Dimensionai Magnetic Field

Determination ushg a Scalar Potential," IEEE Transactions on Magnetics, Vol. 13,

1977, pp. 1649-1656.

M. Gyimsi and J.D. Lavers, "Generalized Potentid Formulation for 3-D

Magnetostatic probkms," IEEE Transactions on Magnetics, Vol. 28, No. 4, July

1992, pp. 1924- 1929.

M.V.K. Chari, P.P. Silvester, A. Konrad, and M.Ac Pairno, b'Three-Dimensional

Magnetostatic Field Analysis of Electrical Machinery by the Fiite-Elernent Method,"

IEEE Trunsactions on Power Apparatus and Systems, Vol. PAS- 100, No. 8, August

198 1. pp. 407-4019.

N.A. Demerdash, T.W. Netil, F.A. Fouad, and O.A. Mohammed, 'Three Dimensional

Fite Element Vector Potential Formulation of Magnetic Fields in Electrical

Apparatus,'' IEEE Transactions Power Apparatus and Systems, VOLPAS- 100, No. 8,

Aupst 1981, pp. 4104-4113.

O. Biro and K. Preû, ''On the Use of the Magnetic Vector Potential in the Finite

Element Analysis of Three-Dimcnsional Eddy Cumnt." lEEE Transactions on

Magnetics. Vo1.25, Na. 4, July 1989, pp.3145-3 159.

R. Wang and N.A Demerdash, "A Combined Vector Potential - Scalar Potential

Method for Computation of 3D Magnetic Fields in Electrical Devices with Iron

Cores, " IEEE Transactions on Magnetics, Vol. 27, No. 5. September 199 1, pp. 397 1 - 3977.

Igor Tsukerman, Tomparison of Accuracy Cnteria for Approximation of

Conservative Fiel& on Tetrahedra," IEEE Transactions on Magnetics, Vo1.34, No. 5,

Sepemkr 1998. pp.3252-3255.

P.Alotto, A Castagnini, P. Girdinio, P. Fernandes, 'Enor estimation and Adaptive

Meshing m 3D Electrostatic and Magnetostatic Probkrns," IEEE Transactions on

Magnetics, Vo1.34, No. 5, Septemôer 1998, pp.3260-3263.

Fc W. Grover, Iductmm Calculations, Dover pubtications, 1962, P. 143.

J.C. Nedelcc. 'Mxed tinite elements in R~," Nunier. Meth, Vo1.35, pp. 3 15-341,

1980-

A. Bossavit and J.C. Verite, "A rnixed FEM-BIEM method to solve 3-D eddy current

problems," ZEEE Transactions on Magnetics, Vol. 18, March 1982, pp.43 1 -435.

A Kameari, Wculation of Transient 3D Eddy Cumnt &ing Fiiite Elements,"

IEEE Trunsuctions on Magnetics, Vo1.26, No. 2, March 1990, pp.466-469.

O. Mur. 'The fallacy of edge elemnts." IEEE Transcctions on Magnetics, Vo1.34,

No. 5, September 1998, pp.3244-3247.

N.A Dernerdash and R. Wang, 'Theoretical and Numrical Difficulties in 3-D

Vector Potential mthods in F i e Element Magnetostatic Computation," IEEE

Transuctions on Magnetics, Vol. 26, No. 5, September 1990, pp. 16564658.

J. Jin, The Finite Element Method in Electromagnetics, New York: Wiley , 1993.

P.P. S ilvester and R.L. Ferrari, Finite Elemen ts for Electrical Eng ineen, Cambridge

University Press, 1990.

J.E. Akin, Application and Implementation of Finite Eiement Methods, Academic

Press, 1982.

A.R. Mitchell and R. Wait, The Finite Element Method in Parrial Differential

Equations, John Wiky & Sons, 1977.

O.C. Zienkiewicz, C. Ernson and P. Bettess, "A Novel Boundary Infinite Elemtnt,"

Intematrond Journal for Numerical MethodF in Engineering, Vo1.19, pp.393-404,

1983.

Hui Li, Sunil Saigal, Ashraf Ali and Timothy P. Pawlak., "Mapped Infinite Elements

for 3-D Vector Potential Magnetic Problems," International Journal for Numerical

Methodc in Engineering, Vo1.37, pp.343-356, 1 994.

S-RH. Hoole. Computer-Aided Analysis and Design of Electromugnetic Devices,

Elsevier Science Publishing Co., 1989.

J.T. ûàcn, Finite Elements of Nonlinear Continua, McGraw-Hili, 1972.

G. Mahinthakumar and S.R-H. Hook, "A Paraüeîized Element by Element Jacobi

Conjugate Gradients Algorithm for Field Problems and a Cornparison with ûther

Schemes," Applied Electromgnetics in Materiah, Vol. 1, pp. 1 5-28, 1990.

NOTE TO USERS

Page(s) not included in the original manuscript are unavailable frorn the author or university. The

manuscript was microfilmed as received.

This reproduction is the best copy available.

UMI

Appendix B

Cl006 Characteristic Cumes

Figure B. 1: The B-H characteristic curve of Cl006 used in the M N magnet.

Figure B.2: The v-B' characteristic curve of Cl06 used in the MRI mgnet

Magnetostatic F i ~ t e Element IvIethod in 3D

The Finite Ekrnent Method is a numerical technique for obtaining approximate solutions to

boundary value problems in mathematical physics. The principle of the method is to replace

the entire continuous domain of the problem by a number of suô-domains in which the

unknown hinction is approximated using interpolation functions with unknown coefficients.

Thus the solution of the whok system is approximated by a finite numkr of unknown

coefficients.

The theory of the Fuite Element Method has ken studied and reported in the üterature.

Software packages arc also avaüabk that effectively impiement the mthod. ANSYS is one

of the leading FEM software ptogram and is used exclusively in this thesis. The main

objective of this chapter is to introduce and detd the 3D limar and nonlinear magnetostatic

Fîîte Ekment Method cornpmhensively.

There are two reasons for inclusion of this appendix. First, to use a method to solve a

problem correctly and effectively, the mtthod itself has to be understood. The Finite Element

Method incorporates rnany steps that one has to understand in order to effectively impkmnt

and solve a problem. The use of a FEM package to solve a problem seems very trivial at tirst;

however, as soon as a difficulty appears in any stage approaching the final and comct

solution of the probkm, the reasons for the faüun should be sought. A clear and

comprrhensive knowltdge of the mcthod and its seps are necessary to trace and to

understand the source of the method faüun. This knowledge is more criticai when d&g

with ditiicult pmbkms that require the utmst capabilitks of the FEM application. The MRI

mgnet has k n show to be a dinruit probkm, which is causeci by nonlmarity and the

high saturation level of the magnet. the huge size of the magnet, a very large air-gap, and a

high fkinging of the mgnetic field to the surroundhg free-space regions. Thus dl the FEM

steps are detailed in this appendix.

The second nason for the inclusion of this appendix relates to the 3D nonlinear FEM

miplemntation steps. They have been discussed in the literature and in the software

manuals; however, some steps are discussed in detail while other steps in the modeling

process are often omitted. References are usuaiiy made to multiple sources in the literatures,

and the FEM user does not obtain a complete pictun of the method without an exhaustive

search of old and new titerature in darerent disciphes and with different terminology. Thus

the implementation steps. numerical sub-steps. and iterative routines are clearly detaikd in

order to provide enough mfonnation and guidelines for a person to write a program and to

solve a problem.

The FEM analysis of a boundary value problem incorporates the following main steps:

Definition of the boundary value problem.

Formulation of the problem in t e m of Ritz's (variational) or Gderkin's approach.

Definition of the problern domain and its boundaries.

Discretization or suMivision of the domain by ekments and nodes.

Selection of the interpolation hinctions for each category of the used elemnts.

Formulation of the system of the equations in linear or noniinear t e m .

Solution of the system of equations.

Evaluation of the h a 1 results.

The proper Urpkmentation of each step is necessary, although there is not a clear-cut

division ktween the steps. Information as to how these steps affect each other, even at the

stages very close to the final solution, is not widely available. The whole mode1 and all the

steps may k modincd at any stage due to a faihire of the FEM to achieve an acceptable

solution.

This appendut focus is on the whole forxnulation and numerical sequence of the FEM,

and thus, each of the FEM steps is discussed in one separate section. It should be noted that

aii of the FEM steps arc impkmnted by software. however, some steps are un&r the control

Apmndix C. Mametostatic FEM in 3D 133

of the FEM user. These steps (step 1,3,4,5) wen detded in Chapter 2 and are briefly

âiscussed here.

The formulations are customaed for the purpose of this thesis: th=- magnetostatic analysis

of the MRI magnet. This thesis uses both hear and nonlinear analyses of the MRI magnet,

and therefore. the formulation and method is generally given and discussed for both hds.

The solution to a nonlinear probkm is given separately by the Newton-Raphson mthod in

the system solution step.

C. 1 Boundary Value Problem Definition

Mathematical modeling of physical systems typically yields boundary value problems. Any

boundary value problem cm k defined in terms of:

1. the boundary conditions on the boundary rthat enclose the problem dornain D. and

2. a governing differential equation in the domain D as

Lu = f , (CA 1

where L is a diffenntial operator. u is the unknown quantity, and f is the excitation or forcing

fùnction. The form of the Werential equation and unknown are probkm dependent.

Analytical solutions to boundary value problems are availabk for oniy a few cases.

Therefore, approximation methods have ken developed to numricaily obtain the unknown

quantity. The Rayleigh-Ritz and the Galerkin's mthods are usually used to approximate the

solution. It should k noted that both methods yield a sirnilar system of equations for a

problem. The Rayleigh-Ritz mtthod that is preferred in this thesis is discussed in Section

There are msinly two mthods for denning a FE boundary value problem in

ekctromagnetics. The difference is in the rlection of the unkwwn quantity u when posing

the boundary vahie probkm. In one method, components of the magnetic field are used to

de& the probkm This nvthod uses edge or face based elements. It uses a vector b i s or

vector elements and assigns degrecs of fkeedom to the edges rather than to the nodes of the

elemuits [18-201. The edge ekments have not been used and discussed in this thesis, and the

detailed discussions are given in the literature [21].

In the second method, magnetic potentials are used to define the boundary vkue problem.

Selecting the unhiown quantity u as the magnetic potential has an advantage kcause it can

be chosen to be continuous across interfaces between different media Therefore, mculties

that rnay arise duc: to the discontmuity of the magnetic field or the magnetic flux density

across those surfaces cm be avoided. This method. which assigns the degrees of fieedom to

the nodes rather than to the edges of the elemcnt. is nfemd to as r node based Fiite

Ekment Method. The thesis uses the nodal FEM based on the Magnetic Vector Potential

(MW) [13]. This is discussed in the following subsections after a review of Maxwell's

equations. The FEM based on the Magnetic Scaiar Potential (MSP) were discussed briefly in

Section 2.4.1.

In electromagnetics, the Merential operator L, the unknown quantity u, and the forcing

function f are ail obtained fiom Maxwell's equations, and the selection of u dehes the final

form of the differential operator L. The m e n t i a l forms of Maxwell's equations descriiing

the magnetostatic field are

VxH = J (Ampere's law), (Ca

V-B = O (Gauss's law), c . 3 )

where H is the magnetic field intensity vector, B is the magnetic flux density vector, and J

is the cumnt density vector. Using p for the permeabüity and v for the reluctivity of the

medium, the constitutive relation is writtcn as

B = pH or H = vB.

The boundary conditions at the outer bounâacy f are

M = O (normal field on rH), B-fi = O (paiallel flux on TB),

Appendix C. Magnetostatic FEM in 3D 135

where ri is the unit vector normal to the boundary, and f = TH u f0 . The field continuity

conditions at the interface between the two media are

&(a+ -H- ) = 0 (tangential field continuity), (c.7)

&(B' -B- ) = 0 ( n o d flux continuity), (C-8)

where superscripts + or - refer to Merent sides of the discontinuity interface.

C.l.2 Magnetic Vector Potential

The FE modeling of the static magnetic fiekt is traditionally defmed in terms of magnetic

potentials. The problem can be defined in tenas of the magnetic scalar potential [9-101 or the

magnetic vector potential [Il-131, or the combination of the scalar and vector magnetic

potential [14]. The magnetostatic FEM based on the magnetic vector potential is used in this

thesis and detailed in the following. The reasons for not using the scalar and combined

potential formulation FEM ace given in Subsection 2.4.2.

Maxwell's Equations. given in (C.2) and (C.3), govern magnetostatic field problems.

Since the magnetic flux density B is a divergence free (or solenoidal) field, it can be

represented in terms of a magnetic vector potential A as

B = VXA. C.9)

Substitution of (C.9) in (C.2) with the aid of (C.4) yields the second order differential

equation

Vx(vVxA) = J, (C. 10)

which defines the magnetostatic boundary value problem on its domain D. Equation (C.10)

does not determine a unique solution of A. If A is a solution to (C. 10). then any function that

can be mitten as A'= A+Vf is also a solution of (C. 10). That is due to the vector identity of

Vx(V f )=0. A gauge condition can k imposed on the divergem of A in order that (C.10)

determines A uniquely. One of the most used gauges is the Coulomb gauge defied as

V-A = 0 (Coulomb Gauge). (C. 1 1)

The necessity of enforcing the Coulomb gauge has been a very controversial subject (221.

However, if the ultimate goal of the probkm study is to compute the magnetic flux density B

Appcndix C. Magnetostaiic FEM in 3D 136

or the energy associated with the pmblem domain, the uniqueness of A does not affect the

uniqueness of the magnetic field or energy [23]. There is not a final resolution on the

dilemma of the gauge problem f i e grnerai definition and fornulahon of thé problem is

conventionaiiy done with the use of the Coulomb gauge [13,23] and is outlined below.

C.1.3 Problem Description

The boundary value problem c m be described by ushg vector potential A that satisfies the

vector differential equation of

Vx(vVxA) - V(vV-A) = J, (C. 12)

which is obtained by enforcing the Coulomb gauge on to (C.10). The boundary conditions at

the outer boundary Tof the problem domain D are of two kuids

&A = P Dirichlet condition on ri, (C. 13)

&(VxA) = O Neumann condition on f i , (C. 14)

which are applicable at planes of symmetry and f = ïl u T2 . The continuity conditions

applied at the interface betwetn two media of Werent pemeability or nluctivity are

v'iixVxA' = v'&VxA' Continuity of tangentiai 8, (C. 15)

fi- VxA ' = 2, V x A' Contuiuity of normal B , (C. 16)

iixA' = AXA- Continuity of A. (C. 17)

where superscripts + or - refer to different sides of the discontinuity interfxe Sd.

C.2 Variational Formulation

Two of the rnost cornmon approximation mthods for solving a boundary value problem - which was defined by (C.12) to (C.17) - are Galerich's mthod and the Ritz method.

Galericin's method belongs to the W y of weighted residual methods. which seeks a

solution by weighting the rcsidual of the dfirential equation (C.1). The best approximate is

the solution that d u a s the residual to the kast value over the problem domain [23].

Appcndix C. Magnciosiatic FEM in 3D 137

The Ritz method is a variational mthod in which the boundary value problem is

formulated in terms of a variational expression or tùnctiond. The minimum of the hinctional

corresponds to the governing di&nntial equation under the given boundary conditions. The

approxhïite solution is obtained by minimizing the functional with respect to its variables.

This method is included hem to complete the theory of the magnetostatic FEM.

According to the variation phciples, the solution to (C. 1 2)-(C. 1 7) can be obtained by

extremizing the modified energy functional [11,23]

F( A ) = + ~ V ( V X A ) - ( V X A ) W - ~ v J - A N + + J Y V(V.A)'~V , (c. 18)

under the condition given by (C. 13) and potential continuity (C. 17). The first and second

tenns of (C. 18) are standard energy hinctions, and their integrals yield the stored energy in

the volume. The last terni of ((2.18) is the enforcernent of the Coulomb gauge. The ikst

variation of the fiinctional is used to fhd

Using the vector identities

X*VxY = V*(YxX)+Y - V x X . (C.20)

aVeX = Ve(aX)-&Va, (C.2 1)

for the first and third te- of (C. 19). respectively, yRWs

S F ( A ) = ~ v V * ( ~ A X V ( V X A ) ) ~ V + [ v 6A - V X V ( V X A ) ~ V - / v JbSAdV

+/'v*(vv*A G A ) ~ V - 1 v S A - V ( V V * A ) ~ V . (C.22)

Applying the divergence theorem.

to the £kt and fourth te=, and then rcgrouping the second. third and ffih terms of (C.22)

SF(A)=J'[VX(VV xA)-V(VV .A)-J]iTAdV (C.24)

+IP, n^-(GAx(vVx A))& +fi &(vV AGA)&. St

Using the vector idmtity of X -(Yx Z) = -Y *(XX Z)and X- uY = UX Y yields

The surface integrals, which result from application of the divergence theorem (C.23). should

be taken carefuliy over the surface of each media in the dornain. Assuming two volumes of

different reluctivity in the domain. where their interface discontinuity surface is Sd . the

integral dornain S, cm be ananged as

S&cml Volwncs = SEwmu~l V I + ~ ~ t r r n i t r l V Z = fi+& = r- Since Dirichlet's conditions (C. 13) are prescribed over ï l , f i x a vanishes on i'' and ((2.25)

changes to

Then the solution is obtained when the stationary requirement @(A)=O is iniposed for any

arbitrary variation of 6A. It is then evident that A must satisfy:

The boundaiy value probkm as (C. 12) Vx(vVxA) - V(vV-A) = J ,

The continuity of the tangentid H (C. 15) on Sd: v'îkVxA+ = v-rû<VxA-,

The Neumann boundary condition (C. 14) on r2: &(VxA) = O,

The Coulomb gauge on Sd: v'V-A' = v'V-A-,

The Coulomb gauge (C. 1 1) on T: VeA = 0.

The continuity of the n o d B on Sd (C.16) is aiready satisfied from the continuity of A in

(C. 17). The Dirichlet condition on ri (C.13) should k forced on the solution. Thereforc, the

approximatc solution of the boundary vahie problem - which is defintd in Section C. 1 -3 - can

bc obtaincd by mbimizbg the variationai functional (C. 18).

A ~ ~ c n d i x C. Maenetostatic FEM in 3D 139

Domain Definition

. . The domain in most electromagmtic studies contains the device and the sunounding fm-

space. The surrounding space should be defincd propcrly, depending on the type and nature

of the problem and the objective of the analysis. It is impossible to mode1 an infinite outer

surrounding space, and therefon, the dornain should be restricted within a reasonable space

around the device. A rnethod to compensate for the lost infinite space should be considered

beforehand. The so-calkd far field or infinite elements are usualiy employed to take the

infinite space hto account. Failure of the FEM to converge to either a solution or an accurate

solution necessitates the modikation, of both the domain definition and the domain

discretization. A complete discussion of the dornain definition was given in Subsection 2.1.4.

Domain Discre tization

The discretization of the domain is the most important and the most timeconsuming step in

any FE analysis. The manner in which the domain is discretized wiil affect the total element

and node numbea, and the cornputer storage requuemnts. It also defuies the computation

tirne, the software requirements, the convergence of the analysis to a solution, and the

accuracy of the numerical results. The efficiency or even propriety of the discretization

cannot k completely judged till the results are obtained and verified.

The problem domain is typicaliy discretized using one or more of the foiîowing

techniques:

1. Cornputer-aided discretization or frcemeshing,

2. SemLcomputer-aided discntization or extrusion,

3. Adaptive discrctization, and

4. Non-adaptive discrctization and modification.

Tbis thesis uses a combination of the extrusion and k-mshing techniques in order to non-

adaptivcly discrttiwd the MRI magnet problem These metho& were discussed and

àemonstrated m Subsection 2.1.5, Subsection 2.4.4.

Interpolation Functions Selection

The boundary value problem is defined in tenns of the magnetic vector potential A in Section

C.1, and then the proper variational formulation is introduced in Section C.2. The probkm

domain, which was dehed in Section C.3, is subsequently discretized into srnall elemnts of

hexahedral, prism and tetrahedral elements. The next steps of the FE analysis are, f~s t , to

select interpolation functions that provide an approximation of a scalar function withh each

element, and second, to choose a proper elemental coordinate system.

The interpolation or shape functions are usually selected to be polynomial of the first

order (lincar), the second order (quadratic), or a higher order. The iinear hterpolation

hinction is used in this thesis due to the ease and availability of its formulation,

representation, and appiication. A typical interpolation function for an elernent e having n

ial

where the continuous function f W approximated - at any point inside the elernent - by the

values off at the nodes of the elernent f;: and the continuous interpolation functions Nj.

Therefon, the hinction f can be differentiated or integrated with respect to global 3D

coordinates, as it is required in functional formulation (C. 18). Thus a local coordinate system

should be dehed to facilitate the dfierentiation and integration within the element. The

intrinsic coordinate systcm is the most commonly used system in the FEM.

A reference nomiaüzcd elemuit in the local intrinsic 3D coordinate system is defined by

thme coordinates (r,s,t) which are nomiPlized in the range of [-1,1]. The interpolation

functions are then defuicd in t e m of the intrinsic coordinates as &(r,s,t). Then,

dinerentiation w.r.t. global coordinates cm be carried out in terms of intrinsic coordinates.

Two kinds of ekmnts w m used in the analysis of the MRI magnet: the hexahedral elemnts

for the interior region, and the infinite elemnts for the exterior region. These ekmnts are

detaikd m the foiiowing discussion in order to help clarify the terminology that is used in the

calculation of the system of equations m the next step of the FE formulation.

Appcndix C. Magnecostatic FEM in 3D 141

. , Dflerent elernents of the modcl, which are t

defined in the gloM coordinate system, 3

should be mapped to the normaiizcd reference 8

elemnt in the intrinsic coordinate system.

This rnapping is ais0 necessary to express the

globai coordinates in t e m of the nitrinsic

local coordinates to facilitate the intcgration

of the variational fùnctional (C.18) over the

volumc of each elemtnt. The same family of Figure C. 1 : Eight-node hexahedral

interpolation hinctions that is u r d to element in intrinsic coordinates

a p p r o h t e the potential functions can also be used to express the ekment shape and

coordinate transformation. Such an ekment system is cded an isoparametric element system

[24-261.

An eight-node hear isoparametric hexahedral ekmtnt is show in Figure C.1. The

global coordinates and local inthsic coordinates of a typkai corner node, i, are (xi, Yi, zi) and

(ri, si, ri), respectively. The local interpolation hinctions or shape functions for this ekment

are

( N * ] , = ( ( 1 + r,r)(l+ s,s)(l + t i f ) +8 ) ; for i=l: 8, ((2.28)

whete ri, si, and tj are +1 or -1 for the node i. Thenfore, (C.27) can k written as

when the scdar function f is any quantity of interest such as the global coordinates x, y, z, or

any components of magnetic vector potential Ar, A , or A,. The isopararnetric shape functions

(C.28) sptisfy the nlations

A~~endix C. Maenetastatic i%M in 3D 142

Along the edges of the ekment, where -1, e l , or e l , the interpolation functions

(C.28) becom linear. Then, the function f e is defmed by the two comsponding nodal values

on that edge. Since the function f ' varies linearly on the edge of the element, the elemnt

with interpolation functions (C.28) is called a hear isoparamtric hexahedral ekment,

although the interpolation hinctions are trilinear inside the element.

Although the global coordinates x, y, and z of an isoparametric elernent are defued in

terms of local coordinates r, s, and t (C.28). a unique inverse transformation defining r, s, and

f in terms of x, y, and z is not usualiy needed. However, the relation bctween derivatives in

two coordinate systems is necessary. The Jacobian matrix relating two coordinate systems is

generally written as

Symbolically, the relation between the derivatives of a quantity, such as f, is also written as

Therefore, to evaluate the global and locai derivatives, the matrices J and J " should k

obtained. In practical appiications, these two quantities are evaluated numerically. For an

isoparamctnc element, the global cwrdinatt xi from (C.29), is

Applying (C.33) on the nrSt mw of the lacobian motrix yieids

Generaiizing (C.34) yields the ~acobian rnatrix as

- - - Lat at atJ,

The partial derivatives of the interpa

{"'a; S."}'

{ " s s . t , } *

dation functions

Jm

, which are also the mapping functions.

with respect to the local coordinate system are obtained in the matrix fom of

L 4

Thus (C.35) defines the Jacobian matrix J, at a local point inside a typical elemnt in ternis of

the spatial gbbai coordinates of the element nodes [{xe}, (ye), {?HP and the local derivatives

of the interpolation hnctions [A&, s, t)] (C.36). Therefore, at any point of interest, such as

the numtrical integrating points in an element, the matrices J, J -', and 1 JI cm be

numerically obtained.

C.5.2 Inlinlte Element

The mapped infinite brick elements are used to

discretae the exterior region of the MRI rnagnet

do&. The basic idea is to rnap the domain of a

replar finite elemcnt to an infinite element [27-28).

CA Node 1 Node2 Nodeid r=-1 r=O r=+ l

r

Figure C.2: Ont dimensional infinite element

In the one-dimensional case of Figure C.2, the coordinate transformation yields the

mapping hinction as

Then, the scalar hinction f e can be approximated with a second order function as

An eight-node linear innnite hexahedral element is shown in Figure C. L with the local

coordinate r toward uifinity. The global coordinates and local coordinates of a typical corner

node i are (xi, yi, U) and (ri, si, fi), nspectively. By extension of (C.37) for the 3D infimite

element. the coordinate transfomations or mapping functions are

and where si and ti are +1 or -1 for the P term. Substitutions of x with y or z yield the

rnapping functions of y or z global coordinates.

Approximation of the potential A with second order hinctions in the direction of infinity,

r, yield the interpolation functions of

( r 2 - r ) (1 + si)(1 + t ) i = 1.2,3,4 where N: (r, s* t ) =

+(1-r2)(l+sis)(1+tit) i=5,6,7,8'

where s and ti are +1 or -1 for the term, and f is any one of the magnetic vector potential

components A,, A,, and A,.

The Jacobian ma& is obtaimd, similar to the isoparametric elements, as

The partial derivatives of the rnapping functions with respect to the local coordinate system

are obtained in the matrix form of

Thus (C.43) defînes the Jacobian matrix I. at a local point inside a typical element in ternis of

the spatial global coordmates of the element node [(xe}, {ye), (f)], and the local derivatives

of the mapping hinctions [A&=, S. t)] (C.44). Therefon. at any point of interest, such as the

numrricaf integroting points in an element. the matrices J, I -', and 1 J 1 can k numricaiiy

O btained.

For hiture use, the partial derivatives of the interpolation îunctions with respect to the

local coordinate system are obtained in the matrix fonn of

Apptndix C Magnttostatic FEM in 3D 146

C.6 System Formulation

The domain of the problem is first discretized into M elernents, where each element has n

nodes. The interpolation functions of elements are then selected. The next step of the FE

analysis is to apply the variational method to formulate the system of equation (23.241. The

foliowing discusses the impkmentation of the van-ational fomlation and the difKerent

techniques that are necessary to obtain the finai form of the systern of equations.

C.6.1 Variationai Implementation

The variational principles and formulation are given in Section C.2 and (C. 18). Sorne

preiiminary steps are necessary to arrive at the fmal form of the rnodified energy hinctional

(C. 18). The energy functional for each element is wrinen in the fotm of

F ~ A ' ) =J w ~ A ~ ) dv -1 J' +A' IN. v* v* (C.46)

where W(Ae) is the energy density associated with the solution A' given by

The components of the magnetic vector potential {Ax, A,, A,) and current density ( J x , Jy, Jz)

in elemnt e are approximated as

where the nodal values of the potential Ap are {A;), , the nurnber of nodes is n=8, and the

interpolation functions arc either isoparamctric elernents (C.28) or innnite elemcnts (C.42).

The minmwation of the huntional with respect to the nodal values of the A yields

The h t part of the integral (C.49) is derived fiom (C.47) as

Applying the chain rule of diffenntiation to (C.50) yields

Substitution of (C.51) in (C.49) yields

By substitution of B= VM, (C.52) becornes the FE form of the minimized energy hinctional

(C. 18) without the Coulomb gauge enforcing tem. The minllnuation of the hinctional (C.52)

yields the rnatrix form of

where the element degrees of fnedom vector @le) is

The element load vector (be ) is

The application of the curl operator on the shape fundion maaix VA] yklds

where the vector (N} is the sbape function vector as de- for each elemnt.

To complete the fomwlation of the stifhess matrix. the Coulomb gauge condition is

included fiom functional (C. 18) as

where the divergence of the shape function rnatrk [NA] yields

Finaiiy. the element siiffnea matrix becomes *

[ K e l = [ K ; I + r q l

and the system of equations becomes

The detached npnsentation of the element system of equation (C.6 1 ) is

where the{b:)~ load vector is given by (C.55). and the [ K,'IeM elernent sub-matrices are

given by

where the term s-1 is included to show the te= which an generateû from the divergence

part of the functional F (C. 18). It is concludcd that [Ka, = [KmT, = [Kae], , whkh is

the result of the Coulomb gage enforcement in the functional (C. 18).

The element matrices and load vector are calculated numerically. These quantities an

obtained from the volum integral of integrands that contain global derivatives of

interpolation functions. Thenfore, first, a numerical intcgration method should'be selected,

and second, the global derivatives of interpolation hinctions should be obtained.

(2.6.2 Gausïian Eiement htegration

The rnost accurate numerical method is the Gauss method, in which a typical integral of

is approximated by

where the integration points (ïgi, s , th) and weighting factors Wgi, Wd, Wgk are given by the

Gaussian quadrature formula. ANSYS sofiware uses the Gaussian quadrature formula of the

second order. n=2. for hexahedral elemental caiculation in which the integration points (r,i.

sd, tg,) are sekcted at M.577350269189626, and the weighting factors are defmed at Wgi, r l'hus the numricai integration (C.68) is written as w,, W&.

*

The integral definitions of t k ekmnt matrix (C.63) to (C.66) require the global derivatives

of interpolation hinctions {M(r, s, t) ), which are hinctions of local coordinates (r, s, t). By

using (C.29), the local denvatives of quantities of mterest, such as A., , are related to the

nodal values of the quantity A, by

where [ ~ ( r , s . o L=[J]& [ ~ , ( r , s . t ) ] , (C.74)

The matrix D, which relates the global derivatives of the quantity Ar to the quantity's nodal

values, can k partitioned as

a { ~ ' (r, S. r )r -

Appendix C. Magnetostatk FEM in 3D 151

Therefore, each row of rnatrix D, which is n u m e ~ d y avaiiable at any point of interest such

as the Gaussian integration pomts inside the elemnt, represents a derivative of the

interpolation functions with respect. to one global cooidinate direction. '

To obtain the element rnatrix, aU the element sub-matrices [ KPJaxa are to be calculated. The

element (f, m) of rnairix [ Ka is obtained by evaluation of the matrix elements as

The Jacobian matrix (C.43) is used to change the integration h m global coordinates to local

e hrnent coordinates. As a result. the volume integral takes the form of

The integrand function can then be written as

hla (rd 9 SII , ttk ) =v' (& Dxm + D ~ l & + D ~ i Dz, )I J I (c.79)

The proper value of v e , and application of the Gaussian integration (C.69) yield -

Since vectors Dx, Dy. Dz are not changing at each integration point within each ekment. the

element mat* cm be obtained by calculating

Element subrnatrix [ KJ is numericaliy obtained by the same approach as

It should be noted that since the integration points within an ekmnt are constant, the same

vectors Dx, Dy, Dz of (C.8 1) are used in (C.82).

C.6.5 Load Vector Calculaaon

To calculate the load vector for each element with n=8 nodes, (C.55) is written as

(C.83)

(C. 84)

Applying the Jacobian relation between global and local volumc dfierential yieMs

kgL =M,. { ~ ' ( r , s, ?)kt (r, s, $1 ~ ( r . S. tlldrdrdt. (C.85)

To apply the Gaussian numerical integration (C.69). the integrand function shouM be

evaluated at the Gaussian points (r,, sd, tg&). Therefore, for an isoparamtric ekment with

n=8 nodes, and for each Gaussian mtegration point, the following quantities should be

calculated:

[Adrgi, su. &k)lug from (C-36).

[J(rgi, S . t,k)]3%3 h m (C.33, and then 1 J 1 .

Consequently, the Gaussian intcgration (C.64) yields the [a matrix in the fom of

Fially, the load vector {b:) is obtained fiom (C.83).

C.6.6 System of Equatiom Assembly

Mer calculating al1 of the ekment mavices and element loads, the system of equations is

then obtained by assembling al1 the elements, by global n u m b e ~ g the nodes, and imposing

the stationary condition @=O. The final form of the equations is

The Bounàatv Condition:

The boundary conditions should be appiied at this stage of the FE procedure. The total

exterior surface of the problem domain can ôe divided into syrnmetry planes and the extcrior

surface that fwes the W t e . The boundary condition at the infinite exterior surface is

accounted for by using the hfmite ekments. The symmetry planes of the geomctry are

usudy characterized as either normai or pardel magnetic f ~ l d planes. These conditions

shouM be enforced on üie set of equation (C.87).

For the normal mgnetic field plane of symmetry, the normal component of the magnetic

vector potential must k set to zero. Thus the degrees of freedom of aii the nodes in such a

plane of symrmtry are reduced by one, and for each node the relative row of the element

ma- and the relative elcmmt of the load vector are nmoved frorn the set of equations

(C.87).

Apptndix C. Magnctostaîic FEM in 3D 154

For the tangentid magnetic field plane of syrnrnetry, the in-plane or tangential

components of the magnetic vector potential must be set to zero. Therefore, the degrees of

b d o m of aii the nodes in such a i>lane of symmtry are reduced to ohy one. ~ h u s for such

a node, the relative rows of the element matrix and relative elements of the load vector are

nmoved from the set of equations (C.87).

The Final Set of Eauations:

The system of equations can finally be written as

or symbolicaliy [d{~k@)- (C.89)

Matrix [KI can be tenned the global coefficient matrix, Dirichlet matru<, or stiffhess rnatrix.

Vector { u ) is the non-zero global vector of the unknown quantity or degrees of fieedom

vector, and vector {b) is the global load vector or forcing hinction vector.

C.7 System Solution

To obtain the solution to the boundary value problem, defined in Section C. 1.4, the system of

equations is fomlated and then assembled into one form of (C.89). If al the materials in the

domah of the problem are iinear materi*, then the solution of (C.89) yiclds the solution to

the problem The magnetic flux density of the probkm domain is then obtained from the

appropriate use of (C.9) and (C 57).

Existence of a noniincar magnetic material in the dornain of the problem leads to a

nonluicar stifhiess matnx [a of (C.89). The nonlinearity arises from dependency of the

magnetic matcriai penmability on the magnetic field intensity. Therefore, the B-H curve of

any nonlintar matcriai in the probkrn domain should be discussed accordingly.

There aie two important considerations for the material pcoperty specifïcation. First, to

insure the u~queness and solvability of the FE approach, the pemabüity of materials

Appcndix C* Magnctostatic FEM in 3D 155

should k single-vahxed and monotonie [29]. Second. to increase the efficiency of the

calculation. the B-H c m of the material is usuaiiy converted to a sphe fit curve of v versus

B' , fiom whkh the v e is evaluated for each ekmnt.

A simple iterative method can k set up to solve the noniinear equations of (C.89) [24].

Starting with a zero magnetic vector potentiai, a guess set of reluctivity values for the

elements is assumed. The stifhess matrix is calculated, and then the linear KU=F is solved

for U. The new set of U is used to obtain a new set of element reluctivity values in whkh

relaxation methods can be used. The iterative calculation of K, U, and v is terminated after

t wo successive solutions agree within an acceptable tolerance. This approach converges very

slowly. The preferred method of solving the nonlinear set of equation (C.89) is the Newton-

Raphson method.

C.7.1 Newton-Rapbson Method

The Newton-Raphson method has proven itself to be a very stable and fast converging

iterative method [24,30]. Conside~g the comct solution of (C.89) as vector (A,,'), and

the iteration solution as vector (Ae) , then the distance to the solution or displacement vector

is

The multidllncnsional Taylor's series expansion of the gradient of the FIA) functional n a

(A') yields

Minimization of the functional (C.52) requires that at the exact solution, (Ae)={Amc:), all

the gradients of the functional must vanish. Approxirnating the Taylor series with its fwst two

terms, and equating the kfk-hand-side with zero, yields the matrix form of (C.9 1) as

{Vc} +[Hg](&'}=O. (C.92)

where [m is the elemait Hessian or Jacobian matrix and ( is the element gradient vector.

Appcndix C. Mapetostatic FEM in 3D 156

The element gradient vector or residual vector {VI is obtained fiom the non-zero

hinctional (C.52) and the Coulomb gauge enforcement as

(V')=[Ke]{A'}-(bel , (C.93)

where [K.] is given by (C.60).

To evaluate the element Hessian matrix, (C.49) and (CS 1) are used to obtain

By application of B= V>c4, and writing A in terms of nodal values of A' as (C.54), the element

Hessian matrix (C.94) is written as

where [Ki] is the element stiffness matrix given in (C.56). and [Khe] is

The dope of reluctivity is obtained fiom the iteration value of the magnetic flux density

The curl of the shape function rnatrix (vw]' ) is given in (C.57). The element Hessian

matrix is numericdy calculated in the same fashion as the stiffness mat* outlined in

Section C.6.3.

The global numbering and assernbly of ail the related matrices and vectors kad to the

system stiffncss ma& the Hessian matrix, and the gradient vector.

The iterative Newton-Raphson method is stasted with the initial guess vector . Then. in each P iteration,

2. me elemnts rcluctiviîy from iterotim value ofpo tentials.

2. the snmess [WJrom (C.891,

3. the gradicnt vector (VIi f (C.93).

4. the Hessian mirLI [Hli f i (CM),

5 the new displucement or correction vector /ali frorn (C. !U),

6. the new drgrees offreedom vectot from {A} i+I = [Ali + /ali, are calcuiated. The iteration continues till a convergence or an enm mesure is satisfied.

The convergence criterion uses the norm of the displacernent or distance vector { & ) i or.

more logicdy, that of the gradient or residual vector {Vjc In the prefernd latter case, the

ümit is expressed as a percentage of the norm of the extemal load or forcing function vector

IIm l l l ~ ~ l l ~ ~ ~ l l 9 (C.99)

when the n o m of a vector is defined as Il{x)II = ( { x ) ~ {x) ) , and E is selected at O. 1 %.

In step five of the Newton-Raphson mthod iteration (C.98), a set of iinear algebraic

equations of the Ax=b fonn has to be solved. The diffennt solution techniques are prirnarily

divided into direct mthods and iterative methods. The direct mthods comonly use

decornposition techniques to arrive at a solution. Their main disadvantage is an increase in

both the mmory storage and processing the , coupkd with an increase in the number of the

probkm unknowns, in this case the degrees of &dom associated with ail nodes. The

terative mthods have proven themselves very enicient with respect to the processing t im

and the mernory requirements for the solution. Among various iterative methods. the Jacobi-

Conjugate-Gradient, KG. is widely used. In this thesis, the software ANSYS employs this

rnethod as outlined in [3 11.

C.8 Evaluation of the Final Results

The F i t e E i e m t formulation and its immediate results are given in terms of magnetic

vector potentials, which are mathematical fwrtions and not physical quantities. Therefore,

the primary nsults of the FEM carmot be used. or compand with mcasured data. or verificd

m their original format. Different quanti- of interest must be obtaimd in order to analyze

the device, to estimate the e m in the solutions, and to ver@ the resuhs. Quantitics of

interest include the magnetic flux density B, the magnetic &Id htensity 8, the magnetic flux

through a closed surface @, the magnetomotive force dong a contour line mm5 and the

stored energy in a part of the model. It should be noted that the pst-processing of the . .

primary data introduces a new level of error in the solution due to numerical dierentiation

and integration of the primary data The derived data are usually calculated after the

convergence of the Newton-Raphson method while elernental data are still available. The

final solutions - which are descnid in the foliowing subsections - are caiculated by the

ANSYS program

C.8.1 Magnetic Field

The magnetic flux density solution to a boundûry value problem is sought for the foliowing

Rasons. First, B is generally the primary physical quantity of interest when considering the

design of a device and should be obtained within a tolerabk accuracy. Second, it is important

to compare the approximation solution with the measured data when such data are available

for the device. This would provide the besî general quality criterion for the solution

verification. Third. Arnpere's law (C.2) is used as an effective global accuracy criterion. Its

calculation requins 8, w hich is obtained from B and the B -H curve.

The magnetic flux density B is defincd as the curl of the magnetic vector potential A. The

nodal magnetic flux density of each elemcnt can then be obtained Born numerical

di&rentiation of nodal values of the vector potential [A). However, a mon accurate

approach is to usc the elemnt interpolation functions, which yields

(~'l=(vxrN,IT)(~'l, (C. 100)

where the curl of shape functions is given in (C.57). To increase the accuracy of the

calculation, the field is computed at the integration points of each elemnt. It should be noted

that the curl of the shape functions has h a d y k e n cdculated at each integration point

within the ekment. The integration points' values are then extrapolated to the ekment ndes

to yield the ekment nodal magnetic flux dcnsity Bt' . The extrapolation is bascd on the

elemit shape iùnction, or its SUnpKed fom as

B(r,s,t) = a , r ~ t + u ~ r s + ~ ~ ~ t + a ~ t r + u ~ r + u ~ ~ + u , t + a , . (C. Ml)

where the hexahedrd element is. considered. Because of the elemntal calculation. the

elementai magnetic flux density is not continuous from each ebment to the adjacent

elements, while the primary nodal magnetic potential is continuous. The averaging of the

elemnt nodal field is usualiy employed to obtain mon acceptable and continuous lookuig

results where there is no material discontinuity between ekments. The element nodal

rnagnetic field intensity, H L , is then calculated from the B-H relation of the element

material. The magnetic field data are used to assess the quality of the results if the masured

data are available in ngions of intenst.

Contour plots of the field data are usually drawn at places of interest to graphically study

the rnagnetic field data 3D contour plots can be viewed on color rnonitoa that make them

very difficult to reproduce on paper for later analysis. Contour plots are usually drawn for a

2D slice of the pmblem domain at a region of interest. The fint and best location for contour

plots is at the planes of synmietry on which the tangentid or normal component of the

magnetic field vanishes. The out-of-normal, or tangent. field can ais0 be drawn to view the

error due to the numericd pst-processing of degrees of frcedom The second good location

for contour plots is at the planes of material discontinuity. where the boundary conditions can

k verified. Continuity and the rate of fkld change, closeness of the contour ünes. and the

location of a very high and low ûeld are used to evaiuate the study model.

It is always more desirable to assess the quality of the results from primary data than from

denved data. The total magnetic flux passing through a closed surface can be calculated

directly nom the magrtetic vcctor potentiais. By definition. the magnetic mix passing through

the closed surface S is writtcn as

a=/ S B.&=[ S ( V X A ) . ~ ~ . (c, 102)

Applying Stokes's thcorcm yields

(C. 103)

when 1 if the bounding contour line of the surface Siand iis the unit tangent vector to line i.

The nodal values of magnetic vector potential ( A } are then used to approximate the rnagnetic

flux through a closed surface as

a= I ( A } + ~ ~ L I , (c. 104)

when the integraiion is carried out numericaily. Fiux calculations are used extensively by

design engineea to analyze different parts of a magnetic device. Leakage of the magnetic

flux from the magnetic materiai is also used to evaluate the efficiency of the rnagnetic circuit

design.

C.83 Magnetomotive Force

The integral fom of Ampere's law (C.2) k written as

f, w î d l =J ~ * d s = ~ , , S (C. 105)

where Is is the total current passing through the surface S, 1 is the bounding contour üne of

the surface S, and fis the unit tangent vector to contour line 1. Then, (C.105) is useâ to

approximate the magnetomotive force drop, mmf, along a con tour line, 1, as

mmf, = J ~ ~ l - ~ t l d i r (C* 106)

where 1 is the bounding contour line of the surface, and vector(H} contains the nodal values

of magnetic field intensity. The magnetomotive force drop along a path provides a valuabk

global accuracy masure for verikation of the FE results with the rnagnetic vector potential

formulation.

The total energy storcd in the pmbiem domain is oîten used to assess the quaiity of the FE

nsuhs. The stored encrgy in an ekmcnt is

where the energy density is given by

(C. 107)

(C. 108)

The energy density (C.108) is calculated either d k t l y fkom the B-H curve of the element

material, or fkom the spline fit function of the material B-H curve. The Gaussian integration

(C.69) is then used to evaiuate the stored energy as

(C. 109)

The stored energy in different parts is then calculated From the ekments' stored energy. The

total stored energy is used to calculate the inductance of the magnetic circuits. It was used in

the selection study of the FE application.

C.9 Conclusion

This appendix has detaikd the F i t e Elemnt Method for magnetostatic 3D probkms: fkom

definition of the problem, to the solution of the system of equations. Should the Newton-

Raphson method (C.98) fail to converge to a solution, the F i t e Ekmnt approach must k

modified properly. Then, the ha1 converged solutions of the FEM should be verified

because of the numrical nature of the mthod. This hm ken discussed in Chapter 2.

Appendix D

Axisymrnetrîc Finite Element Method

The axisymmetric Finite Ekment rnodeling of the MRI rnagnet is summacized in the

following steps:

1. The boundary value problem was dehed in te= of the magnetic vector potential A in a

cylindricd coordinate frarne ( r, #, z ). The r and z components of A are zero, and there is

only one degree of fmdom for each node denoted as A&, z).

2. The variational formulation was used [Z, 241.

3. The problem domain was defined in the rot coordinate plane. The azirnuthal coordinate

is the symmetry plane and is not used in the domain definition and representation.

4. Due to the specific symrnetry of the MRI magnet configuration, only one quarter of the

M'RI magnet probkm domain was modeld. The exterior ünes of the domain divided into

the normal field, the tangentid field, and the far field lines. The extent of the domain

definition should be verified by a seleciion approach. The B-H characteristic of the

magnetic material may be specified in this step.

5. Two element types were used: the linear isoparamtric quadrilateral elements, and the

Lagrangian isoparamttrk idmite ekmnt. The shape functions and the mapping

functions an the s l l n p l i W and modifieci notation of the thne-dimcnsional hexahcdral

and W t e elemaits givea in Appendk C.

6. The docnain of the p b k m was properly disccctasd The rcfiemnt of the e k m t site

and shapc is used to okain the opiinuun mode1 and final discretization of the probkm

domain. . .

7 The ekment c n a t ~ s were calcuiatcd and assemMed into the system of equations. The

boundary conditions of A@ wcn appW to the nodcs on the tangcntial flux exterior

h. The n o r d bouadary condition is sathfiai by the probkm fortllulation. The proper

cumnt density was appüed to the c o l Fnaly, the noniincar system of equations was

obtaimd.

8. The Newton-Raphson mthod was uscd to solve the noiilinear equations.

9. The Magnetic vcctor potential nsults were post-proassed to obtain the magnetic field

data The rewihs w e n then studicd t o obtain the proper conclusioh

10. The total magnetic fhu passing a circdar surf- normil to the roz plane is given as

when R is the radius of suc& d 8 c t . Then the average magnetic flux density at such a

circular cross-section was o û t W as

Equation (D.2) was usai to calculate the average mgnetic flux &&y at the core centrd

plane, the pok and air-gap mterfacc pianc, Md the air-gap cenvpl plane.

11. The average magnetic hix dcnsity dong the a n t a line of thc air-gap was obtplied h m

Finally, the accurg, d the solutions shaild be examined.