Finite Elements, Electromagnetics and Design

S. Ratnajeevan H, Hoole fEd.) vii EDITOR'S PREFACE I have already undertaken a fairly well-received textbook on comput...

Author: S.R.H. Hoole

125 downloads 2345 Views 31MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

S. Ratnajeevan H, Hoole fEd.)

vii

EDITOR'S PREFACE I have already undertaken a fairly well-received textbook on computational electromagnetics at the beginner's level: S. Ratnajeevan H. Hoole

Computer-Aided Analysis and Design of Electromagnetic Devices, Elsevier, New "fork, 1989. Subs~uently it had been put to me by many of my colleagues that I should follow up on the success of the book by authoring a more advanced text on computational electromagnetics. However, advanced topics on field computation are so complex that it not really within the province of any one person to treat them adequately. Thus I was convinced that I should take up the more serious, advanced and current topics of research in field computation, and go directly to the experts with their intimate knowledge for effective exposition. And this book is the product of that exercise. So as to avoid the usual problems with edited books - - lack of continuity, different notations, repetition of references, vawing formats and so on and so forth I have made changes in the submitted texts with appropriate cross-references, introduced a single collected set of references at the end and used the same word processor to type-set the text again. Nonetheless, the reader will note that some times there is an overlap, for example in coupling circuit models to field computation. But this was done intentionally to provide the reader with the different perspectives of independent research groups racing towards discovery. I trust that this book would serve the purpose for which it is written - - a greater understanding and appreciation of the beauty, power, and effectiveness of computational techniques in engineering elech-omagnetics. Finally, my thanks to Srisivane Subramaniam, my loyal graduate student, for painstaking help in the type-setting process. My thanks also to Harvey Mudd College and the National University of Singapore for use of their extensive facilities. S. RatnNeevan H. Hoole Singapore, May, 1994.

S. Ratnajeevan H. H ~ l e (Ed.)

LIST

xv

CONTRIBUTORS

Professor Abd. A. Arkadan Department of Electrical and Computer Engineering, Marquette University, Milwaukee, WI 53233, U. S. A.

Mr. B. A. A. P. Balasuriya Department of Electrical and Electronics Engineering, University of Peradenya, Peradenya, SI~d LANKA.

Dr. Gary Bedrosian General Electric Corporate Research & Development, 1 River Road, Schenectady, NY 12301, U.S.A.

Dr. John R. Brauer MacNeal-Schwendler Corporation, 9076 N. D-eerbrook Trail, Milwaukee, ~YI 53223-2434, U.S.A.

Dr. Mark DeBo~oli Magsoft Corporation, 1223 Peoples Avenue, Troy, NY 12180, U.S.A.

Dr. Madabushi V. K. Chari Department of Electric Power Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A.

Dr. K. Fujiwara Electrical Engineering Department, Okayama University, Tsushima, Okayama 700, JAPAN

Dr. P. Ratnalnahilan P. Hoole Department of Electrical and Electronics Engineering, University of Peradenya, Peradenya, SRd LANKA.

Professor S. Ratnajevan H. Hoole Department of Engineering, Harvey Mudd College, Claremont, Ca 91711, U.S.A.

Dr. S. Kalaichelvan Bell Northern Research Ltd., P. O. Box 3511, Station C, Ottawa, Ont KIY 4H7, CANADA

xvi

Contribug9~

Mr. T. Kirubarajan Department of Electrical and Electronics Engineering, University of Peradenya, Peradenya, SRI LANKA.

Mr. Dhammika Kurumbalapitiya Department of Engineering, Harvey Mudd College, Claremont, Ca 91711, U.S.A.

Professor Adalbert Konrad Electrical and Computer EngmeerLng Department, Idniversity of Toronto, 10 King's College Road, Toronto, Ontario M5S 1A4, CANADA

Dr. Bruce E. MacNeal The MacNeal-Schwendler Corporation, 815 Colorado Boulevard, Los Angeles, CA 90041, U.S.A.

Dr. Gerard Meunier Laboratoire d'Electrotecb_nique de Grenoble UR~a~CNRS 355, EcoIe Nationale Superieure d'Ingenieurs d'Electriciens de Grenoble, BP 46, Domaine Universitaire, 38406 St. Martin d'Heres, FR~.NCE~

Professor Takayoshi Nakata Electrical Engineering Department, Okayama University, TsushLrna, Okayama 700, JAPAN

Dr. Florence Ossart Laboratoire d'Electrotechnique de Grenoble URA CNRS 355, Ecole Nationale Superieure d'Lngenieurs d'Electriciens de Grenoble, BP 46, Domaine Universitaire, 38406 St. Martin d'Heres, FR~a~NCE. Correspondence: Department of Electrical & Computer EngLneermg, Carnegie-Mellon Universi~¢, Schenley Park, Pittsburgh, PA 15212, U.S.A.

Dr. Gilbert Reyne Laboratoire d'Electrotechnique de Grenoble URA CNRS 355, Ecole Nafiona!e Superieure d'Ingenieurs d'Electriciens de Grenoble, BP 46, Domaiane Universitaire, 38406 St. Martin d'Heres, FRANCE.

S. RatnajeevanH. Hoole(Ed.)

xvii

Professor Jean-Claude Sabonnadiere Laboratoire d'Electrotecba~ique de Grenoble, Ecole Nationale Superieure d'~genieurs d'Electriciens de Grenoble, BP 46, Dornaine Universitaire, 38406 St. Martin d'Heres, FRANCE.

Professor Sd. Salon Department of Electric Power Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A.

Dr. C. J. Slavik MartLn-Marietta Corporation, Schenectady, NY 12301, U.S.A.

Ms Srisivane Subramaniam Department of Electrical and Computer EngLneering, Marquette University, Milwaukee, WI 53233, U. S. A.

Professor Norio Takahashi Electrical Engineering Department, Okayarna University, Tsushima, Okayama 700, JAPAN

Dr. Igor A. Tsukerman Electrical and Computer Engineering Department, University of Toronto, 10 King's College Road, Toronto, Ontario M5S 1A4, CANADA

S. Ratnajeevan H. Hoole

Chapter 1 !

[

I!1111 !11 IIIIII IIII !!111

.......

lJ/JJIIIIN Ill/

ELECTROMAGNETIC FIELD COMPUTATION

1.1

The Need for Computer Assisted Analysis and Design

Many scientists and engineers today deal with closed form solutions to electromagnetic field problems; i.e., with solutions that are expressed through exact mathematical formulae. This is the historical approach, in keeping with the development by great physicists and mathematicians of the past. The advantage is a clear exposition and understanding of the behavior of fields - but this is so only where we are in a position to obtain a closed form solution. If engineers and physicists are to use electromagnetic fields to advantage, it is necessary to gain some intuitive feeling for their behavior, and this is difficult where the mathematical expressions are complicated or, for that matter, just not obtainable because of the complexity of the problem. Moreover, the ability to solve problems in closed-form requires simplifying assumptions on geometry and physical behavior, in order to make the mathematics tractable, i.e., amenable to solution. The limitations of this approach must be recognized by the discerning engineer closed form solutions, while being truly clever, are trivial For example, if we take the development of the analysis of the transmission line, we will see how, from very simplifying assumptions, as the accuracy required it, cleverer and cleverer techniques were introduced: First, the single long, cylindrical conductor had a neat solution of both electric and magnetic fields; then the effect of the earth was accounted for by reflections; and following this, conductors of large radius had special techniques for deriving point like images. But then, that was it. However clever and elegant, these image methods do not work: i. For conductors of other cross-sections ii. When the each has finite conductivity. Now the surPace of the earth will not be an equipotential line. iii. When the earth is not flat.

..................

Chapter 1" Electromagnetic Field Computation

iv. When the transmission line is not infinitely long. The purpose of pointing out the limitations of these methods is not to ridicule or decry these methods. These were historical developments in the progress of mankind. They represent attempts by our predecessors at solving problems growing in complexity a step at a time, ever approximating the true world more and more closely. But now, with the availability of the computer, a quantum leap is suddenly possible in our pilgrimage towards modeling the real world. However, it requires a different approach to surrender the expectations of closed form analytical solutions and to seek numerical field values directly. Recent advances in computational power, in terms of both computer hardware and the software that drives it, have brought computational techniques to the fore. This approach has reached a state of development in which the modern engineer and scientist should understand how to use it to advantage. Although numerical methods by definition are approximate, high degrees of accuracy are now possible, so much so as to allow us to call these solutions exact. With these methods few simpli[ying assumptions of geometry are necessary. In comparison, "exact" c!osed form solutions which require simplifying assumptions, are trivial, limited and approximate. Furthermore, methods of graphically presenting solutions on the computer, through such means as contour plots for potential and arrows and colors for field density, make the solutions comprehendible and attractive. Experience has demonstrated that the ability to visualize and to gain intuitive feeling for field behavior is frequently better through these techniques than with some older, closed fbrm methods. And the ability and promise of being able to solve classes of prc,¢blems which were heretofore hopeless, has stirred interest and progress.

1.2 The Governing Equations In short, computational electromagnetics is about computing electromagnetic field values, and the associated performances numerically. In this chapter, the basic methods of field computation will be descried, and then, in subsequent chapters, we shall use those to our advantage in looking at applications and deeper issues associated with the methods of computation. But first, what equations do we solve ? Whatver equation we solve in field computation, it is always derived in some way fi-om Maxwell's 4 equations. Maxwell's equations are in fact generalizations of cruder laws of physics, coupling magnetic and electric fields. In the original form, the electric field intensity E, and the magnetic flux density B were independently defined. E, generated by charge clouds of density ~ C/m 3, was defined as the force on a unit charge, and given by Coulomb's law: E(x,y,z) = ~

r3 Sources

S. R, H. Hoole

3

where the vector r is the position vector from an elemental charge pdR over which the integration is perfo~ed, to the fixed field point (x,y,z) where the field strength is being evaluated. Similalrty, the magnetic field strength H is sourced by current densities J AJm 2, and is evaluated using a form of the Biot-Savart law: H(x,y,z) = ~

r3

(1.2.2)

Sources ~ e generalized Maxwell's laws in differential form relating these vectors are: OD VxH = J + (1.2.3) at aB VxE = - at (I.2.4) V-B = 0 (1.2.5) V.D = 0 (1.2.6) where H is the magnetic field strength (or intensity) and B is the magnetic flux density, which are related to each other through the permeability t~ in the constitutive equation B = g H

(1.2.7),

and E is the electric field strength and D the electric flux density related to each other through the permittivity e: D = EE

(1.2.8).

Another constitutive relationship relates the conduction current density J to the the electric field intensity E through the conductivity (~: J = cE (1.2.9). The Maxwell equations 1,2.3-6 also have their integral forms of physical laws, from which, indeed, they were historically derived. These are first, the generalized form of Ampere' s law relating the cyclic integral of H around a closed loop 1 to the current crossing a surface S enclosed by 1,

ij,

,s

2,0>,

second, Faraday's law of induction stating that the voltage induced in the loop I is the rote of change of flux passing through the surface S (1.2.11);

third, the statement that magnetic flux lines must close on themselves in the absence of a monopole or, in other words, the nett magnetic flux out of a closed surface S must vanish

f j- o..s:o

,:,.2.,2),

and, finally, Gauss' law that the flux out of a closed surface S is the charge enclosed within the volume R bounded by S

Chapter 1: Electromagnetic Field Computation

4...............................

Side 2 ~ ' ~ S Side I a.

b~

Fig. 1.3.1 Continuity of Vector Components

f

D-dS = f

pdR

(1.2.13).

1.3 Simplified 1.3.1: Electric and Magnetic Field Vectors The laws described mathematically above, are general laws involving vector unknowns; often the field values undergo discontinuities at material interfaces, thereby making them multi-valued on the interfaces, thereby complicating data handling. To see the discontinuity of vector components, let us apply the integral equations to the cylindrical pill-box and loop at a material interface - - shown in Fig. 1.3.1 - - as their dimensions in the n o d a l direction to the surface vanish. Before applying eqn. 1.2.I3 to the cylindrical pill-box, we first note that that i. D.dS = DnS where D n is the normal component of the vector D; ii. the normal direction n m the interface is opposite to the outward n o d a l to the cylinder at side I, but the same on side 2; iii. The volume charge, for a charge density of P C/m 3, pdR --) 0 as the height of the cylinder limits to zero, while the surface charge is crsS where Os is the surface charge density at the interface, and S is the cylinder cross-section; and iv. As the cylinder height limits to zero, so does the flux out of the curved wall of the cylinder by virtue of the vanishing surface area of the wall. Proceeding: Dn2S - Dnl S = crsS (I.3.I) or

Dnl + Cs = Dn2 (1.3.2) Similarly, applying eqn. 1.2.12 to the cylinder, we have Bnl = Bn2 (1.3.3) Now, applying eqn. 1.2.9 to the loop I, for a surface current density of Js Amperes per metre length in the direction perpendicular to the plane of Ng. 1.3. lb, Ht2L - HtlL = JsL (1.3.4) or Htl + Js = Ht2

(1.3.5)

However, in working with eqn. 1.2.1I, noting that there is never an infinite surface flux density B, we obtain Etl = F~2 (1.3.6)

S. R. H. Hoole

5

H t2 Htl

Bn

Bn2

Region I lRegion Figure 1.3.2: Continuity Considerations at a Corner Although we have derived the continuity conditions at an interface, the reality is even more complicated than seems from the equations. For instance, referring to Fig. 1.3.2, at a simple corner, applying these conditions assuming no surface charge - - or current-density - - we have along the horizontal edge Htl = Ht2, while on the vertical edge Bnl = Bn2. But, if the point of application is moved to the corner from both sides, in the limit, the tangent on one edge becomes the normal on the other. As such, it is not quite clear as to whether it is the field intensity component Ht or the flux density Bn that is continuous. Thus we see the complexities of working with vector electric and magnetic fields. Besides these discontinuities, an additional complication is on account of these general vectors having three components, as a result of which the number of unknowns increases. And where matrices need to be solved for these vectors, the storage requirements associated with 3 times as many unknowns, is not 3, but 9 times as large. In the most general situation, it is these equations that we must solve. However, in many down-to-earth problems, we are fortunately not always called upon to solve for the vector unknown directly. As we do with closed form solutions, even with numerical methods, we try to simplify the problem description as much as possible.

1.3.2: The Vector and Scalar Potentials We have seen the inconvenience of dealing directly with the electric and magnetic field vectors because of their multiva!ued nature at interfaces and especially at corners. Be that as it may, a mathematical entity known as the potential, provides us a way out. We say mathematical entity because it is derived from mathematical abstractions of the governing equations, rather than any physical inte~retation. The value of the potential has no physical meaning, but the way it changes does, as we shall see. There are two kinds of potential, the vector potential and the scalar potential and they are complementary concepts. The magnetic vector potential A is derived from a comparison of the Maxwell eqn. 1.2.5 with the vector identity V.V× = 0 (A5). Thus we say that the flux density B must be of the form

B = V×A Now using this in eqn. 1.2.4,

(1.3.7)

6

Chapter I: Electromagnetic Field Computation

VxE - - ~B _ . ~~t7 x A = - 7 x ~A bt bt and comparing with the vector identity VxV = 0 we must have

(1.3.8)

(A4),

E = - 8 . 4 V, (1.3.9), Ot since on taking the curl of both sides, the t e ~ 0 would drop out in view of the identity A4. 0 is called the electric scalar potential - - or the electric voltage to the electrical engineer. Thus the electric field E is made up of an induced component that depends on the way the magnetic field changes, and another component that is governed by changes in the electric scal~ potential: J = Ji + J0 (1.3.10) Besides these physical interpretations, others are available though the integral identities. Applying Stokes' theorem .1" A.dL = l" jr (VxA),dS

(alI)

to dete~ine the magnetic flux ~ crossing a surface S bound by a closed loop L,

Similarly, noting that the static electric field strength E is defined as the force on a unit charge, the work done in moving a unit charge from point I to point 2 along a path L, dropping the time derivative of eqn. 1.3.9 to allow for stationarity, and using the identity V = U x ~ + U y ~ + Uzoz-2

(AI),

2

W = f E - d L = - Jr" [ UX~x+Uy0y+UZ ~ ~ ~ ] .[uxdx+uydy+uzdz ,~ ] 1

I

2 -

2 ~3xdX+bydy+~zd~ =-

1

d0 = 01- 02

(I.3.12)

1

If we wish to describe the system by the pair (A,~), what equation should we solve ? For the magnetic vector potential, substituting eqns. 1.3.7 and 1.3.9 into eqn. 1.2.3, we get: v x l V x A = J + ;OD = c y E+OeE 3t = [ - ; -OA V0] (1.3.!3). This is insufficient, however, for a unique solution. As shown in Appendix C, both the curl and dive~ence of a vector need to be specified, whereas so far we have

S. R. H. Hoole

7

specified only the curl of the vector potential in eqn. 1.3.7. And we are free to give any value for the divergence. This is known as the gauge condition. A common gauge is the Lorentz gauge: V.A = ErtV°~-~r

(1.3.14),

and another is the so-called Coulomb gauge, which is the Lorentz gauge under static conditions: V.A = 0 (1.3.15). For solving the electric scalar potential, substitution of eqn. 1.3.9 in eqn. 1.2.6, yields: V.D=V-~E = V.~ [ a-A ~_v,]= - V . ~aA - V-~V~ = 0 (1.3.16) Here we have encountered the magnetic vector potential-electric scalar potential pair, (A,,), to describe the totality of fields. An alternative description relies on the pair consisting of the electric vector potential and magnetic scalar potential (T,~). The concept relies on yet another physical law stating that current flow is continuous ~ of course assuming no charge accumulation. That is, for a closed surface S

f

f

J.dS = 0

(I.3.17)

Applying Gauss' law

f,jt'A.dS=I"I']r(V.A) dP,.

(A12)

to eqn. 1.3.17, we get

and since this applies to every closed volume R V.J = 0 (1.3,19), But by the identity V.Vx = 0 (A5) J must be describable by a vector potential J = VxT (1.3.20) Applying this to the Maxwell eqn. 1.2.3, and allowing J to be the total current density incorporating the displacement current term ~D/3t, VxH = J = VxT (1.3.21), we have, using the identity VxV~ 0, H = T - V~2 (!.3.22), where f~ is the magnetic scalar potential. As a caution it is note~ that ~ can be multi-valued and this is the important difference between f~ and the electric scalar potential ¢. For instance, in a current free region outside conductors, since eqn. 1.2.3 yields v x H = 0, by the identity v × v ~ 0, we have H = - v o . When we integrate the electric field E = -V~ round a closed contour (1 and 2 are the same in

8

Chapter l: Electromagnetic Field Computation

eqn. 1.3.12), the result ' I - , 2 is zero - - that is, it is a conservative field. But when we do this to H = - v~?, Ampere's law, eqn. 1.2.10, demands that the result ~21-~2 be the current passing through the contour. As such we must make geometric cuts and allow for these discontinuities. And we will see more of this in chapter 2 where we will take this issue up in greater detail.

1.3,3: Static Fields While the potentials offer us some simplification in describing fields, the static situation offers the greatest simplification in many problems of field computation. With all derivatives in time being zero then, the differential laws of eqns. 1.2.3 and 1.2.4 simpilify to VxH = J (I .3.23) VxE = 0 (1.3.24), while eqns. 1.2.5 and 1.2.6 remain the way they are. Thus, the equations for solution are, from eqn. 1.3.13, (1.3.25), Vx~ VxA = J = - ~ V , where the right hand side is known in many practical problems. The associated gauge is of course the Coulomb gauge of eqn. 1.3.5. Similarly, eqn. 1.3.16 reduces to the well known Poisson equation: - V-~V~ = p (1.3.26) Similar simplifications are associated with the integral laws.

1.3.4: Two Dimensional Systems Some of the greatest simplifications are afforded where 2-dimensional approximations are possible - - especially so in magnetics. While electric field problems can be reduced to 2-dimensions, the u n k n o w n , is still a scalar and as such, the reduction is only in dimensionality. But in magnetics, the vector potential A, although still a vector in 2-dimensions, turns out to be a single component vector pointing in the direction in which no changes occur. Two common forms of 2-dimensional systems may be identified: i. Translationally symmetric systems, where no changes occur in one of the Cartesian directions, usually taken to be the z-direction and ii. Axisymmetric systems, where no changes occur in the 0 direction in the cylindrical system (r,0,z). Taking up the translationally symmetric system first, we have 0/3z, and we observe from eqns. 1.2.9 and 1.3.9 that E and A must be in the direction of the current density J. Again, eqn. 1.2.3 together with eqn. 1.3.7, tells us that taking the curl of A twice must take us back to the direction of the vector A. Thus the vector potential must be uzA. This leads to great simplifications since A now has only one component. Similarly, in the axisymmetric case also, A has only one component, ueA,again pointing in the direction of no change.

1.4 Differential and Integral Methods Computational electromagnetics, in essence, requires the solution of the equations governing the electromagnetic device we are attempting to study. As we have

S. R. H. Hoole

9

already seen, these equations come in integral and differential forms. Integral equations express action at a distance. That is, given the charges that are the sources of electrostatic fields, integral equations express the effects of these at often far-off spatiN points, the integration representing the superposition of the effects of elemental point-like sources. If we are given the totality of sources then, integral equations allow us to determine easily the field at a point in space. A good example of an integral law we have encountered is: E(x,y,z) = ~

r3 dR

(1.2.1)

Sources It is quite clear from inspection that we may employ this relationship without reference to what happens at other field points. As a result, integral methods are particularly efficient when we wish to know the field at a few limited points of space. Moreover, the integral expressions have the far-field boundary conditions implicitly contained in them. For instance, it may be observed that E tends to zero as r goes to infinity. However, when inhomogeneities are present, difficulties arise on account of induced secondary sources. For example, in the problem of a transmission line over the ground, charges may be induced on material interfaces such as between a conductor and a dielectric. The employment of integral methods in situations of multi-regions therefore, as a prerequisite, requires us to determine these secondary sources and this poses several difficulties and f o ~ s a large area of study. Differential equations on the other hand, express the relationship of the field at a point to that at its neighbor. The differential counterpart to eqn. 1.2.1 is the well known Poisson equation, eqn. 1.3. I6, which in a homogeneous region reduces to: - 8 V2,

= 9

(1.4.1)

This differential equation expresses how the potential 0 at a point changes in the vicinity of that point. Thus in a rather simplistic view, it expresses how the field close to a source is related to that at a point very' close by. Then going to that point, we may determine how it affects the field at a point a little further beyond. And progressing in this manner, we may determine how the effect of a source ripples out to a field point of interest. Consequently, while the equation tells us how electromagnetic effects travel out from the source point to the field point, we cannot treat each field point in isolation - - that is, we must solve for all the field points together so that the number of variables to be determined is many. On the other hand, by the very nature of differential equations, the discretization of the differential equations will relate the field at a point to that at a few others close by so that the matrix equation for solution is sparse; as a result much of the losses suffered on account of the numerous unknowns may be recouped using efficient storage and solution techniques for sparse matrices. Particularly for problems involving geometric detail (that is discontinuities), differential methods are far superior. And among the many differential methods that are there, the finite element method stands out.

10


~= lOOV 10m

t x

~=OV Figure 1.5.1: Capacitor with Charged Gas 1.5 The Finite Element Method - A Simple Presentation

1.5.1 The Finite Element Method in 1-Dimension The finite element method is a general technique fbr the solution of differential equations, and is presently the most advanced of the methods for the solution of electromagnetic field problems. In its precise mathematical form the method involves complex concepts which give it generality and power. Here in this text, however, we adopt a simpler apwoach taken by early workers, since their method affords greater understanding. We shall deal initially with Poisson's equation for electrostatic fields, the energy of the system, and first order triangular elements. These concepts can be generalized later to different systems in electromagnetics. The finite element method for solving differential equations and what it means are best demonstrated by a simple example from electrostatics in one dimension. Through this example we will try to bring out the essential ingredients of the finite element method, which go to make it what it is - - the finite element method. These ingredients may be summarized as: a. Division of the solution region into elements or subdomains b. Postulation of a trial function with free parameters c. Identification of an Optimality Criterion to specify the free parameters of the trial function d. Solution of a set of linear equations relating the free parameters e. Reconstruction and post~processing of the solution from that at discrete points. Consider the simple problem configuration of Fig. 1.5.1, where we have two long parallel plates I0 meters apart, at voltages 0 and I00 V with a charge of constant density equal to the permittivity in betw~n. This problem in a more generalized tbrm with a jump in permittivity is of considerable interest to the oil industry where the long plates will really be the walls of a pipe, the lower part of the capacitor will consist of a liquid and the upper part of charged vapor. Here, by virtue of the large size of the plates, any changes in potential can take place only in the x-direction, going from plate to plate. Since we have seen that the electric potential obeys the Poisson equation, eqn. !.4.1, for this problem in 1-dimension with O/0y ~ 0, 0/~Jz ~ 0, the governing equation becomes:

S. R. H. Hoole

I1

dx 2 = p

(1,5,1 a)

with boundary conditions x = 0 ~ , = 0; x = 10 -->, = 100 (1,5.2) obtained from the plate potentials. O f course this is a trivial problem with a closed form solution and needs no recourse m approximation schemes. But the puwose of selecting this example is to demonstrate the finite element method in 1-dimension, which, once we have grasped the essential ingredients of the method, may be generalized to complex equations which have no closed form solution, such as when we have arbitrary material j u m p s and charge distributions as in the oil industry problem. The closed form solution also allows us to compare the approximate solution with whatever we may obtain by numerical techniques. To get the exact closed form solution, first note that for our example o = ~ so that eqn. 1.5. la reduces to - dx2, = 1

(1.5,lb)

Integrating eqn. 1,5. I b twice we get , = - 2 x2 + ax + b

(1.5.3)

where a and b are constants of integration. Putting in the boundary conditions of eqn. 1.5.2, we get b = 0 from the first and a = 15 from the second so that: I2 0 = - ~x + 15x (1.5.4)

120

120

..................................................................................

100

100 (D

8O

80

6O

60

4O

40

E

o x

20

j¢¢"

.... • ....

Approximate

20

/ ':

0

I

2

........ " . . . . .

I

~'

4

I

......

6

~ .......... ' |

8

~

'"

I

10

..... w

........

12

X Figure 1.5.2: Exact

and Approximate

X

0

12


is the exact solution as shown in Fig. 1.5.2. "We will now see how a problem such as this is solved approximately and the sources of error in our approximations. In the variational approach approach to finite elements, we first identify some functional (i.e., a function of the unknown function ,), which is at its minimum at the point of solution. It is easily shown that 2

,f [62

=

e[~,]

-

200]

(1.5.5a)

1 is that functional which at its minimum over the interval 1 to 2, satisfies eqn. 1.5.1 a, provided that e i t h e r , or its first derivative is fixed at each end of the solution region, as shown in Appendix B. This is required to obtain a unique solution with the two constants of integration coming from a second order differential equation pegged down. It is also pointed out that this functional is 2 2 L[¢I = ~

D.E dx -

¢odx

(1.5.5b)

1 1 since E = - v , = - uxd,/dx and D = EE = - euxd,/dx. The first term is in fact the energy stored in the external field. And so is the second term which is referred to as the co-energy, and is the work done in moving the charges to the conductors on which they reside. That we would expect it to be at an extremum is natural. The functional minimizes the difference between the two. For our particular example of eqn. 1.5.Ib, with ~ = p, this functional reduces to: 2 L[,] = ~e J ("

[ ~d , ] 2 - 2 , ] &

(!.5.5c)

1 To see the validity of L of eqn. 1.5.5a, l e t , take a small excursion to ,+5,, about the exact solution. If L is truly a functional satisfying the differential equation 1.5.1a at its minimum, then 8L ought to tend to zero when the differential equation holds, as we shall show: 2 2 1 ; [d ]~ 8L = ~ e ~ (0+8*)] 2 dx (,+8,)0 dx 1

,;d

I

2

-~

e[~,l

1

2 2 dx

Cpdx

+

1

S. R, H. Hoole

13

100

x

"i

~-

2 [~ ~ ~

8~]dx -

1

5~p dx

2

fd d ~

1

neglecting (~)2

1

2

=

x

Figure 1.5.4: General Element from xi to xj

Figure 1.5.3: Variational Parameters for Problem 2

j

~ldx-

2

f

~8~dx1

8~pdx

t

2 d d ff d2 = [ ~ ~ ;5~]2 - [ ~ ~ ~ ] l - J 8,[~ ~ ~ + p]dx

(1.5.6)

1 where the subscripts 2 and t respectively refer m the value of the quantity within the square brackets evaluated at the end and beginning of the interval of integration, and the chain rule has been used. These will naturally be zero for our problem since is fixed at the limits x = 0 and x=10 so that 8~ = 0 at those points. Indeed, these terms will be zero even if d~,/dx had been alternatively zero at the limits of integration. Thus it is seen that if the differential eqn. 1.5.la is satisfied, then 8L is zero so that L has to be at an extmmum. We may look at it in another way that is relevant to our numerical scheme. Let us assume a trial function ~ which satisfies such boundary conditions that make the boundary terms of eqn. 1.5.6 go m zero. If we put any such @into L and extremize L (that is, set 8L/8~ to zero) with respect to the variational parameters, then all the terms of eqn. 1.5.6, except the last will be zero. This term too then is zero, since 8~ is arbitrary. That is, the differential eqn. 1.5. Ia will be optimally satisfied in the solution region.

!_4_........


Let us apply this theory by constructing the simple trial function shown in Fig. 1.5.3. Here, we have divided the solution region into three parts, our finite elements, with 4 inte~olation nodes at x-~, 3.333, 6.666, and 10. The values of the potential 0 at these nodes are *1, *2, 03 and 04. The first and last of these are known and are given by the boundary conditions. We need to determine 02 and 03 and these are the variational parameters which may vary to satisfy the condition that the functional L should be at a minimum. Now our functional involves an integration over the whole solution space going from x = 0 to x = 10. This integration may be replaced by summing the integrals over the three finite elements. In general, over an element from x= x i to xj shown in Fig. 1.5.4, we have assumed a linear variation of 0: 0 = a + bx (1.5.7a). Since the continuity of 0 is necessary from element to element, it is preferable to write a and b in terms of the end potentials 0i and ,j. This would then allow us to impose continuity easily by using the same 0i for 0j of the adjacent element to the left and the same Cj for 0i of the a~acent element to the right. Thus at x i 0i = a + bx i (1.5.8) and at xj ,j = a + bxj (I.5.9). Solving for b first by subtracting eqn. 1.5.8 from eqn. 1.5.9: b = ~ - 0i xj ~ x i Now from ec_ln. 1.5.8

(1.5.10)

a - 0i - bxi = ~i - ~ " *i xi xj - x i Substituting in eqn. 1.5.7 0i 70i ~ x x-xi 0 = 0i - xj - xi xi + xj - xi = ~i + (0j-0i) ~_xi

(I .5. ! I )

so that d

~7'i

~ 0 = xj- xi

(1.5.12)

(1.5.I3)

Putting these into eqn. 1.5.5b, the contribution to the functional made by an element i beginning at node x = xi is: xi xi

xj I(*i-~) 2 1 = ............ 2 ~-x i - ~0i+@(xj-xi) Therefore summing for the three elements L=L 1 +L2+L 3

(I.5.14)

S. R. H. Hoole

15

i (t~2-~1) 2 1 1 (~3-~2) 2 1 --2 X2-X1 "2(~I+~2)(X2-XI)+2 X2-Xl " 2 (o2+o3)(X3"X2) 1 (,4-,3) 2 1 + 2 x4-x3 - ~(~4+~3)(x4-x3)

(1.5.I5a)

In this expression, , t and *4 being known, only ,2 and *3 are variational parameters. We have seen in eqn. 1.5.6 that it is when aL tends to zero and the boundary conditions are satisfied, that the differential equation is satisfied. To make aL go to zero, we have to extremize eqn. (I .5.I5) with respect to the free variables:

aL

'2"01

I

~-'3

1

3*2 - x2_xl - 2 (x2"xl) + x3-x2 " -2 (x3"x2) = 0

(1.5.16a)

OL '3-~ 1 '3"'4 1 3-,3 - x3"x2 - ~ (x3-x2) + x4-x3 " 2 (x4-x3) = 0

(1.5.I7a)

Putting in the values for the coordinates and for *I and *4, we get: 3 3 I0 02- ~ *3 = T 3 3 100 " lO *2 + 5 *3 = ~

(1.5.t6b) (1.5.I7b)

Solving we get, *2 =400/9 = 44.444 (1.5.16c) a,3cl *3 = 700/9 = 77.778 (1.5.17c) To compare with the analytical solution of eqn. 1.5.4, 02 = -(1/2)(100/9) + 15x(I0/3) which is 44.444 and *3 = -(1/2)(400/9) + 15x(20/3) = 77.778! Although we have seemingly got the exact solution, this is not so. What we have got is the best possible for the trial function of Fig. 1.5.3. The heights of the graph at the interpolation nodes have been variationally set and the solution is a straight line variation from one interpolation node to the next, as seen in Fig. !.5.2. That the values at the interpolation nodes coincide with the exact values, is a mere fortuitous accident. The finite element solution however, is different from the exact solutiom

1.5.2:

The Ritz Solution

The Ritz solution (of course due to Ritz) is similar to the procedure that we just saw and, in a sense complements what we did. Just now we divided the domain into small elements over each of which we assumed a simple variation - or trial function - of the unknown ,. In the Ritz scheme, the whole domain is treated as one, but now more complex trial functions are in order to model the actual solution better. This is best demonstrated by example, as before: Solve the problem posed by eqns. 1.5.1 and 1.5.2 using the trial function = a + bx + cx 2 (1.5.7b) over the whole interval [0,10]. Proceeding to solve this, we first note that the constants a, b and c are the three degrees of freedom our trial function has. For this trial function, applying the boundary conditions of eqn. 1.5.2, the first boundary

I6


condition gives a = 0 and the second one gives b = 10 - 10c. This reduces our trial function to = 10(1-c)x + cx 2 (1.5.7c) with only one independent variational parameter, or 1 degree of freedom in c. Our functional of eqn. 1.5.5b therefore becomes 10 10 I f[lO(l_c)+2cx]2dx_ j-[lO(1.c)x~x2]d x L=20 0 =I

I 3.10 2 [lO0(l-c)2x + 20(1-c)cx2 + 3 c2x3] t(0 - [!O(I-c)x2 + ~cx- 10

= 500( 1-2C+C2) + l O'(,h3(C-C2) + 2~ 0 ~ c 2 - 50()(1_c). "1000 ~ C 500 c2 + 50O ~ c = 3 Extremizing this with respect to the only free variable c, we get OL ! 000 500 ac 3 c +--f- =0 so that = - 1/2, and, correspondingly 1 = 15x - ~ cx 2

(1.5.15b)

(1.5.15c)

(1.5.7d)

a solution that exactly matches the analytical solution of eqn. 1.5.4 everywhere in the solution region! Interestingly we have only one degree of freedom in the trial function in c and yet we have obtained a better solution than from the trial function of Fig. 1.5.3, where we have 2 degrees of freedom in ~2 and ~3 and therefore more work. And what does this teach us ? It is that a trial flanction has m be judiciously chosen and the finite element method gives us the best possible shape for that trial function we c h ~ s e .

1.5.3 Symmetry and Natural Boundary Conditions In the hand worked example of the parallel plate capacitor above, we had Dirichlet boundary conditions with the potential, fixed at both ends x = 0 and x = 10 of the domain or interval of solution. These conditions we imposed through the trial functions by saying et = 0 and ,4 = 100. Such boundary conditions which are forced to be satisfied exactly through the trial functions are said to be strongly imposed. We saw in eqn. 1.5,6 that it was necessary to enforce these boundary conditions exactly so that the residual [~d2~/dx 2 + p], of the governing Poisson equation shall vanish. B a t is, 6~2 and 8~| being zero, when the functional L of eqn. 1.5.5a is extremized making 8L is zero, the residual of the differential equation 1.5.1a we are solving, disappears everywhere in the interval so that eqn. 1.5.1a is satisfied. Likewise, when we are solving a differential equation with a Neumann boundary condition with d~/dx vanishing at one end of the solution interval, we may force the boundary condition through the trial function. For example, let us

S. R. H. Hoole

17

°l

100

-

*3 5'

10

X

Figure 1.5.5: Forced N e u m a n n Condition at x = 10

suppose that we are solving eqn. 1.5.1b subject to the new boundary" conditions, in place of the old conditions given in eqn. 1.5.2: d x = 0---), =0; x=10~,=0 (1.5.18a) To solve this problem by the variational principle of eqn. 1.5.6 we may postulate the trial function of Fig, 1.5.5 in place of Fig. 1.5.3. In the new trial function we have forced the Neumann condition d~/dx at x4 by making *3 = *4 so that the graph is compulsorily made to be flat. This according to eqn. 1.5.6 will again make the residual vanish because now the term [e(d~/dx)So]2 vanishes because (d,/dx) 2 is zero and [~(d,/dx)50]l vanishes because 5,1 is zero. Unfortunately, however, this strong imposition of the Neumann condition not only makes d~/dx zero at x4, but it also makes it zero throughout the last element from x=x 3 to x=x4. As a result, the satistaction of eqn. 1.5.Ib in that interval becomes poor, True, as we refine the mesh into finer and finer elements, the last element becomes negligibly small so that ultimately the finite element solution will converge towards the exact solution with mesh refinement. However, the Neumann condition may be weakly imposed through the extremization of the functional 1.5.6a. That is, if we take as our trial function the graph of Fig. 1.5.6a where the '3 and 04 are completely free and the Neumann boundary condition is totally ignored, and put the trial function into the functional of eqn. 1.5.5a and extremize with respect to the tYee parameters 02, ~3 and ,4 of the trial function, we would have a simultaneous satisfaction of the difI2erential equation in the interval of solution and the Neumann condition at x4. In other words, in eqn. 1.5.6, when ~L vanishes because of extremization and ~,! because o f , l being fixed, the gradient

18

Chapter 1' Electromagnetic Field Computation

Fixed

0

01

4

xI

x2

x3

x4

a.

0

Fixed

0 A

xI

x2

x3

x4

x5

x6

X

7

Figure 1.5.6: Symmetry and Natural Neumann Conditions. (d,/dx)2 and the residual [ed2(~/dx2 + p] will both be zero. Regrettably this is not obvious from eqn. 1.5.6. In fact all that we can say from eqn. 1.5.6 when we exkremize L of eqn. 1.5.5a ignoring the Neumann condition, is that [ ~d,

5,]2 - J(" 5,[~ ~22 + 01dx = 0

(1.5.19)

How then are these Neumann conditions natural to the variational principle? The natural conditions result from the strong analogy that exists between natural

S. R. H. Hoole

19

Neumann conditions and symmetry. To see this, consider a problem defined by eqn. 1.5. la over the interval [xl,x4] with a defined charge distribution p(x) in that interval with bounde - (x, qi) (8.1.6) where {rli} is a chosen set of elements of X (called a projection system or a system of trial j':unctions). In the case when the projecfon system is taken to be the same as the basis system (qq = qi, i=l,Z.n) the moment method is usually called Galerkin's method.

Even a norm and a scalar product were not needed. Obviously, however, convergence and accuracy of the approximate solution can make sense only if a n o n ~ sp~ified.

A. Konrad and I. A. Tsukerman

263

Note one specific choice of functionals f/ in (8.1.4), namely the Dirac 5function. It is a linear functional, not a regular function 4, defined as < x(t), 5a > = x(a) (8.1.7) where x(t) is a function belonging to an appropriate functional space (Shwartz, 1950, 1966), and a is a parameter. The particular case of the method of moments with Dirac functions used as trial functionals is called collocation or pointmatching. The general method of moments can be applied to integral or differential formulations of electromagnetic field problems.

8.1.3

Fredholm Equations of the First Kind

Recall that the Fredholm integral equation of the first kind is defined as K(x,y)f(y)dy = g(x)

(8.1.8)

where G is a one-, two- or three-dimensional bounded domain, K(x,y) is a known function called the kernel, f is an unknown function on G, g is a known fight hand side. The usual restriction imposed on K is

Gx~K2(x,y)dxdy < o~ Unlike Fredholm integral equations of the second kind, the equations of the first kind constitute an ill-posed problem (Baker, I977; Kantorovich, 1982; Sobolev, I963). The solution may not exist; if it exists, it is unstable with respect to small fluctuations of the right hand side. To illustrate some difficulties connected with Fredholm equations of the first kind, consider two simple examples (for more examples and discussion, see (Delves, I985; Baker, 1977).

Example 1: Let G in (8.1.8) be the segment [0, 1] and K(x, y) -= 1. Then (8.1.8) ~comes ! Jf(y)dy = g(x) (8.1.9) 0 The left hand side of this equation does not depend on x; therefore if g(x) ¢ const, the solution does not exist. If g is constant, then clearly any function f(x) with

4

The term "k-functional" would avoid confusion.

264

Chapter 8: Application of Integral and Differential Methods

the mean value equal to g will satisfy (8,1.9), Therefore, there is an infinite number of solutions in this case. Trying to solve (8.1.9) numerically, say, by applying a simple quadrature formula 1

n

Jf(y)dy= f h-h 0 j=l

h=n

one ends up with a linear algebraic system with the singular matrix whose elements are all equal to one.

Example 2: (Baker, I977, p. 636): I F(x) = _ f ~ + if(Y) x 2 y ) 2 dy - 1, a¢~9 This example is interesting because, at least at first glance, the expression on the left hand side has some similarity with the single layer potential (add square root in the denominator and set a ~ ) . As shown in (Baker, 1977), this integral equation has no integrable solutions. Indeed, if the argument x on the left hand side is extended to the complex plane, the integral becomes an analytical function everywhere except for the segments [l+ia, l+/a] and [-1-ia, I-ia]. If an analytical function equals one on [-1, 1], it has to be equal to one everywhere. However, F(x) tends to zero when the complex argument x tends to infinity. This contradiction shows that the solution does not exist, These two examples show that the behaviour of the solutions of Fredholm equations of the first kind can be rather weird. Regrettably, there is not much theory available on this subject. Baker (1977, p. 637), for example, gives the following "rule of thumb." "the smoother the kernel K(x,y), the more illconditioned is the equation of the first kind." From this point of view, the kernels with singularities appearing in electromagnetic problems may be expected to exhibit "better" behaviour than the smoother kernels of examples 1 and 2 above (see (Yan and Sloan, 1988) and references there). Nevertheless, the equation of the first kind remains an ill-posed problem. Unless special regularization methods (Tikonov, 1977) are used, there is always a good chance that the numerical instability will manifest itself. This delicate matter is not considered in Harrington (1967, 1982). Surprisingly, despite theoretical instability, straightforward solution of integral equations of the first kind often yields good results in practice. The quadrature method (approximation of the integral by a suitable quadrature formula) can also be used to discretize Fredholm equations. In the simplest case, the quadrature method yields the same equations as those obtained by the moment method.


8.1.4:

265

Finite Element Method

The finite element method (FEM) is now very widely used in many engineering applications, including electromagnetics+ There are at least two reasons for its popularity. The main one: FEM is a very flexible and general solution tool for boundary value problems. In most cases, the finite element fo~ulation is rather straightforward to obtain. Very often, bounda~ conditions in FEM are natural; it means, loosely speaking, that they are automatically satisfied by the solution and therefore do not require special approximation. The discrete FEM problem is equivalent to the continuous one formulated in a narrower (finite dimensional) space. Hence the FEM problem ordinarily inherits important features of the continuous one. This establishes the other reason of the success of FEM in engineering. For elliptic-type problems, the FEM matrix normally inherits the symmetry and positive definiteness of the continuous operator. In addition, special basis functions employed in ~ M ensure, on the one hand, a good approximation of the solution and, on the other hand, the sparsity of the system matrix. Therefore, one can utilize efficient numerical methods for sparse symmetric positive definite matrices. There exists much literature on FEM. The description of FEM from the engineering point of view can be found in Zienkiewicz (1977), Segerlind (1984), Hoole (I989), Silvester and Ferrari (1990), and Chari and Silvester (I980). The mathematical literature (Strang and Fix, t973; Szabo, 1991; Ciarlet, 1978; Oden ,I983; Babuska and Aziz, 1972; and Oganesian and Rukhovets, 1979) usually requires more mathematical background than most engineers have. Nevertheless, the monograph by Rektorys (I980) on variational methods is clearly written and is relatively easy to read. The paper (Babuska 1989) provides useful guidelines to the mathematical problems of FEM+

8.2. EXAMPLE OF INTEGRAL METHODS: CAPACITANCE 8.2.1: Integral Formulation According to Maxwell's theory, 3B V X E =-3~

(1.2.4)

and therefore in the electrostatic case, using (A4): VXE=0 (1.3.24) ~E=-V~ Using another of Maxwell's equation, V • ~E = p (1.3.16) one obtains V • ~V, = - p (1.3.26) In the case of a homogeneous medium (~ = const+) (I .3.26) reduces to the Poisson ~uation

266


V 2 ~ = - 9£

(1.4.1)

The solution of (1.4.1) in the whole space R 3 can be explicitly expressed as the electrostatic potential created by volume charges 0: ,(x) = ~1 R~ ~I x.y I dSy where x, y denote points in R3 and Jx-yt is the distance between them. The electrostatic potential created (in a homogeneous medium) by a single layer of charges ~located on a surface (or surfaces) S is known to be ~(x) - ~1 S f ~I x-yt dSy

(8.2.1)

Finally, the ~tential of a double layer v is 1 dSy 0(x) = ~1 sdrV(y)~y l x-yl where ny is the outer normal to S at point y. For the rest of section 8.2, we shall consider the electrostatic field E in a homogeneous dielectric 5 in the presence of conducting bodies or surfaces. It will be assumed that there are no volume charges and hence the Poisson equation (1.4.1) turns into the Laplace equation ~=0 (8.2.2) The solution of the Laplace equation may be sought as the potential of a single layer, a double layer or a combination of both. These potentials need not be created by real charges - - one may look for a fictitious distribution of surface charges whose potential is the same as that of the actual field. Since the potential created by surface charges satisfies the Laplace equation (8.2.2) it is sufficient to satisfy the boundary conditions which may have two forms: (i) Given potentials , ISi =,i (8.2.3) where Si is the i-th conducting surface and Oi is the given potential of this surface; (ii) Given total charges.

5

The case of piece-wise homogeneous media can be treated in a similar way (Tozoni, andMayergoyz, 1974)


267

equations (8.2,2), (8.2.3) with unknown ¢i' yield Qi = r J q ~ dF i

(8.2.4)

1

where F i is an arbitrary closed surface containing the i-th conductor and none of the others, and Qi is the given total charge on the i-th conductor. Consider the boundary conditions (8.2.3) corresponding to the known potentials of the conductors. If the solution is sought as a single layer potential, the following integral equations result: f ~ ( g ) dy = ¢i x ~ Si, i = 1,2..n (8.2.5) l x-y I U where each equation corresponds to a separate conductor. If there is only one conductor in a homogeneous medium, the system (8.2.5) reduces to one equation 4=e

1 )¢ dy = ¢ Kc~ - ~ t x-y I

(8.2.6)

where K is an integral operator. This is a Fredholm equation of the first kind which is an ill-posed problem (see section 8.1,3). Solutions of (8.2.5) or (8.2.6) do exist because these integral equations are equivalent to a well-posed boundary value problem, However, numerical instability can generally be expected when the equations (8.2.5) or (8.2.6) am solved.

8.2.2:

Harrington's

Method

The simplest variant of the moment method applied to (8.2.5) or (8.2.6) consists of the following: (a) Subdivide the conducting surfaces Sj into subsections As/; (b) Approximate the charge density ~s by a linear combination of pulse functions f) with yet unknown coefficients ci n

(c)

c = ~ ~rifi (8.2.7) i=1 where 1 on ASi fi = 0 on all other ASk (k~i) (8.2.8) Substitute the approximation (8.2.7) into the integral equation (8.2.5) and, using point-matching, obtain the algebraic equations n

aijcrj = ¢i, i=I or in the matrix form

i= 1,2..n

268


Ac=, (8.2.9) where ~i is the given potential of the conducting surface on which the subsection z~Si is located, and A is an nxn matrix with the elements

aij =

f 4r~rij dS

(8.2.10)

rij = [(x.xi)2 + (y_yi)2 + (z_zi)2 ]1/2 (d) Solve the system (8.2.9) for the coefficients With the coefficients oi known, the approximate charge density is expressed according to (8.2.7); then, with the known distribution of o, the electric scalar potential can be computed as a single layer potential (8.2.1). The capacitance of a single conductor is then computed numerically as n l~lCiASi C Q "= (8.2.11) Note the physical meaning of matrix elements aij. A charge zXqj = c~jA~ located on the j-th subsection creates the potential ~ij = aijAqj at the middle of the i-th "1

subsection. Therefore aij is the mutual capacitance of the subsections i andj. The variant of the moment method formulated above can be inte~reted in two different ways: (a) As a general method of moments (see section 8.1.2) applied to the Fredholm equation (8.2.5) with pulse functions fi (8.2.8) taken as basis functions and with 5-flmctions used as test functionals, ire., with pointmatching (collocation); (b) As a quadrature m e t h ~ applied to the same integral equation (8.2.5). This does not imply, however, that the two approaches - - the method of moments and the quadrature method - - always coincide. It is clear that any choice of basis functions in the method of moments results in a certain quadrature formula; therefore, the method of moments with collocation can always be interpreted as a variant of the quadrature method. However, if more complicated trial functions are used in the method of moments, it becomes essentially different from the quadrature method. At the same time, the quadrature method can employ more complicated quadrature f o ~ u l a e which cannot, at least directly, be interpreted as a variant of the method of moments. It should again be pointed out that we consider the same starting point for both methods, namely, the Fredholm integral equation (8.2.5)of the first kind.

8.2.3: Implementation To implement the moment method, we need a procedure to calculate aij. In the simplest case, the expression (8.2.10) for aij can be computed analytically. When

A. Konrad and I. A. Tsuke~an

269

AS i is a square of size h x h, a g ~ d approximation for the diagonal element aii of the matrix A is given by Ha~ington (1967, 1982): 0.8814h aii re (8.2.12) If i~j, it is sufficiently accurate to treat the charges Aqi, and Aqj as point charges, which yields the f o ~ u l a (8.2.13) aij = 0,282 ~...........A~S-i to be used. The approximation (8.2.12) is valid for two separate subsections of any shape. If the accuracy of the approximations (8.2.12), (8.2.13) is not satisfactory, subsections AS i can be further divided to obtain more accurate estimates of the integral in (8.2.10). This matter is discussed in greater detail by (Ha~ington 1967, 1982). Note that the approximate formulae (8.2.12), (8.2.13) yield a symmetric matrix A if the areas of the subsections are equal, This is not true of the matrix defined by the exact integral expression (8.2. I0). Indeed, suppose that the centres of the subsections i and j are fixed; then the element aji depends only on the shape and orientation of the j-th element, whereas aij depends only on that of the i-th element.

8.2.4: Computational Complexity We can now estimate the computer memory and the number of arithmetic operations required to implement the moment method. The number of subsections of the size O(h) on a two-dimensional surface is obviously O(h2). The dimension of the matrix A (8.2.9) of the moment method is equal to the number of subsections; therefore A is an O(h -2) x O(h "2) square matrix. This is a full matrix, so it contains O(h "4) nonzero elements which have to be kept in computer memory. Thus the moment method requires O(h "4) units of memory. The numerical solution of an n x n system by Gaussian elimination requires 0(n 3) arithmetic operations (Faddeev and Faddeev, 1963). Since n = O(h "2) for the system (8.2.9), the required number of operations is 0(h'6). After the system is solved, one needs O(h "2) arithmetic operations to compute the capacitance using (8.2.11). This number is negligible compared to O(h-6). However, if, besides the capacitance, the potential or field values are needed at m points, this will require O(mh -2) additional operations. As we shall see in section 8.4.7, the finite element method usually requires substantially fewer arithmetic operations and computer memory. This is worth keeping in mind - - of course, as well as the fact that FEM is not free t~om disadvantages either.

8.2.5: Numerical Example: The Capacitance of a Square Plate We illustrate this example from Harrington (1982), with a FORTRAN program figr computing the capacitance of a plate. (Appendix 8.A). The program was

270


Table 8.2:!!.Convergence of Capacita.nce w i ~ Subdivision Capacitance, pic0-Farads Number of S u b ~ u a r e s 1 4 9 16

36 100 ~0 ,,

...........

. . . . . .

31.5

,,

,

35.7 37.4 38.2 38.8 39.1 39.8 40.3 ,,

,

,,,

intended to be as simple as possible, and for this reason, no attempts have been made to optimize the code. To compute the capacitance of a conducting square plate I m x 1m, the plate was subdivided into n x n subsquares of the size h x h, h = l/n. The results, which almost coincide with those presented by Harrington (1967, 1982), are summarized in Table 8.2.1. The capacitance Versus the size h of subsquares is plotted in Fig. 8.2.1. There is no numerical instability, at least as far as the computation of capacitance

"- 42 co

40 38 36

34

32

.

30 0

I ........

I

...........

I ............

I, .......

|

f

L ...........

~-....

....

!

0,5

!. . . . .

1.0 h

Figure 8.2.1: Capacitance [pF] Agamst Size h [m]

-.....L

A. Konrad and L A. Tsukemaan

271

is concerned. The computed values of the capacitance clearly seem to converge to C---41 picofarads as h --~ 0.

8.2.6 Advantages and Disadvantages We can now summarize the advantages and disadvantages of Harrington's moment method, i.e. the method of solution of Fredholm integral equation (8.2.5) of the first kind by point matching.

8.2.6.1 : Advantages

(a) (b) (c)

Simplicity The possibility of handling problems with non-closed conducting surfaces The possibility of solving problems in unbounded domains

8.2.6.2 : Disadvantages

(a)

6b) (c) (d)

It is difficult to solve problems with given charges (as opposed to given potentials, see section 8.2.1) of conductors The m e t h ~ is not well-suited to problems in inhomogeneous or bounded domains Numerical instability is possible (see section 8.1.3) The required computer memory and the number of arithmetic operations grow very quickly with the attempts to increase the accuracy (i.e., when h 0); s ~ s~tion 8.2.4.

8.2.7: Method of Average Potential The method of average potential (Jordan i961) is a simple technique for estimating full and mutual capacitances. The actual physical condition on a conducting surface is that the electric scalar potential should be constant. If the surface is smooth enough, it might be expected that the surface charge density will be more or less unifo~, except for the edges (if the surface is not closed). Hence the approximate average potential method: assuming constant charge densi~ (as opposed to constant ~tential) on a conducting surface, compute the potential created by these charges. K e n , to estimate the capacitance of the surface, divide the total charge by the average potential on the surface. The procedure for the mutual capacitance is similar: assume constant charge density on one surface and compute the average potential created by this charge on the other surface. Computation of the capacitance between two square plates by the method of average potential is considered in (Konrad 1974, 1986; Konrad and Sober 1986). Table 8.2.2 summarizes some of the results. The method of average potential has also been successfully applied by Konrad and Sober (1986) to ceramic chip carrier problems which could hardly have been solved by any other method b~ause of the large number of conductors involved.

272


Table 8.2.2: Comparison of Two Inte jral M e ~ o d s Konrad and Normalized Harrhngton (1967, 1982) Sober (1986) Distance (pF/m) I......................( p F / m ) 103.5 104.9 0.1 61.0 59.7 0.2 45.7 46.3 0.3 39.8' 38.9 0.4 34.5 36.8 ..........o . 5 31.6 0.6 4.5 . . . . 29.5 32.2 0.7 27.9 ..... 30.4 0.8 28.6 26.8 0.9 27.4 25.8 1.0 '

. . . . . . . .

I

. . . . . .

'. . . . . . .

,,,,,.,

.................

,,

Percent Difference

, ,...,,,

~

........

,,

J,

,,,,,,

,,,,,,, .

.

.

,,,,,,, . . . . . . . . . . . . . . .

.

,,11111

,,,,,

, ,,it

....

|

, ,,,,,,,,

. . . . . . . . . . . . . . . . .

,,,

1,

..........

!

,,,,,,

1.3 2.1 1.3 2.2 6.2 8.4 8.3 8.2 6.2 5.8 ,,,, .

.

.

.

.

.

.

................

Although the method of average potential is obviously not rigorous (especially as far as the computation of mutual capacitances is concerned), some accuracy estimates for the full capacitance will be considered below A. M i n i m u m and M a x i m m n Potential One useFal estimate of the full capacitance (Mayergoyz, 1979, p.42) is related to the method of average potential. It is instructive to see how the notions of linear operators in Hilbert spaces (section 8.1.2) can be employed to obtain this estimate. Let e be an arbitrary charge density on a surface S, q be the total charge of this distribution and q~ the ~tential created by these charges. Define the integral operator K (say, in the Hilbert space L2(S)) as in the equation (8.2.6); t h e n , = Ke. Let e* be the actual charge distribution corresponding to the unit potential on the conducting surface: Ko* = 1 on S Since the kernel of K is symmetric, the operator K is self-adjoint with respect to the scalar product (e;,,) = drc~dS Using the self-adjointness of K, one immediately obtains (,,~*) = (Kma*) = (m K~*) = (ml) ~ JodS = q S Assuming that c~* k 0, we can use the mean value theorem for integrals:

A, Konrad and I, A. Tsuke~an

~kn

273

I,,*dS

-< J,a*dS _0) then the following bounds for the capacitance are obtained: q

_ 0. Recall the well known maximum property of harmonic functions: a function satisfying the Laplace equation in a domain reaches its maximum on the N~undary of the domain (and not inside the domain). The potential ¢* corresponding to e* is, by definition, equal to 1 on S and tends to zero at infinity. Tnen, according to the maximum principle, the maximum value of cr* is reached on S; i.e., ** = I on S; 0" -< 1 outside S (8.2.16) It immediately follows from (8.2.16) that (3¢*/3n e) -< 0, where n e is the external normal on S. It is only left now, first, to use the connection between the charge density c~* of the single layer and the leap of the normal derivative and, second, to note that (O¢*/0n i) = 0 , where ni is the internal n o d a l on S: o , .__

> o

4n ~,.~le /,~i] -

B, The Gauss Prmciple Let c~ again be an arbitrary charge distribution on a closed surface S and ¢ be the potential created by this charge. The Gauss principle states that the capacitance C of the surface S can be estimated as C _>........q2 .....

~c¢ ds

274

Chapter 8: Applicafon of Integral and Differential Methods

where q = J'S, ¢r¢ ds is the total charge. The physical meaning of this principle is v e ~ simple. With the total charge q fixed, the minimum of the energy E of the electrostatic field is achieved in the case of el~trostatic equilibrium; that is, o2 o2 ,-,2 C - _a__> '! ...........~ .............. -

2 E * -

2E

-

~

cr,

ds

where the superscript * refers to electrostatic equilibrium. A mathematical proof of the Gauss principle can be found in (Polya and Szego 195I). The Gauss principle can, in particular, be applied when the surface charge density ~ is constant: CE q2 = _ ,-,2 ~ . = ......_ ................... _ ......................... q ...... average(e) a¢ ds ¢r~¢ ds S~!¢ ds This is exactly the expression for the method of average potential. Thus this method can be viewed as a particular case of the Gauss principle; it therefore gives the lower bound of the actual capacitance. Having mentioned the Gauss principle, we would like to draw readers' attention to the excellent book by P61ya and Szego (1951).

8 . 3 EXAMPLE OF INTEGRAL METHODS: ANTENNAS 6

WIRE

8.3.1: Integral Formulation and Harrington's Method We consider the electromagnetic field in a tree space. A linear wire object acts as an antenna if the excitation source is located on the wire and as a scatterer if an externally impressed field is acting as a source. The formulation will cover both cases: antenna problems and scattering. The full electric field is assumed to be decomposed into the sum of the impressed field E i and the scattered field E s. (This is possible because the problem is linear). The impressed field is known, the scattered field is to be found. The problem will be considered in the frequency domain. It fi:~llows from Mar.well's equations that E s = -jcoA- V¢ (8.3.1) where we have preserved Harrington's notation; note that, although A and ¢ do not have sut~rscripts, they refer to He scattered field. The magnetic vector potential A can be expressed as a retarded potential of the current density" J:

We follow R.F. Harrington (1967, 1982) to d~cribe the "moment method" for wire antenna p

A. Konrad and L A. Tsukerman

A(x) = g j e je-jkr(x,y) 4nr(x,y') dVy

275

(8.3.2)

where G is the conducting body; fix,y) is the distance between points x, y; k = (2n/~) is the wavenum~r (~ is the wavelength). The eI~tric scal~ ~tential can also ~ expressed as a retarded integral

,(x)= where

e-jkr(x'Y) P4nr(x,y) dVy

1I

(8.3.3)

1 0=--V.J (8.3.4) jo~ is the charge density. To complete the formulation, the consftuOve relationship between E and J is needed. We shall consider the conductor to be ideal, which means that the electric field is zero inside the conductor and the tangential component is zero on its surface, ~ = 0 on S. Therefore ES = . Ei x "c In case of a thin wire, equations (8.3. I - 8.3.4) are rewritten via linear integrals: j~a. + V~ = E i (8.3.5)

(8.3.6) axis 1 jCa .it-,. (8.3.7) axis 1 dI = -j~d~

(8.3.8)

This is a system of integral and differential equations. A, ~ and rr can be expressed via I and substituted into (8.3.5); this would yield a Fredholm equation of the first ~nd. Following Hamngton, we consider a discretization procedure. The thin wire is subdivided into n segments (Fig. 8.3.1). The ends of the segments are marked by circles, the middlepoints are marked by X-s. The segments are numbered | 11 +, and 522 +. ~ e n derivatives in (8.3.5 -8.3.8) are approximated by finite differences and integrals are approximated by sums over the intervals:

276


j~Al(m) =

, ( m ) - ,(rn +) A1m

(8.3.9)

t A(m) = . £I(n) ] n A

(8.3.10)

1Ea(n+) f ,(m+) =~n A

(8.3.11)

1 I ( m + l ) - I(m) e(m+) = "jo~ Aim+

(8.3.12)

with equations simil~ m (8.3.I 1) and (8.3.I2) for ¢(m) and c(m) . In (8.3.9 8.3.12), AIn is an increment between fi and n+; AI~, Aln+ denote increments shifted one-half segment minus or plus along the axis. According to (8.3.10 - 8.3.12), A and ¢ can be expressed using I's only, This procedure described in detail by Harrington (1967, 1982), results in a system of simultaneous equmions of the form. n+

3*

g3 ÷

1

Figure 8.3.1" A Wire Antenna Divided into n ~ g m e n t s


277

ZI=V where I is the unknown column vector of the cu~ents Ii (i = 1, 2 ..... n); V is a column vector of impressed voltages Ei(i)Ali; and Z is the impedance matrix with elements (Harrington 1967, 1982)

Zmn = j,~,a~ alm~(n.m) 1 [~-P(n+,m+) - ~ ( h , m + ) - ~ ( n + , m ) - 'P(fi,m) ]

+ Jo2~

(8.3.13)

where (see Fig. 8.3.2)

n+ ~(rn.n)-

Lrn = tl

1

j

8nAln _

rm

+ (z'zm)2 qa2+z 2

dz

rn~ m=n

and a is the wire radius. According to Harrington (1967, 1982), reasonably good accuracy for m = n is given by the approximation • (m,n) = 1 ,1 o ~kin - ~ (8.3.14a) 2~kin 4~ and for m#n

Figure 8.3.2: ~tegration over Wire Element n

278


• (m,n) =

........

/.

(8.3.14b)

rmn being the distance from m to n. Better approximations can be obtained by numerical integration. One of the simple numerical integration procedures consists of subdividing each interval Aim into three ~ual subintervals and evaluating ~ n as 1 Znln = @Z21 + Z22 + ~Z23~ m where the subscripts 1, 2, 3 correspond to the subintervals, and (Z21)mn etc., are computed using (8.3.14a) for coinciding subintervals or (8.3. I4b). 8.3.2: Implementation and Numerical Example A commented FORTRAN program illustrating the solution of wire antenna problems by Harrington's method is presented in Ap~ndix 8.B.

8.4 THE FINITE ELEMENT METHOD 8.4.1: FEM as the Method of Moments R~all the description of the moment method as a general mathematical tool (section 8.1.2). To apply it to a linear boundary value problem, for example, to the Poisson equation vZ,=finG (8.4.1) with homogeneous Dirichlet boundary conditions ~[3G = 0 (8.4.2) one, roughly speaking, needs to do the following: (a) Formulate this problem in oF~rator form in a suitable function space X: L0=fiX~ ~L) Choose an appropriate system of basis functions and the projection system of trial functions; (c) Compute elements (8.1.5) of the matrix in (8,1.4) and solve the algebraic system (8,1.4) to obtain the approximate solution (8.1,3). ~ M is a method of moments with a special choice of basis functions (see the following section). As already noted, this is very" often viewed also as a Ritz method (minimization of a functional). However, the Ritz fo~ulation is less general and applicable only to self-adjoint positive definite operators, whereas the moment method can be, in principle, applied to an arbitrary problem. (b)

8.4.2: Nodal Basis Functions The choice of basis functions should be aimed at a better approximation of the exact solution of a problem. One could try, say, polynomials, sinusoidal functions etc.. In fact, this is often done when closed form solutions are sought. For numerical methods, however, this "analytical" choice of basis functions has serious disadvantages.


279

First, it is difficult to match a combination of polynomials with the exact solution in the whole domain, since the solution may "behave differently" in different parts of the domain. In other words, analytical approximations in the whole domain are usually not that good. Secondly, the matrix of the method of moments with "analytical" basis functions is going to be full; i.e., all or almost all the elements are nonzero. Therefore, numerical solution will require extensive computational resources. These two difficulties are solved in ~ M by choosing s~cial basis functions with very small supports. 7 A one-dimensional domain is subdivided into small segments of the size O(h); a two-diniensional domain is meshed into small "elements" (h-iangles, rectangles, etc.); a three-dimensional domain into te~ahedra, rectangular bricks or other 3D elements. Each basis function is "attached" to its node of the mesh, where it is equal to one; the function is nonzero only on a few adjacent elements. A more detailed explanation is found in the following section. Basis functions with small supports solve both the difficulties indicated at the beginning of this section. First, each basis function is "responsible" for the approximation on its support; therefore the solution is approximated locally, which generally yields more accurate approximations than possible with analytical expressions on the whole domain. Secondly, the supports of most pairs of basic functions do not intersect; hence the corresponding entries (8.1.5) of the matrix of the method of moments are zero. The sparsity of the matrix is used to save both computer memory and reduce the arithmetic operations required to solve the problem. Typical examples of a finite element basis function in one and two dimensions are shown in Fig. 8.4.1. Both 1D (Ng. 8.4.1a) and 2D (Ng. 8.4.1b) basis functions are continuous and piecewise-Iinear. The 1D function is nonzero only on two adjacent segments; the 2D basis function is nonzero only on a few (usually about six) adjacent triangles. Each of the functions is equal to 1 at one of the nodes and equals 0 at all other nodes. It should be explained why these functions are picked from many other possible options. First of all, these functions obviously have small supports an essential feature of FEM. Among the functions with small supports, these are one of the simplest to use and manipulate. Some textbooks on FEM treat "shape functions" as being defined only on one individual element; for example, a piecewise-Iinear 1D shape function is depicted as a "half-hat" (Fig. 8.4. l c) instead of the "fail hat" shown in Ng. 8.4. I a. This point of view, albeit possible, is not consistent with the general Galerkin procedure: the "half-hat" (extended to zero outside the element) has a leap at the node and thus does not belong to the relevant Sobolev space of the differential problem and does not qualify as a basis funcfon. One then has to match pairs of half-hats at the nodes, ending up with the full hats anyway. Of course, in any case, boundary, nodes can have only "hahLhat" basis functions corresponding to them.

The support of a basis function is the set of points at "which this function is nonzero,

280


Elcmcr~ I

a) Thc~plegFF.2~ba~qshmctionL,1 ID

b) The s ~ e s t

~l

basis function in 2D

| |

I I |

I

c) The 'half.~' shapef~anction

Figure 8.4.1: Finite Element B a s i s / S h a p e Functions


281

Since the basis functions (as shown in Fig. 8.4. l a, b) are continuous, the continuity of the approximate solution, which is a linear combination of basis functions, is automatically ensured. First derivatives are piecewise-constant, being discontinuous at the nodes and (in 2D) at the edges of triangles. It is legitimate to ask whether this level of smoothness of basis functions is sufficient for pracfcal purposes and whether smoother functions would be desirable. Solutions of practical problems are usually sufficiently smooth; hence it would, indeed, be preferable to take smoother basis functions, which would yield a better approximation of the exact solution. However, functions with small supports and continuous (not just piecewise-continuous!) first derivatives are bulky enough and inconvenient to use even in 1D, so much so that in 2D or 3D the-3,"are completely impractical for use in FEM. This does not mean that the only possible basis functions for ~ M are those shown in Fig. 8.4.1. Second order (quadratic), third or even higher order functions are sometimes appropriate. This subject is studied in detail by Silvester (1969 a,b). These higher order approximations, being smooth within each element (e.g. a segment in 1D, a triangle in 2D or a tetrahedron in 3D), are n o t continuously differentiable everywhere: their normal derivatives at element borders are discontinuous. Potentially higher accuracy of high order elements has to be weighed against additional compumtiona! costs and human efforts required to use these elements. Besides triangles in 2D and tetrahedra in 3D, many other shapes of finite elements are possible. For example, rectangular elements with bilinear shape

l~ecewise-linear I I l !

/ : "

Exact soluti~

• " :

Figure 8.4.2: A Piecewise-LLnear A p p r o x i m a t i o n of the Exact Solution

282


functions are often convenient for 2D computations. Using the so-called isoparametric coordinate transformations, one can adjust the shape of elements to match, for instance, a curved boundary of the domain. Various types of finite elements, including isoparametric elements, are presented in most books on FEM. Getting back to the first order basis functions (Fig. 8.4.1), let us consider if they are smooth enough for practical puooses. An approximation one can get with the piecewise-linear functions is shown in Figure 8.4.2. This approximation, despite its discontinuous first derivatives, can be made as accurate as desired if the element size h is sufficiently small. This assertion is rigorously formulated and proven in the mathematical literature (Strang and Fix 1973; Ciarlet I978; Oganesian and Rukhovets, I979). Approximation only refers to the function itself and its first derivatives. Clearly, second derivatives cannot be directly approximated by piecewise-linear functions, since the second derivatives of the latter are identically zero inside the elements and undefined at the borders between the elements. In electromagnetic problems, the unknown functions that have to be approximated by a combination of FEM basis functions are fields or potentials. Let us assume that a potential is sought. Piecewise-Iinear basis functions allow us to approximate the ~tentiaI itselt, its first derivatives and therefore the field. This is enough for most practical problems, although the derivatives of the field, i.e. the second derivatives of the potential, cannot be determined direcdy.

8.4.3: Edge Elements The finite elements considered in section 8.4.2 were "node elements," which means that the basis functions were related to the nod~ of the elements. A different t y ~ of finite element, where the basis functions are associated with the edges rather than nodes, is now getting increasingly popular. Edge elements were first proposed in 1980 by Nedelec. In 1982-83, Bossavit and Verite used these elemen~ in a new formulation of the eddy current problem. Edge elements can be formed on tetrahedral, brick, or hexahedral elements ( Bossavit and Mayergoyz, 1989; Bossavit, I983, 1988, 1990; Nedelec, 1980; van Welij, 1985; Mur and Hoop, I985) and these edge elemen~ may be of the first or higher o:~-ers (Kameari 1990). "Fine simplest element of this type is formed on a tetrahedron (Fig. 8.4.3). The basis function We corresponding to the edge e is a vector function with the following properties: (i) w e is linear in the tetrahedron; (ii) R e circulation of w e along the edge e is 1 and its circulation along the other five edges of the tetrahedron is zero. We shall see that a function We with these properties d ~ s indeed exist; but first of all, let us address the question why such basis functions are useful. With edge elements, the field H is described via its circulations along the edges of the elements, as opposed to nodal values in the case of node elements. Since circulation is defined only by the tangential component of H along the edge, no restrictions on the norrnal component at the facets are im~sed. Two adjacent tetrahedra sharing a common facet may have different normal components of H,

A. Konrad and L A. Tsukerman

283

while the tangential component is the same. In other words, edge elements allow a discontinuity of the normal component of a field at interfaces, ensuring at the same time the continuity of tangential components. This is a desirable physical property for electromagnetic problems: the tangential components of electric and magnetic fields are always continuous, whereas their normal components have discontinuities on b o u n t i e s between different media. Node elements impose "too much" continuity: all the field components are forced to ~ continuous. Edge elements are connected with some basic concepts of differential geomeh'7 which treats fields as differential forms (loosely speaking, elementary circulations and fluxes) rather than vector quantities. The differential form approach is more general than the conventional vector field analysis, and this explains the greater flexibility of edge elements. For a detailed discussion of this and other related issues, please see (Bossavit and Mayergoyz, 1989; Bossavit, I983, 1988, 1990; Baldomir, 1986). We follow Bossavit and Mayergoyz (I989) and Bossavit (1983, !988, I990) to summarize the main prope~ies of first order tetrahedral edge elements. Nedelec's elements (Nedelec, I980) are part of "Whitney's complex" of elements (Bossavit and Mayergoyz 1989; Bossavit 1983, 1988, 1990) which is related to de Rham's complex of differential geometry. A "Whitney element of order 0" Wn is just a conventional node element: Wn = ~.i(x,y,z) where Xi is a ~alar function which is linear on a tetrahedron, equals one at the node i and ~ o at the other nodes.

o

=0

~

/ - -"

~clge

'e

Figure 8.4.3: The Edge Element on a Tetrahedron

284


Nedelec's edge element (Nedelec, I980) is a "Whitney element of order 1" defined as We = Xi V~ "Xj V~-i where the subscript "e" refers to the edge with end nodes i, j. Note that the element so defined is divergence-free. Mur and de Hoop (1985) define basis functions separately by each of the t e ~ s of the above expression, i.e. the basis functions have the form ;q V;~j. A "Whitney element of order 2" is a "facet element" defined as Wf = 2(xi V~.j x ;~j + .... + ...) where the indices i, j, k co~espond to the facet nodes of a tetrahedron, and terms denoted by do~ correspond to a cyclic pernmtation of i, j, k. The following properties (Bossavit and Mayergoyz 1989; Bossavit 1983, 1988, I990) of the "Whitney's complex" of elements are very important: The value of w n is I at node n and 0 at other nodes The circulation of We is I along e and 0 along other edges The flux of wf is 1 across facet f and 0 along other facets (8.4.3) l Function Wn is continuous across facets I T he tangential component of We is continuous across facets '~ (whereas the normal component is generally not) The normal component of wf is continuous across facets (whereas the tangential component is generally not)

L

(8.4.4) The pro~rty (8.4.3) for we is what the edge elements are most famous for. It allows discontinuity of the normal components of H, J, and A while the continuity of their tangential comDgnents is ensured - - exactly as physics requires. Denoting by W k the finite dimensional space spanned by Whitney's elements8of order k (k=0, 1, 2) one observes the following "exactness property" of Whitney's complex: VW is the kernel of curl in W I VxW 1 is the kernel of div in W 2 (8.4.5) This implies, in particular, that any curl-free field 1,71in Wlcan be expressed as Vf2, where ~ belongs to W 0 (i.e. ~ is continuous in G and linear on each tetrahedron of a given mesh). In a similar way, any solenoidal field that belongs to W 2 is a curl of some vector field of W 1. These properties mean that the finite dimensional Whitney's spaces resemble the properties of spaces of continuous scalar/vector fields wi~ respect to the operators grad, curl and div. Although this assertion seems rather abstract, it has very important practical implications. For example, it is due to the resemblance between discrete and continuous spaces that nonphysical modes in 3D waveguide and cavity 8

The boundary conditions s/nould be specified.


285

computations are eliminated (section 8.6.3). It can be predicted that in the near future, most of 3D numerical modeling in high and low frequency el~tromagnetics will be based on edge elements.

8.4.4: The Galerkin Formulation If the exact solution of an electromagnetic problem is a sufficiently smooth function, it can be accurately approximated by a linear combination of FEM basis functions. The question, of course, is how is such an approximation found. Consider again a simple one-dimensional plot as in Figure 8.4.4. Obviously, Approximation 2 seems to be more accurate than Approximation 1. To give a rigorous sense to this assertion, one needs a certain measure of the accuracy; that is, a certain norm in a functional space to which the exact solution and its approximation belong. A suitable linear space for electromagnetic problems is the so-called Sobolev space, which is studied in detail elsewhere (Adams, 1975; Sobolev, 1963; Rektorys, 1980). If we consider, for definiteness, an electrostatic problem, a natural norm is the energy norm 1 I ~ , I1 = ~ G ~V,V, d V

(8.4.6)

which is a square root of the energy of the field. With respect to this norm, the error of approximation is me~ured as

Approximation 1

xamaUon 2

Exact solution

Figure 8 4.4: ~ac'hich Approximation

is Better: A p p r o x ~ a t i o n 1 or 2?

286


error =

V (~ - $ )V(,- $ )dv

(8.4.7)

where ~. is an approximation of the exact solution ~. If the exact solution ~ were known, we could define a good approximation to it, for example, by simple interpolation; that is, by choosing ~(Xk) = ~(xk)at the nodes xk of a given mesh. However, to find the best approximation of ~ would require some effort even if ~ is known. Our problem seems to be much more difficult: to find an approximation of the unknown solution ~. This can be accomplished by employing the Galerkin method. As mentioned in section 8. 1.2, the Galerkin method is a method of moments with trial funcfionals defined by the basis functions. Therefore, the Galerkin form of equations (8.1.4) of the method of moments for the equation Lu=g is {?n / ajWj , Wi = (g,Wi) i=1, 2, ..., n (8.4.8) j=l where Wk are hhe basis functions and Me approximate solution is sought as n E aiWi " i=I It turns out that Galerkin mahod has the following wonderful property. For linear elliptic boundary value problems (say, electrostatic or magnetostatic problems) it automatically yields the best possible approximation of the exact solution (if the set of basis functions is given). For rigorous analysis and proofs, we again refer the reader to Rektorys (1980). As a simple example, once more, we take the Dirichlet boundary value problem for the Poisson equation (8.4.1, 8.4.2) in a bounded two-dimensional domain G. We want to apply the GalerMn method using piecewise-linear basis functions (Fig. 8.4.!b). The Galerkin equations are given by (8.4.8). The procedure could be straightforward: substituting the chosen basis functions Wi into (8.4.8), obtain the matrix (often called the stiffness matrix) with the entries p~ = (LWj,Wi) (8.4.9) then compute the right hand side of (8.4.8), JgWidS, for the known g and chosen Wi; and finally, solve the linear system (8.4.8) for the unknown coefficients ¢xi. As soon as we start implementing this procedure, however, we shall encounter one difficulty. The operator L in our example is the Laplace operator V2, also often denoted A; so the matrix entries in (8.4.9) are: e~ = (V2~j,~i) ~ j V 2 ~ j ~ i d S

(8.4.10)


287

The difficulty is that the Laplace operator V 2 cannot be applied to a piecewiselinear function Wj because it is not twice-differentiable everywhere in G. V2Wj is not defined on the edges of the elemen~ and is identically zero witNn an individual element. T_his fact seems to undermine the whole idea of employing piecewiselinear basis functions for second order differential operators. One may therefore be tempted to use quadratic basis functions. Although higher order basis functions can, indeed, be used, this does not solve the difficulty indicated above. Even quadratic basis functions are generally n o t twice-differentiable in the whole domain (again, there are problems with their normal derivatives at the borders of the elemen~). Nevertheless, there is another way around which is always used but seldom fully explained in the t~hnical literature. Suppose that instead of W i we chose smooth functions ~ i obtained by slightly rounding off the corners (Fig. 8.4.5 where ID functions are shown for the sake of simplicity). It is clear that ~ i can ~ as close to Wi ~,s as desired (if "closeness" is evaluated according to the norm (8.4.6)). Since Wj is smooth, we can substitute it into (8.4. I0) and apply integration by parts:

22 i jdS:

jaF-2

V jdS:-2 V jdS

8F (8.4.11) The integral over the boundaw 3G is zero because the basis function Wi is taken

f ~cc, ewis~-lincar basis function Sm

~s

ftmc~on

Figure 8.4.5: Piecewme-Lhnear a n d Smooth Basis Functions

288


to satisfy the homogeneous Dirichlet boundary condition (8.4.2)). Thus for the smooth ~ i one can rewrite (8.4.1 I) as n ajL( ~Pj, ~ i ) = - (g, ~i)

(8.4.12)

j=l The symbol L is used to denote L("Pj, ~ i ) = tfi'V ~ i V ~ i d S It will be readily recognized that these expressions for the coefficient matrices P and q of (1.5.49) and (I.5.50) are particular cases of the more general setting provided here. It should be emphasiz~ that for s m i t h functions, the formulations (8.4.8) and (8.4.12) are absolutely equivalent, (8.4.12) being just some what simpler than (8.4.8). Consider now a sequence of ~i that tends to W. The formulation (8.4.12) in the limit ~ i ~ ~/~ turns into n E ajL('~j,Wi) = - (g, Wi) j=l

(8.4.13)

Note that this formulation is valid as long as the W's have first derivatives (not necessarily continuous). As we have seen, (8.4.13) may be viewed as a limit of Galerkin formulations (8.4.8) with smooth basis functions W. Let us now review what we have done from a more general point of view. We started with a functional equation Lu=g (8.4.14) in a certain Hil~rt space H. The operator L was applicable only to a subset of H (smooth enough functions); i.e. H D D(L). In D(L), the equation (8.4.14) can be written in Galerkin form (Lu,'~) = (g,W) (8.4.15) for any D(L) D W, or equivalently L(u,W) = (g,W) (8.4.16) where L now denotes a bilinear form 9 defined by the left hand side of (K4.15). The formulation (8.4.16) is, in fact, an extension of formulations (8.4.14) and (8.4.15) to a wider set of functions. Indeed, the ~uivalent formulations (8.4.14) and (8.4.15) are valid for functions in D(L) (if L = V 2, for twice-differentiab!e A bilinear form L(x, y) in a linear space X is a number defined for a pair (x,y), where x ~ X and y ~ X, so that L(c~x+ ~y,z) = o~L(x,z) + ~L(y,z) for an arbitrary z ~ X and any real numbers o~and ~.


289

functions), whereas the formulation (8.4.16) is valid for any function that can be represented as a limit of a sequence of functions belonging to D(L). Therefore, (8.4.16) is a generalized formulation of the problem (8.4.14). It is also called a weak formulation. One of its implications is the possibility of using a wider class of basis functions.

8.4.5: Principal and Natural Boundary Conditions One of the advantages of the weak forrnulation (and hence of the FEM-Galerkin method) is that some of the boundary conditions do not have to be explicitly imposed on basis and trial functions. Such conditions are called "natural." They follow directly from the weak formulation and therefore are satisfied "automatically." We shall later illustrate this matter with an example of a waveguide problem. A detailed explanation and analysis can be found in most textbooks on variational methods, for example, in Rektorys (1980).

8.4.6:

Implementation

The FEM is now a m~or computational tool in electrical engineering. There are numerous monographs explaining how to implement and use ~ M (Silvester and Chari, I980; Hoole, 1989; Segerlind 1984; Silvester and Ferrari, 1990; Zienkiewich, I977). We give only a brief overview of the main stages of implementation and illustrate it with some practical examples: The 1 2 3 4o 5

of FEM Obtain a weak (Galerkin) formulation of the p r o b l e m Generate a finite element mesh and choose the b a s ~ functions C o m p u t e the stiffness matrix and th e right hmnd side Solve the system of ai~;ebr'aic equations Post-process the results

Technically, stage 1 is usually pertbrrned as integration by parts. The most delicate point in obtaining the weak formulation is to distinguish between principal and natural ~undary conditions. Mesh generation is the most software-consuming stage. Good interactive mesh generators are now built into many commercial finite element packages for 2D problems; in 3D, mesh generation is much more difficult, but 3D generators do exist and are being developed (as given in chapter 16). For a given mesh, the choice of basis functions is not unique. For example, if a triangular mesh is constructed, one can choose first order (piecewise-lineax) elements, second order (piecewise-quadratic) elements, etc.. For high order elements additional nodes on their edges or inside the elements will be required. Computation of the stiffness matrix is normally rather straightforward, Extensive tables for standard element matrices are given by Silvester (1969a, 1969b). If a problem includes non-standard terms, computing the matrix may require some algebra. When analytical formulas for shape function are not available or are too complex (for example, in isoparametric elements), quadrature

290


formulas may be used to obtain the element matrix. Element matrices are assembled into the global stiffness matrix by the standard procedures (George and Liu, I981; Segerlind, 1984; Silvester, 1969a, 1969b; Zienkdewich, 1977). The solution of the algebraic system that results is often the most complicated stage. The main problem in obtaining an efficient solution is in exploiting and preserving the inhe~nt sparsi~ of the finite element matrices. Generally, there are two groups of methods: direct and iterative. The most popular nowadays is an iterative scheme called the Incomplete Cholesky Conjugate Gradient (ICCG) method 10 (Meijerink and van der Vorst, 1977); it requires only O(n) units of computer memory and O(n 1.5) arithmetic operations for 2D problems, n being the dimension of the ~ M system. At the same time, since modern computers have sufficient random access memory, the direct methods of Quotient Minimum Degree (QMD) or Nested Dissection (ND) (George and Liu, 1981) can also be used for 2D problems. For example, problems with 10 - 20 thousand nodes can be solved in a few minutes on SPARC Station2 TM utilizing about 10 - 20 Mbytes of memory. Asymptotic memory required for ND is O(nlogn) and the operation count is the same as for !CCG, i.e. O(nI.5). This is, of course, only an asymptotic estimate in the order of magnitude; numerical ex~riments show that QMD and ND are usually faster for 2D problems than ICCG as long as sufficient computer memory is available. For small problems (up to 1 - 2 thousand nodes) QMD may be faster than ND, but for moderate size problems, ND is preferable, since QMD requires substantially more overhead. For 3D problems, direct methods are not efficient. ICCG can still be used. Other groups of methods, potentially much more efficient than ICCG, are intensively studied (Bramble, 1990). Once the FEM equation system is solved, post-processing of results, in principle, is simple. It may, however, take much effort by system programmers to develop a user-friendly environment for post-processing. Modern FEM packages otter extensive options for post-processing, as may be seen in chapter 16.

8.4.7: Computational Complexity Consider a finite element mesh in a bounded domain. For one-dimensional problems, the mesh consists of line segments; for two-dimensional problems, of triangles, re=tangles, etc.; and, for three-dimensional problems, of tetrahedra, "brick dements" (parallel , hexahedra etc.. For definiteness, let us discuss the 3D case. If the elements have size of order O(h) and the domain is bounded, then there are O(h "3) basis functions; therefore one deals with an O(h "3) x (h "3 )matrix. However, this matrix is very sparse and contains only a few nonzero elements per row. The typical total number of nonzero elements in the ma~ix is O(h'3).

I0

Sometimes also abbreviated as PCG - - for the Preconditioned Conjugate Gradient Method. However, PCG may imply any preconditioning, not necessarily Cholesky precondifiong.


291

The solution of the problem preserving the sparsity of the matrix is a complicated matter (see section 4.6). For elliptic equations, efficient iterative methods with an operation count of O(h "4) or even of O(h "3 log h) (Bramble, 1990) are available. The computer memory required is only of order O(h'3). Comparison with the estimates for the integral equation methods (section 8.2.4) shows that ~ M is potentially much superior in t e ~ s of numerical complexity.

8.5 EXAMPLE OF FEM: HOMOGENEOUS WAVEGUIDES 8.5.1: Formulation We consider a cylindrical homogeneous waveguide with an arbitrary cross-section and an electromagnetic wave sinusoidal both in time and in the longitudinal direction z: E(x,y,z,t)= E(x,y) expj(~z-o~t); H(x,y,z,t)= H(x,y)expj(~z-cot); The field in a dielectric is governed by source-free Maxwell's equations which for linear media might have the form VxE = -j~--I; VxH = j ~ (8.5.1) It follows from (8.5.1) that V , ~H = 0 and V , eE = 0. Since tl and e are assumed to be constant, and also V , H = 0 and V - E = 0. Due to the assumptions, the problem may be stated in 2D. Maxwell's equations are, of course, subject to certain boundary conditions which will be considered later. Now, using the first one of equations (8.5. I), and expressing H via VxE and substituting into the second equation, one obtains Vx(VxE) - k2E = 0, k=0r,i ~ ~ (8.5.2) Since E is divergence-free, Vx(VxE) = V(V.E) - V2E = - V2E and (8.5.2) tu~s into a 3D wave equation V2E + k2E - 0 (8.5.3) The sane tyt~_ of equation can be deduced for H: V2H + k2H = 0 (8.5.4) Due to the sinusoidal distribution of E in the longitudinal direction, for the zcomponent of E, equation (8.5.3) becomes 2 V 2 Ez + (k 2 - b2)Ez - 0 (8.5.5) where V2 is is the Laplace operator in x-y plane: V 2 ~ (~2/3x2 + 32/~y2). Thus the z-component of E satisfies the 2D Helmholtz equation. In the case of ideally conducting walls of the waveguide, the b o u n d ~ condition is Ez = 0 o n ~ (8.5.6) where G denotes the cross-section and 3(3 - its boundary. The electromagnetic field can be composed into two complementary parts: [Hx, Hy, E z] and [Ex, Ey, Hz]. The first part is called the transverse magnetic

292


(TM) field, since H has only components o~hogonal to the z-direction. The second pm-t is called the transverse electric ( ~ ) field. The boundary condition (8.5.6) holds for both TM and ~ modes. It can be applied to the TM solution directly, because it is E z that is sought. For the TE mode, it is convenient to formulate the problem in terms of Hz, so one needs a boundary condition for Hz. Since H has only the z-component, from Maxwell's equation VxH =j0~E one obtains (assuming the continuity of spafal derivatives in

G) 3Hz

= j~on

~

(8.5.7);

that is, H z satisfies the homogeneous Neumann condition on 3G. The waveguide problem is a s~cific one. There are no sources, and therefore the equations (8.5.3) and (8.5.4) and the boundary conditions (8.5.6) and 8.5.7) are homogeneous (zero right hand sides). We are interested in a non-trivial solution, which exists only for special values of 13. To solve this problem means to find these values and the corresponding solutions (modes). '6,,reshall describe the stages of numerical solution using FEM. 8.5.2: FEM for TM Mode We shall follow the stages of FEM described in secfon 8.4.6. The first stage is to obtain a weak form of the problem (8.5.5, 8.5.6). Multiplying the equation (8.5.5) scalarly by a trial field E' (the subscript z of E z and Ez will be dropped for simplicity) and repeating in part the algebra of (8.4.11) for the left hand side of (8.5~5), one obtains:

~EE'dS + ~(k2-~2)EE'dS

-~~E'dF-c~VEVE'dS

+ ~(k2-I~2)EE'dS

= - c~VEVE'dS + ~(k2-I32)EE'dS

(8.5.8)

and therefore (8.5.5) translates into dVEVE'dS- ~(k2-~2)EE'dS = 0

(8.5.9)

Note that the left hand side of (8.5.5) and (8.5.8) is defined only for a field E having at least second derivatives, whereas (8.5.9) is correctly defined for less smooth fields - - the first deriva6ve suffices. For smooth functions, equations (8.5.5) and (8.5.9) are completely equivalent as long as homogeneous Dirichlet conditions are in place. Equation (8.5.9) is the weak fom-'tulation of (8.5~5, 8.5.6). The weak formulation is valid for a wider class of functions and hence may be viewed as a generadization of the original problem (8.5.5, 8.5.6). It should ~ kept in mind that the weak formulation (8.5.9) is equivalent to (8.5.5, 8.5.6) only if homogeneous Dirichlet conditions are satisfied. Otherwise


293

the surface integral which appears in (8.5.8) will not vanish and will be pre~nt on the right hand side of (8.5.8) and in (8.5.9). The trial field E' must satisfy the homogeneous Dirichlet boundary condition. The second stage of implementation is grid generation. To make the explanation that follows as simple as ~ssib!e, we assume that the cross-section of the waveguide is rectangular, although such an assumption is not at all necessary for the actual implementation of FEM. We then consider a very simple mesh (Fig. 8.5. I) with 15 nodes and 16 triangular elements. The mesh is uniform in both the x- and y-directions, mesh sizes being hx and hy, respectively. Such a mesh with a small number of elements can be generated manually; of course, for real problems automatic mesh generation is in order (please see section 8.4.6 for the references). The resulting mesh has to be stored in a suitable format, as we have seen in Fig. 1.5.12. The relevant data corresponding to a mesh includes the total number of nodes and elements; the x- and y-coordinates of the nodes; numbers of vertices for each element; and the material code for each element (Table 8.5.1). The material c ~ e is a certain number used to identify the material in each element and is a simpler alternative to what was presented in Fig. 1.5. I2. In our case, the waveguide is homogeneous and therefore the medium in all elements is the same; the code number 0 in the right-most column refers to this medium. The rest of Table 8.5.1 is self-evident. Now we can pass on to the third stage of implementation of FEM matrix formation. We remember from section 8.4.4 that the entry (i,j) of the FEM matrix is Pij = L(Wi, Wj), where L is the bilinear form co~esponding to the

11

12

13

(,1)-.., ,7

1

2

.....

(.) ... \"~ '8 .....

3

"-~ :9 ........................",.~ 10

4

Figure 8.5.1: A Sh'nple Mesh for the Rectangular W a v e g u i d e Nodes: 1 to 15; Elements: 1 to 16 in Brackets

5

294


problem. The bilineax form for our problem is given by the left hand side of the weak formulation (8.5.9). The ~ M matrix P in our case is a combination of two matrices S and T with the entries Sij = ~ V ~ i V ~ j d S ;

p

Tij = j ' P i ~ j d S

2-k2

The integrals over the whole domain G may be expressed as a sum of integrals over individual elements. In other words, as we have already seen in chapter 1, computations of ~ M matrices can ~ performed on an element-by-element basis. Thus one needs to compute the expressions T a b l e 8.5,1. Data related to the grid in Fig. 8.5.1. Number of nodes = 15; N u m ~ r of elements= 16 Coordi Node

Ililllll/l!. 1

Element Data Element Number 1 2 3 4 5 6 7 8 9 10 11 12 i3 14 15 16

Node1

Node 2

Node 3

1 2 2 3 3 4 4 5 6 7 7 8 8 9 9 10

2 7 3 8 4 9 5 10 7 12 8 13 9 14 10 15

6 6 7 7 8 8 9 9 11 11 12 12 13 13 14 14

.................. ,,,

.....

Material Code 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ......

,,,,,,,

,

,,,,

,j,,,


S~) = (e!IVWiVWjdS;

29 5

T~e)= /~WiWjdS (e')

(8.5.10)

where (e)refers to a certain element. Note that the contributions S~c)" and T~i e)" may be nonzero only for vertices i,j which belong to the same element (e). All the other basis functions Wi, Wj are zero on (e). For example, computing S(I 1) (Fig. 8.5.1), one needs to take into account only the s h a ~ functions W7, ~P8 and W12. To illustrate how this can be done, let us compute S~i 1- and T~i 1- for i g {7,8,12} andj 8 {7,8,12}. Note first of all that the node 12 happens to be located on the Dirichlet boundary; therefore the solution at the node 12 must be zero and the shape function corresponding to this node is not actually included into the finite element basis. (In our example, only three shape functions, namely, those corresponding to the inner nodes 7, 8 and 9, are included into the basis). So, for the element (!I), we have to compute only S~771-''),S{;7181~')," S (11) and 88 ,,(11) the same entries ofT(11) (it is obvious from the expressions (8.5.10) that ~ 8 = S ) and T(181) = 187 . In the local coordinate system {,rt (Fig.8.5.2), the basis functions ~ 7 and ~ 8 are expressed as

,,~12

(11)

8 Figure 8.5.2: Computh-'tg the Element Matrix

296


q.,7=1 - ~h.x hny; ~8=h~xx Indeed, it is easy to check that q*7 equals 1 at the node 7 (~ = 1t = 0) and equals 0 at the node 8 ({= hx; rl = 0) and the node 12 ({= hy; q = 0). The function ~ 8 is equal to 1 at the node 8 and is equal zero at nodes 7,12. (ll) We can now compute, for example, ~ 8 : S (11) 78

=

~(3~7 a~8

(1_fV~7V~P8MS= (1~) \ at

=

at

+

3~7 3 ~ 8 )

an an

dS

1

fdS 1 hx h K _ (1) h 2x 2 - " 2h x The other entries of S (e) and T (e) can be computed in a similar way. Expressions for element matrices are slightly more complex for triangular elements of an arbitrary shape and for other types of finite elements, especially high order elements. Each of the works by Zienkiewicz (1977), Silvester (I969), Silvester and Ferrari (1990), Hoole (1989), and Segerlind (1984) may serve as a good reference. For example, element matrices for the first order shade functions on an arbitrary triangle with vertices i, j, k are (Silvester 1969; Silvester and Ferrari 1990). S(e)=

T(e)

(i00) 8(2111

=~ 12

I -1 -1 1

cot0i+

0 0 0 -1 0 1

(110)

cot0j+ -1 1 0 0 0 0

cot0k

121 1 1 2

where 0i, 0j, Ok are the angles of the triangle corresponding to the vertices id,k; Sa is the area of the triangle. The algorithm for FEM matrix formation in case of homogeneous Dirichlet boundary, conditions can be summarized in Alg. 8.5.1. The nodes on Dirichlet boundaries are "fictitious" (shape functions corresponding to them are not included in the finite element basis). Keeping these nodes in the system of equations is only a matter of convenience: elimination of Dirichlet n ~ e s would require renumbering of the other nodes. Before the last cycle of the algorithm the rows and columns corresponding to Dirichlet nodes are zero; the last cycle sets the diagonal entries corresponding to these nodes to one. The right hand side at Dirichlet nodes must be set to zero. In our homogeneous


29 7

problem the right hand side is zero everywhere anyway. Then, clearly, the numerical solution will satisfy the homogeneous Dirichlet boundary condition. The fourth stage of FEM is the solution of the algebraic system. For the problem being considered the algebraic system has the form Pw = ~Tw (8.5.11) where w is an unknown vector in Euclidean space E n, and P and T are the ~ M matrices ob~ined at the previous stage. Equation (8.5. I I) has a trivial solution w = 0 (no field in the waveguide). Looking for non-trivial solutions is an eigenvalue problem (Gantmaher, 1960; Horn, 1986). Methods for the solution of eigenvalue problems are beyond the scope of this chapter and the reader is referred to the works by Faddeev and Faddeev (1963) and Harrington (! 982). The last stage of FEM is post-processing. The node values of the solution are known once the algebraic problem has been solved. The solution at any point is a combination of shape functions with known coefficients; for piecewise-linear Initialization Set S=0; M=0; Matrix assembly For each element E mesh For each nodel E {vertices of the element} If node1 is on Dirichlet boundary, skip the node1 cycle; For each node2 E {vertices of the element} If node2 is on Dirichlet boundary, skip the node2 cycle: s(element) T(element) Compute (nodel,node2)' (nodel,node2) Update FEM matrices:

-I ........... >

-t . . . . . . . . . . . >

Y-axis Y-axis X-axis Z-axis X-axis (going (com&ng OUT of the page) INTO the page) OUT of the page)

A. Konrad and I. A, Tsukerman

CC C C

313

Note that the above notation is used in the program.:

C C C

CCW direction is taken to be POSI~VE for ALL angles and ~ e y are assumed to be m RADIANS.

INTEGER LIMIT, PLIM, X, Y, Z PARA2~ETER (LIMIT = 300, PLIM = 30)

c

x,Y,Z

Er~U~GER I, J, N P, PINFO(PLIM,2), PSTART, PEND ~ U B L E PRECISION + THETA(PLIM,3), B, C(PLIM), A(PLIM), ORIGFN(PLIM,3), + C (L~¢IIT,3), C E N ~ R ( L ~ I T , 3 ) , L(LIMIT, LIMIT), + ALPHA(LFMIT), SIGMA(LIMIT), G(LIMIT), V(PLIM), + CAP(PLFM,PLIM), Q(PLIM) C C C

Initialize d u m m y array indices X=I Y=2 Z=3 =1 PEN~ = 2

C OPEN (UNIT = 1, ~ L E = 'INPUT1.DAT', STATUS = 'OLD') OPEN (UNIT = 2, FILE = 'MM'TIPLOT.M', STATUS = 'N~W')

WRIt(*,*) (*,*)' WR~(*,*)

DATa,.'

re,EAD (1,*) NU~cIOFP, B VVRj~(*,2) N~,rM OF P ' N u m b e r of plates = ', I3) WRJTE(*,3) 2*B FOI~vIAT(' Side of sub-squares, 2b = ', F7.4) WFa~(*,*) C

314


WRITE(*,4) 4 FO~AT(15X,'2c 2a ThetaX ThetaY ThetaZ Xo + 'Yo Zo V') WRITE(*,*) '

', t

DO 7 I-1,N-U},4OFP READ(I,*) C(I), A(I), THETA(I,X), THETA(I,Y), + THETA(I,Z), ORIGIN(I,X), OmGIN(I,Y), + OmGIN(I,Z), V(I) WRITE(*,6) I, 2*C(I), 2*A(I), THETA(I,X), THETA(I,Y), + ~ E T A ( I , Z ) , ORIGIN(I,X), OI~dGIN(I,Y), + ORdGIN(I,Z), V(I) 6 FORMAT( ' Plate',I3, ' t', F7.3, F7.3, F7.3, F7.3, F7.3, + F7.3, F7.3, F7.3, F7.2) 7C E WRIt(*,*) ' I

+

)

)

wF3~(*,*) WPdTE(*,*) (*,*) '

CALCWL

(.,*)

J=l WR/TE(*,*) '

'

SUB-SQUARES'

WRITE(*,*) WFJTE(*,*) ' ENDS' WRITE(*,*) '-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' 10 I-1, P PINFO(LPSTARA~) = J PINFO(LPEND) = NI't-TF(C(I)/B)*NINT(A(I)/B) + J - 1 W R I t ( * , 8 ) I, PINFO(I, PSTART), PINFO(LPEND) F T(' PIate ~, I3,' I', I9, I9) J= (I,PEND) + 1 10 CON WRITE(*,*) )

)

wed~(.,*) C WRITE(*,*) WRITE(*,*) * CENTERS OF SUB-SQUARES IN LOCAL C O - O R D I N A l S ' WRITE(*,*) WR_ITE(*,*) ' R XC YC ZC'

A, Konrad and I. A. Tsukerman

315

*

WRITE(*,*) '-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' DO 20 I=l, CALL CALCEN(C(I), A(I), B, PINFO(I,PSTART), + PINFO(I,PEND), + CENLOC, LIMIT) 20 CONTIN-13E WRITE(*,*) '-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' WPdTE(*,*) C DO 25 I=I,PINFO(N P,PEND) CEN~ER(I,X) = CENDOC(I,X) CENTER(LY) = CENLOC(I,Y) CENTER(I,Z) = CENLOC(I,Z) 25 CONTINUE C DO 35 I=1, P DO 30 J=PINFO(I, PSTART),PINFO(I,PEND) GO) = V(I) CONTINUE 35 CONTINUE C IF (NL~vIOFP .GT. 1) THEN D O 40 I=2,

*

4-

*

+

*

4-

*

4-

*

+

4-

WRITE(*,*) 37

ROTATION", ROTATION', ROTATION',

',

CALL ROTATE0, THETA(I,J), PINFO(I, PSTART), PD-JFO(I, PEND), CENTER, LIMIT) WRITE(*,*) '- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ',

4-

*

P

DO 37 J=X,Z WRITE(*,*) IF O.EQ.X) W R I t ( * , * ) ' CENTERS A F a R ' X-AXJS' IF (J.EQ.Y) WI~d~(*,*) ' C E N ~ R S A ~ R ' ALONG Y-A,XIS' IF (J.EQ.Z) WRITE(*,*) ' CENTERS A F a R ' ALONG Z-AXIS' WRITE(*,*) W R ~ ( * , * ) ' NldMBER XC YC', ' ZC' WRITE(*,*) '

E

316


WRITE(*,*) WRITE(*,*) ' CENTER OF SUB-SQUARE A ~ E R TRANSLATION' WRITE(*,*) WFJTE(**)' N~dMBER XC YC', ' ZC' WRITE(*,*) '- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ',

+

*

+

*

+

). . . . . . . . . . .

CALL TRANSL(ORIGIN(I,X), ORIGIN(I,Y), ORIGIN(I,Z), PINFO(I,PSTART), PINFO(I,PEND), CENTER, LIMIT) WRITE(*,*) '- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ',

+ + + *

)

+

*

C ENDIF

WRITE(*,*) E

C C Compute the [1] matrix C WRITE(*,*) WR_ITE(*,*) 'COMPLFUNG [I] M_ATR_IX..?

wRJTE(*,*) CALL LMAT(PINTO(NUMOFP, PEI'~), B, L, CEWrEK L ~ , ~ ) C C Solve [1][alpha]- [g] C ~v^,TR_ITE(*,*) (*,*) 'SOL"V~qG FOR [1][alpha] = [g] ...' CALL GAUSS(PINFO(NUMOFP, PEN~D), L, ALPHa., G, LIMIT) C VWRITE(*,*) WRITE(*,*) ' CHARGE DENSITY DISTRIBUTION' WRITE(*,*) (*,*)' SU~SQ. N K J M C/sq. m' WRITE(*,*) '-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' DO 50 I=I,PINFO P,PEND) SIGMA(I) = ALPHA(I)*(2*B)**2 WR/TE(*,45) I, SIGMA(I) F

T(I9, E~.6)


317

50 C WRITE(*,*) '- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '

WRITE(*,*) C C Calculate the TOTAL charge on each plate C DO 60 I=I,NUIvlOFP Q(I) = 0 60 CONTII'-~dE C DO 100 I=I,NL~dOFP DO 80 J=PINFO(I, PSTART),PINFO(I,PEND) Q(I) = Q(I) + SI G M A ( I )

80 CON~NUE 100 CONTINUE C C Compute the capacitance matrix C WRITE(*,*) WRdTE(*,*) ' CAPACITANCE MATRIX'

WRITE(*,*) (*,*)'ELEMENT FA_R_AD' WRITE(*,*) '- ............................... ' CALL COMCAP(.N1_o~vlOFP, Q ,V, CAP, PLDd) WRITE(*,*) '- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '

WI~TE(*,*) C C Exporting charge d e n s i ~ distribution to 'M'MTIPLOT.M' to be C plotted ushng MATLAB later. C WRdTE(2,*) 'CharDen = [ ..' DO 120 I= 1,PINFO(NUMOFP, PEND) WRdTE(2,*) SIGMA(I), ', ..' 120 C E W (2,*) '1;' C EN© C C S U B R O U ~ N ~ CALCEN (C, A, B, P, Q, CENLOC, LIMIT) C C This subroutine CALculates the CENters of the sub-squares m LOCAL

318

C COC


S and stores them m CENLOC.

ER NX, P, Q, I, NPREV, M, X, Y, Z DOUBLE PRECISION + C, A, B, CEI-CLOC(LFMIT,3),XC, YC C X,Y,Z NX = NINT(C/B) =P-I x c =-(c + B) =1 10 I=P,Q YC = (A + B) - 2*B*ROWNUM XC = XC + 2*B CENLOC (I,X) = XC CENLOC (I,Y)= YC CENLGK2 (I,Z) = 0 * (*,5) I, XC, YC, 0.0 * 5 FOR2vIAT(I9,F13.6, F13.6, FI3.6) IF (MOD(I-NPREV,NX).EQ. 0) THEN M+I XC =-(C + B) ENDIF 10 CONTINU~ C R~TURN EI~ C SUBROUTINE ROTATE(AXIS, MET.A, P, Q, C E N ~ R , LIMIT) C C ~ i s subroutine ROTATEs the plate by the angle theta (with respect C to the X-axis) m the X-Y plane. C AXIS, P, Q, I, X, Y, Z, LIMIT DOUBLE PRECISION + ~qETA, CEl'-~I~ER(LIMI%3),XOLD, YOLD X,Y, Z C DO 10 I=P,Q XOLD = CENTER(I,X) YOLD = CENTER(I,f)

A. Konradand I. A. Tsukerman

319

ZOLD = CENTER(I,Z) IF (AXIS .EQ. X) THEN CEN~ER(I,Y) = CG"3(THETA)*YOLD - S F N ( ~ E T A ) * Z O L D CENTER(I,Z) = SIN(THETA)*YOLD + COS(TrtETA)*ZOLD ELSEIF (AXIS .EQ. Y) THEN CENTER(I,X) = COS(THETA)*XOLD - SK'q(THETA)*ZOLD C E N ~ R ( I , Z ) = SIN(THETA)*XOLD + COS(Tt-IETA)*ZOLD ELSEIF (AXIS .EQ. Z) THEN CENTER(I,X) = C ~ ( T H E T A ) * X O L D - SIN(THETA)*YOLD C E N ~ R ( I , Y ) = SIN(THETA)*XOLD + COS(THETA)*YOLD ENDIF WRIT-E(*,5) I, CENTER(I,X), CENTER(I,Y), CENTER(I,Z) FORdVIAT(I9, F13.6, F!3.G F13.6) UE

* * 5 10 C C RETURN ENX) C SUBRO C C C C

E TRANSL(XO, YO, ZO, P, Q, CENTER, LIMIT)

~ i s subroutine TRA2-qSLate the plate according to the position specified by the ORIGIN. GER P, Q, I, X, Y, Z, LIMIT DOUBLE PRECISION XO, YO, ZO, CENTER(LIMIT,3) X,Y,Z

I~O 10 I=P,Q CENTER(I,X) = CENTER(I,X) + XO CEbCFER(I,Y) = CEN~ER(I,Y) + YO CEN~ER(I,Z) = CENTER(I,Z) + ZO * WRITE(*,5) I, CENTER(I,X), CENTER(I,Y), CENTER(I,Z) * 5 F (I9, F13.6, F13.6, F13.6) 10 CONTINUE C R~TURT-q END C C SUBROU~NE LMAT(NTOTAL, B, G CENTER, LIMIT) C

320


C This subroutine calculates the [I] MATrix of the system. C ER M, N, NTOTAL, LEMIT, X, Y, Z DOUBLE PRECISION + L(LIIvlIT,LIMIT), CEI'~R(LIMIT,3), XSA, "~qvl,ZM, + Yd'-q,YN, 7_3",I,B, RMN X,Y,Z DATA EPK PI /8.8541878E-12, 3.141592654/ C

DE) 20 M=I,NTOTAL XM = CENTER(M,X) YM = CEN~R(M,Y) ZM = C E N ~ R ( M , Z ) DO 10 N-M,NTOTAL IF (M ,EQ. N) THEN L(N,N) = 2*B*0.8814/PI/EPS ELSE XN = CEt'-WER(N,X) YN = CENTER(N,Y) ZN = CENTER(N,Z) IF (ABS(XN-XM).LT. 1E-5 .AND.ABS(YN-YM).LT.1E-5 + .AND. ABS(ZN-ZM).LE. B) THEN L(M,N) = 0.282*2*B*(SQRT(I+PI/4*((ZN-ZM)/B)**2) + -SQRT(PI)*ABS(ZN-ZM)/2/B)/EPS ELSE R)v~[N = SQRT((XM-XN)**2 + (YM-YN)**2 + (ZM + -ZN)**2) L(M,N) = B**2/PI/EPS/R2vIN ENDIF L(N,M) = L(M,N) ENDIF 10 COI'-,ITINUE 20 CON~-qLrE C RJ3~JR~ END C ....

C SUBROUTINE GAUSS(N, A, W, B, LIMIT) C C This subroutine uses the GAUSS elimination method to solve a set C of simultaneous equations written in the standard matrix form of


C [AI[W] = [B]. C C This subroutine calls on three o ~ e r subroutines: ORDER, ELIM, C and BACKSB. C C These four subroutines :were taken and modified slightly from: C D. M. Etter, STRUC~JRE FO ~ FOR ENGIi'\~ERS AND C SCIEN~STS 3RD ED. California: The BenjamLn/Cummings C Publishing Company, hnc., 1990, p.4~-485 C C C INrrEGER N, I, J, PIVOT, LIMIT DOUBLE PRECISION + A(LIMIT, LIMIT), W(LIMIT), B(LIMIT) AL ERI?,OR C =1 ERROR = .FALSE. 1 0 IF (PIVOT .LT. N .AND..NOT. ERROR) THEN CALL R(N, A, B, PIVOT, E LIMIT) IF (.NOT. EI~dROR)THEN CALL ELIM(N, A, B, PIVOT, LIMIT) PIVOT = PWOT + 1 ENDIF 10 ENDIF

* * * *

IF (ER~.OR) THEN WRITE (*,*)'NO UNIQUE SOLUTION EXISTS!' ELSE CALL BACKSB(N, A, B, W, LIMIT) WRITE (*,*) 'THE SOLU~ONS ARE:' D O 2 0 I=I,N WRITE(*,*) I, W(I) 20 C UE ENDIF

C END C

321

322


SUBROUTIN~ ORDER(N, A, B, PIVO% ERROR, LIMIT) C C qYnissubroutme reORDERs the equations so that the pivot C position in the pivot equation has the m a x ~ u m absolute value. C GER N, ROW, R2dAX, PIVOT, K, LDAIT DOUBLE PRECISION A(LIMIT,LIMIT), B(LIMIT), ~ M P AL E RMAX = PIVOT DO 10 ROW=PIVOT+I,N IF (ABS(A(ROW,PIVOT)).GT.ABS(A(RMAX,PIVOT))) RMAX = ROW 10 CONTINU~ IF (ABS(A(RMAX, PIVOT)).LT.1.0E-5) THEN ERROR = .TRUE. ELSE

IF (RMAX .NE. PIVOT) THEN DO 20 K=I,N TEMP = A ,K) A(R2vIAX,K) = A(PIVOT,K) A(PIVOT, K) = ~ M P 20

~ M P = B(R~d~X) B(RMAX) = B(PIVOT) B(PIVOT) = TEMP END!F ENDIF C

R~TUR~-q EN~ C SUBROUTINE ELIM(N, A, B, PIVOT, LIMIT) C C ~ i s subroutine ELIMinates the element in the pivot position C from rows following the pivot equation. C ER N, PIVOT, ROW, COL, LIMIT ~ U B L E PRECISION A(LIMIT,LIMIT), B(LIMW), FACTOR C DO 10 P,.OW=PIVOT+I,N

A, Konradand I. A. Tsukerman

3 23

FACTOR = A(ROW,PIVOT) / A(PIVOT,PIVOT) A(ROW, PWOT) = 0.0 EXC)5 COL=PWOT+I,N A( ,COL) = A(ROW,COL) - A(PIVOT, COL)*FACTOR 5 CON~NUE B(RC)W) = B(ROgW)- B(PIVOT)*FACTOR 10 CON TFNUE C RE END C

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C SUBROUTH',IE BACKSB(N, A, B, W, LIMIT) C C ~ i s subroutine performs the BACK-SuBstitution to determine C the solution to the system of equations. C INTEGER N, ROW, COL, LIMIT DOUBLE PRECISION A(LIMIT, LIMIT), B(LIMIT), W(LIMIT) C DO 20 RoOW=N,1,-1 DO 10 COL=N,~OW+I,-1 B(ROW) = B(ROW)- W(COL)*A(ROW,COL) 10 CONTINUE W(ROW) = B(ROW)/A(ROW,ROW) 20 C UE C RE END ~ .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

C SUBROUTINE COMCAP(NIJMOFP, Q, V, CAP, PLIM) C C This subroutine COMputes the CAPacitance matrix of the system. C INTEGER I, J, NUMOFP, PLIM, X, Y, Z DOUBLE PRECISION Q(PLIM), V(PLIM), CAP(PLIM, PLIM) C X,Y,Z C DC) 20 I=1, EnD 10 J= 1,~-,VUMOFP IF (I .EQ. J) THEN

324

3

5 10 20 C


CAP(I,I) = Q(I)/V(I) ELSEIF (V(I).EQ. Vq)) THEN WRITE(*,3) I, J UNDEFI~D') FOR),/IAT(' c', I3, 7, I3, ' I GOTO 10 ELSE CAP(I,J) = ABS(Q(I)/(V(I) - v(J))) ENDIF "v~.I~(*,5) I, J, CAP(Id) FOR2~dAT( ' c', 13, 7, I3,' I ', E15.6) CONTIb~,dE CONTINUE

P3~dRN END


325

Appendix 8.B I]]]]] III ]]

................

HI

The Wire Antenna

C_ C

P

antenna By Jonathan Lo C N, p, numSeg REAL*8 a, 1, deltkn, k, pi, epso, muo, L omega, + !ambda, grid(65,-1:1), ratio, port, volt, + Imgtde(65), Iphase(65) COMPLEX*16 j, Z(65,65), I(65), V(65)

C C

/ d i m e n / a, I /segrrmt/ N, d e l ~ /excit/ k, omega / c o n s t / j , pi, epso, muo

OPEN(UNIT = 2, FILE = 'cur.re', STATUS = 'NEW') j= (0, 1.0e+0) pi = 3.141 epso = 8.85418781~-12 muo = 4*pi*l.0e-7 WR/~(*,*) WPdTE(*,*) 'Antenna Version 1.0 March, 1991' WFJ~(*,*) WNdTE(*,'(A\)') ' Input Wavelength: ' READ(*,*) lambda WPd~(*,*) WRI~(*,'(A\)') ' ~ p u t Antenna Length to Wavelength Ratio: '

3~26


1 1 = l*lambda WRJ~(*,*) WRI~(*,'(A\)') ' Input Antenna Length to Wire Diameter Ratio: ' READ(*,*) ratio a = I/(2*ratio) 10 !¢VRITE(*,'(A\)') ' Input Number of Segments (= 1):' READ(*,*) WRJTE(*,*) N = numSeg + 1 ((numSeg .LT. 1).OIL (numSeg .GT. ~ ) ) GOTO 10 15 p = 1,N V(p) = (0.0d+0,0.0d+0) 15 E WI~d~(*,'(A\)') ' Input Number of Excitation Points:' READ(*,*) n ~ E P WRdTE(*,'(A\)') ' I n p u t Excitation Point(s) and Voltage(s):' DO20 p = 1, nurnEP READ(*,*) port, volt V(port) = dcmplx(volt) 20 CONTII'-~dE WRITE(*,*) C f = 2.9 7e+8/larnbda omega = 2*pi*f k = 2*pi/lambda deltln = 1/ (N - 1) C CALL g e o m ( ~ d ) C

WRIt(*,*) WRITE(*,*) 'Computing [Z] Matrix ...'

WRITE(*,*) CALL Zmat(Z, grid) WRIt(*,*) 'Solving for [Z][I] = [V] ...' WRIt(*,*) CALL cgauss(N, Z, L V, 6.5) ~p=l,N Imgtde(p ) = sqrt(dreal(I(p )*dconjg(I(p))))


Iphase(p) - dangle(I (p))*180 / pi 40 COb~I'-~AE C WRdTE(*,*) 'Current Distributions:' WI~TE(*,*) WRdTE(*,*) ' DIST./I MAG. (A) PHASE (deg.)' WRJ~(*,*) DO 50 p = 1,N WR3[~(*,45) grid(p,-1)/l, Imgtde(p), Iphase(p) 45 FOI~2vIAT(e17.7, e17.7, f17.7) 50 CONTII'-~UE w

(*,*)

C C ~ a ~ u t Result to 'cur.m' C WRd~(2,*) 'I = [1' ~ 7 0 p = 1,N WRITE(2,*) 'I(',p,',1:3) = [..' WRI~(2,65) grid(p,-1)/l, Imgtde(p), Iphase(p) 65 FOR2-,/IAT(e17.7, e17.Z e l Z Z ']') 70 E C STOP FZ-qD C

SUBRO

geom(grid)

ER N , p , q REAL*8 grid(65,-1:1), delthn, hncre C lsegwuntf N, delfhn incre = 0.~+0 ~20p=l,N ~10q= -1, 1 ~ d ( p , ~ = incre incre -hncre + deltLn/2 10 C E hncre = hncre - deltln/2 20 CONTINUE C

327

328


P~JR2-,I F2~

COMPLEX*16 F U N C ~ O N psi(p,q) INTEGER N REAL*8 deltln, a, I, K o m e g a , pL epso, m u o , p, q, + alpha, z, r, rho, I1, I2, I3, I4, + A0, A1, A2, A3, A4 C 16j C C C

/ d h m e n / a, 1 / s e g r r m t / N, delthn / e x c i t / k, omega / c o n s t / j, pi, epso, m u o

alpha = deltln/2 z=p-q r = abs(p - q) rho=a IF (r .LT. 1 0 * a l p h a ) T H E N I1 = lo g ( ( z + a lp h a + s q r t( r h o * * 2 + ( z + a lp h a ) * * 2 ) ) / + (z-alpha+sqrt( rho**2 +( z - a l p h a )**2) ) ) I2 - 2*alpha I3 - 0.5*(alpha+z)*sqrt(rho**2+(alpha+z)**2) + + 0~5*(alpha-z)*sqrt(rho**2+(z-alpha)**2) + + 0.5*rho**2*I1 I4 = 2*alpha*rho**2 + (2*alpha**3 + 6*alpha*z**2)/3 psi = exp(-j*k*r)/(8*pi*alpha)*(I1 - j*k*(I2-r*I!) + 0.5*k**2*(I3-2*r*I2+r**2*I1) + + j*k**3/6*(I4 - 3"r*I3 + 3"r*'2"I2 - r**3*I1)) ELSEIF (r .GE. 10*alpha) T H E N A0 = 1 + 1/6*(alpha/r)**2*(-1 + 3) + + 1/40*(alpha/r)**4*(3 - 30 + 35) A1 = 1/6*(alpha/r)*(-1 + 3) + + 1/40*(alpha/r)**3*(3 - 30 + 35) A2 = - 1 / 6 - 1/40*(a!pha/r)**2*(1- 12 + 15) A3 = 1 / 6 0 * ( a l p h a / r ) * ( 3 - 5) A4 = 1/120 psi = exp(-j*k*r)/(4*pi*r)*(A0 + j ' k ' a l p h a * A 1 + + (k*alpha)**2*A2 + j*(k*alpha)**3*A3 +


329

+ (k*alpha)**4*A4) Eb~IF

RETURN END C

SUBROUTENE Zmat(Z,~id) ER N . u , v REAL*8 deWm, K omega, pi, epso, muo, + rr~-~EG,mXMID,mPG~, pd'-4~G,nMID, nPGS, + ~d(65,-1:1) COMPLEX*16 j, Z(65,65), psi C / ~ g r r m t / N, d_eltln /excit/ k, omega /cop.st/ j, pi, epso, muo

C C C

1202Du = 1, N ~10v=l,N rr'&qEG = grid(u,-1) InM1D = grid(u,0) toPOS = ~d(u,1)

rdX~G = ~d(:¢,-1) rLM~ = grid(v,O) = grid(v,1) Z(u,v) = j*omega*muo*(deltLn)**2*psi(nMID,mMID) + 1/ 0*°mega*epso)*(psi(nPOS,mP~) - p~(I~S-qEG,m_POS) -psi ) +psi )) C E COt"TI~INUE

+ +

10 20 C

RN END C C C

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

INCLUDE 'cgauss,for'

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

330


C See Below C INCLUDE 'dangle.for' C See BelowC SUBROUTINE CGAUSS(N, A, W, B, LIMIT) C C C C C C C C C C C C C C C

This subroutine uses the GAUSS elimination method to solve a set of simultaneous equations written in the standard matrix f o ~ of [AI[W] = [B]. h i s subroutine calls on three other subroutines: ORDER, ELIM, and BACKSB. h e s e four subroutines were taken and modified slightly from: D. M. Etter, STRUCTURE FORTRAN 77 FOR ENGINEERS AND SCIENTISTS 3RD ED. California: h e Benjamin/Cummings Publishing Company, ~c., 1990, p.484-485

Ib~EGER N, PIVOT, LDcI~ COMPLEX*16 + A(LIMIT, LIMIT), W(LIMIT), B(LIMIT) LC~-GICAL EPN,OR C PWOT = 1 ERROR = .FALSE. 10 IF (PWOT .LT. N .AI'-~..NOT. ERROR) THEN CALL ORDER(N, A, B, PIVOT, E LIMIT) IF (.NOT. ERROR) THEN CALL ELIM(N, A, B, PIVOT, LIMIT) PWOT = PWOT + 1 10 ENT)IF C IF (ERROR) ~ I E N WI~d~ (*,*) 'NO UNIQUE SOLU~ON EXISTS!'


c C C C

3 31

EI_SE CALL BACKSB(N, A, B, W, LIMIT) WRITE (*,*) ' H E SOLI~IONS ARE:' EnD 20 I=I,N WRITE(*,*) I, W(I) 20 CONTINUE EN~IF

C PdN END C

................................................

.

C SUBROUTINE ORDER(N, A, B, PWOT, E LIMIT) C C This subroutine reORDERs the equations so that the pivot C position in the pivot equation has the maximum absolute value. C INTEGER l'q,ROW, RMAX, PIVOT, K, LIMIT COMPLEX*16 A(LIMIT,LIMIT), BGIMIT), TEMP LCK3ICAL ERROR C R)2~A_X = PIVOT DO 10 ROW=PIVOT+I,N IF (ABS(A (ROW, PIVOT) ).GT.ABS( A(R2MAX,PIVOT))) + R&I,a~X= ROW 10 CO E IF (ABS(A(I~d~dAKPIVOT)).LT.1.0E-5) THEN ERROR = .TRUE. ELSE IF (RMAdK .NE. PIVOT) THEN ~ 2 0 K=I,N ~ M P = A(R~d~A_X,K) A(I~)¢IAX,K) = A(PIVOT,K) A(PWOT, K) = P 20 C E TEMP = B(R~vIAX) B(RMAX) = B(PIVOT) B(PPi'OT) = ENDIF PdSTUKN

332


EN~ C C

-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

-

.

.

-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

-

.

.

-

.

.

-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

SUBROUTINE ELIM(N, A, B, PIVOT, LIMIT) C C This subroutine ELIMmates the element in the pivot position C from rows following the pivot equation. C INTEGER N, PIVOT, ROW, COL, LIMIT COMPLEX*16 A(LDdIT, LIMIT), B(LIMIT), FACTOR DO 10 ROW=PIVOT+I,N FACTOR = A(ROW,PIVOT)/A(PIVOT, PIVOT) A(ROW,PPv'OT) = 0.0 5 COL= 1,N A(ROW,COL) = A(ROW,COL) - A(PIVOT,COL)*FACTOR 5 C E B(ROW) = B(ROW) - B(PIVOT)*FACTOR 10 CON~I'-~AE C RETUPJ'-,I END C C

_

_

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

.

-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

SUBROUTINE BACKSB(N, A, B, W, LIMIT) C C ~ i s subroutine performs the BACK-SuBstitution to deterrnhqe C the solution to ~ e system of equations. C INrTEGER N, ROW, COL, LIMIT COMPLEX*16 A(LIMIT, LIMIT), B(LIMIT), W(LIMIT) C DO 20 ROW=N,1,-1 DO 10 COL=N,R:OW+I,-1 B(RDW) = B(ROW) - W(COL)*A(ROW,COL) 10 C W(ROW) = B(ROW)/A(ROW,ROW) 20 CONTINFUE C R~TURd-~ END C

.

.

.

.

.

A. Konrad mad I. A. Tsukerman

C C C FUNCTION dan~e(z) C COMPLEX*16 z I~AL*8 x, y, pi C C

pi = 3.141592654d+0 z = x +jy x - real(z) y = imag(z) IF ((x .GT. 0.0d+0).AND. (y .GE. 0.0d+0)) THEN dangle = atan(y/x) ELSEIF ((x .LT. 0.0d+0).AND. (y .GE. 0.0d+0)) THEN dangle = pi - atan(y/abs(x)) ELSEIF ((x .LT. 0.0d+0) .AND. (y .LE. 0.0d+0)) THEN dangle = -pi + atan(abs(y) / abs(x)) ELSE d ~ g t e = atan(y/x) F

C RETURN t~--4D

33 3

3_..3..4


8.C . . . . . .

IMIII

II

II

IIIIII!1

IIIIIIII

..............

IIIII1[!111

. . . . . . . . .

IIIIIII

i

!!!![ll[lUlll

i .......

FEA of TM and TE Modes in a Waveguide ,

,,,,

,,

:

..........

. . . . . . . . . . . . . . . .

IMPLEMENTATION IN MATLAB (Revised) By William Prajogo ~ e following brief description of the Matlab finite element code is intended as a ~ i d e l i n e for those interested in pursuing this topic. The first step is to read in the geometrical information that defines a finite element grid. One way to do that is to use the Matlab load statement to load the data stored in a file named filename.ext to memory and assign it to a variable named filename. Note that the data is always stored in the form of a r e c t a n ~ l a r matrix since Matlab works only with matrices. Here, the data is stored in matl.dat: load m a t l . d a t ; A typical example of a data set in the form of a 16x4 matrix stored in matl.dat is as follows: 15000 1100 2000 3010 4110 5200 6210 7220 8120 9020


10300 11310 12320 13400 14410 15420 A variable n a m e d np is a s s i ~ e d to hold the number in the first row and colu~--mnof mat1, which is the number of finite element nodes. Four 15xl matrices ip,x,y and g each holds ~ e i~dormation stored hn row 2 to 1G columns 1,2,3 and 4 of mat1 respectively. Matrix ip contains ~ e node numbers, matrices x and y contain the x-y coordinates, and g contains the source function values. np=matl(1,1); ip=matl(2:np+l,1); x=matl(2:np+l,2); y=matl(2:np+l,3); g=matl(2:np+l,4); b e 153 nodes correspond to points from a subdivision of a recatangular domain using 17nodes x 9 nodes. Each subrecamgle is then subdivided into two trahngles each, where cuts at alternating nodes are made at 0, 90, 180 and 270 degrees or, the above 4 plus along the diagonals. The source function is set to zero everywhere in the region since Laplace's equation is to be solved. Similarly, a second data set Ln mat2.dat is to be loaded to memory. load mat2.dat; A typical data set in the form of a 17x5 matrix is as follows: 160000 11231 21431 31541 45461 53491 64891 74681

3 35

336


88671 956101 10 61011 1 11101113 1 12111314 1 13111214 1 14121415 1 15 71112 1 16 6 711 1 Again, a variable ne ~ used to hoM the number in row I rand column I of mat2, which is the number of triangular finite elements. Five 16xl matrices ie, a,b,c and e hold the information stored in row 2 to 17, column 1,2,3,4 and 5 of mat2 respectively. Matrix ie contains the triangular numbers, matrices a,b and c contain the vertices of the ~iangles and matrix e contains the relative permitivity of each element which is 1 for all triangles. ne=mat2(1,1); ie=mat2(2:ne+l,1); a=mat2(2:ne+l,2); b=mat2(2:ne+l,3); c=mat2(2:ne+l,4); e=mat2(2:ne+l,5); The last data set to be loaded to m e m o ~ is stored in mat3.dat. load mat3.dat;

A typical data set in the form of a 7x3 matrix would be as follows: 600 120 230 390 4131 514 1 615 1


337

Once again, a variable nb is to hold the number in row I and column I of mat3, which ~ the number of bounda D" nodes. Three 6xl matrices ib,d and p will hold the h-flotation stored in row 2 to 7, column 1,2 and 3 of mat3 respectively. Matrix ib corttains the boundary node sequence numbers, matrix d contains the actual boundary node numbers (as stored fin matrix ip) and finally matrix p contains the potential values associated with each Dirichlet boundary node stored in matrix d. nb=mat3(1,1); ib=mat3(2:nb+ 1,1); d=mat3(2:nb+l,2); p=mat3(2:nb+l,3); Notice that the capacitor plate on the left is grotmded and the one on the r i ~ t is at I Volt. The first order finite element matrices are entered permanently into matrices sa,sb, sc and ta, respectively. These matrices are the preintegrated type given in P. Silvester, "A General High-Order Finite Element Waveguide Analysis program," IEEE Trans. Microwave Th. & Tech+, VoL 17, No. 4, pp+ 204-210, April 1969. sa=[ 000 0 1-1 0-1 I

]; sb=[ 1 0-1 000 -101

]; sc=[ 1-1 0 -1 1 0 000

]; ta=[

338......


211 121 112

]; At this point, one can define the "empty" global finite element coefficient matrices [Sg] and [Tg] appearing in Eq.(3). The~ s~e depends on ~ e total number of node points, np. This step corresponds to the following Matlab statements that initialise two np by np matrices s and t to contain zeros. s(np,np)=zeros; t(np,np)=zeros; For reasons ~ a t "will become obvious later, it is n e c e s s a ~ to create a 3×a-rema~ix rn out of fine three triangIe vertex matrices a,b and c. The first row of m will contain a, ~ e second b and the third one c as follows: m =[a,b " ';c ']; R e following is a loop over the ne elements during global finite element matrix assembly: for i=l:ne R e area of each triangular finite element is computed based on Eq.(6). area=abs((x(a(i))-x(c(i)))*(y(b(i))-y(c(i))) + (x(c(i))-x(b(i)))*(y(a(i))-y(c(i))))/2; Similarly, the cotangents of ~ e interior angles as given in Eqs. (7) through (9) are obtained for each element as follows: ctga=((x(c(i))-x(a(i)))*(x(b(i))-x(a(i))) + (y(c(i))-y(a(i)))*(y(b(i))-y(a(i))))/(2*area); ctgb=((x(a(i))-x(b(i)))*(x(c(i))-x(b(i))) + (y(a(i))-y(b (i)))*(y(c(i))-y(b(i))))/(2*area); ctgc=((x(b(i))-x(c(i)))*(x(a(i))-x(c(i))) + (y(b (i))-y(c(i)))*(y(a(i))-y(c(i))))/(2*area);


One can then compute the local coefficient matrices [S1] and [T1] for each i, as per Eqs. (4) and (5): sl=e(i)*(sa*ctga+sb*ctgb+sc*ctgc)/2; ti=ta*area/12; The necessity" of the 3xne matrix m should now become clear. In order to insert the local coefficient matrices [S1] and IT1] in the right position m global context, it is necessary again to use matrix subscriptmg feature of Matlab+ The global node numberLng scheme for the i-th element is extracted from the i-th column of m, i.e. m(:,i), and is used to d e s i ~ a t e the global position of the local matrix element contributions:

ii=m(:,i); s(ii,ii) =s(ii,ii)+sl; t(ii,ii)=t(ii,ii)+tl; The Ioop g ~ s on to the next finite element in the list: end order to separate the unknown, or free nodes, from the knovcn, or [Y~ichlet nodes, one must fi~t find the node n u m b e ~ of free nodes. This can be done easily h-~Matlab by noting that empty-matrix assignment is the same as deleting that particular element(s) a s s i ~ e d to empty. Initially, a new matrix f is to hold the node numbers stored in ip. Then, one deletes the Diriclnlet node numbers from f by a s s i ~ i n g the correspondmg elements to empty as follows: f=ip; f(d)=[]; Now that column matrices f and d contain the flee and fixed node numbers, respectively, one may proceed to partition the global finite element coefficient matrices [Sg] and [Tg]+ With reference to Eq. (10), fine global coefficient matrices [Sg] and [Tg] are partitioned by employing matrices f and d as subscripts in different combinations:

339

340

Chapter 8: Applicationof Integral and Differential Methods

sll=s(f,f); sl2=s(Ld); t11=t(f,f); t12=t(f,d); Notice that the s21,s22,t21, and t22 submatrices ~ v e n by s(d,f),s(d,d),t(d,f), and t(d,d), respectively, are not needed in the calculations. The global column matrix of potentials, [Vg], can be partitioned into free (vl) and fixed (v2) parts. The free part contains np-nb potentials :which are now initialised to zero. R e fixed part contains nb potentials which are set to the values stored in matrix p. vl(np,nb,1)=zeros; v2=p; The global column m a ~ of sources [Gg] can be partitioned and initialised accordingly. gl=g(f);

g2=g(d); The right-hand side of Eq. (11)with k-0 can now be ob~hned from the MlowLng Maflab statement: rhs=t11*g1 + t12"~ - s12*v2; Eq. (11) with k=0 is given as [s11][v1]=[rhs]. This can be solved using the following statement m Matlab: vl = s l l ~rhs; Matrix vl now contains the potential solution but in the wrong sequence. The coluwa-t m a r x v is used to store the potentials m the correct sequence by employing column matrices f and d as subscripts to accomplish the task: v(np, I)=zeros; v(f)=vl;

A+ Konrad and I. A. Tsukerman

v(d)=v2; The following Matlab statements are used to display Lhe output of tabular f o ~ on the screen. Note that the whole program code is enclosed by fine pair of statement: diary filenarne.ext (on the very+top) - diary off (or,. the very bottom). This is meant to save everything that is displayed on screen to a file n a m e d filename.ext; irt our example it ~ progl.out + fprintf(' IP X Y V',,n'); fprintf(' -. . . . . . ',n'); for i=l:np fprintf(' %2g %2g %2g ',ip(i),x(i),y(i)); fprintf(,%6.2g"m+,v(i)); end To display any of the resulting matrices, one can type the name of the matrix (say s11) and hhen press Enter.

341

Chapter 9 I

II!!1

IIII

III

III

Srisivane Subramaniam and S. Ratnajeevan H. Hoole II!1

III!1

IUIIIII

I[llJ

Jill

. . . . . . . . . . . . .

.......

f

......

EDGE ELEMENTS

9.1 The Finite Element Method In most problems in engineering and science, either the geometry, material properties or some other feature of the problem is irregular or complicated. We can easily formulate the governing equations, related boundary conditions and the initial conditions for these problems, but it is not so simple to find the exact solution which rarely exists in explicit form for complicated systems. Here, we find the necessity for an approximate numerical solution and more often it is feasible to construct an approximation in terms of a finite number of state variables. For an accurate enough solution, although finite, many such state variables are needed and the appearance of the digital computers presenr~s us with the capability of obtaining approximate numerical solutions. The special advantages of the finite element method are the suitability of the equation formation process in a systematic manner and the ability to represent highly irregular and complex structures and material proFerties.

9.2 The History of the Edge-Based Finite Element Method Although the ideas for the development of the finite element method evolved inde~ndently from many people in the fields of applied mathematics, engineering and physics, they began to appear in publications in the form we know the method only from the 1960s. In the engineering community, Hrenikoff (1941) and McHenry (1943) replaced a continuum by a lattice-like assembly of bars. In the early 1950s, Langefors (1952) and Argyris (I954, 1955) establish~ the frameworkanalysis procedures and refo~ulated them into a matrix fo~at. In Turner, Clough, Martin and Topp's work (1956), triangular shape panels were used to model an

S. Subramaniam and S. R. H. Hoole

343

aircraft and introduced the element concept. But the name of fine method, "finite elements," first appeared in Clough (1960). In the late I960s, Zienkiewicz broadened and demonstrated the applicability of the finite element method and in 1970s the method spread to many other fields of application. In electrical engineering, even though the work by Winslow (1967) used all the concepts of finite elements, it was Silvester (1969c), who with his colleagues used and popularized the method in its present I~rm by applying it in various areas. Initially, the finite element method was applied to scalar fields. The unknowns are associated with the nodes and the elements are known as node-based finite elements. For the solution of vector fields, the fields were derived either from a vector potential or from a scalar potential (Simkin and Trowbridge, 1979; Trowbridge, 1981). The magnetic field has been solve~ for directly (Hoole and Cendes, 1985) using node-based finite elements for the three coupled scalar field that are the vector components. There are, however, diLficulties with these nodebased finite elements in imposing the continuity conditions at the material interI~ces and in modeling sharp corners. Further, it has been shown by Hoole, Rios and Yoganathan (1986) that the trial functions over-specify the system so much that accuracy is vitiated. However, as already alluded to in the preceding chapter, these drawbacks can be overcome by using edge-based finite elements, in which the unknowns are associated with the edges or faces of the elements. The edge-based finite elements were introduced first by de Veubeke (1965, 1975) with the relaxation of the continuity of the normal component of the vector variables across the inter-element edges. Raviart and Thomas (1977) used these elements for solving two dimensional problems. Nedelec (1980, I982, 1985) introduced and used two families of three dimensional conforming elements which have tangential and normal continuity and are built on tetrahedrons or cubes. Another set of two families of elements were introduced by Brezzi et aI (1985) in two dimensions. In the electrical engineering community, Bossavit (I981, 1982, I988) derived variational formulations to solve eddy current problems for magnetic field intensity with the finite elements similar to those given in Nedelec (1980). The same elements were used to solve open boundary eddy-current problems by a hybrid finite element - - boundary integral method in Bossavit and Verite (1982, 1983) and Ren (I99(3) respectively for magnetic field intensity and electric field intensity. Bossavit and Mayergoyz (1989) applied edge elements for scattering problems and Bossavit (1990) solved the M~well equations in a closed cavity.

9.3 Rectangular Tangential Vector Elements In the edge-b~ed finite element formulation our main aim is to represent the vector field in terms of the variables along the edges and/or normal to the edges. In most of the applications we solve for vector fields which are divergence free. In addition to being divergence free, some of the fields have constant curl. Therefore, starting with a parametric representation and using tensor methods, Okon (I982) derived

344

Chapter 9: Edge Elements

:

!

I

x=a

•~

Z

| 7--A

x=-a

I I Figure 9.3.1" Parallel conducting thick strips ('Bus-bars') some zeroth-order ve~=torfunctions having zero divergence and constant cud over quadrilaterals and triangles. This approach is almost similar to the derivation of covariant projection elements for 3D vector fields (Crowley, Silvester and Hurwitz Jr., I988; Pinchuk, Crowley and Silvester, 1988). Miniowitz and Webb (1991) applied the cov~iant-proje~tion elements for the analysis of waveguides with sharp edges in two dimensions.

9.3.1: A Simple Demonstrative Problem Analyzing the skin effectdue to the ac. current flowing in a pair of parallel conducting strips ('bus-bars'), is taken as the demonstrative problem here ( 1985; and Pinchuk 1985). The conductors which are separated by a finite dis~nce, are supposed to be very wide and very thick, as shown in Fig. 9.3.1. The plane x = 0 becomes the plane of antisymmetry with the assumption that the pair of conductors are driven by a balanced generator, so only one of the conductors is analyzed. We shall consider the conductor in which the current flows in the positive z-direction. If a Node,_No for i := 1 to Total_No_Of Nodes do begin 1. Allocate memory for IndexColumn 2. Establish the link with the previously made one. { Linked list concept } 3. Set R o w N u m b e r := Node_No 4, Node_Address := Node, Address->Next_Node 5. Node_No := Node_Address->Node N o 6. Node_Add := Starting Address of the Node List 7. New Node3-.~o := Node Add->Node_No 12orj := 1 to Tota!,N'o,Of_Nodes do begin 1. Allocate memory for Element 2. Establish the Links 3. Set Element_No := N e w , N o d e N o 4. Node_Add := Node Add->Nextj"qode 5. N e w N o d e N o := Node_Add->Node_No end end; Algorithm 15.3.3: Steps for Building the Global Matrix.

D. Kurumbalapitiya and S. R. H. Hoole

593

sGlobal Matrix: Our solution technique is based on energy minimization. A local matrix contains information on the energy enclosed within the associated triangular element. Therefore in order to minimize the total energy it is necessary to find that. That is where Type 15.3.5 of data structure described in Figure I5.3.5 for the Global Matrix comes in use. Placing all the local matrices in the global matrix in a proper way will produce a matrix that contains the total energy of the system. The steps in Algorithm 15.3.3 together with Figure 15~3.6 might guide the reader in establishing the global matrix in memory (compare the code with Figure 15.3.6):

Element

!golds E l ~ e n t No P_Value Next Index

Nit

Figure 15.3.6:

The Global Matrix.

The P V a l u e and Q V a l u e slots in the structure are used to put values that are read from the P Matrix and Q Matrix slots in each local matrix. It is appropriate at this point to proceed further considering the last result we obtained, that is eq. (15.3.23). The multiplication a t [ P , Matrix] a causes the coefficients in the P Matrix to couple with the nodes in the triangle. Therefore the resultant 3 x 3

594

Chapter 15: Open Boundary Problems with Finite Elements

Hypothetical Boundary

Conductor 2

Conductorl

Figure 15.4,1:.Two Long Current Carrying Conductors matrix has coefficients that have "directions ". This directs each coefficient of the local matrix where to go and sit in the global matrix. A similar argument is applied to the terms in Q M a t r i x . This hints at what the global matrix looks like. We must be able to follow those "directions " given by any local matrix coefficient within the body of the global matrix, Therefore it must be arranged according to the node sequence. Any element within the global matrix is a unique combination of its nodes and inherit this characteristic from the local matrices. Once the matrix is implemented, the rest of the steps like placing local matrices in the global matrix and minimization follow the same steps as the classical finite element routines.

15.4 Examples 15.4.1. Field Forms Conductors

Around

Two

Long

Current

Carrying

Two examples are worked out in this section to illustrate how to apply the method for some practical situations. 1.2D magnetostatic problem with a scalar potential formulation. 2. An axisymmetric problem with a scalar potential formulation. The simple magnetostatic problem is a very good example fbr illustrating the beauty of the method. This is an open boundary problem. The problem wilt be formulated using a scalar potential function because there is no field variation along the Z - direction. Figure 15.4.I shows the conductor arrangement plus the two domains for the interior and exterior. The semi-circular boundary divides the entire open region into two regions, Win and Wout. Wout is an open region with just one semi-circular boundary on one side of it. The other one, Win is a closed region. Note that both of them share a common boundary which has a semi-circular shape, what we have named the "Hypothetical Boundary" in Figure 15.4. I, as described in the section 15.2.

D. Kurumbalapitiya and S. R. H. Hoole

595

It is always possible to find a transformation that maps the exterior of a circle onto its interior. So the points at infinity will be mapped onto the center of the circle. What does that mean? The open region is now within the circle. This kind of transformation will not modify the circumference. Then the open region, Wou t will be mapped into the closed region with a semi-circular boundary. Let us denote the transformed version of region Wou t as Wtransformed. Therefore it allows us to divide Wtransforme d into triangular elements forming finite elements. How do we make use of the mapped boundary condition of zero potential at infinity? In this case of course the node which is at the center of the transformed region carries the known potential, zero. That is, one and only one node in the node list is at a known potential. Now it is possible to create Node and Element data files according m the formats described earlier. The next task would be to write a routine to find the necessary, coefficients as derived in the last section. They are K I, K 2 and K 3. Let us first look at such a transformation in mathematical terms. X = I ( x 2R2 + y2) ~l x , Y = I ( x2 R2 + Y2)l y

(15.4.1)

will perform such a function. Then x,y represent the domain Win and X,Y represent the domain Wtransformed.. R represents the radius of the semi-circular arc. Now three things must be found; specifically (VX) 2, (VY) 2 and the Jacobian J:

(VX)2 - (Vy)2 = h ard J =

R2

]

(15.4.2)

[ ( x 2R2+ y2) 1

(15.4.3)

Therefore what will happen to the terms (VX)2J and (Vy)2j in our original integral equation, 1

2

2

2

2

L[A] =Z{2o([ba] J'~(VX) JdR+[ca] ~,[(VY)JdR)-atJ~faJdR}

(15.3,19)?

They become one. Therefore the first two integrals are exactly similar to the integrals which we found in case of the natural unmapped triangle. In other words, the coefficients K 1 and K 2 are equal to unity. Therefore the transformation will not affect that part in the local matrix development process. But what about the last integral, f.[aJdR? Expanding this integral we get i[J~aJ dR ~- [ Lf;tJ dR J'~;2J dR ~ ; 3 J dR ] (15.4.4). This is the place where we are going to apply Gaussian Quadrature. The aim is to find .[J[{iJ dR where J = [R2/( x 2 + y2)]. Then {iJ will be equal to the function [~i R2 / ( x 2 + y 2 ) ] . This is a second order function. We shall apply Gaussian Quadratureassuming a first order distribution of the integrand. Thus the integrand must be evaluated at least at three specific points on the triangle.

596

Chapter t5: Open Boundary Problems with Finite Elements

1. Evaluate (x I Yl ) , (x2 Y2) and 2. Set S u m [ j ] : = 0 f o r j : = l to3 3. for j : = l t o 3 d o begin for i := 1 to 3 do begin case i of I : begin 41 := 0; 2 : begin 41 := 0.5; 3 : begin 41 := 0.5;

(x 3 Y3).

~2 := 0.5; ~2 := 0; 42 := 0.5;

~3 := 0.5 end; ~3 := 0.5 end; {3 := 0 end;

Sum[ j ] := Sum[j ] + [ ;jR2 2t; 1,3) x [(Xl{l+X2{2+x3{3)2 + (Yl{l+Y2~2+Y3~3) end; end;

Algorithm 15.4.1: Numerical Evaluation of the Integral Furthermore these three points, we shall take, lie at the middle of each side of the traingle over which the integration is performed. Let us denote the coordinates of these three middle points as (xI,yl), (x2,Y2) and (x3,Y3). The next step is to replace all x and y in the [~iR 2 / ( x 2 + y2)] by (Xl~l + x2{2 + x343) and (Yl~I + Y242 +Y343). We can evaluate the function effectively at three well defined points on the triangle by means of the previous substitution. But how do we know the values of ~I, 42 and ~3 at (x I Yl), (x2 Y2) and (x 3 Y3) at these points? The values are as shown in Table 15.4. I, based on their definition. xi,yi x!,yl

I

~.1...................I . . ,,,0.0

x2,Y2 . . . . . . . .0.5 ........ x3,Y3 0.5

~2 0.5 0.0 0.'5

0.5 0.5 0.0

Table 15.4.1: Triangular Coordinated at Triangle Edge Mid-Points We have explained the necessary background except the main number crunching process. The following formula gives us the result: fj'f(x,y) dR = Z wif(4 I, 42, 43)i for i := I to 3 (15.4.5) where wi = (1/3), and f(x,y) is a function defined on the x,y plane, f(~l, 42, ~3) is the corres~nding function in Gaussian Quadrature.

D. Kurumbalapitiya and S. R. H. H~)le

597

f({l, 42, 13) -[

{jR2 ] (15.4.6). [(Xl(l+X2(2+x3(3 )2 + (yl~l+Y2~2+Y3~3)2 The procedure is fairly straight-forward now and Algorithm 15.4.1 provides the necessary steps. In the algorithm [ Sum[l] Sum[2] Sum[3] ] = .[ [j'Jr~lJ dR ]'f~zJ dR ff~3J dR] ~ K3 (15.4.7) and that is what we wanted. Now this is matrix K 3. All these values may now be passed to the routine which makes the local matrix. 15.4.2.

Field A r o u n d a S o l e n o i d

a n d an Iron R o d

Figure 15.4,2. shows the device a~angement. This is an axisymmetric case, This is actually a 3D problem. But the field is symmetric around the Z axis of the device. Therefore it is possible to convert it to an equivalent 2D problem. Let us write an expression mr energy within an element using the functional described in the previous section. LIA]= it'

~[VA] 2 -JA

}

dV

(15.4.8)

where dV = rdrd0dz using the cylindrical (polar) coordinate system. Then

Z

O=O

X-Section of the Selenoid J O. Figure 15.4.2:

Innner Iron Rod with High ~. Solenoid

- Isometric View

.

598

Chapter 15: Open Boundary Problems with Finite Elements

2r~R

Z

L [ A ] - ~ f2 x(J[jV' JA " ] 2{- J / , . ~

rdrde~.

(15.4.9)

2~ The integral jf dO in the above expression can be evaluated independently, to yield he result 2~, Therefore the functional gets the following form R

Z

L[A] = C}r 0j" 2r~r 1~ 2 x [V.a,]2 - JA } d,~

(15.4.10)

which is again a surface integral. Now, drdz = dR, so the inmgral takes the general form

L[AI=Je ,f2~r{~xv [VA]2-JA }dR

(15.4.11)

It is very important to note here that the operator V is not the same as the previous one. It is defined for the cylindrical polar coordinate system as V = ur ~ + u0

~ + Uz ~z

(15.4.12)

where Ur, u0 and Uz are unit vectors in the respective directions. Since there is no variation in the 0 direction, the term ue

~ has no contribution to the

integral and hence it can be thrown away. The the new V operator will 3 V = u r ~ + u z ~ and it may be observed that it looks very similar to the previous one. We shall now consider the device and how the regions are formed. Figure 15.4.3 shows the 2D equivalent of the device with the circular boundary. The transformation is exactly similar to the previous one. Therefore it is possible to write the new equation in analogy with the previous one: L[A] =

x 2r~v([bal2~j'r(VR)2jdR+[ca]2~fr(VZ)2JdR)-at~fJraJdR

(I5.4.13) where R and Z represent the coordinates in the transt%rmed domain. The transformation affects the entire body of the integral. The coefficients K t, K 2 and K 3 are KI = SSr(VR)2J dR (15.4.14a) K 2 = SIr(VZ)2J dR (15.4.14b) K3 = ffra J dR (15.4.I4)c

D. Kurumbatapitiya and S. R. H. Hoole

599

This time too, ( V R ) 2 J = ( V Z ) 2 J - 1. Then KI = J~J'rdR and K2 = f.fr dR. Therefore we may easily compute K1, K2 and K3 using the Gaussian Quadrature f o ~ u l a e as before.

where

.....

Figure 15.4.3: 2D Representation of the Device

Chapter I

I

IIII . . . . . .

IIIII

I

16 iiii!]]1

I

]1

Abd. A. A r k a d a n IIIIII

IIIU///IJI

. . . . .

....

. . . . . .

[frlHffllfH~

. . . .

ON THE FEATURES OF ELECTROMAGNETIC FINITE ELEMENT ANALYSIS SOFTWARE PACKAGES

16.1 : Introduction The finite element (FE) method for electromagnetic field analysis, introduced over two decades ago (Silvester and Chari 1970; Hoole I989), has become a well established tool for the design and analysis of electromagnetic devices. It is heavily used in both university and industry research. It is the purpose of this chapter to familiarize the reader with some of the features of the commercial software packages for electromagnetic applications that are available in the market. The data presented is based on a survey of commercially available two dimensional (2D) and three dimensional (3D) finite element software packages. This survey is part of a short course which is presented annually at Marquette University (Arkadan 1993). The information presented in this chapter was current at the time of the survey. The packages listed here contain other features not discussed in the survey. For full details, the vendors should be contacted directly. Several finite element based commercial packages for the modelling, design, and perfo~ance evaluation of electromagnetic devices through the numerical solution of electrical and magnetic field problem are readily available in the market. These packages can be used m model and analyze many electromagnetic devices which include, integrated circuits, electromagnetic energy conversion devices, high voltage components, and high frequency devices. Over the past few years, the status of commercially available software packages was discussed in the literature (Cendes 1989; Swanson 1991; Sabbonnadiere and Konrad 1992). In this chapter, the capabilities of some software packages are presented as suggested by the software vendors in response to a survey see Fig. I6.1.I for a list of software packages discussed. A summary of the results of this survey is presented as related to the features of the i. Field Solvers; ii. Preprocessors; iii. Post processors

A. A. Arkadan

60 t

The features of the field solvers, which are finite element based, involves the types of solution, such as static, quasistatic, high frequency, coupled problems. The features of the preprocessor involve the capabilities of the software as related to setting up the problem. The features of the postprocessor involve the processing the results of the field solution such as field plots, graphing certain parameters versus space or time, performance characteristics computations, and solution results verification.

16.2: Field Solvers One may presume that 3-D analysis may give more accurate results because there is no assumption about the model. But in practice the time for a complete

company Ansoft Corp. Ferrari Associates Infolytica Corporation MacNeal Schwendler C0rp. Magsoft Corp.

Platforms

2-D

Workstations and PCs

Maxwell 2D, Maxwell 3D, Spicelink OPUS

Different plattorms Workstations and PCs

Swanson Analysis and Systems, Inc.

Mainframes, workstations and PCs Mainframes, workstations and minicomputer IBM PC 486/386, WS, HP, SGI, SUN, IBM 60(~, DEC Mainframes, Workstations, and PCs

Vector Fields, Inc.

Different Platforms

Structural Research & Analysis Corp.

3-D Maxwell 3D, Spicelink MAGNUS

MAGNETS TB - 2D, MAGI"-OETS TB - 3D MSC/EMAS

MSC/EMAS

MICROFLUX' FLUX2D

FLUX3D, PHI3D

CosmosM/ Estar

CosmosM/ Estar

ANsYS, ANSYS-PC/ MAGNETIC, ANSYS/ED PC-OPERA, OPERA-2d, TOSCA, ELEKTRA

ANSYS, ANSYS-PC/ MAGNETIC, ANSYS/~D

Figure 16.1.1: Software Packages Surveyed

MAGNETS TB - 3D

TOSCA, ELEKTRA

602

Chapter 16: On the Features of Electromagnetic FEA Software Packages

simulation is often 10 or more times shorter in 2-D analysis. In addition, a 3-D analysis may become cumbersome and less accurate in complicated devices which have nonlinear fields or eddy current distribution (Cendes 1989; Sabbonnadiere and Konrad 1992). Even though 3-D solvers based on the FE method are commonly used in general for wave guide and micro strip analysis, 2.5-D solvers based on moment methods seem to be more efficient t~)r planar circuits (Swanson 199 I).

16.2.1: Static Solvers The features of interest for the static solver are listed below and are labeled as features FS1 through FS6. A survey on these features as related to some software packages is given in Fig. 16.2.1. FSI

Linear Magnetostatics: In two-dimensional linear problems B is on the xy plane and is invariant with z, A obeys Poisson's equation. In three-dimensional or axisymmetric problems, the scalar Poisson's equation is not obeyed by B or A (Brauer 1993),

FS2

Nonlinear Magnetostatics: Many magnetic devices are made of steel and the steel has a nonlinear B-H characteristics. Further, in many of these devices, saturation effects play an important role in the t,mrformance. Accordingly, the material permeability has to be accounted for in the analysis. If the ~ e a b i l i t y is not a constant the finite element matrix depends on the magnitude of B (and J), so an iterative procedure is developed and B-H curves are supplied as input material data, as described in previous chapters.

FS3

Nonlinear Magnetostatics with Hysteresis: Ferromagnetic material used in electromechanical applications can be described as nonlinear and irreversible. The irreversibility of the B - H characteristics is represented by hysteresis loops. Further this loop is used to estimate the energy (heat) dissipated in the magnetic material as a result of reorienting the magnetic domains during a magnetization cycle.

FS4

Electrostatics: Electrostatic fields obey Poisson's equation in nonconducting media with the electric scalar potential which is measured in volts, as the state variable.

FS5

FS6

Current Flow: Poisson's equation governs steady electrical currents flows in good conductors. Here the electric scalar potential is considered as the state variable. Current features.

Flow

plus

Magnetostatics:

Combines the two

A. A. Arkadan

603

16.2.2: Quasistatic Solvers The features of interest for the quasistatic solver are listed below and are labeled as features F S 7 through F S I 0 - - see Fig. 16,2.2. A C Eddy Currents: Magnetic fields that vary with time can induce currents in conducting materials. These induced currents may be large enough to alter the original magnetic field and one may want to calculate the induced eddy currents. The fields are assumed to vary sinusoidaly.

FS7

Company A n soft

Software FSI FS2 FS3 Maxwell 2D ...... Y Y Maxwell 3D Y Y Spice Link Y Opus Y Y Magus Y Y Magnets TB-2D .......Y .................Y .... Magnets TB-3D Y Y

I

Corp. ....

Ferrari Associates Infolytica Corp.

MacNeal " 1 Schwendler

MSC / E M A S

Y

Y

Y

FS4 Y Y Y !

FS5 Y Y Y Y

FS6 Y Y Y Y

!

Y Y

Y Y

Y

Y

Y

Corp.

t

Microflux Flux2D Flux3D Phi3D

Magsoft Corp.

Structural Research & Analysis

C

o

Swanson Analysis and Systems,

CosmosM/Estar

r

p

,

........Y .......... Y Y Y Y Y Y Y

Y .......... Y Y Y Y Y

Y

Y

Y

..................~........................................................... ,..................,.................., ................

Ansys Ansys-Pc / Magnetic Ansys / ED

Y Y Y

Y Y Y

Y Y Y Y

Y Y Y Y

Y Y*

Y Y Y

Y Y Y Y

Y Y Y Y

Inc. PC - Opera Opera 2d Tosca Electra * Heat conduction analogy ......

Vector Fields, I n c .

Figure 16.2.1: Static Solvers

Y Y Y Y

Y Y Y

604

Chapter I6: On the Features of Electromagnetic FEA Software Packages

F $8

AC Electric Fields in Lossy Dielectrics

FS9

AC Coupled Electromagnetic

FS10

Time Periodic Nonlinear Eddy Currents: Nonlinearity introduces multiple frequencies; therefore the analysis is different from the previous one.

Fields

16.2.3: Transient and High-Frequency Solvers The features of interest with transient and high frequency solvers are listed as C o m p a n y ........

Software

FS7

FS8

FS9

Ansoft Corp.

Maxwell 2D Maxwell 3D

Y Y

Y Y

Y Y

Ferrari Associates Infolytica ....C o r p . MacNeal Schwendler .......... C o r p .

~us Magus Magnets TB-2D Magnets TB-3D

Y Y

MSC / EMAS

Y

Y

Y


Y Y Y

Y

CosmosM/Estar

Y


Y Y Y

Magsoft Corp. Structural Research & Analysis Corp. Swanson Analysis and Systems, Inc. Vector Fields, Inc.

Opera 2d Tosca Electra

Y Y

Figure 16.2.2: Quasistatic Solvers

FS10

Y

Y Y Y Y Y

Y Y Y

A, A. Arkadan

605

follows and are labeled as features F S l l through FS15: FSI 1

T r a n s i e n t eddy currents: Here the fields vary with time but the variations need not be sinusoidal; therefore the results are obtained by stepping along in time.

FS12

Transient Nonlinear Eddy Currents: The materials are nonlinear, therefore in addition to stepping along in time one has to apply an iterative procedure.

FS13

Real Eigenfrequencies and Eigenvectors: The problem is

FS12

Software F S 11 Maxwell 2D Maxwell 3D Spice Link Ferrari Opus Y Associates Magus Infolytica Magnets TB-2D Y Corp. ....... Magnets ~ - 3 D ...............y MacNeal Schwendler MSC / EMAS Y Corp. Microflux Magsoft Flux2D Corp. Flux3D Phi3D

[__Cg_mp.a.ny

........

FS13

F S I 4 [FS15

Ansoft Corp.

.

.

.

.

.

.

.

.

.

.........................................

,,,,

J

,,,,,,,,,,,,

,,,,,,,,,,

;

Y Y Y Y

Y Y

Y

Y Y Y

Y

Y

Y

Y Y Y

.....

Structural Research & Analysis Corp. Swanson

CosmosM/Estar

Y

Y

Analysis and Systems, Inc.

Ansys Ansys-Pc / Magnetic Ansys/ED

Y(2D) Y Y

Y(2D) Y Y

Y Y

Y Y

PC 21Opera Opera 2d Tosca Elexctra.... + Massive conductors only.

Vector Fields, Inc.

Y

I

Figure 16.2.3: Transient and High frequency solvers

y+

y+

606


solved in the frequency domain with zero-excitation amplitude with the assumption that the displacement term is negligible.

FS14

Complex Eigenfrequencies and Eigenvectors: The introduction of the displacement current term allows one to consider the conductivity and associated ohmic losses.

FS15

Voltage Source Excitation: FE analysis usually assumes that the current is known. However in most practical applications the voltage is what is known.

Company Ansoft Corp. Ferrari Associates lnfolytica Corp. MacNeal Schwendler .........C .. o r p .

,

Vector Fields, Inc.

FS16

FS17

MSC / EMAS

Y

Y

Y

Y

CosmosM/Estar

Y

Y


Y Y Y

Y Y Y

FS18

FS19 Y Y Y Y

"', I


Magsoft Corp. Structural Research & Analysis Corp. Swanson Analysis and Systems, Inc.

Software Maxwell 2D Maxwell 3D Spice Link Opus Magus Magnets TB-2D Magnets TB-3D

Y

Y

Y

Y

.i, I

. . . . . .

,,

PC - Opera ~ r a 2d Tosca Electra

Figure 16.2A: Coupled Solvers

Y Y

Y Y

A. A. Arkadan

607

16.2.4: Coupled Solvers The features of interest I~r coupled solvers are listed below and are labeled as features FS16 through FS19 - - see Fig. 16.2.3. FSI6

T r a n s i e n t Coupled Electromagnetic Fields: Both Electric and magnetic fields exist and are coupled. The material is linear. The fields cannot be calculated separately. Results are obtain~ by stepping along in time,

F S17

Transient Nonlinear Coupled Electromagnetic Fields

FS18

Transient Nonlinear Eddy Rotary Mechanical Motion

.....C o m p a n y Ansoft Corp.

Software Maxwell 2D Maxwell 3D

PR1

Ferrari Associates Infolytica Corp. ' MacNea! Schwendler Corp.

Opus Magus Magnets ~ - 2 D Magnets TB-3D

Y Y Y

MSC / EMAS

Magsoft Corp. Structural Research & Analysis Corp. Swanson Analysis and Systems, Inc.

with

Linear

PR2 Y Y

PR3 Y Y

PR4 Y Y

PR5 Y

Y

Y

Y

Y

Y Y

Y Y

Y Y

Y

Y

Y

Y


Y Y Y Y

Y Y Y Y

Y Y Y Y

Y Y

CosmosM t~star

Y

Y

Y

Y


Y Y Y

Y Y Y

Y Y Y

Y Y Y

Y Y Y Y

Y Y Y Y

Y Y Y Y

Y Y Y Y

PC Vector Fields, Inc.

Currents

Opera Opera 2d Tosca Electra

Y

lLu,,

Figure 16.3.1: Preprocessors

608


FS19

Coupling

to

Electrical

Circuit

Models: This is as

described in chapters 3 and 5.

16.3: When a differential or integral method is used for the analysis of a device, the preprocessor is used to define the geometry of the device and provide the material property for further processing by the solver, In a preprocessor a computer model is created and the data is stored in computer files to be used by the solver or to remodel and reuse in future, This process is known as geometric modeling. The features of interest for the preprocessor are listed below and labeled as features PR1 through PR5 - - see Fig, I6.3.1, P R I Solid Modeling: Solid modeling has its advanced features in Company

Software PO! Maxwell 2D Y Maxwell 3D Y Spice Link ...... Y Opus Y Magus Y Magnets ~ - 2 D Y Magnets TB-3D Y

PO2 Y Y Y Y Y Y Y

PO3 Y Y Y

PO4 Y Y Y

Y Y Y

Y Y

PO5 Y Y Y Y Y Y Y

Y

Y

Y

Y

Y Y Y Y

Y Y Y Y

Y Y Y

Y Y Y Y

Y Y Y Y

CosmosM/Estar

Y

Y

Y

Y

Y

Swanson Analysis and Systems, Inc.


Y Y Y

Y Y Y

Y Y Y

Y Y Y

Y Y Y

Vector Fields, Inc.

PC - Opera Opera 2d Toga Electra

Y Y Y Y

Y Y Y Y

Ansoft Corp,

Ferrari Associates Infolytica C o rp.

MaeNea! Schwendler Corp . . M a gs oft Corp.


.

MSC / EMAS . . . . . Microflux Flux2D Flux3D Phi3D

Y ,

.....

Corp.

..................

Figure 16.4.1: Postprocessors

,,,,,

Y Y Y Y

-

,,,,JIL

.

Y Y Y Y

A. A. Arkadan

609

general applicability with less user interaction. P R 2 Automatic Meshing: First the grid points are placed throughout the region to be meshed and then the Delaunay tesselation algorithm or an octree mesh generator is used for meshing the region. PR3

Adaptive Meshing: The adaptive mesh generator reduces the solution e~or by reducing the size and increasing the number of elements in the areas of the model where the e~or is high.

.......C o m p a n y .... Ansoft I Corp. t

Ferrari t Associates Infolytiea Corp,

•

MacNeal Schwendler

Software Maxwell 2D Maxwell 3D Spice Link Opus Magus Magnets ~ - 2 D Magnets TB-3D

PO6

MSC / EMAS

Y

Y Y

PO7 Y Y ,, ...... Y ,, ,,,j

,,,,,, , , ,

PO8

,,

;

. . . . . . . . . .

~

PO9 [ Y* Y* Y* Y Y Y Y ,

•

PO10 Y* Y* Y* Y Y Y Y . . . . . . . . . . ,,,,,,

Y Y

Y Y

Y

Y

Y

Y

Y Y Y Y

Y Y

y+ y+ y+

y+ y+ y+ y+

Corp. Microflux Flux2D F!ux3D Phi3D

Magsoft Corp.


Y

CosmosM/Estar

Y

Y

Y

Y


Y Y Y

Y Y Y

Y Y Y

y** y** Y**

Corp.

Swanson Analysis and Systems, Inc. , .......................................................

Vector Fields, Inc.

~

........................

f,J,J.

:

PC - Opera O ~ m 2d Tosca Electra

* Local Jacobian + Virtual work *+ Maxwell Stress Tensor or Virtual work

=

,,,,,

,. . . . .

J

Y

.... ,IT++

Y

y++ y++

Y

",tr++

g*+

y*+ y,+ (3D) y++ y++ y++ y++

** Maxwell Stress Tensor ++ Maxwell Stress, Lorentz Force


610


P R 4 Interactive User Interface PR5

Parametric Modeling: This is a feature-based design method in which parts can be described by parameters (Brauer 1993).

16.4: Postprocessor The objective of postprocessing is using the computer to obtain some meaningful quantity, such as energy, flux density, inductance, capacitance or force and display the numerical information in the form of plots, graphs or tables (Hoole 1989; Sabonnadiere and Coulomb 1987). The features of interest for the postprocessor are listed below and labeled as features P O I through PO14, see Figs. 16.4.1 through 16.4.3. PO1

Color Contour Plots

PO2

Color Element Fill Plots

PO3

Color Arrow Plots

PO4

Numerical B and E Values Written on Elements

PO5

Graphing Output Versus Spatial Position

PO6

Graphing Output Ver:sus Time

PO7

Loss

PO8

Animated Color Plots

PO9

Torque Calculation (Method)

Calculations

P O 1 0 Force Calculation on any part (Method) POll

Linkage of any Coil

POI2

Flux

P O 1 3 Energy/Coenergy

Calculations

P O 1 4 Vector Line Integral of H.dl

16.4.1: Plots, Flux and Line Integral The plots of equipotentials, field intensity or flux density provide a quick means of getting an idea of the solution of the electromagnetic fields. An experienced

A. A. Arkadan

611

engineer can judge higher field regions from the plots as one expects that the equipotential lines should be crowded there (Hoole 1989) Color contour plots directly tell us the higher field regions using color and the color element fill plots give us some idea about the finite elements which have higher fields. In the case of arrow plots the size of the arrow shows the higher fields. But the identification of the small variations with the human eye is a very difficult task in analyzing complicated devices. Color arrow plots solve this for a user. In some designs, for example the breakdown of dielectrics or nonlinear magnetics, the analysis is totally based on the magnitude of the B or E field and Company

Software POll PO12 PO13 PO14 Maxwell 2D Y Y Y Y Corp. Maxwell 3D Y Y Y Y .................................... Sp!ceL!nk ....................................... Y ........................Y ....................Y .................y ........... Ferrari Opus Y Y Y Y Associates Magus Y Y Y Y lnfolytica Magnets TB-2D Y Y Y Y .....C o r p . Magnets TB-3D Y Y Y Y MacNeal Schwendler MSC / EMAS Y Y Y Y ............C o r p . Microflux Y Magsoft Flux2D Y Corp. FIux3D Y Y Phi3D

Ansoft


CosmosM/Estar

Y

C o r p , ..........

Swanson Analysis l Ansys and Ansys-Pc / Magnetic Systems, Inc. Vector Fields, Inc.

*

)" (MoV)Hdv V

Ansys / ED

Y Y Y

Y Y Y

y+ y+ y+

Y Y Y

PC - Opera Ot~m 2d Tosca Electra

Y* Y* Y* Y*

Y Y Y Y

Y Y Y Y

Y Y Y Y

+ Line~ized energy only


612

Chapter t 6: On the Features of Electromagnetic FEA Software Packages

the area of further modeling is chosen using these numerical values. In these devices one needs the numerical values of the fields within each finite element and a package with this capability is more useful. One may need information on how the electromagnetic field varies in a particular direction. In that case graphing output versus spatial position provides the necessary information to the user. In transient analysis, the variation of the field quantity with time is studied and one may need the feature of graphing output against time. Ampere's law predicts the current as a vector line integral of H.dl. Thus to calculate the cu~ent as well as to veriPy results, this vector line integral is used.

16.4.2: Calculation of Loss, Torque, Force, Energy Loss, Force, Energy and Coenergy are frequently used in the analysis of electromagnetic devices and frequently the wediction of the efficiency of the device is based on these quantities. Torque and coenergy are well known terms for engineers who works on electric machines.

VECTOR IDENTITIES

The purpose of this appendix is to give useful vector relationships that are repeatedly used in the book, For a proof of the relationships, the more interested reader is referred m Ferraro (1970) and PanoI~ky and Phillips (1969) or any book on vector calculus. O 3 V= Ux~+Uy ~ +Uz~ (A1) A'B = AxBx + AyBy + AzBz AxB = ux(AyBz-AzBy) + uy(AzBx-AxBz) + uz(AxBy-AyBx) VxV,= 0 V,Vx = 0 g20 = g.g0 V2A = uxV2Ax + u V2Ay + uzV2Az Vxgx A = VV.A- Y2A V(0tl,') = 0Vgt + ,4tV, V-(,A) = ¢V.A + A.V,

(12) (13) (A4) (A5) (16) (A7) (AS) (19) (AlO)

I~ I A . d S - J ~ " !f J('(V.A)dl~. Gauss'sTheorem

(112)

Appendix B "

II

. . . . . . . . .

S. Ratnajeevan H. Hoole I]]]q II!11!11

II

]]]

U]

II

IIIII

I

I]

THE UNIQUENESS OF THE SCALAR POTENTIAL

In this appendix, we establish the conditions for the uniqueness of the solution of Poisson's equation - e V2¢ = - e lax2 + 0y 2 j = P

~1)

As will be shown, the conditions are that ¢ should be specified at least at one point on the boundau of the solution region and, at all other points, either ¢ or its derivative in the normal direction should be specified. We shall prove this by deriving the conditions under which any two almrnative solutions to eqn. (B1), ¢I and ¢2, differ by 0; that is, we d e t e ~ i n e the conditions for ~4t = ¢1 - ¢2 = 0 ~2) By putting ¢1 and ¢2 into eqm (B1) which they both, being alternative solutions, satisfy, and subtracting one from the other we obtain: ~V2~," = 0 Now, consider the vector identity (A10) with ~=¢ and V~=A: V. 04,V~,) = [g'q].[V~4t ] + -vjg2~ ~4) The last term is zero in view of Eqn. (B3). Inmgrating the rest of eqn. (B4) over the region of solution R, using Gauss' divergence theorem (A12) on the left hand side t e ~ s of Eqm (B4), we obtain:

where n is the direction of the normal to the boundary S. Since the integrand on the left, being a square, is nonnegative, we may conclude that V ~ is zero everywhere in R, provided that either ~" or 0~4t/0n is zero. The Iauer conditions apply, whenever ¢ or its normal derivative on the boundary of solution is specified; for then ¢I and ¢2 will both have either the same value or the same

]

S. R. H. Hoole

615

normal derivative and therefore ~, their difference, will necessarily be either zero or have zero normal derivative. The specification of ~ along R is known as a Dirichlet boundary condition and that of its normal derivative, as a Neumann boundary condition. What we have shown is that if a Neumann or Dirichtet condition applies on every part of the boundary, then V0 is unique. However, that does not mean that 0 itself is unique; but we may assert that 01 = 0 2 + c ~6) since when taking the gradients on both sides, we would have V~I = V02 only ifc is a constant; for any inconstant function c(x,y) would have a gradient and invalidate the uniqueness of the gradient. Therefore, with Neumann and Dirichlet conditions everywhere on R, 0 is unique within a constant c. But so long as the Dirichlet condition applies at least at one point on the boundary, then ~1 = 02 at that point, making c zero, so that ~ is unique everywhere in R. In conclusion then, Poisson's equation (B 1) has a unique solution so long as the Dirichlet condition applies at least at one point and the Neumann condition applies wherever the Dirichlet condition is not at play.

Appendix C .................

S. RatnaJeevan H. Hoole ............

!lJl!J

IIJ

II!11111

!11!

I

I

[11111111 !11111

!11

THE UNIQUENESS OF THE VECTOR FIELD

When the curl and divergence of a vector field ~ are defined e y e . w h e r e in a region R bounded by a surface S, we shall here determine sufficient boundary conditions to make that vector uniquely determinable. We prove in this section that it is sufficient for the normal or tangential component of that vector to be specified along the boundary to make the vector unique everywhere in the solution region. Consider two alternative solutions V I and V 2 to the pair of vector equations VxV =~ (C1) V. V = s (C2) Proc~:ding as in the previous section, since both the alternative solutions satisfy eqns. (C l) and (C2), their difference U = V I - V2 (C3) satisfies Vx U = 0 (C4) and V. U = 0 (C5) Now, comparing eqn. (C4) with the vector identity (A4) we may say that U = V0 (C6) which when put into eqn. (C5) yields V20 = 0 (C7) We know from Appendix B that this has a unique V~ (and therefore unique U from eqn. (C6) if ¢~or its n o d a l gradient is specified on the boundary S. Clearly, from eqn. (C6) the specification of ~ along S implies that 0~/Ot - which is the tangential component of the vector - field is specified on the boundary, Similarly, the Neumann condition hnpties that the normal component of the vector field is specified. Of course, when the vector field is defined by a vector potential (or stream function), then the specification of ~ is the specification of the normal component of the vector field and the imposition of a Neumann condition is the same as specifying the tangential c o m ~ n e n t of the vector field.

Appendix D u,lu!uu,iHil,!

i!

iii

i!111

i i 1!1!1111...........

I]

IHII .........

!

III

S. RatnaJeevan H. Hoole I!11!

I!!11]!11!

.........

INTEGRATION OF TRIANGULAR COORDINATES

The integration of triangular coordinates repeatedly occurs in finite element analysis, while forming the element matrices as well as during the integration of our approximation f;anctions such as when we need to compute by numerical integration, the stored energy of the system. In this section we give a formal proof of the often required formula used in such integrations:

f f l~i~ ijk3

dxdy = ~i~!k!2IA +k+2)!

(D1)

A where A is the area of the triangle A. To prove this we first require a second Ibrmula for translating from the integrations with respect to x and y to any two of the 4's: dxdy = 2Ad~ld~2 - 2Ad~2d~3 = 2Ad~3d~ 1 (D2) To prove eqn. (D2) consider the triangle ABC, with node numbering in the same order, shown in Ng. D1, where for convenience we have assumed that the side BC is along the x axis. As a result y corresponds to the altitude hl of a point within the triangle, with respect to BC which is our side 1. Also it is seen from Fig. D1 that dh3Sin~ = dx (D3) From the definition of the the triangular coordinates we have therefore Hld~l = d h l = d y (D4) dx H3d~3 = dh3 - Sin~ (D5) From eqns. (D3) and (D4) we theref\gre have dxdy = H3Sin ~ d~3. H 1d~ 1 = B3HId~Id~3 = 2Ad~ 1d~3

(D6)

618

Appendix D: Integration of Triangular Coordinates

B Figure D.I: Elemental Lengths and Triangular Coordinates where B3, the base or length of side 3, has been substituted for H3Sin~ and the geometric definition of the area of the triangle as a half of the base times the height has been used. By symmetry about the coordinates, eqn. (D2) follows. Using eqn. (D2) to evaluate the integral of interest to us we may write r'J r k dxdy = 2A ~ it,~2,.a3

I=

Ij"

l l ~ ~ d~2d~3

(D7)

"

A A We have already seen that only two of the three homogeneous coordinates ~ are independent, with the third fixed by:

Y A=I

B- 2

;~0

C-3

Figure D.2: Limits Integrating over a Triangle

x

S. R. H. Hoole

619 ....

,,

,

,u,,

:

(1.5.37)

41 + 4 2 + 4 3 = 1 Using this we have

1 I=

1-43 (1-42-43) i 4

4 1"~2~3dxdy = 2A

d42d43

~8)

A 434 42=0 where the limits of integration have been obtained by considerations from Fig. 2. We shall attempt to evaluate this difficult integral in a piece-meal fashion. For p = !-43, let us tackle the inner integral by parts: P P [rj+l'] Ii,j=

(p-42)i4 J d42 = 0

(p-42) i d l ~ j 0

{(p.42) i [4J~l]" t

L~JJ

-

"

0

Lj+I A d

0

~. )i-l~j+l - j+i 1 f ( P%2 g 2 d42 = ~ i Ii-1,j+ t

(D8)

0 Repeatedly employing this mcursive relationship, i i i-! i i-1 i-2 Ii,j - j+l Ii-l,j+l = j+l j+2 Ii-2,j+2 - j + l j+2j+3 Ii-3,j+3 = ... - (i+j)! I0,i÷J

(D9)

Evaluating I0,i+j from the definition of eqn. (D8) and substituting 1-43 for p,

1-43 jI4~+j ~(i+j)!

Ii,j--

i,j~

d42 = (i÷j+i)i (1"43)i+j+1

Olo)

0 Before trying to evaluate I, we shall get an addition relationship from eqns. (D8) and (D9) by substituting 43 for 42 and 1 for p, which we shall need recourse to in evaluating I: 1

Ji,j = f 0

. (1-43)i4 Jd43 = (.,+j+~): : it jr: ~;J0,i+j

Dll)

620

Appendix D: Integration of Triangular Coordinates

Now putting eqn. (D10) into the double integral for I, in eqn. (D7), we have 1

i! j!~2A j/~ (1-~3) i~+l ~~A I _- (i+j+l)! ;~ d~3 -(i+j+l)! Ji+j+I,k = .. 0 i! j ! 2& 0 ± j ± ~ \ k !

= (i+j+l)! (i+j+l+k)! Ji+j+i,k J0,i~+k+l from eqn. DI 1

~i÷j+k+2

= 21 (i~+k+l)!i" j[ .......... f ~i+j+k+ Ida3 = 2A (~+j+k+i)ii' i! i+j+k+2 0 = J!_j! 2A (i+j+k+2)! which is the result that we seek.

(DI2)

COLLECTED BIBLIOGRAPHY

Adams, R. A. M., 1975, Sobolev Spaces, New York: Academic Press. Ahn, Chang-Hoi, Lee, Sang-Soo, l~e, Hyuek-Jae, and k,~, Soo-Young, 1991, "A self-organizing neural network approach for automatic mesh generation", IEEE Trans. Magn., Vol. 27, No. 5, 4201-4204. Akin, J. E., 1980, "Application and Implementation of Finite Element Methods," New York: Academic Press. AIbanese, R. and Rubinacci, G., 1988, "Solution of Three Dimensional Eddy Current Problems by Integral and Differential Methods," IEEE Trans. Magn., Vol. MAG - 24, No. 1, pp. 98-I01. Alger, P. L., 1954, "The Magnetic Noise of Polyphase Induction Motors," AIEE Trans. on Power Apparatus and Systems, VoI. PAS-73, Part IIIB, Apr. pp. 118I24. Alger, P. L., 1970, Induction Machines-Their Behavior and ~ e s , Gordon and Breach Science Publishers. Anderson, D., Kuhn, F. P. and G!eason, D., 1986, "A New Nonlinear Tracker Using Attitude Measurements," g E E Trans. on AES, AES-22, Vol. 5, Sept. Anderson, D., Kim, E. T., Schie~an, J. and Kuhn, F. P., 1991, "A Nonlinear Helicopter Tracker Using Attitude Measurements," IEEE Trans. on AES, AES-27, Vol. 1,/an.

622

Collected Bibliography

Argyris, J+ H+, 1954, "Energy Theorems and Structural Analysis," Aircraft Eng., Vol. 26. Argyris, J. H., 1955, "Energy Theorems and Structural Analysis," Aircraft Eng., Vol. 27. Arkadan, Abd+ A., (Ed.), "Finite Elements for Electromagnetic Applications," Sho~-Course Notes, May 1993, Marquette University, Milwaukee, WI., U.S.A. Arnoldussen, T. C,, 1986, "Thin Film Recording Media," Proc.IEEE, Vol.74, No. 1 I, Nov., pp+1526-1539. Athermn, D. L., Szpunar, B+ and Szpunar, J. A., I987, "A New Approach to Preisach diagrams," IEEE.Trans.on Magn., Vol. MAG-23, No. 3, May, pp. 18561865. Babusvka, I., I989, "Trends in finite elements," IEEE Trans. Magn., Vol. 25, No. 4, pp.2799-2803+ Babusvka, I+ and Aziz, A. K., I972, "Survey lectures on the mathematical goundations of the finite element method," in: The Mathematical Foundations of the finite element method with applications to partial differential equations, ed. A.K+ Aziz, Academic Press, New York, 5-359. Babuskva, I., and Rheinboldt, W. C., 1978, "A Posteriori Error Estimates for the Finite Element Method," int. J. Num. Meth. Eng., Vol. 12, pp+ 157%I615. Baker, 1977,The Numerical Treatment of Integral Equations, Oxford University Press Balakrishnan, J., 1986, A. CLosses in Conductor Arrays, M.A.Sc+ Diss., University of Toronto+ Balchin, M+ J+ and Davidson, J+ A+ M+, 1980, "Numerical method for calculating magnetic flux and eddy current distributions in three dimensions," IEEE Proc. Vol. 127, Pt.A, No+I, pp. 46-53. Baldomir, 1986, "Differential forms and electromagnetism in 3-dimensional Euclidean space R3, '' IEE Proc+, 133A(3), 139-143+ Barlow, H.M+, 197I, The Seventh Clerk Maxwell Memorial Lecture: Guided Electromagnetic Waves, Radio and Electronic Eng., Vol+ 4I, pp. 14% 152. Ba~on, M. L., and Cendes, Z. J., 1987, "New Vector Finite Elements for Three Dimensional Magnetic Field Computation," J. App. Phys., Vol+ 61, No+ 8, pp. 3919-.3921, April.

S. R. H. Hoole (Editor)

623

Bastos, J. P . , 1985, "3D Modelling of a Nonlinear Anisotropic Lamination," Trans. Magn, Vol. 21 pp 2366-2369. Bate, G , 1986, " Partlculat " e recording media" G.Bate, Proc. Nov., pp.1513-1525.

, Vol. 74, No. 1 I,

Bathe, K.J., 1982, Finite Element Procedures in Engineering Analysis, PrenticeHall. Bathe, KJ. and Wilson, E.L, 1973, "Stability and Accuracy Analysis of Direct Integration Methods," Int. a: Earthquake Eng. Saruct.Dynam., Vol. I, pp 283-291. Bayliss, and E. Tukel, I980, "Radiation boundary conditions for wave-like equations," Comm. Pure Appl. Math., 33, pp. 707-725. Bedrosian, G., D'Angelo, J., and de Blois, A., 1989, "Parallel Computing for RF Scattering Calculations," IEEE Trans. Magn. Vol. 25, pp. 2884-2889. Bertram, H. N., 1986, "Fundamentals of the Magnetic Recording Process" P r o c u r E , Vol. 74, No. 11, Nov., pp. 1494-1512. Binns, K. J., 1960, "The Magnetic Field and Centering Force of Displaced Ventilating Ducts in Machine Cores," Proc. lEE, Vol. 108, Monograph No. 394, Part C, Aug. pp. 64-70. Binns, K. J., 1964, "Calculations of Some Basic Flux Quantities in Inductions and Other Doubly-Slotted Electric Machines," Proc. I~E, Vol. 111, No. 1 I, pp. 1847I858. Binns, K. J. and Schmid, E., 1975, "Some Concepts Involved in the Analysis of the Magnetic Field in Cage Induction Machines," Proc. lEE, Vol. 122, No. 2, pp. 169-175. Binns, K, L, and Rowlands Rees, G., 1978, "Radial Tooth-Ripple Forces in Induction Motors Due to the Main Flux," Proc. lEE, Vol. 125, No. 1 I, pp. 12271231. Binns, K. J. and Kahan, P. A., I981, "Effect of a Load on the Tangential Force Pulsations and Harmonic ToNues of Squirrel-Cage Induction Motors," Proc. lEE, Vol. 128, No. 4, pp. 207-211. Bossavit, A., 198I, "A Variational Formulation for the Eddy-Currents Problem," Comp. Meth. Mech. & Engng.,

624

.............


Bossavit, A., 1982, "On Finite Elements for the Electricity Equation," in Whiteman, J. R., ed., The Mathematics of Finite Elements and Applications IV, Acadamic Press, pp. 85-92. Bossavit, A., I988a, "Whitney forms: a class of finite elements for threedimensional computations in electromagnetism," lEE Proc., Vol. 135A, No. 8, pp. 493-5~. Bossavit, A., !988b, "A rationale for 'edge elements' in 3-D field computations," IEEE Trans. Magn., 24, No. I, pp. 74-79. Bossavit, A., 1990, "Solving Maxwell equations in a closed cavity and the question of spurious modes," IEEE Trans. Magn., Vol. MAG-26, No. 2, pp. 702705 Bossavit, A., 1992, " Edge-e 1 ement computation of the force field in deformable bodies," Trans. Magnetics, Vol. MAG-28, No. 2, pp. 1263- ! 266. Bossavit, A. and Mayergoyz, I., 1989, "Edge-Elements for Scattering P r o blems, " IEEE Trans. Magn., VoI. 25, No. 4, pp. 2816-2820. Bossavit, A. and Verite J. C., 1982, "A Mixed FEM-BIEM Method to Solve 3-D ~dy-Current Problems," 1EEE Trans. Magn. Vol. 18, No. 2, pp. 431-435. Bossavit, A. and Verite, J. C., 1983, "The Trifou Code " Solving the 3-D Eddy Currents Problem by Using H as State Variable," 1EEE Trans. Magn., Vol. 19, No. 6, pp. 2465-2470. Bramble, J. H., Pasciak, J. E., and Xu, J., 1990, "Parallel Multilevel Preconditioners," Math. Comp., Vol. 55, No. 191, pp. 1-22. Brandl, P., Reichert, K., and Vogt, W., 1975, "Simulation of Turbogenerators on Swady State ~ , " Brown Boveri Review 9. Brauer, John R., 1993, What eve N engineer should know about finite element analysis, Marcel Dekker, Inc. • New York. Brauer, J. R., MacNeal, B. E., Larkin, L. A, and Overbye, V. D, 1991, "New Method of Modeling Electronic Circuits Coupled with 3D E!ectromagneticFinite Element M~els," IEEE Trans. Magn., Vol. 27, Sep., pp. 4085-4088. Brezzi, F., Douglas, J., Marini, L. D., 1985, "Two Families of Mixed Finite Elements for Second Order Elliptic Problems," Numer. Math., Vol. 47, pp. 217235.

S. R. H. Hoole (~itoO

625

Brokate, M., 1989, "Some mathematical properties of the Preisach model for hysteresis," IEEE.Trans. Magn., Vol. 25, No. 4, Jul., pp.2922-2924. Brunotte, X., Meunier, G., and Imhoff, J.F., 1992, "Finite Element Modeling of Unbounded Problems Using Transformations - - A Rigorous, Powerful and Easy Solution," IEEE Trans. Magn., Vol. 28, No. 2, pp. 1663-I666. Bullard, S., and Dowdy, P. C,, 1991, "Pulse Doppler Signature of a Rotary-Wing Aircraft," Vol. 6, No. 5, May. Cangellaris, A. C., Prince, J. L. and Vankanas, L. P., 1990, "Frequency-De~ndant Inductance and Resistance Calculation for Three-Diemensional Structures in High-Speed Interconnect Systems," IEEE Trans. CHMT, Vol. 13, No. 1, pp 154t59. Carey, G. F. and Jiang, B. N., 1986, "Element-by-Element Linear and Nonline~ Solution Schemes", Comm. App. Num. Meth., Vol. 2, pp. I45-I53. Carey, G. F., Barragay, E., Maclay, R., and Sharma, M., 1988, "Element-byElement Vector and Parallel Computations", Comm. App. Num. Meth., Vol. 4, pp. 299-307. Carpenter, C. J., 1960, "Surface Integral Methods of Calculating Forces on Magnetised Iron Parts," Proc. lEE, Vol. 107, Monograph No. 342, Part C, Aug. pp. 19-28. Carpenter, C. J., 1977, "Comp~ison of Alternative Formulations of 3-Dimensional Magnetic Field and Eddy Current Problems at Power Frequencies," lEE Proc., Vol. 124, No. 11, pp. I026-34, November. Cendes, Zoltan J., 1989, "Unlocking the magic of MaxwelI's Equations," IEEE Spectrum, Apr., pp. 29 - 33. Cendes, Z. J., I991, "Vector Finite Elements for Electromagnetic Field Computation," IEEE Trans. Magn., VoI. 27, No. 5, Sept., pp. 3958-3966. Chari, M. V. K., Silvester, P. (eds.), 1980, Finite Elements in Electric and Magnetic FieM Problems, John Wiley, Chichester. Chari, M. V. K., Bedrosian, G., D'Angelo, J., Konrad, A., Cotzas, G. M., and Shah, M. R., I991, "El~tromagnetic Field Analysis for Electrical Machine Design," PIER 4 Progress in Electromagnetics Research, J.A. Kong (ed.), New York: Elsevier, pp. 159-21 I. (In particular, see pp. 189-191.)

626


Chaff, M. V. K. and Csendes, Z. J., 1977, "Finite Element Analysis of the Skin Effect in Current Carrying Conductors," IEEE Trans. ,Magnetics, Vol. 13, No. 5, pp. 1125- I 127, September. Chari, M. V. K., Konrad, A., Palmo, M. A. and D'Angelo, J., 1982, "ThreeDimensional Vector Potential Analysis for Machine Field Problems,", IEEE Trans. Magn. Vol. 18, No. 2, pp. 436-446. Chavanne, J., 1988, "Contribution h la Moddlisation des Systbmes Statiques Aimants Pe~anents," Thfse de Doctorat INPG.

Chavanne, J., and Meunier, G., 1989, "A New Model for Nonlinear Anisotropic Hard Magnetic Material," IEE-E Trans. Magn., Vol. 25, No. 4, pp. 3083-3085, July. Ciarlet, P. G., 1978, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam. Clough, R. W., 1960, "The Finite Element Method in Plane Stress Analysis," Proc. 2d ASCE Conf. Electronic Computation, Pittsburgh, PA., pp. 345-378. Coleman, B. D. and Hodgdon, M. L., 1986, "A constitutive relation for rateindependant hysteresis in ferromagnetically soft materials", lnt.ZEng..Sci., Vol. 24, No. 6, pp.897-919. Coleman, B. D. and Hodgdon, M. L., 1987, "On a class of constitutive relations for ferromagnetic hysteresis", Arch.Rational Mech.AnaL, 99, No. 4, pp.375-396. Collins, R. E., 1968, Mathematical Methods for Physicists and Engineers, Reinhold Book Corporation. Cook, David M., 1975, Ztte Theory of Electromagnetic Fields, Prentice-Hall, Inc., Englewood, Cliffy, New Jersey. Cook, R. D., 1981, Concepts and Applications of Finite Element Analysis, New York: Johns Wiley. Costache, G.I., 1988, "Finite Element Method Applied to Skin-Effect Problem in Strip Transmission Lines," IEEE Trans. Microwave Th. and Tech., Vol. 35, No. 11. Coulomb, J.-L., I98I, Analyse Tridimensionnelle des Champs Electrique et Magndtique par la M~thode des Eldments Finis, Th~se de Doctorat d'Etat USMG INPG, Grenoble.

S~R. H. Hoole (Editor)

627

Coulomb, J. L., 1983, "Methodology for the Determination of Global Electromechanical Quantities from a Finite Element Analysis and its Application to the Evaluation of Magnetic Forces, Torques and Stiffness," IEEE Trans. Magn~, Vol. 19, No. 6, Nov. pp. 2514-2519. Coulomb, L L. and Meunier, G, 1984, "Finite Element Implementation of Virtual Work Principle for Magnetic or Electric Force and Torque Computation," IEEE Trans. Magn., VoL 20, No. 5, pp. t 894-I896, September. Chen, G. L. 1986, "New Longitudinal Recording Media CoxNiyCrz from High Rate Static Magnetron Sputtering System," IEEE Trans. Magn., Vol. 22, No. 5, Sept, pp~334-336~ Crowley, C. W~, Silvester, P. P. and Hurwitz Jr, H., 1988, "Covariant Projection Elemetns for 3D Vector Field Problems," IEEE Trans. Magn., Vol. 24, No. 1, pp~ 397-400. Csendes, Z. L and Silvester, P., 1970, '*Numerical solution of dielectric loaded waveguides: I -Finite-element Analysis," IEEE Trans. Microwave Th. and Teeh., Vol. 18, No. I2. CulIen, A. L~, 1986, "The Eleventh Clerk Maxwell memorial lecture: Electromagnetism: Obscurities, curiosities, and practicalities," Radio and Electronic Eng., Vo!. 56, pp. 281-291. D'Angelo, J. and Mayergoyz, I. D., 1991,"Three dimensional RF scattering by the finite element method", 1EEE Zrans. Magn., Vol. 27~ No~ 5, 3827-3832. de Veubeke, Fraeijs B. X., 1965, "Displacement and Equilibrium Models in the Finite Element Method," in Zienkiewicz, O. C. and Holister, G., eds., Stress Analysis, New York: Wiley. de Veubeke, Fraeijs B~ X., 1975, "Stress Function Approach," World Congress on the Finite Element Method in Structural Mechanics, Bournemouth. Defouny, M~ M. and Scarpa, P. G. 1989, "Electrical and Electromagnetic Applications," Chapter 1 in Topics in Boundary Element Research, edited by C A Brebbia, Springer~Verlag. Del Vecchio, R. M., 1980, "An efficient procedure for modeling complex hysteresis processes in ferromagnetic materials", IEEE. Trans. Magn~, Vol. 16, No~ 5, Sept., pp.809-81 I. Delves, L. M. and Mohamed, J. L., 1985, Computational Methods for LntegraI Equations, Cambridge University Press, Cambridge.

628 -

.

.

.

.

.

.

.

.

.

,,,,

,,

,,

,,,

,.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


Demerdash,, N. A. and Nehl, T. W., 1979, "An evaluation of the methods of finite elements and finite difference in the solution of non-linear electromagntic fields in electric machines," IEEE Trans. Power App. Syst., Vol. 98, No. I, pp. 74-87. De Silva, C. W., 1989, Control Sensors and Actuators, Englewood Cliffs, NJ, Prentice-Hall. di Napoli, A., and Paggi, R., I983, "A Model of Anisotropic Grain Oriented Steel," Trans. Magn., Vol. 19, pp 1557-1561. Djorjevic, A. R., Sarkar, T., K. and Rat, S. M., 1985, "Analaysis of Finite Conductivity Cylinderical Conductor Excited by Axially-Independant TM Electromagnetic Field," IEEE Trans~ Microwave Th. and Tech~, Vol. 33. Doong, T. and Mayergoyz, I. D., 1985, "On Numerical Implementation of Hysteresis Models," IEEE. Trans. Magn., Vol. 2 I, No. 5, Sept., pp. 1853-1855. Duff, I. S.(edited), I981,Sparse Matrices and Their Uses, London: Academic Press. Dwayer, D. L., Hussami, M. Y., Voigt, R. G.(eds.), 1988, Finite Eleraents: Theory a ~ Applicathgns, ed. New York: Springer-Verlag. Fain, Gerald, 1990, Curves and Surfaces Jbr Computer Aided Geometric Design, Second ~ition, Academic Pros, New York. Ellison, A. J. and Yang, S. J., 1971, "Effects of Rotor Eccen~icity on Acoustic Noise from Induction Machines," Proc. lEE, Vol. 118, No.l, pp. 174-184. Emson, C. R. I. and Simkin, J., 1983, "An Optimal Method for 3 - D Eddy Currents ", IEEE Trans. Magn., Vol. 19, No. 6, pp. 2450-2452. Engq~s~,,B.;and Majda, A., 1977, "Absorbing boundary conditions for the hum simulation of waves," Math. Comp., 31 (1~3~),629-651. Eric, Da.valo~and Naim, Patrick, 1990, Neural Networks, Macmillan series. Faddeev, D. K., Faddeeva, V. N., 1963, Computational Methods of Linear Algebra, San Francisco: W.H. Freeman. Fawzi, T. H., Ali, K. F. and Burke, P. E., 1983, "Boundary" integral equation analysis of induction devices with rotational symmetry," IEEE Trans. Magn., Vol. 19, Jan., pp. 36-44.


629

Fawzi, T. H., AIi, K. F. and Burke, P. E., 1984, "Boundary element analysis of multilayer induction problems with rotational symmetry," IEEE Trans. Magn., Sept., pp. 2010-2012. Feller, A., Kaupp, H. R. and Digia-coma, J. J., 1965, "Crosstalk and Reflections in High-Speed Digital Systems," Proc. of Fall Joint Computer Conference, pp 5I 1525. Felsen ,L.B., 1976, (Editor), Transient Electromagnetic Fields, Springer-Verlag. Ferrafi, R. L., 1985, "Complementary Variational Formulation for Eddy-Current Problems Using the Field Variables E and H Directly," Proc. IEE, Vol. 132, Pt. A, No. 4, July, pp. 157-I64. Ferrari, R. L. and Pinchuk, A. R., 1985, "Complementary Variational FiniteElement Solution of Eddy Current Problems Using Field Variables," I~EE Trans. Magn., VoI. 2 I, No.6, Nov., pp. 2242-2245. Forsythe, G. E., 1967, Computer Solution of Linear Algebraic Systems, Englewood Cliffs, N.J.: Prenctice-Hall. Gallagher, Richard H., 1975, Finite Element Analysis Fundamentals, PrenticeHall, Inc., Englew~d Cliffs, New Jersey. Gantmaher, F. R., 1960, The Theory of Matrices, New York: Chelsea Publishing Co. Gardiol, F. E., I987, I~ssy Transmission Lines, ~Lrtech House, Inc., MA. George, A. and Liu, G., 1981, Computer Solution of ~ r g e Sparse Positive Definite Systems, New Jersey: Prentice Hall. George, A. and Liu, J. W. H., 1981, Computer Solution qfL~rge Positive Definite Systems, Englewood Cliffs, New Jersey: Prentice-Hall Series in Computational Mathematics. Girgis, R. S. and Verma, S. P., t979, "Resonant Frequencies and Vibration Behavior of Stators of Electrical Machines as Affected by Teeth, Windings, Frame and Laminations," Trans. PowerApp. Sys., Vol. 95, No. 10, pp. 1446-1455. Girgis, R. S. and Venna, S. P., 1981, "Method for Accurate Determination of Resonant Frequencies and Vibration Behavior of Stators of Electrical Machines," Proc, lEE, Vol. I28, No.l, P ~ B, Jan.., pp. 1-11.

630


Gitosusastro, S., Coulomb, J.-L., and Sabonnadiere, J. C., 1989, "Pertormance Derivative Calculations and Optimization Process," IEEE Trans. Mag., Vol. 25, 2834-39. Givors, D., Lienard, A., de Ia Bathie, R. Perrier, Tenaud, P., and Viadieu, T., 1985, J. de Phys., 46-C6, pp. 313. Golias, N. A. and Tsiboukis, T. D., 1992, "Three-dimensional" " automatic adaptive mesh generation", IEEE Trans. Magn., VoI. 28, No. 2, 1700-1703. Goodwin, G. C., and Sin, K. S., 1984, Adaptive Filtering Prediction and Control, PrenticeHaI1, Inc., USA. Greason, W. D., 1987, "Review of the Effect of Electrostatic Discharge and Protection Techniques For Electrostatic Systems," IEEE Trans. Ind. AppL, Vol. 23, No 2, Mar./Apr., pp 205-216. Gross, D. A., 1988, "Quench Propagation Analysis in Large Solenoidal Magnets," Trans.Magnetics, Vol. 24, No. 2, pp. 1190-! I93. Gurol, H., Hardy, G. E., Peck, S. D., and Leung, E., 1989, "Transient Magnetic Field Modeling in Large Magnet Applications," 1EEE Trans.. Magn., Vol. 25, pp. 2852-2854. Hackbusch, Wotfgang, 1980, "Multigrid Methods and Applications," Springer Verlag, Berlin. Hano, Mitsuo, 1984, "Finite-Element Analysis of Dielectri-Loaded Waveguides," IEEE Trans. Microwave Th. Tech., Vol. 32, pp. 1275-1279. Harrington, R. F., 1967, "Matrix methods for field problems," Proc. lEE, 55(2), 136-149. Harrington, R. F., 1982, Field Computation by Moment Methods, Krieger Publishing Co., Malabar, Norida. Harrington, R. F. and Wei, C., 1984, "Losses on Multiconductor Transmission Lines in Multilayered Dielectric Media," 1EEE Trans. Microwave Th. and Tech., Vol. 32, July, pp 705-710. Henneberger, G., Sattler, P. and Shen, D., 1990a, "Ach'~evement of Analytical Accracy in the Magnetostatic Field Computation by the Finite Element M-e th- o -cL" Conference on Electromagnetic Field Computation, Toronto, Canada, Oct., Paper BA-05.


631

Henneberger, G., Sattter, G. P., Shen, D., !990b, "Force Calculation with Analytical Accuracy in the Finite Element Based Computational Magnetostatics," Conference on Electromagnetic Field Computation, Toronto, Canada, Oct., Paper BA-05. Hodgdon, M. L, 1988, "Application of a Theory of Ferromagnetic Hysteresis," IEEE Tram. Magn., Vol. 24, No. 1, Jan., pp.218-220. Hoole, P. R. P., I984, M.Sc Thesis (Electric Plasma Science), University of Oxgord. Hoole, P. R. P., 1987, D.Phil Thesis (Engineering), University of Oxford. Hoole, P. R. P° and Hoole, S. R. H., 1986, "Finite Element Computation of Magnetic Fields from Lightning Return Strokes", in Cendes, Z.J. ed, Computational Electromagnetics, North-Holland, New York, pp. 229-237. Hoole, P. R. P., and Hoole, S. R. H., 1987, "Computing Transient Electromagnetic Fields radiated from Lightning," J.AppL ~fiys., Vol. 61, No. 8, pp. 3473-3475. Hoole, S. Ratnajeevan H., 1986, "Enhancing Interactivity and Automation Through Finite Element Neighbours", Comm. in App. Num. Meth., Vol. 2, pp. 509517, Sept. !986. Hoole, S. Ratnajeevan H., 1987, "Nodal Perturbations for Adaptive Finite Element Mesh Generation," IE~E Trans. Magn., Vol. 23, NO. 5, pp. 2635-2637, Sept. Hoole, S. R. H., I989, Computer-Aided Anakvsis and Design of Electromagnetic Devices, Elsevier. Hoole, S. Ratnajeevan H., 1990, "Finite Element Electromagnetic Field Computation on the Sequent Symmetry 8I Parallel Computer," 1EEE Trans. Magn.., VoI. MAG- 26, No. 2, pp. 837-840, March 1990. Hoole, S. Ratn~eevan H., 1991, "Optimal Design, Inverse Problems and Parallel Computers," IEEE Trans. Magn., Vol. 27, pp. 4146-4 I49, May. Hoole, S. Ratn~eevan H, I992, "Parallelization of Cholesky's Scheme in Finite Element Electromagnetic Field Computation," Int. £ App. Electromag. Matrls_ VoI. 3, No. 1, pp. 1-7, May, !992 Hoole, S. R. H. and Cendes, Z. J., 1985, "Direct Vector Solution of ThreeDimensional Magnetic Field Problems," £ AppL Phys., Vol. 57, pp. 3835-3837, Apr.

632

CollecvextBibliography

Hoole, S. R. H., Cendes, Z. J. and Hoole, P. R. P., 1986, "Preconditioning and Renumbering in the Conjugate Gradients Algorithm," in Cendes, Z.J., ed., Computational Electromxtgnetics, NorthHolland, pp. 91-99. Hoole, S. R. H. and Hoole, P. R. P., I984, "Building an Expert System for Interactive Finite Element Design," Proc. of the CS Third Annual VCorksl-.,op on Interactive Computing, IEEE Computer Society, pp.90-93. Hoole, S. R. H., Hoole, P. R. P. and Jayakumaran, S., 1986, "A Graphic Library for the Finite Element Modelling of Differential Equations," Koval, D.O. ed. Applied Simulation and Modelling, Acta Press, Calm'y, pp. 207-210. Hoole, S. R. H., Jayakumaran, S., Anandaraj, A. W., and Hoole, P. R. P., 1990, "Relevant Purpose Based Error Criteria for Expe~ Finite Element Mesh Generation," Z Etectromagn. Waves and AppL, Vol. 3, No. 2, pp. 167-177, 1989. Hoole, S. Ratnajeevan H., Jayakumaran, S., and Hoole, N. Ratnasuneeran G., 1988, "Flux Density and Energy Perturbations in Adaptive Finite Element Mesh Generation," IEEE Trans. on Magn., Vol. 24, pp. 322-325, Jan.. Hoole, S. Ratnajeevan H. and Mahinthakumar, G., I990, "Parallelism in Interactive Operations in Finite Element Simulation," IEEE Trans. on Magn., Vol. 26, pp. 1252-1255, July 1990. Hoo!e, S. Ratnajeevan and Subramaniam, S., 1992, "Inverse Problems with Boundary Elements," 1EEE Trans. Magn., Vol. 28, No. 2, pp. 1529 - 1532, March. Hoole, S. Ratn~eevan H., Subramaniam, S., Saldanha, R., Coulomb, J.-L., and Sa~nnadiere, J.-C., 1991, "Inverse Problem Methodology and Finite Elements in the Identification of Cracks, Sources, Materials and their Geometry in Inaccessible Locations," IEEE Trans. Magn., Vol. 27, March. Hoole, S. Ratnajeevan H., Weeber, K., and Subramaniam, S, 1991, "Fictitious Minima of Object Functions, Finite Element Meshes, and Edge Elements in EI~ nefic Device Synthesis," IEEE Trans. Magn., Vol. 27, pp. 5214-5216, Nov.. Hoole, S. R. H., Yoganat.han, S., and Jayakumaran, S, 1986, "Implementing the Smoothness Criterion in Adaptive Meshes," IEEE Trans. Magn., VoI. MAG-22, No. 5, pp. 808-8 I0, Sept.. Horn, R. A. and Johnson, C. R., 1986, Matrix Analysis, Cambridge University Press, Ca_mbddge.

Houbolt, J.C., 1950, "A recurrence matrix solution for the dynamic response of elastic "J. Aero. Sci, Vol. 17, pp. 540-550.

S. R. H. H~le (Editor)

633

Hrenikoff, A., I941, "Solution of Problems in Elasticity by the Framework Method," Z AppL Mech, Vol. 8, pp. 169-175. Huebner, K. H., 1975,The Finite Element Method for Engineers, Wiley, New York, Ch. 4. Hwang, Chang-Chou, Salon, S. J. and Palma, R., 1988, "A Finite Element PreProcessor for Induction Motors Including Circuit and Motion Constraints," IEEE Trans. Magn., Vol. 24, No. 6, Nov., pp. 2573-2575. Imhoff J. F., Meunier G., Sabonnadiere J. C. 1990, "Finite Element Modelling of Open Boundary Problems," IEEE Trans. Magn., Vol. 26, No. 2, Mar. Imhoff, J. F., Meunier, G., Reyne, G., Foggia, A. and Sabonnadiere, J. C., 1989, "Spectral Analysis of Electromagnetic Vibrations in DC Machines through the Finite Element Method," IEEE Trans. Magn., Vol. 25, 1'4o.5, pp. 3590-3592. Imhoff, J. F., Meunier, G. and Sobonnadiere, J. C., 1990, "Finite element M~eling of Open Boundary Problems, Trans. Magn, Vol. 26, March. Istfan, B., 1987, Extensions to the Finite Element Method for Nonlinear Magnetic FieM Problems, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, New York. Istfan, B. and Salon, S. J., 1988, "Inverse nonlinear finite element problems with local and global constraints," IEEE Trans. Magn., Vol. 24, No. 6, pp. 2568-2572, Nov.. Jackson J. D., 1962, Classical Electrody~mics, John Wiley & Sons, Inc, New York. Jarvis, D. B., 1963, "Interconnections Effects on High Speed Logic," IEEE Trans. Electronic Computers, VoI. 12, Oct.. Jahn, J., Elk, K., and Shuman, R., 1987, Z Magn. Magn. Mat. ,Vol. 68, p. 335. Jeffers, F., I986, "High Density Magnetic Recording Heads," Proc. No. 11, Nov., pp. 1540-1556.

Vol. 74,

Jiles, D. C. and Atherton, D. L , 1986, "Theory of ferromagnetic hysteresis," Z.Magn. 3¢Sagn. Materials, Vol. 61, pp.48-60. Jones, D. J., 1964, The Theory of Electromagnetisrn, Pergamon Press. Jones, R. V., 1981, The tenth Clerk Maxwell Memorial Lecture: The Complete Physicist: James Clerk Maxwell, Radio and Electronic Eng., Vol. 51, pp. 527-539.

634. . . . . . . . . . . . . . . .


Joos, G., I934, Theoretical Physics, Blackie. Jordan, E. C, 1961, Electromagnetic Waves and Radiating Systems, New York: Prentice-Hall. Jordan, J.G. and Balmain, K.G, I968, Electromagnetic Waves and Radiating Systems, 2nd Edition, Prentice-Hall. Jordan, E. C. and Balmain, K. G., 1980, "Electromagnetic Waves and Radiation Systems," Prentice Hall Private Ltd. Josmo, K., Tatalias, D., Bornholdt, and James M., 1989, "Mapping Electromagnetic Field Computations to Parallel Processors," IEEE Trans. Magn., Vol. 25, No. 4, pp. 2901 - 2906, July. Kadar, G., 1987, "On the Preisach Function of Ferromagnetism Hysteresis," J.

Appl. Phys., 61, Apr., pp. 4013-4015. Kadar, G. and Della Tore, E. D., 1988, "Determination of the Bi!inear Product Preisach Function," ZAppLPhys, Vol..63, Apr., pp.3004-3CX)6. Kalaichelvan, K. S., 1984, An evaluation of sparce solution techniques for finite element eddy current problems, M.A.Sc. Thesis, University of Toronto, Toronto. Kalaichelvan, K. S., 1987, A Boundary Integral Equation Method for 3Dimensional Eddy Current Problems, Ph.D. Diss, University of Toronto, I987. Kalaichelvan, K. S., 1990, "An Efficient Implementation of the Integro-differential Method to Evaluate High Speed Interconnect Losses," Proc. Canadian Conference on Electrical and Computer Engineering, pp 17.2.1-17.2.4. Kalaichelvan, K. S. and Lavers, J. D., 1987, "Singularity Evaluation in Boundary Integral Equation of Minimum Order for 3-D Eddy C U rrents, " IEEE Trans. Magn., Vol. 23, pp 3053-3055. Kalaichelvan, K. S. and Lavers J. D., 1988a, "A Rezoning Algorithm to Evaluate Singular Integrals in 3-D Boundary Element Method," Symposium on Progress in Electromagnetic Research, Boston. Kalaichelvan, K. S. and Lavers, J. D. 1988b, "On the Implementation of Boundary Element Method for 3D Multiply connected Eddy Current Problems," IEEE Trans. Magn., Vol. 24.

S. R. H. H~te (Editor)

635

Kalaichelvan, K. S. and Lavers, J. D., 1989, "Boundary Element Methods for Eddy Current Problems," Chapter 4 in Topics in Boundary Element Research, Springer-Verlag. Kameari, A., 1988, "Three Dimensional Finite Element Method With A - V in Conductor and f~ in Vacuum," IEEE Trans. Magn., Vol. 24, No. 1, pp. 118-121. Kameari, A., 1989, "Three Dimensional Eddy Current Calculation Using Edge Elements for Magnetic Vector Potential," Applied Electromagnetics in Materials (Ed. K.Miya), pp. 225, Pergamon Press. Kameari, A., 1990, "Calculation of Transient 3d Eddy Current Using EdgeElements", 1EEE Trans. Magn., Vol. 26, No. 2, pp. 466469. Kanmrovich, L. V. and Akilov, G. P., 1982, Functional Analysis, Oxford, New York: Pergamon Press. Kaplan, B. Z., 1967, "Analysis of a Method of Magnetic Levitation," Proc. IEEE, VoI. 114, pp. 180 !- 18(N. Kaplan B. Z. and Regev, D., 1976, "Dynamic Stabilization of Tuned Circuit Levitators," !EEE Trans. Magn., Vol. I2, No. 5, pp. 556-559. Kardestuncer H., 1987, Basic Concepts of Finite Element Method, Chapter 3, Finite Element HandBook, Kardestuncer H. (ed.), McGraw-Hill. Kerruish, N., 1958, "Deflexion of a Thick Cylindrical Ring Due to a Sinusoidal Force," Electrical Energy, Vol. 2, June, pp. 236-238. Kincaid, David R. and Oppe, Thomas C., 1988, "A Parallel Algorithm for the General LU Facmrization," Comm. App. Num. Meth., Vol. 4, pp. 349-359. Kirubarajan, T. and Hoole, P. R. P., 1993, "A Novel State Estimator for Rotation and Tilts in Helicopters," IEEE Trans. Magn., Vol. 29, March. Kirubarajan, T. and Hoole, P. R. P. 1994, "Optimal Estimator and an Adaptive Stabilizer for Low Frequency Vibrations in Satellite Sensors," University of Peradenya Internal Report. Also under review for IEEE Trans. or Aerospace and Electronic Systems. Konrad, A., 1974, Triangular Finite Elements for Vector Fields in Electromagnetics, Ph.D. Thesis, MCGill University, Montr6al, Qu6bec.

636


Konrad, A., 1981, "The Numerical Solution of Steady-State Skin Effect Problems - An Integrodifferential Approach," IEEE Trans. Magn., Vol. 17, No. t, pp. 11481152. Konrad, A., 1982, "lntegrodifferential Finite Element Formulation of TwoDimensional Steady-State Skin Effect Problems" IEEE Trans. Magn., Vol. 18, No. I, Jan., pp. 284-292. Konrad, A., 1986, "On the reduction of the number of spurious modes in the vectorial finite-element solution of three-dimensional cavities and waveguides," IEEE Trans. Microw. Th. Tech., Vo!. 34(2), 224-227. Konrad, A. and Sober, T. J., 1986, "A fast met,hod for computing the capacitance of ceramic chip carrier conductors," Proc. IEEE Workshop on Electromagnetic Field Co~gutation, Schenectady, NY, October 20-21, pp. C41-C47, (IEEE Cat. #87-TH0192-5). IZoonrad, A. and Tsukerman, I.A., 1993, "Comparison of High- and LowFrequency Electromagnetic Field Analysis," J. Phys. 111"France, VoI. 2, pp. 36337I. Kxiezis, E. E. and Cangellaris and E. E., I984, "An integral equation approach to the problem of eddy currents in cylindrical shells of finite thickness with infinite or finite length," Archiv. fur Elektrotec.hnik, Vol. 67, pp. 3 I7-324. Kron, G., 1943, "Equivalent circuits to represent the electromagnetic field equations," Phys. Rev., Vol. ~ , pp. I26-I28. Kwakernaak, H., and Sivan, R., 1972, Linear Optimal Control System,. John Wiley & Sons, Inc. Langefors, B., !952, "Analysis of Elastic Structures by Matrix Transformation, with S~cial Regard m Semimonocoque Structures," .1".Aeron Sci., Vol. 19, No. 7, pp. 451-458. Lavers, J. D., 1982, "Force and sti~ing patterns produced by an electromagnetic mold," IEEE Trans. Ind. App. Sys., pp. 954-960. Lavers, J. D., 1983, "Finite element solution of non-linear two-dimensional ~ mode eddy current proNems," Trans. Mag.n, pp. 2201-2203, Sept. l~an, M. H., Friedman, M~ and Wexler, A., 1985, "Application of the Boundary Element Method in Electrical Engineering Problem," Chapter 9, Application of Bounda~ Element Met,fiods.

S. R. H. H~!e (EditoO

637

Liang, G. C., LIu, Y. W. and Mei, K.K., 1990, "Analysis of Coplanar Waveguide by Time-Domain Finite Difference Method," mEE M ~ - S Internation Microwave Symposium Digest, pp 1005-I008. Ling, D. D. and Ruehli, A. E., 1981, "Interconnection Modeling," Chapter XI in Circuit Analysis, Simulation and Design, edited by A.E.Ruehli, Elsevier Science Publications Inc. Lindman, E. L., 1975, "Free space boundary conditions for time dependent wave equation,",/. Comp. Phys., Vol. 18, 66-78. Liwschitz, M. M., I942, "Field Harmonics in Induction Motors," AIEE Trans. Power Apparatus and Systems, Vol. 61, Nov., pp~797-803. Lowther, D. A., Freeman, E. and Forghani, B., 1989, "A Sparse Matrix Open Boundaxy Method for Finite Element Analysis," 1EEE Trans. Magn., Vol. 25, No. 4, July. Lowther, D. A. and Silvester, P. P., 1986, Computer-Aided Design in Magnetics, New York: Springer-Verlag, pp. 209-211. Lynch, D. R., Paulsen, K. D. and Strohbehn, J. W, 1986, "Hybrid Element method for unbounded electromagnetic problems in hyperthermia," Int. J. for Numer. Meth. Eng., Vol. 23, pp. 1915-1937. MacNeal, B. E., Brauer, J. R., and Coppolino, R. N., 1990, "A General Finite Element Vector Potential Formulation of Electromagnetics Using a TimeIntegrated Scalar Potential," IEEE Trans. Magn.,Vol. 26, Sept., pp. 1768-1770. MacNeaI, B. E., MacNeal, R. H., and Coppolino, R. N., 1989, "Spurious Modes of Electromagnetic Vector Potential Finite Elements," IEEE Trans. Magn., 25, 414I. Magnin, H. and Coulomb, J.-L., 1989, "A Parallel and Vectorial Implementation of basic Linear Algebra Subroutines in Iterative Solving of Large Sparse Linear Systems of Equations," IEEE Trans. Magn., Vol. 25, pp. 2895-2898. Mahinthakumar, G. and Hoole, S. Ratn~eevan H., 1990a, "A Parallel Conjugate Gradients Algorithm for Finite Element Analysis of Electromagnetic Fields," Z of Applied Pho,sics, Vol. 67, No. 9, pp. 5818-5820. Mahinthakumar, G. and Hoole, S. Ratnajeevan H., 1990b, "A ParalIelized Element by Element Jacobi Conjugate Gradients Algorithm for Field Problems and a Comparison with other Schemes," Int. Z App. Etectromag. in Matl., Vol. 1, No. 1, pp. 15-28. Maho, M. M., 1979, Digital Logic and Computer Design, Prentice-Hall, Inc.

638


Mayergoyz, I. D., 1979, Iteratsionnye Metody Raschem Staticheskih Polei v Neodnorodnyh, Anizotropnyh i Nelineinyh Sredah (Iterative Methods for Computations of Static Fields in Inhomogeneous, Anisotropic and Nonlinear Media), Kiev (USSR), Naukova Dumka, [in Russian]. Mayergoyz, I. D., 1983, "A New Approach to the Calculation of ThreeDimensioanl Skin-Effect Problems," IEEE Trans. Magn., Vol. 19, No. 5, pp 21982200. Mayergoyz, I. D., 1985, "Hysteresis Models from the Mathematical and Control Theory Points of View," J;App. Phys., Vol. 57, Apr., pp.3803-3805. Mayergoyz, I. D. and Friedman, G., 1987, "The Preisach Model and Hysteretic Energy Losses," J.Appl, Phys., Vol..61, Apr., pp.3910-3912. Mayergoyz, I. D. and Friedman, G., 1988, "Generalized Preisach model of hysteresis," iEEE.Trans. Magn., Vol. 24, No. 1, Jan., pp.212-217. McFee, S. and Lowtherr, D. A, 1987, "Towards Accurate and Consistent Force Calculation in Finite Element Based Magnetostatics," lEEE Trans. Magn., VoI. 23, No. 5, Sept., pp. 377 !-3773. McFee, S., Webb, J. P., and Lowther, D. A., I988, "A Turnable Volume Integration Formulation for Force Calculation in Finite-Element Based Magnetostatics," IEEE Trans. Magn., Vol. 24, No. 1, Jan. pp. 439-442. McHenry, D., I943, "A Lattice Analogy for the Solution of Plane Stress Problems," J Inst. Civil. Eng., Vol. 21, pp. 59-82. Mee, C. D. 1964, 1986, The Physics of Magnetic Recording, North-Holland: Amsterdam, pp 158-161. Meijerink, J. A. and Van der Vorst, H. A., 1977, "An Iterative Solution Method tor Linear Systems of which the Coefficient Matrix is a Symmetric M Matrix," Math. Comp., Vol. 31, pp. 148-162. Mikhlin, S. G., 1960, Linear Integral Equations, Delhi Hindustan Publishing Co.. Mikhlin, S. G., 1964,Variational Methods in Mathematical Physics, Oxford: Pergamon Press. Miniowitz, Ruth and Webb, J. P., 1991, "Covariant-Projection Quadrilateral Elements for the Analysis of Waveguides with Sharp Edges," IEEE Trans. Microwave T,h. Tech., Vol. 39, No. 3, Mar., pp. 501-505.


639

Moore, T. G., Blaschak, J. G., Taflove, A. and Kriegsman, G. A. 1988, "Theo~ and application of radiation boundary operators," IEEE Trans. Anten. Prop., Vol. 36(12), I797-1812. Mueller, C., 1969, Foundations of the Mathematical ~ e o t y of Electromagnetic Waves, Berlin: Springer-Verlag, 1969. Mur, G. and De Hoop, A. T. 1985, "A finite-element method for computing threedimensional electromagnetic fields in inhomogeneous media," 1EEE Trans. Magn., 2I(1), 2188-2191. Nakata, T., and Takahashi, N., I988, Finite Element Method in Electrical Engineering, (Second Edition) Morikita Publishing Co., Tokyo. Nakata, T., Takahashi, N. and Fujiwara, K., 1988, "Efficient Solving Techniques of Matrix Equations f:or Finite Element Analysis of Eddy Current ", IEEE Trans. Magn., Vol. 24, No. 1, pp. t 70-173. Nakata, T., Takahashi, N., Fujiwara, K. and Ahagon,.A, 1988 ,"3D Finite Element Method for Analyzing Magnetic Fields in Electrical Machines Excited from Voltage Sources," IEEE Trans. Magn., Vol. 24, Nov, pp. 2582~2584. Nakata, T., Takahashi, N., Fujiwara, K., Imai, T. and Muramatsu, K., 1991, "Comparison of Various Methods of Analysis and Finite Elements in 3-D Magnetic Field Analysis," IEEE Trans. Magn., Vol. 27 No. 5, pp. 4073-4076. Nakata, T, Takahashi, N, Fujiwara, K., Muramatsu, Y.and Cheng, Z. G., 1988, "Comparison of Various Methods for 3-D Eddy Current Analysis," IEEE Trans. Magn., Vol. 24, No. 6, pp. 3 t 59-3161. Nakata, T., Takahashi, N., Fujiwara, K., and Okada, Y., 1988, "Improvements of the T - f;~ Method for 3 ~ D Eddy Current Analysis," IEEE Trans. Magn., Vol. 24, No. 1, pp. 94-97. Nakata, T., Takahashi, N., F~iwara, K. and Olsewski, P., 1990, "Verification of Softwares for 3-D Eddy Current Analysis Using IEE J Model," Advances in Electrical Engineering Software (Editor: P. P. Silvester), pp. 349 Computational Mechanics Publications and Springer ~ Verlag. Nakata, T., Takahashi, N., Fujiwara, K., and Shiraki, Y., 1990, "Comparison of Different Finite Elements for 3-D Eddy Current Analysis," IEEE Trans. Magn., Vol. 26, No. 2, pp. 434-437. Nedelec, J. C., 1980, "Mixed Finite Elements in R3," Numer. Math., Vol. 35, pp. 315-341.

640


Nedelec, J. C., 1982, "Elements finis mixtes incompressibles pour requation de Stokes dans R3, '' Numer. Math., Vol. 39, pp. 97-112. Nedelec, J. C., 1985, "A New Family of Mixed Finite Elements in R3, '' Numer. Math., Vol. 50, pp. 57-81. Newmark, N. M., 1959, "A Method of Computation for Structural Dynamics," A.S.C.E. J. of Eng. Mech., Vol. 85, pp 67-94. Oden, J. T., !969, "A General Theory of Finite Elements, I: Topological Considerations, II: Applications," Int. J. Num. Meth. Eng., pp. 205-221, pp. 247260, Vol. I. Oden, J. T. and Carey, G. F., 1983, Finite Elements, volume I-IV, Englewood Cliffs, N.J. Oganesian, G. A. and Rukhovets, L. A., 1979, Variatsionno-Raznostnye Metody Resheniia Ell(pticheskikh Uravnenij (Variational-Difference Methods for the Solution of Elliptic Equations): Izd-vo AN Armianskoi SSR, Yerevan, USSR [in Russian]. Okon, E. E., 1982, "Some Vector Expansion Functions For Quadrilateral and Triangular Domains," Lnt. J. Num. Meth. Eng., Vol. 18, pp. 727-735. Opsahl, T. and Taflove, A., 1989, "Predicting Scattering of Electromagnetic Fields using ~ - ~ on a Connection Machine," IEEE Trans. Magn., Vol. 25, pp. 29102912. Oristaglio, M.L. and Worthington, M. H., 1980, "Inversion of Surface and Borehole Electromagnetic Data for Two-Dimensional Electrical conductivity Models," Geophys. Pros., VoI. 28, pp. 633-657. Osterhaug, Anita, 1987, Guide to Parallel Programming on Sequent Computer Systems, S~uent Computer Systems Inc.. Ossart, F , 1991, "ModElisation numrrique de l'enregistrement magnrtique, " Th~se de doctorat de l'Institut National Polytechnique de Grenoble, France, Frvrier. Ossart, F. and Meunier, G., 1990, "Computation of the readback voltage in digital magnetic recording", Jr. Magn. and Magn. Materials, Vol. 83, N-o.I, Jan., pp.4850. Palma, Rodolfo, 1989, "Transient Analysis of Induction Machines Using Finite Elements," Ph. D. Thesis, Rensselaer Polytechnic Institute, Aug.


641

Parlett, B. N., 1980, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, N.J., 1980. Penman, J. and Grieves, M. D., 1986, "Efficient Calculation of Force in E!~tromagnetic Devices," Proc. lEE, Vol. 133, No. 4, Part B, July, pp. 212-216. Pichon, L. and Razek, A., 1992, "Three dimensional resonant mode analysis using edge elements", IEEE Trans. Magn., Vol. 28, No. 2, 1493-1496. Pinchuk, A. R., Crowley, C. W., and Silvester, P. P., I988, "Spurious Solutions to Vector Diffusion and Wave Field Problems," Trans. Magn., Vol. 24, No. 1, Jan. pp. I58-161. Pinchuk, A. R. and Silvester, P. P., 1985, "Error Estimation for Automatic Adaptive Finite Element Mesh Generation," IEEE Trans. Mag., Vol. 21, pp. 25512554. Piriou, Francis, and Razek, Adel, 1988, "Coupling of Saturated Electromagnetic Systems to Non-linear Power Electronic Devices," IEEE Trans. Magn., Vol. 24, January, pp. 274-277. Pironneau, Olivier, 1984, Optimal Design for Elliptic Systems, Springer Verlag. Plonus, M. A., 1978, Applied Etectromagnetics, McGraw-Hill Book Company, New York. P61ya, G. and Szeg6, G., 1951, lsoperimetric Inequalities in Mathematical Physics, Princeton University Press. Potter, R. I. and Schmulian, R. J., 197I, "Self-consistent Computed Magnetization Patterns in Thin Magnetic Recording Media," LEEE Trans. Magn., Vol. 7, No. 4, Dec., pp.873-879. Preis, K., Magele, C., and Biro, O., 1990, "FEM and evolution strategies in the optimal design of EM devices," IEEE Trans. Magn., Vol. 26, No. 5, pp. 21812183, Sept. 1990. Preisach, Z, 1935, Phys., Vol. 94, p.277. Preston, T. W. and Reece, A. B. J,, 1982, "Solution of 3 - Dimensional Eddy Current Problems • The T - ~ Method," Trans. Magn., Vol. 18, No. 2, pp. 486-491.

642


Railton, C. J. and McGeehan, J. P., 1990, "Analysis of Microstrip Discontinuities using Finite Dif/~rece Time Domain Technique," IEEE M ~ - S Internation Microwave Symposium Digest, pp 1~9-1012. Raizer, A., Hoole, S. R. H., Meunier, G., and Coulomb, J.-L., 1990, "P- and HType Mesh Generation," Z App.Phys., Vol. 67, No. 9, pp. 5803-5805. Rao, S. M, Sarkar, T. K. and Harrington, R. F., 1984, "The Electrostatic Field of Conducting Bodies in Multiple Dielectric Media," IEEE Trans. Microwave Th. Tech., Vol. 32, No 11, Nov., pp 1~1-1448. Rafin~ad, Parviz, 1977, Adaptation de la Methode des E;e, emts Finis a la Modelisation des Systemes Electromecaniques de Conversion d'Emergoe, Ph. D. Theisis, L'Institut National Polytechnique de Grenoble, Jan. Raizer, A., Meunier, G., and Coulomb, J.-L., 1989, "An approach for Automatic Adaptive Mesh Refinement in Finite Element Computation of Magnetic Fields," Trans Magn., Vol. 25, No. 4, July. Ratman, D.V., and Buessem, W. R., 1972, ./".Appl, Phys., Vol. 43, p. 1291. Raviart, P. A. and Thomas, J. M., 1977, "A Mixed Finite Element Method for 2nd Order Elliptic Problems," in Dold, A. and Eckmann, B., eds., Mathematical Aspects of Finite Element Methods, Lect. Notes 606, Berlin, Heidelberg, New York: Springer. Ren, Z., Bouillault, F., Razek, A., Bossavit, A. and Verite, J. C., 1990, "A New Hybrid Model Using Electric Field Formulation for 3-D Eddy Current Problems," IEEE Trans. Magn., Vol. 26, No. 2, pp. 470-473. Reyne, Gilbert, 1987, Analyse Theorique et Experimenmle des Phenomenes Vibratoire d'Oritine Electromagnetique, P h . D . Theisis, L'Institut National Polytechnique de Grenoble, Dec. Reichert, K., Skoczylas, J. and T/irnhuvud, T., 199 I, "Automatic mesh generation based on expert-system=methods", Trans. Magn., Vol. 27, No. 5, pp. 41974200. Reid, J. G, 1985, Linear systems fundamentals, McGraw-Hill, 1985. Rektorys, K., 1980,Variational Methods in Mathematics, Science and Engineering, Dr. Reidel Publishing Co., Dordrecht, Holland Reyne, G., Meunier, G., Imhoff, J. F, and Euxibie, E, 1988, "Magnetic Forces and Mechanical Behavior of Ferromagnetic Materials Presentation and Results on


the Theoretical, Experimental and Numerical Approaches," Vol. MAG-24, no. 1, pp. 234-237.

643

Trans. Magn.,

Reyne, G., Sabonnadiere, J. C., Coulomb, J. L. and Brissonneau, P., 1987, "A Survey of the Main Aspects of Magnetic Forces and Mechanical Behaviour of Ferromagnetic Materials Under Magne t"lsatlon, " " IEEE Trans. Magn., VoI. 23. No. 5, Sept., pp. 3765-3767. Reyne, G., Sabonnadiere, J. C., and Imhoff, J. F., 1988, "Finite Element Modelling of Electromagnetic Force Densities in DC Machines," Trans. Magn., Vol. 24, No. 6, pp. 3171-3173. Ruehli, A. E., 1987, Circuit Analysis, Simulation and Design: VLSI Circuit Analysis and Simulation, Elsevier Science Publications Inc. Russ, S. R. and Alford, C. O , 1990, "Coupled Noise and i~ Effects on Modern Packaging Technology, "IEEE Trans. CHMT, Vol. 13, No 4, Dec., pp 1074-1082. Sabonnadiere, Jean-Claude and Coulomb, Jean-Louis, 1987, Finite Element methods in CAD: Electrical and magnetic fields, North Oxford Academic Publishers Ltd: New "York. Sabonnadiere, Jean-Claude and Konrad, Adalbert, 1992 "Computing EM fields," IEEE Spectrum, Nov., pp. 52 - 56. Sakiyama, K., Kotera, H. and Ahagon, A., 1990, "3-D Electromagntic Field Mode Analysis Using Finite Element Method by Edge Element," IEEE Trans. Magn., Vol. 26, No. 5, pp. 1759 - 1761. Salon, S. J., DeBo~oli, M. J., and Palma, R., 1990, "Coupling of Transient Fields, Circuits, and Motion Using Finite Element Analysis," Journal of Electromagnetic Waves and Applications, Vol. 4, No. 11, pp. 1077-1106. Salon, S. J. and Istfan, B., 1986, "Inverse Nonlinear Finite Element Problems," IEEE Trans. Magn., Vol. 22, No. 5, pp. 817-818, Sept.. Schwartz, L., 1966, Mathematics for Pho~ical Sciences, Paris: Addison-Wesley Publishing Co. Schwartz, L., 1950, Theorie des D&tributions, Paris Hermann Searl, C. W., Frederick, W. G. D., and Garret, H. J., 1973, IEEE Trans. Magn., Vol. 9, p. 164. Segerlind, L. J., 1984, Applied .Finite Element Anaiysis, New York: Wiley.


Shen, D, 1987, Contribution ~ la Modglisation Numgrique de Ph~nomknes Electromagn~tiques, Th~se de D~torat INPG, Grenoble. Silvester, P., 1966, ~'Modal network theory of skin effect in flat conductors," Proc. Vol. 54, No.9, pp. 1147-1151. Silvester, P., 1967, "Network analogue solution of skin and proximity effect problems," IEEE Trans. Power and App. Sys., Vol. PAS-86, No. 2, pp. 241-247. Silvesmr, P. 1969a, "High-order polynomial triangular finite elements for potential problems", Int. J. Eng. Scie., Vol. 7, pp.849-861. Silvester, P., 1969b, "A general high-order finite element waveguide analysis program", IEEE Trans. Microwave Th. Tech., Vol. 21, pp. 538-542. Silvester, P., 1969c, "Finite Element Solution of Homogeneous Wave Guide Problems," Alta Frequen~a, Vo!. 38, pp. 313-317. Silvester, P., 1970, "Symmetric Quadrature Formulae for Simplexes," Math. Comp., Vol. 24, No. 109, Jan., pp 95-100. Silvester, P. and Chari, M. V. K., t970, "Finite element solution of saturable magnetic field problems," IEEE Trans..Power Appar. Syst., Vol. PAS-89, pp. 1642 - 1652. Silvester, P. P. and Chari, M. V. K., 1980, Finiw Element Methods in Electric and Magnetic FieM Problems, John Wiley and Sons. Silvester, P. P. and Ferrari, R. L., 1983, Finite ElementsJbr Electrical Engineers, Cambridge University Press, Cambridge. Silvester, P. P. and Ferrari, R. L., 1990, Finiw Elements for Electrical Engineers, 2nd Ed., Cambridge University Press, Cambridge. Si!vester, P., Lowther, D. A., Ca~enter, C. J. and Wyatt, E. A, 1977, "Exterior finite elements for two dimensional field problems with open boundaries," Proc. IEEE, Vol. 124, No.12, pp 1267-1270. Simkin, J., and Trowbridge, C. W , 1979, " On the Use of the Total Scalar Potential in the Numerical Solution of Field Problems in Electromagnetics," Int. J. Num. Meth. in Eng., Vol. 14, pp. 423 -440. Simkin, J. and Trowbridge, C. W., 1992, "Optimizing electromagnetic devices combining direct search methods with simulated annealing," 1EEE Trans. Magn., Vol. 28, No. 3, pp. 1545-I548.

S. R. H. H~ole (Editor)

645

Smith, N., 1987, "Reciprocity principles for magnetic recording theory," IEEE Trans. M:agn., Vol. 23, No.4, Jul., pp. 1995-2~O2. Sobolev, S. L., 1963, Applications of Functional Analysis in Mathematical Physics, Providence, American Math. Soc. Sorenson, H. W.(editor), 1985, Kalman Filtering: theory and Application,. IEEE press. Stark, H. and Woods, J. W. 1986, Probability, Random Processes, and L~timation Theoo, for Engineers, Prentice-Hall, Englewood Clift%, New Jersey. Stoll, R. L., 1974,The Analysis of EAdy Currents, Oxfo~, England: Clarendon Press. Strang, G. and Fix, G. J., 1973, An Analysis of the Finite Element Method, Englewood Cliffs, N.J.: Prentice Hall. Stranges, Elias G., 1985, "Coupling the Circuit Equations to the Nonlinear Time Dependent Field Solution in Inverter Driven Induction Motors," IEEE Trans. Magn., Vol. 2 I, No. 6, Nov., pp. 2408-2411. Stranges, Elias G. and Theis, K. R., 1985, "Shaded Pole Motor Design Using Coupled Field and Circuit Equations," 1EEE Trans. EClagn.,Vol. 21, No. 5, Sept. pp. 1880-1882. Stratton, Julius A., 1941, Electromagnetic Theory, McGraw Hill, New York. Subramaniam, S, Arkadan, A. A. and Hoole, S. R. H., 1994, "Optimization of a Magnetic Pole Face using Linear Constraints to Avoid Jagged Contours," IEEE Trans. Magn., Vol. 30, Sept.. Subramaniam, S., Kanaganathan, S., and Hoole, S. Ratnajeevan H., 199I, "Two Requisite Tools in the Optimal Design of Electromagnetic Devices," IEEE Trans. Magn., Vol. 27, pp. 4 t 05-4109.. Subramaniam, Srisivane, Feliziani, M. and Hoole, S. Ratnajeevan H., 1993, "Open boundary eddy-current problems using edge elements," IEEE Trans. Magn., Vol. 29, No. 2, Mar., 1499-1503.

Suriva, J., 1988, "Trade-Oft~ in Highspeed Computer Inter-connects," RCA Report. Swanson Jr., Daniel G., 199 I, "Simulating EM fields," IEEE Spectrum, Nov., pp. 34 - 37.

646


Szabo, B. A. and Babusvka, I., 1991, Finite Element Analysis, New York: Wiley. Takahashi, N., Nakata, T. and Fujiwara, K., 1989, "Improvements of 3-D Finite Element Method for Eddy Current Analysis and Its Application to Fusion Technology," b~sion Engineering and Design, Vol. 9, pp. 113. Tarnhuvud, T. and Reichert, K., 1988, "Accuracy Problems of Force and Torque Calculation in FE Systems," IEEE Trans. Magn., Vol. 24, No. I, Jan., pp. 443446. Tavner, P. J. and Penman, J., 1987, Condition Monitoring of Electric Machines, John Wiley and Sons., New York. Tikhonov, A. N., Arsenin, V., I. 1977, Solutions of Ill-Posed Problems, Washington: Halsted Press. Tozoni, O. V. and Mayergoyz, I. D., 1974, Raschet Trakhmernyh Electromagnitnyh Polei (Computations of Three-Dimensional Electromagnetic Fields), Kiev (USSR), Tekhnika [in Russian], Kiev.. Trowbridge, C. W., 1981, "Three-Dimensional Field Computation," IEEE Trans. Magn., Vol. 18, pp. 293-297. Turner, M. J., Clough, H. C., Martin, H. C., and Topp, L. L, 1956 "Stiffness and Deflection Analysis of Complex Structures," J. Aeron Sci., Vol. 23, No. 9, 805823. Vander Heiden, R. H., Arkadan, A. A., Brauer, J. R., and Hummert, G. T., 1991, "Finite Element Modeling of a Transformer Feeding a Rectified Load: the Coupled Power Electronics and Nonlinear Magnetic Field Problem," IEEE Trans. Magn., Vol. 27, Nov. van Welij, J. S., 1985, "Calculation of eddy currents in t e ~ s of H on hexahedra," IEEE Trans. Magn., Vol. 2 I, No. 6, pp. 2239-2241. Vanderplaats, G., 1983, Numerical Qptim&ation Techniques for Engineering Design, McGraw-Hill. Vassent, E., Meunier, G, and Sabonnadiere, J.-C., I989, "Simulation of Induction Machine Operation using Complex Magnetodynamic Finites Elements," IEEE Trans. Magn., Vol. 25, No. 4, pp. 3064-3066, July. Venkataraman, J., Rao, S. M., Djordjevic, A. R., Sarkar, T. K., Naiheung, Y., 1985, "Analysis of Arbitrary Oriented Microstrip Transmission Lines in


647

Arbitrarily Shaped Dielectric Media over a Finite Ground Plane," lEEE Trans. Microwave Th. Tech., Vol. 33, No. 10, Oct., pp 952-959. Walker, J. H. and Kerruish, N., 1960, "Open-Circuit Noise in Synchronous Machines," Proc. lEE, Vol. 107, Part A, pp. 505-5 ! 2. Watters, S., 1980, Electromagnetic Transients and Induction, Prentice-Hall. Webb, J. P., 1993, "E"ag e Elements and What They Can Do, " IEEE Trans. Magn., Vol. 29, No. 2, March, pp. 1460-1465. Webb, J. P., Maile, G. L. and Ferrari, R. L., I983, "Finite-element Solution of Three-dimensional Electromagnetic Problems," IEE Proc., VoI. 130, Pt. H, No. 2, pp. 153-159. Wei, C., Harrington, R. F., Mautz, J. R. and Sarkar, T. K., 1984, "Multiconductor Transmission Lines in Multilayered Dielectric Media," i~EE Trans. Microwave Th. Tech., Vol..32, No 4, pp 439-449. Weeber, K. and Hoole, S. Ratnajeevan H., 1992a, "A Structural Mapping Technique for Geometric Parametrization in the Synthesis of Magnetic Devices," ,Int. J. Num. Meth. Eng. Vol. 33, pp. 2145~2179, July 15 Weeber, K. and Hoole, S. Ratnajeevan H., 1992b, "Geometric Parametrization and Constrained Optimization Techniques in the Design of Salient Pole Synchronous Machines," IEEE Trans. Magn., Vol. 28, pp. 1948-I960, July. Weeks, W. T., 1970, "Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of dielectric interface," IEEE Trans. Microwave Th. Tech., Vol. 18, pp 35-43, Jan. Wei, C., Harrington, R. F., Mautz, J. R., and Sarkar, T. K., 1984, "Multiconductor Transmission Lines in Multilayered Dielectric Media," IEEE Trans. Microwave Th. Techniques, Vol. M ~ . 3 2 , No. 4, pp. 439-449. White, R. M., 1984, Introduction to Magnetic Recording, New York: IEEE Press. Whitney, H., 1957, Geometric Integration Theory, Princeton University Press. Widrow, B., and Stearns, S. D., 1985, Adaptive Signal Processing, Prentice-Hall, Inc., USA. Wiesen, K. and Charap, S, H., 1988, "A Better Scalar Preisach Algorithm," 1EEE Trans. Magn., Vol. 24, No. 6, Nov., pp.2491-2493.

648

Collected Biblio~aphy

Williamson, J+, 1991, "A Brief Treatise on High-Speed Signal Integrity," DOC. BNR, Canada. Wilson, D+ K+, 1990, "Partitioning - The Payoff Role in Telecommunication System Design," Proc. Int. Conf. Advances in Interconnections and Packaging, SPIE Vol. 1390, pp 537-547. Winslow, A. M., 1967, "Numerical Solution of the Quasilinear Poisson Equation in a Nonunifo~ Triangle Mesh," J. Comp. Phys., Vol+ 2, pp. 149-172. Yan, Y. and Sloan, I. H., 1988, "On integral equations of the first kind with logarithmic kernels,"./. 1at. Equations and AppL, Vol+ I, No. 4, pp. 549-579. Yang, S. J+, 1981, Low-Noise Electric Motors, Clarendon Press, Oxford. Yang, S. J., 1988, "Effects of Voltage/Current Harmonics on Noise Emission from Induction Motors," pp. 457-468 in Vibrations and Audible Noise in Alternating Current Machines, Edited by R. Belmans, K+ J., Binns, W. Geysen and Vandenput, A+, NATO Advanced Science Institute Series, Kluwer Academic Publishers, Dordecht. Yang, S. J. and Timar, P+ L+, 1980, "The Effect of Harmonic Currents on the Noise of a Three-Phase Induction Motor," BEEE Trans. Power App. Sys., Vol+ 99, No. 1, 1980, pp. 307-310. Zaky, S+ G. and Robe~son, S. D+ T., 1973, "Integral equation formulation for the solution of magnetic field problems, Part 1: Dipole and current models," IEEE Trans. PowerApp. Syst., Vol+ PAS-92, No 2, pp.808-815+ Zhou, J. M+, Zhou, K. D, Shao, K. R+, 1992, "Automatic generation of 3D meshes for complicated solids", IEEE Trans. Magnetics, MAG-28(2), 1759-1762 Zhu, J. G. and Bertram, H. N+, 1988, "Micromagnetic Studies of Thin Metallic Films," J.Appl+Phys.63, Apr., pp.3248-3253+ Zienkiewicz, O. C., 197 I, The Finite Element Method in Engineering Science, London: McGraw-Hill. Zienkiewicz, O. C., 1977, The FiniW Element Method in Engineering Science, (3rd Edn), McGraw-Hill+ Zienkiewicz, O+C., 1977, "A New Look at the Newmark Houbolt and Other Time Stepping Schemes, A Weighted Residual Approach," ,Int. J. Earthquake Struct. Dynam., VoL 5, pp 413-418.


649

Zienkiewicz, O.C. and Lewis, R.H., I973, "An Analysis of various time stepping schemes for initial value problems," Int. 0I. Earthquake Eng. Struct. Dynam., Vol. 1, pp. 407-408. Zilstra~ H., 1987, 5th Int. Syrup. on Anisotropy and Coercivity in TE-TM alloys, Badsoden, Vol. 17. Zois, Dimitris, 1988, "Parallel Processing Techniques for FE Analysis: Stiffnesses, Loads and Stresses Evaluation," Computers a ~ Structures, Vol. 28, pp. 247-260.

Finite Elements, Electromagnetics and Design

Elements of Electromagnetics

Finite Element Method for Electromagnetics

Elements of Electromagnetics - Third Edition

Structural Analysis with Finite Elements

Damage Mechanics with Finite Elements

Structural Analysis with Finite Elements

Structural Analysis with Finite Elements

Elements of Finite Model Theory

Programming Finite Elements in Java

Mixed Finite Elements, Compatibility Conditions, and Applications

Machine Elements: Life and Design

Discontinuum Mechanics : Using Finite and Discrete Elements

Nonlinear Finite Elements for Continua and Structures

Nonlinear Finite Elements for Continua and Structures

Nonlinear Finite Elements for Continua and Structures

Structural analysis with finite elements

Design of Structural Elements

Elements of Compiler Design

Elements of Compiler Design

Electromagnetics

Electromagnetics

Electromagnetics

Electromagnetics

Electromagnetics

Electromagnetics

Elements of Electromagnetics (Oxford Series in Electrical and Computer Engineering)

Critical state soil mechanics via finite elements

Finite Elements for Nonlinear Continua & Structures

Finite Elements Part 1 (skeletel structures)

Finite Elements, Electromagnetics and Design

Elements of Electromagnetics

Finite Element Method for Electromagnetics

Elements of Electromagnetics - Third Edition

Structural Analysis with Finite Elements

Damage Mechanics with Finite Elements

Structural Analysis with Finite Elements

Structural Analysis with Finite Elements

Elements of Finite Model Theory

Programming Finite Elements in Java

Mixed Finite Elements, Compatibility Conditions, and Applications

Machine Elements: Life and Design

Discontinuum Mechanics : Using Finite and Discrete Elements

Nonlinear Finite Elements for Continua and Structures

Nonlinear Finite Elements for Continua and Structures

Nonlinear Finite Elements for Continua and Structures

Structural analysis with finite elements

Design of Structural Elements

Elements of Compiler Design

Elements of Compiler Design

Electromagnetics

Electromagnetics

Electromagnetics

Electromagnetics

Electromagnetics

Electromagnetics

Elements of Electromagnetics (Oxford Series in Electrical and Computer Engineering)

Critical state soil mechanics via finite elements

Finite Elements for Nonlinear Continua & Structures

Finite Elements Part 1 (skeletel structures)

Recommend Documents