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...

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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

Chapter I: Electromagnetic Field Computation

~= 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

Chapter I: Electromagnetic Field Computation

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_........

Chapter I: Electromagnetic Field Computation

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

Chapter I: Electromagnetic Field Computation

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
20

Chapter 1: Electromagnetic Field Computation

To understand these ideas more clearly, we will solve ~ n . 1.5.1b subject to the boundary condition of eqn. 1.5.18a analytically and then by the finite element method, first using the trial function of Fig. 15.5 and then by that of Fig. 1.5.6a and then compare the solutions to see whether the answers bear out the observations of this section. Although we may do this just the way we did the problem before, we will use a new approach to afford new insights into the algorithms. From eqn.I.5.14 aLi ~-¢j _ 1 1 a#i - ~-xi 2 (xi-xi) = (¢i-¢j)- ~ Li where L i = xj - x i the length of element i. Similarly aLi *i~ I

= xj-xi" 2 %-xi) =

1

(,j-,i)- ~ Li

F~-~I =

[.I

-

-

I

1

ka,jj The matrices -I- [ J l ' l ]

and 1

are known as the local matrices, Differentiation with respect to any other ¢ will give a zero. We will now use these results. Taking up the trial function of Fig. 1.5.6a first, for element I from node I to 2 we have, further noting that *l is not a variational parameter

F&i7 i/ -~j/~ / =

3 o ~o °o

'IoL, I Io ':]I°'Io[o o] ,2

o o

1

,~

I.-. 0~)4.-I

Similarly, for element 2 from node 2 to node 3

[
taL~/

|a*3|

t_~

3

=~

Ix 00 0

1-10

:¢ 2

-I I

*3 "

0~

and for element 3 from node 3 m node 4:

I

S. R. H. Hoole

21

~]

r|oL3/

o o o

[aL3|

0 0 -1

/~OB/_- 3 o o ,

~;3j Y ,

o

1 L04

1

Now I2)r satisfaction of the differential equation, since L = L I+L2+L 3

~L2"l DE3-1

|0L31 3 /~/=]aLt[ + /r/0L2! a,3/+/~l =~

la~21

L~ /k~

,,of

loLz/ t_a04 ~

"1 1 0 o o o o

|aL3|

0 0 0 0

*2 ¢3 *4

t_~_t

3 I° ,-, ilI°,]o2, ,o[j,;

+ 10 0o

"23

-I 1 0 0

° i;fLJl 3[-, 2., i]i l i-,o] ,o[!1

03 : - ~ T 04

o,

=

0 -I

10

*1

2

0

0-1

-

~

-

@3

-

=

312_, oliot =

¢4

-I

1

-1

2-I

0-11'4

2 -

0-1

~3-

+

i =0,

23L1 j

since ¢1 = 0. Solving, we have *I = 250N = 27.77', 03 = 400/'9 = 44.44 and 04 = 50. On the other hand, had we allowed *3 = 04, that is a forced Neumann condition, everything would remain the same except that we would differentiate only with respect to 03 and in element 3, using *3 = *4 in eqn, 1.5.14 1

(04-@3)2

L 3 - 2 x4~x3 so that 0L3 10

1

2 (*4+$3)(X4"X3) " = 13 . ¢3

~¢3-- 3 all other derivatives being zero. Thus 3~2]

I

~LJ

r3cl]

fOL21

FOL3]

= laaL~ +|~02|+]|I [~L=[ |,aL3 ] = La,3J L ~ J

La,3j

,,O[o ] +3[oo -1l - l ]1

23

~[o'

,3 [°1

@2 -

°] L03J 1~[I

']

22

Chapter !: Electromagnetic Field Computation

*1 0.00

Exact

*2 27.77

....

*4

* 3 .......... 44.44

50.00

44.44

44.44

44.44

50.~;9

.

II!ll

27.77

Forced Neumann Weak Neumann .

.

.

0.00

.

Table 1.5.1" Comparison of Weak and Strong Neumann Conditions

00]

+

3 [2-1 - 10 1 _

-

[;]

'

[*¢~1 3

]

E0

Solving we have *2 = 250/9 = 27.77' and 02 = *3 = 400/9 = 44.44'. To compare with the exact solution, integrating the differential equation once, we have d , / d x = -x + a. Applying the fact that the derivative is zero at x = 10, we obtain a = I0, Integrating again t h e n , = -(I/2) x 2 + 10x + b. A p p l y i n g , = 0 at x = 0, we get b = 0. That i s , = -(1/2) x 2 + !0x. Thus at x =(10/3) ,2 = - (100/!8) + (100/3) = (250/9) = 2 .77. At x = (20/3), *3 = (400/I8) + (200/3) = (400/9) 44.44'. At x = 10, *4 = - (t00/2) + 100 = 50. These results are summarized in Table 1.5.I. Evidently, the forced condition makes the boundary condition be satisfied exactly but in the last element the solution is far away from the exact. On the other hand, the weak scheme gives answers close to the exact (but not exact because it uses straight line interpolations); however the boundary condition is seen to be weakly satisfied since at x4 = 50.00-44.44 = 2 dx 3,33 and not zero as required. But it is zero compared to the graph in other parts of the domain.

1.5.4 First Order Triangular Finite Elements We have seen some of the properties of the finite element method in 1-dimension and shall try to extend their, to the more useful 2-dimensional situation. From what we have seen, we may say that the essential ingredients of the method are: a. Division of the Solution region into elements or subdomains: Extending what we saw in 1-dimension, we will use the triangle as the basic element and divide the domain of solution into little triangles as shown in Fig. 1.5.7 for a two-conductor system enclosed by a far away boundary. Curved boundaries may

S. R. H. Hoole

23

be approximated by little straight lines to facilitate the subdivision into triangles. Observe that there is no requirement of a u n i f o ~ mesh. b. Postulation of a trial function with fi'ee parameters: Extending eqn. 1.5.7 we will postulate a linear variation over each triangle as given by , = a + bx + cy (1.5.22) The smaller the triangles the more accurate is the assumption of linear variation. (This is easily seen in one dimension where we can approximate a curve using short straight lines; the shorter the line lengths the more closely is the curve approached.) The only limit on accuracy in this regard is the time and data storage capacity required to calculate on a large number of nodes. Thus, wherever the field varies rapidly we will have a finer subdivision to model the changes better with straight lines. c. Identification of an Optimality Criterion to specify the free parameters of the trial function: Extending eqn. 1,5.5b to two dimensions we assume the functional

4

L[,] = ~ **J

= f

e [V,]2dxdy -

p~xdy

v

.t

~L!kOx)

(3x)

ldxdy-..[" f p0dxdy

(1.5.23)

d. Solution of a set of linear equations relating the free parameters: This follows in just the same way as we did in 1-dimension. As we did in 1-dimension, it is better to write the free parameters a, b and c of the triangle in terms of the vertex potentials so that the continuity of the potential may be maintained automatically from triangle to triangle by using the potentials at shared vertices. To this end we write the expression for the potential in a triangle in terms of the potentials at the three vertices of the triangle. At these vertices

[ ~'--~'~'\|

Lines Far a w a y Figure 1.5.7: Triangular Finite Element Mesh

24

Chapter i: Electromagnetic Field Computation

eqn. (I.5.22) gives *1 = a + bx 1 + cy| *2 = a + bx 2 + cy2 *3 = a + bx 3 + cy3 Solving ~ar a, b and c in terms of the nodal potentials: I a = ~ [¢I (x2Y3-x3Y2) + ¢2(x3y l-xl Y3) + ,3(x 1y2-x2y 1)]

(1.5.24) (1.5.25) (1.5.26)

= a l , i + a2¢2 + a3*3 b=

(1.5.27)

[*I(Y2-Y3)+ *2(Y3-Yl) + *3(Yl-Y2)]

bl¢ 1 + b2, 2 + b3¢ 3 I c = ~ [,l(X3-X2) + ,2(xl-Y3) + ,3(x2-xl)]

(t.5.28)

=

= c1¢ 1 + c2, 2 + c3, 3 (I.5.29) I xl yl 1 1 x2 Y2 (1.5.30) A=~ lx3Y3 A is the area of the triangle provided the nodes of the triangle are numbered anticlockwise. The proof may be found in many textbooks on coordinate geometry. The new variables bl, b2, b3, Cl, c2, c3 have been implicitly defined above in eqns. 1.5.28 and 1.5.29, and for computational convenience may be cast in the form: bi =

(Yil-Yi2)

(1.5.3 l)

{Outputs: b,c: The lx3 first order matrices defined in eqns. (1.5.30) to (I .5.32). A = Area of triangle Inputs: x,y = 3x 1 vectors containing the coordinates of the 3 vertices } Begin Delta ~ x[2]*y[31 - x[3]*y[2] + x[3]*y[1] - x[I]*y[3] + x[ll*y[2] -x[2]*y[l] A ~--Abs(Delta)/2 {eqn. (!.5.30)} Fori~ 1 To3Do il~Mod(i,3)+l {OriMod3+l} i2 ~- Mod(i 1,3) + 1 b[i] ~ (y[il]-y[i2])melm c[i] ~ (x[i2]-x[il])/Delta End

Algorithm 1.5.1" Computing the First Order Matrices b and c ..........

...........

: ......::~

......

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

,

,

,,,~

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

,

. . . . . . . . . .

S. R. H. Hoole

1

Ci = ~

25

(xi2-Xil)

(1.5.32)

where when i is l, i l and i2 are 2 and 3; when i is 2, i l and i2 are 3 and 1; and when i is 3, i l and i2 are 1 and 2. This mapping is available under the modulo 3 function on computers. Note that substituting these values of a, b, c in eqn. 1.5.22 gives us ~, within the triangle, in terms of its vertex values. For a triangle of known vertex coordinates, AIg. 1.5.1 in pseudo-code gives a procedure Triangle that forms the coefficients bl, b2 ..... c3 and the area. At this point, before proceeding further, it is useful to ask what these constants a, b, and c mean. We will see, in the next few lines of text, that they have inherent to them concepts of interpolation and differentiation. Substituting eqns. 1.5.27-29 in eqn. 1.5.22, and rearranging by collecting the vertex potentials together, we will have: 0 = a + bx + cy = alO1 + a202 + a3,3 + x(blo1 + b2~2 + b3~3) + y(cl¢~ + c2¢2 + c3¢3) = (al + blX + c|y)0t + (a2 + b2x + c2y)~2 + (a3 + b3x + c3y),3 = Nt~I + N2qb2 + N3~3 (1.5,33) Thus we see that the functions N i , i = 1, 2, and 3, defined by Ni = ai + bi x + ci y (1.5.34), actually interpolate the vertex values of the potential to the interior. Since we used first order interpolation, naturally, the inte~olation function in eqn. 1,5.34 also turns out to be a first order function in x and y. In fact it can be shown rather trivially that these inte~olation functions are the triangular coordinates depicted in Fig. 1.5.8. These triangular coordinates are also referred to as homogenous

A-I

h

~1=

H1

(x,y) h B-

2

Cartesian Coordinates

1

(0,0,1) C- 3 x "~ Triangular Coordinates

(~I' ~2' ~3)

(x,y) Figure1.5.8: Triangular Coordinates and Interpolation

26

Chapter I: Electromagnetic Field Computation

coordinates. As shown in Fig. 1.5.8, each coordinate of a point (x,y) is defined by L

(I.5.35), where hi is the altitude of the point with respect to the side of the triangle that is opposite vertex i and Hi is the altitude of the vertex i itself. Since we cannot get 3 independent coordinates ;i from 2 coordinates x and y, there must be a dependence relationship between them. This is shown by summing the areas of the three triangles formed by each of the three bases Bi, Bi being the side opposite vertex i, and the point (x,y): 1

A=~Blhl+

1

B2h2+~B3h3

_1 1 - 2I B 1 H I ~ I + 2 B2H2;2+ 2 B3H3;3 = A;1 + A;2 + A~3 (I.5.36), giving us the de~ndency result ~1 + ~2 + 43 = I (1.5.37). Further, the three vertices of the triangle have triangular coordinates (I,0,0), (0,!,0), and (0,0,1), showing that it fits eqn. 1.5.33 exactly at the three vertices. Applying the first order Cartesian coordinates to this interpolation: Xl;1 + x2;2 + x3;3 = x (1.5.38) Yl;I + Y2;2 + Y3;3 =Y (1,5.39). Now solving eqns. 1.5.37-39 for the triangular coordinates, we get ;i - ai + bi x + ciy = Ni (1.5.40), showing us that the interpolations N are the same as the triangular c~rdinates 4. That the b's and c's relate to the differentiation operation is seen by simply differentiating eqn. 1,5.22, and using eqns. 1.5.28-29:

ey

= b - bl• I + b2•2 + b3,3

(1.5.41)

= c = c i , 1 + c202 + c3,3

(1.5.42).

Proceeding now with the finite element formulation, using what we have learnt in I-dimension, we now wish to express L, in each triangle, in terms of the vertex potentials. For this we must treat each term differently. Within a single triangle we have, using eqns. 1.5.41-42: = Uxb + u y c (1.5.43) V0 = Ux 0~ + U y . Ox oy a constant vector, so that eV¢-V, dS = ~ eA(b2+c 2)

(1.5.44)

With " i n a single triangle the second term of the functional, ~J~o* dS, is the integration of a linear function over the triangular area. For convenience we assume that 0 is a constant within each triangle. This may be an exact representation of reality or an approximation that becomes more and more accurate with triangles of

S. R. H. Hoole

27

decreasing area. For the 1-dimensional problem it was the area between the x axis and the curve. Because we assumed a straight line variation, the area was a trapezium with area given by the average height times base. Here, correspondingly, the integral is the volume ~ t w e e n the triangle on the xy-pane and the curve of dO plotted on top of the plane. The evaluation of this integral yields the area of the triangle (the base) times the average height (q~I+,2+003)/3, which is the potential at the centroid of the triangle: l" I" p ~ d S = ~(0t+~2+o03)3

(1.5.45)

Substituting eqns. 1.5.44 and 1.5.45 in eqn. 1.5.23, and using eqns. 1.5.27-29 for a, b and c, we obtain, for the single triangle, . . 2 + 2- 2 -b2+c 2- 2 LI = [02 CzJdO2 + (- 3 3)dO3+ 2(blbz+clc2)dOldO2

+ 2(bzb3~2c3)dO2dO3 + 2(blb3+ClC3)dO ldO3] - PA(dOI+dO2+dO3)(1.5.46) This energy may alternatively be expressed in matrix f o ~ as 1 LI = ~ {0}t [p]{~}. {~}t q

(1.5.47)

where {o0}t= {O01 02 0O3}

(1.5.48)

PrOcedU're'"'"'Firs'torderLocaiMat's (Piqlxly' Eps'Rho) {Function: To compute the first order differentiation matrices Outputs: P = The 3x3 first order Local Element Matrix - eqn. (1.5.49) q = The 3x 1 local right hand side vector - eqn. (1.5.50) Inputs: x,y = 3xl vectors containing the coordinates of the 3 vertices Eps = The constant material value E in the triangle Rho = A constant giving source p in the triangle Required = Procedure Triangle

} Begin Triangle(b, c, A, x, y) Fori~-- 1 T o 3 D o iI ~Mod(i,3)+ 1 {OriMod3+I} i2 ~ Mod(il,3) + I q[i] ~-- A*RJho/3 Forj~ 1To3Do P[i,j] ~- Eps*A*(b[i]*b0]~[i]*c[j] End

{eqn. (1.5.49)}

Algorithm 1.5.2: Forming Local Matrices for First Order Triangle ii

iii ii

iiii iiiii Ill

llil{l!

llll II HilIIIIIJ

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

28

Chapter l: Electromagnetic Field Computation

[WHII[IHII

/I

HlJJllll

I!!

I

III

........

I}// ............

IIIIINIIIII

IIIIIIIIIIHHH

!!1

II

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

I ........

Procedure GlohalPlace(PG, qG,P, q, v, Phi) {Function Adds the local matrices to the global matrices Outputs PG = The global matrix of size NUnk x NUnk qG = ~ e global fight hand side qG of size NUnk x I. Inpu~ P = 3x3 local matrix P q = 3x 1 local right hand side vector NUnk = Number of Unknown Nodes in mesh numbered before the known nodes v = A 3-vector containing the node numbers of the triangle Phi = A vector as long as there are finite nodes. The first NUnk elements are unknown and to be determined by the finite element method and the rest of the elements are known through the boundary conditions,

} Begin For R o w ~ 1 T o 3 D o If v[Row]_< NUnk Then {Else Row Corresponds to a Known; Ignore it} qG[v[Row]] ~-- qG[v[Row]]+q[Row] For C o l ~ I T o 3 D o If v[Col]_
II1! III

II

!

IIII

r22 bl+C i

P=eA

J(

III

[J[

JJJ

[1[

I IIII I!1[

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

blb2+clc2

b2bl+C2Cl b3bl+c3cl

b2+c 2

![J

J! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

blb3+ClC3

]

b2b3+c2c3 2 2 b3b2+c3c2 b 3 + c 3 J

(1.5.49)

and {q}t = A913

1 7]

(I.5.50)

The matrix [P] and and vector {q} are referred to as local matrices and their numerical formation is given in pseudo-code in Alg. 1..5.2. From this point onwards, the reader is urged to keep in mind the techniques used in the example we

S. R~ H. Hoole

29

worked in 1-dimension. Now we come to the key point in finite element analysis. We have the functional expressed as a second order polynomial in the unknown nodal potentials. The functional is the sum of the contributions of the type given in eqn. 1.5.46, from all the triangles. Since the functional must be a minimum, the nodal potentials must be such as to give it its lowest value. Performing the requisite differentiations with r e s ~ c t to ¢ 1 for example ~L 2 2 1 0~ 1--' = EA eA [(b2+b I )*I +(b 1b2 +c t c2)¢2+(b l b3+c I c3)~3 ]- ~ 9A (1.5.51 ) which we recognize as corresponding to the first row of the matrix equation zz~ [PI{¢} - {q} = 0 (1.5.52) where the matrix [P} and the vector {q } are defined in eqns. 1.5.49 and 1.5.50. !1 . . . . . .

iii

i!

!l!t/lllll

i

i

ill

/11 i!!llt . . . . . . . . . . . . . . .

P r o c e d u r e Gauss(x,A,B,n) {Function - Solves the nxn matrix equation Ax=B by Gaussian Elimination Output: x = The solution, an n-vector Inputs A = A positive definite nxn matrix B = The right hand side column vector of the equation n = The size of the matrix

l Begin {Make the Matrix Upper Triangular} F o r i ~- I To n-1 Do {Begin - Scale Row i} F o r j ~ i+l To n Do {Scale row i to make its diagonal 1.0} a[i,j] ~ A[i,j]/A[i,i] B[i] ~-- B[i]/A[i,i] A[i,i] ~- 1.0 F o r j ~ i+l To n Do {Row j -A~,i] x row i (j>i) to upper A} B~] ~ B[j]- A[j,il*B[i] Fork~ i+l T o n D o A~,k] ~-- A[j,k] - A~,i]*A[i,kl Afj,il ~ 0.0 {End - Making Column i below row i zero} x[n] ~- B[n]/A[n,n] {Begin - Back-Substitution } Fori~ 1 Ton-I Do j~n-i

xW ~B[j] F o r k +-- j + l T o n D o

xU] ~ xW-A[j,kl*x[k] End

Algorithm 5.3.4: Gaussian Elimination As we perform the summation over the triangles of eqn. 1.5.52, the vector {¢} will in turn contain the 3 vertex potentials of the triangle whose contribution is being added. The rules for the summation of the energy contributions from the various triangles are easily discerned by adding the polynomials from 2 adjacent

30

Chapter I: Electromagnetic Field Computation

LI = {,}I t [P]{~}I -{*}I t {q}

(1,5.53)

elements with vertices 1, 2, 3 and 3, 2, 4 as shown in Fig. 1.5.9. If the first gives a contribution and the second gives us L1 = {,}2 t [M]{,}2- {,12 t {n} (1.5.54) We recognize that {~}I is {~I ~2 ~3} and {~}2 is {~3 ~2 ~4}. However, in each triangle the vertices map internally with order; i.e., they are numbered internally, 1, 2, 3, in a countercl~kwise direction. We may now expand L1 and L2 into the scalar polynomials they are as in eqn. 1.5.46, work out the summation and differentiate with respect to the 4 potentials. Alternatively. we may by examining the methods ex~unded through the example done in 1-dimension, glean the rules for adding the contributions to L from the two triangles. Either way, we will find that differentiation of LI+L 2 with respect to the 4 potentials gives us the 4 linear equations Pll P2I P31 1

f

P1 2 P22+M22 P32+M 1M3 2 12 0

PI 3 P23+M21 P33+M M3 1

1

0 )[~I l M 2 3 ~2 M3 MI ~4¢'3

I

ql ] q2+n2

(I.5.55)

= Lq3~nlj

3

2 4 Figure 1.5.9: Adjacent Triangles We thus define the global matrix as that matrix of the same size as the number of vertices and in which the local matrices are placed. By the rules of eqn. 1.5.55 the local matrices may be placed in the global matrix as given in Alg. 1.5.3. However, since some of the boundary potentials will be known, no variations will be made with respect to them so that rows co~esponding to them do not appear in the global matrix We thus conveniently number the unknown nodes first, up to NUnk and the knowns thereafter up to NNodes. The columns corresponding to the known nodes are multiplied with the known value of ¢ and shifted to the fight hand side as shown in Alg. 1.5.3 and demons~ated in the example worked earlier. Neumann boundary conditions (specifying the normal gradient of potential to be a zero, curiously turn out naturally in finite elements as we have seen. Therefore we do not bother about them. That is, any boundary on which the potential or its gradient is not set, will automatically have a zero normal gradient in the solution. For this reason, the Neumann condition is called natural to the finite element formulation. Thus, with these 4 algorithms, 1.5.1-4, in pseudocode, one may easily write one's own programs and solve complex problems.

S. R. H. Hoole

31

Sheath 9 = 3.0 C / s q . i n

~¢=

1.0

i

f

14 c.m.

9 = 0.0

e r = 1.0 8 c.m

Figure 1.5.10: A Sheathed Cable The data for a finite element field problem are contained in a file. To explain a convenient f o ~ a t for the data file, we will use the test problem of Fig. 1.5.10. {This is the main finite element algorithm' It relies on Alg. 1.511' Alg' i'5121 Alg. 1.5.3 and Alg. t.5.4}

Begin Readln(File l ,NUnk,NNodes,NTria) PG ~ 0 QG+-0 For i ~ I To NUnk Do Readln(Filel ,Dummy,x[i],y[i]); For i ~ NUnk+l To NNodes Do Readln(File 1,Dummy,x [i],y [i],Phi [i ]); For Tria +- I To NTria Do Readln(v [ I ],v [2],v[3],Eps,Rho); Fori~ I To3Do xv[i] ~- x[v[il] yv[i] ~ y[v[il] LocalMats(P,q,xv,yv,Eps,Rho) GlobalPlace(PG,qG,P,q,v,Phi) Gauss(Phi,PG,qG,NUnk) For i ~- I To NNodes Do Writeln(FiIe2x[i],y[i],Phi[i]); End {Program } Algorithm 1.5.5: The Main Finite Element .........................................

Rnllt[nP,

32

Chapter I: Electromagnetic Field Computation

Besides, in writing our own finite element software, it is useful to have a test problem - which also this problem will serves - so that we may detect any possible errors that may creep into our program. In this test problem we are trying to model the electrostatic behavior of a sheathed cable. By symmetry, it suffices to model a quarter of the system as shown in Fig. 1.5.11, with zero potential on the original grounded boundac,', and Neumann conditions on the two lines of symmetry. Thus, in Fig. 1.5.I 1, the 4 unknown nodes are numered first, and the 5 nodes on the grounded sheath are numbered last.. Fig. 1.5.12 gives the corresponding data. The date file of Fig. 1.5.12 consists of, on the first row, three integers, these being the number of unknown nodes 4, the total number of nodes 9, and the number of triangles 8. These numbers allow us to read easily the rest.of the file The next block of data gives the node data, characterized by the x and y coordinates and the solution if the node is known The last block of data consists of triangle i n f b ~ a t i o n - a series of triangles, each defined by 3 vertices and the values of and P in it. The finite element program, described in Alg.l.5.5, after initializing the global matrix, takes as input the typical data file of Fig. 1.5.12. It reads the number of unknowns NUnk, the Number of nodes NNodes and the number of triangles NTria and then uses these to read into memory the coordinates of all nodes and the known potentials. Thereafter, it reads every triangle and then, from the node numbers, reads the coordinates of the the 3 vertices now in m e m o ~ and forms the corresponding local matrix using Alg. 1.5.2. Alg. 1.5.3 is then pressed into service to place the local matrix in the global matrix before reading the next triangle. Once the global matrix is assembled, the Gaussian elimination solver described in Alg. 1.5.4 is used to find the potentials. The solution is written onto an output file so that standard plotting routines may be called to plot equipotential lines within the device once the vermx solutions are identified and written on file along with the coordinates. When worked out by hand, it will be seen that the local matrices for the first element of Fig. 1.5.10, element 1, 2, 3 are P=|-0.25 I_ 0.0

1.25-1 -1.0 l.OJ

q t = [ l . 0 1.0 1.01 The final global matrix equation is

(1.5.56)

(1.5.57)

o o

-1.0 0 . 0 -0.5 ¢4 I If the program written by the reader is working co~ectly, then the final assembled equation should correspond to the above.

S. R. H. Hoole

33

N9

N8 T6

N4

N7

T8 T7

T5

T4 T3

v N1

X

4 2 Como

N2

N5

N = N o d e s ; T - Triangles; 4 Unknowns, 9 Nodes, 8 Triangles. Figure 1.5.11" Finite Element Mesh for Symmeteric Quarter of Figure 1.5.10 4 ! 2 3 4 5 6 7 8 9 1 I 2 3 4 9 3 3

9 0.0 2.0 2.0 0.0 4.0 4.0 4.0 2.0 0.0 2 3 5 5 3 3 6 7

8 0.0 0.0 !.0 1,0 0.0 1.0 2.0 2.0 2.0 3 4 3 6 9 8 7 8

0.0 0.0 0.0 0,0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

3.0 3.0 0.0 0,0 0.0 0,0 0.0 0.0

Fig. 1.5.12 Finite Element Data File for the Problem of Fig. 1.5.11

34

Chapter 1: Electromagnetic Field Computation

1.6 The Galerkin Scheme What we have seen so far is the finite element scheme presented in its classical variational formulation. What is attractive about the variational formulation - our optimization criterion - is that the functional relates to the physical concept of energy. However, the weakness of the approach is that for every equation we solve, we need to identify a functional that, at its minimum, satisfies the differential equation being solved. We always have, for the difI2erential equation with operator A oFerating on the unknown f A(f) = 0 (1.6.I) over the domain R, (he variational functional L(~= ~ a2dR (1.6.2). J But this is not always a convenient one to adopt. While it would yield symmetric matrices, for the Poisson equation with the t e ~ V2G it requires at least a second order polynomial trial function for the unknown f to ensure that the second derivative of the functional shall not vanish. This difficulty of identifying a convenient functional is overcome by the Galer~n approach to finite elements, d Here we shall encounter a brief exposition of the Galerkin scheme, and in the process will choose to abandon our customary o~isance to mathematical precision so as not to clutter the discussion with all too many distracting concepts. The theory makes recourse to function spaces and their bases. This idea has its parallels to vector spaces and bases. For instance, three dimensional vectors have as base the set of three unit vectors u x, Uy and u z, and any 3-dimensional vector may be created from these by first scaling them and then summing them: XUx + yuy + ZUz. Likewise in the function space not so familiar to engineers, there exist basis functions from which all functions may be generated. For instance, the space of third order polynomials in x may be generated from the basis set I, x, x2 and x3: a + bx + cx 2 +dx 3. How does all this relate to Galer~n's theory? To see this, for the operator eqn. 1.6.1, let us assume that the approximate solution lies in a function space with basis vectors N t, N 2 ..... Nn. For example, within a first order triangle, these could be the three functions of linear interpolation ~1,~2, and 43, in which case the dimension n of the basis set would be 3. Clearly, by choosing a solution from this trial space, we could be making an approximation - for the actual solution need not lie in this space. Taking a general function in this space f* = alN1 + a2N2 + ...+ anNn (1.6.3), and substituting it in the operator equation, eqn. 1.6.1: A(f*) ~: 0 (1.6.4). Indeed, if the actual solution is not in our trial space, A(f*) can never be zero as required.But what the Galerkin method allows us to do is choose the numbers ai of eqn. 1.6.3, so as m make A(f*) as small as possible. With the vectors that we are

S. R. H. Hoole

35

so familiar with, we would say that a vector is zero if the dot (or scalar)product with every basis vector is zero. That is, we might say fUx'(XUx + y u y + ZUz) = x = 0 XUx+ y U y + Z U z = 0 i f ~ u y ' ( X U x + y U y + ZUz) y 0 (I.6.5). kUz(XUx. + yUy + ZUz) z = 0 We have a similar scalar product operation with function spaces called the inner product. For any two fianctions f and g, it receives the notation , and is usually and conveniently defines as: = {" l~" fg dR

(1.6.6).

Thus the basic Galerkin approximation for minimizing the deviation from 0 in eqn. 1.6.4 is, for every i: = [ { N i A(f*) dR = 0 (1.6.7). J .7 Thus for the n different values of i, we will have n equations that would enable us to identify the n unknowns a i of eqn. 1.6.3~ Without stopping here, the Galerkin method goes even further and states that if we were solving two operator equations, eqn. 1.6.1 and B(f) = 0 (1.6.8), both of which need to be satisfied simultaneously, then we need to solve the n equations: + k

= f .,l~r,~iA(f*) dR + k ]" f ~qiB(f*)dR = 0

(1.6.9).

where the number k is a scaling factor that may be used to ensure that the resulting numbers from the two operators are of similar oder so that one equation is not emphasized at the expense of the othjer. Let us now move on to inte~reting these. At the same time we will also see how this method can yield the same answers as the Variational method. To this end, therefore we will take up for solution the Poisson equation recast in the lbrm of eqn. 1.6.1 - V.eV~ - 9 = 0 (I .3.26), subject to the boundary condition 3~ = 0 on S 1 (I.6.10) On where S I is part of the boundary S of the domain R in which we are solving. We will take our trial function f* approximating ~ to be first order linear inte~olafions over triangles of a finite element mesh of R: f* = N l f l + N2f2 + N3f3 (l.5.33), where, we have already seen in the previous section, that the functions N are the same as the triangular corrdinates ~ and the subscripted f's are the values of f* at

36

Chapter t: Electromagnetic Field Computation

the triangle vertices, Applying eqn. 1.6.9 over every triangle and summing over the domain then, for every i: [Ni(-V.cVf*-p)] dR, +

Ni~]dS dn AI

=0

(1.6.1 l),

D

Sl where we have used e to scale the second equation so that coefficients associated with f are of the same size. Now from the identity V,(fA) = ~ - A + A.Vf (AI0) with A = cVf* and f = N i, we have Ni(V.eVf* = V.Ni eVf* - cVf*.VN i (1.6.12). Integrating this identity over R and manipulating using Gauss' theorem

jc jCA.dS=I l"f(V.A)dR

(112)

with 1 = N i EVf*, we have

Nod:

"" t V f * d S is the same as OffOn dS, and using eqn. 1.6.13 in eqn. 1.6.11 : eN i ~

-

D

dS +

S -

eVf*.VN i dR S

Nip dR. +

[ Ni~

<_IS= 0

(1.6.1 1),

SI It is seen that this is the same equation as we had with the variational formulation.

Chapter 2 . . . . . . . . . .

Ill

II/I

Illl}

T.Nakata,

II

Ill[

J/[/

N.Takahashi and K.Fujiwara

Ill{ / I}[{[ . . . . . . . . . . . .

I

I

}

{

II

II[

I

}

SOLUTION OF 3-D EDDY CURRENT PROBLEMS BY FINITE ELEMENTS

2.1 Introduction Meth~s of 3-dimensional eddy current analysis are described here. In 3-dimensional eddy current analysis, very long CPU times are necessary because the number of unknown variables usually becomes extremely large compared with 2-dimensional analysis. Many attempts have been made to reduce the number of unknown variables associated with 3-dimensional analysis. For example, one such attempt has been by defining the unknown variables corresponding m the eddy current only in the conducting region (Nakata, Takahashi, Fujiwara and GS:ada 1988). As such, there are now several kinds of 3-dimensional finite element methods in terms of unknown potentials (Nakata, Takahashi, F@wara, Muramatsu and Cheng 1988) and the kind of finite elements used (Nakata, Takahashi, Fujiwara and Shiraki 1990b). The purpose of this chapter therefore is to describe the various methods that are in use and their relative merit. Although the focus is on eddy current analysis, the discussions apply, where relevant, m static fields as well as to those described by the wave equation. First, we take up the formulation of the A-~ method (Chari, Konrad, Palmo and D'Angelo 1982). Here A is the magnetic vector potential, and the electric scalar potential, as defined in section 1.3. To assist the description of the method in the finite element context, first-order tetrahedral elements are used (Zienkiewicz, 1977; Hoole, I989; Silvester and Ferrari, 1990; and Nakata and Takahashi, 1988). Thereafter, generalizing the elements, we move on to explain the finite element method using various kinds of potentials and finite element shapes. Finally, the features of each method and element are investigated, and the accuracy, and the computer storage and the CPU time associated with each method and the type of finite element used, are compared using a simple model.

38

Chapter 2: Solution of 3-D Eddy Current Problems

2.2 Linear Finite Element Analysis (The A-¢ Method) 2.2.1: The Basic Equations Here we wish to take up the finite element method far analyzing 3-dimensional linear magnetic fields with eddy currents using the variables A-¢ as descriptors. Recalling from the previous chapter, the solution procedure involves the following steps: The identification of the equation to be solved. i. ii. Discretization of the domain into finite elements, over each of which a trial function - - that is a specific function for the variation of the unknown with respect to the coordinates - - is assumed. This we saw was most conveniently done using inte~olations. iii. Using some optimality criterion such as Galerkin's method or functional minimization to determine the unknowns. In this section we will see these three steps in action for the solution of 3dimensional eddy-current problems. Turning now to the first step, that of identifying the governing equation, the basic equation for solution is Vx

VxA=J+~=~E+~=

c+ ~

- ~-

V¢

(1.3.13).

Since we are dealing with eddy current analysis, low power frequencies are assumed so that c~>> j ~ , and the wave t e ~ - - the second derivative term with respect to time - - is negligible. We also saw in the previous chapter how the current density J may be broken up into an externally imposed part and an induced part. In eddy current analysis, it is convenient to break this up into a part Jo that is fi-om the magnetization of the magnetic materials present and a part Je that represents the eddy currents. Thus the representative equation becomes (Hoole, I989; and Nakam and Takahashi, I988): VxvVxA = Jo + Je (2.2. I) where v, the reluctivity, is the inverse of the permeability ~. Je in turn can be expressed ~ : Je = - c

+ V ¢1

(2.2.2)

It is assumed that the reluctivity v is expressed by the tensor of which the nondiagonal components axe zero (Nakata and Takahashi, 1988; see also chapter 6 for models of materials): v =

Vy 0v

(2.2.3)

where vx, Vy and vz are the x-, y- and z-components, respectively, of v. Here, not only are we assuming linear values for the reluctivities - - that is values that are independent of the magnetic field intensity H ~ but also that the coordinate axes are along the principal orientations of the magnetic material. These assumptions are not generally valid. But we, for pu~oses of exposition, wish to take up only

T. Nakata, N, Takahashi and K. Fujiwara

39

one difficulty at a time. Subsequently in chapter 6, more general modeling of magnetic materials will be considered. Eqn. 2.2.1 can be rewritten using eqn. 2.2.2: V x v xVA = Jo - ~

+ V¢

(2.2.4)

Although the number of unknown variables is four (¢ and the three components of A), the number of independent scalar equations of eqn. 2.2A is three. This is because an additional restriction on the vector potential is imposed via the continuity restriction on current: V.J = 0 (1,3,19), which when imposed on the eddy current translates into: V'Je=V" {-~

[

+V0]

=0

(2.2.5)

Thus if we can solve for A a n d , from eqns. (2.2.4) and (2.2.5), all other quantities such as the electric and magnetic fields, may be computed from them.

2.2.2: The First Order Tetrahedral Element We have identified the equations that need to be solved. Now, if we recall the finite element procedure from the previous chapter, we need to discretize the domain into elements and over each element assume a trial function.Preferably this should be an interpolatory function. Recall also that in 2-dimensions, we t~und it the most natural to take the triangle as the basic element, and assumed a variation of the form: , = a + bx + cy (1.5.22),

z

le(xie,

Y ~ e , Z zo)

o!i.............. X/

.............. y ~6

2e( x 2o. 5"2o.

|0

Z3o, Y3o, Z3e) IA_4e 4e(x4,. Y4,. Z4.)

Figure 2.2.1: The Tetrahedral Element

40

Chapter 2: Solution of 3-D Eddy Current Problems

which we then cast in the interpolatory form using the vertex values for interpolation: ¢ = ~I¢1 + 42¢2 + 43¢3 (2.2.6) Figure 2.2.1 shows the first order tetrahedral element, the extnsion of the triangle in 3-dimensions. The node numbers of the element e are denoted by 1e, 2e, 3e and 4 e. The subscript e denotes that the node number is defined only in the element. The unknown variables are defined on these nodes. Such an element is called a "nodal element" (When the component of the unknown vector is defined on the edge of the element as explained in section 2.5.2, the element is called an "edge element") The nodal element is discussed here. The x-, y- and z-components A x, Ay, and A z of the vector potential in the first-order tetrahedral element are approximated by the linear equation of the coordinates x, y and z as follows: (e) c,(e) 0¢(3) ~10 ~-20 a

{A(e) A(e) A (e)}= { l x y z y z

}

-

11 O:(e ' ~(#2) 0~(;:~ 12 (e) ~(2e3) (e) c~13

(2.2.7)

~33-

where ~(e) is the coefficient associated with the interpolations. The superscript (e) means that the value is defined in the element e. By substituting three vertex potentials and coordinates into (2.2.7) in turn, the tVollowing simultaneous equations can be obtained: Axle Ayle A z l e l Ax2e Ay2e Az2e 1 Ax3e Ay3e Az3e[ Ax4e Ay4e Az4e -.l ,J

I I _ -

1 1 1 1

Xle X2e X3e X4e

Yle Y2e Y3e Y4e

-- (e) o~(e) c~(e)-O~i0 20 30

Zle 1 Z2e/ Z3e/ Z4e~

(2.2.8)

~(1~ Ot(e)22~(;~ (e) ~(e) ~(e) -- c( 13 23 33Let us simply represent the inverse of the coordinate matrix by -1 f i xte Yle z I ~ l F b l e b2e b3e b 4 e l X2e Y2e z2 1 | C t e C2e C3e C 4 e | d2e d3e d4e/ X3e Y3e Z3e 6 X4e Y4e z4 L e l e e2e e3e e 4 e J where bie, Cie, die and eie are given by the following equations -

v(e) Idle

(2.2.9)

T. N~ata, N. Takahashi and K. Fujiwara

41

bie = (-l) i {Xje (YmeZke - YkeZme) + Xke(YjeZme- YmeZje) + Xme(YkeZje - YjeZke) } Cie =(-1) i {~e(Zke- Zme) + Yke(Zme- Zje)+ Yme(Zje- Zke)} (2.2.10) d£ =(-1) i { ~e(Xke-Xme ) + Zke(Xme-Xje ) + Zme(Xje-Xke ) } eie =(" I) i {~e(Yke "Yme) + Xke(Yme-Yje) + Xme(Yje" Yke)} with the other coefficients obtained by a cyclic permutation of subscripts in the order i, j, k, m. V(e) is the volume of the element e which is given by v(e) = 61 (x te c le + X2eC2e+ X3eC3e+ X4eC4e) By multiplying (2.2.8) by (2.2.9) in reverse order, we have (e) ~(e) RI0 20 R (3e{~ ble b2e b3e b4e~(e) (e) c (3e~ IlU2I 1 [ C l e C2e C3e C4e ~(e) r,(e)o~(;e~ - 6v(e) t dled2e d3e d4e 12 "~'22 Lelee2e e3e e4e-

F

_

t~-23

(2.2.tl)

Axle Ax2e Ax3e Ax4e

I

Ayle Azte 7 Ay2e Az2el Ay3e Az3e/ Ay4e Az4e-a

(2.2.12) The potential at an arbitrary point (x, y, z) in the element e can be calculated by the following equation, which are derived from (2.2.7) and (2.2.12) { A(e)a (' a(ez) = [ b l e b2e b3e b4el Faxle ayle Azleq 1 [Cle C2e C3e C4el lAx2 e Ay2e Az2el 6v(e ) { 1 x y z } | d i e dye ,. d3e d4el lAx3 e Ay3e az3e | t-ele e2e e3e e4eJ LAx4e Ay4e Az4e~ Using the interpolation function Nke, (2.2.13) can be rewritten as 4 A(e) x = £ Nke Axke k=l 4 A( = Z Nke AYke k=t

(2.2,13)

(2.2.14)

42

Chapter 2: Solution of 3-D Eddy Current Problems

4

A(ez) = Z Nke Azke k=t where Nke is given by (2.2.15)

1 ) ( b k e + CkeX + dkeY + ekeZ ) Nke - 6v(e

2.2.3: Discretization by the Galerkin Method (a) Time Stepping The equation of the finite element methed is derived by using the Galerkin method (Hoole 1989, Silvester and Ferrari 1990). The eddy current term (the time derivative term) is discretized by the backward finite difference formula (Zienl,dewicz 1977), so that the transient phenomena can ~ analyzed. Now by applying the Galerkin method of (1.6.7) to (2.2.4), the following residual equation can ~ obtained

+

Ni • ~ ~dV

+

Niec V¢ dV

(2.2.I6)

where V denotes the volume of the analyzed region. The 3-D interpolation function is defined by N i = uxNi + uyNi + uzNi (2.2.17) V-(AxB)= B.VxA-AoVxB (2.2.18) By using the vector formula of (2.2.18) and Gauss's law, the residual Goi of the first t e ~ in (2.2.16) can be rewritten as shown in (2.2.19). Goi =

v VxAeVxNidV-

Nio(v

V xA xn)dS

(2.2.19) where n is the unit vector normal to the surface S of the region V. If it is assumed that the magnetic field H = v V x A is perpendicular to the Neumann boundary (one component of the vector potential A is fixed as denoted in section 2.4.3), the second term of (2.2.19) can be considered as zero. This term is also zero on the Difichlet boundary. Then, the first and second terms in (2.2.16) can be given by

T. Nakata, N. Takahashi and K. Fujiwara

43

Goi=f ; jlv VxA'VxNidV-l" f .li'~'~iJod'V"

(2.2.20)

Let us discretize the x-component Goxi of the residual equation Goi in (2.2.20). As y- and z-components of Ni are zero in this case, V x Ni is given by the dete~inant I1 X

11

U Z

a ay a . aN i aN i V x Ni = ax ay az = Uyj ~ - u z aY Ni 0 0 By substituting (2.2.21) into (2.2.20), one obtains .,I" jr,

rf

FaAz

raAx aAz~

L az - X j

X "~ ~

c

bdV

(2.2.21)

F~ ~Axll Z L Ix

ay JJ

dV

~J

=; ; .,I,q~' FaA~-aAz~aI' Y ~iFaa'~0AxlONit~d V "/Laz ax Jaz -VzL ax " ~ _ l T y ~ " ,I ~'~J('vr'IiJOxdV aN i aAy rv l" aNi OAx aN i OAx {Vz~ Oy +Vy ~ OZ -Vz ay ax

- Vy &

ax J dV -

NiJox dV

(2.2.22)

From (2.2.22) and the integral formula (Zienkiewicz 1977) of (2.2.23), (2.2.22) can be discretized as shown in (2.2.24).

44

Chapter 2: Solution of 3-D Eddy Current Problems

f ; fN1NbNCNd 2

a!b!c!d! 3 4 dV = (a+b+c+d+3)! 6v(e)

(2.2.23)

4

1

G°xi - 3 6 ~

{ (vzdidke+Vyeieke)Axke" vzdiCkeAyke- vyeiCkeAzke }

V (e) - 4 Jox

(2.2.24)

where V (e) is the volume of the first-order tetrahedral element. The electric scalar potential ,(e) in the element e can be expressed similarly to (2.2.14) as follows 4 ¢(e) = Z Nke Oke (2.2.25) k=l Discretizing the residual Gti of the third term in (2.2.16) by the backward finite difference formula, we obtain

f j' j"v

Gti = c~

Ni

A,

At

dV

(2.2.26)

where At is the time interval step. The unknown vector potential at the instant t (considered instant) is expressed by A. A' represents the value at the previous instant (t~At). Using (2.2.23), Gti can be rewritten as o V (e) + G t i - 2 ~ L ( I + 8ike)(Ak e - A ' k e ) (2.2.27) k=| where Bike is the Kronecker delta. The x-component Ggxi of the fourth term in (2.2.16) is written by Gg x i - 6v(e) °

ffj

NiZCkeCkedV k=|

(2.2,28)

From (2,2.23), 4 Gg xi

24

Cke*ke

k=l The y- and z~components Ggyi and Ggzi are also given by

(2.2,29)

T. Nakata, N. Takahashi and K. Fujiwara

45

4 Ggyi = 2 ~ Z dke*ke (2.2.30) k=l 4 (y Gg zi = ~ 2 eke¢ke (2.2.3 I) k=I From (2.2.24), (2.2.27) and (2.2.29), the x-component of Gxi of the residual equation of (2.2.!6) is given by 4 1 Gxi - ( e ) 2 { (vzdidke+Vyeieke)axke-vzdiCkeayke'VyeiCkeazke } 36V k= 1 v(e) ~v(e) 4 4 4 JOx + Z (|+Sike)(Axke'Adxke) + ~ 4 Z CkeCke (2.2.32) k= i k= l The y- and z-components are obtained in the same way as follows: 1 4 Gy i - 36v(e)k~l { -vzcidkeaxke + (vxeieke+VzCiCke)ayke- ~xeidkeAzke } _

°z, -

v(e) ~v(e) 4 4 4 ~Joy+ ~ Z ( 1 6ike)(Ayke -A yke) + ~ Z dke*ke k=l k=l 4 +

~

e;

(2.2.33)

1

~ e ) 2 { VYCi°keaxkev~d~°keay~e+ (~yC,~ke +~xdidke)A~o}

36V k=i

_

cv(e) 4 4 V(4!J°z + ~ Z ( l + S i k e ) ( A z k e ' A ' z k e ) + ~ 4 Z ekeCke (2.2.34) k=l k=l Now, let us discretize (2.2.5). By applying the Ga!erkin method to (2.2.5), and utilizing vector identities and Gauss's law, the tbllowing equation can be obtained

Odi= J j NidivedV

46

Chapter 2: Solution of 3-D Eddy Current Problems

= j

, i,e.od

.~I 1('.~,(gradNi)" ,.|edV

(2.2.35)

When the eddy current density vector is parallel to the boundary, the first term in (2.2.35) is equal to zero. When the eddy current density vector is perpendicular m the boundary, the value of ~ on the boundary is equal to a constant value (see section 2.4.3). By imposing (2.2.5) into (2.2.35), the following equation can be obtained Gdi = I] ,[" ;

cgradNi'j

dV- Jr Jr" I. ¢~gradNi" grad~dV

(2.2.36) From (2.2.14), (2.2.15), (2.2.25) and (2.2.36), and applying the volume integral formula of (2.2.23), we obtain Gdi =

;;J/"

aAx aNi aAy 3N i 3Az'~a¢. °{ @Ni at + by at + aZ at ) -

ON~a~ + aNi ...... a~z)dV

+ 3y ay

=~

az

4 f aAxke d" 3Ayk~ aA-z'zke"~ ~ , . C i e ~ + le at + eie at )

4 e £ (CiCke+didke+eieke),ke (2.2.37) 36v(e) k= 1 By discretizing the time derivative term using backward finite difference, and multiplying (2.2.37) by -,at to (2.2.37) in order to obtain a symmetrical matrix (see (2.2.39) and (2.2.40), Nakata et al (I988a)), the following equation can be obtained 4 c Gdi = ~ Z { Cie(axkeA'xke) + die(Ayke" A'yke) + eie(Azke'a'zke) } k=l 4 . ~al~ £(CiCke+didke+eieke),k e (2.2.38) -

36V k=i

T. Nakata, N. Takahashi and K. Fujiwara

47

By equatl ~ ~ng G×i, Gyi, Gzi and Gdi of (2.2.32) ~ (2.2.34) and (2.2.38) to zero, the matrix equation shown in Fig. 2.2.2 can be obtained. In this figure nx, ny, and nz are equal to the numbers of unknown values of Ax, Ay and Az respectively, m is the number of unknown values of ~, and ne is the number of to~l elements.

000

xx xy H l n x H lny

xz xd Hl nz Him

000

yx YY Hl nx Hlny

yz yd Hlnz Hlm

zx zy zz zd HI i HI 1 HI 1 H1 1

zx zy Hlnx Hlny

zz zd H lnz H l m

dx dy HI 1 HI I

dx dy Hlnx Hlny

dz dd H Inz Hlm

xx xy xz xd H1 1 H1 1 H 1 1 H1 1 H

~x yy 1 HI1

!

!

dx dy Hml Hml

Hyz 11

yd Hll

dz dd HI 1 HI 1

!

!

dz HmI

dd Hml

= • eee

H

= •

!

= •

dx dy dz dd mm Hmm Hmm Hmm

Kxl Kyl Kzl

Kdl (2.2.39)

Figure 2.2.2: Matrix Equation (2.2.39)

48

Chapter 2: Solution of 3-D Eddy Current Problems

~e te~s

~;x, H~iy' ,, in the

x =

ne

~

e=l

figure are given by

vzdi d. +~ e.e.

oV (e)

36 V (e)

206t

~A

_J+

1 +~Je)

ne

36V (e)

e=l ne XZ

:

Hij

C. ]

36v(e)

e=l (y

ne

H~ x = 36V (e)

e=l ne

•~,~ ej +v~c i ej + o V (e)

36 V (e)

e=l ne

nij

=

(l+~j e) 20at

36V (e)

e=l

ne n ZX

ij

_

-

e=l ne

36V (e)

e=l

36V (e)

ne

H;?-

o V (e)

e

5i e=l

= ~ ~e dx o Hij = ~ ~e

36 V (e)

20At

(1 +~je)

T. N~ata, N. Takahashi and K. Fujiwara

13

H~j"¢ = ~

49

~e

13

H~ =~ ~e Hij

(2.2.40)

13At (ciCje + didje + qeje ) - 36v(e)

where 8e is defined as follows: e [ 1 when node i is in the element e 8i = ~ 0 when node i is not in the element e The t e ~ s Kxi, Kyi and Kzi are given by ne v(e) Axke K'xi = fSie _4 ~x + e=l zu~t k=l ne

v(e)

~1

4

ne

v(e)

Ky i =

Ayke

- JOy+

Kzi = ~1

(2.2.4I)

(2.2.42)

z u a t k=l

- - ~z + 4 zuzxt k=l

+~ke

Azke

!3

( CieAx' k e + dieA;k e + ~eAzke ' ) k=l Equations (2.2.39) and (2.2.40) indicate that the coefficient matrix is symmetrical. Therefore, the ICCG (Incomplete Cholesky Conjugate Gradient;) method (Meijerink and Van der Vorst 1977; Hoole, 1989) can be used in order to solve large simultaneous equations which usually appear in 3-D eddy current analysis. Kdi = 24

(b) Phasor If the linear steady state phenomenon with eddy currents (magnetic field, eddy current: sinusoidal) is analyzed by the time-stepping method, a number of iterations are necessary in order to obtain the final steady state solution (sinusoidal). In the analysis of such a phenomenon, the phasor method, of which the vector potential and the current density are assumed to be sinusoidal in time, is used to reduce CPU time. In this method, the vector potential and the current density are represented by complex v~iables, and the time derivative a,"~ is replaced by j~, where ~ is the angular frequency. The phasor representations of (2.2.4) and (2.2.5) are written as Vx

(

vVx

0"

")

= J 0 - 13 mA+ V 0

(2.2.43)

50

Chapter 2: Solution of 3-D Eddy Current Problems

o

V "Je = 0 (2.2.~) where (o) denotes a complex value. For example, the magnetic vector potential A is written by = A R + jA I (2.2.45) where A R and A I are the values at rot=0° and oat=-90° respectively. By replacing I/At of the third and fourth t e ~ s in (2.2.32) - (2.2.34) by jo~ and removing -Ayke etc., the following equations can be obtained: jox~V(e) 4 xl-

20

•

2 ( l + S i k e ) Axke + ~ k= 1

joz~V(e) 4 e i) _ G° (yl

° (e)

20

•

2 CkeCke k= 1

(2.2.46)

c (e) 4

Z (l+Sike)Ay ke + ~ k=l

joxCv'(e) 4

Gzi -

c (e) 4

o

20 Z( %e)az e+ k=l

d ° Z ke*ke k=l

~(e)

(2.2.47)

e

eke~k e

(2.2.48)

k=l

4 \

~ (e)

CieAxke + 8ieAyke + eieAzke )

di = ~ k=l

4

(

)"

C i t e + did ke + eieke eke (2.2.49) jm36 V (e) k=I As mentioned above, the equations for the phasor expression can be easily obtained by a minor change of the equations for time stepping.

2.3 Nonlinear Finite Element Analysis (A-¢ Method) The Newton-Raphson iteration technique (Hoole I989; Silvester and Fe~ari 1990) is used in nonlinear analysis. The increment 8Axi of the vector potential Axi, tbr example, is given by A(k+l) , (k) ~. (k) (2.3.1) xi = *axi + ~Axi The increments 8Ai and 8¢i are obtained by solving the matrix equation given in Figure 2.3.1:

T. Nakata, N. Takahashi and K. Fujiwara

3Gxl bAxl

51

aAxt 0Gz I 3Axl ~?Gd1 ~Axi

aGxl OAyl aG"~1 bAy I 0Gz 1 ~Ayl ~d t ~Ay1

OGxI 3Azl ~ 3Azl 0Gz I 3Azl 0Gd1 0Azl

OGxl O~l aGy 1 3¢ 1 aGdt 3~I ~d 1 3~ I

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

qlJ

•

•

0

•

•

•

•

~)Gxnx 0Axl 0Gyny ~Axl 3Gznz aAxl /~-~dm 3Axl

aGxnx aGxnx bAy I 0Azl OG.¢ny OGyny 3Ayl ~Az! 0Gznz OGznz O A y l 0Azl 0Gdm 0Gdm 0Ayl 0Azl

aGxnx Oq~l

oeo

o,e eoo oeo

ooo

3Gyny O~! 0Gzn 3~1 ;~Gdm

,,.o

3!~1

•••

•••

OGxl 3Axnx 0Gy 1 3Axnx OGz1 3Axnx 0Gd1 3Axnx

aGxnx OAxnx 0Gyny 0Axnx 3Gznz 0Axnx ~-~dm 0Axnx

OGxl ~Ayny aGy !_ OAyny S~z I OAyny o~Gd1 ~Ayny

aGxnx 3Ayny 3Gyny 0Ayny OGznz aAyny bGdm 3Ayny

aGxl OGxl OAznz O~m aAznz OGzI ~Aznz 0Gd1 ~Aznz

0

~Gxnx 0Aznz 3Gyny ~Aznz SGznz aAznz ~dm OAznz

5Ax 1

-Gx 1

SAy!

-Gyl

5Azl

-Gzl

a~m aGz 1 ~q~m 0GdI 3~m

~Gxnx 3~m 0Gyny 3q~m ~znz 0q~m /~3dm 0~m

-Gdl o

®

• o

o

5Axnx

~Gxn X

5Ayny

-Gyny

5Aznz

-Gznz

5,m

-G~

J

Figure 2.3.1: Matrix Equation (2.3.2)

>

(2.3.2)

52

Chapter 2: Solution of 3-D Eddy Currant Problems

As an example, the discretization of aGxi/bAxj is shown here. By differentiating (2.2.32) with respect to Axj, aGxi I (vzdidje +vyei~e ) + ~ aAxj - 36v(e) 4

idkeAxke

avz

4 k~idiCkeAyke + aAxj "0Axj k=l = ~Vy 4 "1~ ~V(e) _3Axj ZeiCkeAzke| + (l+Sije) 3Vy ZeiekeAxke

(2.3.3)

k:l The magnetic characteristic of the nonlinear material is represented by the v-B2 curve. Then, av/aA is given by av av aB2 (2.3.4) OA - aB 2 aA B =V xA

(1.3.7)

The x-, y- and z-components of B am obtained from (2.2.14), (2.2.15)and (1.3.7) as follows: 1 I~..k=1dkeAzke " k~lekeAyke Bx - 6v(e) k-!4 -) <eAxke. ~CkeAzke

B y 6VI(e) ~

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

(2.3.5) (2.3.6)

A

Bz =

1 (~. 6v(e) .,~=I

- ZdkeAxke k=l

From (2.3.4) and (2.3.7), byz bvz aBz2 OAxi - aBz2 aAxi 4 3-% die Z(dkeAxke. CkeAyke) aBz2 18v(e)2 k=l From (2.3.4) and (2.3.6), aVy OB, 2 OAxi

(2.3.7)

(2.3.8)

OBy2 aAxi 4

_3Vy eie Z(ekeAxke. CkeAzke) - 3By2 18V(e)2 k=l By substituting (2.3.8) and (2.3.9) into (2.3.3), OGxi/0Axj is written,

(2.3.9)

T. N~ata, N. Takahashi and K. Fujiwara

I{

aGx i 1 ~e aA~,j - 36v(e) I

vz +

53

1

1 avy + eieje Vy + 18v(e) 2 0BY2

Method ....,

.

,

.

.

.

Variable

.

A

~z

18v(e)2 aBz2 A

(A =1

4

-E k=l

- ECkeAyke k=l eAzkei~l}1

Basic ~,uation Current Free Current Cam/ing Region (RO) Region (Rj) Vx(vVxA) Vx(vVxA)=0

A-, =-or (\maA _ ~ + V, )

v.{ o(oa, o, + ~o)}

A-¢ - n

U3

=0

Vx(vVxA*) V.(-~V~)=O

A* at

T-~

EE~

V x ( e1_v xT) = a {~(T-Va)} at V.{~ (T - V~) } - 0

Table 2.4.1: Various Methods for 3-D Eddy Current Analysis

54

Chapter 2: Solution of 3-D Eddy Current Problems

ov(e) +~ (1 + 8ije)

(2.3.10)

~ e other t e ~ s aGyi/aAxj etc. can be derived in the same way, and this is left as an exemise for the morn interesmd readers.

2.4 Various Kinds of Methods 2.4.1:

Definitions

of

Variables

Various kinds of methods with different kinds of unknown variables are described here. It is shown that the codes of these methods are the same. The boundary' conditions for the A-¢ method and the T-~ method (See (I.3.22); T: current vector potential, ~: magnetic scalar potential); Nakata, Takahashi, Fujiwara and Okada 1988; Carpenter 1977; Preston and Reece 1982) are shown, and some considerations in applying the T-~2 method to practical problems are described. The number of unknown variables and the CPU time are considerably increased in 3-D eddy current analysis using the A-+ method, because the magnetic vector potential A with three components is defined in the whole region as shown in Table 2.4.1. In order to reduce computer storage and CPU time, the various other methods shown in Table 2.4.1 (Nakam, Takahashi, Fujiwara, Muramatsu and Cheng 1988) are proposed. A*, T, ¢ and ~ are the modified magnetic vector potential, the current vector potential, the el~tric scalar potential and the magnetic scalar potential respectively. A* is defined by the following equation (Emson and Simkin 1983) :

I Met.hod Region '

Used ...... a Equation

R i ......... (P) . . . .

A-+

A

(@

R0 Ri A-+-n

A

(P)

A

(o _(ei

f

i

cl

I

c2

R0 R i

T- n

Ro

-

aA at 0 ,i

G

.....

(Q)

"

0

~

-

(P)

T

aT

vt

at

• - a~2..... , at

I

(~

(Q)

0

~

-

-

Table 2.4.2:

c3

aA

. . . .

A* - ~

I

at

(P)

(~ Ro ..... Ri ..........!.

b

Coefficients for the Respective Methods

ii

ii

T. Nakata, N. Takahashi and K. Fujiwara

C A* - A + J V

~ dt

55

(2.4.1)

In those methods except for the A-¢ m e t h ~ , the vector quantities A, A* and T are defined only in the current carrying region Rj, and the scalar quantity ~ is defined only in the current fi'ee region Ro except the T-~ method as shown in Table 2.4.1.

2.4.2: The Basic Equations (a) The T-~:~method In this method (Nakata, Takahashi, Fujiwara and Okada 1988; Carpenter 1977; Preston and Reece 1982), T and ~ are defined in ~ and only ~ is defined in Ro in order to reduce the number of unknown variables. The basic equations for the T-n method are given by (Nakata, Takahashi, Fujiwara and Okada 1988) V.~(T - V ~)=0 (2.4.2) Vx

('

8VxT

)

= - ~0{

g (T - V ~ ) }

(2.4.3)

where p. is the permeability and ~ is the conductivity. The current vector potential T is defined by J = VxT (2.4.4) The magnetic scalar potential ~:~is defined by H - T = - V~ (2.4.5) where H is the magnetic field intensity. In the conventional method, T in (2.4.4) is equal to the current vector potentiaI T e corresponding to the eddy current density $e as follows: Je = V x T e (2.4.6) Similarly, a current vector potential T o corresponding to the magnetizing current density Jo can be defined as fbllows: Je = V x T o (2.4.7) The basic equations for the calculation of magnetic fields which are produced by the magnetizing currents and eddy currents are obtained by replacing T in (2.4.2) and (2.4.3) with the following equation: T = To+ T e (2.4.8) If T o is obtained beforehand from the magnetizing current, the magnetic field in the region, in which magnetizing currents and eddy currents are present, can be directly calculated by (2.4.2) and (2.4.3). T e and ~ are unknown variables in (2.4.2) and (2.4.3). (b) The A-¢-f2 method In this method, (Kameari 1988) the magnetic scalar potential ~ is defined in the current free region Ro instead of A in order to reduce the number of unknown variables. The basic equations for the A-¢-~:?method are (2.2.4) and (2.2.5) in the region Rj, and (2.4.2) in the region Ro. In this case, T in (2.4.2) is replaced by

56

Chapter 2: Solution of 3-D Eddy Current Problems

~

H

A

S~2

(a) Magnetic Vector Potential

(b) Electric Scala,- Potential

Figure 2.4.1" Boundary Conditions for the A - ~ Method

T o. As the variables (A,,) in the region ~ are different from the variaNe (f~) in the region Ro, the t~llowing continuity conditions of the magnetic field intensity H and the flux density B which connect both potentials ( A , , and f~) are necessary: n x H.= n x H (continuity condition of It) (2.4.9) j o n • B. = n • B (continuity condition of B) (2.4.10) j o where n is the unit vector pe~endicular to the boundary, the subscripts j and o represent the values in Rj and Ro respectively. Hi' ~ ' Ito and Bo can be written as follows using the potentials A and f~: It.l = v V x A (2.4.11) B. = V x A J

(2.4.I2)

grad

~

equipotential l i n e

(a) Electric Vector Potential

(b) Magnetic Scalar Potential

Figure 2.4.2: Bounda@ Conditions for the T-~ Method

T. Nakata, N. Takahashi and K. Fujiwara

H

57

= - Vn

(2.4.13)

= - g Vn

(2.4.I4)

O

B

O

From (2.4.9), (2.4.10) and (2.4.11) _ (2.4.14), the following equations are ob~ned: nx(n V xA)=nx(-Vf~) (2.4.15) n o (V x A ) = n • (. ~t V ~~) {12.4.I6) (c) The A*-a method In the A*-~ method (Emson and Simkin 1983), only one kind of variable, the modified magnetic vector potential A* in (2.4.1), is defined in the region RJ- From (2.2.4) and (2.4.1), the basic equation in the region Rj is given by V x (v V x A*)- =-c~ OA0t

1:2.4.17)

The basic equation in the region Ro and the continuity condition on the boundary are the same with those of the A-¢ -~ method. These basic equations are summarized in Table 2.4.1. For simplicity, the magnetizing cu~ent is ignored. Table 2.4.1 indicates that the equations for all methods can be written in the same style as follows: (P)

V x ( c 1 V x a ) = - c2 ( ~

(Q)

V • { c3(b + Vf) } = 0

+ Vf

(2.4.18) (2.4.19)

where the vector quantities a and b and the scalar quantity f are the unknown variables, and cl, c2 and c 3 are the material constants. These variables and constants for the respective methods are given in Table 2.4.2. This fact shows that a code corresponding to (2.4.18) and (2.4.19) can be used for all methods. In nonlinear analysis using the T-f~ method, however, c2 cannot be replaced by the permeability g.

2.4.3: Boundary Conditions ~ e boundary conditions for the A-, and T-~ methods are described here (Takahashi 1989). (a) The A-¢ method The boundary conditions of A and ¢ are classified as shown in Fig. 2.4.1 corresponding to the directions of flux density B and the eddy current density Je (1) Magnetic vector potential A (i) B o u n d ~ SA 1 perlzcndicul~ to B When B is per~ndicular to the boundary SA 1, A is parallel to S A t as shown in Fig.2 4.1 (a). This means that one component of A in the B direction is zero on the bounda~¢ to which B is pe~ndicular.

58

Chapter 2: Solution of 3-D Eddy Current Problems

(ii) Boundary SA2 parallel to B When B is pm'allel to the boundary SA2, A is perpendicular m SA2 as shown in Fig.2.4.1(a). This means that two components of A within the boundary to which B is parallel are zero. (iii) Boundary" through which the prescribed flux • flows When the flux • passing through the boundary is prescribed, the following relationship between ,t, and A is obtained: A ds = •

~4.~3)

where, s is the unit tangent vector along the circumference of the surface.

I lair ,

_c~

Figure 2.4.3: Model with Winding and Conductor

T. Nakata, N. Takahashi and K. Fujiwara

59

air winding

droP,

JO conductq 1

I l I I t i I

,I t

,/,

~L

TOz

I I I I !

I I I i I

I

I

Figure 2.4.4: Toz Corresponding to Jo (2) Electric scalar potential ¢ (i) Boundary S,! ~rpendicular to Je When Je is perpendicular to the boundary S¢l, V¢ is pe~endicular to S¢1 as shown in Ng.2.4.1(b). Because the direction of Je is the same as that of A, V¢ should also be the same as that of A as denoted in (2.2.2). This means that ¢ is constant on the b o u n d ~ to which Je is perpendicular. (ii) Boundary S,2 parallel to Je When Je is parallel to the boundary S¢2, V0 is parallel to S¢2 shown in Fig.2.4.1(b). This means that ¢ is unknown on the boundary to which Je is parallel.

60

Chapter 2: Solution of 3-D Eddy Current Problems

(b) The T-~ method The boundary conditions of T and fa on the planes of symmetry are classified as shown in Fig.2.4.2. (1) Current vector potential T (i) Boundary ST1 perpendicul~ to J Vv'nen J is perpendicular to the boundary ST1, T is parallel to ST1 as shown in Fig. 2.4.2(a). This means that one component of T in the J direction is zero on the ~undaD' to which J is perpendicular. Oi) Boundary S ~ parallel to J When J is parallel to the boundary ST2, T is perpendicular to ST2 as shown in Fig. 2A.2(a). This means that two components of T within the boundary to which J is parallel are zero. (2) Magnetic scalar potential (i) Boundary Sf~I perpendicular to I{ When H is perpendicular to the boundary Sal, grad~ is perpendicular to Sni as shown in Fig. 2,4.2(b). This means that ~ is constant on the boundary to which H is perpendicular. (ii) BoundaD' S~.2 parallel to H When H is parallel to the boundary 8~2, grad ~ is parallel to Sfn as shown in Fig.2.4.2(b). This means that ~ is unknown on the boundary to which H is parallel. The examples of the boundary conditions of the A- ~ and T- f~ methods for practical problems are shown in Figs.(2.5.4) and (2.5.5) in Section 2.5.3.

2.4.4: A d v a n c e d Considerations on the T-f~ M e t h o d (a) Determination of To from Jo (Nakata, Takahashi, Fujiwara and Okada 1988) In order to obtain T O from the magnetizing cu~ent Jo, the relationship between

h o ) = t,~ ,-n~

,%

:tor ( a l u m i n u m )

,Vc-~.25X I0' Figure 2.4.5: Mode! with a Hole

S/m)

T. Nakata, N. Tak~ashi and K. Fujiwara

61

T O and Jo is examined using an example shown in Fig. 2.4.3. The Figure shows one-eighth of the analyzed model. It is assumed that the eddy current flows only in the conductor. Then, T e is defined only in the conductor (Nakata, Takahashi, Fujiwara and Okada I988). T o is defined in the dotted part which is surrounded by the winding. If it is assumed that the magnetizing current flows twodimensionally, the x- and y-components Tox and Toy of T o become zero. Then, the x- and y-components Jox and Joy of Jo can be represented from (2.4.7) as follows: ~Toz ~Toz Jox - ~y Joy = - ~x (2.4.21) where Toz is the z-component of T o. (2.4.21) denotes that Toz changes with x or y linearly, if the magnetizing current is distributed uniformly in the winding. Therefore, Toz is distributed linearly in the winding as shown in Fig.2.4.4. Toz is constant and equal to IJoI.L in the region surrounded by the winding, because there is no magnetizing current in the region. L is the width of the winding. The value of IJoloL is obtained from (2.4.21). When the shape of the winding is complicated and the distribution of Jo is not known beforehand, the following Laplace's equation for the el~tric potential, should be solved (l',lakata, Takahashi and Fujiwara 1988). V.V, = 0 (2.4.22) Jo can be calculated from tile obtained ,.

cQnductor

m

l"l

q

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

hole

|

z( A

B

C

Figure 2.4.6: Distribution of Eddy Current Vectors (z = 25,4mm)

62

Chapter 2: Solution of 3-D Eddy Current Problems

.~._

.- J e c

..

A

JeB v

",=, x

10 "li

u

10 "lz

10 "=

10"*

1

(:/el/0 C -5

Figure 2.4.7: Effects of (~a/'Cc) on JeB and JeC (y=Omm, z=25.4mm) (b) P r o b l e m s in treating holes in conductors The difficulty in treating holes in a conductor (Nakata, Takahashi, Fujiwara and Okada 1988) is that 1/o in (2.4.3) becomes infinite, because the conductivity of the hole (air) is equal to zero. In order to overcome the difficulty, the holes are treated as conductors with very low conductivity. The effect of the conductivity oa of a hole in Figure 2.4.5 on the accuracy of eddy currents is investigated. Figure 2.4.3 may be thought of as a typical model with a hole. The following sinusoidal flux is applied in the z-direction: B z = 0. I • sin (2n • 50t) (2.4.23) where t is the time (in seconds). The conductivity cc of the conductor is 0.25x I08 S/re. The analysis has been carried out by using the double-precision (72 bits) computer. Figure 2.4.6 shows the distribution of eddy current vectors on the top surface of the conductor. In this case, oa is equal to I S/re. V e ~ small eddy current flows

1

..../j....,

10"* 10"l' -

t . t . / -

I 0 "¢= ! 10 ".6

10"~2

10"

!

i 0"*

J

1

Oa / O c Figure 2.4.8: Effects of (oa/oc) on IJeAJ/MeBI (y=Omm, z=25 4ram)

T. Nakata, N. Takahashi and K. Fujiwara

63

in the hole (for example, Je = 6.23x10"2 A/m2 at a point A). As it is negligibly small compared with the eddy current in the conductor (for example, Je = 1.92x 106 Aim 2 at a point B), this does not affect the accuracy of eddy current. Figures 2.4.7 and 2.4.8 show the effects of ~ c ~ c on JeB, JeC and JeA/JeB, where JeA, JeB and JeC are the y-components of eddy current densities at the points A, B and C in Figure 2.4.4. Figure 2.4.7 indicates that the distribution of eddy current density deviates from the proper one, when ~a/Gc is less than I(Y 12, Figure 2.4.8 proves that ~a/~c should be less than 10-2 under the condition that the eddy current density in the hole is within 1% of the minimum value of that in the conductor~ Therefore, the conductivity of the hole should be in the range of 10-12 < ~JGc < I0-2.

2.5 Various Finite Elements 2.5.1: B r i c k N o d a l E l e m e n t In the element described in section 2.2.2, the potentials are defined at each node. This type of element is called a "nodal element." The other type of element called the "edge element", on which the potentials are defined on each edge, is also used (Bossavit and Verite 1983; Barton and Cendes I987; Albanese and RuNnacci 1988; Kameari 1989). In this section, the brick nodal element and the brick edge element using the A-method are explained. Figure 2.5. l(a) shows the first order brick nodal element. The x~, y~ and zcomponents Ax, Ay and Az of the magnetic vector potential at an arbitrary point in an element e can be represented by using the same inte~olation function Nke~ 8 (e) Ax = Z NkeAxke k=l Z A-t-

,~i e 2e

Axle

Axle

7¢ (a) Nodal

8g

(b) Edge

Figure 2.5.1: First Order Brick Elements

le

64

Chapter 2: Solution of 3-D Eddy Current Problems

8

(2.5.1)

4 e ) = Z NkeAyke k=l 8

A(e)z = Z NkeAzke k=l The components are continuous on the boundary of adjacent elements, For example, Nle of a node le is given by

Nle - 8v(e )I (a(e) + {) (b(e)+ ~) (c(e) + {) 1 +{1 e{) (1 + = g(1

.le q )

(1 + ~le~)

(2.5.2)

where {, tl and ; are

x,xg -

~ ,

z-z~

a(e?, rl = b(e)

¢-

c(e)

(2.5.3)

2a(e), 2b (e) and 2c (e) are the lengths of respective edges as shown in Fig.2.5.1 (a). Xg, yg and Zg are the x-, y- and z- coordinates of the centroid of the element e. In general, the inte~olation function Nke of a node ke is written by Nke =

( I + ~ke~[) ( I + rtkeq) ( I + ~ke~)

(2.5.4)

where {ke, rlke and ~ke are equal to {, 11and ~ at the node ke respectively, and these are given in Table 2.5.1 (a).

2.5.2: The Brick Edge Element (a) General description Figure 2.5.1(b) shows the first order brick edge element (Nakata, Takahashi, Fujiwara and Shiraki 1990; Kameari 1989). The component of the magnetic (a) Nodal

(b) Fdge

Node Number le

{ke nke 1 1 -1 I 1

30

-i { ,i

4e

I 1 -1 -i 1

6e 7e . . . . . . . . . . . . 8e

-I 1 1 -i -1

eke 1 1

1 I -1 -I -1 -I

Edge Number le 2e Ze .....

Table 2.5.1: Local Coordinates of Nodes and Edges

O~ke ~ke I 1 -I 1 i -1 .

-1

.

-1

.

.

T. Nakata, N. Takaln~hi and K. Fujiwara

65

vector potential parallel to the edge is defined on the edge of the element. ~nerefore, the number of unknown variables is 12, which is equal to the number of edges as shown in Fig. 2.5.1. The numbering of the edge parallel to the x-axis is of the order of le, 2e, 3e and 4e as shown in Fig.2.5.1(b). This is in the clockwise direction when observed from the positive side of the x-axis. ~ e edges parallel to y- and z-axes are numbered in the sm'ne manner.

(b) Interpolation functions In the edge element, Axi, Ayi and Azi can be defined by 4 A(e) x = Z NxkeAxke k=l 4 A(e) y = Z NykeAy ke (2.5.5) k=I 4 A (e) z = Z NzkeAzke k=I The interpolation functions Nxk e, Nyke and Nzk e for x-, y- and z-components are different from each other. Let us now define the expanded form of (2.5.4) as Nxk e. We then obtain Nxke = °~lke + C~2ke{ + °t3kerl + a4ke z+ a5ke~rl + a6kerl~ + ~7ke~{ + aSke{q; (2.5.6) where al ke etc., are coefficients. In order to make the vector potential A unique, the condition of V • A = 0 is imposed. Here, the following condition is given aA x bAy aA z = an = ~ = 0 (2.5.7) From (2.5.5) and (2.5.6), bAx/b{ is written as ~A x

~--, , .2I = L ( = 2 k e + °t5kerl + U7ke; + et8ke rl)Axke = 0 k=l

Then, O~2ke = a5k e = a7k e = ask e = 0 As a result, Nxk e can be given by Nxke = alke + a3ke~ + ~4ke{ + ~6keq; For example, the interpolation function Nxl e of the edge le is written as 1

N x l e = ~ ( 1 +rl) (1 +4)

(2.5.8)

(2.5.9) (2.5.10)

(2.5.11)

e~lke etc. are determined using the property that Nxl e should be equal to unity on the edge 1e and zero on the edges 2e, 3e and 4e. Other interpolation functions can also be obtained in the same way. They can be written in the general style as

66

Chapter 2: Solution of 3-D Eddy Current Problems

N x k e = ~1( 1 + ~ k e h ) ( 1 + N y k e = 41( l

~ke~)

+ ~ k e Z ) ( 1 + ~ke{,)

N z k e = X1 ( 1 + ~keX) ( 1 +

(2.5.12)

~ke~l)

{, rl and ~ are equal to those defined in (2.5.3). The values of %e and [3keare given in Table 3(b). Jox (e), Joy(e) and Jox (e) are written using the interpolation functions in the same way as (2.5.5). As can be understood by comparing (2.5.4) with (2.5.12), the interpolation function of the nodal element is different from that of the edge element.

(c) Discretization Equations of the finite element method for the edge element can be obtained by discretizing (2.2.16) in the same way described in section 2.2.3. The fourth term of (2.2.16) becomes zero as the electric scalar potential, is equal to zero in the case of the edge element (Kame~i 1989). The interpolation function N i can be written by N i = uxNxi+ uyNyi+ uzNzi (2.5.13) Let us examine the second term of (2.2.19), the surface integral of Ni-(vVxAx n). If the 2-components of the magnetic vector potential parallel to the edges of the element on the boundapy are chosen to be zero, the tangential component of H is continuous on the boundary of the element (see section 2.4.3). Therefbre, if the second term of (2.2.18) is chosen to be zero, the condition of (2.4.9) is satisfied. In the edge element, the tangential component of A is continuous on the boundary of the element, because the component of A parallel to the edge of the element is defined as an unknown variable. Therefore, the normal component of B is continuous on the boundary. Then, the continuity conditions on H and B denoted

(a) Nodal

(b) Edge Figure 2.5.2: Related Elements

T. N~ata, N. Takahashi and K. Fujiwara

67

in (2.4.9) and (2.4.10), are satisfied when the edge element is used. As a result, the equations finally obtained can be written as follows:

Gxi =

v V x A • V x uxNxidV -

ffc

Nxi J0xdV

jv

(2.5.I4)

coil (I~AT

r z

(a) front view a l u m i - , , , , , ,,~o,,.

Y

~ x

"b (c) "a (d) (b) plan view Figure 2.5.3: Analysed Model (With Hole)

68

Chapter2: Solutionof 3-D EddyCurrentProblems

A x --

_.~g A x - A y = A z = 0

Ax - A y = Az = 0 hq

= Az=0

A

=0

Y

" " - 160,9 Ax = Az = 0

Ay=Az=O ff a

Az=0

Figure 2.5.4: Bounda~ Conditions (A - , Method)

Gyi = /" Jr ;vVxA'VxuyNyidV- j "

J~¢Nyi J0ydV

ili yi Gzi = j'(" f JJ" v V x A * V x uzl~"lzidV-

(2.5.15)

f ..ll-vNziJ~dV

(2.5.16) A little algebra like (2.2.21) yields

T. Nakata, N. Takahashi and K. Fujiwara

o,, =f f;

Vy

69

aA x OAz'~aNxi

(~..y,OA

aAx"]h aNxi

f f If""~'lxiJOxdV+f ](";)"Ixi°OAx~

dV (2.5.17)

By discretizing the time derivative t e ~ by backward finite difference, and from (2.5.5), (2.5.12) and (2.5.17), the final residual equation can be written as Gxi =

ne

,8e

e= 1

4

L Iv(e) t ic(e)2 ~ ( 1 ") | | ~ie~ke 1 + ~ i e ~ k e

k= 1

+ ~~ie~ke b(e)2

1 + ~ ~ie~ke

Axke

. ? c (e) Oqe~ke(1 + ! [SieO~ke)Ayke vv

b (e)

ie ke

I

+ g io ke) azko

+v(e' ~ ~+g oi°Oko)(, +

I

c~

/1

)

Axk e - Axk e At " Jo x ke

(2.5.18)

The detailed derivation of (2.5.18) is left to the more interested reader.

(d) Comparison of the number of unknown variables First, the number of unknown variables is examined for the magnetostatic problem. In the case of the brick nodal element, the number of elements which share a node i is equal to 8 as shown in Fig. 2.5.2(a). The number of unknown vmables per node is 3, and the number of nodes in one element is 8. Therefore, the average n u m ~ r Nn of unknown variables per element is given by" 1

Nn = g x 3 x 8 =3

(2.5.19)

In the case of brick edge element, the number of elements which share an edge i is equal to 4 as shown in Fig. 2.5.2(b). The number of unknown variables per edge is unity, and the number of edges in one element is equal to 12. Therefore, the average number Ne of unknown variables per element is given by

70

Chapter 2: Solution of 3-D Eddy Current Problems

~=0 ~=0

g

~ =0 On

Z Tx = Tz = 0

O~ =0 On

J

Tx=~

t

-

=Tz=0 =0

.-.,,. Tx

Ty=0

T× =Ty =0

Figure 2.5.5: Boundary Conditions (T-~;~Method) 1 Ne = ~ x 1x 12 = 3

(2.5.20)

Equations (2.5.19) and (2.5.20) denote that the number of unknown variables of the brick nodal element is the same as that of the brick edge element when the boundary condition is ignored. Secondly, the number of unknown variables is examined for a magnetodynamic problem. In the case of the brick nodal element, the number of unknown variables per node is equal to 4, because both A and ~ are unknown variables. Therefore, Nn in the magnetodynamic problem with eddy currents is larger than that in the magnetostafic problem. On the contra~, in the case of the brick edge element, Ne for the magnetodynamic problem is the same as that for the magnetostatic problem, because ¢ can be set at zero (Emson and Simkin I983). As a result, the number of unknown variables of the edge element is less than that of the nodal element. Thirdly, the number of non-zero entries which determine computer storage requirements is examined for the magnemstatic problem. ~ e number of non-zero entries for the brick nodal element is 81. This is the multiple of the number of unknown variables per node (equal to 3) and the number of nodes per element (equal to 27 as shown in Figure 2.5.2(a)). The number of non-zero entries fbr the brick edge element is 33 (number of edges per element shown in Figure2.5.2(b)). Therefore, the number of non-zero entries of the edge element is considerably less than that of the nodal element.

T. Nakata, N. Takahashi and K. Fujiwara

71

2.5.3: Comparisons In order to evaluate the most suitable method of analysis (A-, or T-~::~method) and the kind of suitable finite element (nodal or edge element) fbr a given problem, the features of each method and element are investigated here. The accuracy, the computer storage requirements and the CPU time of each method and element are compared with each other for the 3-D eddy cu~ent model (IEEJ model) (Nakata, Takahashi, Fujiwara and Olsewski 1990; Nakata 1991). The flux and eddy current distributions in the models are analyzed, and the results calculated are compared with those measured. Figure 2.5.3 shows a model proposed by the lEE of Japan. The features of the model are described by Nakata, Takahashi, Fujiwara and Olsewski (I990). A rectangular ferrite core is surrounded by an exciting coil. An AC current whose effective value is 1000AT at the frequency of 50 Hz is applied. Two aluminum plates are set on the upper and lower sides of the core. The conductivity of the aluminum is 3.215x107 S/m, and the relative permeability gr of the core is assumed to be 3000. Both cases with and without a hole in the plate are investigated. Figures 2.5,4 and 2.5.5 show the boundary conditions for the A-¢ and T-~ methods in the case of the nodal element (see section 2.4.3). Figure 2.5.6 shows the maximum absolute value IB[ of the flux density along the line at z=57.5mm. The discrepancies between the calculated and measured values are seen to be not so large.

Without Hole A-, T-~2

Item

] i

A-0

With Hole T-fz

Number of Elements 14400 N u m ~ r of Nodes 16275 Number of ....... ......43417 ! 4i(YJ0 I 22844 ! 224i2 I 42885 I 41G6b 1228~ I 22412 Unknowns Number of Non-zero Entries Computer Storage 72.2 28,4 30.7 .... 19.4 70.5 28.4 30.7 19.4 ,,,,,,,,,,,,,

................ g,~)

,,,,,, . . . .

,

.........

Number of Iterations Url H H i|P| iD)| iP~I |1:~111 l i l l l BOil of ICCG Method CPU Time (S) ~H |1|1 | " f l a i l | ~ O ] i d ? l O ] i[irq~J P~Ol#]i | I N Computer Used: NEC Super Computer SX- 1E (Maximum Speed: 285 MFLOPS) Convergence Criterion of ICCG Method: !0 .7 Table 2.5.1" Discretization Data and CPU Time

72

Chapter 2: Solution of 3-D Eddy Current Problems

The computer storage, the CPU time, etc. are shown in Table 2.5.1. The CPU time of the T-f~ method in the analysis of the model without hole is extremely decreased compared with the A-, method. The CPU times of the edge element flgr the A-~ and T-~ methods are about 1/6 and 1/2 of the nodal element. Although the CPU time of the T-~2 method in the analysis of the model with hole in the case of nodal element, it is not so remarkable in the case of edge element. From the viewpoints of the CPU time, the T-O m e t h ~ with edge element is favorable.

F

50

0

150

50

0 z

z

.

0

(a)x=Omm Z:

Y

.

.

.

.

.

.

.

0

nodal

without --~: nodal hole edge % f] method measured

Figure 2.5.6:

.

.

.

.

.

.

I00

x(mm)

150

.

(b) y=Omm --,-.,- : n ~

"edge } A- ~ method

with hole

Z: n~al --~: edge 7"- ~ method .-~: measured

Spatial Distribution of Flux Density (z = 57.5 mm)

Chapter 3 iiii

j

iiii

!!11111

G. Bedrosian and M.V.K. Chari [[1!!

jllll!l!l

ii!llg[![HuJ/i

..........

i

iiiiii

.....

iiiiiiii

COUPLING FINITE ELEMENT MODELS OF ELECTRICAL MACHINERY TO EXTERNAL CIRCUITS

3.1 Introduction Numerical techniques in use for the last two decades - the finite element method in particular - have proven valuable in the design, performance evaluation, and optimization of electrical machinery. The modeling approach has most commonly involved two-dimensional finite element boundary-value problems with specified current sources for the conductors, on the assumption that the sources are unaffected by the finite element solution and external circuitry. Voltages at the terminals of the machine under evaluation are derived a posteriori by calculating flux linkages and their products with angular velocity, or, for some simple configurations, by computing the electric field and integrating over a prescribed path to yield the voltage. In practice, however, electrical machines are coupled to voltage sources - - or to more general external circuit sources - - and not to ideal cu~ent sources. Furthermore, the problem is complicated by the connections of the windings (series or parallel) and the presence of active or passive external circuits. In modern power systems, electrical machines such as generators and motors usually operate in paraIlel, connected to a source with a known voltage. Shortcircuits, system transients and other abnormalities on the power system affect the integrity and efficient performance of all the connected devices. An electrical machine is not completely modeled until its interaction with the rest of the power system is taken into account. For the electromagnetic model to be representative of the physical phenomena that machinery are subjected to in operation, a comprehensive field analysis plus circuit analysis is necessary. Brandt, Reichert and Vogt (I975) were perhaps the

74

Chapter 3: Coupling Finite Element Models m External Electric Circuits

first to apply circuit constraint equations to a turbine-generator to evaluate its steady-state performance. Stoll (1977) applied an integral constraint to a finite difference model of skin effect in conductors. This was followed by Chari and Csendes (1977) who added integral constraints to a finite element model of bus bars. Konrad (1981) introduced an integro-differential equation approach to modeling integral constraints in the skin effect problems for transformer windings. Chari, Bedrosian, d'Angelo and Konrad (1991) developed a superposition method for calculating skin effect in sheet-wound transformers, Bedrosian introduced a generalized circuit model for conductors in electrical apparatus with or without skin effect by means of admittance matrices derived from supe~osition - - this will be covered in detail below. Strangas and Theis (1985) pioneered the coupling of circuit equations to a nonlinear time-dependent solution of inverter-driven induction motors. An integrated approach to coupling problems that include voltage constraints, external circuits, and mechanical motion was presented by Istfan (1987), and later by Salon, DeBortoli and Palma (1990). In this chapter, we will develop several numerical methods for coupling finite element models to external circuits, from a simple energy-based method, suitable for linear problems with few external connex~tions and negligible eddy currents, to a method capable of handling nonlinear materials, rotor kinematics, and nonlinear devices in the external circuit. First, we will briefly review the basics of finite element analysis, ~ applied to electromagnetics, introduced in the first chapter.

3.2 The Finite Element Method 3.2.1 : M a g n e t o s t a t i e s Ampere's law f o ~ s the basis of magnetostatic analysis, and the field quantities are related to the excitation source via the magnetic vector potential. The following quantities are used in the analysis: A B H Js Je M0 F

Magnetic vector potential Magnetic flux density Magnetic field Source current ~ d y current Permanent magnetization Energy-related functional

W Weighting function V Voltage I Current ~ Magnetic permeability v Magnetic reluctivity = ~"! c Electrical conductivity 0 Electrical resistivity = ~-l

In modeling magnetostatic devices, as described in this section, the following assumptions are made: i. The problem is two-dimensional - B, H, and M 0 have only two vector components (in the plane of the model) while A and Js have only one vector component (orthogonal to the plane of the model);

G. Bedmsian and M. V. K. Chaff

75

ii. All quantities are time invariant; iii. The source function is represented by a constant cu~ent density distribution in the conductors, ,Is, and/or permanent magnetization, M0; and iv. The permeability is isotropic and either a constant or a function of flux density - the B-H curves are assumed single valued and hysteresis is neglex-ted. In finite element modeling, the resulting partial differential equation for the field problem is expressed by an energy-relamd functional or by a weighted residual procedure. In the former, the stored energy or the functional is minimized, and in the latter, the residual is set to zero in a weighted average sense. This can be expressed by the following equations: V x H = ,Is (3.2.1) H =vB (3,2.2) B = V × A

(1.3.7)

V × v (V × A) = ,Is

(3.2.3)

F=

v (V x A ) ,

(V x A ) - A , J s

dV

(3.2.4)

,v

(3.2.5)

The unknown magnetic vector potential, A, is approximated within each finite element using A = Z Aj Wj (3.2.6) where Aj are the values at the nodes and Wj are the weighting functions (often called interpolation or shape functions) associated with the respective nodes. Substituting eqn. 3.2.6 into eqn. 3.2.5 results in a symmetric matrix equation with {Aj} as the unknowns. The matrix equation is also sparse, because Aj and Ak are coupled only when nodes j and k share at least one element in common. To summarize, the finite element procedure comprises the following steps: i, Subdividing the field region into sub-regions (elements); ii. Expressing the vector potential and source as linear or higher-order interpolates of their values at the vertices of the elements (nodes) in terms of weighting functions as in eqn. 3.2.6; iii. Substituting these expressions in eqn. 3.2.4 or eqn. 3.2.5 and setting the first variation of eqn. 3.2.4 to zero or implementing eqn. 3.2.5; and iv. Solving for the vector ~gtentiat at the nodes. Setting the required boundary conditions, and choosing the appropriate equation solver are based on engineering judgement and expediency dictated by the computer power available and execution time. For example, setting the outside diameter of the stator yoke in an electrical machine as a zero potential boundary

76

Chapter 3: Coupling Finite Element Models to External Electric Circuits

(flux-line boundary) may suffice to yield the required solution accuracy. But in the case of an electrostatic problem of a support insulator for a power-line conductor with zero field at infinity, it may require uneconomical finite element discretization of free space surrounding the conductor, or an alternative open-boundary formulation for the far field, in order to simulate the appropriate boundary conditions (see also chapter 15) In the case of nonlinear problems, where the material re!uctivity may be a function of the magnetic flux density, an iterative solution procedure is employed. We have already seen this in chapter 2, while chapter 6 takes up different ways of m ~ e l i n g the magnetic material. In the iterative solution procedure we use here, an approximate value of the reluctivity is assumed at the start of the iterative scheme, and the vector potential is computed by the finite element method. The corresponding flux density is calculated and the reluctivity is updated with respect to the B-H characteristic obtained fl-om measurements. A first-order iterative scheme can be expressed as vnew = v°ld + ~ ( v n°w - v ° l d ) (3.2.7) where ~ is an under-relaxation factor usually in the range of 0 to 1, v°ld is the value of v used in the current iteration, v n°w is the value of v obtained from material property curves of v versus IBI = IVxAI, and vnew is the value of v to use in the next iteration. This method yields linear convergence to the true solution from an initial estimate and the error decreases in each iterative step in a linear fashion. Although this procedure yields a unique solution, if there is one, it is slow to converge and, therefore, is not the preferred choice. A second-order or quadratically convergent method is usually employed for machinery analysis, where the error in each step decreases quadratically. The most widely used second-order itemtive scheme is the Newton-Raphson method, which can be summarized in one equation as [A k+l ] = [A k] + [7]"1 [R] (3.2.8) where [A k+l ] is the value of the vector potential at the ( k + l ~ iteration, [A k] is the value of the vector potential at the k ~ iteration, and [J] is the Jacobian matrix of partial derivatives of the residual vector, [R], with respect to the vector potential. If the magnetic flux density at the k~ iteration is given by Bk = V x Ak (3.2.9) and we solve for the difference between successive solutions, A: Ak+l = A k+l - A k (3.2.10) then the following f o ~ u l a results for the Newton-Raphson scheme:

f I

[(v ×

v ( v × Ak+,)

~v (V × W i ) . Bk (V x A k + l ) , B k "

lBk[ OIBI

G. Bedrosian and M. V. K. Chaff

77

(V x W i ) * v B k - W i * Ss I dV = 0

(3.2.11)

Note that this forrn is identical to eqn. 3.2.5 when v is constant, and that both the symmetry and sparsity of the matrix are preserved with the addition of the NewmnRaphson term when v is not constant. In a few applications, axisymmetric models (r,z coordinates) of devices are considered to be better representations of the geometry, or as a more effective means of calculating design parameters such as inductances and power losses. These models differ from the regular two-dimensional models (x,y coordinates) in that the curl of the vector potential when expanded yields a singularity in r and, therefore, either the singularity is avoided in the geometric modeling or special expressions for the vector potential are used. Konrad (1974) developed a formulation of the vector potential such that the new value is r "1/2 times the old one. Lowther and Silvester (1986) used an expression that is r -I times the old vector potential. An isoparametric formulation using Gaussian integration points appears to yield the most convenient and computationally efficient form of integrating expressions involving the curl of A. Since the integration points are located inside the element boundaries, singularities at element edges where r - 0 are avoided. The h~llowing is a brief description of the isoparametric tmnsfo~ation: In this method, the geometrical coordinates (x,y or r,z) are approximated by interpolation using the same weighting functions as those used for the unknown field quantity (vector or scalar potential or any other). Defining the weighting functions as interpoIatory polynomials identically to the expansion for A in eqn. 3.2.6, we can express x and y, the global coordinates, as x = ~ xi W i (3.2.I2) Y = 1~ Yi Wi (3.2.13) where x i and Yi are the cc)ordinate values at the finite element nodes. One can then relate the derivatives of A z with respect to the global coordinates (x,y) to derivatives with respect to the local coordinates ({,q) in each element using the Jacobian matrix of partial derivatives, [J], such that

OA I=

Zxi

l

~ Z xi ~ 0Wi 0Wi ~ Yi ~ ~ Yi The derivatives of A z with respect to x and y are then obtained as

[J]

=

Ox ~x

Oq Oy Oy

|0~| O~ | 0 A z [ = [J] "T k-~--fi-A k Oq A

(3.2.14)

(3.2.15)

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Chapter 3: Coupling Finite Element Models to External Electric Circuits

f

Figure 3.2.1: Flux Distribution in Axisymmetric Model of a Magnetic Resonance Magnet

G. Bedrosian and M. V. K. Chari

--

,,

....

, =~.r~,~.~,

~,,,

Figure 3.2.2: Electrostatic Equipotential Lines for a String Insulator

79

80

Chapter 3: Coupling Nnite Element Models to External Electric Circuits

where the inverse transpose of [J] is [j]-T= Ox 0v - Ox

N~-

~

~x 0x

-N

(3.2.16)

The computational conveniences of this method are twofold. First, since the weighting functions are usually polynomials of the local coordinates in each element (as we will see below for two examples), differentiating with respect to local coordinates is straightforward using eqn. 3.2.6. More importantly, this method allows us to differentiate and integrate in higher-order elements with curved sides and to avoid singularities in the integrals when r = 0. For a first-order triangular element, the local coordinates themselves are the weighting functions and are governed by the relationship +n+x= 1 (3.2.17) where {, rl, and ~. are the weighting functions at the three vertices of the triangle (the same as ~1,~2, and 43 of eqn. 1.5..37). Further, the elemental area, dxdy, is related to the local coordinate values such that dV = dx dy = I J [ d~ dr1 (3.2.18) where I J [ is the determinant of the Jacobian [J]. The ranges of { and rl are 0 ~ { 1 and 0 _
3.2.2: Eddy Currents We shall consider induced (eddy) currents in source-l¥ee regions, which we shall designate as eddy current regions. We shall also examine induced currents in source current carrying conductors. Traditionally, f~quency domain analysis has been employed in both these cases because of its simplicity, ease of computation and fast execution. However the limitations of the method are i. Ferromagnetic materials must be modeled with time-constant permeabilities, although the values can vary with position;

G. B~rosian and M. V. K. Chari

81

ii. The analysis must be ~rfbrmed at one source frequency at a time; iii, The source and solution are harmonic functions of time; and iv. Complex arithmetic must be used. The frequency-domain method is not amenable for analyzing strongly nonlinear problems, yet it is useful for many types of problems, as will be shown in examples later in the chapter. Induced currents in conductors in source-free regions can be m ~ e l e d by a linear partial differential equation and the corresponding functional or by the weighted residual proc~ure as follows: V x v (V x A) + jo~oA = Js (3.2.20)

F=

f

2(V

xA)"(V

xA)+A"

( ~A2

- Js)

dV

(3.2.21)

.~[" , , I [ ( V x W i )

ov(VxA)+Wie(jmcA

~Js)]dV=0

(3.2.22)

where j = (-1) 1/2, ~ is the radian frequency, and all other quantities remain as defined earlier.

3.2.3: The Skin Effect Problem The skin effect - current tending to flow near the outer surface of a conductor as frequency increases - occurs in current carrying conductors (including stranded conductors) when the thickness of the conductor or strand is comparable to or larger than a skin depth. The distribution of the induced currents and the transport current is not known a priori, and only the total current in the conductor is measurable. This imposes an integral constraint on the induced and transport current densities such that the integrated value of the total current density over the conductor crosssection equals the total current in the conductor. Konrad (1981) presented an integro-differential equation formulation for modeling skin-effect in transformer windings. B~rosian (Chari 1991)developed a generalized circuit model based on a supe~osition method suggested by Cendes for applying integral constraints for solving generalized skin-effect problems in multiple conductors. In a later section of this chapter, a more detailed description of integral constraints and their application is included. Eddy current effects and the skin effect are respectively illustrated in Figures 3.2.3 and 3.2.4.

3.2.4: Time-Dependent Eddy Current Problem In many applications, such as thyristor-controlled machines, switched reluctance generators, p e ~ a n e n t magnet motors operating on non~sinusoidal input sources, or electrical machinery operating in a transient mode, a frequency-dependent

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Chapter 3: Coupling Finite Element Models to External Electric Circuits

Figure 3.2.3: Eddy Current Effects on Flux Distribution in a Conducting Stab

G. Bedrosian and M. V. K. Chari

Figure 3.2.4: Skin Effect in a Slot Embedded Conductor

83

84

Chapter 3: Coupling Finite Element Models to External Electric Circuits

/; I1(

Figure 3.2.5:

Transient Flux Distribution in a 'C' Core Inductor

G. Bedrosian and M. V. K. Chari

85

solution will not be adequate. First, nonlinearity, an integral part of the analysis, cannot be taken into account accurately and in a strictly mathematical sense. Second, transient response can be simulated only by combining several harmonic responses by the fast Fourier transI~rm or equivalent methods. A truly timedependent formulation, therefore, becomes necessary. In the following section, modeling of transients in a stationary apparatus such as an inductor or a transformer will be described. The jm terms in the differential equation and finite element equations are replaced by explicit time differentiation: OA V x v (V x A) + c~~ = Js (3.2.23)

/ (a)

(b)

Figure 3,2.6: (a) Flux-Distribution on No-Load for the PM Motor (b) FluxDistribution in the PM Motor on Load

86

Chapter 3: Coupling Finite Element Models to External Electric Circuits

F=f;[v

~(V ×A),

(V × A ) + A .

(eaA -2 -at -"

Js)]

dV

(3.2.24)

j'j'[

(V x W i ) ,

v (V x A) + W i ,

aA (c ~ -

Js)

]

dV=0

(3.2.25)

~ e flux distribution at one time instant in a 'C' core inductor is illustrated in Fig. 3.2.5. The application of the nonlinear time-dependent analysis to rotating machines is somewhat more complex, and requires fuller treatment in a later section of this chapter.

3.2.5: Permanent Magnet Sources In modeling pe~anent magnet devices in two dimensions by the vector potential method, the following assumptions are made in addition to those that have been described in magnetostatic analysis: i. The source function for permanent magnets is representexl by a magnetic moment distribution; ii. For samarium cobalt magnets and ferrites, the magnetic moment is assumed to be unidirectional and uniform over the pole cross-s~tion; and iii. For Alnico magnets, the moment distribution is considered non-uniform, having more than one component. The governing partial differential equation, and the energy related functional or the weighted residuN proc~ures are V x v (V x A - ~t0 M 0) = Js (3.2.26)

F-ff [ f

f[(v

2(V

x A),

x Wi),,v

(V x A - 2 ~0 M 0 ) -

A • Js

(V x A - ~ 0 M 0 ) - W i " J s ] d V = 0

dV (3.2.27)

(3.2.28)

Flux plots obtained by Gross (1988) for the two-dimensional geometry of a pe~anent magnet device is illustrated in Figures 3.2.6. This section has presented a brief description of the finite element method as it applies to typical magnetostatic and eddy current problems in electrical machines. Readers interested in more details can consult (Chm-i 1991) or standard references on the finite element method for electromagnetics.

G. Bedrosian and M, V. K. Chad

87

3.3 Circuit Models of Electrical Machinery In this section, we examine models of electrical machines from a perspective complementary to finite elements. The source of currents (or voltages) in a finite element problem must perforce lie outside the finite element mesh itselt, for to model an entire electrical power grid by one huge finite element mesh is a practical impossibility. Furthermore, there may be nonlinear devices in the power source, such as diodes and thyristors. Consequently, the design engineer must use a circuit model of the machine when he considers the electrical requirements of external connections. The simplest possible circuit model for an electrical device is shown in Fig. 3.3.1. An external power source provides I a m ~ r e s at V volts to an electrical device with resistance, R Ojhms, and inductance, L Henries. V and I are related by the familiar equation V = RI + L b~ 0t

(3.3.1)

Of course, R and L are not known a priori, but must be deduced from the results of the finite element analysis, as shown schematically in Fig. 3.3.2. There are two basic methods for doing this. In the first method, one assumes a current, I, a number of turns, N, and the total cross-sectional area of the circuit in the finite element mesh, A, to arrive at the source current density, J, to be used in the finite element solution by the methods of 3.2 above: NI J = -(3.3.2) A

1 R

Figure 3.3.1: Simple Circuit Model of an Electrical Device

88

Chapter 3: Coupling Finite Element Models to External Electric Circuits

Figure 3.3.2: External Source Coupled to Finite Element Model V is then found by suitable post-processing of the finite element solution. Another approach that is more useful when the source voltage, V, is given instead of the current, is to let the current density be split into two parts, Js and Je, where Js is due to a quasi-static ~tentia! equal to V and Je is the part left over (the "eddy c ~ n t " ) : J = Js + Je (3.3.3) In this formulation, Js is the source term and Je is found by post-processing the finite element solution. Although this m e t h ~ seems to be at first glance merely a mathematical artifice, it is in fact possible to ascribe physical meaning to the separation of J into Js + Je. We will cover this topic in detail in Section 3.5 below. When the finite element model is linear, or can be approximated as linear, one can solve a time-varying problem in the frequency domain where we can represent I and V at one frequency, f, as complex quantities given by I = I0.O~t (3.3.4) V = V ~ °~t (3.3.5) where ~ is 2=f, t is time, and I 0 and V 0 are complex constants. Equation 3.3.1 becomes V = R1 + j0~'LI (3.3.6) The effect of R and L together can be given by the complex impedance: Z= V T = R + j~L (3.3.7) Note that Z is a function of frequency and - - strictly speaking - - only makes sense for linem" m~els. We will also consider multi-circuit (or multi-port) models, as shown in Fig. 3.3.3 for four circuits. The im~dance is now a square, symmetric matrix, [Zl, relating the voltages on each of the circuits, Vi, to the cu~ents on each of the circuits, Ij:

G. Bedrosian and M. V. K. Chaff

89

+ ~-- ~ 4--F--F--t

J._l_ L J _ I _ . L I I I ! I I I "1"-1-- ~ ~ - - ~ T ~ J- -J-- ~- ~ - - I - - 4- -1

Figure 3.3,3: Multi-Circuit (-Po~) Model

Vi =

ZijI j (3.3.8) j=l where N is the number of circuits. The matrix inverse of [Z], the admittance matrix, [Y], gives the currents in t e ~ s of voltages: N Ii =

YijVj (3.3.9) j=l The terminal characteristics of any linear finite element model are completely determined by [Z] or [Y]. In the next sections, we will present methods for calculating [Z] or [Y], using a succession of examples of increasing complexity. In the final example, we will switch from the frequency domain to the time domain to solve a nonlinear problem with a generalization of the impedance matrix concept. The reader should note that the electrical devices modeled in the examples are not meant to be practical - - in the sense of practical things that one would actually build and use - - but only serve to demonstrate the methods.

3.4 Magnetostatic

Model

of a Transformer

In this section, we witl present an energy-based method for calculating the impedance matrix, [Z], for finite element models. This method is suitable for linear models without appreciable eddy current skin effects. (We will describe these effects in detail in Section 3.5 below.) Consider the simple two-dimensional transformer model shown in Figure 3.4.1. In the discussion to follow, we will assume that all quantities are given per unit length in the third (unmodeled) dimension. The region marked +Jl in Fig.

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Chapter 3: Coupling Finite Element Models to External Electric Circuits

Figure 3.4.1: Two-Dimensional Model of a Two-Cimuit Transformer

3.4.1 is the primary winding. Its return is the region marked NII1 J 1 - AI

-J1. Jl is given by (3.4.1)

where N 1 is the number of turns and A 1 is the cross-sectional area. The current density in the secondary winding, J2 is similarly given by N212 (3.4.2) J 2 - A2 There are four complex values in the impedance matrix for this transformer of which two are equal: Zl 1 = RI I + jc0Ll 1 (3.4.3) Z22 = R22 + jmL22 (3.4.4) Z12 =jcoL12 =jcoL21 = Z21 (3.4.5) LI2 is often termed the mutual inductance, while L11 and L22 are called the selfinductances. Because we are neglecting sNn effect, the resistive components, R11 and R22, can be calculated without resort to finite element analysis. Consider the distribution of wires in the primary winding as shown in Ng. 3.4.2. Although the winding is considered to occupy a rectangular area, A1, in the finite element model, the conductors acmally fill a smaller area because of the insulation and geometric considerations. If there are N1 turns of al area each, then we can define a fillfactor, fl:

G. Bedrosian and M. V. K. Chari

91

Nlal (3.4.6) f I - A1 The fill-factor will always be less than one. If the resistivity of the wire material is r 1, then the total resistance of the ~ m a r y winding, considering the return path, is RII -

2NI P 1 2NlP al - f l A l

(3.4.7)

R e resistance of the secondary can be calculated similarly. (Note again that all quantities in this two-dimensional model are per unit length.) The inductance terms are calculated with the finite element method using energy. ~ e stored magnetic energy in the finite element model is given by Energy = ~

vB2dR

(3.4.8)

VoI The numerical value of this integral is provided either in the output report of the finite element analysis program or by the post-processor in most electromagnetic finite element software. The stored magnetic energy as a function of pfima,~y and secondary, currents, I1 and 12, is Energy(II,I2) = ~ L1 II +

+ L121112

(3.4.9)

To find the individual inductance terms, one solves three finite element problems: one solution with 1 Ampere in the primary, and zero in the secondary, one solution

Figure 3,4.2: Wires Fill a Fraction of the Total Area

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Chapter 3: Coupling Finite Element Models to External Electric Circuits

Figure 3.4.3: Flux Plot of Transformer Model

G. B~msian and M. V. K. Chari

93

with 1 Ampere in the secondary and zero in the primary, and one solution with I Ampere in both. The inductance terms are then easily calculated as L11 = 2 Energy(l,0) (3.4.10) L22 = 2 Energy(0,1) (3.4. I 1) 1 1 L12 = Energy(I,1)- ~ L l l - ~ L22 = Energy( I, 1) - Energy(1,0) - Energy(0,1 ) (3.4.12) Fig. 3.4.3 shows the flux lines as plotted by a finite element post-processor for the case with I Ampere in the primary and zero in the s ~ o n d ~ y . The C-shaped transformer in this example is 3 cm wide by 5 cm tall with a 2 mm air gap and g = 1000 g0; the winding regions are each 1 cm square with 100 turns. Using our own finite element code, we calculated the following values: Energy(I,0) = 0.050844 J]m (3.4~13) (3.4,14) Energy(0,1) = 0.050843 J/m Energy(I, 1) = 0.188701 J/m (3.4.15) whence (3,4.16), L11 = 0.101688 ~ m (3.4.17), L22 = 0.101686 I-[/m (3,4.18). L12 = 0.087014 H/m Note the difference in the last digit between L I 1 and L22. This is due to the finite element discretization errors in the model, and is typical of the numerical results one should expect from finite elements. Another quantity of interest to electrical engineers, related to Lij, is leakage inductance. The leakage inductance of circuit 1 with respect to circuit 2 can be defined as the difference between the inductance from flux linking circuit I and the inductance from flux linking circuit 2 when circuit I carries current, corrected for the turns ratio. In effect, this represents flux "lost" from the circuit into surrounding space and it is often expressed as a dimensionless ratio with respect to the self-inductance. In the example above, we have N1 LLeakage = L11 - ~ LI2 = 0.014674 H/m = I4.4 % (3.4.19). This method of calculating inductances can be generalized to multiple-circuit transformers by considering each circuit by itself to get the self-inductance using eqn. 3.4.10, and each pair of circuits to get the mutual inductance using eqn. 3.4.12. Unfortunately, this requires the solution of N ( N + I ) / 2 finite element problems for N circuits. We will develop a method in the next section that can find the impedance matrix for N circuits with only N finite element solutions.

3.5 Eddy Current Model of an Induction Motor with Locked Rotor In this section, we will deal with solid conductors that are large enough for ~ d y currents to modify significantly the distribution of current density in the

94

Chapter 3: Coupling Finite Element Models to External Electric Circuits

aq tS"

17

Figure 3.5.1: Simple 3-Phase Induction Motor

G. Bedrosian and M. V. K. Chari

95

conductors. We will also introduce a computationally efficient method for dete~ining the impedance matrix. The model in Fig. 3.5.1 is a simple three-phase induction motor with a locked rotor. The stator yoke is assumed to be a non-conducting (laminated)linear material with g = 1090 gO. The rotor material is non-conducting and nonmagnetic. The outer radius of the motor is six inches. All conductors are solid copper bars. The stator conductors are wound in two layers with the A, B, and C phases as shown. Each phase is driven by a voltage source at 60 Hz with a peak value of 120 volts (per meter). The rotor conductors are wound in three layers and assumed to be connected in series at the ends. The series connections in this motor present an interesting problem for finite element analysis, because neither the current nor voltage in any given layer are known a priori. Instead, we have circuit constraints on the conducting regions as follows: i. Each layer in each series circuit ca~ies the same total current, but the exact values are unknown until the problem is solved. ii. The total voltage in the A, B, and C phase circuits is 120 volts (per meter) times the appropriate complex phase constant. iii. The total voltage in the rotor is zero (short circuit). We will solve this problem in three steps. First, we will use finite element analysis to determine the impedance matrix for the motor, where each separate conductor is considered individually. (There are eighteen separate conductors: t~ur each for phases A, B, and C and six for the rotor) Second, we will use this matrix to construct a linear system of equations for the four unknown currents. Finally, we will construct the solution as a linear superposition of the trial solutions that were used to calculate the impedance matrix. Let us return our attention briefly to the finite element formulation for eddy currents, with more emphasis on the details of coupling to the electric field, E, than the exposition in the previous section. We will solve for the vector potential, A. (In the two-dimensional model, A has only a z component) The flux density, B, and electric field, E, are B = V x A

(1.3.7)

OA E - - ~t - V V

(3.5.1)

where V is the electric scalar potential (we will discuss V below.) The total current density, J, can be divided into two parts, Js and Je: Js = gE = Js + Je (3.5.2) Js = - cr V V (3.5.3)

The simple motor in this example is for illustrative purposes only. tts winding patterns and geometrical configurations would prove completely unsuitable for a real motor!

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Chapter 3: Coupling Finite Element Models to External Electric Circuits

aA Je = - e at

(3.5.4)

The total current in conductor j, Ij, is found by integration over the cross-section of the conductor:

Ij

= ..IC ond,j Os + Je) * dS

(3.5.5)

If we assume the Coulomb gauge for A, V• A =0 (I.3.I5) then V is the physical voltage difference that one would measure with an instrument (Carpenter I977). Using the normal sign convention of a positive voltage drop for positive current, we have V V V - - length uz (3.5.6) where Uz is the unit vector in the z direction. We will assume unit length. Then Js is given by Js = crVuz (3.5.7) From this, we see that there is a direct correspondence between Js and t e ~ i n a I voltage. If we consider V to be known and A unknown, then the partial differential equation tbr A is aA V x (v V x A) + o ~ = oVuz - Js (3.5.8) Consider the solution of a problem in which conductor i has V = 1 and all other conductors are shorted (a trial solution): 1;k=i Vk = 0; k ~ i (3.5.9) If !j is d e t e ~ i n e d by eqn. 3.5.5 above, then the admittance matrix components are immediately given by Yji = Ij (3.5.10) This process can be repeated for all of the conductors, i = 1 to N, completing the admiuance matrix, [YI. Having [Y], [Z] is found by matrix inversion:

[Z] = [Y]-!

(3.5.1 I),

where, it is to be noted, [Y] and [Z] are symmetric. This completes the first step. With this method, the entire impedance matrix can be d e t e ~ i n e d by N finite element solutions, rather than N(N+I)/2 as in the method of the previous section. In either case, it is the same finite element matrix equation that is being solved, but different right-hand sides. It is best to use a matrix solver that is suited to solving the same matrix ~uation with multiple right-hand sides. The authors have been successful using the nested dissection method for this type of twodimensional finite element analysis (George, 198 I).

G. Bedrosian and M. V. K. Chad

97

The second step is to build a system of linear equations for determination of the currents in the three phases (IA, IB, IC) and in the rotor (IR). Table 3.5. ! relates the conductor n u m ~ r in the model to the circuit. With the circuits defined as above, the four voltages are V A = V 1 + V 2 - V 3 - V 4 = 120 (3.5.12) VB=V 5+V 6-V 7-V 8=120e

3

(3.5.13)

4, J Vc=V9+VI0-VII-V12= 120e 3 (3.5.14) VR = V13 + VI4 + VI5 - VI6 - VI7 - V18 = 0 (3.5.15) The - signs in eqn. 3.5.12- eqn. 3.5.15 above reflect the fact that the return conductors are connected in reverse fi~om the convention of positive voltage drop in the positive z direction. As a matter of convenience, we introduce the notation: signi = +1 or -I, depending on the sign of the connection in Table 3.5.1. Using eqn. 3.3.8 and eqn. 3.5.12, we obtain:

4

Z

j

signi signj Zij I A +

j=l +

~

?

Z

sign i signj Zij IB

j=5 signi signj Zij Ic+

Conductor

Z signi signj Zij I R = V A j=13 (3.5.16) Circuit

1

A+

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

A+ AAB+ B+ EC+ C+ CCR+ R+ R+ RRR-

Table 3.5.1: Circuit Connections

98

Chapter 3: Coupling Finite ELement Models to External Electric Circuits

Figure 3.5.2: Flux Plot of 3-Phase Inductor Motor

G. Be~rosian and M. V. K. Chaff

99

The other three equations are obtained similarly. This linear system is easily solved for the unknown currents, completing the second step. The third and final step is m calculate the actual finite element solution for the currents as determined above, The voltage on each conductor can ~ calculated with eqn. 3.3.8. The finite element solution can then be obtained by a linear supe~osition of the trial solutions calculated for the first step (if they were saved), or the finite element matrix equation can be solved one last time. Fig. 3.5.2 shows the flux lines of the real part of the solution.

3.6 Rotating Grid Model of a Generator As useful as the above methods are, they fall short for many problems relating to the operation of rotating machines connected to external circuits: i. If the analysis is performed in the fi-equency domain, the materials and external circuits must be assumed to be linear; ii. The rotation of the machine cannot be directly modeled, including slotpassing effects; iii. Torque and speed variations cannot be coupled with the analysis; and iv. Transient phenomena cannot be studied without recourse to Fourier transforms. In this section, we will generalize the analysis methods discussed thus far in order to remove the above limitations. Salon et al (1990) have done extensive work in the area of coupling finite element models with moving grids to external circuits. The method presented here expands on this idea to allow coupling to general, nonlinear circuit models and equations of motion without sacrificing the sparseness and well-conditioning of the finite element matrix equation. We will present an example of this method at the end of the section. If A k+l and A k are the vector potential solutions at time steps (k+l) and k, respectively, and if (for simplicity in this discussion) we use a first-order forward difference to approximate the time derivative, bA A k+l - A k Ak -0t = t k + l . t k - A t (3.6.1) where Ak is given in eqn. 3.2.10 above. Combining the Newton-Raphson scheme eqn. 3.2.11 with the time-domain expression for the eddy current problem eqn. 3.2.25 and the expression for permanent magnet sources eqn. 3.2.28, we obtain

[ ( V × W i ) - v (V x Ak+l)

+

0v IBkI~)IBI(V x W i ) - B k ( V x A k + 1 ) , B k + w i . ~ . c Ak+l x Wi),v~0M0-(V

× Wi) o v B k

Wi • Js]dV=0

(3.6.2)

I O0

Chapter 3: Coupling Finite Element Models to External Electric Circuits

Figure 3.6.1: Elements Near Gap Before Stitching

G. Bedrosian and M. V. K. Chaff

Figure 3.6.2: Elements Near Air Gap After Stitching

I0I

102

Chapter 3: Coupling Finite Element Models to External Et~tric Circuits

Note that eqn. 3.6.2 is valid in the frame of reference of the material with conductivity e. If a different reference frame is used, with a local velocity, v, with respect to the material, then an additional t e ~ proportional to v x B must be added to the eddy current. The most direct way to avoid this problem is to use separate finite element grids that are fixed to the rotor and stator, respectively. In regions where e = 0 (like the air gap of a rotating machine), there is no eddy current and so the frame of reference can be arbitrary,. At each time step, the rotor grid is rotated to its new position and a new set of elements is constructed to join the two parts by "stitching" across one layer of the air gap. The grid stitching process is illustrated in Figs. 3.6.1 and 3.6.2, showing details of the finite element grid near the air gap before and after stitching respectively. The element outlines are drawn slightly smaller than the elements themselves ("element shrink plot") so that the outlines of material regions remain visible. Other than providing for a compatible air gap spacing between the rotor and stator grids, the two parts have separate meshes and need not have the same element size or number of nodes along the air gap. Note that while the rotor and stator grids can have quadrilateral elements, the air gap is meshed with triangular elements only. Using triangular elements in the stitch region ensures that no matter how the gap is connected (depending on the relative position of the rotor with respect to the stator), there are the same numbers of elements and nodes in the final mesh at all time steps. This is pa~icularly important in the case of secondorder elements when the nodal potentials at each time step are storeA in one directaccess data file for later post-processing. Having the finite element model of the machine we wish to analyze, the next step is to create the external circuit model. To create the circuit model, we introduce the concepts of devices and voltage nodes, or v-nodes.* A device is a circuit element with two or more leads through which current flows. For simplicity, we will consider only devices with two leads. Associated with each device is one value of current; any current flowing into the device through one lead must flow out through the other. A v-nocle is a connection in the circuit to which one or more leads are attached. The sum of all currents flowing into a wnode must be zero. Each v-node has an electrical potential with respect to ground; ground is arbitrarily assigned a potential value of zero as a boundary condition. If the number of devices is ND and the number of wnodes is NV, then the circuit equation has ND+NV unknowns. There are NV equations resulting from enforcement of zero net current at each v-node. The remaining N D equations are obtained with one equation of state for each device. For example, consider device k

We use *'device" rather than "circuit element" to avoid confusion with "finite elements". Similarly, we use the "v-" modifier to differentiate circuit wnodes from finite element nodes.

G. Bedrosian and M. V. K. Chaff

103

carrying current I k. If the device is a switch connecting v-nodes i and j at potentials Vi and Vj, respectively, the equations of state are

[ .......... t A-

A+ i

Figure 3.6.3: A Simple Three-Phase Generator

104

Chapter 3: Coupling Finite Element Models to External Elex:tric Circuits

Region I Region 2 Region 3

Fq .................i

Region 4 Region 5 Region 6

'

Figure 3.6.4: Rotor Circuit Ik = 0 (open) (3.6.3) V i = Vj (closed) (3.6.4). If device k were a resistor with resistance R ohms, the equation of state would be R Ik = V i - Vj (3.6~5). State equations for other devices are derived using standard circuit theory. In order to couple the external circuit elements with the finite element model, we consider the finite element regions (those that carry current) to be devices as well. There are two basic types of devices in the finite element model: solid conductors and stranded conductors. For a solid conductor, the voltage drop [V in eqn. 3.5.71 can be specified, but not the total current. It must be calculated using eqn. 3.5.5. For a stranded conductor, the s term in eqn. 3.6.2 drops out. The total current density, averaged over the cross-section, is identical to the source current density and is calculated with eqn. 3.4.1 for a given current. The voltage drop, again averaged over the cross-section, can only be calculat~ once A is known: VDr°p = Area E - ~

+ I'-~

ff

dS 1

(3.6.6)

where N is the n u m ~ r of turns, I is the current in each turn, p is the resistivity of the wire material, and f is the fill-factor. The first term in eqn. 3.6.6 is the voltage drop due to resistance, while the second term is the voltage drop (negative the induced voltage) due to flux linkage. Given these properties, our strategy for evaluating the state equations for finite element regions is as follows: i. Set the voltages on solid conductors and currents in stranded conductors at

G. Bedrosian and M. V, K, Chad

105

A

Region 10 A+ Re~ion 11 A

Region 12 B+ Region 13 B-

~" ~

Region 14 C+ Region 15 C-

~

-'~ B Neutral C

Figure 3.6.5: Armature Circuit their values l¥om the last time step and calculate a trial value for A with the finite element method. ii. With the trial value for A, compute the total currents in solid conductors and voltage drops in stranded conductors,. iii, For each conductor in turn, set its voltage drop to one unit (if solid) or its current to one unit (if stranded), Set all other voltage drops (for solid conductors) or currents (for stranded conductors) to zero and solve the finite element matrix equation again for the change in A. iv, With the change in A, calculate the voltage drop changes in stranded conductors and the total current changes in solid conductors. In this way, one can calculate the effect that changing voltage or current level in a finite element region will have on the voltage and current levels in all regions. For a stranded conductor, the state equation will relate the change in the difference in potential at the two ends to the changes in potentials and currents in all of the other conductors. For a solid conductor, the state equation will relate the change in current to the changes in potentials and currents in all of the other conductors. Steps (i) and (ii) are used to set the right-hand sides of these state equations. In general, this part of the circuit matrix will be full while the parts relating to external devices will be sparse. Note that a key feature of this approach, which is similar to the method in 3.5 above, is that the same finite element matrix equation is solved NC+I times tor different right-hand sides, where N c is the number of conducting regions. The authors have found the nested dissection method (George and Liu, 1981) to be suited t::)r this application because it solves multiple righbhand sides quickly after the matrix is decomposed.

i06

Chapter 3: Coupling Finite Element Models to External Electric Circuits

Having the state equations for the conductors in the finite element model, one can add the state equations for the external devices and solve the total circuit matrix equation. It will be smaller than the finite element matrix equation, having only a few hundred unknowns at most rather than tens of thousands in the finite element equations. A direct solver for full matrices using Gaussian elimination is satisfactory. If any of the external devices is nonlinear, the circuit equations can be iterated without solving the finite element matrix again. Nonlinear external devices are handled by iteration of the circuit equations, but what about nonlinear materials in the finite element model? As the algorithm has been stated so far, it is equivalent to one Newton-Raphson iteration per time step in terms of tracking the nonlinear behavior of v. One can easily add an outer loop to this process in order to converge on the "exact" value of v at the end of a time s t e p - but does that make sense? Since v is changing over the time interval from t = tk to t = tk+l, using the value at the end of the interval as a constant value throughout the interval is not strictly correct in any case. The authors have found it better to perform one Newton-Raphson iteration per time step. If the changes in v are too large to track with one Newton-Raphson iteration per time step, it is better to decrease the time step size. To conclude this section, we will apply these ideas to the simple two-pole, three-phase generator model shown in Fig. 3.6.3.* The finite element model has total of 1,082 second-order elements (quadrilaterals and triangles) and 2,993 nodes. Individual finite elements are not shown, only material region boundaries. The region numbers relevant to the external circuit connections are labeled in the figure. The rotor windings are solid copper, while the stator windings are stranded copper wire with I00 turns each. The circuit connections of the finite element regions are shown in Figs. 3.6.4 and 3.6.5. The rotor and stator are made of laminated (nonconducting) magnetic steel whose B-H curve is shown in Fig. 3.6.6. The generator is connected to a diode bridge rectifier with an inductive load to smooth the cu~ent, as shown in Fig. 3.6.7. The following features make this case difficult to solve with conventional lumped circuit models or static finite element models: i. The steel material is heavily saturated; ii. Slot-passing effects are important; iii. The electrical output is rich in ha~onics; and iv. The external circuit is nonlinear. For the same reasons, a nonlinear rotating grid analysis is ideal for this case. One of the more difficult tasks in a time-domain finite element analysis is to set the initial conditions. In our example, we will assume that outputs of the generator are initially open-circuited. We will model this by setting resistors R3 in Fig. 3.6.7 to 100 M~. By running the generator for one cycle, the output comes to a steady state from an initial condition of zero for all quantities except the Once again, the simple generator in this example is for illustrative purposes only and does not represent a practical rotating machine.

G, Bedrosian and M. V. K. Chad

107

rotor field current. The state of the system at the end of the cycle can then be used as the initial condition for subsequent analyses. Having a valid initial condition, we will model a transient event in the generator, in this case switching on the load. The initial values of the circuit elements are listed in Table 3.6.1: After one cycle, we suddenly change the values of resistors R3 in Fig. 3.6.7 m 0,1 f~. We then run the analysis for two more cycles. Before we examine the voltages and currents in the circuit as functions of

A285 s t e e l

1.5

.5

~L 8

____L

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

~

| ........................................................................................................................................................................................................

18888

K~:gnetic Induction (teslo)

28008 vs.

38888

I~gnetic Field (o~-turn/m)

Figure 3.6.6: B-H Curve for Nonlinear Steal

48888

108

Chapter 3: Coupling Finite Element Models to External Electric Circuits

.....I A B

:7

C

-5 R3

R4

b

Figure 3.6.7" 3+Phase Diode Bridge time, several features of the model require discussion. Resistors R3 were included in the circuit not only to serve as switches (of a type), but to prevent short circuits during the analysis. It is possible that during the circuit iterations, enough of the diodes in the bridge can be "on" to allow local current loops+ When the circuit equations finally converge, of course, this will not be the case. But during the iterations, the temporary states of the diodes can cause a singularity in the circuit state equations. Similarly, resistors R4 serve to ground their respective parts of the circuit in case all of the diodes are "off" at some point in the circuit iterations. The diodes in the bridge are controlled by switching them between two states: If a diode is currently off, it switches on when the potential difference across it exceeds a set threshold, in this case, 0+7 V+ If a diode is currently on, it switches off if its current is negative+ The circuit state equations are solved until the diode states converge. If a diode oscillates between circuit solution iterations without converging, it is arbitrarily turned off after a maximum number of iterations to maintain stabi!ity~ In the finite element model, time steps are selected such that the rotor angle changes by five degrees (counter-clockwise) every time step. 72 time steps constitu~ one revolution. If the generator operates at 60 Hz, one time step is 0+23148148 msec+ Each time step requires about 13 CPU seconds (computer time) on a Digital Equipment Corporation DECstation 50C~)/200 workstation, including

G. Bedrosian and M. V, K. Chari

46606r

........

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

i ...........

109

i

~ .............

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

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

2B~Oe

/

-40080 L 8

8.81

6.62

6.83

B.84

Figure 3.6.6: Phase A Voltage (Volt) Versus Time (Sec)

8.85

! 10

Chapter 3: Coupling Finite Element Models to External Electric Circuits

400

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

~

.....

~ ..................... ~ ........................ ; .................

~........................

~ ......

;

#

...........

388 I-" 200

1O0

-168

-2OO

-300

-400 L O

O.0!

0.02

0.03

0.04

Figure 3,6.9: Phase A Current (Amps) Versus Time (Sec)

0.~

G, Bedrosian and M, V. K. Chad

40B

11I

~ .......................... i .........................

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

~

~......

~

....

300

200

lee .................................................. .~.................................................................................

! 0

8.81

/i

e.82

e.83

e.04

F i g u r e 3.6.10: Load Current (Amps) Versus Time (Sec)

0.05

1 12

Chapter 3: Coupling Finite Element Models to External Electric Circuits

Figure

3.6.11: Flux Plot With No Load

G. Bedrosian and M. V. K. Chart

Figure 3.6.12: Flux Plot With Steady Load

113

114

Chapter 3: Coupling Nnite Element Models to External Elect6c Circuits

circuit solution. The total computer time for an analysis of three revolutions is about 45 minutes. Larger models would take more time, of course, but this indicates the viability of the method for routine design studies. We now turn to the results of the analysis. Fig. 3.6.8 plots the phase A voltage (with respect to neutral) versus time. Fig. 3.6.9 plots the phase A current versus time. Phase B and C voltages and currents are similar, but shifted in time. Fig. 3.6.10 plots the cu~ent in the load (through the inductor) versus time. Note the characteristic L/R time constant of 5 msec and the ~ harmonic ripple. Circuit Elements I R1 L1 R2 R3 R4

Value 20,000A 100fl 0.5H 0.I~ 100M~ 1MO

Table 3.6.1: Initial Circuit Values Flux plots of the finite element solutions are also of interest. Fig. 3.6. ! ! is a plot of the magnetic flux lines after one revolution with no load. Fig. 3.6.12 is a flux plot at the end of three revolutions. Note that the magnetic angle lags the mechanical angle by about 45 degrees. Rotor kinematics can also be incorporated into the analysis. The electromagnetic and kinematic equations of the machine are linked through the torque on the rotor. The total rotor torque can be efficiently calculated at each time step using the method of virtual work (Coulomb and Meunier, 1984). Salon, DeBortoli and Palma (1990) connect the kinematic equations directly to the finite element equations and solve one large system. Another approach is to take advantage of the high rotational inertia in most machines to couple the kinematic and electromagnetic equations indirectly. The strategy is as follows: i. Estimate the next rotor position using the present angular velocity and angular acceleration. ii. Solve the finite element + circuit equations as before. iii. Calculate the torque on the rotor and solve the kinematics equations. iv. Correct the rotor position and r e , a t . This strategy works because the angular velocity of the rotor does not change much from time step to time step, although the integrated change over dozens or hundreds of time steps can be large.

3.7 Conclusion This chapter has presented several methods for coupling finite element models of

G. Bedrosian and M, V. K. Chari

115

electrical machinery to external circuits. The simple energy-based method fbr calculating inductances in Section 3.4 is limited m cases where the eddy current effects are negligible and the materials are linear, Its simplicity makes it easy to use with almost any two- or three-dimensional electromagnetic software package. In cases where eddy currents are important but the materials can still be treated as linear and motion can be either neglected or handled by a coordinate transformation, the method of Section 3.5 is computationally efficient and suitable for coupling with linear external circuits by way of the impedance matrix, [Z]. Once [Z] has been calculated and stored, it can be used in an external circuit analysis for any excitation (voltage and/or current sources) without direct reference to the finite element model itself. The general time-domain method of Section 3.6 is applicable to cases with nonlinear materials, motion of the finite element grid, nonlinear external circuits, and switching transients. Although we have taken care to retain the efficiency of the finite element solutions by preserving the sparsity and well-conditioning of the finite element matrix equation, a time-domain analysis will generally consume far greater computer resources than a frequency-domain analysis at one or a small number of discrete frequencies. Not only must the finite element matrix equation be solved hundreds or thousands of times, but even storing the solution is a problem. For example, if the grid has 5,000 nodes and the solution is computed at 200 time steps, storing the nodal potentials in single precision takes 4 Mbytes of disk space. Nevertheless, as high-performance workstations with large disks become commonplace, this type of analysis will take its place as a routine design tool along with the static and frequency-domain techniques. While we have devoted this chapter to two-dimensional analysis techniques, three-dimensional analysis will become increasingly practical in the future. End effects play an important role in the evaluation of machine performance, and these effects are difficult to calculate without some kind of three-dimensional analysis, whether by finite elements, closed-fo~ expressions, or other methods. Estimation of inductances (or capacitances) for three-dimensional models using the static method of Section 3.4 is possible with today's compumr software and hardware. Reliable and practical three-dimensional eddy current analysis of the type in Section 3.5 is a current research topic; applications may be available to users within a few years. Time-domain analysis for three-dimensional finite element models with rotating grids must wait for the next generation of engineering workstations perhaps with massive parallelism - before feasibility. With the rapid growth in power and reduction in cost of computers continuing unabated, it seems inevitable that tba'ee-dimensional analysis of electrical machinery wilt be commonplace by the end of the century.

Chapter 4 ...............

Ill

}

S.J. Salon, C. J. Slavik, M. J. DeBo~oli and G: Reyne /lilll

I

........

I

IIIIIII

Illll

}l

.....

}llll

Ill

NUll

}ll . . . . . . Illl . . . .

{]]]]

I

I

I][I

ANALYSIS OF MAGNETIC VIBRATIONS IN ROTATING ELECTRIC MACHINES

4.1 Introduction The design of electrical machinery requires allowance tor dimensional and manufacturing tolerances on every part; for example, tolerances occur on stator and rotor punchings, frame dimensions, bearing clearances, etc.. The larger the tolerances in the manufacturing process, the lower the cost of manufacturing the machine. Dimensional tolerances, however, can, and often do, impact machine performance, for example, as in operating efficiency, reliability, and production of noise and vibrations. In this chapter, the term "noise" refers to airborne sound, and the term "vibration" refers to periodic structure-borne motion at a specific frequency,. The sources of noise and vibrations within electrical machinery are numerous. The particular machine design determines those which are most important. Some examples are: mechanical vibrations due to dynamic rotor unbalance or rubbing and rolling motions in the bearings; airborne sound due to pressure fluctuations generated directly in the air; and oscillating electromagnetic forces generated within the machine's magnetic and electric circuits. The relative importance of each source depends on the design and size of the machine; however, all sources are present in all machines, to some extent. This chapter will discuss methods available for the analysis of vibrations produced within electric machines by forces of magnetic origin and the insight and understanding of these forces brought about by application of the method of finite element analysis. In the following sections the source of magnetic forces is traced

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

I 17

to the harmonics of the magnetic flux density distribution. The leading methods for analyzing the flux density are described, culminating in the method of finite element analysis, including examples of its application to specific machines.

4.1.1: Magnetic Vibrations Magnetic vibrations, or ele~tromagnetically excited vibrations of the stator core or rotor in electric machines have been actively studied for more than seventy years and continue to pre~nt challenging problems to the researcher and modem machine designer. This is due, in part, to the following: the complexity of the problem of analyzing the magnetic field within electric machines; the evolution of machines with increased power density and hence increased electromagnetic loading; and the continued development of special areas of application where very low levels of vibrations are of critical importance. The sources of magnetic vibrations are the magnetic forces of attraction or repulsion between the rotor and stator. These forces occur in two distinct frequency ranges. The low frequency range includes slip frequency and once, twice and higher multiples of the line frequency. The high frequency range includes frequencies related to the product of the number of rotor slots and the rotational speed of the rotor; typically these frequencies lie above one kilohertz, but may be as low as a few hundred hertz for machines with a large number of poles. In the low frequency range the magnetic vibrations are dominated by the flux flowing between the stator and rotor surfaces producing rotating force waves with twice the frequency and half the pole pitch of the fundamental flux field. In the high frequency range, the air gap flux ripple, caused by the interaction between the stator and rotor winding m.m.f, distributions and the permeance fluctuations due to rotor and stator slot openings leads to the dominant components of magnetic forces and vibrations. Other combinations of stator and rotor space harmonics and saturation harmonics can lead to important magnetic vibrations, as shown in the example given in Appendix 4.B at the end of this chapter.

4.1.2: Magnetic Forces The most important forces for the designer of quiet electric machines are due to the magnetic flux harmonics in the air gap. Magnetostriction and magnetic forces between laminations, which produce significant vibrations in transformers, are usually not important sources in rotating machines (Alger 1954). This is because the dimensional changes produced by the magnetostriction effect are out of phase, except over short lengths of the flux path, and can therefore be neglected for most practical considerations of machine vibrations. Magnetic flux can cross the interface between air and an infinitely permeable material only in a direction perpendicular to the surface of the material. A magnetic pulling force (tension) is developed in the direction of the flux, and its magnitude is given by the following formula (in units of pressure or force per unit area);

I! 8

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

B2 P=~

(4.I.1)

where B is the magnetic flux density in Wb/m 2 and g0 is the pe~eability of free space (4~ x 10-7 H/m). If the material is not infinitely permeable, the situation becomes more complex and the flux density at the interface contains normal (Bn) and tangential (B t) components. These flux density components must be combined, using the Maxwell stress formula (Carpenter 1960; Jackson 1962), to obtain the n o d a l and tangential components of the magnetic force (per unit area) acting on the material: 2 2 Bn - B t Pn 2g 0 (4.1.2) Pt - ta0 (4.1.3) The flux entering a machine's iron surfaces along their boundary with the air gap varies significantly with both spatial position and time because of the effects of rotor and stator slotting, the distribution of smtor and rotor currents, saturation effects, and air gap eccentricity l. The resulting magnetic forces change with spatial position and time, causing the machine to develop vibrations. As an example, the frequency and spatial distribution spectra of the magnetic flux of a cage induction motor are discussed in Appendix 4.A.

4.1.3: Magnetic Field Analysis There are several methods available to the designer to analyze the magnetic fields and forces in the air gap of an electric machine. The leading methods are discussed below.

4.1.3.1: Method of Conformal Transformation If the iron can be assumed to be infinitely permeable, the method of conformal transformation can be used effectively to determine the distribution of the air gap flux ~ound the teeth or poles. The radial magnetic force acting on the teeth or ~ l e s may be found by integration of the normal component of flux density over the tooth surface or pole face and the tangential force may be found by integration over the sides of the tooth or pole. These techniques have been used extensively by Binns and his collaborators (Binns 1964; Binns and Rowlands-Rees 1978) in the analysis of magnetic forces and torque ripple produced in induction and salient pole synchronous machines. The deficiencies of this approach are that such analysis is complicated if harmonic currents flowing in the stator or rotor conductors must be included. In addition, the inability to account for finite pe~eability effects, particul~ly saturation in the tooth tip or pole face, can lead to 4 The asterisk indicates that dF is not necessarily the true local force. I The airgap eccentricity may be due to either rotor offset or the stator being out-ofround.

S. J. Salon, C. J. S!avik, M. J. DeBortoli and G. Reyne

1! 9

unreliable results. Binns, however, claims negligible error if the relative permeability of the iron is greater than 25 (Binns 1960).

4.1.3.2: Method of M,M.F.-Permeance Waves A simpler and more flexible m e t h ~ of determining the magnetic fluxes and forces in the air gap is to calculate the flux distribution from the product of the m.m.f. and the permeance of the air gap boundaries. In magnetic circuits the m.m.f., flux, and reluctance (the inverse of the permeance) have a correspondence to e.m.f., current, and resistance in electric circuits. The distribution of both the m.m.f, and the permeance around the air gap may be represented in terms of a Fourier series. The basic concept of this approach is the implementation of Ampere's law and the assumption that the m.m.f, represents the force developed by the current to drive the flux across the gap. The spatial distribution of the m.m.f, is determined by integrating the current distribution around the air gap boundary and normalizing so that the total flux crossing the gap is zero. This approach yields a harmonic spectrum with a period of twice the pole pitch of the machine. As the stator current varies with time, the m.m.f, it develops is represented by a family of waves, the members of which vary sinusoidally in both space and time (see, for example, Equation (B.2) of Appendix 4.B). The m.m.f.-permeance wave approach consists of a one-dimensional view of the air gap field, consideration being given only to the radial field in the air gap. However, many quantities of importance in predicting electrical machine performance, such as torque, forces, losses, and magnetic vibrations cannot be described accurately by such a "lumped" parameter approach but insmad depend upon two-dimensional distributions of field quantities such as flux density. The magnetic field in an electric machine air gap is not generally onedimensional because the stator and rotor currents are contained in slotted iron boundaries. The actual field in an electric machine air gap resembles a onedimensional (homopolar) model only when considered far away from the current sources in slotted boundaries. In a homopolar region the m.m.f, and permeance waves are of considerably different wavelength (the relevant m.m.f, component changes only slightly over several slot pitches) and the m.m.f.-permeance wave approach is a reasonable approximation to the actual air gap field. However this approximation breaks down when the m.m.f, and permeance waves are of similar order because in these regions the field can no longer be described as homopolar but has ins~ad become heteropolar in form; the one-dimensional condition assumed in the permeance wave approach therefore becomes invalid. As an example of the m.m.f.-permeance wave approach, Appendix 4.B analyzes the magnetic vibrations generated by an induction motor, with saturation effects included. One fu~her point must be mentioned: because the stator and rotor conductors lie in slots which also generate permeance variations, the wavelengths of the flux density ha~onics due to these boundary effects are identical to the wavelengths of the m.m.f, harmonics from the current-carrying conductors. Therefore the m.m.I2permeance wave approach always provides the correct order of flux harmonics. Prediction of the flux harmonic magnitudes, however, is usually where the

120

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

approximation breaks down. The m.m.f.-permeance wave ag~"oach has been used extensively in the literature of both synchronous and induction machine vibrations because of its convenience and flexibility. References to key papers are Alger (1970), Liwschitz (I942), Walker and Kerruish (1960), Yang (1981), and Ellison and Yang (I 97 I). As discussed above, this method is most successful in predicting the frequencies and pole numbers of vibration-producing magnetic force waves and less effective in predicting the vibration amplitudes.

4.1.3,3: Method of Finite Elements The use of the finite element method to preAetennine the vibration characteristics of an electric machine occupies the remainder of this chapter. Using the finite element method it is possible to determine the air gap magnetic field, the magnetic force spectrum in an operating machine, and the spectrum of noise-producing vibrations. The finite element method (FEM) has several advantages over the other methods of analysis. Unlike the other methods, the FEM can model complicated geometries and saturable magnetic materials (both of which characterize electric machines) with relative ease. However, there exist some difficulties. For example, the current distribution induced in the squirrel cage and the effects of the motion of the rotor must be properly accounted for. The following sections discuss how this may be accomplished using the example of a squh"rel-cage induction motor. As mentioned in the previous section, there are several methods available for magnetic field analysis. The finite element method is often the most convenient method, particularly for electric machines, since it can handle complicated geometries and saturable materials with relative e~e. This section will describe a technique for the dynamic electromechanical modeling of an electric machine, using finite element analysis to compute the magnetic fields. First, a general method of finite element mesh generation for electric machines is discussed, and then the mathematical details of the electromechanicaI model are presented.

4.2 Dynamic Electromechanical Finite Element Analysis of Electric Machines 4.2.1: Mesh Generation In the case of a squirrel cage induction motor, the geometry' is regular and periodic. This was taken advantage of in the mesh generation strategy. The mesh was f o x e d using a mapping function which subdivided each region of the motor. The regions are defined in Figure 4.2.1. The innermost region is the rotor core. This region represents the space between two cylinders, one at the motor shaft and the other at the bottom of the rotor slots. The inner circle is divided into a number of equal parts set by the user. Each rotor slot and the root of the tooth is likewise divided into a user defined number of segments. Then the number of intervening layers and weights is specified. The resulting mesh is generated layer by layer from the inner circle to

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the outer circle. The method uses a weighting function (Hwang, Salon and Palma 1988) which refers to Figure 4.2.2. R W = (4.2.1) R+GH/NA where AB and GH refer to the arc lengths, NB and NA are the specified number of segments along these arcs, and R is as shown in Figure 4.2.2. The total number of layers is then

N = I'~NT

I

log

(

(4.2.2)

log W

where NINT returns the nearest integer m the argument. R e length, d 1, is found d1=

(W-I) *R

(4.2.3)

The number of segments on this layer is N I = NA + NII'qr((NB -NA) * d 1 ~ ) (4.2.4) To find the next layer, we recompute W, replace N by N-1 and R by R - d 1 and so on. The first step in predicting the magnetic vibration characteristics of a particular

:::::::::::::::::::::::

S l a l ~ core

slalor slots

air gap " rotor slots r core

Figure 4.2.1: Topological Regions of an Induction Motor

122

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Nil "t

C

Figure 4.2.2: Division of Motor into Layers electric machine is to analyze the dynamic magnetic field acting in the machine. The magnetic field data may then be used to compute forces and force distributions, which are then in turn applied to a mechanical model of the machine and used to predict vibrations. Consider the case illustrated in Figure 4.2.2, with NA segments on the inner circle and NB segments on the outer segment. The interior nodes will all be placed on circles whose radii are completely determined by the number of layers and weights. It only remains to determine the number of nodes (or segments) on each circle. ~ e rotor tooth and slot section exhibits symmetry over each half slot pitch. The geometry is specified in Figure 4.2.3. The number eY layers and weighting is specified as in the case of the rotor core and the mesh is generated for the half slot using F~uation (4.2.4). The mesh is then reflected around the center line of the slot and repeated N r times, where N r is the number of rotor slots. The mesh regions are difterent from the material regions. The material properties are specified independently. With this extra flexibility, the rotor meshing region contains one layer of elements in the air gap as shown above. The reason for this choice will be explained below in the discussion of the moving mesh. The air gap region is an annular region similar to the rotor core and is meshed in the same way. Note that the air gap mesh region does not correspond exactly to the air gap since one layer of the actual air gap is meshed with the rotor and one with the stator.

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~ e stator tooth and slot region (and one layer of elements in the air gap) is treated like the rotor tooth and slot region above. The stator core region is treated like the rotor core region. As the rotor turns, the air gap elements must be continualy remeshed. In the analysis which follows, we show that for electromagnetic purposes, the rotor is modelled in its own reference frame. ~ e rotor core and slot-too(h regions are not remeshed, only rotated. The stator core and slot-tooth regions remain stationary. It is only the air gap region (excluding the part of the air gap attached to the rotor mesh and stator mesh as described above) which is continuously changing. The mesh is generated in such a way that the air gap region lies between two circles with uniform node spacing. The remeshing is accomplished (when needed) without adding any new nodes or elements. The algorithm only reconnects existing elements using the new nodal positions. This is illustrated in Figure 4.2.4 for two instants of time~

I

b5 m

m

J hs4

hs3 ~ ~ !

~¢ "~ ~ hs22 _4~ hs21

!

Figure 4.2.3: Slot and Tooth

IK hsl

124

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

4I

42

43

44

Stator 34

Air Gap

24 14

1 ~"-"~"~-,

41

42

43

Rotor

44 Stator

Air Gap

Rotor t+~t

Figure 4.2.4: Remeshing of the Air Gap Layers 4.2.2:

Mathematical Analysis

R e standard procedure of using finite element analysis m approximate magnetic field quantities within a fixed device or region is well known. The user describes the problem geometry and material characteristics, sets the boundary conditions, and specifies numerically all current densities, which act as the source of the magnetic field. The region of interest is then discretized in space into a mesh, and the finite element field approximation equations are set up and solved. The solution consists of a set of field potentials at each node of the mesh. Such a procedure is inad~uate for a l~ge class of practical problems. Consider the transient analysis of an electro-mechanical device which is activated by a voltage (or current) source such as a motor. The voltage (current) source for such devices is time dependent; therefore, one cannot specify a priori the numerical value of the current density in the conductive regions of the device because skin effect and eddy currents cause the current density to vary with time and position within the conductor. The standaxd finite element procedure, however, requires current density as a known input to the analysis.

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In addition, it may be necessary to attach lumped circuit components, such as resistance or inductance, between the voltage (current) source and the region to be modelled by finite elements. The lumped components may represent the internal impedance of the voltage (current) source, or they may be used to approximate the effects of the parts of the device which are outside the region modelled by finite elements. The corresponding transient circuit equations must be coupled with the transient finite element field equations. Moreover, there may be movable components such as the rotor. Magnetic forces determine the position of these components, and the positions, in turn, affect the magnetic field within the device. Provisions must be made for the transient modeling of such a coupled electro-mechanical system. Therefore, a method for the proper coupling of transient fields, circuits, and motion must be such that: (a) only terminal voltage (or total terminal current) applied to the device is required as an independent variable, and total terminal current (terminal voltage) is then computed; (b) the transient external circuit equations that model electrical sources and circuit components are coupled to the finite element field ~uations; and (c) equations of motion are coupled to the field equations.

4.2.2.1: Electromagnetic and Mechanical Theory The electromagnetic field theory, electric circuit theory', and basic mechanical motion theory, upon which the proposed method is based, are summarized below. The equations necessary for transient coupling of fields, circuits, and motion are summarized below: Field Equation

Vb

aA

V x v V x A = c ~ - ~ at Total Current Equation I =

- - - ~ - er ~

dxdy.

onductor

Series Bar-Coil Equation dIc Vc = {db }T c {Vb ~ +Lest ~ + R e x t l c

Parallel Coil Equation Vs=Rs

{lIT{!

~ + L s {I} T

+ Vc. C

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Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Mechanical Acceleration Equation

d~

J ~+3~

- Xem-Xext

Mechanical Velocity Equation do where v A c Vb 1 t

= = = = = =

I

-

d Vc Ic J

= = = = = = = = =

~. ~em ~ext 0

magnetic reluctivity (reciprocal of permeability) magnetic vector potential electrical conductivity voltage applied to conducting region axial length of the machine time total current in a conductor polarity ofcondutor: +1 or-1 voltage applied to a coil current flowing in a coil moment of inertia angular v e l ~ i t y of rotor rotational damping c~fficient electromagnetic torque developed by machine load torque angular position of rotor

Note that the first three equations are coupled by the voltage applied to the finite element region, Vb; the second, third, and fourth equations are coupled by the conductor currents; and the first, second, and fifth equations are coupled by magnetic vector potential A. The portion of the problem to be analyzed with finite elements must be discretized in space, i.e., meshed. The Galerkin method (see section 1.6) is used to approximate the field and current equations in discretized space.

4.2.2.2:

Time

Discretization

In this section, the field equation, total current equation, circuit equations, and the mechanical equations of motion are discretized in the time dommn~ The method of time-discretization used here is based on the following equation: {" a A ) t + A t aA t {A}t+At~ {A}t (4.2.5) t ~. + (1 - 8) at = ......... At ............... ' The value of the constant 13determines whether the algorithm is of the forward difference t y ~ (13 = 0), backward difference type (13 = 1), or some intermediate type (0 < ~ < 1). Note that if ~ = (I/2), the Crank-Nicholson Method is implemented.

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

127

The goal is to solve for {A} t+At. The derivatives (aA/at) (t+At) and (aA/at)(t) are unknown.

4.2.2.3:

Linearization

The field equation and the acceleration equations am non-linear functions of the vector potential, A, and/or component displacement, 0. These ~uations must be linearized before they can be combined with the other equations of the sysvem in a general global system matrix equation. The linearization of the field and acceleration equations is accomplished using the Newton-Raphson method.

Linearization of the Field Equation The field equation, Vb V × vV x A = ~ T -

aA cry,

(4.2.6)

is nonqinear in A in cases where the reluctivity, v, is a non-constant function of flux density, B (and hence ofthe vector potential, A). To linearize this equation we begin with the space- and time-discretized form of the field equation:

°

{A}t+At- T {Q} V~

= l~At + T {Q}

' ,Ipq~. {A} t Vb

(4.2.7)

where P, T, and Q are matrices commonly encountered in finite element analysis (see section 1.5.4). The P matrix is the finite element stiffness matrix of Equation (1.5.49). The T matrix for each finite element is a symmetric 3 x 3 matrix with whose diagonal entries are equal to one-sixth of the element area and whose offdiagonal entries are equal to one-twelfth of the element area. For each element, the Q matrix is a 3 x 1 vector whose entries are each equal to one-third of the elemnt area. We now apply the Newton-Raphson linearization procedure to this equation, with res~ct to A i, introduce the time-dependence of the [S] matrix, and obtain [[G]+~ATt~{AA}t+At ~ , , , t+At_ k+! - ]- {Q} ~Vb' k +1 o [ T ] ] {A}k+At [ ~ [ T ] c l@vt + I {Q} b where t+At vat [G] = vk +

t ~ , , t [ p ] t ] {a}t (4.2.8)

([p t+At t+At t+At { }t+At T ],~ {A }k ) ([P~ A k )

Note that for linear materials, av/aB 2, and Equation (4.2.8) reduces to Equation (4.2.7).

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Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

The field equation is also a non-linear function of the displacement (rotor position), 0. Applying the Newton-Raphson procedure with respect to 0 to Equation (4.2.8) yields: {AA/t+At ~ AV t+At t + A t [ ~ ] "k+l " ! {Q} b'k+l + Vk =" L['v[t+At P ] t + A tkk + ~ {Q}

+ cr~]pAt $ /A'lk +

~xt -

t+At---t+At {A}k lA0Jk+ !

t[p]t {A}t

Vb

(4.2,9)

Equation (4.2.9) is the linearized diffusion equation.

Linearization of the Acceleration Equation In this section, the acceleration equation will be linearized and combined with the velocity equation. The acceleration equation, d~ J ~ + ~ t~ = "Cem- ~Cext (4.2.10) is a non-linear function of the vector potential, A, because the electromagnetic torque, "Cem, is a quadratic function of A. The Newton-Raphson technique will be used to linearize the acceleration equation with respect to the vector potential at the mesh nodes. Recall the time-discretized form of the acceleration equation: t+At t+At t~ - 13zem = -~z

+

[~tt

] t - (1-~)Z. ~t + (l_~)x em

t (1-~)~ ext" (4.2,I I) Now apply the Newton-Rmphson linearization procedure to this equation and obtain 0~em } ~em -

A~k+ l - ~

_ p t+At+ [ J ext t+At -

~k

~

- (1-~)X

{AAJk+l - ~

]

{AAIk+I =

~t + (l-p),; emt t (I-~)~ ext

t+At + 13~ e m k"

(4.2.12)

To compute the derivatives (3~:em/OA) and (3Zem/30) first note that the electromagnetic force acting on a component is, ~em - - O Iconstant magnetic flux (4.2.13) where Wmag is the stored magnetic energy of the system and ~0 represents "virtual motion" of the component under consideration. It can be shown that, in t e ~ s of familiar finite element analysis quantities, 1

Wmag = ~ v l {A}T [SI {A} Substitution of Equation (4.2.14) into Equation (4.2.13) yields

(4.2.14)

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1

Zem = - ~ v 1 {A

}T

I29

{A}

(4.2.15)

Symbolic differentiation of the stiffness matrix with respect to 0, elaborated in chapter I0 in the context of optimization, is straightforward, since the entries of the stiffness matrix can ~ expressed as simple functions of 0. Numerical values are given to the derivatives by assigning a fiactor between 0 and I to each mesh node. Nodes that are displaced (virtually, in the case of Equation (4.2.15), or physically, in the case of Equation (4.2.9)) in one-to-one correspondence with object motion, such as the nodes fixed to the surface of the object that moves are assigned a factor of I. Nodes which are unaffected by motion, such as those attached to fixed objects, are assigned a factor of 0. Intermediate factors between 0 and 1 may be assigned to nodes in the air surrounding the moving object in such way that the object motion is "absorbed" by the finite elements in the air region. From Equation (4.2.15), then, define new variables C and U as follows: 8~em 3A - - v 1 {A} T = - {cIT1 (4.2.16) 3"Cem

I

|3SZ[j {A} =-U. - - 2 vl {A}TEa02

(4.2.17)

Now substitute Equations (4.2.16) and (4.2.17) into Equation (4.2.12) and obtain - 13~

A~k+l - ~I {C} T {aA}k+ 1 - ~ U A0k+ l = t+At + ext

- (1-~)X

]

~t + (1-fi)~t - (1-fi)~t em ext

t+At t+At ~k + ~'c em k'

(4.2.18)

Next, take the time-discretized f o ~ of the velocity equation, and put it in "Newton-Raphson fi)rm" (where the unknown is the c h a n g e in the variable rather than the variable itself): 1 t+At I 0t+At _ (1.13)~t 1 t [3At~k+l - ~ A 0 k + l - ~ k + l +~ k -~0 or

I

t+At

a~k+I = ~ a 0 k + l - i n k

I t+At ~ +~°k _~

1 mt - ~ 0

t

(4.2.19)

Now substitute Equation (4.2.19) into Equation (4.2.18) to obtain

I a0k+ l ' ~ k

t+At

1

+~a0

k

[,

~/{C} T {A}k+l + ~UaOk+l =-~¢ t (1-~) • e x t -

~t+At k

At. 1~ ~-

t+At + ~ "¢ e x t k

t

(I- ~)X

- ~1 o t 1 ~t + (1-~) ~:t

em

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Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

t+At Note that ~ "ce m =

. t+At t {C} T {AA/k , and re~ange t e ~ s to yield

-

~l {C}T {zXA~+At+ ~ t -2

ll{c}T{a}t+At( -

(~

+ (~t)2)

4.2.2.4:

+

+ ~U

K

(~t)2)0t+kt

At-

fi

Ok+1 =

k

~ +

rot+

t t " ~ ~ ext t+At 0t + (1-t3) "cem" ( 1-~) z ext

(4.2.20)

Global System of Equations

The field, circuit, and mechanical equations are now available in a discretized and linearized form. It remains to assemble these matrix equations into a global system of equations describing the entire problem. From the summary above, it is apparent that, in general, there are five vector unknowns: AA"t+kt : change in vector lyotential of each node tk+l {aVb }k~1t .. t+At

AlJc k+ t V .t+at a C/k+1 }t+At {a0 k+ 1

: change in voltage across each bar : change in cu~ent in each coil : change in parallel terminal volmge : change in position of the rotor

The global system matrix equation may then be set up in the form

[MI {x} = {N}

(4.2.21)

or, in expanded form, M11 M12

MI5

{aA}] t {{N1}' {N2}[I /,

MT2 M22 M23 {aVb}[ T {as} (4.2.22) = {N3}? M23 M33 M34 {N4}! M T34 M44 {Ns}-' MT M55 _ 15 By solving this global system of equations for the unknowns, {x }, at each time step, a full set of data on the coupled transient fields, voltages, currents, and rotor motion is obtained. The transient field information is used to compute the vibration-inducing fbrce distributions in the machine, employing techniques described in the next section.

%1'J

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4.3 Magnetic Force Computation Once the magnetic field inside the machine has been determined, the magnetic forces acting on components of the machine may be computed by a variety of methods. This section will describe and evaluate several common methods, with particular attention to the advantages and disadvantages of each in the finite element context. In general, there are two different types of forces that are of interest in machine analysis: net forces and force distributions. A net force, or "global force", is defined as the single net vector force acting on an object. A force distribution, on the other hand, is a set of forces, known as "local forces", acting at specific locations on an object. For the pur:~ses of magnetic noise and vibration analysis, force distributions are usually of more interest than net forces; therefore, this section will focus primarily on the computation of force distributions and local forces. Unfortunately, the mere existence of a unique, definable local force has been the subject of debate for many years, and the matter remains unresolved. This section will not deal with the theoretical dispute (interested readers are referred to the paper by Carpenter (1960), but will approach the subject of local force computation from a practical standpoint. Various methods of local force computation will be discussed in light of their ability to provide designers of machines with a model of force distribution that is accurate on a macroscopic scale, and can be used to produce a reliable mechanical vibration model of the machine.

4.3.1: Computation of Forces from B 2 R e simplest method for approximating a force distribution (force per unit area), p, about the stator or rotor of an electric machine is to assume that the stator and rotor irons are infinitely permeable and then compute Br2(0) (4.3.1) p(0) = - 2~ 0 Ur where B r is the radial component of the air gap flux density, at angular position 0 in the air gap and Ur is a unit vector in the radial direction. This method is justified as follows. Assume that the stator bore and rotor outer surface are smooth, consisting of an infinite number of opposing pole faces, such as those shown in Figure 4.3. I. (For a smooth surface, h --~ 0). If the iron is infinitely permeable, the flux enters and leaves the pole faces perpendicular to the surfaces, as shown. The energy density in the air gap, Wmag, is then Wmag-

BH

2

(4.3.2)

where t t is the magnetic field intensity, related to B by the constitutive law: B = ~H

(1.2.7)

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Chapmr 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

B Stator

Rotor

Figure 4.3.1: Opposing Magnetic Pole Faces and Associated Flux Densi~ In the air gap, the magnetic permeability g is equal to the permeability of free space, g0, so B = g0H. The total energy stored in the air gap between the pole faces is found by integrating Wmag over the volume of the gap: B2

Wmag = -2~t0 hlg

(4.3.3)

where I is the depth of the pole faces in the z-direction (into the page in F~gure 4.3.1). To find the force of attraction between the pole faces, the standard virtual work approach is used: Fx = OWmag= B2 - Og " 2;0 hl. (4.3.4) In the case of an electric machine with smooth stator and rotor surfaces, the flux density normal to the ficititious "pole faces" is the radial component of B in the air gap, and the force distribution around the air gap is then Br2(0) p(0) = - ~ ur (4.3.1) Using this method to compute force distribution from a finite element field solution is straightforward. Simply take a layer of elements from the air gapand compute p from Equation (4.3.1) at various angular F,3sitions within the layer. The B 2 method has two major drawbacks, however. First, the assumption of infinitely permeable iron is not an accurate one, particularly with electric machines, since they usually operate at some level of saturation. Second, the B 2 method is only capable of computing attractive forces between iron pieces, and cannot account for repulsive forces between currents (the sign in Equation (4.3.1) is always the same). In an electric machine with windings on both the stator and the rotor, for example, a rotor bar carrying current in a certain direction may pass by a stator bar ca~ying current in the opposite direction, and the two bars will repel each other, producing a positive local force. Equation (4.3.1) cannot prc~luce this result.

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4.3.2: Maxwell Stress Method The Maxwell stress method for computing forces overcomes some of the drawbacks of the B 2 method. The assumption of infinitely p e ~ e a b l e iron is not required when using the Maxwell stress method; in fact, the iron can even be nonline~. Both attractive and repulsive forces can be computed with Maxwell stress. This section contains a derivation of the Maxwell stress expressions, and discusses their capabilities and limitations.

4.3.2.1: Overview of the Derivation The object of the derivation is to find a method for computing force on a component of a system, given only the values of the magnetic field outside the component. To achieve this objective, the following step~by-step procedure will be use~t: 1. Replace ferromagnetic material in the component of interest by a distribution of currents, J, such that the field external to the component is not altered. 2. Find an expression for the volume force density, Pv, on the currents, 3, Express Pv as the divergence of some tensor 2, T*: Pv V.T* Then the torn force, F, on the entire volume is the volume integral: =

F = JfvPV dv = J(' V-T* dv. 4, Apply the tensor analog of the vector divergence theorem to transform the volume integral to a surface integral:

F = ,Is T * . d S , The tollowing sections discuss each of these steps in further detail.

4.3.2.2: Replacement of Iron by Currents The first step in this process is beyond the scope of this work, and the result will be stated without proof. It can be shown (see Stratton (I941, p. 242)) that a volume v of ferromagnetic material may be replaced by a distribution of surface currents Mt Js - ~ ut (4.3.5) and volume currents

4 The asterisk indicates that dF is not necessarily the true local force. 2 A tensor will be denoted by an asterisk after it.

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Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Jv -

VxM ~t0

(4.3.6)

where M is the magnetization vector, defined as M = B - ~t0H. (4.3.7) The subscript t denotes the vector component tangential to the surface of the volume v; ut is a unit vector tangential to this surface. The "Ampefian" currents of Equations (4.3.5) and (4.3.6) may be added to any free currents already existing in the component of interest, yielding a net current density, J. 4.3.2.3: Force Density on the Currents Now that the ferromagnetic material has been replaced by currents, the permeability throughout the component is constant at u 0. An expression for the volume force density on the currents in such a region is desired. The empiricallydetermined Lorentz volume force density is appropriate: Pv = J x B. (4.3.8)

4.3.2.4: Force Density as the Divergence of a Tensor The equation for force density, (4.3.8), will be now be manipulated so that it is the divergence of an as-yet-unknown tensor, T*. First recall Ampere's Law, which, neglecting high-frequency effects, states that VxH =J (1.3.23) In the region of interest, the constitutive law which relates B to H is B = goH. (4.3.9) Substituting Equations (4.3.9) and (1.3.23) into Equation (4.3.8) yields

.1

Pv =

V x ~

xB

(4.3.10)

The x-component of force density is then

I (

px=~_

aBx

_ aBX_B Y a x ] "

+Uy~

1 Now add and subtract the quantity ~ 1 [ aBx Px = ~ Ik B x ~ x +

By noting that

aBq

ap Bq2 = 2 Bq ap

aB x + By. ~ -

(4.3.11)

as follows: Bx

aB x

-

(4,3.12) (4.3.13)

where p and q are generic coordinates, the expression for the x-component of force density may be rewritten as Px 1 fl a aBx I a =la7 L2 ~ Bx2 + Bz + By ay - 2 ax (Bx2 + By2 + Bz2) (4.3.14)

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

135

or Px

= l a0

Bx

+ ~ (BxBy) +

- ~ IBt 2

(BxBz) - BxV.B ]

(4.3.15)

Gauss's Law states that V. B = 0, so Equation (4.3.15)reduces to P x = ~1 [ ~

I Bx 2 - ~IBI 2

+ ~~ (BxBy) + ~(BxBz)~ ] .

(4.3.16)

* ¢

This equation is of the f o ~ 1 aTxx ~Tx'r' + OTxz ~ Px-~0 ~x + ~y ~z where Txx = ~1 ( B x 2

(4.3.17)

,Bt 2 )

(4.3.18) (4.3. 19)

Txy = ~ BxBy I

Txz = ~ BxBz

(4.3.20)

We can say that Px is the divergence of a vector Tx: Px = V . T x (4.3.21) where Tx = Txx Ux + Txy Uy + Txz Uz (4.3.22) A similar analysis for the y- and z-components of force density yields py = V • Ty and Pz = V • T z . (4.3.23) The vectors Tx, Ty, and Tz, may be combined into a tensor, T*, such that Bx 2 - ~ IBt 2 1

T* = ~

BxBy 2 1

ByB x

By -2 lBI2

BzB x

BzBy

BxBz ByBz

"

(4.3.24)

Bx2- 2

Now the vector force density, Pv, may be expressed as the divergence of the tensor, T*: Pv = V • T* (4.3.25)

4.3.2.5:

Surface Integral Expression for Force

The total force on the component of interest is then the integral of Equation (4.3.25) over the volume of the component: Pv = J ~ . V ' * T d v

(4.3.26)

Now the tensor analog of the vector divergence theorem can be used to transform the volume integral of Equation (4.3.26) to a surface integral:

136

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

F = .~!s T* • d S

(4.3.27)

where S is a closed surface surrounding the component. Equation (4.3.27) demonstrates that the force on an o~ect may be computed by integrating field quantities on a surface enclosing the object. This result greatly facilitates force computation from numerical field solutions. Appendix 4.C shows how Equation (4.3.27) may be transformed into the familiar Maxwell stress expressions: BnBt Pt* - N0 (4.3.28) Pn* - Bn2 - Bt2 (4.3.29) 2~0 where Pt* and Pn* are "stress-like" quantities directed tangential and normal to the surface of integration, respectively. Although these quantities have the units of stress or pressure, they may not necessarily be actual local stresses3; therefore they are marked with an asterisk to distinguish them from the components of p, the true stress distribution.

4.3.2.6: Notes on the Maxwell Stress Derivation In this section, some of the capabilities and limitations of the Maxwell stress method are discussed in light of the derivation above.

Considerations involving region and material characteristics When using the Maxwell stress method to compute forces, it is not necessary to compute the equivalent current distributions of Equations (4.3.5) and (4.3.6). It is only necessary m know that such distributions exist, and then Equations (4.3.28) and (4.3.29) can be applied in a straightforward manner. The equivalent current distributions are valid regardless of whether the material is homogeneous or inhomogeneous, magnetically linear or magnetically nonlinear, current-free or current-bearing. Thus the Maxwell stress method may be aRDlied to compute forces on any of these types of materials or regions.

Considerations involving the surface of integration When using the Maxwell stress method, it is important to choose a valid surface of integration. The surface must lie in free space surrounding the oNect of interest; it must not pass through (i) any tizrrornagnetic material, or (ii) any region where ferromagnetic material has been assumed to have been replaced by equivalent currents. The reason for (i) is that the permeability in the volume would be a

4 The asterisk indicates that dF is not necessarily the true local force. 3 See below: "Considerations Involving Force Density."

S. L Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

13 7

function of position, and therefore Equation (4.3.10) would not rezluce to Equation (4.3.11). The reason for (ii) is that the equivalent current distributions used to replace the ferromagnetic material preserves the magnetic field external to the material, but not inside the material. Integration along a path passing through the component at any point would implicitly involve the use of field quantities of unknown validity.

Considerations Involving Force Density It is also important to know whether the Maxwell stress method can be used to obtain a force distribution. At first glance, one might conclude that since global figrce computation involves the surface integration of the quantities Pt* and Pn*, then those quantities must be stresses. In a strict mathematical sense, however, this reasoning is not correct. Recall that in order to t r a n s f o ~ the volume integral of Equation (4.3.26) to the surface integral of Equation (4.3.27), the tensor analog of the divergence theorem was used. The divergence theorem is valid only if the surface of integration for the surface integral is closed. Strictly, then, the integrand T*dS cannot be localized to particular points on the surface; all that is guaranteed by the mathematics is that the integral over the entire closed surface is equivalent to the volume integral. Nevertheless, the Maxwell stress method is frequently used to compute force distributions. The results are often in agreement with intuition, theory, or experimental data. Section 4.5 will present examples of force distributions computed using both the B 2 and Maxwell stress methods, and compare them with force distributions derived from theory.

4.3.2.7: Application of Maxwell Stress to FE Field Solutions The Maxwell stress expressions may be applied to finite element field solutions in a straighttorward manner. The path of integration is drawn through the domain and the quantities of equations (4.3.28) and (4.3.29) are computed from the field solution. There has been some discussion regarding the construction of the path of integration. Proposals for an "optimal path" include: (i) a path joining the centroids of neighboring elements, (ii) a path crossing elements by joining the midpoints of two of their sides, and (iii) a path crossing perpendicular to each element boundary. Path (i) is often favored for first-order elements since it passes through the centroid of the elements, the point of highest validity for the field solution. Path (ii), when used with first-order elements produces exactly the same force result as the Coulomb implementation of the virtual work method (see !stfan 1987). Path (iii) may have some advantages in e~or analysis.

4.3.3: The Reyne Stress Tensor Method In the previous section, it was noted that the Maxwell stress method, when interpreted rigorously, cannot yield intormation on local forces or I~rce distributions. In addition, the Maxwell stress method cannot be used to predict forces on surfaces lying partially or completely within a ferromagnetic material.

138

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

To overcome the shortcomings, Reyne (!987) proposes a new method for force computation, based on the work of Rafinejad (1977). An energy approach is used to derive a stress tensor which, when applied properly, can be used to compute l ~ a l forces within ferromagnetic as well as non-fe~omagnetic materials. The method can also account for the change in permeability with material density, p, thereby allowing for the prediction of material deformation due to magnetic t~rces. The components of the stress tensor are as follows, in double subscript notation: BxHx - Z ByH x BzHx ] (4.3.30) T* BxHy ByHy- Z BzHy BxHz

ByHz

BxH x - Z

where H

Z=

[ [g-p

0_K ~p]HdH

(4.3.31 )

0 and p is the material mass density. Reyne et al have used this tensor to study machine vibrations, and have published their findings in a series of papers (Reyne, Sabonnadiere, Coulomb and Brissonneau, 1987; Reyne, Meunier, Imhoff and EuxiNe, 1988; Reyne, Sabonnadiere and Imhoff, 1988). The method is particularly useful for finding force distributions on iron surfaces, such as the teeth or salient poles of electric machines, as shown in Section 5 in Figures (4.5.5) - (4.5.9).

4.3.4: Errors in Force Computation from FE Field Solutions Forces compu~d from ~ field solutions are prone to error, that can sometimes be severe. This section will describe some of the sources of error and discuss briefly some methods proposed for reducing the error.

4.3.4.1: Sources of Error in Computed Forces Since t~rces are computed from field quantities, errors in the computed forces may be attributable to errors in the field solution. The FEM, while minimizing the error for total system co-energy, does not guarantee a minimization of error locally, at each point. Thus some of the nodal potentials provided in the FE field solution may have considerable error, while others may ~ quite accurate. The geomet~ of the problem is one factor that can influence local field error. Near a sharp corner of a magnetic object, for example, the actual magnetic field intensity can approach infinity. The FEM cannot produce this result. This problem is particularly evident when computing forces around the air gap of electric machines. Since the air gap is usually very narrow, any path of integration for force computation taken through the air gap will pass close to many "corner points" (the corners of the teeth on the stator and/or rotor).

S. J. Salon, C. J. Slavik, M. J. DeBo~oli and G, Reyne

139

In addition, the nature of force computation methods is such that only a small fraction of the total number of finite elements in the problem (those in the path of integration) are used to compute the force. This tends to magnify the effects of local field errors. Errors in the finite element field solution are passed on to the computed force in such a way that the errors may be magnified. Say, for example, that the FE approximation to the magnetic vector potential solution is AFE. The approximation to the magnetic flux density, BFE, is then found by numerical differentiation of A ~ : B ~ = V x AFE. (4.3.32) Now, since B ~ results from numerical differentiation of an approximate quantity, it is likely that BFE will not be equal to the true magnetic flux density, Btrue, but will contain some error, ~: BFE = Btrue + ~. (4.3.33) The computation of force requires the square of B ~ : BFE 2 = Btrue 2 + 2Btrue e + E2. (4.3.34) Thus, when computing force, any error present in the flux density solution is doubled (neglecting the ~2 t e ~ ) .

4.3.4.2: Reduction of Error in Computed Forces The difficulties of magnetic force have been investigated in some detail, and several methods for improving accuracy have been proposed. This section will describe some of those methods. Many of the techniques proposed for increasing the accuracy of computed forces involve estimating and/or reducing the error in the FE field solution. A common way of analyzing field error is to employ the magnetic interface conditions. From electromagnetic field theory, these conditions are (see section 1.3.1) 1. The component of magnetic flux density normal to an interface (Bn) is continuous across the interface. 2. The component of magnetic field intensity tangential to an interthce (Ht) is discontinuous across the interface by an amount directly proportional to the surface current at the interface. When the finite element method is used to solve field problems, the solution is guaranteed to satisfy one, but not both, of these conditions. If the magnetic vector potential is the unknown in the FE problem, the solution satisfies condition (1), but not necessarily condition (2). Conversely, if the magnetic scalar potential is the unknown in the FE problem, the solution is guaranteed to satisfy condition (2), but not necessarily condition (1). An analysis of the interface conditions at the edges of the finite elements gives some measure of the accuracy of the field solution (Hoole, Yoganathan and Jayakumaran, 1986). For example, if the two elements shown in Figure 4.3.2 are part of a magnetic vector potential problem, and there is no surface current at the

140

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Figure 4.3.2: Adjacent Elements in FE Field Solution common edge of the elements, the quantity Htl - Ht2 might be indicative of the quality of the ~ field solution in the vicinity of the two elements. This type of error analysis has been used in various ways to aid in the computation of forces. McFee and Lowther (I 987) use the discontinuity in Ht as a measure of quality for comparing force integration paths. McFee, Webb and Lowther (I 988) derive a generalized formulation for force computation and suggest using the discontinuity in H t as an estimate of local field error for tuning parameters in the formulation. Henneberger, Sattler and Shen (1990a) propose that discontinuities in Ht be used to compute fictitious surface currents to be placed at the element edges. The fields due to these currents are then added to the finite element field solution. They report increased accuracy in forces computed from the modified field solutions (Henneberger, Sattler and Shen 1990b). Tarnhuvud and Reichert (1988) analyze a "non-conforming" element to improve adherence to the interface conditions. Penman and Grieve (1986) propose that the finite element problem be solved twice, once with the magnetic vector potential as the unknown to satisfy interface condition (I) and once again with the magnetic scalar potential as the unknown to satisfy interface condition (2). The authors note that the system energies obtained from the two solutions are guaranteed to bound the true system energy value, and suggest that force computations from the two solutions may similarly bound the true force value. By using this two-solution method, an error estimate for the computed forces can be obtained, and the true force may be more closely apwoximated~

4.4 Vibrations of the Stator Core Using the finite element method discussed in the preceding sections it is possible to calculate the air gap field of an ac machine and to determine the distribution of magnetic fl9rces on the stator and rotor surfaces at any instant of time. By applying the force distribution to the stator bore, the defiectional response at the exterior surface of the stator core can be determined in terms of displacements from the undisturbed case. Accelerations at the exterior of the stator surthce can also be predicte& With this information, the propagation of vibrational energy throughout the structure surrounding the stator core, such as the motor shell and

S. J. Salon, C. J. Slavik, M. J. DeBoaoli and G. Reyne

141

supp~ting ribs, or the medium surrounding the motor, can be predicted in terms of structureborne, airborne or fluidborne vibrations. However, many difficulties are involved in such a calculation, including • Obtaining reasonable estimates for the stiffness and damping parameters of a wound stator core. • Determination of the natural frequencies of the machine structure and their contribution to the vibrational levels at the operating frequency and harmonics. The following are several approaches for determining the stator core deflections and resulting accelerations caused by magnetic force harmonics. 4.4.1:

Simple

Beam

Model

The first model to consider for determining the radial deflection of the starer core and the resulting accelerations generated by the magnetic force h a ~ o n i c s , is the "simple beam" model discussed by Alger (1954). In this model the laminated stator ring is assumed to be "rolled-out" or developed into a straight line and treated as a continuous beam. This beam will bend into a shape determined by the sinusoidaI wave of the magnetic force harmonic being considered, with a node at each zero ~ i n t of the force wave. Each node may be considered a fi~ee support, the beam length taken to be half the wavelength of the magnetic force harmonic, and the deflection, d, of such a freely supported beam under a sinusoidal force distribution can then be dete~ined: W L3 (4.4. t) d - 2n 3 E1 where L = the distance between beam supports W = the magnetic torce amplitude (per unit of core length) of the force harmonic under consideration E = modulus of elasticity = moment of inertia of the beam about its center line I Using _ [mean circumference of stator core) . . . . L (nodes of core deflection due to force loading) riDs 4P and h3 (4.4.2) I = 12 we have for the radial deflection of the starer core 3 3W (4.4.3) d 4E where h = the radial depth of the starer core behind the slots (starer back iron) 2P = the number of ~le-pairs of the magnetic force harmonic being ~

......

142

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Ds

considered = mean diameter of the s~tor core = Dg + twice the stator slot depth + h

with Dg = the air gap (stator bore) diameter Finally, the acceleration, in d~ibels, produced by the stator core deflection is given by A(db) = 20 I°gI0[0.557_ x I0"3~

(4.4.4)

relative to l0 -5 mJsec 2, where d is the zero-to-peak deflection amplitude.

4.4.2: Thick Cylindrical Ring Model Kerruish (1958) derived the deflections of an electrical machine stator core by treating the core as a smooth (no teeth) thick cylindrical ring influenced by either a 2 pole-pair or 4 pole-pair force wave on its inner surface. These waves are typically the most important sources of magnetic vibrations in large, two-pole machines and also produce significant vibrations in machines with higher pole numbers. Table B.I in Appendix 4.B shows that these force waves are also common in squi~el-cage induction motors and are probably the strongest sources of radial vibations in these machines, if rotor/smtor asymmetry is excepted. Kerruish's formulas are given in Appendix 4.D. The resulting accelerations can be obtained from Equation (4.4.4).

4.4.3: Effect of Mechanical Resonance and Damping Yang (1981) provides a method for incorporating the effects due to the proximity of a mechanical resonance in the dynamic model for the stator core deflection. Yang's equations also account for the eft?cts of mechanical damping of the stator core. Yang's formula for the stator core deflection is

d d{E! where d' d c,~

8

2]2

2} - 1/2

= stator core deflection, including effects of mechanical resonance and damping = deflection, without resonance and damping effects = the radial frequency of the magnetic force harmonic being considered = the radial frequency of the mechanical resonance of the stator core in closest proximity to the frequency of the magnetic force harmonic being considered = the logarithmic decrement parameter, determined by Yang in Figure 4.4.7 of (Yang I981).

(4.4.5)

S. J. Salon, C. J. Slavik, M. J. DeBo~oli and G, Reyne

143

Both Alger (1954) and Yang (1981) provide methods t~r determining the resonance frequencies of a stator core. The simplest relation is given by Alger: ~0 = 2~. 36,700 m(m2~!)h~ (4.4.6) Ds2q,m2+ 1 where m is half the number of nodes of core deflection and D s and h are as defined above. More complex, and presumably more accurate, formulae are given by Yang (1981), and by Girgis and V e ~ a (1981).

4.5 This section contains two examples of machine vibration analysis using the technique described above. The first example illustrates the analysis of an AC induction motor with the full transient machine model presented in Section 4.2. Forces are then computed from the field data generated by the model, and vibration ch~actefistics are derived l¥om the forces using the analytical techniques of Section 4.4. In the second example, the vibratory behavior of a DC machine is studied using a combined electromagnetic and mechanical finite element analysis approach, based on the Reyne stress tensor method.

4.5.1" Vibration Analysis of an AC Induction Motor 4.5.1.1: Model of the Test Motor The three-phase AC induction motor studied in this section has 48 stator slots and 36 rotor slots. Figure 4.5.1 shows the cross-section of the motor, and general motor data are given in Table 4.5.1. This motor is to be modelled using a transient, finite element-based analysis program that employs the techniques described in Section 4.2 The input data for

Parameter

Value

Units

440 Phase Voltage V(~s) frequency 60 resistance/phase mr2 0.82 inductance/ph~e 28.68 /aH Winding Coils DC resistance f2 0.0123 lead in inductance 20.0 gH winding type double layer pole pitch slots 5/6 16 number of turns turns Rotor end ring inter-b~ resistance 21.0 g in~er-bar inductance 0.04 gH Rotor bars electric conductivity 2.43. 107 Table 4.5.1: Electrical and Mechanical Parameters of AC Induction Motor Voltage Source

144

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Figure 4.5,1: Cross Section of AC Induction Motor to be Analysed the computer program are the motor geometry, material characteristics, and winding configuration, as well as the waveform of voltage applied to the windings. At each time step, the program computes as output the magnetic field distribution within the machine, stator and rotor currents, and the rotor position and torque. Information required for vibration analysis of the motor will ~ determined from these output data.

4.5.1.2: Air Gap Flux Density Analysis The first step in the vibration analysis of the motor is to extract the air gap flux density distribution from the magnetic field solution provided by the finite element program. Figure 4.5.2. shows several plots of the radial component of air gap flux density vs. angular position in the air gap. The model is simulating rated speed, rated load operating conditions, at "steady state" (no DC component in stator currents), These plots may be considered as "snapshots" of the flux density distribution taken at four instants in time. The four pole pattern is clearly evident, but the distribution is far from purely sinusoidal. The waves contain a significant

S. J. Salon, C. J. Slavik, M. J. DeBoaoli and G. Reyne

NOI

B1 B2 B3 B5 B6 B7 B8 B9 B10 Bll BI2 B13 BI4

Space time pole freq. pai:rs ~ ~:z) ............ 2 60 46 60 50 60 94 60 198 60 10 60 I4 60 6 180 46 180 50 180 34 980,4 14 980,4 38 1 I0'0.4 10 1100.4

direction

forwa~ backw~ forward backward forward bac~ forward backwa,rd back,vaN

back',v~

145

Amplitude Origin in % o f fund~ental 100 I P0- mt t7.8 (S-P)0 + ,~t 14.5 (S÷P)0- mt 2.5 I (2SzDo + ~t 2.0 (2S+P)O- o~t 0.6 5PO+ o~t 0.4 7P0- cot 2.6 3PO- 3~t ! .4 (S:P)o + 3rot 1.0 (S+P)0 - 3o~t 4.3 (R-P)o + (m - PdN)t 1.3 (S-R+P)O - (co- RN)t ... 4.3 )o - (co+ RN)t 1,8 (S-R-P)o + (co+ RN)t

Table 4.5.2: 2D.FFT Analysis of Air Gap Flux Density

number of spatial harmonics, which could be found by applying the standard Fast Fourier Transform ( ~ ) to the waveforms of Figure 4.5.2. A spatial harmonic analysis presents only pan of the total picture, however. As the plots illustrate, the flux density distribution varies with time as well as with position, and most likely contains time harmonics. The distribution may be considered as the aggregate of sinusoidal traveling waves, as shown mathematically in Appendix 4.B, Equation (4.B.4). In this equation, if the number multiplying the spatial variable 0 is of the opposite sign to the number multiplying the time variable t, the wave travels forward (in the same direction as the rotor). If the numbers are of the same sign, the wave travels backward. The individual traveling wave components can be determined by a twodimensional (space and time) FFT, or 2D-FFT. Table 4.5.2. shows the results of a 2 D - ~ analysis of air gap flux density over one cycle of the supply frequency. The symbols used in the table are as follows: P = number of machine ~ I e pairs S = number of stator slots R = number of rotor slots N = rotor s p ~ in r a d i a n s / s ~ n d co = radial supply frequency (2rff) o = spatial position in air gap t = time

146

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

o

Figure 4.5.2: Air Gap Flux Density Distribution at Several Points in One Cycle of Supply Frequency

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

I47

Wave B 1 is the fundamental, forward-rotating wave of flux density, produced by the fundamental MMF. Waves B2-B5 are produced by the interaction of the fundamental MMF with the variation in air gap p e ~ e a n c e resulting from stator slots. Waves B6 and B7 reflect the 5-th and 7-th "phase belt" harmonics of the MMF. The effects of saturation produce waves B8-B 10, as indicated by the third time harmonic. Waves B I I-B 14 are produced by rotor effects. Their time components reflect the fact that the rotor is creating a time- and space-dependent variation in air gap permeance: R slots in space are moving at angular velocity N in time. Spatial influences of rotor slotting are evident in each of the waves Bt l-B14. Waves B 12 and B 14 illustrate that the total air gap p e ~ e a n c e is influenced by both the rotor and stator slotting.

4.5.1.3: Force Distribution Analysis The force distribution about the air gap of the machine can now be computed from the data in Table 4.5.2 using the methods given in Section 4.3. These force distributions will then be applied to the mechanical model of the stator presented in Section 4.4. Since the mechanical models only approximate the geomet~ of the stator, they employ general patterns of force distributions, as opposed to precise

..........N01 ........ Space pole pairs Fi ....... 0 F2 48 F3 96 F4 4 F5 44 F6 52 F7 92 F8 48 F9 8 FI0 8 ..... F 11 12 F12 32 FI3 t6 F14 12 FI5 60 .~

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

time flex!. (Hz) 0 0 0 120 120 120 120 120 I20 240 360 920.4 920.4 1040.4 1040.4

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

direction

Amplituce in flux density % of wave pairs fundamental (Table 4.5.2) 212.0 ..... ~ ...........(! .,_l) 66.0 ........(!,.2), (.!, 3) ....... t2.8 (1,5) 100.0 ( 1, I) 34.2 .(! .,.2) 27.5 (1,3) 7.0 .........(2,2): (1~4)......... 5.4 (1,9) 1.6 {I,6) 7.6 ([.,8) 1.6 (8,8) 8.0 (I, I 1) 3. 4 (!,12) .. 8.9 (1,12), (1,14) 2.3 (2,12), (3,14) .

forw~ b~kward fbrward backward backward b~kward fop,yard forward l~br-w:~] backward backward backward ,,.,,J,,

.

.

.

.

.

i

J

,.,,

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

,.........-..,

.

.

.

.

.

.

.

.

.

.

.

,,

Table 4.5.3: 2D-FFr Analysis of Air Gap Force Distribution

148

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines 0~6 -v

0~5 A

:

!

0.4

S 0.2 e

0

"6 °

0

2

4

~ulor l~n~iti~ (~aian#)

0~6

0~5

0~3 0~2

0

-0/~ t --0.2

. . . . .

0

1'"*

.......

~

i

2

.........

~

4

. . . . . . . .

I .....

6

(b) Figure 4.5.3: (a) Spatial Distributions of Air Gap Force Density at Starting, Computed using the Maxwell Stress Method (b) Spatial Distributions of Air Gap Force Density at Stating, Computed using the B 2 Method

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

149

t.5 t 1.4 !,2 o

i 0.9 n

0,8

0.6

0,4

E

03

z

0~2 0 -0.1 -02

4 /~ngu~r polition (rodion=)

t.5 1.41.2 L1 0.9

0~8 0.7 0.6 05 04 03 0~2 0.1 0 -0.1 -0.2 0

2

4

6

(b) Figure 4.5.4: (a) Spatial Distributions of Air Gap Force Density at Rated Speed, Quasi-Steady State Conditions, Computed Using the Maxwell Stress Method (b) Spatial Distributions of Air Gap Force Density at Rated Speed, QuasiSteady State Conditions, Computed Using the B2 Method

150

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

force density values for s~cific locations on the stator. Therefore, the methods of error analysis and correction described in Section 4.3.4 are not applied. These techniques may be necessary, however, when using more detailed mechanical models, such as those presented in Section 4.5.2. For force distribution computation, the B 2 method is most convenient, but, as indicated in Section 4.3, it is not able to compute repulsive forces between the stator and the rotor, and therefore may not be reliable under certain circumstances. Repulsive forces result from relatively high levels of "tangential" (peripheral) flux density in the air gap (see Equation 4.3.29). Under starting conditions, tbr example, the rotor currents are high and may produce enough tangential flux density in the air gap to bring about repulsive forces between the rotor and stator conductors. Figure 4.5.3a shows the spatial distribution of force density in the air gap, computed with the Maxwell stress method using magnetic field data from a simulation of starting conditions. Noticeable negative (repulsive) forces are seen in the plot. When the B 2 method is used to compute the force distribution from the same data, the plot of Figure Space ~le pairs

Wave No. (Table

time freq.

Magnitude of B 2 (Telsa2)

0 0 0 120

0,8171 0,2545 0.0494 0,.3854 0.1317

Deflection (m)

Acceleratio n (dB)*

6.903x I0 "I3 8.366 x 10-15 2.168 x 10 -8 5,06I x 10-13

0 0 58.80 -33,83

5..~3) ........ F1 F2 F3 F4 F5 F6 F7 F8 F9 FIO FII F12 FI3 FI4

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

FI5

0 48 96 4 "~ 52 92 48 8 8 12 32 l 16 12

~:

~

~

_

~

~

60

!

i

. . . . . . . . . .

I20 120 120

...................6~i666 .............. 2,087 0.0270 5.421 0.0209 5.683 120 0.~63 240 0.0294 1.032 360 0.0060 4,166 920.4 0.0310 920.4 0.0129 1040. 0.0343 4 1040. 0.5388 4 ,..,,,,

. . ~ ~ ,,L,w,J,~:~JJJ~

w.

J,

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

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

, , ,JL,JJ ,JJJ. . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

~ .

.

.

.

.

.

.

.

.

.

.

. . . .

x 10 "13 x 10 "15 x I0-14

-41.53 -73.24 -52.83

x 10 -10 x I0 -I 2

24.40 3.56

.

.

.

.

.

.

.

Table 4.5.4: Mechanical Response of Stator to Magnetic Force Waves

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

t 51

4.5.3b is obtained. The shortcomings of the B 2 method under these conditions are apparent from a comparison of the two plots. Under normal, rated speed, quasi-steady state operating conditions, the tangential flux density in the air gap plays a smaller role. Figure 4.5.4a shows the spatial distribution of force density in the air gap, computed with the Maxwell stress method using data from a simulation of rated speed, quasi-steady state conditions. The forces are purely attractive. Figure 4.5.4b is a plot of force distribution from the same magnetic field data, but using the B 2 method. At rated conditions, the B 2 and Maxwell stress methods are seen to give very similar results. Table 4.5.3 lists some of the traveling waves of force distribution, computed using the B 2 method, that are present when the machine is simulated at rated speed and rated load operating conditions. The rightmost column in the table shows the pair(s) of flux density traveling waves that combine (multiply) to form the force density wave. (See Appendix 4.B, Equation (B.5)). Waves FI-F3 are stationary in time. The interaction of the fundamental flux density wave (B!) with itself produces wave F1, which has no poles. This wave represents the constant, radially-directed attractive pull between the stator and the rotor. The interaction of the fundamental flux density wave with the flux density waves associated with starer slotting pr~uces waves F2 and 1=3. The fundmnaental force density wave is wave F4, which is also created by the interaction of the fundamental flux density wave with itself. Stator slotting effects play a rote in waves F5-F7. Waves F8-FI 1 result from the interaction of saturation-induced flux density waves with the fundamental flux density wave (waves F8-F 10) or with themselves (wave F 11). Waves F t2-F14 result from the interaction of the fundamental flux density wave with the flux density waves caused by rotor motion and rotor slotting. Stator slotting, rotor slotting, rotor motion, and the fundamental flux density wave each play a role in producing wave F16.

4.5.1.4: Stator Mechanical R e s p o n s e The mechanical reponse of the stator m the fi)rce distributions of Table 4.5.3 can be found using the methods of Section 4.4. Using the simple beam model proposed by Alger (1954), for example (see Section 4.4.1), the mechanical responses shown in Table 4.5.4 can be computed. These deflection and acceleration data are the ultimate goals of the analysis; they characterize the vibratory and acoustic pe\rformance of the machine.

4.5.2: Combined Electromagnetic and Mechanical Finite Element Analysis In some cases, where the machine geometry or material is unusual, or where a high degree of accuracy is required, it may not be sufficient to extract field data from a circular path through the air gap and compute l~rces, deflections, and accelerations using a "beam" model of the stator. It may be necessary to compute forces acting at precise locations on the stator (e.g., a pole face in a D.C. machine)

152

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

and then apply these local forces m a highly accurate mechanical model of the stator. The I~a! forces can be computed using the Reyne stress tensor approach, discussed in Section 4.3.3. The mechanical implementation of the finite element method is well-suited for accurate mechanical modeling of the stator. It can be applied to the dynamics of a stator core, including the mass and damping effects of teeth, windings, insulation and laminations as well as stator core bracing bars, core stack end plates, stator shells, bracing fibs and frames. A discussion of the details of such a complete modeling approach is beyond the scope of this chapter; however, several examples are discussed below. The interested reader is also referred to the work of Girgis and Venna (1979, 1981). An example of coupled finite element electromagnetic and mechanical analysis of electric machines is shown in Figures (4.5.5) - (4.5.9). This work was performed with a two-dimensional electro-mechanical finite element software package, FLUXMECA, developed at the Laboratoire d'Electrotechnique de Grenoble in France. Further details of this work are provided in Imhoff, Meunier, Reyne, Foggia and Sabonnadiere (1989). Figure 4.5.5a shows a finite element analysis of a DC machine with 4-poles and thirty-three rotor slots, illustrating the distribution of magnetic force over the surface of the pole tips. The arrows in Figure 4.5.5a show the direction and relative magnitude of the magnetic force at the surface of the poles. The modulation of the force distribution, produced by the permeance effects of the rotor slotting, is clearly seen in Figure 4.5.5a. Figure 4.5.5b shows the spaceharmonic spectrum of the square of the magnetic flux density, which is proportional to the distribution of the magnetic force over the surface of the ~les. The large thirty-third harmonic, and the sidebands at 33±4, shown in Figure 4.5.5b, are the result of the rotor slotting effects. These harmonics lead to disturbing vibrations during operation of this machine. This is illustrated in Figure 4.5.5c where the mechanical deformations of the pole tip are displayed and compared with the initial (undeformed) finite element mesh. These harmonics are significantly reduced (Figure 4.5.5d) by reduction of the rotor slot o~nings, as shown in Figures 4.5.8a - 4.5.8c. The following figures show the sensitivity of the magnetic force at the pole face, to the variation of several design parameters. Figures 4.5.6a - 4.5.6c show the variation of the pole-face force distribution with the ratio of the slot width (s) to the air gap length (g). Figures 4.5.7a and 4.5.7b show the effect on the pole face force distribution of magnetic saturation. Figures 4.5.8a - 4.5.8c show the effect on pole face force distribution of the degree of closure of the rotor slots, as discussed above. Figure 4.5.9a shows the computed force distribution across a pole of a synchronous machine and, Figure 4.5.9b, the corresponding mechanical deflection due to the computed force distribution. The vibration level produced by this machine can be determined by applying Equation 4.4.4 given above and using the deflections determined in Figure 4.5.9b.

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

15 3

Figure 4.5.5a. Distribution of Magnetic Surface Force Density on Pole Tip of a DC Machine. The Modulation of the Slotting of the Rotor is Evident.

154

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

I, 4,

t1,I

I. ll.

............... i . I

II

Figure 4.5.5b: Space-harmonic Spectrum of the Square of the Magnetic Flux Density, which is Proportional to the Distribution of the Magnetic Force Over the Surface of the Poles.

S. J. Salon, C. J. Slavik, M. J. DeBoaoli and G, Reyne

P o l e Tf

Figure 4.5.5c: Mechanical Deformation of the Pole Top.

155

156

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

8,S

9.4

8.3

8.2

R Q

28

~

$8

Figure 4.5.5d: Space-harmonic Spectrum of the Square of Magnetic Flux Density, with Reduced Rotor Slot Openings.

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

. . . . . .

,......

157

i

j llllll

i~11

!u

!

Figure 4.5.6: a. Modulation of Magnetic Surface Force on a Pole Tip of a DC Machine with Variation of the Ratio of the Air Gap Width to the Rotor Slot Width (g/s).a.(g/s) = 0.25.

158

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

_] Figure 4.5.6: b, Modulation of Magnetic Surface Force on a Pole Tip of a DC Machine with Variation of the Ratio of the Air Gap Wi~h to the Rotor Slot Wi~h (g/s), b.(g/s) = 0,30.

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

159

Figure 4.5.6: c. Modulation of Magnetic Surface Force on a Pole Tip of a DC Machine with Variation of the Ratio of the Air Gap Width to the Rotor Stot Width (g/s). c . ( ~ s ) = 0.50.

160

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

t

Figure 4.5.7:

a. Variation of Magnetic Surface Force with Material Properties, on a Pole Tip of a DC Machine a. Linear Magnetic Material.

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

161

Figure 4.5.7: b. Variation of Magnetic Surface Force with Material Properties, on a Pole Tip of a DC Machine b. Non-linear Magnetic Material,

162

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Figure 4,5,8: a. Variation of Magnetic Surface Force with Closure of Rotor Slots, on a Pole Tip of a DC Machine. a. Rotor Slot Closure of 30%.

S, J. Salon, C, J. Slavik, M. J, DeBortoli and G, Reyne

163

Figure 4,5.8: b. Variation of Magnetic Surface Force with Closure of Rotor Slots, on a Pole Tip of a DC Machine. b. Rotor Slot Closure of 50%.

164

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Figure 4,5,8:

c, Variation of Magnetic Surface Force with Closure of Rotor Slots, on a Pole Tip of a DC Machine. c. Rotor Slot Closure of 80%.

S. J. Salon, C, J. Slavik, M, J. DeBortoli and G. Reyne

. . . . . .

I .... ~ _

-

165

I

....

a. Magnetic Surface Force Density

b. Stator Deformation Figure 4.5.9: Deformation of the Stator Core due to Magnetic Surface Forces in a Synchronous Machine

166

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

4.6 Conclusions As shown in the case studies presented in Section 4.5, finite element analysis is the first method to develop a consismnt model for both the electromagnetic and mechanical vibrational aspects of the electric machinery noise problem. The advantages of the finite element method are its ability to handle the most general type of machine geometries, the relative motion effects due to the movement of the rotor, and nonlinear iron saturation effects. Advanced finite element techniques provide the first complete models of the complex interaction of fields, circuits, tbrces, and motion present in electric machines. The main limitation of the finite element method for this application is that complem, coupled, transient electric machine models are at present limited to two dimensions, due to practical constraints of computing resources. Theret~re, threedimensional phenomena such as smtor winding end turns, rotor cage end rings, slot skewing, and axial flux cannot be modelled precisely. End turn and end ring effects are presently approximated by lumped circuit elements. Slot skewing may be simulated by dividing the rotor into several "slices," offsetting the slices according to the skew, and connecting the slices via circuit connections. The two~ dimensional model is then solved for each rotor slice. Axial flux effects are currently neglected in the two~dimensional models. While unimportant for most cases, axial flux may be significant in machines with offset rotors. In addition, currents flowing in stator or rotor laminations and bar-to-bar rotor currents are presently neglected in the two-dimensional models. The computing resources necessary to achieve a solution also limits the usefulness of the finite element m e t h ~ as a practical design tool. A large amount of computer time is required to reduce the starting DC transients so that a "steadystate" situation can be simulated. Methods at decreasing the time spent computing the DC transient are being investigated. One idea is to start the transient

Machine Type Squirrel motor

cage

Factors Which May Lead to Failure induction

Winding or insulation defects. Large currents & winding forces at starting. Stator end turn bracing problems. Stator core defects. Wound rotor induction motor Offset rotor, stator or rotor out-of-round. High stresses on rotor winding over?hang. Rotor winding defects. Stator core defects. Winding insulation defects. Synchronous motor Forces on excitation winding and poles. Defects in excitation system. End winding forces during transients. Smmr core defects. Winding insulation defects. Table 4.6.1: Causes of Rotating Machine Failure

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

I67

simulation with a solution produced by a sinusoidal, steady-state model. Finite element models of machines such as the ones discussed in this chapter for vibration analysis can be used I~r other applications as well. The area of conditioning monitoring of machines, tbr example, contains many possible applications. Finite element models may be used to analyze and simulate factors leading to failures for several different types of machines (see Table 4.6.1). For a more thorough review of these applications, the interested reader is referred to the work of Tavner and Penman (1987).

168

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Mac:hines

Appendix 4.A ~,,,~,~, ,,,

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

~,,~

,

, ........

~. . . . .

~,~,

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

,,

.,,,,,,~m,,,,~,,,,,~,

,,

,,,

,

.........

i . . . .

i

,i

iii1,

...........

Frequency and Spatial Distribution Harmonics of the Magnetic Flux Field of a Cage Induction Motor

Inte~retaion of the resulting magnetic field harmonics of a finite element analysis of a cage induction motor can ~ difficult. To assist in this task reference should be made to the space and time distributions of the magnetic field in a three-phase squi~el-cage induction motor as presented by Binns and Schmid (1975), from which this discussion is taken. If the rotor of a cage induction motor moves at a rotational speed of Ns(l-s), where N s is the synchronous speed and s is the slip, then the frequency of the magnetic flux components in the air gap observed fi"om the stator are the I2.~llowing: The stator magnemmotive force (m.m.f.)waves due to the applied stator winding currents at the line frequency f. The time frequencies of these flux components are all t, the supply frequency. These components do have different distributions in space and different spatial mode numbers, given by the series: +1, -5, +7, -11, +13, -17, where the sign in front of each number represents the direction of rotation of the flux wave relative to the fundamental field, which is taken as + 1. .

The slot-ripple harmonics modulated by the stator m.m.£ with carrier frequencies of NsnR(1-s), which equals 2fn(R/p)(l-s); where p is the number of poles of the machine, R is the number of rotor slots, and n is the harmonic order of the slot ripple. The slot-ripple frequencies measured from the stator are given by the following: fsr = In(2RJp)(1-s)± I lof, for n= 1,2,3 ..... (4.A. 1) Since the harmonic amplitudes converge quickly with n, the flux harmonics of this family above n=3 are very small. These harmonics

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

169

appear as pairs of "side frequencies" located on either side of of the slotripple carrier frequencies given above. .

.

The rotor m.m.f, waves, due to the currents flowing in the rotor cage. The resulting flux harmonics are speed-dependent. If a motor is operated under no-load conditions the amplitude of group (3) components is v e ~ small. Note that for cage motors, flux h a ~ o n i c s of groups (2) and (3) have identical frequencies.

Significant harmonics occur due to magnetic saturation of the main flux paths, usually encompassing the tooth tips. These flux components reveal themselves with characteristic frequencies shown below and in the slot-ripple sidebands extending to either side of the pairs of side frequencies in group (2). The frequencies of the most important components, relative to the stator, are: fsat = kf (4.A.2) with k = 3,5,7,9 .... where k is the harmonic order, and tsa t = In(2R/p)(1 -s)_+klof (4.A.3) for n = 1,2,3... and k = 3,5,7,9,.. Some of these harmonics can be relatively small. Additional m.m.f, harmonics will occur if the current phases are not balanced, such as if the motor is not symmetrical as in the case of a nonuniform air gap. In addition, stator m.m.f, harmonics will occur if time harmonics exist in the input current waveform. Note that the frequencies of the harmonics listed under the different groups may coincide and therefore their existence may undetectable. Flux harmonics observed from the rotor fail into different f'requency groups and are not discussed here; reference should be made to the original article by Binns and Schmidt (1975).

170

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Appendix 4.B III!!IHIII

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

III!lJ ..............................

II

II

II

IIIII

II1!

IIII

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

IIII

II

Relationship of 2f, 4f, 6f and 8f Induction Motor Vibrations to Air Gap Magnetic Flux Harmonics

The magnetic flux density in the air gap of the motor is obtained by taking the product of the magnetic circuit permeance and the magnetomotive force (m.m.f.) developed by the currents in the windings. Ignoring the slotting on the stator and rotor, which do not affect the 2f, 4t, 6f or 8f magnetic forces (f being the supply frequency), the air gap p e ~ e a n c e can be considered a constant, P0. The first several flux harmoncs, due to the stator winding m.m.f., called the space or phase-belt harmonics, are (4.B.I) B = P0(m.m.f.) = Po[F1- I cos(po-~-oOt) + F5- Icos(5po+o~t) + F7-1 cos(7po-~t) +,. (4.B.2) where = number of pole-pairs of the motor P = displacement along the stator bore in radians 0 =2rcf = electrical supply frequency f = time in seconds t Fa-b = a-th space harmonic of the stator winding m.m.f, due to the b-th time h a ~ o n i c of the current. (a=l denotes the fundamental.) Under conditions of saturation the following magnetic flux harmonics must be added m the above flux density field (Liwschitz 1942): B = B3cos(3p0_3eh3t ) + B5cos(5p0_5e~0t) + B7cos(7p0-70~9 t) + .... (4.B.3) These saturation harmonics travel 3,5,7 .... wavelengths when the fundamental travels only one wavelength. The saturation harmonics move in "lock-step;" they do not change their position relative to the fundamental or each other. The total magnetic field in the air gap is then: B = P0[Fi~tcos(p0-o~0t) + F 5.Icos(5pO+~o0t) + F7-1cos(7p0-~0t) + B3cos(3p0-3o~ot ) + B5cos(5p0-5~t) + B7cos(7p0-7~gt) + higher order terms (4.BA) As shown in the previous section, the magnetic force distribution can be approximitated B2/(2g0), where g0 is the ~rmeability of the space.

S. J. Salon, C. J, Slavik, M. J. DeBortoli and G. Reyne

171

Table 4.B.1" Origin of Magnetic Force Terms .... Freq. (Units of 0

'

M.M.F. Harmonics

Force Pole Number

F1 - 1

o

F5-1

Sat. Harmonics

F7- I

B3

B5

-44

2f* o 2f 0 2f 2f* 4f 4f 6f 6f

............ ",N.................................................................... : ..... 4 4 4p " q -j 8p 2p 4p 4p 6p 6p

q 4 4 4 ,,l q

6f

8p

'J

4 4

1 .............

M.M.F. Harmonics Force Pole Number

F I- 1

F5-1

o

Sat. Harmonics

. . . . .

F7, t

B3

B5

B7

44 .

.

.

.

.

2f

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

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

,/

Frex+ (Units of t3

o 2~ 2f 4t"* 4f 6f* 6f 8f*

B7

I

....

.

.

.

.

.

.

.

.

.

,,

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

.

lOp

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2p 8p 2p 10p Op 12p 2p

-¢ q 4 4 ~ -,/ 4 4

I I

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

.

.

4 ,,' . . . . .

I .........

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~

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.

.

.

.

.

.

.

.

.

.

"

.

.

.

q 4

172

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Sat. Harmonics

M.M.F. Harmonics ...............

Freq. (Units of f)

Force Pole Number

F 1- 1

F5-1

B3

F7-1

o

B5

,44 .........

2f 2f 4f 4~ 6f 6f 8f

"

.........

,

Freq. (Units of D

14p 4p 10p 2p 12p Op I4p

:1

.

.

.

.

.

.

!

,,,,,,t,,,,,

.

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.........

,,

~,,,

t ....

~ .

~./////J,

.

..... ......

.

q `4

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'4 `4

.

.

Sat. Harmonics

M.M.F. Harmonics Force Pole F5_ ! F7.1 FI-1 Number.

0

,

,

i B5

B3

B7

,u

""

6f 2~ 8f 4f I0f

6p 2p 8p 4p !0p

f

4

II

i

ii

: ............ "4

.4 "4

4 '4

Sat. Harmonics

M.M.F. Harmonics ,

Freq. Force (Units Pole o f ~ ...... Number 0 lOf lOp 2f* 2p ,,,

12f

B7

t,,

12p

FI-t

,!,,,t, . . . . . . . .

F5-1

F7-t

B3 ,,,,J,

.........

t

...........

| . . . . . . . . .

J?

','

B5

B7

`4`4 44 4 4

4 4

v

......

.....

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

173

M.M.F. Harmonics Freq. (Units of 0 0 14f

Force Pole Number

FI-I

F5-!

F7-1

Sat. Harmonics ,,

B3

B5

B7

I4p

The squaring process to obtain the force produces the square of each individual term and the cross product of each pair of terms. A total of 42 terms results from ~ u a t i o n (4.B.4) for B when the relationship I

cos(a)cos(b) = ~ [cos(a+b) + cos(a-b)]

(4.B.5)

is used. Table 4.B. 1 lists each of the resulting 42 magnetic force harmonics, their frequency (in multiples of f, the line frequency) and their force pole number (in units of p, the pole-pair number of the motor). The table also identifies the origin of each of the magnetic force terms (i.e., the combination of flux terms produces each force component). Separating the most important force components: Force = c2f cos(2p0+2o~0t) + o4f cos(2p0+4o)0t) + cr6f cos(2p0+6c,~t) + c~6fcos(2p0+8o~0t) + other terms (4.B.6) where e is the amplitude of the corresponding magnetic force component. Equation (4.B.6) shows that the strongest vibration frequencies, with saturation harmonics included, are the following: 2f (120 Hz.), with 2p, 4p, 8p, 10p and 14p force pole pairs; 4f (240 Hz.), with 2p, 4p and 10p force pole pairs; 6f (360 Hz.), with 0, 6p and 12p force pole pairs; 8f (480 Hz.), with 2p, 8p and 14p force pole pairs; for a supply frequency of 60 Hz. Table 4.B. 1 shows the origin of these terms, indicated by an asterisk *. The table is read as follows: a {- mark indicates the two flux density components which combine to produce the force component described in the left two columns. Two { - marks in the same box mean that the resulting force component is produced by the square of the indicated flux density term. If rotor and stator slotting, time harmonics from the supply, offset rotor and rotor and/or stator out-of-round effects were taken into account additional force components would also appear. Information on the analysis of these effects is provided in references Alger (1954; 1970), Ellison and Yang (197I), Yang and Timar (1980) and Yang (1981; 1988). The 0-pole pair force wave at 6f is unique as it is produced by two magnetic flux waves traveling in opposite directions. In the radial direction, this force wave develops a "breathing" or unil~Jrm expansion/contraction mode of deflection. However, the more important implication of the 0-pole force wave is the production of torsional oscillations, due to the alternating poles of the oppositely traveling waves from the saturation and space harmonic fluxes.

!74

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

Appendix 4.C . . . . . . . . . . . . . . . . .

Illlll!

IIIIII

IIIII

I ..........

JJMIII

II

I

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

IIII

...........

IIIII

II

II

III

!!!1111 IIIII

Extension of Stress Tensor Equation to Standard Maxwell Stress Expressions It has been shown (see Section 4.3) that the force F on an object may be computed as follows: F - ~T*.dS (4~C. 1) S where T* is the Maxwell stress tensor and S is the surface completely su~ounding the object. In this Appendix, Equation (4.C.1) will be used to obtain the commonly-used Maxwell stress "force density" expressions • BnB t P t - la0 (4.C.2) B2 •

Pn -

B2

n"

t

2rt0

(4.C.3)

where t and n denote components of vectors tangential and normal to the surface S, resp~tively. For the sake of simplicity, consider a 2-dimensional domain, where "surface'" S is actually a liner Unit vectors tangential and normal to the line are ~ t = Sx~x + Syi"~y

(4.C.4)

l~n =-Sy~x + sxl~y

(4.C.5)

Then dS = ~n dl, where dl is a differential length along the line of integration. Now the integrand for (4.C. 1) can be written as follows: dF* = T* - dS (4.C,6) d = ~

Bx - ~BI

BxBy

ByB x

By - ~[BI

-Sy

where dF* is a quantity with the units of force 4 (per unit length, in a 2D domain). The x- and y- com~nents of dF* are • ~ 2_ I t_12hl

S, J, Salon, C. J, Slavik, M, J. DeBoaoli and G. Reyne

dF~ = gd~oo[--Sy ByBx + S x (

175

-

(4.C.9)

The components of dF* tangential and normal to S are dFt = dF*°ID t = dFxs x + dFysy

= ~

~x- Sy +SxSy

Bx - B

(4.C.I0)

and dFn

=

dF*o~ n :e

,

= -dFxsy + dFyS x = ~t0 B~Sy + B ySx The components of flux density tangential and normal to the surface of integration aim

Bt Bn

= B°~t = Bxsx + Bysy = B'~n = -Bxsy+ Bysx

(4.C. 12) (4.C. 13)

Ken 2 BtBn =-BxsxSy

+

"S2y)+SxSy

= BxBy( s2 (B 2

_

BxBys x2 - BxBysy+B;~sy 2 2

B t2)

-2~By,sy+

-B2x)

(4.C.14) 2 + 2BxBysxSy + B 2ySy).

BySx'22

22 22 Now add and subtract BxS v_ + Bys x to (4.C.!5) and obtain

-B~)

(4.c.i5)

= 2B2s~-2BxBySxSy+2B2s2 + 2BxBySsy + B 2ysy2 + B2xsy2 + B ySx2 2)

-

Bx s x + Sy

-

s x

+

Sy

+ 2Bysx22 _ 4~BysxSy " IBIz (4.C. I6) By comparing (4.C.14) with (g.C.10) and (g.c.16) with (4.C.1 I), it is seen that =

2Bx2@

176

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

* BnBt dFt= 2 B2 * Bn t dFn2,0 d/ or, per unit length along the surface of integration , BnB t P t - ,0 2 2 , B n - B t Pn 2,0 which are the well-known Maxwell stress "force densi ty" expressions.

(4.C.17) (4.C. 18)

S. J. Salon, C. J. Slavik, M. J. DeBortoli and G. Reyne

177

Appendix 4.D .

.

.

.

.

.

Kerruish's Formulae for the Deflection of Smooth Thick Cylindrical Ring Influenced by 2-Pole-Pair and 4-Pole--Pair Force Waves

Kerruish (1958) has derived the deflection of a smooth thick ring acted upon, at its inner surface, by 2 pole pair and 4 pole force waves. The assumption of a smooth cylindrical ring means that the effect of stator teeth has been neglected.

D.1 Deflection due to 2-Pole-Pair Force Waves Maximum deflection in the radial direction at the inner radius (q) of the cylinder: V?ri ( 5 : V ~ ; ~ ; 3 , , 1 0 n 3 + ISn 2 + 30n + 6 - Zc(n-1) 3) (4,D.I) ot~tn 1) where r i = Cylinder inner radius r 0 = Cylinder outer radius n

-

W E

= Amplitude of Radial Magnetic Force = Modulus of Elasticity = Poisson's Ratio

t.ri J

Maximum deflection in the radial direction at the outer radius (ro) of the cylinder: 8=

(-)

24n 2 + 16n + 24)

(4.D.2)

Maximum deflection in the tangential direction at the inner radius (ri) of the cylinder: (4.D.3) I'1 - 6E(n- 1)318nZ(n+3) - 2e(n- 1)3] Maximum deflection in the tangential direction at the outer radius (r0) of the cylinder: Wro (8 rt=6E~)3t n + 24) (4.0.4)

178

Chapter 4: Analysis of Magnetic Vibrations in Rotating Electric Machines

D.2 Deflection due to 4-Pole-Pair Force Waves

Maximum deflection in the radial direction at the inner radius (ri) of the cylinder: 5=Wril'~9n7+ 9 n 6 + 9n 5 +25n 4 +55n 3 + 7 n 2 + 7n + 7) ~ ] EL r -

(4.D.5) where F= I5E(n-l)3(n 4 +4n 3 +10n 2 + 4n + I).

Bruce E. MacNeal and John R. Brauer

Chapter 5 II

III

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

II

II!llll!Jl!J

ELECTRICAL CIRCUITS AND FINITE ELEMENT FIELD MODELS: A GENERAL APPROACH

5.1 Introduction Numerical models of elexztficaI, electronic, and electromagnetic devices are generally either circuit models or full geometry field models. Circuit models contain ideal circuit components - - inductors and capacitors - - connected to circuit nodes. The physical location of the nodes and the geometry and material properties of the components are of no importance; only the discrete component values and connection topology are significant. Circuit models are often analyzed by a computer program such as SPICE, which may also contain specialized models of transistors, diodes, and other active, nonlinear devices. Field models of devices are usually based on some so~ of discretized geometry. The device and the space surrounding it are first divided into small, nearly regular pieces, called elements. Then the elements are used m obtain solutions to the field equations in the m ~ e l e ~ volume. This chapter is concerned with the finite element method, which relies on a discritization of tku"ee-dimensional space. Electrical engineers often use both finite elements and circuit techniques to model devices. The natural excitation for finite element models is usually a current (or current density). If this current (usually unknown) is supplied by an external circuit and is unknown, then some means of coupling the circuit to the finite element m ~ e l must be developed. The first method developed for coupling circuit models to finite element models was based on adding current variables as unknowns to the finite element matrix equation. The unknown currents coupled the finite element model to the circuit model. The theory and several examples of this current coupling technique

180

Chapter 5: Electrical Circuits and Nnite Element Models: A General Approach

have been described in several papers (Piriou and Razek, 1988; Nakata, Takahashi, Fujiwara and Ahagon, 1988; Salon, DeBortoli and Palma, 1990). A more general approach (Brauer, MacNeal, Larkin and Overbye, 1991; Vander Heiden, 1991) is to model circuits with "zero-dimensional" finite elements inserted directly into the finite element computer program. Extra unknown currents are not needed. This chapter reviews the zero-dimensional finite element approach. It begins with the general formulation of field behavior and shows that circuit behavior is contained as a subset. It then applies the approach to examples of a buck voltage regulator, a circuit-driven wire loop, and to two different transformers.

5.2 The Finite Element Formulation The formulation adopted for combined circuit and field analysis should meet two requirements. First, the formulation should be general. It must represent all classes of field behavior--including electrostatics, magnetostatics, eddy currents and wave propagation. If the formulation is general, then it follows that circuit behavior must be described as a subset of the same tbrmulation. R e second requirement is that the separation between circuit behavior and field behavior be relatively straightforward, and involve only a few assumptions. A simple connection without awkward constructs helps to maintain generality, and make the capability easier to implement and use. The formulation proposed here meets these requirements. The formulation (MacNeaI, 1990) begins with Maxwell's equations in their general form and derives an equivalent variational principle (virtual work). An equivalent matrix equation is then derived for discrete finite element solution variables. Variables which satisfy the matrix equation also minimize the energy of the system and, therefore, satisfy Maxwelrs Equations. Solution Variables Field components make relatively poor solution variables. The six components of E and B represent too many unknowns; in fact they are redundant because they are connected through Maxwell's equations. In addition, these vectors are often discontinuous at material boundaries and may cause convergence problems. Also, separation of field and circuit behavior using field variables appears to be relatively difficult. The choice of potential functions as solution variables is driven by the need for a symmetric matrix fonnulation, and by the need emily to separate circuit behavior from field behavior. We choose the conventional vector potential, B = V x A

(1.3.7)

as one of the solution variables. The time-integrat~ scalar potential is chosen to produce a symmetric matrix equation: E =- V _dA (5.2.1) dt The use of a function that is closely related to the electric scalar potential ~ ( d~g / dt = ~ ) also makes it relatively easy to separate circuit behavior from field behavior.

B. E. MacNeal and J. R. Brauer

181

It is well known that Maxwell's equations do not uniquely define the vector potential A. In numerical work this produces a singular matrix equation (MacNeal 1989). To circumvent this problem an artificial energy is assigned to the divergence of A. This penalty method (Cook 1981; Akin 1980) forces the divergence of A to be zero in the solution, so the solution obeys the Coulomb gauge. By substituting the potential forms of Eqs.(1.3.7)-(5.2.1) into Maxwell's equations, two equations are produced: V. ~

+

= -p0 + V - P 0

-

Vx(v(VxA)

=-~

I dt'

+

(5.2.2)

e~

e e

+ dt2)

+ V x M0 (5.2.3) The first equation is Amwre's Law written in pomntial form, including the effects of forced currents, eddy currents and displacement currents. The second equation is Gauss's Law, which describes the origin of electric fields in terms of charges. Included are the effects of an initial charge distribution and the effects of charge accumulation due to current flow. The other two Maxwell equations (divergence-free B and Faraday's Induction Law) are satisfied by the form of the potential substitution and do not appear explicitly. These two equations, which are equivalent to Maxwell's equations in their complete and general form, are the equations solved using the finite element method. Variational Principle The following variational principle is equivalent to Eqs. (5.2.2) - (5,2.3):

dv

~W =

Ii {/ dt

8Iv

]+oj.

vol

dv vol

I

t? + dA ¥

182

Chapter 5: Electrical Circuits and Finite Element Models: A General Approach

dv

rt; It;

dt { 5 ( V x A ) .

v (VxA)}

Ltl

vol

dv

dt { a6 (V.A) (V.A) }

J

]

ktl voI

dv

[i2 dt

!i

5

:)(oo

tl vol

+j It2 dv

f dt {SA,(VxM0)} Ltl

]

vol

~A.(vB* x n)-5v(n. surf

su?f The first, second and third terms represent energy due to electric fields, ohmic heating and magnetic fields, respectively. The fourth term is the artificial penalty energy. The fifth and sixth terms represent volume excitations due to imposed charges and cu~ents. The remaining terms represent natural boundary conditions. This variational form serves as the starting point for the application of finite element methods. It is straightforward to show that the variational principle of Eq, (5,2.4) is equivalent to MaxwelI's equations, Eqs. (5.2,2) - 5,2.3). Variations are first integrated by parts to obtain only direct variations with respect to the functions (5(v) instead of 5(v~). The coefficients of the variations are then found to be the desired Maxwell equations. ~ e r e is one slight difference, however. Because we use

B. E. MacNeal and J, R. Brauer

183

the potential ~, the second equation, Eq. (5.2.3), is split into two parts: the charge continuity, condition (first time-derivative of Eq. (5.2.3)) and an initial condition (Gauss's L w ) . Ampere's L w is reproduced exactly. Finite Elements In the finite element method (FEM) the analytic functions A(r,t) and v(r,t) are expressed as the product of shape functions, which vary only with position, and discrete values (or degrees-of-freedom, DOFs) at predetermined grid point locations. The space integrals are evaluated once, ahead of time, leaving on the timedependence of the DOFs to be determined in the actual FEM calculation. The solution volume is subdivided up into small, semi-regular regions called finite elements. Within each element, the spatial integrals are ~rformed numerically, and then added together (or assembled) into large matrices representing the entire model. The resulting matrices embody the geometric and material propeaies of the model, but do not depend on time. Elements with different dimensionality are obtained by making assumptions about the spatial dependence of the DOFs. Two-dimensional elements are obtained by assuming that all components of the potentials only vary within the plane of the element, and not through its thickness. These elements are useful for modeling thin, conducting sheets and thin air gaps. One-dimensional elements are obtained by assuming that all potentials vary only along the length of the element, and not within its cross-section. These elements are useful in modeling conducting wires. Axisymmetfic elements are obtained by assuming that the potentials do not vary in the circumfrentiaI (0) direction of some cylindrical (r-0-z) coordinate system. These are useful in modeling axisymmetric devices. Zero-dimensional (also called scalar) elements are generated by inserting terms directly in the matrices. They may be connected between any two DOFs. The meaning of these elements is application dependent. The circuit components described below are a special class of scalar elements connected to ~4tDOFs. ~ r o - , one-, two- and three-dimensional elements may all be connected together in the same finite element model. Zero-dimensional and axisymmetric elements may also be used together. Excitations and Boundary, Conditions The natural excitations of the formulation are dictated by the functional form of the variational principle. Remanent magnetization and the rate of charge evolution represent natural excitations inside the problem volume. In the special case of el~trostatics, the natural excitation is charge density,. Remanent magnetization excitations may be generalized to any time-dependent, divergence-free current distribution, including point currents, line currents and slot windings. The natural surface excitations (Neumann boundary conditions) are to s~cify the tangent component of #-1 B. The normal components of A are usually left unloaded and unconstrained (to avoid forcing the divergence). The natural surface excitation for v DOFs is the normal component of the current entering the model,

184

Chapter5: Electrical Circuits and Finite Element Models: A General Approach

J + dD/dt This ~ DOF excitation provides the connection to circuit models (see below) In the special case of electrostatics, the natural surface excitation is the normal component of EE The natural Dirichlet boundary conditions is to constrain the tangential components of A to specified values, and to constrain

The Matrix Equation Once terms are assembled into matrices a single matrix equation is obtained (MacNeal I990):

[M] [dt2 jT [B]

{ ~ } + [K]{U}= {P}

(5.2.5)

The matrices in this equadon are banded, symmetric and sparse This allows the use of efficient matrix solving algorithms. It is useful to rewrite this matrix equation in partitioned form:

=

(5.2.6)

The top A partition corresponds to Ampere's law The displacement current is represented by [M]; eddy currents are represented by [B]; the [K] matrix represen~ the curI~curl operator acting on H, and current excitations are represented by the A partition of the excitation vector {P} The ~ttom partition represents the charge continuity condition. The [M] matrix represents the time rate-of-change of charge density while the [B] matrix represents the divergence of current The ~ pa~ition of the excitation vector represents injected charge The formalism generates a ~parate initial condition: [M'qn~l

d~,

= {P0}" [M"gA] / (5.2.7) dt which represents Gauss's Law. The initial values of ~ must be (electrostatically) consistent with ~ e initial distribution of electrical charge. Thereafter, the flow and accumulation of charge is accounted for by the charge continuity condition in Eq. (5.2.6).

5.3 Circuits And Finite Elements Any general fo~ulation of electromagnetic field behavior must necessarily contain within it a description of electrical circuits We adopt the view that an "electrical circuit" is an approximation of field behavior in which all field effects external to idealized circuit components are ignored Fields outside the components are assumed to be zero, and all detailed knowledge of field behavior inside the components is suppressed. Component properties are defined only in terms of

B. E. MacNeal and J. R. Brauer

18 5

current/voltage transfer relationships between various connecting points. Because there are no induction effects a unique voltage may be assigned to each point within the circuit. Voltages need only be computed at component connections. Circuit behavior is independent of the physical geometry and material properties of the components; it depends only on the connection topology and the discrete properties ~signed to the components. Circuits can be modeled using the finite element formulation described above. In each circuit region all three components of the vector ~tential are constrained to zero. Thus Eq. (5.2.5) has the following f o ~ : ~fd2-~'/..,. + [K] {,¥} = {V~} (5.3.1) [M] [dt2 j , [B] { ~dt The t e ~ s in the [M], [B] and [K] matrices have dimensions of capacitance, inverse resistance (conductance) and inverse inductance, mspectivdy. The t e ~ s in the excitation vector {P} have dimensions of current. Grid points containing only 'q DOFs are placed at nodes connecting circuit components. The matrices associated with zero-dimensional circuit elements have a simple form. Capacitors only contribute terms to the ~ - ~4tpartition of the [M] matrix: [Mcap]=C [ _~ -1] l

(5.3.2)

where C is capacitance. Resistors contribute only to the [B] rear-'ix: i !] (5.3.3) [Bres] = R L - 1 l where R is the resistance. Finally, inductors contribute only to the [K] matrix: [ 1-I] [Kind] = -1 I (5.3.4)

where L is the inductance. In the field analysis part of the model the N - ~I/ partition of the [K] is always null. It is only when circuits are added to the model does this part of the [K] matrix contain non-zero envies. The circuit model defines the way in wNch current enters and leaves the field part of the model at common grid points. Thus the circuit provides the natural boundary condition on ~ DOFs. Boundary conditions for A DOFs at connection points are implemented in the usual way; i.e. by constraining A DOFs or by specifying tangent values of H. From Eq. (5.2.4) the natural excitations of the circuit are in terms of currents entering and leaving the model. Such excitations correspond to the ideal current sources often used in circuit analysis. The natural Dirichlet conditions for the circuit are to constrain N (or d~g/dt ) to some specified function of time (typically

zero). Voltage excitation is not a natural circuit excitation. However, ideN voltage sources can easily be constructed by placing a large ideal current source (a natural excitation) in parallel with a small resistor To the extent that the impedance of the rest of the model (including the field model) is large compared to the small resistor, the voltage across the resistor will remain constant. Thus ~ e combination acts as an ideal voltage source.

186

Chapter 5: Electrical Circuits and Finite Element Models: A General Approach

While the concept of voltage is well defined in the circuit model, it is not uniquely defined in the finite element field model. The true voltage (i.e., the line integral of E) is path dependent in the field model whenever induction effects axe present. But when a circuit is connected to a field model a specific "voltage" is defined by the circuit. This voltage represents the "electrostatic voltage" due to differences in ( d r / d t ) , and does not include effects from ( dA/dt ). Thus the true voltage in the field model may or may not be equal to the circuit voltage. Circuits are not limited to linear passive components: resistors, capacitors and inductors. The finite element program u s ~ in this work has the ability to model nonlinear components using nonlinear excitations called NOLINs. A NOLIN cu~ent may be an arbitrary nonlinear function of any (V) DOF or its first time derivative anywhere in the model. A NOLIN current may also ~ propo~ional to the product of any two DOFs (or their first time derivatives). With this capability it is possible to model a wide variety of nonlinear circuit com~nents, including transistors (see example below). The program also has a transfer fianction capability. A transfer function is a general, linear cons~aint relationship ~t;-veen any number of DOFs and their first and second time derivatives. Such constraints are useful in modeling complicated individual components or for representing entire circuits through their transfer properties. The program also contains a multipoint constraint capability (MPC). A mulfiI~int constraint equation is a general linear constraint relationship between any number of DOFs located anywhere in the model. In their most simple application, MPCs are used to make the value of one DOF equal to the value of another DOF l~ated somewhere else in the model.

5.4 Example Problems The finite element formulation described above has be~n incorporated into the field analysis program, MSC/EMAS. This program has general static, transient, sinusoidal, eigenvalue and nonlinear analysis capabilities incoworafing all aspects of field behavior. The program uses conventional, node-based finite elements in various forms, including 3D, 2D, 1D, zero-dimensional, axisymmetric and circuit elements. Materials may be linear, nonlinear (magnetics), or anisotropic and complex (sinusoidal and eigenvalue analysis). Several example problems are included to illustrate circuit modeling techniques. The first example, the buck voltage regulator, is strictly a circuit model (no field analysis) which illustrates the use of passive circuit components, nonlinear excitations and transfer functions. The second example, a circuit driving a wire loop, is a 3D example involving a conducting sheet (2D conducting elements) embedded in 3D space driven externally by an RL circuit. The laminated t r a n s f o ~ e r example models a load resistor in the secondary of a t r a n s f o ~ e r operating under both linear and nonlinear conditions. Finally, the model of an axisymmetric transformer includes primary and secondary conductors explicitly, and compares results with a T-equivalent model based on static result.

B. E, MacNeal and J. R. Brauer

187

RL ~I.

L=145.8gH L

ControlCircuit

Vo = +I2V

Vo

D

~c=0At67

C = 200pcF~" ..........i.............

Figure 5.4.1" Buck Regulator Circuit Buck Voltage Regulator Circuit Model This first example represents only a circuit m ~ e l ; no finite element field model is attached. The buck voltage regulator of Figure 5.4.1 consists of BJT power transistor, a diode, an inductor, a capacitor and a resistor. The buck voltage regulator is use~ to step down a constant 12 V source to a lower, nearly constant voltage across the load resistor R E The 12-volt source Vc is turned on and off using the transistor. The duty cycle of the transistor (controlled by the external control circuit) determines the eventual output voltage. The sawtooth wave coming out of transistor is rectified by" the diode D and then smootheA through the low p ~ s LC filter. A nearly-constant voltage appears across R E All features of the circuit including the nonlinear action of transistor and diode can be modeled using advanc~ features described above (see Fig. 5.4.2). Voltage source/transistor action - - The net effect of the voltage source and transistor action is a 12 volt square wave signal with a duty cycle of 0.4167. This is simulated by a time-de~ndent ideal current source (natural excitation)

1

2

3

4

NOLINI~ TF IND=145.8gH

RES=I.0fl k~

RI RES~~ ! 4.01

~ CAP = 2C~tF 7

RE

TM

T .... Figure 5.4.2: MSC/EMAS Model for the Buck Regulator

188

Chapter 5! Electrical Circuits and Finite Element Models: A General Approach

in parallel with a 1 ohm resistor. The square wave time history of the excitation is input in tabular form. Diode action - - A nonlinear load (NOLIN) is used to apply a current to node 2 which is a function of the voltage at node 1. For voltages V 1 greater than O. 17 volts, a current of 1 2 ~ arnps is applied to node 2. For voltages less than 0.17 volts, the applied current is -75 amps. A ~ansfer function is used to transfer the value of~g at node 2 to the time derivative of v at node 3. Passive circuit elements ~ The passive circuit elements are modeled as RES, IND and CAP elements with appropriate values. The behavior of this circuit is nonlinear, so the corresponding time history is obtained using the Newmark-Beta algorithm (implicit time integration). In this

44-

43-

41 !

,

.

.

.

,

"

f

!

r..r. ( ~ )

Figure 5.4.3: Computed Output Voltage of the Buck Regulator.

B. E. MacNeal and J. R. Brauer

189

case the algorithm solved for 125 complete cycles of the circuit (5 m s ~ ) in order to obtain a steady-state response. Output was requested at one microsecond intervals. Figure 5.4.3 shows the output voltage appearing across the load resistor RL, while Figure 5.4.4 shows the voltage appearing across the diode. The average output voltage of 4.37 volts compares well with the output computed using the SPICE circuit analysis program (4.46 volts, a 2.2% error). The peak ripple of 2 I mvo!ts also compares well with the SPICE results of 20.5 mvolts (a 2.4% error). Circuit Driving a Wire Loop A small, quarter-symmetry model of a wire loop driven by an RL drive circuit under steady-state conditions is shown in Fig. 5. 3D brick elements are used to represent free space, with 2D quadrilateral elemen~ embedd~ in 3D space represent the conducting loop (conductivity = copper e=v=0). Constraints are imposed on A at the boundary to make the normal component of B zero.

I~GENO I4".

!2-

m

108642-

O'

-2'

,,:

.

.

.....

.

49O

495

~-OS

Figure 5.4.4: Computed Diode Volmge in the Buck Regulator Circuit.

190

Chapter 5: Electrical Circuits and Finite Element Models: A General Approach

The external drive circuit consists of a sinusoidaI ideal current source in parallel with a 5 ohm resistor. Current passes through a 1 ~tHenry inductor and then another 5 ohm resistor before entering the finite element field model, The v potential on the front and back plane of the loop are constrain~ to the same value using MPCs. ~ (aswe!I as d,~/dt ) is also constrained to zero at one end of the loop to f o ~ a ground connection.

/

Y

L

L.x z

92

93

94 5~

0.2 Amperes AC

1.0~H 5f~ ~

.....

Figure 5.4.5: Model of Circuit Driving a Conducting L~p.

B. E. MacNeal and J. R. Brauer

191

Frequency (Hz) 60 60 x 10 3 600 x I03

FE Results (miliamps) 83.33 - i 0.005 82.93 - i 5.13 59.36 - i 32.2

Circuit Theory (miliamps) 83.33 - i 0.C~31 83.00 - i 5.21 59.74 - i 37.5

Table 5.4.1" Loop Current (milliamps) at Various Frequencies Table 5.4.1 shows the excitation frequency and associated loop current obtained with the finite element formulation compared to approximate results obtained from simple circuit theory. A separate magnetostatic analysis of the loop (using a ~ model) shows that it has an inductance of approximately I gHenry for this quarter-symmetry model. The resistance of the loop is 2 ohms, so the impedance of the circuit, including the drive circuit, is I2 + i 2rff (2 x 10-6). Considering the approximate circuit parameters used in the simple circuit theory the agreement with finite element results is considered good. Figures 5.4.6a and 5.4.6b show the current flowing in the loop and the result magnetic fields surrounding the loop.

jl~f~.~

Figure 5.4.6a: Arrow Plots of Calculated Fields in Wire Loop a) Current Density J

t 92

Chapter 5: Electrical Circuits and Finite Element Models: A General Approach

6.1 e-08 - 6.6e-08 5,6e-08- 6 . 1 ~

~'--'~'~J.I_r.,~ ~ 4,0e-08- 4.5e-08 3.~4.0e-08 2.~08

-

p p

Z

X

f/j

Figure 5.4.6b:

Arrow Plots of Calculated Fields in Wire Loop b) Magnetic Flux Densi~ B

Laminated Transformer with Secondary Resistor: Linear Operation Figure 5.4.7 shows one quadrant of a transformer with its primary winding energized with 60 Hz current and its secondary winding connoted to a load ~resistor. Figure 5.4.7 also shows the finite element model of one quarter of the transformer containing one-turn primary and secondary modeled using conducting 1D elements. The secondary resistance is modeled by a 0.086 ohm RES circuit element. The steel is modeled with 3D finite elements using a permeability of 2000, along with zero conductivity (laminated steel).

B. E. MacNeal and J. R. Brauer

193

~ujel

~d~5 ~~d po~ 99

"~.=0.~i ?

Figure 5.4.7: Transformer with a Secondary Load Resistance, including the Finite Bement Model.

194

Chapter 5: Electrical Circuits and Finite Element Models: A General Approach

rim

-

~7 1

0~9~

(b) Figure 5,48 Waveforms in Saturated Transformer a) B in Saturated steel, b) I in S~ondary A~xisy~e~ic Transformer

B. E. MacNeal and J. R. Brauer

195

The excitation on the primary is l amp. Finite element calculations show that the magnetic field in the steel is 7.9 x 10~4 Tesla, and that the secondary current is 0.7 A at -45 degrees. The finite elements results can be checked as follows. Ampere's Law applied to the closed steel path yields: )B! = . I

1

(5.4.1)

where I is the sum of the primary and secondary current: I = (1 - 0.5 + i 0.5) = 0.707 at 45 degrees (5.4.2) The length 1 is approximately 2.25 m, so 1B1 = 7.9 x 10 -4 Tesla, in good agreement with the finite element results. The secondary current may be checked by using the T-equivalent circuit of the transformer. The magnetizing inductance is calculated using IBI = 1.14 x 10"3 Tesla from an analysis with zero secondary current. Thus the flux linkage is : = N [Bt (area) = 1 x (1.14 x 10-3) x (0.2m x Im) = 0.228 x 10-3 Webers (5.4.3) The definition L = ~ / I gives a 0.228 mHenry magnetizing inductance. At 60 Hz the magnetizing reactance is therefore 0.086 Ohms, and, ignoring leakage reactance, the secondary current should therefore be 0.707 A at -45 degrees. Thus the secondary current computed in the finite element model is verified, including the effects of leakage reactance. Laminated Transformer with Secondary Resistor: Saturated Operation The t r a n s f o ~ e r of the preceding section is excited with a much higher current, 10,000 A peak. The steel is now driven well into the nonlinear regime, so the Newmark-Beta nonlinear transient algorithm must be used to predict behavior. A nonlinear B ( H ) curve for SAE I010 steel was used. This material becomes nonlinear in a magnetic field of approximately 1.5 Tesla. The calculated finite element results are shown in Figs. 8a and 8b. As expected the magnetic field is clipped above 1.5 Tesla, so the resulting secondary current is highly peaked whenever 01Bl/Ot is large. These results appear to be plausible, but no quantitative check was attempted. Axisyrvanetric Transformer Figure 5.4.9a shows a simple axisymmetric transformer. It consists of an inner core of soft ferrite on which a one-turn secondary copper wire is wound. The outer shell of soft ferrite contains a one-turn primary winding. The inner ferrite core and outer ferrite shell are separated by a 1 mm radial air gap. Figure 5.4.9b is a schematic description of the transformer circuit. The primary winding is excited by a voltage source of amplitude V1 and a frequency f in series with a resistor RI = 2 Ohms. R e secondary winding has a resistor R2 = 200 Ohms connected across it. The voltage across R2 is to be determined over a frequency range of approximately 1 kHz to 1 ~ MHz. In the analysis performed here, the ferrite is assumed to have a constant relative permeability of 60. The ferrite is also assumed to have conductivity of

196

Chapter 5: Electrical Circuits and Finite Element Models: A General Approach

q¢

II

I2

Ferrite

Ferrite

X

r

1

2

3

3.1

4.1

5.

(a)

L~ l

,~"~;)

L ~a2

V2

9

RI

~"'~I

R2

l

(b) Figure 5.4.9: Axisymmetric Transformer. a) Dimensions in cm, b) T-Gircmt Model of Transformer with Primary and Secondary Resistances, zero. The conductivity of the primary and secondary is assumed m be that of copper (5 x 107 Siemens/m) and is the only nonzero conductivity in this model. In actual transformers, the ferrite or steel cores may have significant eddy currents due to finite conductivities. Actual transformers also may have many more prima~ and secondary windings than does the simple transformer analyzed here. Static Analysis An approximate analysis of the transformer assumes that there are no currents except in the primary and secondary windings and calculates inductances of an equivalent T-circuit. The required inductances are the magnetizing inductance and the leakage inductances of the primary and secondary.

B. E. MacNeal and J. R. Brauer

Frequency

(Hz)

.

.

Ixl0 3

lxlO4 lxlO 5 lxlO 6 !xlO 7

ixlO9

.

.

197

Circuit

FE

0.0020 ........ 0.02~ 0.194 0.750 0.807 0.664

0£~020 0.0200 0.194 0.750 0.807 0.664

T -

.

Table 5.4.2: Ratio of Output to Input Voltage at Various Frequencies The magnetizing inductance can be computed from a magnetostatic analysis with only the primary energized. In this linear (constant permeability) case the inductance is given by 2W L i2 (5.4.4), where W is the stored magnetic energy computed using static analysis and I is the current input. The inductance seen for this case of no secondary current is the magnetizing inductance Lmag plus the p r i m l y leakage inductance Lleakl. Static analysis gives: Lmag + Lleakl = 0.782 gHenry (5.4.5) The leakage inductances can be computed from another magnemstatic analysis in which the secondary is energizecl equal and opposite to the primary. This gives no current or energy in the magnetizing inductance, and all currents and energies in the leakage inductances. The total leakage inductance can be calculated from Eq. (14.16), but this total result does not indicate how to divide the leakage between the primary and secondary. To appo~ion properly these two leakage inductances, the following formula is used: )-i Ai dli Lleaki- I I (5.4,6) where A i is the vector potential of winding i that dete~ines its flux linkage ~q. Here the magnetostatic result gives Lleakl = 0.145 gHenry, (5.4.7) and Lleak2 = 0. I05 gHenry (5.4.8) The T-equivalent transformer circuit of Fig. 9b was analyzecl as a pure circuit m determine the ratio of V2 to V1 as a fianction of frequency. The results are shown in the T-circuit column of Table 2. The table shows low output voltages at low frequencies due to current passing through the magnetizing inductance. At high frequencies the output voltage is higher. The T-equivalent circuit model is probably adequate over a fairly large frequency range for the simple transformer analyzed here. Eddy currents and a multiplicity of primary and secondary windings preclude a simple equivalent circuit for many realistic transformers designs.

198

Chapter 5: Electrical Circuits and Finite Element Models: A General Approach

Finite Element Analysis The finim element model was modified by attaching primary and secondary circuit resistor elements. The resulting axisymmetric finite element model is shown in Fig. 5.4.10. The resistors are attached to three additional grid points that are placed out of the xz plane of the axisymmetric finite elements. Also attached to the grid points are one-dimensional line elements representing the primary and secondary circuits. The extra three GRIDs must be placed at y = 2nr, where r is either the radius of the secondary or primary winding. Resut~ obtained with the coupled finite elemenffcircuit model are listed in the FE column of Table 2. Note that the agreement with the T-equivalent circuit results is excellent at all frequencies.

5.5 Conclusions A method is proposed for coupling circuit and finite element field analysis into a single matrix problem. This is achieved by adopting a general formulation f-or finite element analysis, one which contains circuit behavior as a subset. By applying simple restrictive assumptions, circuit behavior can ~ rea'aized in selected parts of the model. ~ e theory allows for nonlinear circuit components, such as transistors and diodes, as well as the usual passive components: resistors,

Figure 5.4.10: Finite Element Model of Axisymxnetric Transformer with Resistor in its Secondary.

B. E. MacNeal and J. R. Brauer

199

capacitors and inducmrs. Ideal current sources are a natural excitation of the circuit part of the model. Ideal voltage sources are constructed by placing a large current source in parallel with a small resistor. Initial applications have been to model simple drive circuits and load impedances; but the theory extends to any complexity. The formulation is also valid for all types of analysis, including statics, transient, steady-state sinusoidal response and eigenvalue analysis.

Chapter 6 G. Meunier and J.-C. Sabonnadiere .

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!

Jl Jill IIIIlllill!

....

IjIItLUI!

[1! .......

[! . . . .

U] ............

JJl[ .........

MODELING MAGNETIC MATERIALS FOR FINITE ELEMENT SIMULATION

6.1 Introduction Most electromagnetic field computation programs are generally able to take into account the ch~acteristics of magnetic materials of two kinds: --

Materials with linear characteristics which may be isotropic and possibly with residual magnetization. Soft isotropic nonlinear materials

The implementation of these models in a finite element package allows us to compute the magnetic field in a large number of electric devices. However, many devices are in reality made of materials with more complex characteristics. Especially those exhibiting anisotropic nonlinear phenomena like in oriented grain laminations, cannot be analysed using presently available models. This paper presents two models of this kind of material in the framework of a magnetostatic analysis. They can be used for the computation of magnetic fields in configurations involving nonlinear permanent magnets and oriented-grain laminations. Furthermore we present a survey of the finite element formulations for this kind of problems with special attention to the methodology of implementation. And finally some examples of the applications of these models are presented. These examples verify the Woposed models by replicating on the computer the results of a practical experiment and demonstrate our new ability to model and predict complex magnetic phenomena. The use of numerical methods for electric and magnetic field computation needs a good physical representation of electric or magnetic materials. This representation

G. Meunier, and J.-C. Sabonnadiere

20I

may be expressed as the set up of macrosxzopic relations like B(H) or J(E), which are then integrated into the general model for field computation described by Maxwell's equations. As a matter of fact, we work at a macroscopic scale because the phenomenon under analysis is the field distribution inside a structure the size of which is several orders of magnitude larger than the size of the geometry in which these phenomena appear on a microscopic scale (like crystal anisotropy or the creation of magnetic domains). Under these conditions it is perfectly possible to compute the field distribution with accuracy by solving Maxwell's equations ass~iated with macroscopic constituent relations for materials' description. It may be pointed out, that these relations in thct e m ~ d y a great part of the information which exists at the microscopic level. For instance, in a ferromagnetic material, the various associated energies, like exchange energy, crystal energy, magnetoelastic energy, are in fact implicitly translated in a macroscopic relation which describes the magnetization vector M as a function of the excitation field H. Finally we have to make clear that these relations link intrinsic physical quantities and therefore one of the interests in numerical methods is because of their ability m rake implicitly into account the effects of demagnetizing fields. The heart of this chapter is devoted to the methodology fi9r the setting up of models of magnetic materials and their implementation in finite element software. We shall set out first a model adapted m permanent magnet materials and then, second, one for grain oriented laminations. In both cases the materials under study are anisotropic and exhibit n o n l i n e a r b e h a v i o u r as a function of the excitation magnetic field.

6.2 Setting-up of the B(H) Relationship The setting-up of the intrinsic relation that links the flux density B to the excitation magnetic field H, is generally done from experimental plots because it turns out that it is very difficult to take into account all the phenomena present at the microscopic level by a theoretical approach. This is specifically the case of the models used in presently available finite element software. They are used first in the materials with linear characteristics. In fact these materials are used in a range of flux density in which B(H) is a linear function of the excitation field H. We mention below three kinds of materials in which the behaviour is linear and two types in which it is nonlinear. These materials need to be modelled in a manner suitable for use in field analysis.

6.2.1: Soft Isotropic Linear Materials In soft, isotropic, linear materials the magnetization vector M is expressed as a linear Iqanction of the excitation field H and is always in the same direction as that of the vector H. Therefore we can write : M =xH (6.2.1) in which x is a scalar quantity which represents the magnetic susceptibility of the material. The flux density B which is expressed by the general equation

202

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

B = No (H+ M); rto = 4re x 10-7 can be written : B = g H

(6.2.2) (1.2.7)

where g is a scalar function which represen~ the magnetic pe~eability which is equal to ~to(1 + X ).

6.2.2: Soft Linear Anisotropic Materials Now the magnetization vector is still a linear function of the excitation field H but there is a tensor relation between the vectors M and H: M = [g] H. When it is expressed in the main characteristic axes of the material, the permeability tensor Is], given by laoI+ [X] where I is the identity matrix, takes the simple form [~]

0!l

= 1 0 gy (6.2.3) L00g Here ~x, ~y, ~z are the permeabilities along the three characteristic axes of the material. In this way the description of this kind of material needs merely the knowledge of these three scalar quantities which are generally provided by the manufacturer.

6.2.3: Linear Permanent Magnets Linear permanent magne~ are materials in which the magnetization may be written as:

M = [N] H + M r (6.2.4) in which [ 1I ] is a tensor which is independent of H and M r is the residual magnetization. Generally the structure of magnets is characterized by uniaxial magnetization which defines the direction of easy magnetization. If we choose a system of coordinates in which the direction of e ~ y magnetization lies along the x-axis, the relation (6.2.4) becomes: M = t01~ ± +0 (6.2.5) 00N± In this situation the magnet will be described through the three scalar representative quantities (1t H, If±, Mr). It may be noticed that in many permanent magnet characteristics, the values of 1~II and N ± are the same. This allows us to describe the susceptibility of the magnet by the single value N.

6.2.4: Nonlinear Isotropic Soft Magnetic Material These materials may be modelled by a scalar function B(H) because the vectors B and H are always in the same direction. It is thus necessary to provide a mathematical representation of this function by many values of B and H obtained by measurements. It is usual m describe the material by the relation B(H) : B = g(H) H (6.2.6) in which g is the magnetic permeability of the material.

G. Meunier, and J.-C. Sabonnadiere

203

6.2.5: Nonlinear Anisotropic Soft Magnetic Materials Generally the processing of magnetic materials whose characteristics are more complex than the previously considered models is rather rare in field computation software for two reasons : It is not easy, even experimentally, to set up B(H) characteristics for such materials. The introduction of a complex B ( H ) relation inside a numerical computation package is not an easy task. The difficulty of the problem comes from anisotropic nonlinear materials, in which the hysteresis phenomenon superimposes a new dimension of difficulty. Hereunder we shall f,~us our discussion on materials for which the representation can be done by unique B(H) relations, which allow the processing of anisotropy associated with the nonlinearity. In the most general case the vector B(H) relation is described by three families of plots Bx(Hx, Hy, Hz), By(H x, Hy, Hz), Bz(Hx, Hy, Hz). Obtaining the experimental relation which enables the description of this family of plots is a very delicate problem because it implies the measurement of the strength

B(kG O0

20

20

15 70 °

I0

30 ° 0

30 °

1

10

H (Oe)

1 O0

1000

Figure 6.2.1: Magnetization Plots (the Angles which are reported show the direction between H and the roiling directiofi).

204

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

and direction of flux density as functions of the excitation field H. For instance an Epstein device is well adapted to the measurement of the strength of the flux density B fbr various angles of H (Figure 6.2.I).On the other hand it is to this day, very difficult to get the direction of the flux density B. If we assume that these characteristics are provided it is necessary to put them into software in order to enable it to make all the interpolations needed during numerical processing. Furthermore in order to use a Newton-Raphson method and thus to take the nonlinearities into account, it is necessary to define a tensor of incremental permeabilities [bB/bH T] which will be consuming of computer time. (The term bB/OHT is important in 3-D magnetic field computations using the magnetic scalar potential, as we shall soon see. The term bv/bB 2 (v is the reluctivity of the material) which is useful is 2-D computations using the magnetic vector potential has no meaning in anisotropic situations and is therefore not considered in this paper.The vector potential formulation is in fact the dual of the scalar ~tenfial formulation. For these reasons the setting up of a model is based on simpler mathematical representations coming only from the significant characteristics of a given material. For example, for an anisotropic material, the methodology will consist of expressing the vector relation B(H) from the three characteristics Bx(Hx), By(Hy), Bz(Hz) in the three main directions of the material and performing the most representative interpolations. In fact there is no general methodology for solving this problem and each class of magnetic material must be analysed separately. Besides, a good physical knowledge of the material is of strong help in judging the correcmess of our approximations of the characteristics. In the next sections we shall present the models developed for two types of materials: i. Permanent magnets and ii. Oriented grain laminations. We will also, develop the methods of implementation of these models in a finite element package used for field computation. As a first step, we shall exhibit a mathematical finite element formulation that is able to take into account nonlinear anisotropic materials; as a second step we shall describe the specific models.

6.3 Finite Element Formulations The formulation we shall describe uses the magnetic scalar potential. As we shall show, this formulation needs to know the B(H) relationaship and the incremental permeability tensor bB/0H T. If we want to use the vector potential (which is better in a two-dimensional problem) we should know the H(B) relationship and the bB/~hq" tensor. For a magnetostatic problem Maxwell's equations may be written as : V.B = 0 (1.2.5) VxH=J (1.3.23) in which J represents the source current density. The magnetic flux density B is related to the excitation field H by the constituent relation (6.2.2). In a current free

G. Meunier, and L-C. Sabonnadiere

205

device, the equation V x H = 0 allows the definition of a scalar magnetic potential V such as that H = - VV

(6,3.1)

For a problem with boundary conditions V = V o or B n = 0 on the boundaries of the domain, the solution of the problem is obtained by the minimization of the coenergy functional : H

F=

BdH dn

(6.3.2)

0 The finite element method consists of the use of a family ~i of functions piecewise defined on each of the elements into which the domain is divided. On an element V(x,y,z) = Z ai(x,y,z) V i (6.3.3) The solution of the problem is then provided by the minimization of the function F(V I , V2_.V N) by solving the equations : OF 0V i = 0 ; V i = 1,2 .... N (6.3.4) In the functional F, B is a function of H only. Thus we can write 0 3V

BdH

=

BT OH 0HT .B • 0V i OVi

(6.3.5)

Consequently : ~F j ~ OHT OV = Z B d~ Using the discretization we get : H=-VV=-y~V~i Vi i OH oVi = - v ~

(6.3.6)

(6.3.7)

(6.3.8)

Nnally, using these expressions, the equation of minimization becomes OF _ I[~" v a T B da OVi

(6.3.9)

This leads to two kinds of system according to the linear or nonlinear nature of the constituent relation of the magnetic material, as described next.

206

Chapter 6: Modeling Magnetic Materials for Nnite Element Simulation

6.3.1: Linear Materials For materials which exhibit linear behaviour we write M as : M = [Xr] It + M r (6.3.10) where M r is the pe~anent magnetization, assumed to be independant of H and [Xr] is a susceptibility tensor which is also independent of H. M e flux density then becomes : B = go (H + M(H)) = go ( [gr] H + Mr)

(6.3.11)

where [gr] is the relative magnetic permeability tensor. Note thant [gr] = I + [Xr], as already defined is section 6.2. If we report the material characteristics using the main axes, [gr] becomes a diagonal tensor [~tx 0 0 1 [grlm=/0 gy 0 t (6.3.12)

Lo o g,j

In order to move the relation from the local reference frame to the global reference frame we use a rotation matrix [Rt], to give us the flux density in the laboratory reference frame: B = ~o[gr] tt + B r (6.3.13) with [larl = [Rt] T [grlm [Rt] (6.3.14) Br = [Rt]T go Mr (6.3,15) The minimization equation then becomes : ~V i

e~i go [larlE Vaj Vj dO-

a iBrd~q

(6.3.16)

The minimization of the co-energy functional leads to the solution of a linear system of N equations for the N unknowns Vi : (6.3.17) [S] {V} = {Q} where Sij = fi/I v a T ~0

Qi=j

['r I Vc~j da

v a T B r df~

(6.3.18)

(6.3.19)

6.3.2: Nonlinear Materials When we model nonlinear materials the matrix [S] depends on the tensor [gr] which is itself, influenced by the solution. It is thus necessary to use an iterative method of solution. Among the numerous iterative methods available the Newton

G. Meunier, and J.-C. Sa~nnadiere

207

Raphson algorithm (Coulomb I981) is to be preferred because it is very efficient for the solution of nonlinear problems as we have seen in previous chapters. This method consists of the improvement of a preliminary approximate solution by an iterafive scheme:) [V]n+ I = [V] n + [AV] n (6.3.20) [AVln = [.0vT ~V n"

n

(6.3.21)

Using the previous relations, the coefficients of the Jacobian matrix [32F/oVT3V] may be expressed by : 32F

3ViaVj

--

(

d

V T OB _,

~i~z

(6.3.22)

with LO,,jj =

=

• V~

(6.3.23)

where (OB/aHT) is a tensor that defends on the excitation field H. The linear system of equations to be solved at each iteration of the Newton-Raphson procedure is

Linear Case: H=0 Construction of Matrices [S],[Q] or [JI,[-R]

Non-Linear Case: H B, [dB/dHl

Resolution of Linear System

V

~ ....

Dete~ination of B and [dB/dH] on the integration points

Problem

Linear Problem

Final Solution

]

Figure 6,3.1: Flow chart of the Solution Process

208

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

thus : (6.3,24)

[J]. {z~V} = {R} with ,V~ dfl

(6.3,25)

R i = ~ VRl B(H)d~

(6~3,26)

It is pointed out that for linear materials the tensor [3B/bHT] becomes ~o [~t]rand that when H = 0, the flux density B is equal to Br. This result suggests a global method of setting up the matrix equation for linear and nonlinear problems. The implementation of a process in which B and [bB/bHT]are systematically computed fi-om H allows us to use the mechnism illustrated by Figure 6.3.1~

6.4 Model of Permanent Magnets In almost all the software for CAD in electromagnetics (see the last chapter), permanent magnets (such as SmCo5 and NdGeB) are generally modelled by a scalar permeability. However, a new model presented recently by J. Chavanne and the authors (1989) is capable of taking account both the anisotropy and the nonlinear behaviour of permanent magnets in magnetostatic models.

6.4.1: Magnet Manufactured from Powder Metallurgy Those magnets that are built by a process of powder metallurgy may be considered YA

~H

C M(H)

" Z i j Figure 6,4.1: Ir~drluenceof H on an Assembly of Grains.

G. Meunier, and J.-C. Sabonnadiere

209

in a first approximation as an assembly of independent particles. All the grains have a direction of easy magnetization C, oriented around an x-axis which will be named the macroscopic direction of average magnetization. Recent investigations show that this distribution has an axisymetric structure around the x axis (Fig. 6.4.1) and may be represented by a Gaussian function. However another function that may be used (Jahn et al, I987) is : P(01 ) = (2n + I) (2rt) cos 2n 01 (6.4.1) If we choose n = 0, P(01) represents an isotropic distribution. For usual magnets the typical value is n = 7.

6.4.2: Analysis of Linear Anisotropic Behaviour Let us consider first an isolated grain in a magnetic field. The use of the model of consistent rotations which minimize the energy of the system (Chavanne I988) leads, for small angular variations of the magnetization, to the classical model which describes the reversible linear behaviour of a single crystal structure. In this case we can write, in a reference associated with the grain : Ms = ~

01

H+ 0

(6.4.2)

in which m s is the grain magnetization and H a is the field on account of crystal anisotropy. We consider now an assembly of grains in an excitation magnetic field H, the direction of which makes an angle OH with the x axis (Fig. 6.4.1). The behaviour of this assembly is s u p ~ s e d to be modelled by the supewosition of the behaviour of all the grains considered individually. Taking into account the distribution function of the grains it is possible after integration to express M in the form M(H) = M r + [Xr] H (6.4.3) where Mr is the remanent magnetization oriented along the x axis and is such that : + 1 M r = M s ~zn n + 2 (6.4.4) in which M s is the macroscopic magnetization of the magnets at saturation, and [Xr] is the susceptibility tensor defined by

[Zr] =

°° 1

~ 0 ~-3 n + 21

(6.4.5)

If the grains are perfectly oriented (n - ~ ), this tensor is identical to the relation obtained for a single crystal. Another exciting result is the fact that the ratio r between the parallel and transverse susceptibilities is independent of Ms/H a and is equal to r = (2n+2)/2. Thus the values of xl~ and x± allow the computation of n which characterizes the ratio of non-orientation. This result will be of special interest in the analysis of nonlinear models, as we shall soon see.

210

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

l

Saturation

1

Nucleation

Propagation

Figure 6.4.2: Mechanism of Magnetization in a Massive Material,

6.4.3: Analysis of Nonlinear Behaviour Nonlinear behaviour is directly related to the value of the coercive field of the magnetization. Two models, the Stoner-Wolt~,rth model (Mee 1964; 1986) and the Kondorsky model (Chavanne 1988), axe available to describe this behaviour. In this section we show through experimentation that the Kondorsky model gives a more accurate description of the behaviour and is therefore to be used in field analysis. The model of coherent rotation (Stoner-Wolfarth) gives a theoretical value of the coercive field Hc equal to the field of crystal anisotropy H a, while experimental data show that H c is weaker by one or two orders of magnitude. This phenomenon may be explained by a process of "nucleation" of a magnetization domain in the grains for a value of the excitation field H which is much smaller than H a. This process, related to the structural faults of the material, may not be quantified easily. The extension of the domain is then propagated in the whole volume of the grain for values of the field much smaller than the anisotropy field because the movement of a wall needs an amount of energy smaller than that needed for turn over by coherent rotation. (Fig. 6.4.2). In order to analyse the influence of a field of ordinary direction, it is useful to know the angular variation of the coercive field of the magnetization as a Pdnction of the angle 0 of the field H with the grain axis C. To this end, two different models may ~ proposed,: When the magnetization turn over is generated by a coherent rotation When only the component of the field along the magnetization is acting (the Kondorskym~el (Chavanne 1988): He(0) -

I%(O)

(6,4.6) cos 0 These two models are compared on Fig. 6.4.3. Experimental results obtained on isolated particles like SmCo 5, show that the angular variation follows the Kondorsky law. With the demagnetization plot of the permanent magnet along ox, its macroscopic direx-tion of easy magnetization Mlt(H), and if we lnow the value of the transverse susceptibility assuming it to be constant, it is possible to generate the demagnetization plots for various values of the angle OH, that H makes with the x axis. This result is obtained by taking into account the variation of the coercive field on an assembly of grains, each of them following the Kondorsky law. The magnetization along H may be written (Chavanne 1988): MI(H) = MII(HI) cos oH + x±H sin2 oH (6.4.7) For the magnetization orthogonal to the field H we get :

G. Meunier, and J.-C. Sabonnadiere

211

II!J

0

~!lll! II

45

I

90 e

Figure 6.4.3: Angular Variation of Coercive Field a) Goherent Rotation b)Kondorsky Low. M2(H) = MI~(H2) sin eH+ x±H sin e H cos e H

(6.4.8)

with HI = Hcos (e~ ell) H 2 = (H cos f~OH) / X

(6.4.9) (6.4.I0)

2 ~(n) = ~ cos"

3 2n+3

(6.4.11)

2 2n+l

(6.4.12)

2 p(n) = ~ cos" I k(n) = "~,

,/

(6.4.13) 2n+l However, this model is valid only for fields H which have a strength less than Hc(0)q[(2n+3)/3]; that is, for linear transverse behaviour. Experimental data have been got for various kinds of permanent magnets (SmCo 5, Fe~ite, NdFeB). In all cases excellent agreement has been found with this model as shown in Figure 6.4.4. Specially, it may be pointed out that this model is far better than those obtained by a direct interpolation between the two plots Mll and M L. This latter model corresponds to the infinite value of n of Fig. 6.4.4 where ~ = ~ = !.

212

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

H~ FERRITE

I t2 N- $ 1/, 2, N = O

300

@

32"

200 100

m'¸ ~ ~d/,,~,,~

~B~,,~,

"~

"~°~"

/,*

,'o -100

-200

- 72"

'1

#

I

o

,. ~-~

- - - - CA LC

~o¢

• o

EXP T: 300 K

-400

-200

0

H {kAlm)

Mt kA/m) ~000

NdFeG

.....CAL.C I N : B )

M/I

500

MA(Sa" H )

N=o

-500

.

,.~

-t000 ~

T= 310K .............

--!0()0

- - i

.........

......

"

J

- S00

....

!

........

T

0

H| k A / m )

Figure 6.4.4: Measured and Gomputed Magnetisation Plots for Ferrite and NdFeB Ma~e~s.

G. Meunier, and J.-C. Sabonnadiere

213

6.4.4: The Finite Element Method of Formulation The previous relationships of equations (6.4.9) to (6.4.13) are simplified to be used with the approximate values X = 1 and a = t~. This approach neglects the effects of demagnetization on the components orthogonal to ox, because of the small width of the distribution function of the C axis around the x axis. (This assumption is, for example, valid for n = 7). In the reference frame of the magnet the components of the flux density may be written : B x = Bll(Ht) + li rto (H x - HI) (6.4.14) By = go gr-LHy (6.4.I5) B z = la° ~tr.l.Hz (6.4.16), where BH(H) is the plot of flux density versus excitation field in the x direction. In the laboratory reference frame the flux density is+ (6.4.17) B = [RtlT Bm([Rt]H) The tensor [OB/aHT] in a reference related m the magnet may be written " aBx aBx aBxq

F

(6.4.18)

:I 0"°'r+0 / LO 0 + t o g r ± J with: aBx aHx - ~o [cos(~0H) cos0H+ Rsin(~0H) sin0H] • [,rll(H1 ) - 1]) + go oHy

= go [ sin(c~0H) sin0H- c~cos(~0H) cos0H] • grll(Hl)

E + l Hy+H z

+ bBx bHz

-

go [ sin(a0H)sin0H - o.cos(a0H) cOS0H] • rtrll(Hl)

2 Hy+H z

10~

Ml +v

25mm NdFeB

6ram FERRITE

Figure 6.4.5: G~metric Description of Magnets.

214

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

I

where far II(H) = ~ and bBlt((H)/bH Finally, in the same laboratory reference frame the tensor [bB/OHT] is expressed by the relation " -

6.4.5:

[Rt]T

a Rt

(6.4.19)

Application and Verification: Analysis of a Nonlinear Problem

The experiment which is proposed in this part is an approach which is of interest for the validation of the proposed nonlinear model. A laboratory experiment is first performed on the switching of the direction of magnetization. The computer simulation and, hence, prediction of this phenomenon have not been possible before. Using our material models in finite element analysis, we then simulate the experiment, thus, not only verifying the model, but also showing the power of the new capabilities that these models give us. We consider two permanent magnets, one being of ferrite and the other of NdFeB (Ng. 6.4.5). The characteristics of each of them provid~ by laboratory measurements are: ...........

~. . . . .

~. . . . . . . . . .

~. . . . . . . . . . . . . . . . . . .

~

*

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

..~

..........

.......

~

.

~

........

.

=

i

I P~es,,~e s Int~loti~

H

{~!M

-I~00

- 0 , 4 .

i

-0.6.

Y .,i ~d

F i g u r e

-0.8.

6.4.6: Bli(H) for Ferrite Magnet ($pline Interpolation).

G. Meunier, and L-C, Sabonnadiere

Nd Fe B Magnet"

Fe~ite Magnet '

215

Br

= goMr = I. 19 T

gr I1

= 1.03 = 1.I7

gor .1. gor ±

= 1.2

The B(It) plot along the direction of easy magnetization is represented in figure 6,4.6. The two magnets described above were oriented such that their magnetizations were in opposite directions (Fig. 6.4.7a). The distance between them was then gradually reduced (Fig. 6.4,7b). After initial repulsion, a process of attraction that holds the ferrite in contact with the NdFeB magnet was observed (Fig. 6.4.7c).

Mt

(a)

(b)

(c) Figure 6.4.7: Experimental Gonditions,

216

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

\

I/

~mm

FERRITE ~ A D=2cm

/

/ Hl

Figure 6.4.8: Flux Densi~ Measurement Near the Ferrite Magnet. This phenomenon is specific to the process of demagnetization.. The measurement of the normal component of the flux density in the vicinity of the surface of the ferrite along the segments AA' and BB' (figure 6.4.8) was made in the laboratory, and then compared with the result of the computer simulation. Figures 6.4.9a and 6.4.9b show the flux density calculated along the segments AA' and BB' shown in figure 6.4.8. The comparison with the measured values shows very satisfactory agreement. We have in addition plotted the values provided by an ideal isotropic model. The representation of the magnetization inside the ferrite magnet allows us to analyse the behaviour of the material as a function of the orientation rate. Figures 6.4.10 and 6.4.11 show the distribution of the magnetization on the middle plane of the ferrite magnet for the values n = 7 and n ~ ,~ respectively. The comparison of these two figures shows clearly the influence of the orientation rate on the framed area.

6.5 MODEL OF SOFT MAGNETIC MATERIALS 6.5.1: I n t r o d u c t i o n In this section we shall describe a method for computation in three-dimensional anisoh-opic nonline~ materials. ~ e time dependence of phenomena (eddy currents and hysteresis) is not taken into account. The model of the magnetic circuit of a three phase power transfb~er is used in the finite element method. In almost all power transfo~ers the magnetic circuit is made of grain oriented laminations which are specifically anisotropic. In order to take these properties of crystal anisotropy into account, an elliptic representation is usod. The consequence of the

G. Meunier, and J.-C. Sabonnadiere

I ax(T)

0.151'

2t7

0.15"I

......... Measured

%\

. . . . . . . Calculated with n=7 .....

1

Calculated with n=~O

0.10

~Immm BB"

o. ~,~[

AA' Measured

0.05

0.05

......

Calculated with n=7 Calculated with n=(~

0.00

.

0

.........

5 x(mm)

Figure 6.4.9. Measured and Computed Values of Flux Density" Along Segrn~ts ~_A' and BB'.

l

|

L !

J ~m

No

~

,too

J

O

~

4

,~Q

~

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~

~m~

~

q

~

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t

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t

q~

qm

~

~

~

O

4P

O

e~

~

~

b

a

9t

O

O

qam

~

N

Figure 6.4.10: M a ~ e t i z a ~ o n Distribution in the Ferrite Magnet ( n = 7).

218

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

architecture of the laminations is to show different magnetic characteristics in the plane of laminations and in the transverse direction. In the latter direction we have thus a property of pseudo-anisotropy. The method of homogenization of the magnetic characteristics is used to avoid describing the laminated structure through a grid which would need a large number of finite elements. A tensor describes the permeability through its components which depend on the excitation magnetic field.

6.5.2: Model of Crystal Anisotropy 6.5.2.1: General Points Oriented grain laminations are polycrystaI materials, whose c~staI elements belong m the cubic centered system. The crystals are arranged in order that the directions of easy magnetization are in the plane of the sheets oriented along the lamination sense (Goss orientation). Thus there exist three axes of different magnetization (Figure 6.5. I). The numerical modelling will rely on the use of a macroscopic model that is able to represent the phenomenon of anisotropy related to these laminated sheets.

6.5.2.2 Two Dimensional Models Anistotropic Media Representation A first approach may consist in conducting an analysis on the lamination plane. The macroscopic model used is a simplified model in which the easy and difficult directions of magnetization are the axes of an ellipse. ~ i s model is operated from data from magnetization plots in both directions of the sheet. The trajectory of vector B on the x-y plane is an ellipse for a given field H intensity (Bastos 1985; Shen 1987 ) as shown on Figure 6.5.2. The relation between the components of flux density and field intensity is thus 2 2 2 2 2 = Hf; with Hf = Hfx + Hfy (6.5.1) where gfx and ~fy are the longitudinal and transverse permeabilities of the lamination. The flux density is then defined by the following equations : B=

Hf

(6.5.2)

[gfx cos0] 2 + [lafy sin,] 2 ¢=tan "1 {~-~tfx tan-c }

(6.5.3)

where ¢ and ,~ are the angles of the direction of laminations with B and H respectively. The permeability tensor is thus : I-gfx 01 [ ~ = L0 ; f y ] (6.5.4)

G. Meunier, and J.-C. Sabonnadiere

219

.I !

I

!

!

22"

7~ "77

~/m

~

D

~

d

~

D

m

.

.

.

m

Q

Q

D

Q

e

.

.

.

.

q

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.

.

.

e~

~

t

~

t,~

q

~

~,

b

qm

.

.

.

.

.

,e~

il/ii . . . . . . . . . . . . . .

Figure 6.4.11: Magnetization Distribution in the Ferrite M a n e t ( n = ~).

Isotropic Media Representation In this case, the magnetic properties of the material are the same in all directions. The permeability tensor which can be represented by a scalar value is of the general fo~: [~t0 =

(6.5.5)

The flux density components are : Bfx = ~f Hfx B ~ = ~tf Hfy and thus ~ and "care equal.

6.5.2.3:

Three

Dimensional

(6.5.6) (6.5.7)

Models

Modelling anisotropic media When the field is distributed on a plane orthogonaI to the laminations (transformer's joints) it is necessary to implement three dimensional models. On fine plane of the laminations the macroscopic elliptic model is used. The relation between the components of the flux density and the magnetic field strength is thus: 2

+

[~12 2 H2 2 2 gfy.l = H f ; w i t h f=Hfx+Hfy+

H2 fz

The orthogonal component of the flux density is comput~ by Bfz = ~fz H~

(6.5.8) (6.5.9)

220

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

ection

Figure 6,5.1: Gross Orientation.

and the permeability tensor is written ptfx 0 0"] [~f] = I 0 ~tfy 0t hO 0 ~fz-I

(6.5. I0)

Model of Isotropic Media

Isotropic media are considered as a particular case of anisotropqc media as described previously. In these materials the magnetic characteristics do not depend on the field direction but only on its strength. In a general way it is always possible to consider a tensor, the diagonal elements of which have the same value. Thus we shall have :

F"f 0 07 [~d = | 0 ~f 0 | L0 0 ~fJ This leads to the same value for the angles ~ and z and to the relations: B x = ~fHfx By = ~fHfx

(6.5.11)

(6.5.I2) (6.5.13)

Model of Laminated Materials

The direction orthogonal to the sheets is now assumed to be laminated. The modelling of the laminated structure leads us to compute the homogeneous permeabilifies ~x, ~ty, ~z as functions of I~fx, ~fy, ~tfz. The flux density and the field strength in the equivalent homogeneous block are thus noted: Bx, By, B z and Hx, Ity, Hz. The components of the flux density are given by : [~]2 ~y.l ar'~l

=

H2; with H 2

=

H2 2 2 x +Hy+H z

(6.5.14)

G. Meunier, and J.-C. Sabonnadiere

Transvers( Direction

221

By H

~H

B~

Rolling Direction

Figure 6.5.2: Elliptic Model. B z = ~tz H z

(6.5.15)

6.5.3: Three Dimensional Model of The Iron-Air Region The modelling of the laminated structure of a transformer is done by a compromise which consists of analyzing the structure as though it were of an equivalent homogeneous material. We shall name as homogenization the mental process that consists of substituting for the inhomogeneous material (stacked sheets) a homogeneous one with equivalent magnetic behaviour (Figure 6.5.3). The conservation of the normal components of B and tangential components of H at the iron/air interface allows us to deduce the terms of the homogenized permeability tensor. This calculation is done assuming that the field is uniform in the laminations' cross-section. Thus, the permeability tensor I~ is anisotropic because of the pseudo-anisotropy due to the laminated structure. The sheet permeabilities ~tfx, rtfy, and gfz have various forms depending on the anisotropic models (such as crystal anisotropy). From the crystal structure of oriented grains (Figure 6.5.1) we consider the z direction as an axis of average magnetization just like the x axis.

6.5.4: Application to The Finite Element Method The model of the magnetic circuit of the transformer is made assuming that there is neither leakage flux nor cu~ent density in the circuit, in order to use the scalar potential formulation (Coulomb, I98I). The numerical evaluation of the Jacobian iron: gfx, lafy,gfz air: ga=go

relative quantitie 1-e relative quantitie e

Figure 6.5.3: Iron - Air ~mination and Equivalent Homogeneous Block

222

Chapter 6: Modeling Magnetic Materials for Finim Element Simulation

matrix needs the calculation of differential terms aBi/aHjT which take various forms according to whether it is 2D or 3D analysis that is being performed and whether the media is considered as isotropic or not.

6.5.4.1: Two Dimensional Analysis Isotropic Media in 2-D Analysis ~ e B-H plot must be interpolate& Differents ways are possible: a spline interpolation a approximation by the use of an analytic interpolation. For latter for instance, we can used Marc,cco's function (Vassent et al. t989). In this case we have (13~)27 v (B 2) = u + i~2.t+~) (6.5.I6) Here v = 1/U is the reluctivity and the coefficients ~, 1~,and 3' are computed by the least square method from experimental data. In the present problem the function ~t(H2) comes from v(B 2) and is: (6.5.17)

~(~tH) 23' + ~

This equation is solved by an iterative method like the Newton-Raphson method. Spline interpolation or an analytical function would allow finally the computation of ~tf and alaf/aH2f from the strength of the field H and thus the differential terms yield :

Bf + 2Hfx all2

2 ~ y Hfx aU~ (6.5.18)

2Hfy Hfx aHf2

~f + 2Hfx aH~

Anisotropic Media in 2-D Analysis For anisotropic media it is necessary to define two plots, a B-If plot in the longitudinal direction and a B-H plot in a transverse direction. Each of them is approximated like in the isotropic case from which we deduce the calculation of gfx, gfy and the derivatives agfx/aH~ and agfy/aHf 2. We can thus express the differential term :

G. Meunier, and J.-C. Sabonnadiere

223

aq-x

. a~x

aH~ ......

g f x + 2H~x

2 ~ x ~ Y ;Hff (6.5.1,9)

J

2nfx nfy anf

2

.fy + 2Hfy

aH 2

6.5.4.2: Three Dimensional Analysis Isotropic Media in 3-D Analysis One B-H curve is then necessary. The differential terms similar to the 2D case become: OBx 2 alaf ..... 2Hx~ aHx - ~f + aHf

aBx

a~.___Lf

aB x 01af aHZ = HxHza-~ aB~ a~tfy aHx = HxHy ........2 aHf 2 a~f ally = ~f + 2 HV~__2 -OH f

(6.5.20)

aB z artf aH x - 2 HxH z aH 2 aB z Oaf ally - 2 Hy H z aH~ aBz 2 auf an z = gf + 2"ZbHf2 Anisotropic Media in 3-D Analysis According to the crystal structure of oriented grain, rolled laminations we consider the x direction as an axis of average magnetization. From two experimental plots

224

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

in the longitudinal, and transverse directions, the interpolation is made like in 2D. Finally the differential terms aBi/aHj may be written:

~Bx + S 2 ~'fx aH x - g f x 2 XaH~ aBx a~fx ally - HxHy 8H2 1

OBx agfx aHz - HxHza~f OBv

0gfy

- 20gfy ally = .fy +2 HYitl~

(6.5.21)

l

a"fy 0Hz =

HyHZaH~

aBz ;H x = 2 HxHzaHf aBz 0~fz aH~ =2 H e H z a ~f x,.f(aBz,~Hz)= gfz + 2H'~a(2,z)kf(Op-fzOH'~(2,f)) Anisotropic Laminated Media in 3-D Analysis

The finite element methods needs the evaluation of the incremental permeability [aBi/aHi T]~ from average fields H x, Hy, H z and thus the solution of the system of equations: 2 2 2 2 gO Hf= Hx + Hz [g0(1.e)+gfz e ]2 (6.5.22)

~fz f(U~) =

The function f is obtained from an inte~lation of the characteristic B(H) function of the material. Thus, for the solution of the latter system of equations (6.5.22), we can use an iterative method such as the Newton Raphson method. The differential terms aBi/a~ T are:

G. Meunier, and J,-C. Sabonnadiem

aB x

225

"g fx

aHx = lax+ 2 aBx .............=2 aHy

3 bgfx 231afx

aB x

~=2 aH z

aBy

30,fy

=2

aH x

lamOH2 (6.5.23)

ally = gY + 2 aBy - 2 _

aH z

~~=z 2

aH x

0Bz = 2 aHy

2alafv Hzlarn g0aH ~ 2 3gfz HxHz g0lam3H2 f 231afz Hy Hzlam la0aH~

4 aBz 1-e. 2 la03lafz aH~ = gz +2 " ~ - ' - . H x ~ 3 22 a 2 with gm= la0(1-E)+gfzEand A= lam+2 Hzg 0 alafz/ Hf The flow-chart for solution is the following: From the values of Hx, Hy, Hz solve system (6.5.22) by an iterative method. - - Knowing H~ compute gfx' gfy' agfx/0H2 and 0gf~,faH2 from the approximated B(H) functions in the longitudinal and transverse directions. From gfx' lafy' lafz evaluate lax' lay, laz Use be elliptic model to compute B x and By From laz and Hz calculate Bz = gzHz Compute the differential mrm of the Jacobian matrix [0Bi/aHjT].

226

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

BM ^

realcurve

~=linear

intermediate curve s

H (a,m) 0

~

40000

Figure 6.5.4: Longitudinal Modified B-H Plots. 6.5.4.3: Implementation When the two dimensional model of crystal anisotropy was applied to a transformer, some convergence difficulties ~curred because of the s h y n e s s of the saturation knee+ In order to improve the convergence rate the classical under-

Segment 1 a) Initial mesh

b) Auto-adaptive mesh

Figure 6.5.5: ~ l f Adaptive Mesh Generation.

G. Meunier, and J.-C. Sabonnadiere

227

relaxtion method was replaced by an "accelerated Newton Raphson" method (Shen 1987). The principle of this method is based on a convergence speed criterion. The rate of convergence depends not only on the level of the nonlinearity of the materials, but also on the initial guess. If the initial guess is close to the solution, the convergence is fast. Therel~gre the initial guess is computed through a modification of the nonlinear B-H magnetization function in order that its nonlinearity level should be lower than the actual one. In our problem in which the level of nonlinearity is very high (in the longitudinal direction), we built a sequence of modified, smoother B-It functions which converges to the actual function (Figure 6.5.4). Each intermediate solution associated with a modified function does not need to reach convergence and the number of iterations can thereby be controlled. During the calculations, strong flux density variations appeared around the joints. This phenomenon is due to the misfit between the finite element mesh and the solution algorithms. In order to increase the mesh density in the area of strong discontinuities of fields, an auto-adaptive mesh generator has been used (Figure 6.5.5). The principle of this mesh generator is based on checking the continuity of the normal component of B between all bordering elements (Raizer, Meunier and Coulomb 1989; Hoole, Yoganathan and Jayakumaran, 1986).

(a) Isotropic Case

(b) Anisotropic Case

Figure 6.5.6: Flux Density Lines.a) Anisotropic Material b) Isotropic Material.

228

Chapter 6: Modeling Magnetic Materials for Finite Element Simulation

(deg) ¢ :(Oy.B)

t"

(Oy,H)

0

l Y

-10

-40"

*

;

a) Isotropic Case

(deg) 0 - (Oy~B) O_A............

• " (Oy,~

I

T D

-10

-20 r,q~

I

.30

b) Anisotropic Case Figure 6.5.7: Directions of B(,) and H (~) along Segment No.1. 6.5.4.4:

Results

The preliminary two dimensional analysis in which the nonlinearity and the crystal anisotropy were taken into account has been made and checked against experimental results. In Figures 6.5.6, 6.5.7 and 6.5.8 we have plotted the flux density lines, the flux density (along the segment N°I) and for the two cases of isotropy and anisotropy, the directions of the fields H and B. The transverse component is weaker in the anisotropic case because of the axis of difficult magnetization. The anisotropic analysis reveals the non-coincidence of the directions of the field and the flux density in specific areas like the joints. In cores and yokes the field intensity and flux density are parallel and along the direction of easy magnetization.

6.6

Conclusion

Through electromagnetic field computation software, and the development of macroscopic models of permanent magnets and rolled crystal laminations, we have provided a physical representation of phenomena which is more consistent than that from the isotropic models used hitherto, An experiment done in the laboratory on the reversal of magnetization predicted by computer simulation using the

G. Meunier, and L-C. Sabonnadiere

(T)

229

(T)

J

.5

-1

-2.0-

r~"

-2.0-

(a)

(b)

Figure 6.5.8. Flux Density Component along Segment 1 a) lsotropic Material b) Anisotropic Material.

proposed model for permanent magnets, gives a satisfactory validation of the model. For grain oriented magnetic laminations, a finer representation of the flux lines afforded by simulation using our model, suggests a means for the modification of geometric shape in order to minimize the no-load losses in transfo~ers. Future developments of models of hysteresis phenomena in permanent magnets, and the three dimensional representation of the magnetic circuit of a transformer will provide a better understanding of these complex physical phenomena.

Chapter 7

F. Ossa~, G. Meunier and J.-C. Sabonnadiere JlLIIIW[ ...........

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FINITE ELEMENTS IN THE ANALYSIS OF MAGNETIC RECORDING DEVICES

7.1

INTRODUCTION

Among the various information storage technologies, magnetic recording realizes the best compromise between cost and performance. Thus it has - - and probably for a long time will have - - a preponderant place, Numerous studies show that this technology has not reached its limits and progress is still realized in the design of heads as well as in the development of new materials. For such studies, a good simulation tool is needed to optimize the geometry as well as to analyze the effect of using new materials. This tool has to be general to allow the study of various shapes of the devices under study, but it also has to be able to take into account very small details of geometry and the behaviour of various materials, especially the hysteretic behaviour of the recording media. The analytical models currently used to describe the magnetic recording process do not have these capabilities. Simple and easy to use, they are helpful for a first analysis of the phenomenon, but the restrictive assumptions they are based on do not take into account its whole complexity: the geometry of the head is very simplified as well as its magnetic properties and the hysteresis of the recording media is hardly taken into account. The finite element method avoids such simplifications and allows us to develop the general tc~l needed. In this chapter, we present results obtained by modelling the magnetic recording wocess using the finite element meth~. The difficulty that is addressed is the introduction of the hysteresis of the recording media° Indeed, magnetic hysteresis is a complex phenomenon depending on the history of the material for which there is no well established model convenient for use in electromagnetic computation codes. Furthermore, it leads to a highly nonlinear system difficult to solve.

F. Ossa~, G. Meunier and J.C. Sabonnadiere

231

This chapter will start with a brief description of the magnetic recording process. Then we shall study the most exciting models of hysteresis considering their advantages and drawbacks according to the accuracy against experimental data and their ability to be merged into finite element electromagnetic computer codes. The third part will show how the hysteretic behaviour is introduced in the computation of magnetic fields in order to make a reliable simulation of the write process in a magnetic recording device.

7.2 The Magnetic Recording Process 7.2.1: Principle of the Magnetic Recording Magnetic recording (White 1984) uses the hysteresis of a magnetic medium to memorize int3grmation. We will discuss hereunder only longitudinal digital recording on rigid disk.

7.2.1.1: The Writing Process Figure 7.2.1 shows a device for magnetic recording. The head is a high permeability gapped magnetic circuit with windings around a part of the core away from the gm~. The recorder consists in a magnetic medium deposited on a nonmagnetic substrate. During writing, a signal current Iw applied to the windings magnetizes the head and causes a flux which fringes from the head core due to the presence of the air gap. Since the gap is adjacent to the medium, the fringing field magnetizes the medium. During its motion, the medium stores the writing current variations. In digital recording, only two values _+Imax are applied. This results in a succession of cells of opposite magnetization called "bits". The information lies in the sign of the magnetization. The area between two adjacent bits, where the sense of the magnetization changes, is called the "transition." This transition is an

MF

Figure 7.2.1: Magnetic Recording - the Writing Process

23 2

Chapter 7: Finite Elements in the Analysis of Recording Devices

important characteristic of the device: the sha~er it is, the higher is the density that can be achieved. The transition length depends on the medium itself, but also on the writing field.

7.2.1.2:

The Reading P r o c e s s

During reading, the recorded patterns of magnetization provide a flux source, a part of which circulates through the head, avoiding the air gap. The motion of the medium creates a time dependent flux which induces a reading voltage across the coil. The reading voltage consists of a succession of pulses caused by each transition moving under the gap (Figure 7.2.2).

7.2.2: The Recording Media The material used for the recording media needs a hysteresis cycle as big as possible : a high coercive field to prevent erasure of the information - but the writing head has to generate a writing field high enough to write on it - and a high remanent magnetization to achieve the higher reading voltage as possible. The ideal would be a perfect square cycle. Two families of media exist : the particulate media and the thin film media.

7.2.2.1:

Particulate

Media

The particulate media used in recording (Bate 1986) are made of small magnetic particles (between 0.04 and 1.0 microns) in a nonmagnetic binder. The needle shape of the particle insures strong anisotropy and gives the direction of the magnetization. The coercive field depends on the material used and on the size of the particles. Any value can be obtained between 100 and 3000 Oe. The remanent magnetization is limited by the number of particles by unit of volume produced without excessive dispersion of size and direction. In practice a saturation

Figure 7.2.2: Magnetic Recording - the Reading Process

F. Ossaa, G. Meunier and J.C. Sabonnadiere

233

magnetization 4riMs of 8000 Gauss is a maximum.

7.2.2.2: Thin Film Media More expensive and difficult to produce than particulate coatings, thin film media (Arnoldussen 1986) have the advantage of a greater available signal amplitude. The source of this advantage is the lt~0 % packing of magnetic material, compared to 20-40 % in particulate media, which results in a higher magnetization. The same amount of magnetic flux can be squeezed into a thinner coating, bringing the whole storage layer closer to the recording head and resulting in more efficient devices.

7.2.3: The Recording Head Various kinds of head exist (Jeffers 1986), according to the application and the performances n~ded. Their shape and the material they are made of depend on the manufacturing process. Our purpose is not to give here an overview, but rather to descri~ some of their i m p l a n t features.

7.2.3.1: Ferrite Heads The classical ferrite heads are the cheapest ones. They look like the head designed in figure 7.2. ! and their permeability is very high (gr = 5000). An improvement is made by depositing a thin film metal of very high saturation field on each side of the gap to avoid saturation at the gap corner (the metal in a Gap head). But the use of bulk ferrite generates eddy currents at high frequencies and Ba.rkJaausen noise ( due m the motions of numerous domains walls in the head). 7.2.3.2: Thin Film Heads Thin film heads, an exemple of which is shown on figure 7.2.3, are produced

conductor

magnetic circuit

Iw

media Figure 7.2.3

• Thin Film Head G e o m e t r y - Poles Thickness = 2-5 microns, Head Heigth = 50-100 microns

234

Chapter 7: Finite Elements in the Analysis of Recording Devices

using microelec~onics technologies. Their flat geometry prevents eddy currents and Barkhausen noise and leads to better behaviour at high frequencies. The main drawback is that they are difficult to manufacture, needing heavy equipment. The trend is to achieve always higher recording densities. Thus to avoid selfdemagnetization, media with higher coercive fields are needed, which implies the need for higher writing fields to magnetize the media. This is achieved by making gaps smaller and the flying height lower. For recording on thin film media (rigid disks) with a coercive field Hc of 1200 Oe and a saturation magnetization 4~Ms of 1 0 0 ~ Gauss, the gap length is currently between 0.3 and 1.0 microns, with a flying height of about 0.3 microns.

7.2.4 Modelling Magnetic Recording As we have seen, recording devices are made of the recording head and the media. To simulate the whole device, we have to compute its magnetic state with each part properly modelled. Modelling the head (Bertram 1986) alone in two or three dimensions does not cause any theoretical difficulties: the finite element method allows us to represent any shape and it is also possible to include the saturation of the soft material constituting it. But in fact, practical difficulties arise in meshing the geometry because of the very small size of the gap compared to the whole device. Since high accuracy is needed in this region, no approximation can be made and this leads to the use of numerous elements. The real difficulty of the work arises when the hysteretic behaviour of the recording media has to be included in the simulation. Modelling hysteresis is not easy because it is a dynamic phenomenon depending on the history of the material, for which there is no welI established and reliable model convenient for use in electromagnetic computation codes. Furthe~ore, this highly nonlinear behaviour results in a computation process where convergence is hard to achieve. At the present stage of our work, a two dimensional modelling of longitudinal recording on thin film is considered. The very small thickness of the medium (0.07 microns) insures that the vector magnetization keeps on the plane of the disk; thus a scalar model of hysteresis is enough, as we will see later. The next section of this chapter is a survey of some interesting scalar models of hysteresis.

7.3 Modelling Numerous models of hysteresis have been published in the literature. Some of them have ve~ exciting but sometimes contradictory prope~ies. For instance, all models derived from Preisach theory do not simulate the minor loops reptation while other simpler models emphasize this phenomenon. The test between the behaviour predicted by the models and the experiment has not always been made in a reliable way and the choice of one among this models is not obvious. In this section we shall analyze the most interesting models with a theoretical, numerical and finally experimental point of view.

F. Ossart, G. Meunier and J.C. Sabonnadiere

235

HE

M Mr

J

07

M....J= 0

or

or

...

Figure 7.3.1" Hysteresis loop, definition of Hc, Mr and Ms ; exemNe of domains configurations corresponding to H=0 and M=0.

7.3.1 : I n t r o d u c t i o n Hysteresis is a very complex phenomenon which is tightly linked to an irreversible processes. Thus it remains very difficult to model. The problem arises from the fSact that at each instant of time, the magnetic state of the material depends not only on its intrinsic properties, but also on its history. The structure of the material as domains which determine the reaction of the material to an applied field, is the result of complex interactions which happen during the "birth" of the material as well as on the application of a time dependant fields during its life (Zhu and Bertram 1988; Jiles and Atherton 1986) Magnetic materials are characterized on the (H,M) map by their hysteresis loop (Fig.7.3.1a). This loop maps an area in which all points may be reached: there is no unique M(H) curve but an infinite number of curves. Let us assume a material in the initial state (Ho,Mo). Various domain arrangements are possible in this initial state according to the way in which it is reached; thus the data of Ho and Mo is not enough to define this state. The magnetization curve followed from this state cannot be forecast accurately from this single data. The history of the material has to be known. For ideal soft magnetic materials, the theoretical distribution of domains is provided by the minimal energy of the system. This energy is built around four components: the exchange energy issues from the variation of dipole directions between neighbouring elements, the anisotropy energy which acts like a spring to draw the vector M back along easy magnetization direction, the demagnetizing energy which arises from nonzero divergence variations of M, and, finally, the external energy which tends to keep M along the external field. The theoretical computation of these terms is possible and the structure of M everywhere can be determined by a minimization of the energy of the system. However, the

236

Chapter 7: Finite Elements in the Analysis of Recording Devices

tremendous volume of computation allows only the analysis of very small applications and needs special computers. For actual materials, the motion of the domain walls is complicated by obstacles like grain impurities or flaws. These faults bring about an energy dissipation similar to friction. In addition to the complexity of the computation, we found a shortage of models. Moreover, the vector nature of the phenomenon and its anisotropy make the phenomenon difficult to observe with classical equipment and increases the difficulty of modelling it. The micromagnetic physical approach therefore appears presently to be too heavy in computation time and memory requirement for practical use, especially in our case where we are interested only in macroscopic effects. For this reason only phenomenological models, which represent the phenomenon without any dependence on its origin, are considered here.

7.3.2 Theoretical Analysis of Models Among the various models available in the litterature, two are presented here for their exciting features. The first one is derived from the well known Preisach model, the second is based on the link established between the magnetic field and the flux density by a differential equation. More elementary models are also considered because they have been frequently used during the past. After a short presentation of the models, their results will be checked against experimental data in order to draw some conclusions on their validity.

7.3.2.1:

Elementary

Models

We shall start by the analysis of two very simple models based on an analytical description of the hysteresis loop. It is obvious that such simple model can not be very accurate, but it may be useful to compare them with experimental data.

7.3.2.1a: The Model of Potter This model (Potter and Schmulian 197I)has been extensively used in magnetic recording. Tlne magnetization curves are described by: M(H,~) = Ms{ sgne~ -~[ l+tanh(1 -sgn~HffIc)tanh-1S] } (7.3. I) in which sgn~ is the sign of the real a, Hc is the coercive field, Ms is the saturation magnetization, Mr the remanent magnetization and S=Mr/Ms is the squareness parameter of the cycle, ~ is a characteristic of the loop used at a given instant of time. After each extremum Hm of the applied field, the change in the magnetization curve is made by calculating a new value of a by: 2sgnc~ - a [ 1+tanh( 1-sgno~ .Hm/Hc) tanh" 1S] a =

(7.3.2)

t

[ 1+tanh(1 +sgnex Hrn/Hc) tanh- 1S] This model does not represent closed cycles.

F. Ossart, G. Meunier and J.C. SaN3nnadiere

237

11

Figure 7.3.2 a.The Polynomial Model - a. Polynomial Approach of the Major Loop, For H<_-Hs,M(H)=-Ms ;for -Hs
7.3.2.1b The Polynomial Model We propose a simpler expression (Ossart 1991) for the magnetization curves by the use of new data: the field Hs for which the loop closes. This model assumes a hard material with a slope equal to zero ~ y o n d the closure point. The ascending branch of the major loop is approached by the quadratic polynomial functions P1 and P2 respectively in the intervals [-Hs,0] and [0,Hc] and by the cubic polynomial function P3 on the interval [Hc,Hs]. The parameters of these functions are dete~ined by insuring the continuity of the major loop and its first derivative at the points H=0 and H=Hc and a slope equal to zero at the points H ~ H s (Figure 7.3.2a ). Minor loops re(H) (Figure 7.3.2b ) are derived from the m~or loop M(H) by the specific transfo~ation: Ms - m(H) = ~(Ms-M(H)) (7.3.3) in which o~ is computed by writing that the turning point is on the new

P J

H

Figure 7.3.3: a.Elementary Hystersis Operator Yab b. Step Oprator Ia

238

Chapter 7: Finite Elements in the Analysis of Recording Devices

magnetization curve.

7.3.2.2: The Mayergoyz Model This model is a complex one, but it is based on concepts of great interest and it follows a very rigourous and scientific aproach. The range of use of this model and the ext'~rimental data needed for its identification are ~rfectly define.

7.3.2.2.a: The Classical Preisach Model The model proposed by I.D.Mayergoyz (Mayergoyz 1985; Doong and Mayergoyz 1985) is an extension of the well known Preisach model (Preisach 1935), built from elementary hysteresis operators g ~ characterized by their switching values c~ and ~ (Figure 7.3.3a). The basic idea is to build any hysteresis loop by a weighted integral of these operators:

(7.3.4) The description of the loops is included in the function ~t(a,~) which links a magnitude at each elementary loop. This function may be derived from experimental loops. This model has been analyzed in numerous papers (Atherton, Szpunar and Szpunar 1987; Del Vecchio 1980; Wiesen and Charap 1988; Brokate 1989; Kadar 1987; Kadar and Della Tore 1988). We shall deal hereafter only with the user's view upon its main characteristics. i) This model detects and accumulates the extreme values of the applied field. It also reproduces the erasure of past extrema by fields of larger magnitude (Figure 7.3.4a). ii) Minor loops corresponding to the same values of extremaI fields are stable and congruent: they can be superimposed by a vertical translation (Figure 7.3.4b). iii) The identification of the model is made from first order reversal curves defined in Figure 7.3.5. Such data are directly used in the following algorithm: let us call (ai) and (l~j) the sequences respectively increasing and decreasing of the stored minima and maxima of the field and Aid the function: HI

Figure 7.3.4: a.PartiaJ Storage of the E~rem Values of the Field: H4 and H5 are Erased by H4' b.Congruency of Minor Loops.

F. Ossart, G. Meunier and J.C. Sabonnadiere

239

Figure 7.3.5: First Order Reversall Curves - Definition of Ms and M ~

AM(a,~) = 1~+~ - ]tv~

(7.3.5)

It may be proved that, if we start from negative saturation and finish with a d ~ r e ~ i n g applied field, the magnetization is equal to: n

M(H) = - M s + ~ { AM(ai,~i_l) - AM(oq,~ i) } + AM(H,13n) i=l Similar formulas are used for other initial and final conditions.

(7.3.6)

7.3.2.2b Extension of the Preisach Model In order to extend the classical Preisach model to a better representation of hysteresis loops, a totally reversible operator Xccis introduced and the magnitude of the elementary loops is made dependent on the applied field (Mayergoyz and Friedman 1987, 1988). The magnetization is then given by : dVOO

M(H) =

ff

,v

~(a,~,Hh'a~(H) d~ d[3 +

s

v(c0X~(H)do~

(7,3.7)

~¢*O

a->13 The geometric interpretation is the same but the behaviour of the model is slightly medified: i) The storage of the extreme value is not changed. ii) Minor loops corresponding to the same extreme values of the field are no more congruent, but for a given field their thickness is the same (Figure 7.3.6a). This extends the applicability of the Presisach model.

240

Chapter 7: Finite Elements in the Analysis of Recording Devices

iii) Identification requires additional data: the second order reversal curves, defined in Figure 7.3.6b. The function AM is now defined by: (7.3.8) AM(a,~,H) = Mct~ - Mc~H and the magnetization is given by the computation of: +

n

(7.3.9) M(H) = M H (H) + ~ { AM(ai,t3i.i,H) - AM(ai,~i,H) } i=l This model is able to represent loops with a nonzero slope after the closure point and may be applied to the flux density B as well as the magnetization M.

7.3.2.3:

Hodgdon's

Model

The Hodgdon model (Hodgdon 1988; Coleman and Hodgdon 1986, 1987) is very different from the previous one. It assumes a relationship between B and H defined by a differential equation: = cx

[f(B) - H] + g(B)

(7.3.10)

in which a is a real constant, sgn is the sign function, dB/dt is the time derivative of B and f and g are characteristic functions of the material. A correct choice of and of these functions allows this model to describe hysteresis loops of various materials. The choice of f and g follows some considerations: i ) the graph of f is the inverse of the anhysteretic curve - - the anhysterctic curve being the ideal magnetization curve that the material would follow if there were no hysteresis. ii ) in intervals where g(B) = if(B) the ascending and descending branches of the major loops coincide so that the cycle is closed. Hodgdon suggests the following expressions for f and g:

Figure 7.3.6: a.relaxatien of the congruency of minor loops. +

b.second order reversal curves, definition of Ms, Mctl3and M ~ u

F. Ossart, G. Meunier and LC. Sabonnadiere

241

;is curve

Figure 7.3.7: a.Experimental Data for the Identification of the Hodgdon's Model b.Reptation of the Minor Loops.

A I tan(A2B) f(B) = {

if IB[
A1 tan(A2Bcl) + ( B -

if B_>Bcl

-A1 tan(A2Bcl) +

if B-
g(B) = { f(B)

F IBl "1 I-A3 exPt-A4.i~Bci ~ IBI)] }

t'03)

if IBI<-Bcl

(7.3.11a)

(7.3.I Ib)

if IBI~Bcl

where a = 1, Bcl is the value of the flux density at the closure point and I~clis the slope of the loop beyond this ~int. The parameters A I to A4 are computed from experimental data measured on the major loop at full magnetic remanence, at the coercive point, and at the closure ~int, as shown on figure 7.3.7a. The most exciting properties of this model concern the minor loops. Indeed, it was proved that for any couple of values (Hmin, Hmax), there is a single closed curve on the plane (H,B) which has the prope~ies of a stable loop. When H is oscillating between Hmin and Hmax, the resulting loop converges to this stable loop for any initial state (figure 7.3.7b). Furthermore, infinitely small stable cycles are localized on the graph of the function f, thus representing the anhysteretic curve. This behaviour is consistent with the theoretical understanding of the reptation phenomenon: for any applied field there is a state of minimal energy corresponding to the anhysteretic curve.

242

Chapter 7: Finite Elements in the Analysis of Recording Devices

7.3.3: Experimental Analysis

7.3.3.1:

Experiments

We applied the above models to represent the behaviour of a thin metallic recording medium used for high density recording on rigid disk. This medium is obtained by sputtering a CoNiCr alloy with chxomimum as a base layer and carbon as a protective top layer. The chromium underlayer induces a hexagonal structure with the c-axis on the disk plane, which results in a strong horizontal anisotropy, chromium impureties being the origin of a strong coercive field (Chen I986). The behaviour of the material has been investigated using a Vibrating Sample Magnetometer (VSM). With this equipment, a longitudinal field is applied to a sample of the material and the resulting magnetic moment is measured to determine the flux density B. The shape of the sample makes demagnetizing fields very weak and the applied field is the actual field. The measurements have provided all the data needed for each model: the major loop, and the first and second order reversal curves. Some other measurements were done in order to compare the models and the experimental data for various operating conditions.

7.3.3.2:

Identification

The identification of a model consits in deriving its parameters from experimental

nt~

Figure 7.3.8: Experimental data: First (a.) and Second (b.) Order Reversal Curves

F. Ossart, G, Meunier and J.C. Sabonnadiere

243

data. It is the crossroad between a theoretical study and the practical use of the model. This procedure may ~ more or less easy according to the type of model. 7.3.3.2a: Mayergoyz's Model The identification process is rigourous and without any surprise, which is a substantial advantage. The difficulty is mainly due to the great number of experimental points needed. The accuracy of the model de~nds on the number of reversal curves measured and of their sampling. In the measurements reposed here,

|O00C

-I0000 -2O00O -4000

-2000

0

2000

.~

0

2000

4000

.IG.~

-4000

S ( 0..~. )

l

cx H ( lOers~d ) 0

2O00

4OOO

Figure 7~3.9: Experimental and Computed Major Loops

244

......

Chapter 7: Finite Elements in the Analysis of Recording Devices

I5 sampling points have been generated between extremal values of the loops which need 625 experimental values of the magnetization (Figure 7.3.8). However this limit only depends on the experimental equipment and not on the model itself.

7.3.3.2b:

Hodgdon's Model

For this model, the experimental data are some points or slopes measured on the major loop. There is no particular problem in measuring H c, B r, or B s but it may be difficult to extract the slope of the loop especially when it is steep at the coercive point and flat at the closing point of the loop and beyond. Moreover, the model is very sensitive to these errors and the measured values needed to be fitted after several tries. The choice of the value of the parameter o~ is also tricky and must be done after several tests since this parameter is not determined from ex~rimental data. As we can see the identification of the model seems to be easy, but relies on some empirical skills. It may not be extended to all the materials, but there is no criteria to set up its limits.

7.3.4: Comparison with the Experimental Behaviour 7.3.4.1: Major Loop Various tests were made to compare the models with the experimental behaviour of our material. The major loop is perfectly reproduced by Mayergoyz's model, which is not su~rising, since the model uses them as data! It is m be noticed that the simple polynomial functions we defined are enough to give a good description of this major loop (Ngure 7.3.9).

7.3.4.2: Minor Loops A fundamental difference between the models which were studied concerns the behaviour of the minor loops: is there reptation or not? The answer comes from experiments: our material does not exhibit this phenomenon and minor loops are stable. Experimental loops are not represented here because they superimpose towards Mayergoyz's model. The stable loop reached by Hodgdon's model corresponds m the experimental one, but the loops predicted by the analytical models do not stabilize before they are on the major loop. (Figure 7.3.10).

7.3.4.3: Demagnetization of the Sample The sample is demagnetized by application of an alternatively dex:reasing field. The induction measured at each turning point and the induction predicted by the models are plotted on Figure 7.3.11.

7.3.4.4: Application of an Alternative Field The demagnetized sample is then submitted to an alternative field of random magnitude. The observed and computed induction are plotted on Figure 7.3. I2. In this situation as in the previous one, Hodgdon's model forecast a too low induction, except in the region of the demagnetized state.

F. Ossaa, G. Meunier and LC. Sabonnadiere

7.3.4.5:

245

Discussion

Those tests show that simple analytical models may give interesting resul~ when used with an alternative field. Because of their simplicity, they may be useful in obtaing a preliminary solution to a problem wifi'a hysteresis, but more complex models have to be considered as soon as one needs to reproduce properly the behaviour of the minor loops. A general observation is that the induction predicted by Hodgdon's model is lower than the measured one. Furthermore, the rep~tion of the minor loops, which is the main advantage of the model, does not exist in a significant way for the material considered. Thus, in this case, Mayergoyz's model is the best one. But other arguments have to be considered. The fact that the results predicted by Mayergoyz's model follow very well the experimental behaviour comes from the numerous data this model uses. This amount of data to measure and to manipulate may a p ~ a r as a drawback, but in fact it is also an advantage: the model is able to integrate as many data ~ wanted to improve its accuracy. It can be seen as a method of interpolating magnetization curves from experimentally obtained minor loops.

Figure 7.3,10: Computed Minor Loops

246

Chapter 7: Finite Elements in the Analysis of Recording Devices

S ( 0:.

=)

......

i

i

• Experimental & MlyCr~oyz

Q Hodgdoa

-I0000 t - - -

.20C.-'30+-" .40"~

1 .2000

0

'

14 ( O¢¢sted ) 2000

4000

i

i

e ( O=.s= )

• Expcrim©ntal 1 o l'oly I & slanh

.200004-" -~300

t

i

.2~

0

20OO

.2~

0

2000

......i

H ('Ocrsted) ~. . . . __j

4,O0O

lOfX)O

.I0000

-20000 .4030

Figure 7.3.11: Demagnetization of the Sample

F. Ossart, G. Meunier and J.C. Sabonnadiere

247

2000 I0~

.2000

o

5

IOC~O.

.20000

0

5

!o 1ooo0,

ICwX)O

Ioooo

...... . loc~o

N

-20000 0

5

1o

~2~

Figure 7.3.12: Application

0

of a n A l t e r n a t i n g

1o

5

Field

The good performance of the model is balanced by the complexity of the model and its implementation. The control of the memorized turning points has to be done properly, as well as the control of the numerous data. But the final formula is simple to compute; it requires only aIgebric operators of low computational cost. On the other hand, Hodgdon's model is much simpler from the theoretical point of vew as well as from a practical one. But it requires a

248

Chapter 7: Finite Elements in the Analysis of Recording Devices

numerical integration in which expensive functions (exp, tan) have to be evaluated at many intermediate points.

7.3.5: Conclusion Various models of hysteresis are proposed in the literature. Among these various theories, Mayergoyz's model and Hodgdon's have interesting and complementary properties - - the f o y e r because of its well defined domain of validity, the latter because it models the reptation phenomenon which is observed in some materials. An experimental study is necessary to measure the data used for the identification of the models and to compare the behaviour predicted by the models with experimental data in the case of a particular material. This study was realised with a sample of thin film medium used for longitudinal high density recording on a rigid disk. The reptation of the minor loops does not exist in this material and the model developed by Mayergoyz appears to be the most reliable. Although often criticised for its complexity and the fact that it does not model the theoretical reptation of the minor loops, this model constitutes a good reference. This conclusion, having only an experimental foundation, should be revised in the case of any other material, especially in the case of a soft one, in which the reptation phenomenon may be more pronounced. Morover, in closed loops, the experimental data used for Mayergoyz's model axe not very well defined, but the slopes needed for the identification of Hodgdon's model become easier to measure than on open loops. Thus an experimental study would be interesting for soft materials.

7.4 Computation of Magnetic Field Inclu d~mg Hysteresis 7.4.1: The Problem to S o l v e The next stage of this work of modelling is to include the hysteresis of the media to calculate the magnetization patterns written by the recording head. The problem arises from the demagnetizing field effects taking place around each transition and acting against the writing field. The actual field is different from the applied field and depends on the magnetization profile, but the magnetization also depends on the actual field. Thus, the magnetic stability is difficult to compute, especially around the coercive field where the interdependance between field and magneti~tion is very pronounced. Improvements in computation time as well as in memory capacity of the computers allow us now to have accurate and significant results.

7.4.2: Assumptions Despite its generality, this computation is based on some assumptions which have to be known in order to appreciate its validity.

F. Ossa~, G. Meunier and J.C. Sabonnadiere

249

7.4.2.1: 2D Modelling First of all, this is a two dimensional modelling of a three dimensionnal device; thus, we do not take into account its finite width and end leakage. This assumption shall cease to be valid soon because of the trend to achieve always narrower tracks. Nowadays, the track width is about 20 microns, but it decreases towards 10 microns and the goal is 5 microns or less. Thus side-effects will have to be considered and a 3D modelling will be necessary.

7.4.2.2. Modelling of Materials 7.4.2.2a Macroscopic Approach Magnetism may be studied at three levels. Its origin is on the atomic scale. Then, at the microscopic scale, one is interested in domain configurations, magneti~tion rotation and wall motions. Last, the macroscopic level concerns the classical magnetization curve M(H), the result of the former phenomena. In magnetic recording, the device is small enough m have significant microscopic effects in the head as well as in the recording media. Experimental studies are undertaken to observ and understand those phenomena, but their mechanism is not understood well enough to ~ modeled. This difficult problem is not considered here and a macroscopic modelling of the behaviour of the materials is used. In the recording media, the fundamental L o ~ of the constitutive law is associated with a model of hysteresis M(H): B = go(H+M) and M(H) (6.2.2) The soft material of the head is modelled through its isotropic reluctivity v expressed as a function of the induction: H = v0 Vr B and vr(B 2) (7.4. I)

7,4.2.2b: Scalar Model of Hysteresis In our 2D assumption, we work on the plane [x,y] (Figure 7.4.1). The current density is pe~endicular to the plane of the figure, while the magnetization is on it. Furthermore, we study longitudinal recording on thin film media. Because of the very low thickness of the medium ( 0,07 microns ) the magnetization stays on the plane [x,z] of the disk. Thus only its longitudinal component M x is modelled and is assumed to obey to the longitudinal component of the applied field Hx according to a model Mx(Hx). Those are the data that were studied in the former section. The complex phenomena of micromagnetism which were evoked, are present by their average effect in the measured magnetization curves.

Figure 7.4.1: 2D Modelling of a 3D Device

250 .

.

Chap~r 7: Finite Elements in the Analysis of Recording Devices .

.

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.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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.

.

.

.

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.

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.

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.

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.

,,,,

,,,,,,

7.4.2.3: Magnetostatic Modelling The device is a dynamic one in which the magnetic field sources are the writing current in the windings of the head and the media magnetization acting against the writing field thxough the demagnetizing field. The frequency of the writing current is rather high, up to 20 MHz, but to simplify the problem, dynamic effects are not taken into account and a succession of magnetostatic states is assumed. This assumption is valid for a thin film head in which the relative permeability of the head is not very high (about 1000). Thus, for a l 0 MHz frequency, the skin depth is 2.25 microns, the sam~ magnitude as the pole thickness and eddy currents do not really disturb the running device. During the writing process, time also operates through the motion of the medium and the remanent magnetization. The geometry of the device is not changed; thus, it allows us to take into account this motion with a single meshing, simply by propagation of the magnetization values from one element m its neighbour (Figure 7.4.2). This process is made the easier by the use of a perfectly regular mesh to avoid distortion of the magnetization patterns. The smaller step motion is given by the mesh.

7.4.3: Modelling of the Writing Process 7.4.3.1: Equation to Solve Using the vector potential A, the equation of the magnetostatic system to solve at each step of time is: V x (v0 vr V x A ) = J + V x M(H) (7.4.2) in which J is the current density. In the head, the reluctivity vr is used and the magnetization M is equal to zero ; in the m ~ i a , vr is set to 1 and M depends on the actual magnetic field H, In two dimensions and following our assumption of scalar hysteresis, the projection of this equation along the third dimension gives a scalar equation that is much simpler to solve: GAP dx n

Propagation" M(x+dx,t) = M(x,t-dx/v)

Figure 7.4.2:

Motion of the Media: Propagation of the Values of the Magnetization

F. Ossart, G. Meunier and J.C. Sabonnadiere

251

(7.4.3)

- V • (v0 Vr V A z ) = Jz + V x M x ( H x )

7.4.3.2: Principle of Computation This equation is highly nonlinear and is solved using the Gauss-Seide! process schematized in figure 7.4.3. This algorithm is the same as the so-called "selfconsistent'' one currently used in the area of magnetism. More powerfull methods exist to achieve convergence in problems with classical saturation. The Newton-Raphson method and its derivatives, for exemple, are currently used. But such methods need to derive twice the magnetization curve, which is not easily done with any model of hysteresis. The process exposed here does not depend on the model of hysteresis chosen;

Initialization of Mo and Ho ( histo~-, of the media )

Solution of "rot ( v rot A ) = J + rot Mi-1 ", in which J and Mi-1 are given, using the 2D finite element method ..............................................

"J"J .....

J..........

I

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

'" .............................................

1

Computation of Hi = v rot A - Mi-1 Ponderation : Hi = Hi-I + w . ( Hi - Hi-I ) Smoothing of Hi

(

i

iiiii

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

I

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

Computation of Mi ( Hi ) using the model of hysteresis

)

no

yes Storage of Mi and Hi ( history of each point ) Displacement of the media New value of the current J

Figure 7.4.3: Scheme of the Gauss-Seidel Method

252

Chapter 7: Finite Elements in the Analysis of Recording Devices

thus it allows us to use any model directly. But to realize the computer implementation, it is wiser to ~ g i n with the simplest one. The "polynomial" model, proposed in the section 7.2.2.1b, is numerically very simple and gives good results for a preliminary solution; thus it is a good one to adjust the computation and test its consistency.

7.4.4: Practical Difficulties and Solutions 7.4.4.1: The

Mesh Generation

As usual, the mesh has to be very fine where important variations of the magnetic field are expected and where high accuracy is needed. In the case of magnetic recording, one focuses on the region of the gap and of the medium, which is very small compared to the whole device. This tong-shaped region of small area has to be very well meshed, which results in numerous elements. Furthermore, a perfectly regular mesh in the medium simplifies its motion simulation by propagation of the magnetization t?om one element to its neighbour. An efficient way to respect these constraints is through the use of rectangular elements to mesh the medium and the air between it and the head and triangular elements anywhere else. An exemple is shown on Ngure 7.4.5b.

7.4.4.2:

Under-relaxation

Numerical convergence of the computation is not easy to achieve. Bits of opposite magnetization are written, which generates strong demagnetizing fields around the transitions. When writing a new transition, the demagnetizing field acts against the writing field, resulting in an actual field lower than the applied one. One tries to approach this phenomenon by applying a damped field H'.1 given by the f o ~ u l a (7.4.4) instead of the computed field H i . H i = Hi_ 1 + m(H i - Hi. l)

(7.4.4)

This underre!axation prevents the field from too strong variations between successive iterations. The choice of the damping factor ~ depends on the expected variation of magnetization between the initial and final states. When the Hx

l

t it.1 • Hi' = I/3 ( Hi-1 + Hi + Hi+I ) it.2" Hi" = I/3 ( H'i-I + H'i + H'i+l ) I

I

•.. i-t

I

1 .

.

i+l .

.

.

.

.

Figure

... X .

II

7.4.4: Smoothing of the Field.

F. Ossart, G. Meunier and J.C. Sabonnadiere

253

magnetization is completely reversed - - writing of a new transition - - a strong underrelaxation is necessary and the value o = 0.3 is a good one. When the magnetization does not change significantly - - the motion of the medium with the same writing field - - a strong damping generates too many iterations and the value o~ = 0.5 is enough to insure the convergence of the computation. In practice, if the field computed during inte~ediate iterations is over the stable field, numerical oscillations appear, preventing the convergence of the computation.

7.4.4.2: Smoothing of the Field Underrelaxation improves the approach towards the solution but is not enough to achieve convergence. Indeed, the field H, calculated by substraction of the flux density B and the former magnetization Mi_ 1, has to be perfectly smooth. If not, discontinuities will appear in the next profile Mi(x) resulting in additional demagnetizing fields sufficient to produce nonconvergent oscillations. To solve this problem, the damped field is smoothed by a damped average of the field at the considered node and at its neighbours (Figure 7.4.4). The basic computation consists of a simple weighting of the field at three consecutive nodes following the X axis. If the result H' is not smooth enough this algorithm is applied again. This smoothing process is very efficient to erase small discontinuities of the field without excessive distortion of the general shape. Associated with the underrelaxation process, it allows us to achieve convergence of the Gauss-Seidel process within about 15 iterations when the magnetization is reversed and much less when the medium is moved without change in the writing field. Any difficulty we talk about could be the subject of a deep study to be perfectly controlled, but this is beyond the scope of this chapter and the next section is an example of computation including the hysteresis of the magnetic recording.

7.4.5: Some Results 7.4.5.1: Device to Model We applied this method to simulate a head with infinite poles. The head is very simplified and reduced to the region of the gap. Figure 7.4.5 represents the device geometry, its physical properties, the boundary conditions and a zoomed view of the mesh around the gap. The coil is not modeled and the writing field is caused by a difference in potential between the upper (A = 0) and lower boundaries (A = As). The mesh consists of 2355 elements and 5372 nodes.

254

Chapter 7: Finite Elements in the Analysis of Recording Devices

,. dMdn=O-,. .m

11 I I

I

A-O

" -..

POLES u = I(>00

GAP=0,6um

|

"~.| |

M E D I A " e = 0.07 u m H f l y " 0.2 u m

Hc = 12C.3@ Hs = 20C~3Oe 4 p i M r = 85(XtG 4piMs = i

/

A = Asource (a)

Figure 7.4.5: a. A.Simplified Infinite Pole Head, b. Mesh in the Gap Region

F. Ossart+ G. M e u n i e r and J.C. S a b o n n a d i e r e

255

|

.......... H o = O

Ho=-Hs S

im

~'~

_

Figure 7.4.6: Initial Magnetization of the Medium

seee

...... i

+

+

i

!e..

"

;

~~

i.

:.

:

:

:

•

•

:

•

.

:

:.

-lllle

Figure 7.4.7: Writing of a Transition a. Applied Field b.Resulting Magnetization c. Actual Field, Sum of the Applied and of the Demagnetizing Fields+

:

256

Chapter 7: Finite Elements in the Analysis of Recording Devices

7 . 4 . 5 , 2 : C o m p u t a t i o n of a T r a n s i t i o n We initialize M and H as shown on figure 7.4.6: for x _> 0 , M 0 = ( - M s ) and H 0 = (-Hs) (such points have reached saturation under the gap ) for x < 0 , M 0 = 0, H 0 = 0 (such points are in a demagnetized state before writing ) Then we apply a positive field, writing a positive bit after a negative one. The resulting transition is shown on figure 7.4.7, along with the writing field and the actual field.

7.4.5.3: Motion of the Medium In the next stage of the computation, the medium is moved 1.5 microns to the left in steps of 0.05 micron to write an entire bit. R e curve a of Fig. 7.4.8 shows the resulting magnetization. The saturated region under the gap is larger than before because the demagnetizing field of the transition does not act anymore against the

11000

)

!

$IOO

:'

:

i

i

:.....

./

.

C

.............. i

'i. i

-see9

i

! I

I

Figure 7.4.8: Magnetization after moving the media a. Hw > 0 b. Hw = 0 c. Hw< 0

I |

d

F. Ossart, G. Meunier and LC. Sabonnadiere

257

writing field. The curves b and c of Figure 7.4,8 represent the remanent magnetization when the writing field is removed and the writing of a new transition when the writing field is reversed. Those results are consistent with the experimental behaviour observed for such devices and with the results of classical analytical modelling.

7.5 Conclusion In this chapter, the application of the finite element method to the modelling of the magnetic recording process has been described. The difficulty of this problem arises from the hysteresis of the recording medium, with two points to be considered. First, an accurate model is needed to reproduce the behaviour of the media, and then this model has to be merged into the finite element computation of the magnetization profile. The complexity of the phenomenon of hysteresis forbids a physical approach and leads us to use phenomenological models. The choice of one among the many models proposed in the literature is done after an experimental study comparing the prope~ies predict~ by each model with the actual properties. In the case of thin film media for longitudinal recording, Mayergoyz's model appears to be the most reliable. The polynomial model is also an interesting one because of the good compromise between accuracy and numerical cost. The Gauss-Seidel algorithm allows us to solve the highly nonlinear equation of magnetostatics at each step of time under hysteresis, The motion of the medium is simulated by propagation of the magnetization from one element to its neighbour. Underrelaxation and smoothing are necessary to achieve convergence. The results are consistent with the expected behaviour of a recording device. The reading process has not Ken reviewed here. It is to be noted that the hysteresis of the recording media d ~ s not interfere in a significant way and the use of the reciprocity principle (Smith I987) avoids the need to solve the problem step by step (Ossa~ and Meunier 1990).

C h a p t e r , .8

A. Konrad and I. A. T s u k e r m a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........

i

i

iii

[l[N!!l!l

APPLICATION OF INTEGRAL AND DIFFERENTIAL METHODS TO ELECTROMAGNETIC FIELD PROBLEMS Notation E

Electric field Electric scalar potential

E

Permiuivity

P

Volume charge density Surface charge density (single layer)

D

Surface charge density (double layer)

P

Pe~eability

B

Magnetic flux density

A

Magnetic vector potential

J

Current density

0)

Angular frequency

k

Wavenumber Wavelength

<,> (,)

A functional in a linear space A scalar product in a linear space .................................................

J

Conventional notation for electromagnetic quantities is used through the text. Vector entities are not specially marked, unless there is a possibility of confusion. Primed symbols denote trial functions in weak (Galerkin) formulations (not time derivatives). Dots are often omitted in Euclidean scalar products of vectors in E 3. The partial time derivative is denoted by Ot.

A. Konrad and I. A. Tsukerman

259

The estimate of the form E = O(h0~), where E is an estimated quantity and o~ is a number, means that for sufficiently small h, Cl hc~ ~ ~ <_ c2 h ~ with some constants c 1, c2.

8.1. SOME MATHEMATICAL PROBLEMS 8.1.1 : Introduction

TOOLS

FOR FIELD

A computer makes as many mistakes in two seconds as 20 men working 20 years make. Murphy's laws on technology 1 This chapter is intended as a practical introduction to integral and differential methods in electromagnefics. These two classes of methods are rarely considered together; most books deal either with the finite element method ( ~ M ) which arises from differential formulations, or with the method of integral equations. It is useful, however, to compare the advantages and disadvantages of these methods and to consider their common background. Konrad and Tsukerman (1993) have demonstrated the common origin of these methods. The intelligent use of these methods requires at least some grasp of the underlying mathematical ideas, not just the know-how and recipes for implementation of computational techniques. Formulations and solution techniques for electromagnetic field problems have deep roots in different areas of fundamental and applied mathematics. However, it is quite common for engineers to overlook the mathematical structure of methods and to consider a method acceptable as long as it works. One could take such an approach (somewhat similar to driving a car without knowing where the engine is) if one does not have any problems using the method; but this would be very unusual - - the epigraph partly explains why. The two basic types of formulation of electromagnetic field problems are: (a) Integral equation formulations: Volume and/or surface charges or currents are taken as the unknown functions. Since the electromagnetic field can be expressed via the integral of these unknown functions, one obtains Fpedholm integral equations either of the first or second Nnd. (b) D i f f e r e n t i a l e q u a t i o n f o r m u l a t i o n s : Fields, or more frequently, their potentials, are taken as unknowns. The resulting differential equations are usually solved using finite-difference or finite~element methods. Integral and differential equation formulations may be combined to form hybrid techniques (section 8.7.3). Applying the method of moments to differential and integral equation fommlations, we can now obtain a Lew basic solution methods (Fig. 8. I. 1). FEM is a special case of the method of moments applied to a differential problem, as we saw at the end of chapter t. Most frequently, the FEM employs the Galerkin method, which is a particular case of the method of moments. The method of 1

Copyright Harvey Hurter & Co., Inc. Mr. Murphy licensed by JEB Corp.

260

Chapter 8: Application of Integral and Differential Methods

moments applied m an integral formulation with Fredholm equation of the first kind is known as Hal~ington's method. Harrington's method is often referred to as simply "the moment method" (without further specification). However, the ~ M advocates may consider this

equadon form~ations

FOI~2vIUR.ATIONS

)!

. . . . . . . . .

) L, - - . L_-_

[--

H~n~on's

ME'~rIODS

moment method

- : , 3

r

r

-

-

- .-

,,Ij

-.

¢1

I

*l

met.hod

The general method of moments

GalerP..in'smethod

.... I

~CALTOOLS

|

t!

Figure 8.1.1: Formulation + Mathematical Tool = Method

A. Konrad and I. A. Tsukerman

2 61

terminology ambiguous, since the FEM is also a particular case of the general method of moments. FEM is often viewed as a special case of the Rayleigh-Ritz method. The latter is based on the fact that for certain classes of problems, the differential formulation is equivalent to the minimization of a functional. Although the interpretation of the FEM as a Rayleigh-Ritz method is useful, Galerkin formulation is more general and applicable, in principle, to any woblem. For problems equivalent to functional minimization, Galerkin and Ritz forms of the ~ M coincide. From this point of view, it is sufficient to deal only with the Galerkin formulation.

8.1.2:

Linear Spaces, Functionals and the Method of Moments

To understand the nature of computational methods used in electromagnetics, the sources of possible difficulties and the links between different approaches, we first recall the basic mathematical notions of a linear space and a linear functional. Using only these notions, which form an essential part of the "engine" of most computational methods, one can formulate a very general numerical t ~ l called the method of moments. We first recall some basic mathematical notions related to functional spaces. A linear space is a set X endowed with a binary operation of addition of its elements, with an operation of multiplication by a (real) number and with a zero element, such that for any x, y, z m X and any real ~,~. x+y=y+x ~(x+y)=ocx + 0~y; (0c+~)x = ocx + ~x; ~(~x) = ( ~ ) x ; x+0=x, 2 0x=0; lx=x; In this general definition, the nature of elements of a linear space is not specified; they may be, for instance, numbers, vectors or functions. As an example, one can ke.ep in mind the line~ space C2 [0, 1] of twice differentiable functions on the segment [0, l] with naturally defined addition and multiplication by a real number. An operator L in x is a mapping D(L) --~ R(L), D and R being subsets of X. An operator is called linear if D(L) is a linear space itself (that is, for any x, y in D(L) and any real e~,~ the linear combination ~x+~y is also in D(L)) and L(~+By) = ~Lx +BLy The next important notion is a linear,&nctional. Mathematically, a l i n e ~ functional f i s a linear mapping X ~ RI; it means that for each element x in D(j') there is a real numberJ(x) (often denoted also as (x,f) such that <~x+f~y,f> = ~<x,f> + ~ t~r any x, y in X and any real numbers ~, 8. We can now consider the formulation of the method of moments for the solution of the linear equation Lx = g, g ~ R(L) (8.1.1) 2

Note that zero on the left side is the null element of X, not a real number!

262

Chapter 8: Application of Integral and Differential Methods

where L is a linear operator, x is the unknown and g is a known fight hand side. If x is the exact solufon of (8.1. I), then obviously for any linear functional f = (8.1.2) Seeking x as a linear combination of a finite number of basis elements (Wi), (i=l .... n) n

x = Z aj~j j=l

(8.1.3)

and choosing a system of n linear functionals {fi} (i=l,2_n), one can attempt to satisfy (8.1.2) at least for these functionals, i.e. n

L ~aj*Fj,fi ~ = , (i=1,2, ..., n) (8.1.4) j=l This system of n simultaneous equations (the unknowns are ~j) constitutes the general fommlation of the method of moments. The elements of the matrix of this system are: (8.1.5) The moment method thus formulated requires surprisingly few mathematical tools: only the notions of linear space and linear functional were used to construct the system (8.1.4) 3 One can then try to solve this system for ~j and to obtain an approximate solution of (8.1.1) as a linear combination (8.1.3). Of course, in such a general setting, there is no guarantee that the algebraic system (8.1.4) has a solution. Even if it does, this "approximate solution" may be worthless, and bear no resemblance to the exact solution. Much more specific info~ation concerning the space X and the linear operator L has to be provided to ensure that (8.1.4) constitutes a useful method. A detailed analysis can be found in the mathematical literature (Mikhlin, 1960, 1964; Sobolev, 1963; Rektorys, 1980; Kantorovich, 1982). Our purpose here is only to present a brief sketch clarifying the terminology. In most applications, the functional space X is endowed with a scalar product (,) (and therefore with the norm induced by this scalar product). Then the functionalsfi in (8.1.4) may" (but do not have to) be taken as <x,fi> - (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.

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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)

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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

Chapter 8: Application of Integral and Differential Methods

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

Chapter 8: Application of Integral and Differential Methods

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 _<*max S**dS (8.2.14) S S S where the max and rain operations are taken over the surface S. The expression in the middle of this inequality is just the scalar product o f , and ~*; also, by definition of o*, fa*dS = C S Therefore the inequality (8,2.14) is rewritten as COmin -< q -< C¢rnax If ¢ _:20 (which is ensured if cr _>0) then the following bounds for the capacitance are obtained: q

(8,2,15)

In particular, these bounds for the capacitance hold in the case of the uniform charge density assumed in the method of average potential. Hence one can a posleriori (after applying the method of average potential) obtain the lower and upper bound of the capacitance just by substituting the minimum and maximum potential values into (8.2.15). Clearly, the more unit%rm the computed potential distribution, the closer the bounds that are obtained for the capacitance. We now only need to prove the assumption that c~* >_ 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

Chapter 8: Application of Integral and Differential Methods

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

A. Konrad and I. A. Tsuke~an

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

Chapter 8: Application of Integral and Differential Methods

• (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.

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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

Chapter 8: Application of Integral and Differential Methods

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.

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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)

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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 ~.

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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

Chapter 8: Application of Integral and Differential Methods

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.

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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

Chapter 8: Application of Integral and Differential Methods

(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

A. Konrad and I. A. Tsukerman

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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

Chapter 8: Application of Integral and Differential Methods

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,,,

A. Konrad and I. A. Tsukerman

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

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Chapter 8: Application of Integral and Differential Methods

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

A. Konrad and I. A. Tsukerman

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: <(element) S(nodel,node2) = S(nodel,node2) + ~(nodel,node2) = T(element) T(nodel,node2) T(nodel,node2) + ~(nodel,node2) end of node2 cycle; end of nodeI cycle; end of element cycle Diagonal entries for Dirichlet nodes for each node E Dirichlet nodes set S(node,node ) = 1, T(node,node ) = 1 end of node cycle Algorithm 8.5.1: The Basic Finite Element Algorithm

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Chapter 8: Application of Integral and Differential Methods

shape functions, it is just a linear inte~olation of the node values. It is a matter of g ~ d system programming to view the results in a visible graphical form.

8.5.3: FEM for TE Mode The only difference between the formulation of the problem for TM and ~ modes is in the boundary conditions. Instead of the homogeneous Dirichlet bounda~ condition for Ez in the TM mode, one has the homogeneous Neumann condition for Hz in the TE mode.Thus the weak formulation of the problem can be obtained in the same way as (8.5.9):

~V H VH'ds - ~_(k2-~2)H H ' d s = 0

(8.5.12)

Again, the subscript "z" of H z is dropped, as there is only one component of H in this problem. The question now is this: what boundary conditions should be imposed on H and H' to ensure that the weak formulation (8.5. I2) is equivalent to the Helmholtz equation (8.5.4) with homogeneous Neumann conditions? It is clear that we can proceed from (8.5.4) to (8.5.12) in the same way as it was done for the TM mode problem; the only difference is that this time no b o u n d ~ conditions are needed for H' (the boundary integral will vanish anyway). Thus the weak formulation follows from the Helmholtz equation. What about the opposite assertion? If we want to go backwards from (8.5.12) to (8.5.4) performing integration by parts, it looks as if we had to impose Neumann boundary condition on H to get rid of the boundary integral, Strange as it may seem to those who deal with FEM for the first time, the Neumann boundary condition does n o l have to be explicitly imposed. To prove this assertion, which is an additional advantage of the weak formulation, one needs to exploit the freedom in choosing the trial function H' • First, choosing H' to be an arbitrary function which vanishes on the b o u n d a ~ OG, one can proceed from (8.5.12) to (8.5.4) with the boundary integral eliminated. This will ensure that H, if H satisfies the weak formulation (8.5.12), satisfies the Helmholtz equation (8.5.4). Next, one can take a trial function H' that does not vanish on the boundary and again proceed from (8.5.12) to (8.5.4). Then the integral over G disappears because H was proved to satisfy the Helmholtz ~uafion, and only the boundary integral is left. Since H' is arbitrary, this boundary integral can be zero only if the solution H satisfies the homogeneous Neumann condition. This is, of course, only a sketch of a proof; a detailed discussion can be found, for example, in (Rektorys 1980). We emphasize the main conclusion: the Neumann boundary condition is a natural boundary condition, that is, it does not h a v e to be imposed u ~ n either the trial functions or the solution itself. ~ e weak formulation per se ensures that natural conditions are satisfied. Of course, natural conditions, in principle, can be imposed explicitly; however, the implementation of the FEM becomes much simpler if at least some of the boundary conditions are automatically taken care of by the weak formulation.

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Boundary conditions involving normal derivatives are usually natural, and the others are principal (i.e., must be explicitly imposed). This is only a guideline, not a strict rule. For more complicated problems, the transition from a boundary value problem to its weak formulation and back should be carefully examined to determine which conditions are natural. Getting back the the TE mode problem and the simple mesh of Fig.8.5. I, we note that, unlike for the TM mode, all boundary nodes (I-5,6,10,11-15) are now "active," i.e. the s h a ~ functions corresponding to them are included into the ~ M basis. Thus in this example there are I5 basis functions (as opposed to 3 basis functions for TM). Of course, for a real problem the difference in the number of "active" n ~ e s would not be so remarkable, because a denser mesh would have only a small fraction of its nodes on the boundary. Apart from the boundary conditions, there is no significant difference in the irnplemenmtion of the FEM for TM and TE modes. Clearly, however, different boundary conditions will lead to different solutions.

8.5.4: Numerical Examples Implementation of all stages of FEM for TM and TE modes is illustrated in Appendix 8.C with a MATLAB© program.

8.6 Example of FEM: Inhomogeneous Waveguides A more complicated electromagnetic problem arises when a waveguide is composed of a few homogeneous materials. In the longitudinal direction it is still assumed to be uniform; but the cross-section is now piecewise-homogeneous.

8.6.!: Ez-Hz Formulation Throughout this section we shall utilize the approach proposed by Csendes and Silvester (1970). Their formulation is obtained by writing Maxwell's equations component-wise: 3Ez= 0y j~Ey - j ~ H x (8.6.1) 0Ez 3x = j~Ex - j~,~I'-Iy

(8.6,2)

aEx aEy jwuHz = ay " 3x

(8.6.3)

0Hz Oy - jl3Hy + jo~Ex

(8.6.4)

0Hz Ox - jl3Hx + jo~Ey

(8,6.5)

jo,~tEz = .~Hv Ox'" °Hx 3y

(8.6.6)

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Chapter 8: Application of Integral and Differential Methods

Both E and H have all three components x, y, and z. At the same time,the x- and y-cornponents can be expressed via the z-components. Indeed, (8.6.2) and (8.6.4) is a system of two equations for E x and Hy which yields Ex

j~ 0Ez -i°~ aHz = d ax d ay

Hy=~-

aHz jE~eaEz d ~

(8.6.7) (8.6.8)

where d = m2e~t - ~2 Analogous expressions for Ey and Hx can be ob~ined from (8.6.1) and (8.6.5): ~aEz ~aHz Ey = d ay " d ax Jk alia j~* aEz Hx = d ax " d a y

(8.6.9) (8.6.10)

In each of the homogeneous regions the longitudinal components of H and E still satisfy the Helmholtz equation; the problem is only with the interface conditions between two regions. Consider again the transition from the boundary value problem (8.5.5, 8.5.6) for TM mode to the weak form (8.5.9) of this problem. For the piecewisehomogeneous waveguide, an additional integral over interface boundaries will appear:

f•E'

z dF

(8 ,6. I I )

F where F is an interior boundary between two regions with different parameters. Each inner bound~y must be taken into account twice, as the integral (8.6.11) is a result of integration by parts in each of two subregions sharing the boundary F. Since (aE/an) is not continuous on F (because of different material properties on the two sides of F), the boundary integrals contributed by the two subregions do not cancel. To simplify these boundary integrals, we can use (8.6.t0) to express the normal derivative (aE/an). In order to do so, note that the axes x and y could be chosen arbitrarily; hence if the coordinate system 0:,n,z) - - where ~cis the tangent at a certain point of F - - is used instead of (x,y,z), the expressions (8.6.7- 8.6. I0) will have the same form. Substituting ~ for x and n for y into (8.6.I0), one obtains aE z 1 ~ aHz d ~ - j~0H ~ - ~ at (8.6.12) The two terms on the right side of (8.6.12)can be substituted instead of the normal derivative of E in (8.6.11). Since H is continuous across the interface boundary, two boundary integrals corresponding to this term will cancel. The second term being substitumd into the boundary integral yields

A. Konrad and I. A. Tsukerman

3 01

t= ~ | VHz x VE' u z dS cod

(8.6.13)

d

where Uz is a unit vector in z-direction. A similar expression corresponds to the equation for Hz. Thus the weak formulation of this problem contains, in addition to the integrals (8,5.9) and (8.5.12) of homogeneous TM and TE problems, integrals of the form (8.6,13). With a set of basis functions ~ i chosen, the matrix eigenvalue problem obtained from the weak formulation looks as follows (Csendes and Silvester 1970): Vu = o,2Tu where V=

£

1

subregions

T= subre~ons

ETki 0

1 (" e P k i ~l--Jkj) _ ~ 1 5k. " ~Uki "Pki

(8.6.14)

O)

U (h:);

Pki = ~[ V~kVq~i dS

T k i - [j' e k ~ i d S

3~i 3 ~ k'-~ Uki=Gjf" ((.~k~-'t'i~)dS

As could be expected, E z and H z are coupled in this system. One of the disadvantages of the Ez-H z formulation described above is clear from the system (8.6.14). There is a singularity whenever 82is equal (in practice close to) eg for one of the mamrials in the waveguide. This corresponds to the case when the Helmholtz equation in one of the media turns into the Laplace equation. An alternative approach considered in the following section avoids this difficulty; in addition, symmetry and positive definiteness (for real e and g) of the continuous operators will be preserved in the algebraic problem.

8.6.2: Three-component Field Formulation Konrad (1974) proposed to keep all three components of E or H in a formulation. In this case one starts with the curl curl equation Vx(e'lVxH) - c02taH = 0 (8.6.15)

302

Chapter 8: Applicafion of Integral and Differential Methods

and obtains the weak form multiplying by a trial field H' and integrating by parts: j ' ( V x (E'IVxH)-oa2gH)H'dS = J " ( E ' I V x H VxH') dV-~m2~HH' dV

j

o

- ( n x e-lVxH)H'dS = 0

(8.6.16)

where r represents the domain ~undary ~ and all the in boundaries ~tween different materials. The term n x e'lVxH is proportional to E~: and therefore continuous on the inner boundaries and zero on ~ . Hence the surface integral in (8.6.I6) vanishes, and the following weak formulation co~esponds to (8.6.15) with the boundary conditions for Ex being natural: I (e'lVxH VxH') dV- d0~2~HH ' dV = 0

(8.6.17)

Choosing a suitable set of basis vector functions ~ i - ~ix Ux + ~iy Uy+ ~iz Uz for H, one obtains an algebraic eigenvalue problem Sh - ~2Mth = 0 where S is the "stiffness" matrix with the entries Siam=

jF

E"1 VxWi VxWm dV

and M is the "mass' matrix 1VI4m-Oi gqJi~ m clV It is important to note that the algebraic problem inherits the essential features of the original continuous problem. If ~ and g are real parameters (lossless media), then S is positive semidefinite and M is positive definite, both matrices being symmetric. Another advantage of this three-component formulation is its generality. In particular, it can deal with truly inhomogeneous (as opposed to piecewisehomogeneous) and/or anisotropic media. The actual computation of the stiffness and mass matrices is a rather tedious job, especially for high order finite elements. Computations of these preintegrated FEM matrices (Konrad, I974, 1986a, 1986b; Silvester, 1969a, I969b; Silvester and Ferrari, 1990) accomplished these computations for finite elements up to order six.

A. Konrad and I. A. Tsukerman

303

8.6.3. Spurious Modes Unfortunately, almost all numerical models of waveguides and cavities yield, along with physically correct solutions, a large number of nonphysical modes often referred to as "spurious" or "extraneous" (see Konrad, 1974, 1986a, 1986b and references therein). For a long time, since the beginning of 1970s, this has been an annoying problem for many researchers. Mathematically, the existence of "spurious modes" means that at least some of numerical solutions do not converge to exact solutions. This also raises doubts about the other, "physical," modes: if part of the solution is incorrect - - that is, some of the modes are not valid - - is there any guarantee that the others are accurate? Although a complete mathematical study has not appeared yet, now there is an understanding of the reasons causing nonphysical modes in FEM models and, even more impo~ntly, there is a known method for avoiding these modes. It has been shown by Bossavit and his co-workers (Bossavit, 1988a, 1988b, I992; Bossavit and Verite, I983; Bossavit and Mayergoyz, 1989) that edge elements do not yield spurious modes. Practical results reported in one of the recent publications (Pichon and Razek, 1992) confirm this conclusion. "Spurious modes" are characterized by not being divergence-free. Therefore it is very important to investigate whether or not a given numerical method ensures, at least in weak forrn, that the divergence of the numerical solutions is zero. Let us get back to the weak formulation (8~6.17). Suppose one can choose a trial function H' as a gradient, H' = V~. Then (8.6.17) becomes

~'~tHV~dV = 0

(o~~ 0)

It can easily be checked (integrating by parts) that this equation is precisely the weak form of V • ~tH = 0. In fact, taking H' = V~ is the weak form equivalent of applying the div operator to the original equation (8.6.15). Thus, to ensure that the numerical solution is divergence-free, one needs to have gradients in the set of trial functions. The conventional set of nodal basis functions for Hx, Hy, and Hz does not generally contain gradients. As regards edge elements, due to the exactness property of "Whitney's spaces" W m, m = 0, 1, 2, 3 (section 8.4.3), the space W 1 of edge elements contains VW0, which is the property desired. Thus the problem of "spurious modes" in three-dimensional waveguide and cavity field computations seems to be solved by using edge elements.

8.7. Example of FEM: Scattering 8.7.1 Differential Formulation In section 8.3. I Harrington's method for antenna and scattering problems was considered. This method is well suited to geometrically simple objects located in homogeneous space. In more complex cases, e.g. for multilayered media,

304

Chapter 8: Application of Integral and Differential Methods

Harrington's method may become inefficient. An effective tool for modeling complex structures is FEM. In this section we shall consider applications of FEM to scattering problems. FEM, in its turn, has a disadvantage: it can deal directly only with bounded domains. Scattering problems, however, are unbounded: the scattered wave is not confined within any natural physical region. Two ways of overcome this difficulty are blown in using FEM: (1) The computational domain is artificially bounded, and special A b s o r b i n g B o u n d a r y C o n d i t i o n s (ABC) are imposed. These conditions allow the scattered wave to leave the domain almost without reflection, and the real unbounded problem is thus approximated. Clearly, the smaller the reflection off the boundary', the higher the accuracy of this approach. (2) The computational domain is bounded to include all inhomogeneous materials. Then the field outside the domain is expressed analytically using Green's function. The FEM is used inside the domain. The technique of coupling FEM and Green's function solutions is known as a hybrid Finite E l e m e n t M e t h o d - Boundary E l e m e n t M e t h o d

(FEM-BEM). The following subsections deal with ABC and FEM-BEM in greater detail. But first we consider the formulation of the scattering problem. Equations for H mad E in a source-free domain follow directly frc,m Maxwell's equations: Vx iEV x H . ~ H = 0 1 Vx ~ V x E - o:~2~E= 0

(8.7.1) (8.7.2)

The boundary conditions will be considered later. To obtain the weak form of (8.7. I), one can scalarly mulfply this equation by a trial function H'

J

I V x H - o~2~H~}H'dV = 0 Vx ~-

and integrate by parts:

F( VxHW.,) v-o, ...,dV-

x H]H' = 0 (8,7.3)

where n is the outer normal to the domain G.

A. Konrad and I. A. Tsukerman

305

8.7.2: A b s o r b i n g Boundary" Conditions Several types of ABC have been proposed. A good survey made by Moore, Blaschak, Taflove and Kriegsman (1988) may be used as a reference, so we only summarize the main results here.

A. Far-field Mode Annihilation This group of ABC proposed by Bayliss and Turkel (1980) utilizes the far field expansion of the form u(r,0.,,t)

=

exp(jkr) Z fi(0,,) r ......r. i ........ i=1

(8.7.4)

where u is the solution to the scalar Helmholtz equation VZu + kZu = 0 There is a sequence of operators Bn (Moore, Blaschak, Taflove and Kriegsman, 1988) each of which, when applied to the expansion (8.7.3), annihilates the first n terms of this expansion. The first operator in this sequence is given by a 1 Bl=~-jk+-r It eliminates the first term of the series (8.7.4). The second operator is B2-

(i r

-lk + r

Further operators B i can be obtained by a recurrence relationship (Moore, Blasch~, Taflove and Kriegsman, 1988; Bayliss and Turkel 1980). This sequence of operators gives

Onu=O)

Replacing the right side of this relationship by zero, one gets an approximate boundary condition that can be used to truncate the computational domain: Bnu = 0 The ABC method used by Bayliss and Turkel (1980) has some disadvantages. First, these conditions are asymptotic with respect to r; therefore the artificial boundary of the ~ M domain must be located far enough from the scatterer to ensure good accuracy. Secondly, the operators Bn, especially for n ~ 2, can be efficiently implemented only for the spherical boundary, which may also be inconvenient and may result in increased size of the finite element mesh.

B. Approximation of Perfectly Absorbing Conditions Our description will again be brief; we refer the more interested reader to Engqa~ist and M~da (1977), Moore, Blaschak, Taflove and IG'-iegsman (1988), and Lindman (1975) for detailed explanation.

306

Chapter 8: Application of Integral and Differential Methods

The procedure of obtaining the ABC has two stages (Engquist and Majda, I977). Theoretical, perfectly absorbing conditions are first derived; unfortunately, these conditions are impractical for numerical implementation. Perfectly absorbing conditions are then transformed into practical form by suitable approximations. To obtain perfectly absorbing conditions, Engquist and Majda (1977) consider the wave equation

A F~rfectly absorbing condition for a plane wave w = exp j {,'~{2 -~2x + ~t + my with {2. co2> 0, { > 0 and (o~,{) fixed, has the form

lx:o--° For a wave packet w(x,y,t)

!I

exp j (

+ ~t + my) p(e,c0)w(,~,m) d;dm

a perfectly absorbing boundary' condition is given by dvq

+ my)] ja]~2~-~2-p(e,o)~'e(0A,m) d;dm

dxI f

These perfectly absorbing conditions are non-local, that is, in a time-stepping procedure the',, require the information from all previous time steps on the whole boundary. Lindman (1975) proposed a quasi-local approximation of perfectly absorbing conditions. The Lindman conditions require the solution on the boundary from the previous 3 time steps to obtain reflection coefficients less that 0.01 for a range of incident angles from 0 ° to 89°. Engquist and Majda (1977) have obtained a hierarchy of local ABC. The first two of these conditions at the boundary x = 0 look as follows (Engquist and Majda, 1977):

First approximation: Second approximation: {'32

tY; t

32 +1 ~

+132" G

2 oz2y =°

It is shown by D'Angelo and Mayergoyz (I991) how to incorporate these boundary conditions into the FEM formulation (8.7.3). Consider the scatterer em~dded into a hexahedron as shown in Fig. 8.7.1 (D'Angelo and Mayergoyz 1991).

A. Konrad and I. A. Tsukerman

307

The second order (Engquist and Majda, 1977) condition on the surface Sx+ in the frequency domain assumes the tbrm

3H,z

3x = jk&Iy +

3Hz = jkk-Iz + ,zHz 3x Substituting these expressions into the boundary integral in (8.7.3), for the surface Sx+, one obtains (D'Angelo and Mayergoyz, I991). l"f ' 3 H x

¢,

.|(nxVH)H'dS = .'\~-'IIHY~+ Sx+

dS

Sx+

-jk

+ Hz

dS

Sx+

f(

Sx+

;

)

VyzHy VyzH + VyzH z VyzH z

d

I I I

/

I I

p

f

f

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

• ¢••••••~+

I |

J

dS

s

f

p, Figure 8.7.1: A Scatterer Embedded in a Hexahedron (D'Angelo and Mayergoyz, 1991)

308

Chapter 8: Application of Integral and Differential Methods

('f ' 3Hv 2k J t , H Y~3p Sx+

+

'3Hz',, I Hz~f~t

The last integral in this formula is over the contour of the surface Sx+. Edge elements (section 8.4.3) may be used for the implementation of this approach.

8.7.3: Hybrid FEM-BEM Formulation The weak form of the Helmholtz equation in a bounded domain is given by the equation (8.7.3):

VxHVxH'

dV-~

~tHH'dV-

Gx7Vx

=0

a where nG is the exterior normal to G on ~ . To couple the field inside the domain G with the field outside the domain, note that VxH on the boundary aO depends on the values of H on aO via the solution of the Helmholtz equation outside the domain. The connection between H and VxH stems from the following relationship which expresses a field through its values on a closed surface (D'Angelo and Mayergoyz, 1991; Lynch, Paulsen and Strohbehn, 1986; Stratton, 1941): ~H(x) =

( n Q x VH(x+)g(x, x' )) d S ' a

nQxH(x+)

x

g( , x ) d S -

nQH(x+))x

g( , x ) d S (8.7.5)

3Q 3 where Q is a domain, nQ is the exterior normal to Q, g is the scalar Green's function for the He!mholtz equation in Q and g is a factor which is unity for a point x inside Q and 1/'2 for a point x on a smooth boundary 3Q. Let Q in (8.7.5) be the exterior region for 3G (Fig. 8.7.2). Note that nG ~ -nQ. Since Q is homogeneous (all inhomogeneous materials are included in G), the Green's function in 3D is exp(jkr), g(x,x') = ........4~....... r = Ix,x') and in 2D g(x,x') = ~ Ho(k [ x-x' [ )

(8.7.6)

where Ho is the Hankel function of the first kind, order zero (and is not to be confused with the magnetic field !).

A. Konrad and I. A. Tsuke~an

309

The FEM equation (8.7.3) and the BEM equation (8.7.5) are two simultaneous equations with the unknowns H (in the FEM domain G) and n x VxH (on the b o u n d ~ 3G). The domain G is meshed, and a set of basis functions Wi related to the mesh is introduced. Basis functions corresponding to the nodes on the surface then provide a natural basis for n x VxH. This hybrid method employs FEM and BEM in accordance with their capabilities: ~ M is used in an inhomogeneous bounded domain and BEM is used in the exterior homogeneous unbounded domain. At the same time, the common disadvantage of integral methods cannot be avoided: the unknowns corresponding to the boundary become coupled in the algebraic system, yielding a large full submatrix. This is barely acceptable for three-dimensional problems, and therefore we shall limit further discussion to 2D problems and, even more specifically, to the TE mode. Equation (8.7.5) then is simplified: H has only the z-component; so does nxVxH which can be replaced by (3HdOn) Thus (8.7.3) and (8.7.5) become

~

~

E1-HH,dV. m2 SgHH'dV + G

~H-

n-~

H'dS =0

OG

(8.7.7)

3H ~ S = 0

~HdS+ 3G

n-=ng

3G

Exterior Domain Q

Pl

i ~ ,

....

,"

,

|,.~

~--%

~ I

do~a~u) ~

.'n

~.

/

~M

I

---...

Figure 8.7.2:

I

~ /

f

G

%

s

t

s

p s

,

-"

, I

/ /

-

Expressing the Field at a Point x Via field Values at Boundary Points x'

310

Chapter 8: Application of Integral and Differential Methods

where g is the Hankel function (8.7.6). As usual, the magnetic field H is considered as a sum of the known incident field and the unknown scattered field Hs. With a set ~Pi of basis functions for Hs chosen, the basis functions {rn for (OH~On)can be taken simply as ~i = ~i on 3G; node i ~ 3G Substituting the expansions Hx = ~ i W i , i

aH ~ =m ~n

{m

into (8.7.7), one obtains a partitioned algebraic system of the forln

IF 1[~]= f

(8.7.8)

In (8.7.8) F is a finite element matrix with the entries Fire =

jr1

FVWiVWm dV - 052

j

~tWiWm dV ;

N is a matrix representing the boundary integral in the first equation of (8.7.7):

h

- J ~t"m~i dS k

j

and T is the matrix related to the first two t e ~ s in the second equation of (8.7.7): Tmi= ~ -

J~WidS

d and C corresponds to the third term in the second equation of (8.7.7):

Cn'a=a!g i as The right hand side f in (8.7.8) is obtained by extracting the incident field r-ore (8.7.7). Some applications of the hybrid FEM-BEM technique are given by D'Pmgelo and Mayergoyz (1991) and Lynch, Pau~en and Strobehn (1986).

8.8 Conclusion The examples of integral and differential methods considered in this chapter do not cover all possible approaches to the solution of electromagnetic problems. For example, an important group of methods based on the Fredholm equations of the second type (Mayergoyz, 1979) remains out of the scope of this survey. Finite

A. Konrad and I. A. Tsukerman

31 1

T a b l e 8.8.1: S u m m a r y of the A d v a n t a g e s and Disadvatages of the Meffiods Considered Methods Integral ~M Hybrid The Problem FEM-BEM Harrington equations of the Znd t kind Method Bounded ± ± i

,,,,,,

+

+

gl

+

+

In

C l o s e d Surfaces Ill

Non-Closed Surfaces + Well suited ± Not well suited

- Not advisable to ttse

difference methods and FEM in the time domain have not been dealt with either. It is hoped, however, that the material included in this chapter will be helpful to those who start implementing and using computational methods to solve electromagnetic problems. We have considered formulations based on integral and differential equations. The discretization of integral equations may be performed using the method of moments or quadrature formulas; the discretization of boundary value problems for difI~rentiaI equations may utilize finite difference or finite element methods. The hybrid FEM-BEM method (section 8.7.3) is a combined integrodifferential approach. The finite element method is generally preferable for problems with inhomogeneous media in bounded domains. The formulation based on integral equations of the second kind may be more efficient for u n f u n d e d homogeneous problems with closed conducting surfaces. Harrington's method, i.e. the solution of integral equations of the first kind, is convenient for homogeneous problems with non-closed surfaces. To avoid possible numerical instability, the so-called "regularization methods" (Tikhonov and Arsenin, 1977) can be used. If regu!arization is not explicitly employed, the "level of ill-posedness" can practically be monitored by estimating the condition number of the matrix (Forsyte, 1967). The FEM can also be used for problems with non-closed surfaces; however, singularities of the solution at the edges require additional care. Table 8.8.1 summarizes some of the advantages and disadvantages of the different approaches. It is also important to keep in mind that the computational complexity of integral equation methods grows much faster than that of FEM as the mesh is refined in order to increase accuracy (see sections 8.2.4, 8.4.7).

312

.

Chapter 8: Application of Integral and Differential Methods

Appendix 8.A . . . .

iiii1!

i

i iiiiiiiii . . . . . . . . . .

Capacitance .

C

.

.

.

.

.

.

_

.

.

.

.

.

.

.

.

.

tm,

.....

,,,,

. . . . . . . . .

by ,,

,

Harrington's ,,

Method

,

.

P ROG R.,aSM MM~IX C C C C

VERSION 1.0 BY JO

LO

~ B . 8, 1991

CS RY C C This program, uses the METHODS OF MOMEI'
C C C

-

I. . . . . . . . . .

X-axis

>

-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

Chapter 8: Application of Integral and Differential Methods

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

Chapter 8: Application of Integral and Differential Methods

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)

A. Konrad and I. A. Tsuke~an

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

Chapter 8: Application of Integral and Differential Methods

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

Chapter 8: Application of Integral and Differential Methods

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

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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

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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 ~ .

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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

Chapter 8: Application of Integral and Differential Methods

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

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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 (<= 64, >= 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))))

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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 +

A. Konrad and I. A. Tsukerman

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

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INCLUDE 'cgauss,for'

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330

Chapter 8: Application of Integral and Differential Methods

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!'

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

EN~ C C

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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

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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

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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

Chapter 8: Application of Integral and Differential Methods

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

A. Konrad and I. A. Tsukerman

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

Chapter 8: Application of Integral and Differential Methods

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

A. Konrad and I. A. Tsukerman

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......

Chapter 8: Application of Integral and Differential Methods

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);

A. Konrad and I. A. Tsuke~an

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 << 8, then the inductive effect of the magnetic flux ~etv,,~fi the lir~es~Sis~egligible (Ferrari I985) and the magnetic field along the ydirecti~n is ~ in the air region (Ferrari and Pinchuk 1985).

Conductor a

~

Z

Figure 9.3.2: Section of 'bus-bar' model

S. Subramaniam and S. R. H. Hoole

345

and~

(

i !,5 21~ 1 ~ X' Hx =0 Figure 9.3.3: Boundary" conditions for H-solution with five finite elements (x'=x-0.5, y'=y) One of the conductors, the air space and the line of symmet~ are shown in Figure 9.3.2. (Ferrari and Pinchuk I985). A current of I0 A m ~ r e s ~ r meter width of the conductor flows in the z-direction and ~ e well known quasi-static solution, Hy = - I 0 in the air (9.3.1a) Hy = - I 0 e"C(x-a) in the conducting region (9.3.1 b) (Fe~afi 1985) shows that the source current creates a uniform magnetic field, H with the y-component, Hy = -10 Am "1 in the air region 0 _<x ~0.5m. For simplicity's sake, the skin depth is taken as 8 =1 and Hy is forced to zero at x = 2.5. Even though it is a two dimensional example (Ferrza-i 1985), ~ c a u s e the conductor is very wide and very thick any change in the field intensity can take place only in the x-direction. This simple problem is governed by the Maxwell ~uations, 0B V x E = - 0t (9.3.2a) VxH =J (9.3.2b) and the flux density B is divergence free, V•B - 0 (9.3.3). Here E and J are the electric field intensity and the source current density. In a linear conducting medium the constitutive equations are given by the equations, J =cE O.3.4a) B = ~tlt (9.3.4b) and equations (9.3.2a) and (9.3.2b) can be rewritten OH V x E =- g~ (9.3.5a) V x H = e E. (9.3.5b) For a quasi-static source with frequency o~, the complex phasor form of equation (9.3.5a) is V x E = -j~gtt. (9.3.5c) Taking the cud of equation (9.3.4b) and substituting for V x E from equation (9.3.5a) gives the curl-curl equation, Vx(1V

xH

)+jo~gH=0.

(9.3.6)

346

Chapter 9: ~ g e Elements

Considering equations (9.3.1a) and (9.3.Ib), the model shown in Figure 9.3.3. can be used for the finite element analysis of the conducting medium with linear transformations, x' = x - 0.5 and y' = y. Here we have defined the cud and the divergence of the vector tt and the tangential components Hx and Hy along the boundaries of the finite element model. As such, this mathematically described problem has a unique solution (Hoo!e 1989). The finite element technique, in its variational form, needs a functional to be minimized to obtain the golution (Hoole 1989; Gallagher 1975). The functional, L(H)=

[ [ ( V x~H)2 + jo~gH2] ~

(9.3.7)

which corresponds m the functional derived by Webb, Maile and Ferrari (1983) using the variational principle for nonselfadjoint electromagnetic problems, was suggested by Ferrari (Ferrari 1985; Ferrari and Pinchuk 1985). One should note that the source current is not used in the functional because in the formulation the current is considered as an external one and it is implicitly represented by the boundary condition of eq. (9.3.!a). The stationmty of the functional is analyzed by considering the perturbation gL(H). Let It be the exact solution; the perturbation 5L(H) can be evalua~d from the equation, L(H+~H) = b(H) + 5L(H). 8L(H) = L(H+SH)- L(H) ~

[{Vx(H+SH)} 2 (Y

+ jo~ (H+SH) 2 ] dx dy

[ ( V x H ) 2 + j~I~H 2 ] dx dy

=2

j +2

5H •

[

j~gH + V x

(v x.)] c

•

dx dy

" V , ( V x H x ~H ~ dx dy ~3 (neglecting the higher order terms)

=2

" V . ( V x H x ~H )I dx dy a (since tI is the exact solution.)

s. Subrarnaniarn and S. R. H. Hoole

f( x. ~ x

~H

347

)

°dr (applying Gauss's Theorem)

=2 ~ r (E x 5H) on dr

=2

I (SH , ( E x n ) ) d r = 2

(using eq. (9.3.5b))

I (H ° ( n x ~ E ) ) d r .

(9.3.8)

If we consider the region R as a finite element mesh and the boundary S as made up of element edges Si, then (9.3.9a)

i where 5L(H)i = 2 S!" (8H.

• (E x n ) ) d r = 2 S!r (E.

° ( n x ~SH)) dr

(9.3.9b)

In finite element analysis, the object is to ensure that each contribution in eq. (9.3.9b) either vanishes or cancels with the corresponding contribution from an adjacent element. In our problem, since the region is homogeneous, we need not have to consider the discontinuities due to any sha~ jumps in the material properties. If the component of H tangential to Si is prescribed, then the transverse components 8H vanish and correspondingly the t e ~ (n x SH) in eq. (9.3.9b) vanishes. Note that in this case, no restrictions are specified on the normal components (Ferrari 1985). In this problem, since we have specified the tangential components along the boundary, the specification ensures that the stationary value of the functional is zero. In two dimensions, H = H x u x + Hyuy, (9.3.10a)

VoH = ~~x H x + ~~y

(9.3.10b)

VxH=uz

(9.3.10c)

and X.~x - ~y 7"

Since the problem is simplified as a one dimensional one and we already assumed that any change can occur in the x-direction, 3/~y ~ 0:

348

Chapter 9: Edge Elements

Hx =0

Figure 9.3.4: The five elements with the variational parameters, Htl ,.., Htl 6 along the edges 3Hx V ° H = ...........= 0 Ox

(9.3.1~)

a~ V x H =Uz;) ~x . (9.3.1~) One may then conclude that Hx is independent of x and y, and that Hy is a function of x only. In the finite element model shown in Figure 9.3.4, the region R is divided into five rectangular elements with 16 edges and the tangential component of H along these 16 edges are denoted by Htl,.. Htl6 respectively; Htl .... Ht6 are parallel to the y-axis and two of them have known values: Ht6 = -10 and Ht5 = 0. The other ten are parallel to the x-axis and all of them take the value 0. So, we need to determine the four variational parameters, Htl .... Ht4 which v ~ to satisfy the condition that the functional L(H) should be at a minimum. L(H) involves an integration which can be replaced by summing the in~grals over the five elements. Now we shall consmJct the trial functions to i n t e ~ l a t e Hx and Hy within each element. In general, over the element shown in Figure 9.3.5, we assume a linear variation of ~ along the L-direction: = ~ + (H2-H4)

(9.3.1 la)

Transforming the interpolation to the Cartesian coordinate system,

-1

4 -I..

(xl/yl)

HI

H2

,i /

........ - -

(x2,yl)

.

- ~ x

Figure 9.3.5: Parametric Coordinates for a Rectangular Element

S. Subramaniam and S. R. H. Hoole .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

349

.

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

~

.

.

.

.

.

.

.

.

.

iiJJiiiiJuJu

x-xi ~'~1- x2-x Hy = H4 + (H2-H4) i\x2_x!) - (x2_~ i~4 + (x2_xl i~2

[x x x xLlr.4

= x2-x 1 x2-x 1 H2 (9.3.11b) Since Hx is independent of both the coordinates x and y, it can simply be interpolated ~ follows in the Cartesian coordinate system: He = [ I 1 ] [ ] (9.3.1Ic) X Combining ~uations (9.3. I lb) and (9.3. I Ic) together gives an interpolation for H within an element

if--

o i o

1

0 ~'t 0 (9.3.12) "/ H3 ~ . _ t,.XE-X!j ~x2-xi j H4 It can be easily verified that this interpolation satisfies the non-divergence condition. Within an element, the curl is given by V x H =Uz,,, ~x -

~U Z

[ 0

[

=Uz ~, x2-xl j

.

I

0

x2-x I

-I

x2-x 1

H2 H3 H4

(9.3.13)

and ~i f (V x H) 2 dx dy E "

F

1

x2-xl 0 =[H1 H2 H3 H a i l -1 x2-x 1 F

0 x2-xl

o x2-x~ 1 ]I

0 x2-x 1

o x2-x~ I ]I

~dxdy Elt

I

I = [ H I H2 H3 H 4 ] / x2-x 0 -I x2-x I

L

AElt

350

Chapter 9: Edge Elements

= [ H I H2 H3 H 4 ] P

[H11 H2 H3

AElt

n4

(9.3.14a)

where 0

2

0 P=

0

-i°

0

0

0

2

0 0

(9.3.i4b)

0

0

and AEIt is the area of the element. The other term of the functional, H 2dxdy = [ H I H2 H3 H 4 ] T

(9.3.I5a)

H3

H4 where '*

T

C" - 1

0 (X-Xl ".I 0 /.X2_X1)

=

1

0 --

I

0

) ~X2-Xlj -

I

I

0

0

1

0

Cx-x~- 1 0 (x~-x -,/ ~.x2-xlJ

dxdy (9.3.15b)

~,x2-xlj

1 0I 1 0i l o~o

=

10 1

0

AElt

0

To evaluate the functional, perfov,'mthe integration over each element and add them. For the homogeneous region the functional is given by: 0.3.7) ] [ (W x It)2 + j ~ g a n 2 ] dA Conductor Consider the element with vertices having the coordinates, (0,0), (I/3,0), (1/3,I) and (0,t). For this rectangle, (x2-xl) =1/3, (Y2-YI) =I and AElt=I/3; therefore L(H) =

S. Subramaniam and S. R. H. Hoole .......

........

,,,,,,

, ,

.

.

.

.

.

.

.

.

3 51 .

.

.

.

.

.

.

.

.

No. of Unknowns = 4, No. of nodes = 12,

.

.

No. of edges = 16, No. of rectangles = 5 (a)

E l t # L nodel Eltl 1 EIt2 2 Elt3 5 Eft4 7 ~ ! t . 5 ........ 9

ea~

~ge4

I 2 3 4 5

8 I0 t2 14 16

6 1 2 3 4

vertex2 known(-i)/dnk(O) ....rattle if kn0Wfi 4 -I 10 + j 0 3 0 6 0 8 0 10 0 12 -t 0 + jO 2 -1 0 +j0 3 -1 0 +j0 5 -1 0 + j0 6 -1 0 + j0 7 -1 0+j0 8 -I 0 + j0 9 -1 0 + j0 I0 -1 0 + j0 I1 -1 0 + j0 t2 -1 0 + j0

Node #

X

Y

I 2 3 4 5 6 7 8 9 10 11 I2

0 1/3 1/3 0 2/3 2/3 1 1 1.5 1.5 2 2

0 0 1 I 0 I 0 1 0 1 0 1

node2 2 5 7 9 I1

node3 3 6 8 10 12

node4 . 4 3 6 8 10 ,,

eagel ~

.......

7 9 11 13 15

(b) ~ge # I 2 3 4 5 6 7

10 11 12 13 I4 15 16

vertex1 I 2 5 7 9 I1 I 4 2 3 5 6 7 8 9 10

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

1, , , , , ,

.

.

.

.

.

.

.

.

.

, ........

(e) Table

9.3.1:

1 LI(H) = 3 [ H t 7

+~[Ht7 3

(d)

Data for the bus-bar problem(a) General information (b) Rectangle data (c) Edge related data (d) Nodal coordinate data

Htl Ht8 H t 6 ]

Htl Ht8 Ht6 ]

io o o o ji.t7 l 0 1 0

0 0

0.1111 0

0 0

-0, I l l l 0

Htl Ht8

0

-0.1111

0

0.1111

Ht6

0.3333 0 0.I667

0 1 0

0.1667 0 0.3333

| H t I (9.3.16a) l Ht8 Ht6

352

Chapter 9: Edge Elements

If all the elements are numbered in the same manner as we did the first element, the local matrices P and T of eq. (9.3.16a) would still be valid except for the parameters, Htl, Ht6, Ht7 and Ht8, the non-zero elements of the P matrix and the area of the elements. So the node and edge orderings and element data are similar to those given in Table 9.3.1b, the data related to the exlges and fine nodal data are given in Table 9.3. Ic and Table 9.3. Id respectively. The general information given in Table 9.3. la is provided for computer applications. For these data, we have 5 L(H) =

Li(H ) i=l

0 0 0 0

0 0 0 ] Ht7 0.0370 0 -0.0370 Htl = [Ht7 Htl Ht8 Ht6 ] 0 0 0 Ht8 -0.0370 0 0.0370 Ht6 I 0 1 Ht! +jo~o[Ht7 Htl Ht8 Ht6] 0 0.1111 0 0.05556 1 0 I 0 Ht8 0 0.05556 0 0.1111 Ht6 0 0 0 [ Ht9 0 Ht2 + [Ht9 Ht2 Htl0 Htl ] 0 0~0370 0 -0.0370 0 0 0 0 Htl0 0 -0.0370 0 0.0370 HtI 1 0 1 Ht9 0 Ht2 +jcop~[Ht9 Ht2 Htl0 Htl ] 0 0.1111 0 0.0556 1 0 1 0 Htl0 0 0.0556 0 0.1111 Htl 0 0 0 0 0 0.0370 0 -0.0370 + [ H t l l Ht3 Htl2 Ht2] 0 0 0 0 0 -0.0370 0 0.0370 Htll] 0 0.0556 Ht3 +jo~cs[HtllHt3Htl2Ht 2 0 0.1111 0 1 0 1 0 Htl2 0 0.0556 0 0.IIII L Ht2 0 0 0 0 0.125 0 -0.125 + [ H t I 3 Ht4 Htl4 Ht3] 0 0 0 0 0 -0.I25 0 0.125 1 0 1 0 0.0833 +jo~¢o[Htl3 Ht4 Htl4 Ht3] 0 0.1667 0 1 0 I 0 0 0.0833 0 0.1667

ll[ i

o1[ 7

o

l

o]

S. Subram~iarn and S. R. H. Hoole

353

+ [Htl5 Ht5 Htl6 Ht4

0

0

0

0 0 0

0.125 0 -0.125

0 0 0

01r -0.125 0 0.125

J[

1 0 1 0 ]I +j~olao[Htl5 Ht5 H t l 6 H t 4

0

0.1667

0

0.0833

0

0.0833

0

0.1667

1

0

1

]

0

(9.3.16b) Manipulation of eq. (9.3.14b)gives us a global ~lynomial in the global vector, H = [Htl, Ht2, Ht3, Ht4, I Ht5, Ht6, Ht7, Ht8, Ht9, HtI0, Htll, Htl2, ~a t

Htl3, Htl4, Htl5, Htl6] = [ uk , H~]

(9.3.17)

where H~k and H_~ are the unknown and known parts of H. The local polynomials given by eq. (9.3.15) can be transformed to a global polynomial which is given in Figure 9.3.6. ~ e block matrix form of the global polynomial can be written as,

where Pg and V are symmetric matrices. Because the elements of ~ are known, they are fixed therefore the elements of ~ k adjust themselves to attain a minimum value of the functional. The differential equation would be satisfied when the perturbation dL is zero and the ~undary conditions are satisfied; but dL becomes zero only at the stationary point. Extremization of L with res~ct to the elements of ~ak gives the mah'Jx equation, 7

~L

- [ Pg U ]

[ =0

(9.3.t9a)

J and substituting ~ o with (2/~2 = 2) gives the following system of equations which corres~nds to eq. (9.3.19a)

354

Chapter 9: Edge Elements

L(H) = 2a

-a

-a

-a

2a -a

-a a+b - b

-b 2b -b

-b 2b

-a

H

Ht

+ j~rtc 2/9 1/18

1/18 2/9 1/18

I/I8 1/18 5/18

1/12

1/12

2/6

1~'i'2

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

1/12 i/6 1/6

1/18

I 1

Ht

1 1

H I 1

1 I

I 1

! 1 1 1

1 I 1 1

where 1 a = ~

and

b =

1

Figure 9.3.6: Global Matrix

1 I

S.

Subramaniam and S. R. H. Hoole

0.074+ j0.222 0L 0Huk -

355

-0.037+ -j0.0556

-0.037+ j0.0556

-0.037+ 0.074+ j0.0556 j0.222 -0.037+ j0.0556

I-I=9

-0.037+ j0.0556 0.162+ j0.2778

-0.125+ j0.0833

-0.125+ 0.250+ j0.0833 j0.333

-0.125+ j0.0833 m

(9.3.19b) In eq. (9.3.19b), the matrices U t and V were not used, so it is not n e c e s s ~ to form them. All the elements of ~ are known, and therefore eq. (9.3.32) can be rewritten as, PgHuk = - U Hk and the corresponding system of linear equations becomes

(9.3.20a) m

0.074+ j0.222

-0.037+ 0.074+ j0.0556 j0.222

m

0.370 - j0.556

-0.037+ j0.0556 -0.037+ j0.0556

-0.037+ 0.162+ j0.0556 j0.2778

Huk =

-0.125+ j0.0833

-0.125+ 0.250+ j0.0833 j0.333 (9.3.20b) This tri-diagonal symmetric complex system of linear equations can be solved by Cholesky decomposition or other suitable schemes. The solution is -2.4936 +j0.9815 1 0.5180 - j0.4852 Huk = -0.0472 + j0.1647 " (9,3.20c) -0.0332 - j0.0376 The y-component of the magnetic field is evaluated at several points in the solution region using the approximation of the tangential vector elements and the exponential decay of the field intensity as the magnetic field penetrates inside the conductor is shown in Fig. 9.3.7.

356

Chapter 9: Edge Elements

|

I

i

I,

1

f

!

I iO:i

I !

& ................ 0.6

0~8

! ,0

1.2

i, a

1.o

1.0

8.0

2.8

a.4

x

(a)

12-

h~ ~-

10

~

1.0

Cond

1 ]~

----~ ...... ,H(rea,),

0.8

IH(Jmag)!

m

E

"'_~

"-"-_:1: 0.4

~

0.2

, - - ~ " . l ....... 0

,

,

1

x

0.0

•

,

2

3

(b) Figure

9.3.7:

The eddy current field Hy a.Real part b. The magnitude of the real and imaginary parts

Z.a

S. Subramaniam and S. R. H. Hoole

357

In this simple problem we restricted the vector functions for a vector which flows parallel m two opposite edges of the quadrilateral. In general, the vector need not have to ~ parallel to any of the edges, To address this need, in the next section we shall develop vector trial functions for rectangular edge dements.

9.3.2: First-order Rectangular Tangential Elements Developrnent of the divergence free tangential vector trial functions for the most generalized case of the quadrilateral elements is troublesome, so in this section we restrict it to the case of rectangular elements. For the rectangular elements (Figure 9.3.5) with the edges parallel to the axes of the Cartesian coordinate system, any arbitral3' vector can be written as V = Vxu x + Vyuy (9.3.2I) and the divergence of V is V.V =

+

(9.3.22)

For ~ro divergence vecm~ eq. (9,3,22) becomes ay J - 0.

(9.3.23a)

Rearranging, aVx _ . aVy (9.3.23b) ax ay In node based finite elements the variational parameters are the nodal values of the field variable and the components Hx and Hy can be writmn in terms of Hxi and Hy i (i = 1.... 4) resp~tively (Hoole 1989). With the assumption of a linear vmation of Hx and Hy within a rectangular element, ]

=

-

[ Vxl VX2 VX3 Vx4 Vyl Vy2 Vy3 Vy4

[ Vxl Vx2 Vx3 Vx4 LVyt Vy2 Vy3 Vy4

L2(x)LI(y) L2(x)L2(y) LI(x)L2(y)

INI N2 N3

(9.3.24)

N4 where (. t2-t ). Ll(t) = ~ 7

L2(t)

NI= LI(x)LI(y),

N2= L2(x)LI(y),

/',/4= Ll(x)L2(y)

= (t-t 1 ) Vt2_tlJ N3= L2(x)L2(y)

358

Chapter 9: Edge Elements

One of the advantages of using these edge based finite elements is in implementing the nondivergent condition, V.V=0 directly (Sakiyama et al, 1990). For edge based finite elements, the variational parameters are related to the edges (Fig. 9.3.8b). In electromagnetic field computation, the field intensities E and H satisfy only the tangential continuity at the material interfaces (Hoole 1989). So we need a tangential vector element which has its variational parameters along the tangential directions of the element edges as shown in Fig. 9.3.8b to get rid of the unwanted normal continuity which is implicitly imposed by the nodal based finite elements. Here the variational parameters are V t 1, Vt2, Vt3 and Vt4. An inm~olation for the edge based finite element can be derived from the node based vector element by equating the tangential component of the vector field at the nodes which make the edge (Sakiyama et al. I990); that is, Vxl = Vx2 = Vtl, Vx3 = Vx4 = Vt3, Vy2 = Vy3 = Vt2, Vy 1 = Vy4 = Vt4. (9.3.25a) Substituting these in equation (9.3.24) gives,

[Vx _ i v . Vy

-

0

o

Vt2

o

0

Vt4

L2(y) Ll(x)

(9.3.25b)

and dift2~entiafion matrices are given by I aVx aVx J

=[vtl

aV v aV v

o v,3 o I

0

ay"

Vt2

0

Vt4 _1

0

1

hx

o1

0

3. - hx

(9.3.25c)

-y 0

A similar approximation was used by Hano (1984) to analyze a dielectric-loaded waveguide. It can be seen that the components Vx and Vy are linear functions of y ~y (Vx3,Vy3) 4

AY

4 Vt3 y2 I- --l-...........

I

y 1 _i~¢~ 1

1,Vyl)

xl

yl

(V: (a)

Figure

x2 -x 9.3.8:

it: 1 xl

3

............. Vtl

,2

(b)

x2

(a) Rectangular node based element (b) Rectangular edge based element

x

S. Subramaniam and S, R. H. Hoole

359

and x respectively; therefore the approximation satisfies the nondivergence condition within the element and the curl is given by

{~Vv

v x v

-

,

= Uz

[Vt2 - Vt41 - (Y2-Yl) [Vt3-Vtll

=Uz [_ hy where h x = (x2-x 1)

and

hx

hy

hx

hy = (Y4-Y1)

) (9.3.2 6a)

(9.3.26b)

9 . 3 . 2 . 1 : The Magnetic Field inside a Long, Rectangular Bus-bar Consider the long, rectangular bus-bar shown in Fig. 9.3.9 (Hoole 1989), and ca~ying a known AC current I0 = 10 A. The cable is of infinite length, so any change in the field intensity can take place only on the xy-plane which is pe~endiculax to the cable. For a vector analysis, no unique solution is possible without specifying the curl and divergence of that vector. We know that the magnetic field intensity around a line conductor satisfies Ampere's Law, VxH = J, and in a magnetic charge free region H obeys the divergence free equation, V.H = 0. For this problem in two dimensions with O/0z = 0, the governing equations become, ~Hy ~Hx ~x - ~y - Jz (9.3,27) ~Hx ~x +

=0

(9.3.28)

where H = (Hx, Hy), Now we need a functional to solve the problem using the variational method with finite elements, A functional which uses H alone inside the conductor region was suggested by Ferrari (Ferrari 1985; Ferrari and Pinchuk 1985). In a nonconducting region the divergence free magnetic field becomes irrotational. Therefore it can be represented in terms of a scalar potential W as H = -V~4t. Along the conductor-air boundary, the tangential components match: H t = -(Vw) t = 10. In case one is not interested in the solution in the air region, the air region can be removed from the solution region by using this Difichlet boundary condition along the conductor-air boundary'. One can use the same functional (eq. 9.3.7), which was used for the demonstration problem of the previous section L(H) =

t /

[ (V x H) 2 + jmlaoH 2] dA

Cond~actor

(9.3.7)

360

Chapter 9: Edge Elements

In this approach, it should be noted that the source is considered as an external one and the effect of that external source on the magnetic field intensity is specified by its tangential component along the conductor-air boundary. The following Dirichlet boundary' conditions specified on the conductor-air interface implicitly represent the source current, I0 =10A: x = 0.5 -+ Hy = 10; x = -0.5 --) Hy = -I0; y = 0.5 ---) H x = -10; y = -0.5 ~ H x = 10. (9.3.29a) Using the symmetry of the magnetic fields along the axes of the Cartesian coordinate system, the solution region can be reduced to the first quadrant with the boundaEv conditions x = 0.5 ---) Hy = 10; x = 0 --) Hy = 0; y = 0.5 --~ H x = -10; y = 0 --) Hx = 0. (9.3.29b) The non-divergence of the magnetic field in a homogeneous region (eq. 9.3.28) is not accounted for in the above mentioned functional because the trial functions are chosen in such a way that the approximation satisfies the non-divergence condition within each element. Further simplification of eq. (9.3.15) gives,

1

OHy ~HxX2

Ox- " '~y ,)

L(H) = ~ f

dxdy+

jo~o jr" f 2

(9.3.30) The integrand of the first term of the functional can be approximated using eq. (9.3.26) as

H t = 10 0.5

o

(b)

t Ht--01

0.5m

(o,o)

0.5

~X

Conductor o

8

(c)

-0.5m Air

17

(a)

Figure 9.3.9: a. Rectangular Bus-bar b. Problem reduction c. Mesh

S. Subramaniamand S. R, H, Hoole

361

F

kykx -k2 -kykx Y k -kykx = [Htl Ht2 Ht3 Ht4]/kykx -kykx k2 kykx / Y ~-kykx kykx

Htl "] Ht2 | Ht3 | Ht4 _1 (9.3.3 la)

where 1

1

(9.3.31b)

kX=hx and k y = ~ , Therefore, the approximation of the first term is [ Htl ] =

Ht3 ' Ht4

where !

hx hy P=

h~ hy

I

hx -1

hx --hy hx -hy

-1

hx 1

(9.3.31c).

hx hx In a similar way, Holt can be approximated within an element as,

['Htl l |Ht2 = [ Ht I Ht2 Ht3 Ht4 ] T | Ht3 ' 1_Ht4

H.H

where I T=

Ll(y)2 LI(y)L2(x) El (y)L2(y) 1 L2(x)LI(y) L2(x)2 L2(x)L2(y) LI(y)LI(x) L2(x)LI(x) L2(y)L1(y) L2(y)L2(x) L2(y)2 L2(y)LI(x) L2(x)LI(y) Ll(x)L2(x) Ll(x)L2(y) Ll(x) 2

(9.3,32a)

(9.3,32b)

and the second tema of eq, (9.3,30) becomes ,

(

Hy)dxdy = [Htl Ht2 Ht3 Ht4 ] T 1,I |Ht2 | Ht3 , k Ht4

(9,3.32c)

362

C h a p t e r 9: E d g e E l e m e n t s

0.5

0,4

0,3

0.2

0,1

0.0 0.0

0.1

0.2

0,3

0,4

0.5

0~6

0.6

0.5

-

0~4

\

/

i

f

/

0,3

0.2

0.1

0~0

\

-I"

.

0.0

0.!

0.2

0.3

0.4

0.5

0.6

Figure 9.3,10: The eddy current field H a.Real part b. Imaginary pad

S. Subramaniam and S. R. H. Hoole

363

Ht= 3,19 0.5 ,-

(b)

0.5m

~" II

O

(0,0)

Conductor cr

(c)

-0.5m Air (a)

Figure

Ht=0

0.5

2~

8

2.~

7

2'

6

9.3.11: a. Rectangular Bus-bar b. Problem reduction e. Mesh

where 2010] 0201 TI,I = (9.3.32d) 1020 0102 Within an element therefore, an approximation to the functional is given by

I

L(tt) = 0.5 [Htl Ht2 Ht3 Ht4] ( P +j0~gcT 1,1 ) Ht2 Ht3 Ht4 = 0.5 H t ( P + jo~geT 1,I ) H (9.3.33a) where ttt= [ Htl Ht2 Ht3 Ht4 ] (9.3.33b) This functional is then minimized and solved for H, with the fight hand side of the equation being given by the known components of H being moved to the right. Fig. 9.3.10 presents the solution within the homogeneous conductor. Be it noted that the above mentioned approach cannot be applied to DC analysis because the real part of the global matrix remains singular even though the appropriate boundary conditions are specified.

364

Chapter 9: Edge Elements

9.3.2.2: The Magnetic Field of a Long, Rectangular Bus-bar In practice, the problems are inhomogeneous and considering the source as an external one and imposing it implicitly through the boundary conditions cannot be done in general. In this section the source term is introduced into the functional (H~le I985; 1989) using the well known least squares aoproach and the same busbar problem is revisited as the inhomogeneous one shown in Figure 9.3.11. In the conducting region the magnetic field satisfies (Bossavit 1988) Vx(1V

x It ) + j o ~ g t t - - 0

(9,3.6)

and in the air region It is governed by V x H = Jo (9.32b) Again with the assumption that the non-divergence property is implicitly satisfied within each element by the approximation, in the conducting region eq. (9.3.7) can be used as the functional for the finite element analysis and for the non-conducting region a least squares functional is added:

L(H)=

o " J' x [ (nV x 'H )s2 +0J ~ °] ~ °2n 2 d] d AA +mA!Ie [ vd i u m C nductmg (9.3.34) Expanding the integrand of the s~ond term with the representation, J0 = Jz Uz,

(OH,,,

[V x H - J0] 2 =~. Ox " Oy

"

Jz

~Hx"~2

)2 +

(9.3.35) The approximation for the first term of eq. (9.3.35) is already derived in eq. (9.3.31). In most of the applications, the source Jz is also a variable, and we may, as a good approximation assume it to vary linearly within a rectangular element: Jz(x,y) = [Jzl Jz2 Jz3 Jz4]

-._

(9.3.36a)

N4 Here we evaluate Jzl, Jz2, Jz3 and Jz4 at the vertices (xl,YI), (x2,Y2), (x3,y 3) and (x4,y4) from the distribution and use them in eq. (9.3.37). A similar interpolation was used to analyze dielectric-loaded waveguides (Hano, 1984; Miniowitz and Webb, I991). Considering the second term,

S. Subramaniam and S. R. H. Hoole

365

Nlhy N2hy i(0Hy 0HxN Nlhx N2hx ~k.0y ) J z =[Htl Ht2 Ht3 Ht4 1 _Nlhy .N2hy -N 1hx -N2hx F

TJzl1

N3hy N4hy N3hx N4hx ~Sz2 .N3hy -N4hy I1 Jz3 -N3hx -N4hx Jk Jz4 (9.3.36b)

N2 N1N2 NIN3 N1N4

['Jzl

]/NIN2 N2 N3N2N~2

/ Jz2

2 =[Jzl Jz2 Jz3 Jz4 2 N4N3 Jz / N1N3 N2N3 N3 2

|Jz3 kJz4

(9.3.3~)

L N1N4 N2N4 N3N4 N4 Equation (9.3.36c) shows that the approximation to j2Z does not contain any variational parameters; therefore the integration of this term does not contribute anything to the extremization procedure of finite element analysis. Therefore an approximation to the integral of this term is not derived. The approximation to the integration over an element of Figure 9.3.8b is thus: f

FJzl1

,[ ( 0 H_Ox y 0~3-yX)Jzdxdy =[Htl Ht2 Ht3 Ht4 ] TI'0 [~Tlz~

(9,3.37a)

LJz4 where 4212 T1,0 2 4 2 1 = 12 4 2 " (9.3.37b) 2124_ Summarizing the results, within an element, in the conducting region the approximation takes the form L(H) = 0.5 H t ( P + jo~tcrT1,1 ) H (9.3.33) and in the non-conducting region it becomes (ignoring the term Jz'Jz), L(H) = 0.5 H t (P H - 2 T 1,0 )Jz (9.3.38a) where Jzt = [Jzl Jz2 Jz3 J ~ ]. (9.3,38b) The magnetic field distribution of the real and imaginary components for this problem are shown in Figure 9.3.12.

E 1

9.3.2.3: Continuity Condition At the interfaces, the magnetic or electric field intensity satisfies the condition that its tangential component is continuous. The trial functions of the tangential edge elements are chosen in such a way to satisfy this tangential continuity implicitly. But the flux density, B = gtt is normally continuous Un • B1 = Un " glH1 = Un " g2H2 = Un " B2 (9.3.39)

366

C h a p t e r 9: E d g e E l e m e n t s

0.6

\

O.5

\

-.~

0.4

\

\

\

\

\ \

~3

\

0.2

\

o.o 0.0

t O.l

0.2

0.3

0.4

0,5

0.6

0.6

0.5

0.4

--

\

//

i

f

/

0~3

0.2

0.1

\

-'-

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 9.3.12: The eddy current field H a.Real part b. Imaginary part

S. Subramaniamand S. R. H. Hoole

367

Ht6

BnI~ BI ~Unl

H~ r

Bn2~

d,yi) Ht2

B2

+Un2

(xj,yj) .2

B1

Htl (a)

(xi,yi)

(b)

(c)

Figure 9.3.13: a. Continuity condition on B b. Implementing the continuity condition c. Definitions

where Un is the unit normal. The continuity condition can be implementedby adding the followingintegral te~ in the least squares sense to the functional of eq. (9.3,34): F~ ( ' I H n l - B 2 n n 2 ) 2dl=

B! Hn! . 2,1 ~t2Hnl nn2 +B2Hn2

dl

=IJI'(P~I Hnl [~1 H n l ' B 2 Hn2]) dl

+ J r ( ~ 2 Hn2 [~ Hn2-BI Hnl])~ (9.3A0a) Considering the small portion of the interface F. which is between Eltl and Elt2 of Fig. 9.3.13b: .f(~tlHnl-B2Hn2)2dl =FIr I~1 Hnl [B1 Hni- ~t2Hn2])dl F1 F4t'(~2 Hn2 [~ Hn2- /al Hnl])~ (9.3.40b)

368

Chapter 9: Edge Elements

From Figure 9.3.13c, one can write (Hoole 1989) the outward unit n o d a l , u n to the line segment connecting the points (x i, Yi) and (N, yj) as: un = Ux sin0 - Uy cos0 (9.3Ala) where cos0 =

(xj- xi ) (9.3.41b) -xf[(x j . xi )2 + ( y j . Yi )2] At the mid-point of the edge, which passes through the ~ i n t s (xi, Yi) and (xj, yj) of the rectangul~ element, the field H can be approximated using eq. 9.3.25b [Hx] Hy

=

[Htl 0

0 Ht3 0 Ht2 0 Ht4

L2(x0) L2(Y0) LI(x0)

(9.3.4!c)

where (YJ2 Yi) Y0 = .......

and

x 0 = ( ~ 2 xi)

(9.3.41d)

Considering EltI and Elt2, Hnl = H * Unl = Hx sin0 - Hy cos0 1

L i(Y0) sin0 -I~(x0) cos0 - Htl PI

= [ H t l Ht2 Ht3 Ht4]

L~(y0) sin0 1 _ -~Ir (x0) cos0 -

(9.3.42a) and Hn2 = H * Un2 = - H * Unl = - (Hx sin0- Hy cos0) m

m

-L~(y0)sin0 = [ Ht3 Ht6 Ht5 Ht7 ]

L~(x0)cos0 _L~(y0)sin0

= H~ P2

_ L2(x0)cos0 _ (9.3.42b) w~re t

H 1 = [HtI Ht2 Ht3 Ht4 ], H 2t = [ H t 3 Ht 6 Ht 5 Ht 7], p;=[

]

1 . -LI1(x0)c°s0 _1 L 1i (y0)sin0 -l~(x0)cos0 L 2(Y0)Sm0

S. Subramaniam and S. R. H. Hoole

369

1

P2t = [ 2-L1(y0)sin0 L 2(x0)cos0 -L~(y0)sin0 Ll(x0)cos 2 0 .1

[btl Hal " bt2Hn2] = ~Htl P I " ~ H 2 P2 =

H 1 H2

(9.3.42c)

_ g2P2

The inte~and of the first integral of eq.(9.3.40),

t ][ 1]

"gl ~ PI P2

2 (9.3.42d) In the same way, the integrand of the second t e ~ is given by bt2 Hn2 [bt2Hq2 _ ~ H1 ] = [ H 2t ] [ - ~ 1 bt2P2Ptl ~22P2P2t (9.3.42e) Substituting into eq. (9.3.40b) results in I ( ~ l H n I - ~ t 2 H n 2 ) 2 at FI gl

-,,P1PtE ~t22 p2p~

where

(9.3.42f)

I= "~ [ ( x j - xi) 2 + ( y j - Y i ) 2 ] . The continuity condition, eq. (9.3.39), can be imposed along the interface by repeating the above procedure to all the elements along the interface. Fu~her, eq. (9.3.420 can be modified to impose the Dirichlet boundary condition, H n = 0, which is commonly used along conductor surfaces in waveguide analysis, along the arbitrary boundaries in open boundary problems and along the symmetric or anfisymmetric boundaries when the solution region is reduced using symmetry.

3 70

Chapter 9: Edge Elements

Subprogram rectangle(x, y, b, c, a); Inputs: x, y = 4 x 1 vectors containing the x, y coordinates of the vertices of the rectangle respectively. = 4 x 2 vector containing derivative of the x component. = 4 x 2 vector containing derivative of the y component. = area of the rectangle rectangle(x, y, b, c, a)

Outputs:

b c a

hx = x[2] - x[ 1]; hy = y[41 - y[l]; a = hx * hy; b[l,1] = 0; b[2,11 = 0; b[3,1] = 0;

1

b[1,2] = - hy'

b[2,2] =0;

b[3,2]

1

hy '

!eq. (9.3.26b) b[4,11 = O; b[4,2] =0; !eq. (9.3.25c)

}

1

c[I,11 =0;

c[2,1] = ~

c[ 1,21 = 0;

c[2,2] =0;

Algorithm 9.3.1:

;

c[3,1] = 0 ;

c[4,11 = - ~

c[3,21 = O;

c[4,21 = O;

1

;

Computing First Order Differentiation Matrices for Rectangular Tangential Edge Elements

9.3.3: Algorithms for the first-order Rectangular Tangential Elements 9.3.3.1: Computing First-order Differentiation Matrices To aid the reader who might wish to implement these methods for the solution of problems, Algorithm 9.1. I presents the methodology fi?r computing the first order differentiation matrices.

9.3.3.2: Computing first-order local matrices Similarly, we also give in Algorithm 9.3.2, the procedure for computing the local matrices. Placing local matrices into the global matrix, is just as with nodal elements and treated in chapter I.

9.4 Triangular Tangential Elements 9.4.1 : Preliminaries Cendes (1991) proved that for the triangular elements with the t~ee variables along the three edges the basis functions can be chosen in a way to obtain a unique solution for the state variable of the curl equations. First we develop the local matrices for the zeroth-order triangular tangential elements and then apply them to solve the example. Earlier we chose the tangential vectors along the axes of the Cartesian coordinate system, but in the case of triangular elements it is impossible.

S. Subramaniam and S. R. H. Hoole

371

Subprogram first-order rectangular tangential |oca!mats(x, y, J, P, T1, I, Q); Inputs: x, y = 4 x 1 vectors containing the x, y coordinates of the

J Outputs: P Tl,l

Q

vertices of the rectangle respectively. = 4 x 1 vector giving source J at the four vertices, = 4 x 4 vector first-order tangential r e c t a n g u l a r local element matrix. = 4 x 4 vector first-order tangential r e c t a n g u l a r tensor element matrix. = 4 x 1 l ~ a l right hand side vector

External subprogram: rectangle first-order tangent rect locmats(x, y, J, P, T 1,I, Q) { rectangle(x, y, b, c, a); f o r i =1 t o 4 d o

{

}

il = i m o d 4 + l ; i 2 = i l m o d 4 + l ; i 3 = i 2 m o d 4 + l ; Q[i] = a * (4 * J[i] + 2 * J [ i l ] + J[i2] + 2 * J[i3])/36; ! eq. (9.3.37b) Tl,l[i,i] = a * 2/.6; T l , l [ i , i l ] = 0; Tl,l[i,i2] = a/6; Tl,l[i,i3] = 0; ! eq, (9.3.32d) d[i] = c[i,1] - b[i,2];

f;ari = 1 t o 4 d o

{ for j = 1 to 4 do

{

} } } Algorithm 9.3,2:

! eq. (9.3.31c)

P[id] = a* (d[il * d[j]);

Computing First Order Local Matnces for Rectangular Tangential Edge Elements

Consider a triangular element which has its vertices at ( x l , y I), (x2,Y2) and (x3,Y3). The unit vector along the i-th edge (Fig. 9.4.1) ti

= ~ [(xi2-Xil)Ux +(Yi2~Yil)Uy] =

[CiUx - biuy]

(9.4.1a)

and the unit vector normal to the i4h edge ni

= t i x uz = ~ ~ [(Yil-Yi2)Uy + (xi2-Xil)Ux] = -

where i l = i mod 3 + 1, i2 = i l mod 3 + 1 bi = (Yil-Yi2)

[biux + CiUy]

(9.4.1b), (9.4.2), (9.4.3),

372

Chapter 9: Edge Elements

I (xl,yl)

tJy~ .....~:

1 (xl.vl)

12 (edge 2)

2

(x2,y2)

-

tl

2 ~/.eage I) (x3, y:3) 3 (x2,y2)=(0,1,0)

(a)

,)~

~1=0 (b)

(x3

3

= (0,0,1)

Fig. 9.4.1: a. Unit Vectors (Tangent and Normal) b. Triangular Coordinates ci = (xi2-xi 1)

(9.4.4),

li = (xi2-Xil) 2 + (Yi2-Yil) 2 = b2 + c 2

(9.4.5)

and Ux, Uy and Uz are the unit vectors along the ~ e s of the Caaesian coordinate system. Equations (9.4.1 a) and (9.4.1 b) can be rewritten in matrix t~nns as

Etll=l 0 o1i,3 b'l EarlI"100lib1 Cl][-x]=_C-~ c2 I"l En'I

IT} = t2 t3 and {N}=

n2 =" n

1;1 0 1-1

c2 c3

0 I;I 0 0 0 I; 1

-b2 .b 3

b2 b3

= L -1C {x}

(9.4.6a)

Uy

-y

B {xl

(9.4.6b),

c3

where the vectors {T}=

t2, {N}=

t-t3-1

n2 a n d { x } = n3

IIx

.

Uy '

and ~ e ma~ces L=

[,lOO] tblclI Ec,.b,l 0 12 0 , B = 0 0 13

b2 b3

c2 c3

andC=

c2 c3

-b2 -b3

.

In two-dimension any vector V can be expressed in terms of its x- and ycom~nents, v x and Vy ~ , V = VxUx + VyUy = [Vx,Vyl {x } (9.4.7) and substituting for { X } either in terms of { T} or in terms of { N }, using (9.4.6a) and (9.4.6b) gives the tangential and normal forms (Cendes, 199I) of V as

S. Subramaniam and S. R. H. Hoole

373

V=[vx,vy] ( L "1 C ) ' I { T } =[Vx,Vy](C "1 L ) { T }

(9.4.8a)

V = [Vx,Vy]( L "1 B ) "1 {N } = [Vx,Vy] (,t,B"1 L / { N }

(9.4.8b)

ard respectively, provided that the inverses of B and C exist. In general, finding the inverse of a 2 x 3 matrix is not so easy and it is well known that the inverse need not be unique. Can we then modify equations (9.4.8a) and (9.4.8b) in such a way as to obtain unique expressions for the tangential and normal forms of a vector? Cendes (1991) expressed {T} and {N} in terms of the v~tor {X} t = [0,Ux,Uyl as,

E,, o

= t2 = {T} t3 Inll

{N}=

0 121 0 0 0 131 II;l

0

0 n3

0

0 1

0 0 1;1

aI C l - b l ] [ , ] L" 1 E {X } a2 c2 -b2 Ux = a3 c 3 -b 3 ][.yJU

(9.4.9a)

ata2 b,b2 c2c,;FOy 1 Ux =- L ' I D {X }(9.4.9b), a3 b3 c3

u

where (9.4.I0) ai = (Xil Yi2 - xi2 Yil), i = 1,2,3 and the matrices, I al Cl -bI ] al bl Cl D = a2 c2 -b2 andE= a2 b2 c2 a3 c3 -b3 a3 b3 c3 It should be noted that E is the matrix corresponding to the affine transformation that maps the Cartesian coordinates onto the triangular coordinates of a triangle (Hoole 1989; Cendes 1991; Farin 1990). It is well known that any affine transformation is invertible and therefore the corresponding matrix is also invertible (Farin 1990; Whitney I957). Using fundamental matrix theory we can simply say that D is inverdble because E is inve~ible. With the assumption that the scalar v0 can be determined, using equations (9.4.9a) and (9.4.9b), the tangential and normal form of the vector, V = v00 + VxUx + VyUy = [v0,vx,vy] {X} = {X }tvX (9.4.1 la) can be written in a unique way as -1 {T} = ( v X ) t ( E ' I L ) { T } = {T}tV T (9.4. I 1b) V = [vO,vx,v ar',d V = [v0,vx,vyl ( where

-1

{r,,r } - (vx)t(D

L)IN} =

(9.4. I 1c)

374

Chapter 9: Edge Elements

Vx

=

I

t

[v0,Vx,Vy]t,v T = ( E " L ) V X a n d V N = ( D ' l

L ) t V x.

Rewriting V X in terms of VT and V N as (Cendes 1991),

vX=(E~'I)vT=(DtL'I)v N gives the value, 3

(9.4.12a)

3

v0 =

a& v T ax ViN (9,4.12b) Ii-i = li i=l i=l for the scalar v0. Because the column vectors V T and V N do not mean anything physically, it is not easy to prove that the values given in equation (9.4.12b) are well defined. We shall find the value of v0 in terms of the components of the vector V along the tangential direction to the edges of the triangle (Barton and Cendes, 1987; Crowley, Silvester and Hurwitz Jr., 1988; Miniowitz and Webb, 1991; Webb, 1993): vti = ti • V 0=I,2,3) (9.4.13). Rewriting the equation (9.4.13), using 2 . (9.4.1 lc) gives the matrix form, Vvtl]

vt =/vt21={T} .V= {T}.{N}tVN LVt3-1

={T}o{N}t(D'I L)tvX={T}o{N}tLt(D-I)tv Tnerefore, VX = OtL -1 ( { T } • { N l t ) "1V t.

X

(9.4.14a).

(9.4.14b)

That the matrix {T} ° {N} t is invertible implies that the expression for v0 obtained from eq. (9.4.14b) is unique. Cendes (1991) has expressed the dot product [ 0 -sin03 sin 02 ] { T } . {N}t= sin 03 0 -sin 01 (9.4.15) -sin 02 sin 01 0 in terms of 0i (i = 1, 2, 3) which is the included angle at vertex i, as shown in Fig. 9.4.2. The rows of the matrix, {T } • {N }t are linearly dependent and the matrix is not invertible; therefore eq. (9.4.14b) does not give a unique expression for V X in terms of V t. If Hi is the altitude to the vertex i (Fig. 9.4.2) then sin 0i takes the simple form sin 0i= iiH~2= Hil i and equation (9.4.15) can be rewritten as (Cendes 1991),

(9.4.16,

S. Subramaniam and S. R. H. Hoole

375

12 \

Figure 9.4.2: Included Angles and the Altitudes

{T}'{N} t =

where H=

0

_H~ 12

H~

0

11 H3 " 11

H3 12

[.,0ol

H1 13 H2 13

= HSL 1 = L'ISH

(9.4.17a)

0

i0l ,j

0 H2 0 a n d S = I 0 -1 . (9A.17b) 0 0 H3 -1 1 0 Next we shall find the inverses of the matrices D and E considering the affine transformations. In two dimension, the triangular coordinates, ~i (i = 1, 2, 3) already described in Fig. 1.5.8, are defined by (Hoole, I989; Whitney, I957), ~i = ~ i

(9.4.I8)

and the Cartesian Coordinates a~,'erelated to each other by the affine transformations (Hoole 1989; Cendes 1991): =

Xl x2 x3 Yl Y2 Y3

~2 = ~ 3

a2 b2 c2 a3 b3 c3

~2 43

(9.4.19a)

(9.4.19b) y

where 3 3 A = 2 xi bi = ~ Y i cii=l i=l

(9.4.20)

376

Chapter 9: Edge Elements

!,. (xl,yI) (1/2,0,1/2) xl (x2,y2)

(0,1/2,1/2) ~1-~ (x3,y3)3 (a) (x l,y 1) x30

z31

2 "7 ....

(0,1/2,112)

°

xl ! (x3,y3)

(b) Figure 9.4.3:

The tangential interpolation coefficients a. For a zeroth-order element b. For a

first-order element

~quadons (9.4.19a) and (9.4.19b) give ai bl C l ] -! [ I t 1] E ' I = a2 b2 c2 - A Xl x2 x3 a3 b3 c3 Y! Y2 Y3 1 I 1 a! bl Cl X l x2 x3 a2 b2 c2 (9.4.21a) Yl Y2 Y3 a3 b3 c3 and the application of the simple column operations on matrix E evaluates the inverse of D

I

D "1 = A

Yl Y2 Y3 . -Xl -x2 -x3

(9.4.21b)

S. Subramaniam and S. R. H. Hoole

377

9.4,2: Zeroth-order Triangular Tangential Elements To derive the trial functions for tangential v ~ t o r finite elements, we shall express a two dimensional vector in terms of normal unit vector ~ (Cendes, 199I) V = VxUx+ VyU~= { N } t V N (;I, ;2, ;3) (9.4.22) Next the function V is approximated in terms of variational parameters (xl, x2, x3) along the edges of the triangles as shown Fig. 9.4.3: vN ( ;1, ;2, ;3) = H -1 F ( ;1, ;2, ;3) x (9.4.23a) where z t = (Xl = (xI), "c2= (x2), x3= (x3)) for the zerotth order element x t = ("cI= ('Cl0,Xl 1), x2 = (~20,x21), x3= (x30,'c31))

for the linear dement (9.4.23b) and F ( 41, 42, ;3) is a matrix with the desired vector trial functions. But V t can be expressed in terms of V N as (using eqs. (9.4. I4a) and (9.4.17)) FvtI1 V t = | v t 2 | = { T } " V = { T } , {1'4} t v N = L - I S H V N (9.4.14a). t-vt3J Substituting eq. (9.4.23a) into eq. (9.4.14a) (9.4.23c)

Vt = L-1SH V N = L ' I s F ( ;1, ;2, ;3) x. Eq. (9.4.23a) can be written in the mat"ix form VN =

0 0

H; 1 0 0

(9.4.24a)

Y21 Y22 ]'23

H31

"¢31 '(32 ~'33

and the product S F becomes SF

=

= whe2,e

1 0 -1 -I I 0

Y31 -Y21 T 3 2 - Y 2 2 YII-Y31 YI2-T32 Y2! "YI 1 T 2 2 " T 1 2

I

= [5)~]

721 Y22 723 Y3I Y32 Y33 Y33"Y231 Y13-Y33 723"Y13

J

(9.4.24b)

for a zeroth-order element

ar~J Yij = tt~0ij Yiij I for a first,order element. Combining eqs. (9.4.23c) and (9.4.24b), we get

(9.4.24c)

378

Chapter 9: Edge Elements

Vt=

1 0 0 0 L21 0 0 0 L31

Y31 -Y21 Y32- Y22 Y33 "Y23 Y l l ' Y 3 1 Y12-Y32 Y I 3 - Y 3 3 Y21 Y1 1 Y22 Y12 Y23 YI3

Xl x2 x3

(9.4.24d) Considering the variables associated with side 1, vti must be approximated in terms of the variational parameters Xl which are associated with side 1. This can be done by imposing the necessary conditions along side 1 (Cendes 1991), Y3I " Y21 = LI 9(~2, ~3),

along ~1 = 0

Y32 - Y22 = 0,

along 4I = 0

and Y33 "Y23 = O, along ~I = 0 (9.4.25a) where ~ is a vector containing inte~olation polynomials along side 1 ( ~1 = 0). Similar equations can be written for the other sides as follows: side 2: Y11 "Y31 = 0 along ~2 = 0 Y12" ]'32 = L2 ~(~1, 43)

along ~2 = 0

Y13 - 2(33 = 0

along 42 = 0

side 3: Y21 "YI 1 = 0 Y22 - YI2 = 0

along 43 = 0 along 43 = 0

"113- Y33 = ~ 9(41, 42)

along 43 = 0

(9.4.25b)

(9.4.25c)

The solution of eqs. (9.4.25a), (9.4.25b) and (9.4.25c) gives the interpolation polynomials for the tangential vector elements. But the solution of these equations is not unique (Cendes 1991). We shall choose the first column of the matrix F as (Cendes 1991), F1=

,[0 01 0

- 43

42

0

(9.4.26a).

Thereupon, considering the first-order interpolation, one can write

l l , VlIll = [0, 0] Vll = rt^t0 21 21 V21 = [Y0 ' Y1 ] = LI [0,- 431 31 31 Y31 = [% , Y1 I = q[42,01,

(9.4.26b)

and these polynomials, Y! 1, Y21 and Y31 satisfy the equations "t31- Y21 = ~ [ 4 2 , 43] = L1 9(42, 43),

along 41 = 0

S. Subramaniam and S. R. H. Hoole

3 79

711 - 731 = 0,

along 42 = 0

Y21 - 711 = 0,

along 43 = 0.

In a similar way one can show that the matrices, q , F 2 and F 3 satisfy all the equations (9.4.25): F2 - ~

0 0

o1

(9.4.27a)

l

(9.4.27b)

0 -41

aqd F3 =

4! 0 . 0 0 Therefore, for first-order elements, F takes the form F

[

0

-L

0

_

= [ q F2 F3]

=

0

0

L243

0

0

LI43

0

0

L142

0

0

L241

0 - L341 0

- L342

(9.4.27c)

and for the zeroth-order elements F can be derived by setting '1;10= 1;1I, x20 =/:21 and x30 = x31 with the assumption that the field vaiable is constant along the edge of the triangle and equal to the proj~tion'of the field along the edge at fine midpoint:

- t q r2

=

[

0

0

- L342

- LI43

L3;1

el;2

L2~ 3 0

- L241

j

(9.4.27d)

Now we can find a zerohh-order approximation for a two dimensional vector V in terms of the variational parameters x, using eq. (9.4.27d). Combining eqs. (9.4.22) and (9.4.23a) gives, V = VxUx + VyUy= {N} t H -I F ( 41, 42, 43) x, (9.4.28). But {N} = - L "1D {X}. (9.4.9b) Using this fact in eq. (9.4.28) gives, V = - { X } t D t L "l H "I F ( 41, 42, 43) x, =_ L {X}t D t F ( 41, 42, 43) % ,5

380

Chapter 9: Edge Elements

L2(al~ 3 - a3~ I) L3(a2~ 1 - al~ 2) ] I L I (a3~2 " a2~3) = . 1 {X}t L1 (b3~ 2 _ b2~3 ) L2(b ! 43 - b3~ 1) L3(b2~l " bl~2) Ix A L1(c3~2 - c2~3) L2(Cl~ 3 - c3~1) L3(c2~ 1 - c1~2) _ (9.4.29a), which in turn can be rewritten as (Subramaniam, Feliziani and Hoole, 1993)

rVxl

I."'vy J

1

Ll(b3;2- b2;3)

L2(bl;3" b3;1)

L3(b2;l" bl;2) ]

A

Ll(c3~ 2.c2~3 )

L2(cI~ 3.c3~I )

L3(c2~ 1_c1~2 ) (9.4.29b)

ra-Vx" [

J

Taking derivatives with respect x and y, ~ l

ILl(b3b2"b2b3)=.~

t-.by j

L2(blb 3 - b3bl)

L3(b2bI - blb2) Ix

L2(c 1c3 - c3c 1)

L3(c2c I - CLC2)

L1(c3c2 - c2c3)

]

(9.4.29c)

(_Vx'-~ [ I ~Vv/--

-L3 L lA

k~x )

L2A L3A

(9.4.29d)

The right hand side of 2 . O.4.29c). ~ o m e s zero ~cause each element of the ma~ix is zero and this confirms that the @proximation given by eq. (9.4.29b) implicitly satisfies the divergence free condition within each triangle. The curl of V is given by IV x Vl =

='a2--[ L1 L2 L3 ]1:.

(9.4.29e)

Next we shall derive the local matrices corresponding to these elements. Reconsidering the functional used for the eddy current analysis of the previous section (eq. (9.3.7)), L(H) =

o + jco~tH2 dA (9.3.7). Con Now the approximations to the terms (V x H) 2 and Ht.H have to be derived. Considering the first term first,

S. Subramaniamand S. R. H. Hoole

381

(9.4.30)

and the approximation is integrated over an element to get (9.4.31a),

Etl (VxH) 2dA =Aelt xtP whem

4

[

L~

L!L 2 L1L3

]

(9.4.31b) 2 L3L1 L3L2 L3 The approximations of the x- and y- components of H (given in 9.4.29b) can be modified as follows: Hx =.

xt

_~b 3

Lib 3 - ~ b 2 l 0 L2b 1

L3b2

-~b 1

I°

0

1 7

(9.4.32a)

LlC3 -~c2 q ny = -

! xtI

X

0

"L2c3

0 L3c2 -I3c I

L2Cl I 0

(9.4.32b).

Therefore

E:IHt°H dA = ( Hx+

)dA

E = Aelt I:t T 1,1

(9.4.32c)

where (T l,l)i, j I2A2

+ i, j, k = 1, 2, 3, i ~ej, i ~ k and j ~ k,

(9.4.32d)

382

Chapter 9: Edge Elements

2L2 [b21 . b i l ~ 2 + b 2 2 2 ] (T 1,1)i,i- 122~2 i2 + c i I " bil~2 + c i 2 , i = 1,2,3, .(9.4.32e) il=im~3+l, and i2 = il rood 3 +1. We have already observed in section 9.3 that, to impose the source cu~ent term, one has to use the fiJnctional given by: i" t' / [(V x H ) 2 +jmgerH2]dA + /

L(H)=

[V x H - J 0 ] 2 d A

Art

Co nduct"in g medium

(9.3.34) To apply this functional, the integral of (V x H)toj has to be evaluated with the assumption that the constant term jtoj has no contribution in the extremization procedure of this edge element analysis. As before the source term jtz = [Jzl Jz2 Jz3] can be interpolated in terms of its values at the ve~ices of the triangle:

(V x H ) t ' J - - ~ ~t

41 42 43

A

J2 J3 (9.4.33a)

arid r E~'jt (V x H)t.J dA

= Ael0: t Q J

(9.4.33b)

where

Q=-

L 2 L 2 L3

(9.4.33c)

L 3 L2 L3 One can follow the steps of section 9.3.2.3 to impose constitutive relationship, Bnl - Bn2 across the magnetic materials. An open boundary problem with two conductors ca~ying the current in the opposite directions (Hoole 1989), shown in Ngure 9.4.3 is taken analyzed with the open boundary condition H n = 0 along the ~Ntrary boundary. R e sNution region is reduced using the antisymmetric property of the magnetic field with the

S. Subramaniam and S. R. H. Hoole

383

boundary condition Hn = 0 and the distribution of the real part of the H field is shown in the same figure.

j~

/

oo

o

o

-J

J

_Z

3

0

2

5

2

0

1

5

I/I/f"

.

.

.

.

.

.

.

.

.

"

"

.

.

.

.

.

/ f z -

0

}

0

}

I

t

~

,

,

,

,

,

-05

-2,5-2.0

Figure

-1.5

-1.0

-0,5

0.0

0.5

1.0

!.5

2.0

2,5

9.4.4: a. Analysis of the H distribution around two conductors b. Field distribution of the real part of H

384

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

Chapter 9: Edge Elements

Subprogram triangle(x, y, b, c, 1, a); Inputs: x, y = 3 x 1 vectors containing

the x, y coordinates of the ve~ices of the rectangle res~ctively. Outputs: b = 3 x 1 vector containing a vector proportional to the derivative. c = 3 x ! vector containing a vector propo~ional to the derivative. 1 = 3 x I vector containing the length of each side. a = area of the rectangle triangle(x, y, b, c, 1, a)

{

For i - 1 to 3 do

{

} }

iI-imod3+l; i2=ilmod3+l; b[i] = (y[il] - y[i2] ); c[i] = (x[i2] - x[il] ); l[i] = sqrt[b[i] 2 + c[i]2]; a = a + x[i] * b[i];

! eq. (9.4.3) ! eq. (9.4.4) ! eq. (9.4.5) ! eq. (9.4.20)

a=a/2;

Algorithm

9.4.1"

Computing the Differentiation Matrices for

Triangular Tangential Edge Elements

9.4.3:

Algorithms for Zeroth-order Triangular Tangential Elements 9.4.3.1" Computing First-order Differentiation matrices Just as we presented the algorithms for rectangular tangential elements, we give here the corresponding algorithm 9.4.1 for computing the differentiation matrices for triangular tangentiN elements.

9.4.3.2: Computing zeroth-order local matrices Likewise, Algorithm 9.4.2 presents the procedure for computing the local matrix for triangular tangential elements.

9.5 Compatibility and Unisolvence 9.5.1" Compatibility As we mentioned before, the main purpose of using the edge-based finite elements is to improve the direct field solutions by avoiding imposing improper continuity conditions in addition to the proper ones. The trial functions of the node based finite elements force the variational vector parameters to satis~ both tangential and normal continuity and this leads to spurious solutions.

S. Subrarnaniana and S. R. H. Hoole

Subprogram

Zeroth-order

385

triangular tangential I o c a l m a t s ( x , y,

J, P, T 1,I, Q); Inputs:

Outputs:

x, y = 3 x 1 vectors containing the x, y coordinates of the vertices of the rectangle respectively. J = 3 x 1 v ~ t o r giving source J at the four vertices. P = 3 x 3 vector first-order tangential rectangular local element matrix. T !,1 = 3 x 3 vector first-order tangential rectangular tensor element matrix. Q = 3 x I l ~ a l right hand side vector.

External subprogram: triangle zerotb-order tangent triangle loemats(x, y, J, P, T I,I, Q)

{

triangle(x, y, b, c, I, a); for i =1 to 3 do

{

il = i m o d 3 + l ; i 2 = i l m o d 3 + l ; Q[i] = - i[i] * (Jill + J [ i l ] + J[i2])/3;

! eq. (9.4.33c)

![i].* ![i] Tl'1[i'i] = ...... 24a ..... ( b [ i l ] * b [ i l ] - b[iI]*b[i2] + b[i2] * b[i2]) +

i[i] * ![J.] 24a ( c[iI]*c[il] - c[il]*c[i2] + c[i2] * c[i2]); !eq. (9.4.32e)

F o r j = 1 to 3 do

{

P[i,j] = -

a

;

! eq. (9.4.3Ib)

k=0; while ((k=i) or (k=j)) do k:=k rood 3 +I; T l,l[i,i] .....![~]* ! [ i J ( b [ i ] , b [ k ] . 2*b[i]*b[j] 48a + b y ] * b[k] - b[k]*b[k]) + ![i] * l [ i ] 48a (e[i]*e[k] - 2*c[i]*eU] + eU] * e[k] - e[k]*e[k]); ! eq. (9.4.32d)

} Algorithm 9.4.2:

Computing First Order Local Matrices for Triangular Tangential Edge Elements

386

Chapter 9: Edge Elements

The elements are said to be compatible if they satisfy the proper continuity conditions. For example, scalar potentials are continuous in both senses; that is vectors E and H represented by the gradient of a node-based scalar potential are tangentially continuous as required and m"e not in the normal direction. Therefore node based finite elements are said to be compatible for scalar potential solutions. But the same node based finite elements are not compatible for direct vector solutions. The tangential vector elements which are developed in sections 9.3 and 9.4 are compatible for vector fields that are tangentially continuous like E (and H in the absence of surface currents), but not for those that are normally continuous like the magnetic flux density B (and D in the absence of surface charges). In other words t h o ~ edge based finite elements cannot be applied for flux density solutions and it is one of the disadvantages of edge based finite elements. In general the elements that satisfy tangential continuity are referred to as 1-form elements; those elements that satisfy normal continuity are known as 2-forms and those that satisfy both are called zero-forms (Whimey, 1957; Bossavit, 1988; Cendes, 1991 ). In sections 9.3 and 9.4, tangential vector finite elements were developed to satisfy the tangential continuity of the vector fields m be represenmd, but we have not yet proved that they give a unique solution in applications.

9.5.2:

Unisolvence

In finite element analysis, only linearly independent trial functions lead m unique solutions (Hoole 1989; Cendes 199I). The term unisolvence refers to the linear independence of trial functions. Therefore, in addition to compatibility, one has to look for the linear independence of the trial functions. A solution H a of the curl equation, V x H = J need not be unique because H a + V~palso satisfies the curl equation for any scalar function ~. Linear dependence of the trial functions may give solutions H ~ 0 and H ~ V~ to the homogeneous curl equation, V x H = 0 (Cendes, 1991). Consider a linear variation of ~ within an element. V~ has a different constant value on each side of an edge, and therefore the vector field can ~ represented by a zeroth-order tangential element. The curl of a zeroth-order tangential element is a constant within each element (eq. 9.4.29e). With first-order nodal interpolation, a function ~ can be approximated within a triangle as (Hoole 1989) 3 = Z ~Pi ~i (9.5.1a) i=l and its gradient 3 3 V~ = u x Z bi~i + Uy E ciq~i (9.5.Ib) l=l i=I with three variational parameters ~1, ~P2 and ~3 at the vertices of the triangle. ~ r e e independent variables are required to represent H = H a + V~p such that V x H is a constant within the element. Nedelec (1980) and Cendes (1991) have proved

S. Subramaniam and S, R. H. Hoole

387

that a complete zeroth-order tangential element generated with the three terms, one along each side, is complete. Following the same steps one can show that the rectangular elements developed in section 9.3 require four independent variables to represent H = H a + V~ such that 7 x H is a constant within the element.

9.6 Normal Edge Elements The magnetic and electric flux densities B and D axe continuous along the normal direction of material interfaces with jumps along the tangential directions (See section 1.3.1). If H is chosen as the state variable, solved with tangential elements and then, from such a solution of H, the flux density B is computed using the constitutive equation B = ~H, then X7.B = 0 holds only in weak form (Bossavit, I988). The state variable B satisfies Faraday's Law in strong form and Ampere's theorem in a weak sense. One cannot expect both the divergence and curl to be

a

s

#

#

a

B

o

..

.

*

*

*

3

l

a

~

f

t

J

o

~

t

.

,

.

¢

,

,

t~

It

tf

'

'

*

~

g

|

{

'

,

*

t

t

•

~,.. .

t

t

t

t

~

%

'~.

/ / ' * ' ~

~

.

.

--

.

.

.

.

.

.

.

.

.

. .

.

.

Figure 9.6.1: A Conductor in Front of a Block o f Steel: B Field from the Computed H Field

.

"

388

Chapter 9: Edge Elements

satisfied in strong form (Hoole, 1989). This is evident in Figure 9.6.1, which shows the flux density B computed from the magnetic field intensity H computed as the finite element state variable. ~ e figure depicts a system consisting of a conductor along the center-line and to the left of a rectangular block of steel. Thus one has to select the state v ~ a b l e depending on the application at hand. Keeping this in mind, in this section we develop the trial functions for the vector elements that satisfy the normal continuity along inter-element edges. 9.6.1: Zeroth-order Triangular Normal Elements To derive the trial functions for normal vector finite elements (Fig. 9.6.2), we shall express a two dimensional vector in terms of the normal unit vector as (Cendes 190 I) V - VxUx + VyU3~= { T } t V T (;1, ;2, ;3) (9.6.1a) Next the function V is approximated in terms of the variational parameters (v l, v2, v3) along the edges of the triangles as shown Fig. 9.4.3: vT( ;1, ;2, ;3) = H -1 F ( ; I , ;2, ;3) x (9.6.1b) where v t = (v!= (Zl), v2= (x2), v3= (x3)) (9.6.1c) for the zeroth order element and F ( ;1, ;2, ;3) is a matrix with the desired vector trial functions. But V n can ~ expressed in terms of V T as (using eqs. (9.4.14a) and (9.4.17))

[Vtl] Vn =[vt2/= IN 1. V = {N}. {T}tVT=L'ISHVT t.vt3A substituting eq. (9.6.1b) into eq. (9.4.14a) V n = L-1SH V T = L ' I s F ( ;1, ;2, ;3) v. Following the same steps, one would end up with the same F: V = VxUx + VyUy = {T} t H -1 F ( ; I , ;2, ;3) v But substituting {T} = - L "1E {X}. in eq. (9.6.2a) gives, V = - {X} t E t L " I H - 1 F ( ; I , ; 2 , ~ 3 ) v, 1

v3

(xl,yl)

v2 112,1/2,0) (1/2; (0,l/2,1/2)

(x2,y2) vl

{1~

(x3,y3)3

Figure 9.6.2: A Zeroth-order Normal Vector Finite Element

(9.4.14a) (9.6.1d) (9.6.2a). (9.6.2b)

S. Subramaniamand S. R. H. HoNe

389

=- 1 {x}t Et IF'( ~I, ~2, ;3) V, A 1, {X}t I Ll(a3;2- a2;3) =-A L L1(c3{2"c2{3)

L2(al~ 3 - a3~l)

L3(a2~ 1 - al~ 2) [

L2(Cl~ 3 - c3{ 1) L3(c2~ t - cI~2) Iv

Ll(b2~ 3 - b3{ 2) L2(b3~ ! - bl~ 3) L3(bi~ 2 - b2~ 1) _ (9.6.3a), which can be rewritten as L1(c3{2- c2{3) L2(Cl;3- c3;I) L3(c2~1 " cl~2) lv

LI (b2~3- b3~2) L2(b3; 1 - bl;3) L3(bl~ 2 - b2~ l)

[~::~] =-

J

(9.6.3b) Taking derivatives with respect to x and y, 7

L3(c2bl " Clb2) Iv Ll(b2c3 - b3c2) L2(b3c! - blC3) L3(blC2 b2cl)

9Vv

ay )

.1 [ LIA ='A 2

-LtA

L2'5 L3L~ 1 -ha -~a v

J

(9.6.3c)

faVx-,~ o

0

(9.6.3d)

t.ax ) The irrotafional condition is implicitly imposed within each triangle by eq. (9.6.3d). The divergence of V is given by (xl,yl) x30 i ,/~x~ x21

v2 7:3

xlO '2

(x2,y2) Figure 9.7.1: A

(0,I/2,1/2) ~l=01'v1!h

'-~"_'~-,~,. 20 "ell (x3,y3)

first-order tangential vector finite element

390

Chapter 9: Edge Elements

(V'V)

3Vx u 3Vy

-Ux~+

y~;

L2'5 =-A

L3A Iv.

(9.6.3e)

-LIA

9.7: Higher Order Elements 9.7.1: First-Order Tangential Elements When we were deriving the trial functions for zeroth-order tangential elements in eq. (9.4.27c), the inte~olafion functions were derived for a linear variation along the edges and then, along an edge of the triangle, we set the vector to a constant, this constant being equal to the pr~ection of the vector along the edge at the midpoint. The interpolation polynomials derived in eq. (9A.27c) satisfy the compatibility condition which is known as the groper continuity condition, but do not satisfy the unisolvence condition (Cendes, 1991). First-order complete tangential elements need eight terms to satisfy the unisoIvence condition. Therefore two terms have to be added to eq. (9.4.23b). In other words, the matrix F of eq. (9.4.27c) has to be augmented by two additional columns. These additional basis functions need to contribute to the tangential components of the field along the edges of the element (Cendes, 1991). Cendes (1991) added two extra trial functions which do not contribute to the tangential components of V(x,y) in such a way that the compatibility condition on the field and the linear variation of the tangential component are not affected by the addition of these new parameters. One of these parameters is perpendicular to the first edge and the other is pe~endicular to the second edge as shown in Figure 9.7. I. Even though these two parameters interpolate the normal along these two sides, one should note that the interpolation functions do not totally inte~late the normal components of V(x,y). The advantage of using these elements is in directly imposing the continuity relationship Bnl = Bn2 without adding any integral terms as mentioned in section 9.3.2. The new variational parameters are ~t = (~1 = (~10,,C11), 't:2= (~20,~21), '1:3=(~30,~31) , Vl, v2)

O.7.1a)

and the interpolation matrix is formed by augmenting the matrix F by two additional columns F4:

r

= [ =

r2 r3 0 0 - LI; 2

0 - L1; 3 0

L2;3 0 0

0 0 - L2;I

0 - L3~2 L3;I 0 0

0

4~2;3 0 0

0 -l 44143 0

_

(9.7.1b)

S. Subramaniamand S. R. H. Hoole

St =

391

b3L142

c3L142

b3L 1c2

c3L 1b2

-bzL143

-5L143

-b2Llc3

-~Llb3

blL243

Cl L243

bl L2c3

ClL2b3

-b3L2;1

-c3L241

-b3L2cl

-~L2bl

bzL341

c2L341

b2L3Cl

c2L3bl

-bL3c2 -c b2

-hL3 2 4b 1;243

Wt=

4c 14243

4blC243

4Clb243

+ 4ClC342 + 4Clb342 4b2~143 4c24143

462c143

4c2b143

+

+

4blC341

4Clb341

Figure 9.7.2: The Matrices S and W

One can clearly see that the function in column 1 of ~ is zero along edges 2 and 3 of the triangle and the other one is zero along edges l and 3. Substituting into equation (9.4.28) V = VxUx + VyUy= {N }t H -1 F ( 4t, 42, 43) ~, (9.4.28) and using {N} = - L -! D {X} (9.4.9b) gives Vx - S1 x , (9.4.10) Vy]2 and He derivatives

[vxI =

Oy

-

W~,

(9,4.1 !)

392

Chapter 9: Edge Elements

where the matrix S is given in Figure 9.7,2, That the field is divergence flee can be simply verified as before and the curl can be derived from eq. (9.4. I 1). For these inte~olations, the local matrices are given by: p_

T-

A * 4*A*A ~1"I1 I1"11 11"12 11"12 I3"11 13"I1

11'1I 11"1I 11"12 11"12 11"13 11"13

11"12 11"12 12"I2 12"I2 13"12 13"12

11"12 11"I2 I2"12 12"12 13"12 I3"12

11"13 I1"13 I3"12 I3"12 13"13 I3"13

11"13 11"13 12"13 12"13 13"13 13"13

0

0

0

0

0

0

0

0

0

0

15 30 30 15 15 I5 6 6

15 30 30 15 15 15 6 6

15 15 15 30 30 15 3 6

~1 1! I2

II 11 2

!1 11 12

12

12

12

13 0 -1

13 1 0

13 -1 1

!

720*A*A

,

1

O0

~ 12

30 15 I5 15 30 30 6 3

6 6 6 3 6 6 2 1

0" 0 0 0 0 0 1 12 2

!2

*

30 15 15 I5 15 30 6 3

T10=

0 0 0 0 0 0 2 ~ !2

6

15 15 15 30 30 30 3 6

and T 11 = A[D b T D b] + D c T D c] where D b and D c ~ e diagonal matrices with The diagonal terms of D b = {b3* 11, -b2* 11, b I * 12, -b3* 12, b2* 13, -bl*13, bl*4, b2"4}

3" 6 6 6 3 3 1 2

S. Subramaniam and S. R. H. Hoole

393

The diagonal terrns of D c = {c3" 11, -c2" 11, c 1* 12, -c3" 12, c2" 13, -c1"13, c1"4, c2"4} As an application (as much as demonstration of the theory), we revisit the example of Figure 9.6.1 where we analyzed a system consisting of a conductor to the left of a block of steel - - the It field distribution in front of a block of steel is analyzed and the B field is obtained from the constitutive relation B = p:H. The H field distribution is shown in Figure 9.7.3. What was presented in Figure 9.6.1 is the B field distribution computed from this using the constitutive relationship. This confirms that Ampere's law and the Faraday's law cannot be satisfied at the same time. Therefore one has to make a choice between the two.

, , / , , , 1 , , , . . . . _ - - . - . . - . , . . . , .

~~ I

l

i

l

,

,,

-

,

'

I

\

\

\-,.--.'--

.

.

.

.

.

..

,.

,.

,

'

'

,

,

,

|!

!I

|I

II

I

I

t

!

e

B

t t

I t

I t

a t l e

'

\

,

'

'

'

'

'

/

t

t

I

p

,

,

,

,

,

Figure 9,7.3: A Conductor in Front of a Block of Steel: The H Field The B Field Computed from this is in Figure 9.6.1

10

S. Ratnajeevan H. Hoole

SYNTHESIZING DEVICES THROUGH FINITE ELEMENT PERFORMANCE MODELING

10.1 Inverse Problems: Synthesis and Analysis The analysis of electromagnetic systems is now a well developed and demonstrably accurate art. As shown in Fig. 10.1a, when a device is given, using such well known methods as finite elements or boundary elements (Hoole, 1989), it may be analyzed for its electromagnetic fields; the fields then are used to predict the device's performance such as defined by forces, voltages, flux linkages and so on. For example, static, two dimensional magnetostatic fields, governed by the Poisson equation relating the permeability It, vector potential A and current density J, -LV2A = J

(I0.1.1),

may be solved for A, given the distribution of g and J everywhere in the solution domain. Throughout this chapter, we will use this equation as an example to demonstrate the solution of inverse problems. Once we solve for the vector potential A, its curl may be used to find the flux density B and from it, forces, stresses and other practical pertbrmance measures of interest may be evaluated Returning now to the issue of the direct solution, real engineering rarely calls upon the engineer to solve the direct problem only. Real engineering, indeed, is not based on devices, but rather on needs, in turn described by the performance of the device. For example, the question in engineering is not "What force will this motor produce ?," but rather, "What motor will give me this force ?'" This is depicted in Fig. I b and is the classic Inverse Problem.

S. R. H. Hoole

DEVICE

395

The Direct Problem

PERFORMANCE

a

PERFORMANCE

The Inverse Problem

DEVICE

b Figure 10.1.1: Direct aa~.d Inverse Problems The solution of the inverse problem, often not formally stated to be as such, is done iteratively as shown in Fig. 10.1.2. Historically these iterations were carried out by cut-make-and-test operations which rook months for each iteration or test. As each design is found wanting, a new one is tried, but the changes in the critical, device-descriptive, design parameters are often a matter of guess work. Where it is not, much experience is required of the designer to know correctly what parameter is to be changed and by how much. In time, sophisticated analysis methods such as the finite element method allowed us to test quickly the performance of a design without making it. Thus in Fig. I0.I.2, the testing,

L I .....Pre.i2 ..

[ Performance [ - [ by analysis or lab ![ test

Compare with desired Performance

Change Design Down Object Function Figure 10.1.2: Iterative Solution of the Inverse Problem

396

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

instead of being in the laboratory, is now on the computer through analysis. Such testing is accomplished in a matter of hours. And yet, the iterations from test to test are painstakingly slow because there is no systematic method of changing the parameters and the finite element mesh has to be generated afresh for every design change. In recent years there has begun to emerge a methodology for formalizing this iterative process (Oristaglio and Worthington, 1980; Pironneau, 1984; Gitosusastro, Coulomb and Sobonnadiere, 1989; Istfan and Salon (1988); Salon and Istfan, 1986; Hoole, Subramaniam, Saldanha, Coulomb and Sabonnadiere, 1991; Vande~laats, 1983; Hoole and Subramaniam, 1992; and Weeber and Hoole, 1992a, 1992b). Although imperfect, it is yet an important beginning. This chapter is about explaining this important development to the reader, to the extent, it is hoped, that he or she might be able to use the method to synthesise to advantage devices from required ce criteria. Indeed, we describe here only the powerful gradients based technique of optimization developed in finite element analysis in electromagnefics by Salon and Istfan (1986), Istfan and Salon (I988) and Gotosusastro et al (I989) because, in our experience, it is the most powerful of the methods available. Be it noted however, that there is an independent development based on the statistical optimization method called simulated annealing (Simkin and Trowbridge, 1992), and another based on the search method called the evolution strategy (Preis, magele and Biro, 1990) or the genetic algorithm. These latter two classes of method depend heavily on generating a finite element solution J;or each of a large set of device descriptive parameters, while the gradients based technique requires extensive coding to compute the gradients as each parameter is changed. Thus the advantage of statistical and search strategies is that, albeit with extensive computation, optimization may be accomplished with very' little new coding beyond that already available through standard finite elements packages and off-the-shelf optimization packages.

10. 2

The State of the Art of Inverse Problem Solution in Finite Element analysis

Recent research efforts in electromagnetic field computation have yielded the important, formalized technique of inverse problem solution in electromagnetics. It follows the flow-chart of Fig. 10.1.2, with the performance prediction by making and testing being now replaced by analysis. Classically, in finite element field computation in electromagnetic systems governed by eq. (10.1.1), the solution region comprising the device and its surroundings is discretized into elements and the vector potential A is computed from the matrix equation [P]{A} = {Q} (10.2.1) where, for a first order triangular finite element mesh, the matrix [P] and the vector {Q}, for a traingle I23 with coordinates (xl,Yl), (x2,Y2), and (x3,Y3), are assembled from the element matrix, restating eqs. (1.5.49) and (I ~5.50):

S. R. H. Hoole

397

-IA / - ~ e bzbl+C2Cl

Lb3b I+c3c 1

b2 2 2+c 2

b2b3+c2c3I 2 2 b3b2+~c2 b 3+c3J

(10.2.2)

and element column vector { ~ } J ae -3

{:I~

(10.2.3)

Here, a constant current density J has been assumed in each triangular element and Ae is the area of the triangle, which when numbered counter-clockwise is given by the determinant

A = 2Ae =

1

xI

Yl

1

x2

Y2

1

x3

Y3

In eq. (10,2.2)

,

bi = X

1

c.=--A

Yil " Yi2 x i 2 " Xil

)

)

(10.2.4).

(10.2.5) (10.2.6)

and i,il and i2 are cyclic modulo 3 (see eqs. (1,5.31) and (1.5.32)). That is, they assume the values (1,2,3) or (2,3,1) or (3,1,2), The unknown {A} is a vector of the vector potentials at the nodes of the mesh (Hoole, 1989). Thus the procedure is a direct ana|ysis procedure, Given a device, we may compute the vector potential and, from it, the magnetic flux and thereby predict the performance of the device. Subsequently, this analysis was extended to the inverse problem using an object function F, For example, if the goal is to have a certain amount of flux through a coil, this flux being the difference in vector potential between the two conductors of the loop, we will define the error based object function

1

F = ~ (A1-A2) 2

(10.2.7)

Here 1 and 2 are the two conductors (represented by points in the two-dimensional system). In three dimensional extensions to the research, the object function would be an integral of the vector potential round the loop so that it would be a sum of terms as in eq. (10.2.7), but weighted by the length of each interval of numerical integration. When our performance requirement is satisfied, the object function, as shown in Fig. 10.2.1, will be zero and at all other times it would be positive. Thus the satisfaction of our performance requirement has been reduced to that of

398

Chapter.......!0:Synthesizing Devices through Finite Element Performance Modeling

F

t I

I

I

I

I

I

i .

! I

P

Po Figure 10.2.1: Object Function, Gradient and M m i m u m minimizing this t;anction. We therefore start with the best guess of the parameters of description of the device that will yield the performance. The so called parameters of description could be for the example of the flux out of a coil, the permeability of the material within the coil, the dimensions of the coil, and the current density in the windings. Since these values are initially guessed, the co~esponding value of the object function would be ~sitive rather than zero. To move towards the minimum of this object function, we must change the parameters of description of the system against the slope of the object function F as seen in Fig. 10.2.1, so that we move to the trough or minimum: i+I p

i 3Fi = p ~k Op

(10.2.8)

The different gradient methods of optimization such as Newton's, conjugate gradients and so on (Pironneau, 1984; Gitsusastro et al, 1989; Vanderplaats, 1983), rely on this gradient and differ in the choice of value of the constant k. To this end, the slope of the object function must be computable. In the early stages of the development of the method, it was computed by finite difference (Pironneau, 1984). That is, every parameter p was varied infinitesimally at a time and F was recomputed through a field computation for the new set of parameters and the ratio of the changes in F and p was the gradient with respect to each parameter p. However, this was costly since, for m parameters of description, at

S. R. H. Hoole

399

each iteration of the design cycle, the electromagnetic fields had to be computed m+l times. In time, there came to be an important advance in the method, whereby the gradients were computed without a second field solution (Gitosusastro et aI, 1989; Salon and Isffan, 1986; Hoole and Subramaniam, 1992). This was accomplished using the differentiability of the finite element and boundary element matrices (Hoole and Subramanaiam, 1992; Coulomb 1983). To determine the slope of F with respect to a parameter p, we first note that for the example object function of eq. (10.2.7), OF= (A1-A2) (aA1 3A3p2"~I (10,2.9) )' For programming on a computer, it is more convenient to deal with general expressions. More generally then, if we regard the object function F as being a function of a parameter p and the finite element field solution vector {A} whose components are the values of the vector potential A at the mesh nodes: F = F(p,{A}). (10.2.10), then dF OF OF 0AI OF 0A2 OF 3A3 d p - 3p +3A 1 3p + 0A2 3p + 3A 3 0p +''" OF OFt 0{6} = 0 7 + O{A} 3 p (10.2.11) The term a F,'ap is easily computed from the known form of the object function. To see how the expression evaluates, turning to our example of eq. (10.2.7), it is zero since no parameter explicitly enters it. The term OF/O{A} is also easily computed from the known form of F. For our example, it is given by OFt J "OF OF OF ... 0 2 ] > O{a}-[OAI 0,

3,

"* "n) ~q')(AI-A2) 0(A l-A2) O(A I-A2)

t;

= (Ai-A2) {1 -1 0 0 ...0}

sq(AI-A2)"l

(10.2.!2)

Thus,

3F 3p - 0 +

(AI-A2) {1 -1 0 0 .,.0}

= (AI_A2) i(3A1 A2) \ 3p " O3p

(10,2.13)

400

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

as required. Thus, to complete the computation, we need 0Ai/0 p for i =I and 2. This is obtained by differentiating eq. (10.2. I): {A} + [P] 3tA1 ~ (10.2.14). 0p Op = Op Unfortunately, we need to compute the gradient of every nodal potential whereas only two values are required. Now, since the matrix [P] and vector {Q} are expressed in terms of the physical geometry of the model, their derivatives with respect to any physical parameter may be expressed just as [P] and [Q] are. The detmls of this differentiation are taken up in Section I0.3 next. Therefore, noting that {A } is already comput~ from eq. (10.1.1), the solution of the equation [p]~ ~ ~ ap = ap - O[PlsaOp t.-, (I0.2. I5) will give us the gradient of every nodal potential Ai. In solving eq. (10.2.15), because [P] is sparse and symmetric, it is conventional wisdom to use a conjugate gradients algorithm to solve for {A}. This is usually found to be more and more effective for the commonly encountered matrix sizes larger than 1000. However, in solving inverse problems of this nature, the Cholesky scheme (Hoole, 1989) is to be preferred. For, it is m be noted that the coefficient matrix of eq. (10.2.15) is the same as that of eq. (10.2.1). As such we use the Choleky decomposition method in solving eq. (10.2.1), so that in solving eq. (10.2.15) for the many gradients with respect to all the different parameters p, each vector gradient might be computed quickly with forward elimination and backsubstitution only (Hoole, 1991, 1992). That is, in solving eq. (10.2.1), following Cholesky's scheme, we first factorize [P] into a lower triangular factor [L] and upper triangular factor [U] (which happens m be [L] t because the finite element matrix [P] is symmetric for the problems under consideration): [el = ILl[L] t

(10.2.16)

Thus, P{A} =ILl ILl] t {A} = {Q} may be solved by forward elimination from the equation [L] {z} = {Q} where ILl t {A} = {z}

(10.2.17) (10.2.18) (I0.2.19),

and now, having computed {z}, the unknown {A} may be compumd from eq. (10.2.19) by back-substitution. It is noted that most of the work is in the factorization of [P] into its upper- and lower- triangular factors. Once it is computed, the work expended in forward elimination and back-substitution is trivial. Thus, eq. (10,2.15) can be solved tor the several gradients with respect to the parameters rather quickly. Further, this approach allows us to use parallel computation techniques to speed up the synthesis further (Hoole, 1991). Please see the accompanying Chapter 11 on parallel computation for a discussion of Cholesky's solution scheme.

S. R. H. Hoole

401

10.3 Differentiation of the Finite Element Matrices

10.3.1 Background We have seen that the solution of eq. (10.2.15) gives us the gradients that are so important in determining the direction in which the parameters ought to be changed for us to reach the minimum of the object function, at which point our permormance requirement is realized. But to solve eq. (10.2.15), we must first set it up. And to this end, the gradients 3[P]/Op and 3{Q}/Op must be computed. This section is about how this is done. In describing an electromagnetic field problem, there are three possible kinds of critical descriptive parameters p: i. The material/a that occupies certain parts of the device, ii. The excitation of the system given by the current density J in conductors and iii. Geometric parameters such as critical dimensions of the device. Thus, in performing the diffi3rentiations of [P], which might involve material and dimensions, and {Q} which might involve excitation and dimensions, we must be able to allow for any of these parameters for programming generality,. If this is done, then our computer program may have different subroutines (or procedures or functions) that deal with the different cases and can be called as appropriate. At the same time, we shall also treat the second derivatives of these matrices required for high order optimization techniques. 10.3,2 Differentiation With respect to a Material Parameter p = g

This is one of the simpler cases. An examination of eq. (I0.2.2), shows the material p. of an element is involved only in the first scalar factor g" I and not in the matrix factor that appears next. Therefore clearly

a~t

--

Fb~+c~ hib2+Cl ~ bib3+Clc3-1 Ae[ 2 2 2 b2b l + ~ c I b2+c2 b263+c2~ bt

b3 bl +~Cl

b3

2 b 23+¢3J

bl~+Cl~

bt b3+Cl~]

(10.3.1)

and

3~Pe ] 3t.t2

rb2+c~ -2

b2bl+C2Cl g

hb3bl + ~ c I

b 22+c22

b 2 b3 +c2 ~ ] 2 2 b3 b2 +c3 c2 b 3+c3J

(10.3.2)

Similarly, the righthandside column vector {Q} involves no material g and therefore:

402

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

(10.3.3)

~0

3g a2Qe Og2

(10.3.4)

-0

A note of caution is in order. In a synthesis problem where there are many regions involving different steels and conductors, air and so on, let us suppose that what we seek is the material g that needs to be used in a certain part of the device. Then the parameter p that we seek is the permeability of that part. As such, the derivatives with respect to g for an element matrix arising from a triangle within that part is as given by the above equations. However, the derivatives for matrices outside this region are zero since their pe~eabilities are not being varied so as to make the object function go down - - throughout the iterations of Fig. I0.1.2, they would be at fixed values.

10.3.3 Differentiation with respect to an Excitation Parameter p = J Differentiation with respect to a current density would be required when we are trying to determine the amount of current that would be required to excite the system. We note first that the expression fbr [P] involves no current and that, assuming a constant current density in each triangle, the expression for {Q} has a factor J appearing in it. As such

0p

e ~=o

e

Oj2

(~o.3.5)

(I0.3.6)

-0

Af:}

(10.3.7)

o

(~o.3.8)

3J - - 3

-

1

10.3.4 Differentiation with respect to a Geometric

Parameter

10.3.4.1 Division into Subproblems Differentiation with respect to geometry is much more complex than with respect to material or excitation. We therefore seek to do this by dividing the problem of differentiation into subproblems so as to simplify the process, as well as to m ~ e it suitable for computer programming. In particular, we will seek to reduce

S. R. H. Hoole

403

i

ii

111

iv

Figure 10.3.1: Splitting an Arbitrary Parametric Change into x and y Changes differentiation in any direction to that in the x~direction only and this again, we shall classify into three different types of movement with respect to a triangle.

10.3.4.2 Derivatives in Any Diroction Geometric changes may involve changes in dimensions in any direction. For example, in Fig. I0.3,1, the side marked to be of length p of a triangular element may be along the edge of an object whose length is a critical description. Thus, as that length is optimized, p would change. Because the edge of that object is not along one of the principal axes, the derivatives of the matrix [P] and vector {Q} are more difficult to obtain since p does not appear explicitly in the expressions for these element entities. This analysis is therefore best done through classification into x and y changes since these appear explicitly in the equations for [P] and {Q }. The general case of a movement 6p in any direction in a parameter p may handled for dif~3rentiation by writing d [ ] d[ldx ~ = dx ~ + dy @ (103,9)

y

X-y

x

Y=

-x

Figure 10.3.2: Deriving y-coordhnate expressions from x-coordinate Expressions

404

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

Thus, as shown in Fig. 10.3.1, the actual change in parameter from Fig. 10.3.1 i to ii is handled by two conceptual changes - - from i to iii involving a change in y only, and the change from i to iv involving a change in x only. The terms dy/dp and dx/dp are indeed the s ~ a n t and cosec respectively of the angle between the direction p and the x-axis thus, from expresions for the derivatives with respect to x and y, that with respect to p may be computed. Therefore it suffices for us to consider the derivatives with respect to x and y only, that with respect to any p being obtained from these two derivatives, depending on the actual direction of p. A further simplification is possible if we allow for the fact that the expression for y may be obtained by manipulating the expression for x (or vice versa) using the anti-symmetric relationship between them. To see this, referring to Fig. 10.3.2, consider two coordinate systems (x,y) and (X,Y), the latter being 90 degrees away from the former. Since expressions in terms of x and y are independent of the actual alignment of the axes, to obtain a derivative with respect to y - X, we take an already worked out dertivative with respect to x, re-write X Ior x and Y for y to get the derivative with respect to X and then substitute, looking at Fig. 10.3.2, y for X and -x for Y.That is, once we work out by hand a derivative with respect to x, the derivative with respect to y may be obtained from that with respect to x by substitution and sign changing of the differentials to allow for right- and left- handed systems.

l & & 4.3 Nodal Movements We have seen that it suffices to confine ourselves to obtaining the derivatives with respect to x only, that with respect to y being obtainable from this. And this is what we shall do. That is 8p = 8x. To determine the derivative with respect to x, we must resort to three further classifications. In classifying changes in a triangular element as a parameter changes, we must allow three types of changes: W

D

~C

A

tl

Figure 10.3.3: P a r a m e t r i z a t i o n of a Rectangular Device

S. R. H. Hoole

405

1, 2 or all three vertices of a triangle moving as the geometric parameter changes. Consider Fig. 10.3.3, where a rectangular region ABCD exists within a device. This might be for instance a conductor, whose position and size need to be determined within a system so as to yield a certain force somewhere. This has therefore received 4 geometric parameters of description: location (u,v) of the lower left corner, and height h and width w. In Fig. 10.1.2 depicting the design iterations, it is these that will be varied from computation to computation. Let us say that we wish to get derivatives with respect to the width; that is 8w = 8p = 8x. The derivative itself will depend on how we change the finite element mesh as the parameter changes. Let us say that as the width is changed, the mesh is changed for the next analysis of the loop of Fig. 10.2.2 by shifting the edge BC of Fig. 10.3.3 to the right by the required amount 8w. Triangles 2 and 4 of Fig. 10.3.3 therefore will have only one of their vertices moving as BC moves right. This type of change is depicted below in Fig. 10.3.4 from i to ii. On the other hand, triangles 1 and 3 having two of their re,ices on the edge BC, will have two vertices moving. This is as shown in Fig. 10.3.4 from i to iii. Thus the changes in the element matrices will be different for these two different types of changes and have to be accounted for differently. Similarly, when computing the derivative with respect to the location parameters u and v, the mesh for the next iteration would be unchanged within ABCD since the whole rectangle shifts to a new location and all triangles within the rectangle, have all three of their vertices moving together by the same amount. This is depicted in Fig. 10.3.4 from i to iv. There being no change in triangle shape or area, we would expect a zero derivative of the matrices [P] and {Q}, but

& x coord = p

(0,0)

i

&

(0,0)

ii &

(0,0)

iii

(0,0)

iv

Figure 10.3.4: Distortion of a Triangle i: Original Triangle ii: D e Vertex Moving iii: Two Vertices M o v m g iv: All Three Moving

406

Chapter I0: Synthesizing Devices through Finite Element Performance Modeling

for completenes we will treat it as a separate case. Again, a fu~her classification is necessary for the situations where 1 vertex is moving or 2 vertices are moving. For the one moving vertex may be the first, second or third vertex of the triangle. Similarly, when two vertices move, it might I and 2, 2 and 3 or 3 and 1. But we will confine our analysis to only vertex 1 moving when one vertex is moving, and vertices 1 and 2 moving when two vertices move. The other general situation can always be obtained by substitution of subscripts. That is, when 1 vertex moves, say in the x direction, dp = dx 1 where we have conveniently said that the moving vertex is the first vertex. Similarly, when 2 vertices move, we may set dp = dxI= dx2 where the moving vertices are conveniently taken to be the first two. Proceeding thus to perform the differentiation on the expressions Ibr [P] and {Q} we have, skipping the working out of these final expressions which might be easily verified by the interested reader: One Vertex Moving, p = Xl

d ~ = 2 dA dxl ~ =(y2.Y3) = blA

(10.3.I0)

d2A

I 3P e - ~ b l 3x!

~+

2~

0 c1

Cl

-Cl 1

2c 2 - ~ + ~ /

-c 1 - ~ + 5 - 2 5

3x 2 - 2~t

dx 1 = 3

! i

/

(I0.3.12)

A (10.3.13)

dxl

65° - o

(!o.3.15)

Two Vertices .Moving, p = Xl = x2 dA

dxl

- 2

= - (Yl-Y2) =" b3A

(10.3.16)

S. R. H. Hoole

d2A

407

d2A

dx:-2

(10.3.17)

-°

a,.:-

L- 5

3x 2

(I0.3.18)

Pe + 1c:-c2

2g

dQe J [il?dA dx I - 3 dx 1

0

c3

il

(10.3.19)

(10.3.20)

d]~e -0

(10.3.21)

10.4 An Example and Mesh Considerations We have seen how, in inverting an electromagnetic performance to find the corresponding device, starting with a guess of the descriptive parameters, they may be iteratively changed using the derivatives that have been worked out until the bottom of the object function and, with it, the design realization, are achieved. However, there are practical ~W.cts to be considered carefully. Turning to Fig. 10.4.1, let us suppose that the width of a rectangular region is an important parameter that is being determined so as to yield a certain performance. How then is the mesh to be changed as an increase in p is dictated by the optimization ? As shown in Fig. 10.4.1, this may be done in one of many ways, some of which are: i. The rightmost edge of the rectangle may be moved, so that only triangles 6, and 12 will have two vertices moving while triangles 5 and 11 will have one vertex moving. This is, however, not advisable since the mesh distortion will be heavy if a large change in p is wa~anted, although the simplicity of the scheme is that only a few triangles will undergo changes.

408

Chapter I0: Synthesizing Devices through Finite Element Performance Modeling

A

p

[~

A

2 .......

B

p

A

A

B

p

p

B

B

Figure I0.4.1: Distribution of Parametric Changes

S. R. H. Hoole

409

OB,,~CT FUNCTI~ ~

6-

~°

its G ~ I ~

ME~HI~

!, __/ ~J0

. . tj j

_

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

SJ5

........................... 6..'~"'"'"'".... " ............

p~a, mt~ ~lue.

OI~ECT F~-WCTION ~

6-

(I~LMIZJ~TION with ~

¢oe ~

~

~.s'""

'''~............

~.e'

(~¢ p o l e ~

its (~IENT For GEOI~.,cf OPTIMIZ~TI~ w|~ ~

!-

~I~

i

5~8

5.5

p

~

G.e

6.5

7.0

v~lue - I j - ~ x ~ o¢ pole ¢ ~ e

Figure 10.4.2: Object func~on and Gradient a: I~rge Jumps ~ Obj~t F!&nc~on from Crude Mesh

b: Small but Numerous Jumps in Object Function from Fine Mesh

410

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

Figure 10.4.3: Meshes Corresponding to Figures 10.4.2 a and b

S. R. H. Hoole

Line of

411

Symmetry

pn

Fig. 10.4.4: Problem and Parametrization for Constant Flux Density in Airgap

412

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

ii.

Holding the left edge fixed, if the location of the rectangle is fixed by its lower left corner, the change in p, 8p, may be distributed over the three triangles so that 5x for vertices on the first vertical line will be zero, on the next line 8p/3, on the next 25p/3 and on the last 5p. How then are the derivatives to be evaluated when all three vertices move, but be different amounts ? As an example, considering triangle 4, its leftmost vertex has moved by 8p/3 and the other two by 28p/3. That is, relatively, two vertices have moved by 8p/3. For that triangle then d[Pel d[Pe dx 1 d[ee] - ~ 2 vertices,moving ~ - 3 ~ 2 vertices,moving (10.4. I)

iii. If the location of the rectangle is fixed by its mid ~ i n t , the left most edge has to be moved left by 5p/2 and the rightmost edge shiftged right by 5p/2; and these changes have to be distributed along the intermediate nodes because of the considerations above. Thus for nodes on triangle 1 for 1

example, 5x = - ~p/2 for the leftmost two veaices and dx = - ~ x 8p/2 for the other, this being a third of the way along from the middle to the leftmost edge. The relative movement of the leftmost two nodes with respect to the other then is dp/3 to the left. The derivative then involves a negative sign this time:

d[P]

dx 1 d[Pe ] dx 2 vertices ~ = - ~ ~ 2 vertices (I0.4.2) moving moving Along the same considerations, ol~ject oriented programming is now a fairly common approach to mesh generation. It might therefore be tempting to define the system by the boundaries of the various sections as allowed in many commercial programs. For example, in the programs Flux2D and Maxwell 1, the system is specified by boundaries and then the Delaunay algorithm with its associated Vomnoi tesselation is used to discretize the system into an optimal mesh (Hoole, 1989). This temptation must however, be avoided. As seen in Fig. 10.4.2a, (with corresponding meshes in Fig. 10.4.3), the object function undergoes many discontinuities and this is underscored in the plot of the derivatives of the object function from the problem depicted in Fig. 10.4.4. In this problem, only the symmetric half of which is shown in Fig. 10.4.4, the pole-face is being optimized so as to yield a constant flux density in the airgap. Thus the object function is given by n d[Pe]

-

F=~

Bi c .

(10.4.3)

i=l

Respectively by CEDRAT from Grenoble, France, and Ansoft Cor~ration from Pittsburgh, Pa, USA.

S. R. H. Hoole

413

where Bic is the flux density computed by the finim element solution at evaluation point i (of a total of n points) along the line where the constant flux density Bo is desired. Now, since in a first order finite mesh the evaluated flux density is constant in a triangle, from the triangle containing the evaluation point i, using the derivate approximations of eqs+ (1.5.4 I) and (I .5.42)

Bic:

+

:'%./[ {b,t{ Ae,)2+

{c}t{ Ae,)2]

[{Ae't( {b,t{b, + {c, t{c,){b'~}]

=~

:--,J {Ae/'~ellA~/]

(I0.4.4)

.

from eq, (10,2.2) ior [~}. Therefore, n

dF 1 - 2

B)2

Bic i=l

n

, =~

(

2 Bic - Bo

c

i=l n

I

=

(Bic - %

2

i=t

4[Ix

{Ae}t[Pe]{A}] *

n

-2; Oic-,,[ -

B. lc

2 Ix. 2{%}t[Pe 3p

i=I

.

•

{A}

n

=

Z

Bic-% B.

m

t[
[

+

o. 3p

1 2 . . {Ae} t[p e] ~

x {Ae}t[Pe]{,a~}

e + .

{Ae}

]

,rL-GJIAe} ,l

2

i=I (t0.4.5)

414

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

since in this case, the parameters are geometric and, as such, the pe~eability ~t cannot depend on them. The gradients can then be computed from the triangle containing the point of evaluation and the gradient of the vector potential computed from eq. (10.2.15). Now returning m the problem of discontinuities in the object function of Fig. 10.4.2 that we were discussing, why do they occur? This is clearly seen from eq. (!0.4.5), where the flux density Bic is evaluated from the triangle in which the evaluation point i lies. But when free meshing is used, the point can belong to a different triangle and we know that in first order finite elements, along the line of Fig. 10.4.4, where the flux density is specified, the solution approximates the density by straight flat lines, each straight portion belonging m a different triangle

STAT~

~IR

~

AIR PIECE COIL

Jz z:

-2

~'*

~7 1~I ¢~ .............

.......

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

Figure 10.4.5: Boxed Region for Und~turbed Mesh and Force Mapping of Parameters

S, R. H. Hoole

415

(Hoole, Weeber and Subramaniam, 199I). Thus, in free meshing, when t;rom one iteration to the other a point changes from one triangle to another, a spike in the object function is seen. In minimizing an object function with these spikes, the minimization algorithm sees these as local minima and, as a result, powerful, but slow and expensive schemes like tunneling (Subramaniam et al, 1991) have to be used to get around these artificial minima when in fact, they do not exist physically. Again, it is tempting m say that since these errors in the object function are because of the mesh, powerful adaptive mesh refinement techniques may be used to overcome these (Hoole, Jayakumaran and Yoganathan, 1986; Hoole, 1987; Pinchuk and Silvester, 1985; Babuskva and Rheinholdt, 1978; Hoole, Jayakumaran and Hoole, I988). But alas, no. Indeed, as seen in Fig. 10.4.2b, with refinement of the mesh by adaptive schemes, the discontinuities become smaller; but, on the other hand, they also become more numerous. What it means is that the optimization algorithm sees more and more local minima and takes ever so much longer to find what it perceives as the absolute minimum, which physcially is often the only minimum. Clearly then, to overcome this deficiency, we must work with a fixed mesh where the object function is evaluated. This region is shown boxed in Fig. 10.4.5. As changes in geometric parameters are warranted, the mesh is distorted as though it were a piece of elastic. This is seen in Fig. 10.4.6, where any change in object function would be gradual. Fig. 10.4.7 presents the mesh corresponding to the geometry of Fig. 10.4.4 with these considerations in mind. The corresponding object function is perfectly smooth.

Figure 10.4.6: Elastic Distortion of Mesh to Make Changes in Object Function Gradual

416

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

i

~.

®NNNz~YAtNI !

Fig, 10.4,7: Mesh for Problem of Fig. 10.4.4 with Considerations of Fig. 10.4.6 in Mind

S. R. H. Hoote

417

100

a

?

a

g

|0

Ic, J

Figure 10.5.1: Correct but Impracticable Realization of Object

418

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

10.5 Force Mapping of Parameters When the system of Fig. 10.4.4 was optimized so as to yield a flux density of B o = 1.0 Tesla, the resulting configuration is as presented in Fig. 10.5. I. Clearly, the object has been realized within 2%, but the profile of the pole-face is not manufacturable. What is called for are constraints or some other means by which the changes in pole-face profile will be smooth. The method of overcoming this is to map the parameters on to forces that deform the geometry under change as shown in Fig. 10.5.2 (Weeber and Hoole, 1992a, 1992b). That is, instead of the problem of Fig. 10.5.2a as we did before, we solve a second problem, a structural problem as in Fig. I0.5.2b. The idea underlying this is that when pNnt forces act on the surface of a body, the resulting deformation is smooth so long as the stiffness of the system is high. The solution of the displacement vector {u} resulting from applied forces {f} to a structure is described by the stiffness matrix [K] and is obtained from the equation (Zienkiewicz, I977) [K] {u} = {f} (10.5.1) The vector {u } contains the spatial displacements x and y of the nodes within and on the surface of the shape that is being changed through optimization. It gives the coordinate l~ations {x} of the deflev~:d structure as {x} = {x0} + {u} (10.5.2) Thus some of these coordinates are the parameters of design so that eq. (I0.5. I) is actually our mapping between the artificial force parameters {f} and our actual parameters contained in {u}, shown in the first box of Fig. I0.5.2b. Using the

Dimensions~..~]

Optimization

Performance

lI Deforming

[

Structural Deformation

b Figure 10.5.2: Mapping of Dimensions to Forces

S. R. H. Hoole

419

applied forces as the design variables p in the optimization, the calculation of the gradient of the magnetic potential OA/Op, essential to the calculation of the object function's gradient, is obtained as the solution of the modified form of Eq. (!0.2.15): 3{A} ~ O[P] [P] Op ~p -ap {A}

=

0x i

0p

-

0x i 0p {A }

(10.5.3)

Alternatively, the structural problem may be "sourced," instead of by forces, using displacements on the surface of the changing shape, a subset of the elements of {u}. These known elements of {u} move to the right hand side and act as the source. In this event, the question is: What are the surt3ce displacements necessary to change the shape of the object under design so as to produce the perfo~ance we want? The details are provided by Weeber and Hoole (1992a; 1992b). The terms ~xi/Op are obtained by differentiating eq. (t0.5~ 1) for the structural finite element solution under applied forces:

a{x} ~Kl{u} (10.5.4) [KI alu} . . . = O{f} ....... ap = [KI . ap ap - ap The structural solution of eq. (10.5.2) provides additionally a vector-space of displacement vectors {u }, used m obtain the deformed finim element meshes during a line search - - the line search being described in section 10.6 - - of the optimization routine m the trough by linear supe~osition. For the cost of an extra structural solution per line search for the trough of the object function with a substructural model containing just the regions of varying geomet U, the required smoothly changing discretization error is gained. Furthermore smooth shape outlines are achieved even with only a few optimization parameters, as in Fig. I0.4.5. The procedure will be especially useful for geometries of high complexity and it will also be applicable to any given finite element mesh and wilt allow us trivially to account for magnetic nonlinearities. Alternatively, the coordinams xi of all nodes and their gradients with respect to the applied forces axi/Op may be found as a solution to eq. (I0.5.1) if we set those xi that are parameters of the electromagnetic system to 1; thus since ui is dx i, Oxi/Op is ui itself. As a preliminary test of the idea, the method was applied to the problem of Fig. 10.4.4. The shape optimization process of a pole piece to obtain a given constant value of the flux density in the adjacent air gap is shown in Fig. 10.5.3. Forces, as indicated in Fig. I0.4.5a, are applied to the initial geometry in order to obtain the deflected structure, which ultimately gives the desired field distribution of Fig. 10.4.5b and the airgap flux density in Fig. 10.4.5c. That the method is successful is evident from the plot of constant flux where the flux density is within 1 to 2% of the required 1 T in the airgap.

420

Chapter I0: Synthesizing Devices through Finite Element Performance Modeling

10.6 Mathematical In optimizing in the previous section, the object function F was a function of all the parameters Pi. This wocess by which we d e t e ~ i n e the set of parameters {p} that would give our object function its minimum value is mathematical optimization. In multivariabte optimization, the minimum of the object function F(Pl,_Pn) = F({p}) is approached in iterative searches. Each search tries to minimize the function along a given direction {S} in the space of parameters. Thus the new parameter vector results from the previous one by finding the appropriate value of the scalar factor ~, that minimizes the function F along the search direction {S }: {P}k = {P}k-1 +o~ {S}k_ 1 (10.6.!) This search is effectively a one-dimensional search M the single scalar value of and is therefore generally referred to as a line search. In doing a line search, we know that we want to change {p} in the direction {S} (which is ~OF/O{p} in the simple steepest descent algorithm), but we do not know the value of R which d e t e ~ i n e s by how much we change {p }. This is the reason why the term "search" is used. Should too small a value of o~ be used, convergence would be slow, whereas should the value be too big, we would overshoot the minimum and therefore fail to achieve convergence. To the end of determining the correct value of ~ to be used, we give it a small value, change {p} accordingly, and then evaluate the corresponding F. This evaluation takes a finite element solution for the new value of {p} and so long as our initial value of o~ was small enough, the value of F would have gone down. Next, we increase o~ by a factor of 1.1 to 1.5 and repeat this exercise again and again, until we find F to be increasing. Because of the exploding nature of the powers of 1.5, it is guarantted that the minimum would be reached soon. At this stage we know that the correct value of o~ to be used is between the present value and the immediately preceding value. To determine the exact value, using the latest 3 values R1,R2, and R 3, and the corresponding values of F, F I, F2 and F 3, we can model F as a quadratic polynomial F(o~) = kl + k2o~ + k3 0~2. The 3 constants kI, k2, and k 3 can be determined from the three equations F i = F(Ri) =kl + k2~i + k3ofi 2 for i = 1, 2 and 3. The correct value of o~, co~esponding to the minimum of F then is found from dF/dc~ = k 2 + 2k3 R = 0 or o~ = -.k2/(2k3). This then gives us the proper adjustment to {p }. The art in multivariable optimization consists now of the choice of the search vector {S } so as to render the iterative proceAure reliable and efficient. The simplest approach is to search, as mentioned earlier in reference to Fig. 10.2.1 and eq. (10.2.8), directly in the direction of the negative gradient of the object function (steepest &scent search). In this method the successive search directions are orthogonal, resulting in a high number of required search procedures to find the minima that are "hidden" in narrow and winding valleys. D e convergence rate is improved by the use of conjugate gradient methods, where the minimization along one search direction is not spoilt by a subsequent minimization. The concept is

S. R, H, Hoote

421

based on expanding the object function F({p}) in a Taylor's series at the minimum and neglecting the terms that are higher than second order F({P}}=F({P0})+

Z

0F ~Pi

I Z + ~ ..

l

02F pipj + . . . 0pi0~

U

= {c} + {g}t{p} + 2{P}t~]{p }

(10.6,2)

with the gradient vector {g } and the Hessian matrix [HI as self-evidently defined as above. Subsequent search directions of the conjugate gradient method are said to be conjugate with respect to the Hessian matrix [H] or H-conjugate: t

{S}k[H]{S}k. 1 = 0

(10.6.3)

rendering them non-interfering directions of minimization, The determination of the search directions is possible without the explicit computation of the Hessian

IO

0

1~

20

Steepest Descent

o

~

lo

L~

2o

!

Conjugate Gradien~

Figure 10.6.1: Paths to the Optimum by Conjugate Gradients and Steepest Descent

422

Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

matrix, if the initial search vector is taken as the steepest direction of descent down the object function and all subsequent directions are computed as (Vanderplaats 1983) {g}k {S}k = - { g } k - I + {S}k-1 (10.6.4) It is seen that the new search direction consists of a term equal to the steepest descent direction updated by an additional term based on the information of the previous line search. With these conjugate search directions the minimum of F(Pl,-'Pn) is theoretically found after at most n steps. However, in reality the object function F is hardly ever exactly quadratic, so that a restart of the conjugate gradient method is recommended after approximately n searches. Figure 10.6.1 compares the paths to the optimum using conjugate gradients and the method steepest descent. To be noted are the fewer iterations with conjugate gradients, and the movement normal to lines of equal value of object function with the method of sttepest descent.

10.7 System Identification To demonstrate the effectiveness of optimization methodology beyond mere design to system identification, we have chosen as example a nondestructive evaluation problem. That is, inside an enclosed, inaccessible region we need to identify if there is i) an object such as a hole or crack in a device, and ii) current source inside the device. If so, we need to determine it by exciting the system electromagnetically from outside. Let us suppose, as in Fig. 10.7.1, that the object function quantifies the difference between measured values of potential and those given by the current synthesis in identifying the location of a Poissonian source within an inaccessible region (Hoole, Subramaniam, et al, 1991): 8 F=

~ic - ~ic

(10.7.1)

i=l Thus the current "synethesis" postulates that the system consists of a source at a given parametrized location and of certain parametrized dimensions. The eight measuring points are used on the 900mx900m boundary of the system within a 1 0 ~ m x 10~%9mfinite element domain containing the Poissonian source at position x o = 470, Yo = 480, as shown in Fig.10.7.I. Fig. 10.7.2 shows the the variation of the object function for a similar problem with the coordinate locations Xo and Yo- There are 8 peaks to be observed at the measuring points and a zero minimum at the exact location of the source where measured and computed values correspond exactly. Fig. 10.7.3 gives the results showing the convergence of Y0 by the classical approach,

S. R. H. Hoole

423

! 000

950

-7 I I I I I I I I I I

v ¸ jy )

0

W

50 I w

5O

950

I000

Fig. 10.7.1: G e o m e t r y for the Identification of an Inaccessible Source

×

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Chapter I0: Synthesizing Devices through Finite Element Performance Modeling

400

3,50

300

H

~

H

I

H

I

H

I

H

250

200

150

I00

H 50

0

%00

aO0 XO

300

400

Fig. 10.7.2: Object Function Changing with Identified Location

S. R. H. Hoole

425

800

-800

700

700

600

~600

500

40O

400 0

2

4

6

Iteration

8

10

12

Count

Fig. 10.7.3: Convergence to Location Yo of Source

10.8 Constrained Optimization In section 10.4 we examined the problem of shaping a pole face so as to achieve a constant flux density in the airgap. The problem is formally restated in Fig. 10.8.1: What should be the shape of the pole face so that a constant flux density of 1 T may be achieved along the line shown in the airgap? However, as seen in Figure 10.5.1, we saw that the solution of this synthesis problem resulted in an impracticable realization of the object. To overcome this, the structural mapping technique of section 10.5 was introduced. However, the problem with structural mapping is that a second, expensive solution of a structural problem is required. Here we present a constrained optimization technique that allows many engineering constraints to be incorporated as part of the object function (Subramaniam, Arkadan and Hoole, 1994). Indeed, most engineering problems do have constraints. Examples are that a length cannot be negative, an airgap has a minimum value, currents in a coil cannot exceed a value, machined parts must be within certain tolerances, and so on. The general constrainex] optimization problem may be defined as minimize

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Chapter 10: Synthesizing Devices through Finite Element Performance Modeling

F({p},A{p}) subject to gj ({p},A{p}) N 0 hk ({p},A{p})=0

the object function j=l ..... m inequality constraints k=l ..... l equality constraints i=l ..... n side constraints

PL,i -< Pi -< Pu,i We note that the latter side constraint imposing a range for a parameter with upper and lower limits, may be converted to a single inequality constraint (PL'P)(Pu-P) PL -< p G p u

-~

g(p) =

(pU.PL)2

_< 0

Therefore the side constraints do not warrant separate consideration. How then are the constraints enforced? It is by adding a Penalty Function P to the object function when in violation of the constraints: m 1 P({p}) = Z[max(0,gj({p}))]2 ) + Z[hk({p})]2 j=l k=l The function max takes on the value of the larger of the numbers it operates on. Thus whenever g exceeds O, it adds to the value of the object function - - that is, in regions where the constraints are in violation, the new augmented object function F + P takes on such a high value that the steep gradient will provide a "mat-slide" down which we would quickly exit the objectionable zone. Similarly, h also penalizes us whenever it is not zero. Returning to the problem of enf:orcing the smoothness of the pole-face, as

YA

i

Varying Geometry

X

Figure 10.8.1: Constraints on the Jaggedness of the Contour

S. R. H. Hoole

427

depicted in Figure I0.8.1, we may r~uire that adjacent segments of the contour of the face should not differ by no more than, say, 18°, or some other suitable value as required by circumstances. This is shown as two separate constraints, one for the left-most node i with a segment only to its right --10-

< atan ] < /xj-xiJ -10

and for more general nodes k, with a segment on each side: - - ~ < atan [ - - atan F < 10 k Xk-Xj3 kxI-XkJ - 1 0 With this constrint, the pole face was optimized and the result is shown in Figure 10.8.2 where the smoothness of the resulting object shows the effectiveness of constrained optimization.

Figure 10.8.2: Pole Face Optimized with Constraints.

Chapter 11 [11

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PARALLEL ALGORITHMS AND MATRIX SOLVERS IN FINITE ELEMENT FIELD ANALYSIS

11.1 Introduction The best perfo~ance criterion in crafting general purpose finite element field analysis programs is the convenience or ease of using them with precision (Hoole, 1989). Thus, we may identify, besides capability of analysis, the areas of user friendliness and speed as areas for concentration of fuaher research (H¢~.~le, 1986). The I2ormer is being addressed partly through adaptive schemes of mesh generation and interactive, menu driven graphics capabilities (Hoole, 1986; Babuskva and Rheinboldt, 1978; Pinchuk and Silvester, 1985; and Raizer et al, 1990). The particular aspect of increased speed of solution is being addressed through

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Figure 11.1,1: Beyond-The-State-Of-The-Art Hardware Capab!ities through Software

S. R. H. Hoole

429

the introduction of new hardware that does the computation faster as well as new algorithms that carry on these computational processes more efficiently (Ida, 1985). However any gains in speed through hardware are limited by the hardware presently available. Alternatively, as shown in Fig. ! 1.1.1, dedicated computers operating in parallel can result in better p e r f o ~ a n c e than what is available through the best conventional hardware. The new breed of parallel machines span both methods of improving computational speed - - hardware and software. They represent new machine hardware in that they possess multi-processors. But to exploit the hardware, algorithms have to be redesigned to run on these machines. Fig. 11.1.2 shows how this is done. A computational process that can be divided into two independent tasks has each task assigned to a different processor so that, ideally, the two-processor machine can do the job in half the time. In practice, with a pprocessor machine, we do not reach a speed-up by a factor of p because of the overheads in putting the results together as well as in one processor waiting while another is operating on a variable it needs (Osterhaug, 1987). To give a simple example, let us say that two persons need to add 50 numbers together. While if 1 were doing it, he or she would add them one after the other, in this case, the two would share the work by each person adding 25 of these numbers, yielding two independent sums. That is, they would not work by adding to the same sum since it means that while one person is adding, the other would idle. At the end, each person would, if necessary, wait until the other has finished and then, they would each put their own sums together to form the required result.

Task 1 by

Proc 1

Task 2 by

..... i ....

Proc I

TaSkby2 l Proc 1

m

a. Serial Execution

b. Parallel Execution

Fig. 11.1.2: Concept of Parallel Computation

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Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

This is exactly what happens when two or more processors share the work. There is parallelization of the algorithm as the two added to different sums. That is all operations are as much as possible split into independent parts. And then there is overhead in waiting for the other to finish and in keeping an eye on what the other is doing. In doing this, the algorithm chosen may not be the most efficient by n o d a l standards, but the parallelized form is faster with many computers. That is, curiously, parallel algorithms can sometimes even be efficient in being inefficient. As seen in Fig. 11. 1.3 fi'om the example of the sparsity computation that precedes matrix formation in finite element analysis (Hoole, I989), although 1 processor takes more time to implement the paraIlelized algorithm than the sequential algorithm, when many processors work on the parallel algorithm, it is faster. To explain, let us say that the classical algorithm takes time t and the parallel algorithm by one processor takes time 2t. But because the parallel algorithm allows the tasks to be broken up into independent ones, when we have p processors, the parallet algorithm would take the time 2t/p which is less than what the classical algorithm would take if p>2. 2000

4000

! 0, 0

2000

4000 6000 M a t r i x S|za

8000

t 0000

12000

Fig. 11.1.3: Comparison of Sparsity Times for Sequential and Parallel Algorithms As a simpler example of a parallel algorithm being less efficient by sequential standards and yet faster, consider matrix solution by Gauss's scheme of the equation [2 ~]

X2} { 2 X= l

I}

(11.1.1)

The slower Gauss iterations starts with a guess of x 1 = 0 and x 2 = 0 and then goes through the following steps, using row 1 to evaluate x! and row 2 to evaluate x 2. In the following sequence, the superscript refers to the iteration number: Iteration l: 0 I

1- x 2

Xl-

2

1-0

- 2 -0.5

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431

0 1

2-Xl

x2 -

2

2-0 2

- 1.0

Iteration 2: 1

2 1- x 2 1-I.0 Xl 2 2 -0.0 1

2 2-Xl x2 2

-

2-0.5 2

- 0.75

Iteration 3: 3 Xl-

2 I- x 2 1-0.75 2 2 -0.625 2

3 2-Xl x2 2

-

2-0 2

=1.0

x 1 = 0.0 and x 2 = t .0. The Gauss-Seidel algorithm on the other hand, m a k e s use o f the latest value of the unknowns to evaluate a number using a row, whereas in the Gauss algorithm, in evaluating xl 2, we used x 1 although Xl I had already been computed:

Iteration 1: 0 1

1- x 2

Xl-

2

1-0 2 -0.5

1

1 2"Xl x2 2

-

2-0.5 2

- 0.75

Iteration 2: 2 Xl-

1 1- x 2 1-0.75 2 2 -0.625

2 2 2 - x 1 2 - 0.625 x2 2 2 -0.6875

Iteration 3: 2 3 Xl-

1- x 2 1-0.6875 2 2 -0.15625 3

3 2-Xl x2 2

-

2-0. I5625 2 =0.921875

Except for the initial meandering, it will be found that the modified Gauss scheme converges more rapidly towards the actual answer. But, unfortunately, if this latter

432

Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

"efficient" scheme is to be employed, then while a processor is working on, say Xl 2, a second processor cannot work on x22 because this requires that processor 1 should have completed evaluating Xl 2. Not so, however, the older Gauss scheme. Here since all envies of the unknown vector are improved using values from the previous iteration, they may be independently evaluated. Therefore although it is slower in convergence as measured by the number of iterations to reach the required accuracy, it is faster in time because two processors are available to work on the problem, whereas in the Gauss-Seidel scheme, although it is faster, the availability of numerous processors cannot be used with profit. Likewise, in electromagnetic field analysis researchers have sought means of parallelizing, that is breaking up into independent parts, the many algorithms that are involved in analysis (Ida, 1985; HoNe, 1990; Josmo et al, 1989; OpsahI and Taflove, 1989; Bedrosian, et al, 1989; Mahinthakumar and HoNe, 1990 a & b; HoNe, 1991; Hoole and Mahinthakumar, I990; Magnin and Coulomb, 1989; Carey and Jiang, 1986; Carey et al 1988; Zois, I988; and Kincaid and Opp, 1988). The purpose of this chapter is to review the work that has been done in this increasingly important area and suggest areas in prallelizafion to which new research effort may be directed. It is noted that this chapter deals specifically with the process of paralleIization and not with which particular type of parallel machine is better suited to which algorithm. This aspect of the issue is considered by Josmo et al (1989). But broadly, there are two classes of parallel architectures shared and distributed memory system which need different considerations in implementing parallelization. Shared memory machines have all processors working off the same meory. While the advantages are obvious, so far it has been difficult m build machines with more than 32 processors. With distributed memory machines, thousands of processors are possible, but they work off different memory segments, thus making them better suited to working on different and independent data sets, such as a computer used by the government that needs to go through the income tax returns of millions of persons.

11.2 Coding Considerations In learning anything newly, one of the greatest impediments to starting the process is an intrinsic reluctance to overcome the initial difficulties. Such difficulties are often artificial and created by a lack of knowledge as m whether a lot of effort or little effort is required before a learning momentum is acquired. With parallel programming, such fears deserve to be dismissed out of hand. For anyone who knows programming, with just a few extra system commands - - numbering fewer than 5 commands - - one could be on the road to programming parallel computers with confidence. These commands have m do with assigning processors to a task, locking other operations on a variable while it is being worked on by another process and so on. Two programs, Programs 11.1.1 and 1 I. 1.2, are listed as examples for computing the dot product of two vectors. One is for the Sequent machine and the

S. R. H. Hoole

433

c This is an example program for computing dotproduct of two v~tors using c the fortran parallel instruction set on sequent symmetry 81. Sequent is a c shared memory scalar-parallel machine dotproduct is a Blas I level operation C

C SEQUENT EXAMPLE - D O ~ R O D U C T C

Program Dot_Prod c dotproduct alpha=a*b is computed Parameter (n=1000) Real a(n), b(n) c Initialize vectors a,b Do i=l,n a(i)=2. b(i)=3. enddo alpha=0. c set number of pr~essors to be used (=3) nprocs=3 call m set p r o c s ( n p r o c s ) c begin parallel loop by setting sh~ed and local variables c local variables are local to each processor, ie., each processor has a c copy of its own shared variables are common to all processors any c change in a shared variable is known to all processors. More than 1 c processor may not try to operate on (by operate on, we mean modify or c change - reading or accessing is fine) a shared variable at a time c c The next is not a comment line, but a parallel command and is c understood by the compiler as such because of the $ sign

c$doacross c

c c c

share(a,b,n,a!pha),

locai(slocal,j)

Do i=l,nprocs slocal=0. inner loop is local to each processor Do j=i,n,nprocs slocal=slocal+a(i) b(i) enddo any element of an array shared variable or a scalar shared variable sould not be Ol~,rated on by more than 1 processor at a time. The m lock command guarantees this in a parallel loop

call m lock alpha=alpha+slocal

call

m_unlock

enddo write(0,*)'Dot Product = ',alpha end

Program 11.1.1- Dot Product on the Sequent

434

c c c c c c

Chapter 1I: Parallel ALgorithms and Matrix Solvers in Finite Elements

~ i s is an example program for computing dotproduct of two vectors using the fortran parallel instruction set on the Al!iant FXJ80. Note that Alliant FXJ80 has a CISC instruction called DOTPROD which could be used directly. Each CE of the Alliant FX/80 is a 32-register vector processor Alliant FX/80 can have up to 8 CE's. (CE = processor)

c

C ALLIANT EXAMPLE - D O ~ R O D U C T c

Program Dot_Prod c dotpr~duct alpha=a*b is computed Parameter (n=! 030) Real a(n), b(n) c Assign Values to Vectors a,b. Vector notation could be used c a=3. or a(I :n)=3. or (a(i)=3,i=l,n) are all ~uivalent a=3. b=2. lk=0 alpha=0. c start parallel loop with cdoall c numproc0 gives the number of CE's being used. This could c be set as an option during compilation. c by default all undeclared variables are shared

Cdoall i=l,numprocO c declare l ~ a l variables j, stocal integer j real s l ~ a l slocal=0. loop Do j=i,n,numproc0 slocal=slocal+a(j)*b(j) End do c lock command is similar to re_lock in sequent, this prevents c two or more CE's updating alpha simultaneously

call lock(!k) alpha=alpha+slocaI

call unlock(lk) End cdoall write(O,*)'Dot Product = ',alpha end

Program 11.1.2: Dot Product on the Atliant

S. R. H. Hoole

c This is an example program for matrix-vector multiplication c using the fortran parallel instruction set c on sequent symmetry" 81. Mat-Vec (Blas2 level operation) c could be implemented by a series of Blas I level operations c (ie., dotproducts or vector-triads) c for a distributed memory machine like the sequent, c the vector-triad implementation is more efficient C

c SEQUENT EXAMPLE - MatVe~ c Program Mat Vec c matrix-vector multiplication b=A*x is computed Parameter (n=1000) Real A(n,n), x(n), b(n), blocal(n) c Initialize Variables A,x,b Do i=l,n Do j=l,n a(id)=2. enddo x(i)=3. b(i)=0. blocal(i)=0. enddo c set number of processors to be used (=3) nprocs=3 call m s e t _ p r o c s ( n p r o c s ) c begin parallel loop by setting sh~ed and local variables c$doacross share(A,x,b,n), local(blocald,k,m ) Do i=l,nprocs c inner loops are local to each processor Do j=i,n,nprocs Do k= 1,n blocal(k)=blocaI(k)+A(k,j)*x(j) enddo enddo call m _ l o c k Do m=l,n b(m)=b(m)+blocal(m) enddo call m _ u n l o c k enddo write(0,*)(b(i),i= 1,n) end

Program 11.1.3: Matrix-Vector Multiplication on the Sequent

435

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Chapter II: Parallel Algorithms and Matrix Solvers in Finite Elements

c This is an example program for matrix-vector multiplication c using the fortran parallel instruction set on the Alliant FX_/80. c Note that Alliant F.Yd80 has a CISC instruction called MatMuI which c could be used directly. Each CE (Computational Element) of the c Alliant FX/80 is a 32-register vector processor. c Alliant FXJ80 can have up to 8 CE's. (CE = processor) c We implement the matrix-vector oFeration as a series of vectorc triads (ie, c~+alpha*b, where alpha=scalar and b,c=vectors) c

c ALLIANT EXAMPLE- MATVEC c

Program Mat_Vec c matrix-vector multiplication b=A*x is computed Parameter (n= 1000) Real A(n,n), x(n), b(n), blocal(n) c Initialize Variables A,x,b using Alliant array notations A=I. x=l. b=0. Ik=0 c start parallel loop with cdoall c numproc0 gives the number of CE's being used. This could c be set as an option during compilation. c by default all undeclared variables are shared

Cdoall i=l,numprocO c declare l ~ a l variables j,bl~al integer j real blocal blocaI=0. loop c Alliant array notations could be conveniently used in local loop c A(:d) = jth column of A Do j=i,n,numproc0 blocal=blocal+A(:,j)*x(j) enddo c lock command is similar to m lock in sequent, this prevents c two or more CE's updating alpha simultaneously call lock(lk) b=b+blocal

call unlock(lk) End cdoall write(O,*)b end

Program 11.1.4: Matrix-Vector Multiplication on the Alliant

S. R. H. Hoole

437

other for the Alliant. Both the Sequent and the Alliant are shared memory machines. In the program 3 processors are used. Thus each processor adds every third element of the array to the variable slocal. On the Sequent, a variable declarexl as local makes every processor (or computing element CE) have a copy within itself and a variable declared as global is shared by all. Thus after adding the portion of the dot product assigned to it, the numbers are added to the global variable alpha under the protection of the locking command that ensures that additions are not lost while two processors do it simultaneously. This could happen when for example, the numbers 3 and 4 have to be added to a variable x initially at zero. Thus, from shared memory, processor I reads x at zero and adds 3 m it. Processor 2 does not know this and reads x from memory where it is still at zero. While processor 2 is doing this, say 1 has finished adding and writes its result 3 to the location of x. Now, processor 2, writes its result of 4 back where the number is stored, the result would be 4 instead of the required 7. The programs on the Alliant are slightly different in that they use an extended version of Fortran, Fortran 8x which allows vector variables. Two additional programs, programs 11.1.3 and 11.1.4 are also given on Matrix-vector multiplication, it is left as an exercise to the reader to read them and understand what they are doing.

11.3

Electromagnetic Field Analysis

Electromagnetic field analysis broadly falls under integral and differential methods as described in chapter I and again in detail in chapter 8. Under the former is the famous boundary element (or moment method) and under the latter are the finite element, and finite difference methods. Other methods too exist but are not as widely used as these and are therefore not taken up in this paper for discussion. In field analysis by any of these methods, the stages of analysis are preprocessing where the problem is defined, formation of the equations as defined by the numerical scheme under use and solving them, and, finally, postprocessing the results which is the task of examining the solution and using it to extract engineering performance predictions of the device under analysis. The mid-part of solving the equation takes by far the most amount of computational time, whereas the pre- and post-processing parts are not intensive in computation but require much time of the user. Much of the research effort in parallelism in field analysis has therefore focused on parallelizing the matrix computation. We shall examine below, the para!telization of these segments of finite element analysis.

11.4 Pre- and Post-Processing Te benefits of prallelization being the most in the solution part of field analysis, one of the earliest efforts at bringing parallel algorithms into the pre- and post-

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Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

processing stages of solution is described by Hoole and Mahinthakumar (1990). The theme behind doing this is as follows, although the cpu seconds gained in doing it is trivial compared to those gained in matrix solution: First that so long as there is a way of computing anything more efficiently, then it must be done that way and, second and most importantly, the delays at the pre- and postprocessing stages are much more irritating whereas delays in the solution stage may ~ overcome by planning another activity. Hoole and Mahinthakumar (I 990) give schemes for identifying a node or triangle pointed at by a user, as well as for drawing the equipotential lines and the outline of the device while working on interactive operations. Additional areas to which our efforts at parallelization may be fruitfully directed are the many other areas of pre- and post-processing such as in computing the neighbors for Delaunay mesh generation, the process of Delaunay optimization itself, going through the elements and changing the materials of a region and adding points to a mesh. The difficulty to be overcome is the need for locked execution. For example, after the user specifies a point on the screen to be added to the mesh, if one processor is assigned to read such specified points and all others to adding the point, what happens when the user specifies the next point while the previous one is still being added? In the interim, what kind of mesh is to be displayed on the screen through parallel drawing for the user to decide on the next point to be added? Even the determination of the triangle to which the new point belongs cannot be accomplished until the previous point has been added to the mesh. It is expected that with the introduction of massively parallel systems, these will no longer be issues since the computation time would be negligible compared to the time the user takes to give an instruction. But for now, it is a problem that needs addressing.

11.5 Eauation Formation In the finite element and boundary element methods, the numerical scheme results in a matrix equation. The matrix equation in finite elements is computed from the local matrices, each of which is computed from an element and then added m the global matrix (Algorithm !.5.3). The operation therefore is easily parallelizable and has been ~dressed by Hoole (1990). In the boundary element scheme, each coeffcient of the matrix results from an integral and thereti)re the computation of the coefficients is naturally parallel that is, parallelization is not required. Unlike with the finite element method, no kx:ked operations will be required on account of two processors wishing to operate on the same coefficient of a shared memory" system. A possibility for extending the parallelism in the boundary element scheme is in the computation of the integrals that make up the coefficients of the matrix. The integrand involves a Green's function and, often, no closed form solution to the integration exists. Therefore the integration involves the use of Gaussian quadrature f o ~ u l a s and, associated with it, the computation of the integrand at the integration points. Thus a different processor may be assigned to compute the

S. R. H. Hoole

439

integrand at a different Gaussian point. Moreover, another issue in the computation of these coefficients is the sufficiency for pu~oses of accuracy of the order of integration used. This is addressed by incrementing the order of integration by 1 and verifying that the result has not changed significantly or, alternatively, subdividing the region of integration into smaller ones, particularly close to singularities and verifying that the result has not changed. These tasks then are easily paralIelized by assigning one set of processors to perform the integral and another set to perform the improved integral; when using a massively parallel system with no limit on the number of processors available, within each set of processors the task of integration may be performed in parallel as described above.

11.6 Matrix Solution 11.6.1 Nature of the Matrices and Renumbering Once the matrix equation is assembled, it has to be solved. "When dealing with the boundary element method, for many open boundary problems in homogeneous space, the matrix is symmetric, positive definite and full. Cholesky's scheme is therefore well suited to the solution of the equation. When using the finite element method also the matrix is symmetric and positive definite, but it is sparse. Two of the more popular schemes of solution for finite element problems are Cholesky's scheme with banded or profile storage and the conjugate gradients algorithm for sparse storage. In profile storage (or skyline), a symmetric half of the matrix to the right of the first non-zero term of each row is stored. Profile storage is always done after renumbering so as to reduce the storage requirements by ensuring that a node and those connected to it have numbers close to each other. In sparse storage we store only the non-zero terms of a symmetric half of the matrix. Renumbering is also used with the Incomplete Cholesky Conjugate Gradients Algorithm (ICCG) which uses sparse storage because it improves the approximate Cholesky factor that is used for preconditioning. The process of renumbering involves searching through the mesh data and finding the n ~ e s to which each node is connected. Thereafter, on the basis of this infbrmation, new numbers may be assigned to each node. This task is easy to prallelize by getting each processor to search through the finite element mesh to determine the connectivies of a different subset of the nodes. Moreover, since the connectivities of the nodes are required for making up the data structures for the sp~se and profile storage schemes, the work done in renumbering is quite useful in cutting down the time for the latter. The allocation of the data structures can also now be done in parallel, particularly for sparse storage, allowing each processor to assign the column numbers associated with a different row of the matrix.

11.6.2 Cholesky's

Scheme

Cholesky's aglorithm for matrix solution involves a three part process. For solving the matrix equation [P]{x} = {Q} (11.6.1)

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we first the Cholesky factofization [P] = [L][U] (11.6.2) where ILl is a lower triangular matrix and [U] is upper triangular so that [L][U]{x} = {Q} (11.6.3) may be solved by first writing [U]{x} = {z} (11.6.4) and solving [L]{z} = {q} (11.6.5) by forward elimination. Now, knowing {z}, we obtain {x} by solving eq. (11.6.4) by back substitution. Much of the computing time is spent finding L and U. Kincaid and Oppe (1988) give a parallel implementation of the LU factorization into the 2 independent tasks of finding L and U. In finite element electromagnetic field analysis, the matrix [P] is symmetric so that [U] = [Lt]. In this section we describe a parallel algorithm for finding L using the parallelism inherent to the work of Kincaid and Oppe (1988) and, additionally, give parallel algorithms for forward elimination and back substitution for the special c~es of full-storage, profile storage and sparse storage (Hoole, I991b).

11.6.3. Cholesky's LL t Factorization The means to parallelizing this algorithm, naturally suggests itself when we look at the factorization ~uation ooooo L21 LZ2 0 0 0 0 ... / L31 L32 L33 0 0 0 . . . 1 L41 L 4 2 L 4 3 L 4 4 0 0 . . .

L

Pll P21 = P3I P41

I

.:

J

P21P31P41...] P22 P32 P 4 2 . - . | P32 P33 P 4 3 . - . | P42 P43 P 4 4 . . . J

0

L22 L3Z L42 0 L33 L43 0 0 L44 •

(11.6.6)

Thus we see that equating the term at row I column I of [P]:

(1 1.6.7) LIgl = Pl 1 we have the diagonal term L 11. This is a single arithmetic operation and cannot be done in parallel at the high level of the user. But having computed the diagonal term, observe by equating the terms down column 1 of the nxn matrix [P]: L2IL11 = P21 L31L11 =P31 L41LI I = P41 (I 1.6.8) LnlLll =Pnl

S. R. H. Hoole

441

ZEROES COMPUTED

0

S T E P 1, D I A G O N A L BY 1 PROCESSOR S T E P 2, C O L U M N IN PARALLEL BY ALL P R O C E S S O R S TO BE COMPUTED

Figure 11.6.1: Sequence for Parallel Cholesky Factorization Thus, all the elements of column 1 may be computed independently since L 11 is now known. These operations can therefore be done by different processors. Now, we assign 1 processor to compute L22 from the equation for row 2 column 2 of P in eq. (11.6.6), using the fact that all columns to the left of the diagonal have been already computed as pictorially depicted in Ng. 11.6.1: L~I +

=P22

(11.6.9)

Now, equating the terms of eq. (11.6.6) under the second diagonal term of [P] L31L21 + L32L22 = P32 L41 L21 + L42L22 = P42 L 5 I L 2 I + L52L22 = P52 (11.6.10) L n l L 2 1 + Ln2L22 = Pn2 Each equation involves only 1 unknown, the coefficient of [L] under diagonal 2. Thus the column under the diagonal on row 2 of L may be computed. Fig. 11.6.1 depicts picmrially the order of computation in the generalization of this algorithm. Be it observed that once a term of P, Pij is used in computing Lij, it is no longer required. Thus L is stored in the space assigned to P.

11.6.4. Forward Elimination and Backsubstitution Forward elimination is the process by which we solve the lower triangular eq. (11.6.5). From row 1 we compute Zl and now, knowing z 1 , from row 2 we compute z 2 and so on. This may be parallelized by shiffing the colunm under diagonal 1 to the right in parallel after computing zI and so on, The description of this is thus:

442

Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

A.START WITH:

fc ooooo lti:l tli ca, L~ o o oo...

~

L31 L32 L33 0 0 0 . . . L41 L42 L43 L44 0 0 ..

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ql Zl =LII

ooooo

q

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=

PROCI:q2 = q 2 - L 2 I z l PROC2:q3 =q3 -L31zl PROC3:q4 = q4 - L4Izl.

°.,

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Z

' Q

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TO BE COMPUTED

Figure 11.6.2: Alternative Forward Elimination in Parallel

S. R. H. Hoole

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200

300

400

Matrix

500

600

7( ~0

Size

Fig. 11.6.3: Computation time and speed-up in fully-stored Cholesky Solution - a. Computation time b. Speed-up

444

Chapter !I" Parallel Algorithms and Matrix Solvers in Finite Elemen~

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.....

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Fig. 11.6.4: Computation time and Speed-up in Profile-stored Cholesky Solution. a. Computation time b. Speed-up

S. R. H. Hoole

445

The same scheme may be extended to backsubstitution. An alternative scheme was tried out for forward elimination. Here, if we had p processors, the first p unknowns are computed sequentially by the usual means. From now on, in general, after computing k values of z (k being a multiple of p), the operation k qk+j = q k + j - ~ Lk+j m Zm forj = 1 to p (11.6.11) m=l is done in parallel and is depicted in Fig. 11.6.2. Now the little triangular portion of the block of rows form k+ 1 to k+p between columns k+ ! and k+p is done by I processor just as we did the first p rows. The particular usefulness of this scheme is in solving profile-stored systems and will be discussed in that section below.

11.6.5. Application to Special Storage Schemes 11.6.5.1 Full Storage In numerical electromagnetic field analysis using the boundary element or moment method, the coefficient matrix that results is full and symmetric. Fig. 11.6.3a gives the solution times with matrix size for 1, 2 and 3 processors of a Sequent Balance 81K Parallel Computer. The s p e ~ up with 2 and 3 processors, compared to a single processor, is shown in Fig. 11.6.3b. Observe that the ideal 100% efficiency of time/number of processors is closely approached, but not completely because of the overhead time in managing the processors. Be it also noted that a matrix size larger than 6 ~ is not attempted because of the inordinate storage load that would be placed on the system. For this equation system, there was no substantial difference between the two alternative schemes for forward elimination and backsubstitution; the first scheme where we shift a column to the right was marginally faster because it requires less waiting time compared to the second where there is more single processor operation. But in terms of total solution time, this difference was insignificant because the Cholesky factorization p~t with full storage is very time consuming. (See Fig. ! 1.6.3a).

11.6.5.2 Profile Storage When doing electromagnetic field analysis by finite elements, the resulting system of equations is not only symmetric, but also sparse. ~ i s is what allows us to overcome the storage restriction of full matrices that is evident in Fig. 11.6.3 where we have been unable to exceed a matrix size of 600x6C,0. But the Cho!esky factor has fill-in; fill-in is the effect of zeroes of [P] becoming non-zeroes of the factor L. This occurs only to the right of the first non-zero element of P. As a result, it is insufficient to assign storage only to the non-zero terms. We, in fact, assign storage l~ations to all terms to the right of the first non-zero term of each row. For example, the lower triangular symmetric half matrix 1.0 1.2 2.0 0 1.1 3.0 2.2 0 0 4.0 0 0 1.5 0 5.0

446

Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

is stored by allocating spaces for the numbers marked by x to the right of the first non-zero on each row: X

X

X X

X

X

X

X

X

X

X.

The order is row by row to the diagonal so that the matrix is stored as the vector

V = [1.0, 1.2, 2.0, I.l, 3.0, 2.2, F,F,4.0,1.5,F,5.0] (I 1.6.I2) where F stands for the locations where fill-in occurs. Two additional vectors, FC containing the first column of the profile of each row and DIAG, giving the location of the diagonal term of the coefficient vector V, completely define the data structure from which the original matrix may be retrieved: FC = [1,1,2,1,3] (11.6.13) DIAG = [1,3,5,9,12] (11.6.14) Thus in computing the term Lij of the Cholesky factor, if j is less than FC[i], it is evident that it is outside the profile, and therefore does not exist and has not to be computed. In forward elimination too, while doing the column shifting scheme, whether Lij exists to be shifted right is similarly ascertained from the data structure. Although searching is required, it involves only the comp~ison of the two numbers j and FC[i]. But because the numbers down a column of L are not stored in sequence, some increased page faults are also experienced. But in the second scheme of forward elimination described earlier where a block of rows is shifted to the right as in eq. (I 1.6.11) and Fig. 11.6.2, the sequence Lk+j m as m goes from 1 to k is in the same order as we have stored L in eq. (1 !.6.12). As a result page faults are reduced and searching is obviated, and this scheme shows itself to be better by approximately I0-20% in the range of matrix sizes from 2 0 ~ to 10,000 On the other hand, in back substitution, the equivalent of the first scheme would involve, upon computing a term at row m, shifting all numbers of the column of the upper triangular [U] above the diagonal But then, [U] is [L t] and therefore these numbers are stored along the row of ILl. This operation then may be performed rapidly from the beginning of the profile of row k of L to the term before the diagonal in rapid succession with minimal page fault. Therefore forward elimination is done by row-block shifting and back-substitution by column shifting. Figure 11.6.4a gives the solution time for a finite element problem using first order triangul~ elements with 1, 2 and 3 processors and Fig. 11.6.4b gives the speed-up for 2 and 3 processors compared to using 1 processor. While a similar speed up is obtained as before, the solution time for an 8000x8000 matrix with profile storage is much lower than that for a 600x6~ matrix with full storage.

11.6.6.

Cholesky's Scheme, the Conjugate Gradients Algorithm and Sparse Storage

The incomplete Cholesky, preconditioned conjugate gradients algorithm, or ICCG, is now the most widely used solution method for large sparse, diagonally dominant

S. R. H. Hoole

447

matrices in commercial software packages for large electromagnetic field problems. It employs sparse storage where the matrix demonstrated above h ~ only the nonzeroes stored in the vector V = [1.0, 1.2, 2.0, 1.1, 3.0, 2.2,4.0,1.5,5.0] (11.6.15), along with DIAG which is now modified because of the absence of filHns: DIAG = [1,3,5,7] (11.6.16) and a vector COL that gives the column locations of the non-zeroes: COL = [I,1,2,2,3,1,4,3,5] (11.6.17) Thus ILl is computed approximately, ignoring the fill-ins and, thereafter, preconditioning the system to converge faster requires the use of forward elimination and backsubstitution at every iteration. In this case, since L is computed only once and forward elimination and backsubstitution have to be done repeatedly during every iteration, the latter assume greater importance. The computation of L now ~ c u r s by the same algorithm, except that COL is used to find out if it has to be computed or not. This is done by asking, in the computation of Lij, if j exists in COL between locations DIAG[i-I]+I and DIAG[i]. This searching is of little account because the approximate L has only a few terms and, as already rem~ked, the computation is done only once. In forward elimination and backsubstitution however, this inefficiency on account of searching cannot be brushed aside because the cost builds up with iterations, particularlry when dealing with many gradient boundary conditions and other causes of ill-conditioning such as drastic changes in material permeability from region to region as well as nonlinearities within a region on account of saturation. It is emphasized that, in using the column shifting procedure of forward elimination described above, we have to first determine if a number under the column exists or not, whereas only a very few of them do in a sparse system; this cost is significant in an iterative process. A scheme devised to circumvent these difficulties was to exploit the highly sparse nature of first order triangular finite element matrices. These have only about 5 non-zero coefficients per row when the mesh is properly constructed in an optimal sense such as governed by the Delaunay criterion (Hoole, 1989). Thus this scheme uses a boolean vector Done, corresponding to the unknowns. Done is initially set to False and each term is set to True as the corresponding variable is solved by forward elimination. We also have an integer variable RowsDone which is the number of rows already done or assigned to a processor. Each processor, as it finishes doing its row, increments RowsDone by 1 and starts working on row RowsDone+l. COL is modified to the jump from the first non-zero column of a row to the next. Thus COL of eq. (I 1.6.17), it is modified to COL = [I,1,1, 2,1,1,3,3,2] (11.6.18) This facilitates moving rapidly down the non-zero elements of a row of L. At Lij, if zj is computed, it is shifted right or, else, the processor waits until it is done and proceeds up to the diagonal where it computes z i, sets Done[i] to True and takes on an additional row. It is emphasized, that tbr large matrices, over 2000x20~)0, rarely does the processor have to wait because a term is not computed.

448

Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

2OO A

tD

v

+-............o---.......

O

i

E ¢:

Sequential Parallel Method 1 Parallel Method 2

1 O0

0 m 3 X I,U

100

140 440

720 1 6 5 6 4 6 8 0 Matrix Size

11800

Figure 11.6.5: Comparison of Forward Elimination Methods Method 1- Row by row; Method2: Column Shifting

In doing back-substitution, the proc~ure described above cannot be employed because of our decision to store the matrix row by row in sequence as in eq. (11.6.17), whereas the shifting right in the backsubsfitution is along a row to the right of the diagonal of the upper triangular [U] which is [Lt] so that the numbers exist in a column below the diagonal of [L]. These numbers are not in s~uence and therefore have to be searched for, Searching in a sparse system is much more costly than in a profile stored system where we saw that it only required the comparison of 2 integers. This cost builds up with iterations. We have found it quicker by a factor of 2 per iteration to reorder the matrix [L] column by column before embarking on a backsubstitution. It turns out that the time spent on reordering the matrix column by column and then putting it back in the original order for the forward elimination of the next iteration takes less time than the time expended in searching for the non-zeroes. Figure ! 1,6.5 compares forward elimination by this scheme (method I of the figure) and by the column shifting scheme described earlier (method 2 of the figure). The advantage is apparent.

11.6.7 Conjugate

Gradients

The conjugate gradients algorithm is ideally suited for matrix solution where sparse positive definite, symmetric rna~ices are involved and is therefo~ the choice of many scientists in finite element analysis, particularly because it is sufficient to store only the nonzero coefficients of the matrix. However, it has been generally held that some kind of preconditioning is necessary to make the scheme profitable. A preconditioning matrix [M] is usually used to modify the governing equation in such a way as to cluster the eigen values of the new equation so a~ to make the conjugate gradients algorithm converge faster. The requirement of symmetry on the coefficient matrix confines the preconditioning matrix to a diagonal matrix in Jacobi preconditioning or a

S. R. H. Hoole

¢

449

200"

,9

Z6 ~00"] //~11

Z

lncomple~CholeskyCG Jaco~CG

1 J7

0 t ...... ,

0

,, ~,,,,,,,,,,~ w 2000 4000

•

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Figure

,

6000

•

~

8000

,

~ ....

• -;

10000 12000

Size

11.6.6: C o n j u g a t e G r a d i e n t s : Iterations w i t h Matrix Size

symmetric matrix [L][L t] in the Cholesky preconditioning where L has the same sparsity pattern as the coefficient matrix. In reality, the equation is not explicitly modified since the use of these preconditioners would reduce the sparsity of the new coefficient matrix. For this reason, equivalent operations are carried out that involve the solution of a secondary equation [M]{p} = {q }. By far, the incomplete Cholesky preconditioner has b ~ n the most effective in preconditioning. But the solution of this equation necessarily involves forward elimination and backsubstitution which process we have seen above, is time consuming for sparse systems. On the other hand, the less effective Jacobi preconditioning, as seen from Figs. 11.6.6 and 11.6.7, requires more iterations for convergence, but takes less time per iteration. It has been shown by Mahinthakumar and Hoole (1990b) that for large matrices, the Jacobi scheme dominates when the element matrices are stored so that they do not need to be recomputed at every conjugate gradient iteration. But more importantly for concepts of parallelism, it is shown that the element by

450

Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

element

80

......' ' " ~ ..........~

60

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o x

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0

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2000

a. Matrix Sizes Up to 1656

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----t'_r-----0---

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1000

0

2000

4000

6000

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8000

10000

12000

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a, Matrix Sizes Up to 11800 Fig. 11.6.7:

Conjugate Gradients: Time with Matrix Size

scheme with Jacobi preconditioning is best for parallel implementation. The element by element scheme was first devised for use on personal computer environments with memory limits. Here, the coefficient matrix is never formed, but rather, the equivalent operations on it are performed through operations on the element matrices. The time saving therefore comes from avoiding the expensive

S. R. H. Hoole

451

computation of sparsity associated for large matrices and the avoidance of the forward elimination and backsubstitution schemes by using Jacobi preconditioning. Mahinthakumar and Hoole (1990b) have established the approximate matrix size after which the element by element Jacobi scheme is ascendant. Such conclusions however, were based on the study of a Possonian system with Dirichlet boundary conditions using first order triangular finite elements. It is necessary however, to check these conclusions for matrices with bad convergence properties (such as when Neumann bound~ies or drastic changes in material values are present in the problem) for which stronger pr~onditioning measures are in order. But it is likely that these conclusions would continue to hold since I~r large matrices, because of limitations on renumbering, the incomplete Cholesky approximation is poor. Moreover, for higher orders, the storage of the element matrices also becomes expensive. But because these are intuitive statements, quantitiative studies are a must.

11.7. Adaptive Mesh Generation In adaptive finite element mesh generation, the mesh is refined iteratively on the basis of error estimates. The advantages are that i) The solution is accompanied by an error estimate so that one has a me~ure of confidence in the solution, ii) The solution is cheap as a result of n ~ e s being added only where they serve to enhance accuracy and iii) It allows the uninitiated to use these sophisticated programs without any t~re-knowledge of the esoterics of element properties or refinement strategies. An important aspect of adaptive mesh generation and error estimates that has never been discussed by those who advocate it, is speed and simplicity. Adaptively generated solutions can take a long time compared to a one time generated highly refined mesh created without considerations of error estimates. The philosophy behind the latter, the traditional one, is that when we use a highly refined mesh, although some nodes may be unnecessary, a reliable solution (albeit without an accompanying e~or estimate) is got quickly because adaptive error estimates i) Involve several matrix solution cycles of increasing matrix size, ii) Require much effort in extra coding and iii) Consume cpu time in estimating the errors. Those who argue for adaptive mesh generation would state that since matrix solution time goes up exponentially with matrix size, the many cycles of solution are justified because each cycle involves only a small matrix compared to that one large matrix with u n i f o ~ mesh generation. While both sides of the argument have their merit, there clearly must be a cross-over point where for large matrices the adaptive scheme might be preferable (or so this author believes). No quantitative study exists on this important issue and work must be done on this. For now however, what is certain is that the accompaniment of an adaptively sought solution by an error estimate, enhances its virtue.

452

Chapter I1: Parallel Algorithms and Matrix Solvers in Finite Elements

As far as this discussion is concerned, what are the implications of parallelism to the merit or otherwise of adaptive mesh generation? Hoole (1990) gives means of solving adaptive problems in parallel by getting each processor to estimate the error at each of a different set of nodes of the mesh. Such methods in a massively parallel system would surely tilt the debate in favor of adaptive mesh generation. It is also important to consider what would happen when the number of processors is unlimited - - in such an event, in the ideal case of 100% efficiency in parallelization, whether by adaptive strategies or a once and for all solution of a highly refined mesh, the total solution time would be cut down by the number of processors so that that approach that is best for sequential operation would continue to be best for parallel operation. Other aspects of parallelism that have not b ~ n studied in adaptive mesh generation axe a. Its application in boundary element analysis. The same procedure as in finite element analysis may be applied, with each processor being responsible for error estimation and mesh refinement in a different subset of the mesh. b The use of the existence of an approximate solution in speeding up an iterative scheme of solution such as Gauss's method which is ideally suited to parallelization. c. p-type adaptive mesh generation where the order of the trial functions is raised on the basis of the e~or estimates. Here again, the procedure would be the same as in the h~type parallel refinement.

11.8 Inverse Problems Inverse problem solution is increasingly being used m answer the industrially more relevant task of predicting or synthesizing a device given its performance, rather than the direct problem where we have to predict the t~rformance, given the device (See chapter 10 and Hoole, 1991). In solving these inverse problems, all the previously discussed methods of parallelized matrix solution, mesh generation and so on may be used. But more can ~ done in paraltelization. "We have seen in section 10.2 that in the process of optimization or inverse problem solution, the gradient of the unknown, the magnetic vector potential in the case of magnetics, with respect to every parameter has to be computed by solving [p] ~ ~p = { a{Q} 0p - O[PI{A }

(10.2.15).

That is several equations like eq. (10.2.15) need to be solved. They all have the same coefficient matrix [P], but differ only in the right-hand side. Combining these ~uations for the different parameters P[xl,x2 ..... Xn] = [bl,b2 ..... bnl (11.8.1) has to be sNved for [x 1,x2 ..... Xn] at each iteration of the synthesis cycle. Here the

S. R. H. Hoole

453

500 A e

e E

m

300

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m

3

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100

m

0

1000

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4000

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F i g u r e 11.8.1" Execution Time with Matrix Size by Cholesky's Scheme and Conjugate Gradients: a. Sequential b. Parallel

454

Chapter 11: Parallel Algorithms and Matrix Solvers in Finite Elements

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S. R. H. Hoole

455

xis are the gradients of the vector or scalar potential in 2-dimensional Poissonian problems. The two approaches investigated by Hoole (1991) are i) Doing the Cholesky factorization P = ILl [Lt] in parallel as outlined earlier and then assigning different subsets of the processors to the task of solving a different equation LLtxi = bi and ii) Using different sets of processors to do a conjugate gradients solution of Pxi=b i. Figures 11.8.1 and 11.8.2 from a 3 processor machine give the obvious advantages of taking this approach. Figure 11.8.1 shows the adavantages of the nontraditional Cholesky scheme over conjugate gradients and Fig. 11.8.2 shows the step changes in gains in parameters in multiples of the number of processors. That is, form 1 to 3 parameters, with three processor the same amount of time would be taken. But with 4 to 6 parameters, the three processors can only compute simultaneous the gradients with respect to the first three parameters and then those with res~ct to the remaining I, 2 or 3.

11.9. Multigrid Algorithms Multi-grid algorithms (Hackbusch, 1980) rely on the fact that if the true solution of a partial differential equation is expanded as a Fourier series in tern-as of the coordinates, then the finer meshes capture the higher terms better (Hackbusch, 1980). The process therefore involves starting with a crude mesh, solving and then using a refined mesh to solve for the residual consisting of the higher order t e ~ s . The process therefore allows for a natural capture of the accuracy by comparing the residual from the fine mesh, with the total solution consisting of the sum of the solutions from the previous meshes. The multi-grid algorithm is naturally prallelizable and h ~ been described in its finite-difference and parallel I~mnl. What needs m be done then is using our new massively parallel processor capability to see if multi-grid refinement can be tied up m error estimams and whether finite element mesh generation may be used in parallel multi-grid algorithms.

11.10. Coupled Problems Coupled problems are another area where parallel methodology may be used with profit to cut down the inordinate time delays attandant u ~ n solving two or more field problems simultaneously. For instance, in an electroheat problem (Gurol et al, 1989) as shown in Fig. 11.10.1, starting from time zero, it may be required to find the temperature rise of a system heated through the Joule losses in an eddy current system. Thus, the eddy current system defined by permeabilities,

By Chan and Tumimoro of the University of California at Los Angeles. Reference not available.

456

Chapter I I" Parallel Algorithms and Matrix Solvers in Finite Elements

Define Init. Condns.

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

t--t+At .........L

. . . . . . . . . . .

i

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

using ae(t-At)

Find T(t) using E(t)

t=t+At

Update ~e(t)

No

Fig. 11.10.1: Sequential Etectroheat Solution

S. R. H. Hoole

457

Define Init. Condns.

t=t+At

J

Find E(t)

i

~:

using o e(t-A t)

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

V

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

Find T(t) using E(t-At)

t=t+At

Update oe(t)

Figure 11.10,2: Parallel Electroheat Solution

458

Chapter 1l: Parallel Algorithms and Matrix Solvers in Finite Elements

conductivities and forcing currents will first have to be solved. Thereafter, the Joule rate of heating d e t e ~ i n e d from this solution will be used to d e t e ~ i n e the temperature rise of the thermal system over a little time step. The new temperature distribution gives new values to the materials of the eddy current problem, particularly the conductivity, so that the new electric fields will have to be solved for another time step and so on. The above described process presented as a flow-chart in Fig. 11.10. I, is easily parallelized as shown in Fig. 11. I0.2. The only loss of accuracy is in the fact that the temperature field is computed from an electric field value out of date by 1 step.

11.11 Systems with Limited Processors When a system is massively parallel, we have as many processors as we might wish and we proceed to parallelize all the algorithms we want to parallelize. Thus, once an algorithm is broken up into independent jobs, we seek ways to break up each independent job into other independent jobs. But often, the situation is that we possess only a finite number of processors and we would need to choose which algorithms are to be broken up into parallel segments and which run serially as wont. Generally speaking, on current machines with limited processors, the algorithm at the highest level should be run in parallel and each of its minor parts in sequence because of the increas~ed overheads in managing processors that work in parallel within parallel loops. But the study of such allocation takes on an added dimension of complexity when a task can be broken up into t independent parts and we have p processors with p smaller than t and not an integral factor of t. The natural approach is to do p of the t tasks at a time and finally, with t-ip (
Chap:te r ........1 2

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

S. Kalaiche!van .................................................................................

[1!11111

[

NUMERICAL METHODS FOR CHARACTERIZING HIGH-SPEED INTERCONNECTS IN DIGITAL CIRCUITS

12.1 Introduction This chapter introduces the characterization of high speed interconnects using electromagnetic concepts and numerical methods. The interconnects considered in this work are encountered in the digital design of Integrated Circuits (ICs) and Printed Circuit Packs (PCPs). ICs and PCPs are used in every part of electrical engineering such as computers, telecommunication equipment, cars, and consumer electronics. The ~rformance of such digital systems is measured by the appropriate functional design of the devices and the timing of the signals between them. Interconnects are one of the key elements in digital design to ensure that the required timing is achieved with minimal signal degradation. In the past, as the operating speed of the signals was low, the effects of interconnects were ignored. In recent years, the demand for high-speed technology and high-density circuitry is growing in digital design. To estimate the signal integrity of high-speed signals, the effec~ of interconnect media now have to be considered. In order to consider these effects, the interconnect medium is characterized in electrical terms. The chm-acteristics can be described using electromagnetic concepts and estimated using numerical methods such as the Finite Element Method, the Method of Moments, the Boundary Integral ~uation Method, etc.. This chapter is organized as follows: In section 12.2, an overview of the high speed digital design issues, related to interconnect effects, will ~ outlined. Section 12.3 outlines the electromagnetic concepts which describe the interconnect medium electrically. Section 12.4 outlines various relevant field formulations for interconnect characterizations. In section 12.5, various numerical methods will be outlined to derive the electrical characterization of the interconnects. In particular, the boundary element method applied to electrostatic problems and the integrodifferential method applied to skin eff:ect problems will be outlined. In

460

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

addition, numerical results of the interconnect characterization will be presented. Conclusions will be presented in section I2.6.

12.2 Digital System and Signal Integrity 12.2.1:

Digital Technology

Digital technology is one of the key technologies that have made many scientific advances in the modern world. Digital systems resulting from that technology ~ r f o r m a wide variety of information-processing tasks. Typical examples of digital systems are digital computers, telecommunication systems, calculators, and other consumer products such as electronic toys. The principle behind a digital system is the processing of information which is discrete in nature as opposed to continuous (Maho 1979). For the cases where the physical process is discrete, the data (or information) can be directly used by a digital system without any significant preprocessing. A typical example of a physical process, which is of discrete nature, is the airline reservation system. The infiormation required consists of the time, date, flight number, name of the passenger and so on. This information has an inherent discrete property since the date and time are numerics, the flight number can be alphanumeric, and the name is alphabetic. Such discrete information is processed by the digital system in its natural form. Conversely, in situations where the physical process is continuous, the information has to be quantized in a discrete manner to be used by a digital system. The human voice is a typical example of a continuous physical process. If such continuous information has to be processed using a digital system, then the information has to be quantized using a preprocessor such as an analog to digital (A~q)) converter. The generalized form of discrete information used by the digital system is known as signals. The signals are the electrical quantities of voltage or current. When the voltage or current takes a set of pre-determined values, we say that it is quantized. In general, present digital systems contain two discrete or binary, values for the signals. (Low, 0 or High, 1). The combination of these discrete values results in signals containing the info~ation and the processing of such signals to achieve the desired goal, results in a digital system.

12.2.2: Digital Design A printed circuit board is an example of a result from a typical digital design. Another example is a chip-to-chip interconnect of the board represented in a computer-aided design system. The interconnect is defined as a physical link carrying electrical signals between electrical components (Wilson 1990). Typical examples of interconnect media are wire, cable, printed circuit paths, optical fibers, connectors etc.. In recent years, interconnects have come to play a key role in the functionality of the design. As the interconnects c a m the signal ~tween two t y ~ s of devices (driver and receiver), a minimal signal degradation can be tolerated to maintain the functionality of the design.

S. Kalaichelvan

46 !

(a)

Interconnect Media

Driver(s)

Receiver(s)

(b) Figure 12.2.1: A Chip_to-Chip Interconnect To outline the effects of interconnects, let us consider the simple example of a chip-to-chip interconnect, shown in Figure I2.2.1. The schematic representation of the highlighted signal is shown in Figure 12.2.1a. The various components present in the system are as follows (Fig. 12.2.1 b): Drivers • Receivers • Interconnexctmedia Let us assume now that there is an inherent delay in a device (driver or receiver).. Consider a source injecting a signal at the input of a driver at time equal to zero (Fig. 12.2.2a ). The signal will be delayed because of the inherent delay of the driver, and the signal arrives at point B at tdl (Fig. 12.2.2b). In an ideal situation, where the interconnect effects are not present, the signal leaving B should arrive insmntaneously at C. However, due to the interconnects and the driver and receiver impedances, the signal at C is as shown in Fig. I2.2.3. Assuming the device switches to high at Vth, the continuous wavetorm (Fig. 12.2.3a) can be represented in a digital sense, as shown in Fig. I2.2.3b. Not only is there an additional delay due to the interconnects, but the quality of the signal has been degraded from the ideal case. The degradation of the signal may cause a false trigger

t=O (a)

t=O t=tdl (b)

Figure 12.2.2: Signals Delays at Different Positions of Fig, 12.2.1

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Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

vq_ _t,

!

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

0.0

td! t ~

............ | ..........

t~

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

r--

u

t=tdl t=td2

Figure 12.2.3: Degradation of Signal a) Actual Signal b) Digital Representation affecting the function of the design. There are four major phenomena that cause signal degradation (Williamson 1991): o Reflection,~inging • Crosstalk_/coupling • Ground Noise ° Losses Reflection/Ringing: This is caused by improper terminations at the driver and receiver with reference to the interconn~ts. If the receiver is not terminated with an impedance matching the characteristic impedance of the interconnect, not all of the signal will 19,zabsorbed by the receiver, and some of it will be reflected back to the driver. If the driver is also improperly matched, part of the reflected signal will, in turn, reflect at the driver back to the receiver. Continued reflections of this sort are a prime cause of signal degradation. Due to the limited scope of this chapter, the interested reader is referred to Williamson (I991), Feller, Kaupp and Digiacoma (1965) and Jarvis (1963) for more these phenomena CrosstalkJCoupling: This is due to the coupling of signals from adjacent tracks and the ground plane (Jarvis, 1963). Signals coupled to adjacent tracks are undesirable since, in some instances, they can cause false triggering. Ground Noise: This is a phenomenon in which the ground potential at various points of the ground structure will have different voltage levels (Russ, 1990). These voltage levels may cause signal degradation and, in some instances, may be the cause of false triggering. For example, if a device is supposed to receive a low input, it recognizes a low because there is virtually no difference in potential between its input and ground. If the ground potential suddenly shifts to a potential of reference for high, for instance, the device will detect a high at its input and cause a false trigger.

S. Kalaichelvan

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Losses: These can be of two types, namely, due to conductors and due to dielectrics. Such losses can cause damping on ~ e signal and reduce the tinging effects. In addition, it may also alter the voltage level which, in turn, will affect the switching of devices (Kalaichelvan 1990). To account for the preceding phenomena in the design of an IC or PCP, it is necessary to characterize the interconnects electrically. This electrical characterization can be used to predict the signal integrity and, in turn, the functionality of the design.

12.2.3: Signal Integrity Estimation Issues In estimating the signal integrity, there am two issues that have to be considered: (i) the electrical parameters (such as resistance, inductance etc.) ass~iated with the interconnects and (ii) the electrical representation (lumped or distributed) to predict the signal integrity. The appropriate choice of the electrical parameters and representation depends on many factors such as the signal wavelength and physical parameters. The details of selecting the appropriate choice can be obtained from elsewhere (Ruehli, 1987; Suriva, 1988). However, the key factors of the electrical characterization and the representation of the interconnects depend on the ratio of signal wavelength to the physical length (Suriva, 1988). In a digital system design, the physical length varies from a few microns (gin) to a few meters and the frequency of operation varies from a few MHz to a few GHz. In low-frequency system design, where the wavelength is long, signals could be appropriately represented by lumped circuit components. However, if the interconnect's physical length is short and carries a high-frequency signal, then the ratios and products of capacitance and inductance become key factors. It is important for the digital designer to be aware of the effects of the controlling parameters on signal integrity. The appropriate electrical and mathematical

l/~

Range (Relative)

Representation

representation

0

(DC)

Resistive

Arithmetic

0 < O,OC~J1

Low

Predominantly resistive

Algebraic

0.0001 < 0.05

Medium

Lumped L & C

Ordinary differential

0.05
High

Distributed

Partial differential

Table 12.2.1: Electrical and Mathematical Representations Corresponding to the Ratio Device-Length:Wave-Length

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Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

representations corresponding to the ratio of the electrical wavelength m the physical length are shown in Table 12.2.1 (Suriva, 1988). This table is presented in its simplest form: other physical parameters such as width, thickness etc. are not considered. This table can be used as a guideline for the selection of electrical representation.

12.3. Electromagnetic Field Equations 12.3.1: Statement of the problem The system we will consider consists of multiple conductors in a multiple dielectric region above or ~tween a ground plane or planes, as shown in Figure !2.3.1 or Figure 12.3.2. This system is of finite volume in the space R 3, and the problem region will have a boundary at infinity. The conductors have an arbitrary cross-section and are of finite length. The ground plane, in practice, will be of finite area. The electrical conductivity of the ith conductor is assumed to be ~i and the ~ i t t i v i t y ofjth dielectric is assumed to be Ej. The digital system consists of connecting drivers at one end and receivers at the other end of the conductors. The circuit representation, on the y-z plane, assuming all the interconnects are of the same length, is shown in Figure I2.3.3. The signals travel along the z direction, from driver to receiver(s). Depending on the characteristics of the interconnects and the driver/receiver impedances, the signals get distorted. The interconnect problem considered in this chapter is the electrical characterization of the generalized interconnect system shown in Figure 12.3.1 and

Figure 12.3.1: An Interconnect System with One Ground Plane

S. Kalaichelvan

465

Figure 12.3.2. In the scope of this chapter, the characterization of two and three dimensional interconnects will be outlined.

12.3.2: Maxwe!l's Equations The electrical characterization of interconnects can be derived from electromagnetic field theory concepts. Electromagnetic fields are governed by the basic laws of electromagnetism, which are known ~ Maxwell's equations. The laws that apply to the signal travelling through the interconnects are as follows: Faraday" s Law • ds

~E . d l =

(I2.3.1)

Gauss's law

(12.3.2)

J'E" dS = J'9. d v S Ampere' s law

j'H. dl

~D

=

gf3 .

1

÷-

d,

(12.3.3)

%1

"Ground Plane

Figure 12.3.2: An Interconnect System with Two Ground Planes

466

Chapter 12: Numerical Methods for Characterizing High-SpeeA Interconnects

v

Dn" nth Driver

Rn • nth Receiver

Z

~

• Interconnects

Figure 12.3.3

Magnetic flux ~B • d s = 0 (12.3.4) S T'tle contour, surface, and volume integrals are with reference to a generalized threedimensional object space. Maxwell's equations can also ~ written in terms of differential operators, in local form. The differential forms corresponding to the integral, forms, as we have s ~ n in chapter 1, are:: V x H = J + aD at

(1.2.3)

V x H = - aB at V •B = 0 V , D = p

(1.2.4)

12.3.2.1:

Constitutive

(1.2,5) (1.2.6)

Relations

In addition to the pre~eeding basic laws, there are constitutive relationships that describe the material properties. In general, for interconnect dielectric materials, the electric field E and the displacement field D are proportional by the permittivity as follows: D = EE

(1.2.8)

where e = eoer (12.3,5) eo is the permittivity of free space, and er is the relative permittivity of the dielectric. The proportionality of the above relation is applicable for lossless

S. Kalaichelvan

467

dielectric media. However, in recent years, the dielectric material used in interconnect media is lossy in nature and the proportionality constant (e) takes a complex value. The electric field and displacement field, for such cases, are related as follows: D = Ec E (12.3.6) and ac = e'- je" = e'(1 - J tan ~ ) (12.3.7) where ec : Complex permittivity tan Ge : Loss tangent The second constitutive relation is between the B field and the H field. These are related by the proportionality constant I~, the ~rmeability, as follows: B = ~tH (1.2.7) where g = ~o ~r (I2.3.8) ~o is the permeability of free space, and ~r is the relative permeability with respect to free space. In most interconnect problems, it can be assumed that the relative permeability of the material has a unit value. The third constitutive relation is between the electric current density J and the electric field E. ~ e s e are related by Ohm's law as follows: J = cr E (1.2.9) where z is the conductivity of the conductor.

I2,3.2.2: Boundary Conditions When there axe two different media, the following boundary conditions apply at the interface ~ t w e e n them: u n x (E 1 - E2) = 0 (12.3.9) Un x (HI - H2) = Js (12.3.10) Un • (DI - D2) = rs (12.3.11) Un • (B1 - B2) = 0 (12.3.12) where Js : Surface current density rs : Surface charge density' Un : Unit vector normal to the surface from medium 2 to medium 1 Note: The subscripts 1 and 2 refers to two media with different properties.

12.3.2.3:

Propagation

Mode

The solution to Maxwell's equations for the interconnect problem shows a wave in propagation. The general solution can assume many propagation modes. An interconnect system which has two conductors supports one propagation mode. These conductors are called transmission lines. In general, if there are N+I conductors (including one ground reference), there are N propogation modes (Gardiol, 1987). The mode in which signals propagate is called the dominant mode.

468

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

® (a)

(b)

Figure 12.3,4: Two Transmission Line Types 12.3.2.4:

TEM/Quasi-TEM

mode

A transmission line (Figure 12.3.4) when surrounded by a homogeneous and isotropic medium, carries a Transverse Electric and Magnetic (TEM) wave. A ~ M wave is an EM wave which has only components transverse to the direction of propagation. In Figure 12.3.4, the wave travels along the z direction, the H field is present in the y direction, and the E field is in the x direction (Jordan and Balmain 1980). In the interconnect system shown in Figure 12,3,1, the conductors are not always surrounded by a homogeneous medium. In some cases, there may be many different dielectric media surrounding the conductors and the dominant mode cannot be assumed to be a true TEM wave. In such systems, the wave propagation is assumed to be in Quasi-TEM mode since the longitudinal components are much smaller than the transverse components. Typical interconnect configurations with multiple dielectric media are shown in Figure 12.3.5.

(b)

i ¸¸

Figure 12,3.5: Interconnect Systems with Multiple Dielectrics

S. Kalaichelvan . . . . . . . . . . . . . . .

.,n

469 Hu

rH

u'

J"

rru

,,

uH

12.3.3 Transmission Line Equations 12. 3. 3.1: Generalized Equations As discussed in the previous section, interconnect systems are like transmission lines supporting a Quasi-TEM mode. In this section, the circuit equivalents of the transmission lines are derived from electromagnetic concepts using Maxwell's equations. In deriving the circuit equivalents, let us assume that the transmission lines carry true TEM modes as oppossed to Quasi-TEM modes. For example, consider two parallel transmission lines, as shown in Figure 12.3.6 (Jordan and Balmain, 1980). The width of the transmission lines is b and the distance between them is a. The TEM wave is propagating in the z direction such that E = Ex Ux and H = Hy Uy. ~ t us draw a contour (closed path) ABCDA on the x - z plane.and applying the electromotive force equation (Fm-aday's law (12.3.1)) : ~E ° d l =

• ds

(I2.3.1)

We now have the following: VAB + VBC + VCD + VDA= - ~ a Az

(12.3.13)

However, the potential drop along BC and DA, in terms of the surface current density (Jx) and surface impedance (Zs), is given by VBC = VDA= Jx Zs Az (12.3.14) Equation (12.3.13)can be re-written using (I 2.3.14) as VCD - VBA = . dBv a Az - 2Jx Zs Az

(12.3~15)

Assuming the distance between A and D (or B and C) as dz and expressing it in differential form, we have dV dB v ~ = - a 2J x Zs (12.3.16) In equation (I2.3. I6), By and Jx can be expressed as: By = gHy = gJx = g~

(I2.3.17)

where I is the current flowing in the transmission line and g is the permeability. Using (12.3.17), equation (12.3. !6) can be written as if'g- g a d Z'I ~-" bdt" where Z ', the series impedance, is given by Z' 2Zs =

(I2.3.18)

(12.3.19)

470

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

It[

!I :I

l Figure 12.3.6: Two Parallel Transmission Lines

S. Kalaichelvan

471

The inductance of a parallel plane of width b and spacing a has the following inductance (Jordan and BalmaJn, 1980): a

L = gg

(12.3.20)

Equation (12.3.18) can be written in terms of the inductance as follows: dV dI ~ = - L ~ ~ - Z'I

(12.3.2t)

The preceeding equation, can be written in terms of circuit components as: dV = . (L+L')dI ~ - RI (12.3.22) where R is the series resistance per unit length and L' is the internal inductance w r unit length Let us now derive the circuit equivalence of the magnetomotive force ~uation. ~ t us draw a contour FGHIgda" on the y< plane and apply Ampere's law (12.3.3):

fn.al= J J+ aD.ds

(I2.3.3)

to obtain: aEx bHFG - bHKH = eb ~ Az + c Exb Az

(12.3.23a)

~ u a t i o n (12.3.23a) can ~ rewritten as aEx bHKH - bHFG = - (eb ~ + e Exb) Az

(12.3.23b)

By rea~anging equation (12.3.23) in differential form we have & --

~b

+ e Ex

The electric field component in the x direction can be written as V Ex = -a

(12.3.24)

(12.3.25)

Using (12.3.17) and (I 2.3.25), equation (12.3.24) can be written as dl _ _ Eb aV _ eb V (12.3.26) dt a at a The capacitance and conductance, per unit length, of a parallel plate having width b and a separation a can be evaluated, respectively, as (Jordan and Balmain, I980) Eb C=-- a (12.3.27a) cb G = -a(I 2.3.27b) Equation (I2.3.26) can be written in terms of the capacitance (12.3.27a) and conductance (12.3.27b) as follows: dl dV = - C ~ - GV (12.3.28)

472

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

Equations (12.3,22) and (12.3,28) are commonly known as transmission-line equations. In estimating the signal integrity at the receiver end (Figure 12,3.3), these equations should be solved with the given source and load. R e transmission line equations deriveA above are in the time domain. The same can be written in the frequency domain by simply replacing the time derivatives by jc0, as follows: dV = -[R + jo~(L + L')] I (12.329) = -[G + j ~ ] V

(12.3.30)

The above equations represent a generalized Iossy transmission line system. However, there are cases, where certain assumptions can be imposed to simplify the equations, One such case, commonly used, is the lossless transmission line approximation.

12.3.3.2: LO Tra ion Lines In the case where there axe no losses in the interconnect system, it can be assumed that the transmission lines are perfect conductors and the dielectric is also perfect. In such cases, the series resistance and the conductance of the interconnects is zero. This approximation can be used for the situations described in Table 12.2.I. The transmission line equations can be written for lossless transmission lines, with the above assumptions, as: &at ~=

-(L + L ' ) ~

cdV &

(I 2,3,3 I) (12.3.32)

In the lossless case; the velocity of propagation along the line is given as (Wei, Harrington, Mautz and Sarkar, 1984; Venkataraman, Rao, Djordjevic, Sarkar and Naiheung, 1985): I 0 - ,f-~(12.3.33) The velocity of propagation is independant of the cross-section of the interconnects and it can be stated that for lossless transmission lines, the inductance and the capacitance have a reciprocal relationship. In a later section (Section I2.5), this property will be used to evaluate the inductance from the capacitance.

12.3.4: E l e c t r i c a l P a r a m e t e r s In the preceeding discussion, it was shown that interconnects can be characterized using Maxwell's equations and that the transmission line equations can be used to estimate signal integrity. From the circuit equivalent transmission line equations ((I2.3.22), (12.3,28)), note that there are four electrical parameters describing the interconnects as follows: o Capacitance (C) ° Inductance (L) • Resistance (R) . Conduc~'ace (G)

S. Kalaichelvan

473

For a true characterization of interconnects, the above parameters need m be determined as a function of frequency. However, approximations can be introduced in evaluating the parameters m reflect the interconnect behavior. These parameters can be obtained by various formulations of Maxwell's equations. Three formulations will be discussed in the following sections which can be usecl to derive all of the preceding par~eters: 1. Electrostatic Formulation 2. Magnetostatic Formulation 3. Skin-effect Formulation

12.4 Field Formulations 12.4.1:

Electrostatic

Formulation

The electrostatic formulation is used to evaluate the capacitance and the conductance of the interconnects. Section !2.5 will explain how the high-frequency limiting inductance can also be obtained using this formulation. In an electrostatic formulation, the field distribution is governed by the capacitive effect. It is also implicitly assumed that the magnetic effects of the system are neglected. Then Maxwell's equations for the electrostatic formulation reduce to V xE =0 (1.3.24) V oD=p (1.3.16) D = e E

(1.2.8)

Equation (1.3.24) lets us define the electric scalar potential using the vector identity (A4), which states that when the curl of a given field vanishes, then that field can be expressed as the gradient of a scalar function. The scalar potential is defined E = -V@ (12.4.1) Using equations (1.2.8) and (I2.4.1), equation (I.3.16) can be written as V • (eV@) = - p (1.3.26) Using vector identities, equation (1.3.26) becomes V • V ~= -P(5.4.2a) E Equation (12.4.2a) is known as Poisson's equation. When the charge density is not present, equation (12.4.2a) reduces to Laplace's equation as follows: V o V ,= 0 (12.4.2b) The continuity laws between two different materials determine the interface conditions. For two different dielectric materials, the interface conditions are, as derived in section 1.3. I, V s x E = Un x (El-E2) = 0 (12.4.3) V s ° D = u n ° (elEl-e2E2) = 9s (12.4.4) When surface charges axe not present, the interface conditions can be simplified to Etl = Et2 (t.3.6) elEnl = e2En2 (12.4.5)

474

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

E t and En denote respectively the tangential and normal components of the electric field (E) respectively. 12.4.2: Magnetostatic Formulation The magnetostatic formulation is used to evaluate the inductance of the interconnects. In the magnetostatic formulation, the field distribution is governed by" the magnetic effects and the time dependency need not be taken into account. Let us assume that there is no magnet present. MaxwelI's equation for the magnetostatic formulation can be written as (Defouny and Sca~a, 1989) VxH =J (1.3.23) V oB = 0 (I.2.5) B = ~t H

(12.7)

Similar to the electrostatic formulation where we defined the electric scalar potential, here we will define the magnetic vector potential (A) using equation (1.2.5) as B = VxA (1.3.7) Using (1.2.7) and (I.3.7), equation (1.3.23) b ~ o m e s V x (~V x A) = J Equation (1.3.25) can be rewfiuen as

(1.3.25)

-V x (V x A) + ~t J = (V x A) x ~ Applying Coulomb's gauge condition, it follows that

(12.4.6)

V2A + ~ J = (V x A) x ~

(12.4.7)

As the interconnec~ are linear, ~uation (12.4.7) reduces to V2A + ~ = 0 (12.4.8) The continuity laws between two different materials determine the interface conditions. For different materials, we have at the interface V s x H = u n x (HI-H2) = Ks (12.4.9) V s • B = u n • (~lHl-rt2H 2) = 0 (12.4.10) When surface currents (K s) are not present, the be~andary conditions are simplified to Htl = Ht2 (12.4.11) ~tlHnl = ~2Hn2 (12.4.I2) Ht and Hn denote respectively the tangential and normal components of the magnetic fidd (H).

12.4.3:

Skin-effect

Formulation

The skin-effect formulation is used to evaluate resistance and inductance as a function of frequency. The resistance and inductance vary as a function of frequency. The change in these values is due to the skin-effect phenomenon (Plonus, 1978). The skin-effect formulation is similar to the magnetostatic formulation; however, the effects due to the t i m e - h ~ o n i c field are also taken into account. In

S. Kalaichelvan

475

interconnect problems, as we are using this formulation to evaluate resis~nce and inductance, it is assumed that the displacement current is negligible. Therfore, Maxwell's equations for the skin effect fo~ulation reduce to: VxH =J (1.3.23)

bB

V x E = --3t

(1.2.4)

v • B = 0

(1.2.5)

B = rtH (1.2.7) J = cE (1.2.9) The total current density can also be expressed as: J = Je + J s (12.4.13) In equation (12.4.13), ,Is is the forced current density (Defouny and Scalpa, 1989) and J e is the eddy current density. Assuming time-ha~onic effects, equation (1.2.4) can be rewritten by replacing the time d~ivatives by jo~ VxE = -jo~B (I2.4.14) Using the definition of the magnetic vector potential (1.3.7) and (1.2.4), the ele~ztric field can be written as bA

E = -V,- ~

From equation (1.3.9) and (12.4.13), we have 0A Je = - a 3t Using (I.3.9), equation (1.3.23) becomes V x (1 V x A) = - c; 0A ~ + Js

(1.3.9)

(12.4.I5)

(12.4.16a)

In two-dimensional problems, the crV, factor is included with the forced currents (Js)- However, in 3-dimensional problems, this is an unknown factor and a scalar equation must be solved along with (I2.4.16a). Using vector identities, equation (I 2.4.16a) can be reduced to I 0A V2A - cr ~ = -Js (12.4.16a) The interface conditions are the same as in magnetostatic formulation (12.4.9)(12.4.I2).

12.5 Numerical Solution There are numerous methods available for reducing the continuous system to an equivalent discrete representation. The method of obtaining the discrete representation becomes a crucial factor due to computational limitations. The attractive features of widely used methods are reviewed in this section. The evaluation of each method includes the aspects of replacing the two- or threedimensional solution domain with discretized space without affecting the definition of the problem.

476

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

Lumped network model The lumped network model involves a representation of the continuous field in terms of lumped electric and magnetic circuit components. One of the main advantages of using a lumped network model is that physical insight is retained at least from an engineering point of view. The transformation of the Maxwell's electromagnetic equations into a lumped network was done in the early 1940's (Kron 1943). In the past, two- and three-dimensional problems, have been solved successfully (Silvester, 1966) using network analogues. In fact, Balchin and Davidson (1980) have solved a three-dimensional problem using the network model. Although many researchers are involved in solving electromagnetic field problems using network models, a general solution method using the network model is difficult to obtain. This is due to the physical complexity of most practical three-dimensional problems. These issues restrict the use of network models for solving generali~d two- or three-dimensional interconnect problems.

Difference method The basic principle of this scheme is to approximate the global differential operator a local difference operator. The s~ution domain is divided into a finite h u m o r of mesh points known as nodes. The derivative at each point is replaced by a difference approximation using the neighboring nodes. There are numerous schemes available using this approximation. Recently, time domain finite difference has been attempted by many (Liang, Liu and Mei, I990; Railton and McGeehan, 1990), to solve full-wave equations. Such formulation can be used for interconnects encountered at microwave frequency. Some of the advantages and disadvantages of the finite difference method are listed below: Major advantages The numerical formulation is simple. It is easy to implement for geometries with simple contours. - The system matrix is sparse. Major disadvantages Assignment of the nodal material properties and sources is not unique and that is an important factor in solution accuracy. - The boundary conditions have to be explicitly imposed. This method becomes cumbersome for complex systems, especially systems with complex contours. In many problems, the increase in the number of nodes does not improve the solution accuracy (Demerdash and Nehl, 1979). In most of the cases, the system matrix is not symmetric. - The discrete problem is often solved using Successive Over Relaxation (SOR), and the selection of the convergence schemes becomes an important factor in some problems where the problem space is not homogeneous and non-linear. -

-

-

-

S. Kalaichelvan

477

The exterior field requirement is attained by introducing an artificial boundary,

-

Finite.element method Unlike the finite difference method, where the domain is replaced by a set of discrete nodes, this method divides the domains into subdomains referred to as finite elements. The unknown field quantity is approximated over the subdomain as a trial function which is piecewise continuous. As in the finite difference method, the finite element approach is conceptually simple, but in contrast, mathematically rigorous. In general, there are two approaches in the mathematical formulation of the finite element method: (i) the variational and (ii) the residual formulation. In the variational formulation, the energy functional is minimized; whereas in the residual formulation, the residual function is minimized. In both cases, the differential equation is transformed into a set of simple algebraic equations. To date, a large number of one - and two - dimensional electromagnetic field problems have been solved successfully using these formulations (Silvester and Chari, 1980; Lavers, 1983; Kalaichelvan, 1984). The high reliability and the stability experienced in solving two-dimensional problems using the finite element method have prompted many researchers to extend this method to three dimensions. However, certain computational limitations have caused slow growth in solving three-dimensional problems using this approach. The major advantages of the finite element schemes are This rrm:thod can be used for a non-homogenexms medium This scheme is proven to be stable for many two-dimensional problems and is expected to be stable for three-dimensional problems. The c~fficient matrix is sparse and, in most cases, symmetric. Higher order polynomials can be used for better representation of complex contours and for better solution accuracy. The homogeneous Neumann condition is implicit. The major disadvantages of this scheme are In three-dimensional problems, to represent the geometry a large number of volume elemen~ are re~quired. This becomes a limitation in essing and memory requirements. The representation of the exterior field space is not natural. With finite elements, either an artificial boundary has to be imposed or some special t~hniques, such as ballooning (Silvester, Lowther, Carpenter and Wyatt, 1977), must be implemented. The number of unknowns, especially for three-dimensional subdomains, requires a large amount of input data preparation. However, a preprocessor can be u s ~ for such purposes. An efficient sparse solver is required. -

-

-

-

Integral method For this method, in most cases the problem space is replaced by an equivalent source distribution, and this source distribution simulates the actual problem. In

478

Chapter I2: Numerical Meth~s for Characterizing High-Speed Interconnects

this approach, the unknown field quantity can either be a virtual source or the direct field quantity, defending on the type of formulation used. The integral methods can be broadly classified as volume integral methods and surface integral methocls. The surface integral method has been widely used to solve two-dimensional problems (FawN, Ali and Burke, 1983, t984). Few researchers have attempted to solve three-dimensional problems using volume integral methods (Layers, 1982; Kriezis and Cangellaris, 1984; Zaky and Robertson, 1973). However, for threedimensional linear problems, the surface integral method has more advantages as compared with the volume integral and differential schemes (Tozoni and MayeNoyz, 1974; Mueller, 1969). The main advantages of integral schemes are The exterior field requirement is implicit in the process of obtaining the mathematical formulation. In the surface integral formulation, the dimensionality of the problems is reduced by one. This is especially useful when solving three-dimensional problems. The field at any single point can be obtained as opposed to the distribution for the whole region for s~c~y differential schemes. The required field quantities can be derived from the primary state variables using inmgraI operators as opposed to differential operators. Simple schemes are sufficient to solve the system of equations as opposed to the sparse techniques to be used in finite element or finite difference methods. The main disadvantages of this method are The evaluation of certain integrals needs special treatment. - The type of Gauss integration scheme affects the solution accuracy. The system matrix, in most cases, is dense. Certain ~'-mulafions are not numerically stable. - In exploiting symmetry, the computational requirement to form the system matrix is not reduced as much as in the finite element method. -

-

-

-

-

-

12.5.1: Boundary Integral Equation Method The purpose of this section is to outline the basic formulations that have been used to develop the Boundary Integral Equations (BIE) for electrostatic problems and to outline a boundary integral equation method to solve the interconnect problem.

12.5.1.1: Direct Formulation In the direct formulation, the field quantities of the Laplacian operator (1.4.1) and (5.4.2b)) are themselves the unknowns. Many such formulations have been developed in the past (Lean and Friedman, 1985; Defouny and Scarpa, 1989), focusing both on two- and three- dimensional applications. The advantage of the direct fi3rmulation is obvious, since the field quantities of interest are immediately obtained instead of through post-processing. In some cases, such as threedimensional eddy-current formulations (Kalaichelvan and Layers, 1989), direct formulations are disadvantageous since the number of unknowns is increased.

S. Kalaichelvan

479

The derivation of the inmgral equations for the direct formulation can be quite complex (Lean and Friedman, 1985; Defouny and Scarpa, 1989). 12.5.1.2: Indirect Formulation Rather than fo~nulate the electrostatic problem directly using the surface field (,), it is possible to formulate the problem indirectly in terms of a fictitious source distribution over the surface and interfaces. The integral equations are formulated in terms of these fictitious sources and solved for them. The electromagnetic field quantities are derived from these fictitious sources. For example, for the electrostatic problem, the potential (,) within the region can be obtained using an equivalent electric charge density (e0 as (H~ington and Wei, 1984):

*(;) = ~

(;,n)e(n)d% (12.5.1) s where rrl and r{ are the source and observation point. The Boundary Integral Equations that must be solved for ~J are described in the next section. The obvious disadvantage of any indirect formulation is that once the BIEs have been developed, discretized and solved, post-processing is required to obtain the electromagnetic field quantities. However, the advantages can be significant in some areas of 3-dimensional problems where the number of unknowns is reduced compared to the direct formulation. Such a reduction is useful for large scale 3-D problems where a large computational effort is requirex.t. An indirect formulation for the electrostatic problem applied to interconnect characmfization is discussed in the next section.

12.5.2: BIE Method for Interconnect Problems

12.&2.1: Integral Equations Let us consider N conductors in tel dielectric regions. Using the indirect forrnulation, assume that there are sources on the interfaces. Let us also assume that a total charge density' (c) per unit area is present on the conductor and dielectric interfaces. The potential ~ at any point x due m the charge distribution is (12.5. I): 1 jrG ~,(¢) = ~ (;,n)~(n)ds n s Green's function (G) for 2-dimensionaI problems is given by: In

(! 2.5.1 )

(12.5.2)

and Green's function ((3) for 3-dimensional problems is given by: I

(12.5.3)

The a ~ v e equation is subject the following interface conditions: Conductor to diel-ectric: ~(~) =¢i,

i = 1,2,..N

(12.5.4)

480

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

Dielec~"ic to

e 1E I (~)'Un = e2E2(;)°Un (12.5.5) The above integral equation along with the interface condition is solved for 2 and 3-dimensional problems. The boundary integral equation (I 2.5. I) for a pa~icular field point { on the interfaces is the basis for obtaining the virtual source distribution. It is necessary to solve the continuous integral equation over the interface surface S. The analytical solution of such an integral equation is not always possible, and it is impossible for complex geometries such as the one defined in the statement of the problem. In such cases, the integral equation has to be solved for a discretized source distribution as ~posed to a continuous source distribution. Let us simplify our system by considering two types of interfaces to accommodate the interface conditions: (i) dielectric to conductor and (ii) dielectric to dielectric (Wei, Harrington, Mautz and Sarkar, 1984; Rao, Sarkar and Harrington, 1984; Harrington and Wei, I984). Before discretizing the surface S, let us ~sume that S' is the surface along the conductor-to-dielectric interface and S" is the surface along the dielectric-to-conductor interface such that: S= S' + S" (12.5.6) In order to solve for the discretized source distribution, first let us assume the surface S' consists of m interfaces, and surface S" consists of n interfaces. Using the interface conditions (12.5.3) and the integral equation (12.5.1), we have m+n Z I~(n)G(~,n)ds n - *i ,/

(12.5.7)

j = l Sj

In order to apply the interface condition (12.5.5) on the dielectric to dielectric interface, first let us consider the electric field E E(;) = V~;) (12.5.8) If we substitute the expression for 0 in (12.5.1) in equation (12.5.8), we can obtain the electric field at any point (not on the surface) m+n E(;) = Z [~n)G(;.n)ds n

j=l sjd

(12.5.9)

The E field on the surface is the limiting case of equation (12.5.9) as ~ ~ S. Taking the limit, the electric field on the surface is given by: m+n E1,2(~) = Z ~(n)G(~,r0ds n + n ~ j=l s] - 2e0 Now the interface condition (12.5.5) can be written using (12.5,10) m+n [e(T1)G(;,rl)dsn - n + el Z 2c0 1 j = I s3

(12.5,10)

S, Kalaichelvan

481

m+n fe~ e2~({) = e2 £ 1-(n)G(~'n)dsn ° n - 2e0

>1

(12.5.1 I)

Rearranging ~uation (I 2.5. t 1) and dividing by (~1 "g2) we have: m+n _ (el+eZ)a(~) + Z 2(el-e2)e0 j=l sj

j~(n)G(;,n)dsn • n = 0

(12.5.12)

Equations (12.5.7) and (12.5.12) are the integral equations to be solved f;ar the unknown charge density (c). In high-speed design, it is preferred to have a single ground plane (microstrip) or two ground planes (stripline) to minimize noise. In such cases, equations (12.5.7) and (12.5.12) can be modified using the method of images to simplify the computation and this has been proven to be more stable. Let us assume that there is a ground plane present in the interconnect system as shown in Figure 12.3.!. For such situations, equations (12.5.7) and (I2.5.12) can be m~ified as m+n

Z

(12.5.I3)

[~.rl)[G(;,rl)-G(;,rl')]dsrI = ~i

,t

j=l s'j m+n (e 1+e2)~(~) + j=lz 2(el-e2)e0

)]d srl • n = 0

(12.5.14)

where s' represents the position of the image located equi-distant from the ground plane as the source ~int. For the case of two ground planes, there exists an infinite number of images. In such cases, we can treat the upper ground plane as a conductor with zero potential.

12.5.2.2: Parameter Circuit Representation Solving equations (12.5.13) and (12.5.14), we obtain the charge density distribution along the interfaces. As stated earlier, to characterize the interconnects electrically and to estimate the signal inte~ity, it is required to evaluate the circuit equivalent parameters. This section outlines the evaluation of the capacitance and conductance using the above electrostatic fo~ulation. In addition, a high frequency limit of the inductance matrix will also be derived.

Capacitww_e A parallel-plate capacitor is the simplest form of a capacitor. It consists of two plates with an area A and spacing D between them. There is a uniform dielectric medium with permittivity E between the two parallel plates. The capacitance of such a system is expressed as eAJd. However, in practice, the interconnects are multiple conductors with an arbitrary shape in a multiple dielectric medium. The

482

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

Figure 12.5.1:A Homogeneous Dielectric with n Conductors of Arbitrary Shape capacitance of such a system can be represented in terms of classical circuit theory using a capacitance coefficient matrix. Consider a homogeneous dielectric medium with N conducting bodies of arbitrary shape as shown in Figure t2.5. I. Assume that the zero potential point is at infinity. The total charge Qi on the ith conductor can be written in terms of the applied potential, using supe~osition, as follows (Ling and Ruehli, 198 I): N Qi = z cij fi (I2.5.I5) j=l where C is the NxN capacitance coefficient matrix. In the above, Cij is the charge on the ith conductor, when the jth conductor is raised to a unit potential and when all the other conductors are grounded. The following are the properties of a capacitance coefficient m a r x : 1. cij = cji 2. Cii is positive or zero 3. C~ (i <> j) is negative or zero N 4. The sum Z cij is positive or zero j=l There is a relationship between the capacitance coefficients and classical circuit theory. The classical ~uivalent circuit for a three-conductor system with a reference is shown in Figure 12.5.2: R e preceding circuit can be expanded for a general N conductor system. It can be proved for a general N conductor system that: N Cij = Z lCU" (I2.5.16) 3~

Cij =-cij (12.5.17) For the interconnect characterization, the above relationship can be used to derive the equivalent circuit from the capacitance c~fficient matrix. Having understood the concept of the capacitance coefficient matrix, it is necessary to evaluate the same using the charge density distribution obtained by solving the electrostatic formulation.

S. Kalaichelvan

4 83

!I

-• ....

3

~0

T

w

Figure 12.5.2: Equivalent Circuit of a 3-Conductor System The charge density obtained by solving using (12.5,13) and (12.5,14) can be related to the free charge density distribution as follows::

~F(;) = ~

~(;)

(12.5.I 8)

The capacitance coefficient (cij) can be evaluated by setting the potential of the jth conductor to unity and all the other conductors to ground potential. The expression for the capacitance coefficient is as follows: c~j = J - - ~ a ( ; ) dS (12.5.19) si The above procedure is carried out for N conductors to obtain the NxN capacitance coefficient ma~ix.

484

Chapter !2: Numerical Methods for Characterizing High-Speed Interconnects

Conductance The conductance matrix can be obtained using the electrostatic formulation (Harrington and Wei, I984). In the electrostatic fo~ulation described in the earlier sections, the dielectnc constants had a real value as opposed to a complex value, In evaluating the conductance matrix, we replace the real dielectric constants by a complex dielectric constant e~ = ~' - je" = ~'(I - jtan 5~) (12.5~20) where ~c : Complex permittivity tan ~e : Loss tangent The loss tangent tan ~ = ~ (12.5.21) Using the complex dielectric constant, we can evaluate the complex capacitance matrix [c'] for the interconnect system. The complex capacitance matrix is defined as:

jo~[c'] = [a+j0~] The conductance matrix can be obtained as [G] = Real [j~'] The capacitance matrix for the lossy case can be obtained [c] = ReN[c]

(12.5.22) (12.5.23) (12.5.24)

The high-frequency limit of the inductance matrix can be obtained using the electrostatic formulation. At high frequencies, the skin effect dominates and the internal inductance is assumed to be negligible. In such cases, only the external inductance should be considered. For two dimensional cases (Wei, Harrington, Mautz and Sarkar, 1984), the high-frequency limiting inductance is related to the free space capacitance coefficient matrix by 1

ILl = ~

[Co]-1

(12.5.25)

where u : Velocity" of light in free space Co : Capacitance c~fficient matrix of free space The capacitance coefficient matrix of free space is obtained by replacing the dielectrics with relative permittivity greater than unity with free space and by solving for the capacitance c~fficient matrix.

12.5.2.3: Discretization In solving equations (12.5.7) and (12.5.12), the surface of the interfaces has m be divided into subelements. In 2-dimensional problems, the surface elements will be one-dimensional line elements. Conversely, in 3-dimensional problems, the elements will consist of two dimensional surface patches. Let us consider dividing each interface into elemen~ l

S. Kalaichelvan

485

m+n

S=S1 +S 2=Z Ii i=l Using (12.5.26), equation (12.5.7) and (12.5.12) can be rewritten m+n/j t' Z Z ](ffr0[G(O1)-G(~,n')]dsn = q~i j=l k=lsj1

(~ I +E2)cffO m+nlj

(12.5.26)

(I 2.5.27)

ff

+ Z Z J~(n)fG(~,n)-GGng]ds n • Un = 0 2(al-E2)e0 j=l k=ls)

(I2.5.28)

where Ak represents the kth element. As stated earlier, for 2-D problems, Ak represents the line element, and, for 3-D problems, it represents surface patches. The same procedure can ~ adopt~ in solving eqs. (1Z5.13) and (12.5. I4) for the case where the ground plane is present. Equations (I2.5.13) and (12.5.14) are evaluated for each element, and a global assembly of equations is perth.~ed. This global assembly results in a set of algebraic equations which are in matrix f o ~ [A]{c}={V} (12.5.29) where the vector {eq is the unknown charge density, and V is the known right-hand side vector in terms of known potentials. This equation is solved using standard linear equation solvers. In assembling the coefficients of the matrix (A) of equation (12.5.29), it is required to evaluate the following integrals over the elements.: I1 = S s(rl)G(~,rl) dS~1 Ak

(12.5.30)

12 = ~ s01) VG(Grl) dSrl (12.5.31) Ak The following section describes the parametric representation of each element to the integration.

12.5.2.4: Parametric Ftepresentation of Geometry and Function 12.5.2.4.1: One DimensionalRepresentation The one-dimensional representation of geometry results from discretizing the boundary of a 2-dimensional problem as shown in Figure 12.5.3. These elements are called line elements. Equations (12.5.30) and (12.5.31) can be written in a general f o ~ as AI =

~{f(xI,x2,x3)U} dI(xl,x2,x3) (!2.5.32a) zM(Xl ,x2,x3) where x 1,x2,x3 are position coordinates, f(x l,X2,X3) is a function (G or V) of the position vector, and u is the source distribution (c) over the line element. To

486

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

Line Element

Dielectric

Figure 12.5.3: Discretization of Boundary into Elements simplify matters, assume the source distribution u is uniform over each element; in which case, u can be moved out of the integral in equation (12.5.32a). . 1 1 1 222 Consider a hne element I with coordinates (XI,X2,X3) and (Xl,Xz,X3) as shown in Fig. I2.5.4a. Although the line element has three-dimensional coordinates, its true dimension is only one. For that r e , o n , let us transform the line element from the three-dimensional global coordinate system to a one-dimensional local coordinate system. This will enable us to perform the integral operation in one dimension. The transformation can be accomplished, for a linear elements, using the nodal coordinates and basis functions as follows: 2 x i - Z Nk(~Z)x k (12,5.33) k=l where N 1 = ~(1-~)

(12.5.34)

N 2 = ~I+K)

(12.5.35)

2 X2 X~) (XI' 2'

(x: , X :2, x~)

1

2

-1

+1

Figure 12.5.4: Coordinate Transformation of a 1-D Element a) Three-Coordinate De~ription b) One-Coordinate Description

S. Kalaichelvan

487

Similar transformations can be accomplished for higher order elements. The preceding transformations (12.5.34 - I2.5.35) let us pertorm the integration over the limit -1 to +1 for equation (12.5.33a) as follows: +1 AI = Sf0c)u dl(~) (12.5.32b) -1 12.5.2.4.2: Two-Dimensional Representation The two-dimensional representation of geometry results from discretizing the boundm,'y of a 3-dimensional problem. For the pu~ose of the present discussion, discretization of the surface S is into triangular elements. This will therefore require the numerical integration of a function f over an elemental triangular patch as follows: AI =

~{f(xI,x2,x3)U} dS(xl,x2,x3) (12.5.36) AS(xl ,x2,x 3) where Xl, x2 and x 3 are position coordinates, f(xI,x2,x3)is a function of the position vector, and u is the source distribution on the triangular element. As explained in the earlier section, a unitbrm source distribution is assumed over the element. To simplify the integration of any function over an element, a transformation of the elemental patch to a planar triangle is required. A patch or 2-flat is a twodimensional element having three-dimensional coordinates (Kalaichelvan, 1988). The transformation involves representing the global coordinates of a point in the original triangular patch - - the latter having arbitrary orientation in space - - in terms of the intrinsic nodal c~rdinates and shape functions of the planar element. 1 I 1 2 2 2 Consider a triangular element having nodal coordinates (xl,x2,x3), (xl,x2,x3), . 3 3 3

and (xl,x2,x3) as shown in Fig. 12.5.5a. The triangular patch can be mapped to a planar triangular element in terms of the intrinsic coordinate system ~:l, ~2 and ~3 (Fig. 12.5.5b). For linear elements, the mapping between the global coordinate system (x l,x2,x3) to the intrinsic coordinate system (~:1,~2,~3) (cal. the triangular coordinates of Fig. 1.5.8) can be accomplished by the following: 3 xi = ~ NI(KI,K2d¢3) Xi1

for i=I to 3

(12.5.37)

1=1

where NI = ~:1 (I2.5.38) N2 = ~2 (12.5.39) N3 = ~:3 (12.5.40) Using the pr~eding equations (12.5.37 - 12.5.40), the differential area dS can be written as dS=lTId~l d ~ (12.5.41 ) where ITI is the transformation Jacobian

488

Chapter12: NumericalMethods for Characterizing High-Speed Interconnects

X2

2 i~r'1~2 ~

1' 2'

t~S

3 X3 X3. ~1" 2' 3)

X v X 2• "2

r~l

X3

Figure 12.5.5: Mapping of Global Coordinates (Xl,X2,X3) to Intrinsic Coordinates (K !,K2,~:3) ., /

2

2

2

ITI= ml l+m22+m33 while m ! I, m 12 and m 13 are the minors of I

OXl 1 Ox2 1 Ox3 I

(12.5.42)

l

[M]=

0~1 O~l O~:l (12.5.43) 0Xl 0x2 ~)x3 0~2 OK2 0~2 The integration over the polynomial in terms of triangular coordinates can be perfo~ed using the following: tfi~:1 ~J2 " -,c~dS= (i+j+k)! 2AS (I2.5.44) z~ where i, j, and k are the powers of the intrinsic coordinates ~1, ~2, and ~3 respectively. This is proved in Appendix D at the end of the book. There are

S. Kalaichelvan

489

various computationally efficient integration schemes available for triangular and quadrilateral patches (Kalmchelvan, 1987; Kataichelvan and Lvers, 1988b).

Higher order elements: The typical higher order elements are the quadratic and cubic polynomial representation of the elements. When using triangular elements, the mapping of global to local coordinates for quadratic variation can be accomplished using 6 x i = LNl(~:l,~:2,~:3) x I for i=l to 3 (12.5.45) 1=1 where N1 = ~:I(2~:1 -1) 1 = 1 to 3 (12.5.46) N4 = 4~:I~2 (12.5.47) N 5 = 4~¢2~:3 (12.5.48) N 6 = 4~3~4 (12.5.49) A similar transformation can be p e r f o ~ e d for cubic polynomials and higher order representations. These mapping techniques and the corresponding transformation can be obtained from elsewhere (Konrad, 1982). The higher order representation requires a greater number of nodes than the linear case and the implementation becomes cumbersome. Nevertheless, these elements are useful for complex geometries where the true representation of the geometry can be easily achieved. Until now, the discussion has been restricted to the representation of the geometry assuming a constant source density over each element. Although there are numerous ways of representing any source variation, the simplest variation is constant over the element. This approximation is easy to implement and allows one to define the source point at the centroid of each triangle. Other types of source modeling include linear and higher order variations and the approximation/transformation (similar to the geometrical transformation) is as follows: n

U = ~ NI(~:)U1 (12.5.50) 1=1 In the above, n is 3 or more for linear or for higher order representations while U 1 is the nodal value of the source density. Linear or higher order variation involves nodal and/or intere!ement boundary values as opposed to the centroid value. This poses a problem in the formulation where linear or higher order polynomial variation is used since sharp corners or edges do not have a unique normal. In such cases, an approximation is required to define the normal at the nodes. For our purposes, a constant variation is assumed for the source density and linear elements are used for geometrical discretization. The development of the software is modular, and an extension to linear variation of the source density can be implemented without difficulty.

490

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

Figure 12.5,6: Two Spheres in a Homogeneous Dielectric Moreover, it is interesting to note that the normal vector of the triangular patch can be easily obt~ned as follows (Kalaichelvan, 1987): A

^

^

A

ml 1 Ul + m12 u 2 + m13 u 3

(12.5.51)

A reduction in the computational requirement is achieved using equation (12,5.51) since the transfo~ation is a necessary r~uirement for surface integration, 12.5.2.5 Numerical Results

The Sphere Problem The classic problem of two conducting spheres (Fig. 12.5.6) in a homogeneous dielectric provides a convenient means of verifying the 3-dimensionaI Boundary Integral Equation method for the electrostatic formulation. This problem has a known, closed-fo~ solution (Greason, 1987).

Figure 12.5.7: Boundary Element Mesh on Sphere Surface

S. Kalaichelvan

491

Analytical

3-D BIEM

45.2

43.28

C22

18.9

17.93

C12

-7.7

-6.93

Capaci~ce .......Coefficient pF Cll

.........

TaMe 12.5.I: Comparison of Solutions to the Sphere Problem For the tests, no attempt was made to exploit the symmetry; tests were p e r f o ~ e d for various number of elements distributed over the f:alI surface of the spheres as shown in Fig. 12.5.7, These represent a relatively crude discretization of the surface. The test spheres have a radius of 0.38 m and 0.156 m with a separation of 0.4 m. The problem was solved for the surface charge density by setting the potential of each sphere at unity and setting the other to ground potential. The capacitance coefficients were calculated for the two-sphere problem and compared against the analytical solution as shown in the Table 12.5.1.

Microstrip Configuration This test problem was an interconnect configuration that consists of a pair of coupled microstrips over a dielectric and a ground plane as shown in Fig. 12.5.8. The dimensions were s=2, w=3, h=l, and t=l. The relative pe~itfivity of the dielectric was 2.0. This problem was solved using 2 and 3-dimensional boundary integral equation methods. In the 3-dimensional case, the length of the conductors was assumed to be 20.0 as in (Rao, I984). The capacitance coefficients are re~rted in Table 12.5.2. The results are seen to be in good agreement with the available results (Wei 1984; Weeks 1970). The various tests showed that an increase in the number of elements produced an improved rate of convergence. In this test case, the method of images was used by removing the ground plane.

u

h w

A

Figure 12.5.8: Coupled Microstrip over a Dielectric and Ground Plane

492

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects

Capacitance Coefficient pE/m CI1

Weeks .....................~I970) 92.24

Wei

3-D BIEM

(!984)

......

......

91.65

94.61

C!2

-8.50

-8.22

-7.98

C21

-8.50

-8.22

-7.98

C22

92.24

.

.

.

.

.

.

.

91.65 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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

94.61

Table 12.5.2: Comparison of Capacitance Solutions to the Coupled

Microstrip Line This problem was also solved by replacing the dielectric slab with free space for the capacitance coefficients. Using the capacitance matrix and equation (I2.5.25), the high-frequency limiting inductance matrix was evaluated and is shown in Table 12.5.3.

Stripfine Configuration The test case, a stripline, is a typical interconnect configuration encountered in digital circuit design. A stripline has a dielectric layer or multiple layers sandwiched betw~n two ground planes. The signal-carrTing conductors are present between the ground planes as shown in Fig. 12.5.9. If the conductors are located at the middle of the distance between the ground planes, it is known as a symmetric str~Mine configuration. The dimensions used in the test case were: s=2.0, w=3.0, t= I.O, h= l.O and b=5.0. In this test case, since two ground planes are present, using the method of images would result in an infinite number of images. In order to prevent this, the top ground plane was assumed to be a conductor with zero potential and of infinitesimal thickness. With this assumption, the method of images can be applied by removing the lower ground plane. The capacitance coefficient matrix is shown in Table 12.5.4. Wei

Inductance nHJcm (1970) ........ 1.98

,,,

LI1

3-D BIEM

(I984) 1.86

1.78

L12

0.29

0.24

0.25

L21

0.29

0.24

0.25

1.86

1.78

L22

......

1.98

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

Tble 12.5.3: Comparison of Capacitance Solutions to the Coupled

Microstrip Line

S. KaIaichelvan

493

| E~

Figure 12.5.9: Conductors of a Stripline Between Two Ground Planes Further tests were performed to check the performance of the method for multiple conductors and satisfactory results were obtained.

12.5.3: Integrodifferential

Finite Element Method

~ i s section discusses a numerical method fbr evaluating the time harmonic losses. There are various numerical methods available for evaluting the time harmonic losses for 2 and 3 dimensional structures (Ka!aichelvan, I987; Konrad, 1981; Costache, 1988; Balakrishnan, 1986; Djorjevic, Sarkar and Rao, 1985; Cangellaris, 1990). The appropriate field formulation, which includes the time harmonic effects, is the skin-effect formulation outlined in Section 12.4.3. Among the various numerical methods for skin-effect problems, the integrodifferential method (Konrad ,1981 ) is attractive for the 2-dimensional case and the Mayergoyz method (Mayergoyz, 1983) is attractive for the 3-dimensional case. In this section, the integrodifferential method applied to 2-dimensional problems will be discussed. The Mayergoyz method and its related implementation can be obtained elsewhere (Mayergoyz, 1983; Kalaichelvan and Layers, 1987, 1989). The integrodifferential method is based on finite elements, and the Mayergoyz method is based on BIEs. Capacitance Coefficient pWm CII

"Weeks

<1 970)

Wei (,! 9 8 4 ) ................

63.07

62.64

' 67.48'

C12

-58.66

-5.72

-58,57

C21

-58.66

-5,72

-58.57

C22

63.07

62.64

67.48

3-D BIEM"

Table 12.5.4: Capacitance Coefficients for Stfip!ine with Two Conductors Two Ground

494 .

.

.

.

Chapter 12: Numerical Methods for Characterizing High-Speed Interconnects .

.

.

Using the skin effect fo~ulation outlined in section 12.4.3, we can replace the two unknowns (the magnetic vector potential A and the source current density Js) with a single differential equation with one unknown. The skin-effect formulation results in the following differential equation as described in Sec. 12.4.3: ! V 2 A - ~ aT O A = -Js

(12.5.52)

Equation (12.5.52) can be writmn in terms of a time-harmonic representation as follows: 1 . aA V2A - j m ~ = -Js (12.5.53) In equation (12.5.53), there are two unknowns A and Js- To simplify, let us consider the tomI current I ~ (Konrad, 1982) I = I s + Ie (12.5.54) where Is : Fictitious source current Ie : Total eddy current The source current is given by Is = j' Js * da = Js " a

(12.5.55)

s

where a is the cross-sectional area of the conductor. The eddy current is given by Ic = - j~g ~ A • d s

(I 25.56)

S

Using (12.5.54 - 12.5.56), we have Js = ~+"jma~ ~ A • d s

(12.5.57)

s

Substituting (12.5.57) in equation (12.5.53), we obtain the integrodifferential equation (Konrad, 1981, 1982) 1

I

VZA-jmc;A +jo~-j " a • ds = - as

(12.5.58)

a

In the preceding, the unknown is the magnetic vector potential (A) and the total measurable current is I. The details of the th~ry behind this formulation are outlined by Konrad (198I, 1982). It is required to solve the integrodift:erential equation (12.5.58) for the magnetic vector potential A.

12.5.3.1: Implementation Let us consider using the GalerkJn's method (Huebner, 1975) along with the finite element theory. The unknown magnetic vector potential, for 2-dimensional problems, has the longitudinal component along z-direction. The arbitrary crosssection of the conductor lies along the x-y plane.

S. Kalaichelvan

495

The continuous system is discretized using triangular (or quadrilateral) elements. For the purpose of this discussion, let us consider triangular elements. In the two-dimensional finite element method, as in the present case, the discretized elements are two dimensional. This is different from the Boundary Integral Equation method where the discretized elements axe of one dimension less than the solution region. Assume that the field varies according to equation (12.5.50) within each element. The procedure fbr handling the triangular elements that was discussed in Sec. 12.2.3.I can be applied to this implementation. Using equation (12.5.50), the magnetic vector potential at any point in the triangular element can be expressed as tollows: a

A = Z AjNj(~:) (12.5.59) j=l where Nj(•) is the interpolation polynomial, and a is the number of interpolation nodes. For a linear variation, the number of interpolation nodes is 3 which can be considered as the vertices of the triangle. For higher order polynomials, the number of interpolation nodes will correspondingly increase. The integral of the field component (A) over a triangular element can be expressed using the Newton-Cotes quadrature f o ~ u l a (Silvester, I970): a

J'A" ds = Z s J =IAjWj(~)

(t2.5.60)

where Wj are the weights, and Aj are the magnetic vector potential values at the nodal points. Using (12.5.59) and (12.5.60), equation (12.5.58) can be written, for a single triangular element as: { [P] + j ~ o [ T ] - j 0 ~ [ ' v V ] } {A} = [T]{J} (12~5.6I) where: [P]ij ~

J'VNi ° VNj dS

i,j =I,2,3 .... o~

(12.5.62)

S

,fN . Nj dS = fN . r' k dS S

[W] ~ Wj {A}i ~ ~ I {J}i ~ - a

S

i, j, k = 1,2,3 .... ~ i =1,2,3 .... o,: i =1,2,3 .... ~

(12.5.63) (12.5.64) (!2.5.65)

i =1,2,3 .... ~

(12.5.66)

In the above, a is the number of interpolation nodes of each triangle. Equation (12.5.61) can be assembled for each triangle in a global f o ~ as: [Sc]{A} = {b} (12.5.67) where S c : Complex coefficent matrix A : Unknown magnetic vector potential

496

Chapter I2: Numerical Methods for Characterizing High-Speed Interconnects

b : Known right-hand side vector It is required to solve the linear system of equations to obtain the unknown nodal magnetic vector potential A. Unlike the BIE method where the system matrix in most cases is dense, in this case Sc is sparse. There are special techniques available to solve such a system of equatioin (Kalaichelvan, 1990, 1984).

12.5,3.2: Parameters for Circuit Representation The magnetic vector potential dis~ibution over the solution region can be used to determine the frequency-dependant resistance and inductances required for interconnect characterization.

Resistance: The loss density for a conductor is given by: (12.5.68) P(x,y) = IJ(x,y)!2 c where J(x,y) is the total cuurent density. The total current density Js is given by: J(x,y) = Js + Je(x,Y) =

j0.gA(x,y)

The total power loss ~ r unit length is given by: V = ~V(x,y) dxdy

(12.5.69) (12.5.70)

S

~ e ac-dc resitance is given by: J'P(x,y) dxdy

R~:. =s

(12.5.71)

P,.~ I2/cra This ratio is a measure of the increase in resistance of the conductor for a given geometry and the frequency with res~ct to the DC value,

Inductance: The inductance of a conductor can be obtained using the stored magnetic energy, The stored magnetic energy in terms of the inductance is given by: W = ~I L Ill2 (12,5,72) The stored magnetic energy per unit length in terms of the magnetic vector potential and the current density os given by: W = Real ~1 [P..(x,y) J*(x,y) dxdy (12.5.73) s The inductance per unit length can be written using (12.5.72) and (I2.5.73) as: L = ,.~5 JeA(x,y) J*(x,y) dxdy I 1 1:- S

(12.5.74)

S. Kalaichelvan

497

In the preceding J* is the complex conjugate value of the current density J. The integrodifferenfial method is valid Iq3ra multiconductor problem, and the frequency dependant inductance matrix can ~ easily obtainS.

12.5.3.3: Numerical Rest'Its Single conductorproblem For test pu~oses, a long conductor of square cross section and consent electrical conductivity was chosen. Although symmetry can be exploited to solve this problem, to maintain the generality of the software, the full cross-sectional area of the conductor was solved. The AC resistive loss, normalized with the DC resistive toss, for various values of frequency parameter p2=2a/p82 is dete~ined. Two-dimensional linear finite elements were used for discretization. The results were in good agreement with the finite difference scheme implementation (Balak_rishnan, 1986). To check the validity of the dimensional independence of the frequency parameter, tests were perfo~ed by changing the cross s~don and conductivity. Slot Embedded Prob&m A test was performed to check the validity of the implicit Neumann boundary condition at the surface of the conductor. A slot embedded problem was chosen as shown in Fig. I2.5.11. The magnetic vector potential and current-density distribution over the x-y coordinates are compared against Konrad's[ (1981, I982) implementation and were found to be in good agreement.

7.0

1.0

Figure 12.5.10: The Slot Embedded Problem

498

Chapter12: Numerical Methods for Characterizing High-Speed Interconnects

12.6 Conclusions In this chapter, an introduction to the application of numerical methods to the characterization of high speed interconnects has been presented. The digital design issues and the interconnect problem have been outlined. The basic electromagnetic concepts related to the interconnect problem have l~een described along with various relevant field formulations. The electrical characterization of the interconnects in terms of the circuit theory has been outlined to determine their electrical characteristics. The electrostatic fo~ulation has been proposed for the evaluation of the capacitance, conductance and high freeqauncy limiting inductance. The 2-and 3-dimensional boundary element method applied to the electrostatic formulation has been outlined along with relevant numerical examples. The integrodifferential method applied to the 2-dimensional skin effect formulation has been outlined to evaluate frequency dependant resitance and inductance. Specific numerical examles have also been presented.

Chapter 13 iiii

I[wll[lll

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

T. Kirubarajan and P. Ratnamahilan P. Hoole IIII

IIIIIII

IIIIIllllfl

Ill

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

I

IIIIIII

IIIIH

I[IJIIIIIIWJ

....

IIIII

II

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

ELECTROMAGNETIC SIGNAL PROCESSING: A CASE STUDY

13.1 Introduction Amongst the electromagnetic field computation research areas growing in importance, we may include electromagnetic signal and image processing, electromagnetic sensors for automated systems, passive and active remote sensing and electromagnetic neural networks. In this chapter we summ~ize some of our work in the area of electromagnetic field-based state estimation, tracking and adaptive control. This subject-- the merging of electromagnetic field computation with signal processing techniques - - is of vast importance, and holds much promise for aerospace and medical electronics. We propose a system for runway plane based measurements of the approach angle of an aircraft, the relative ~sition of the aircraft plane with respect to the runway plane and the flight path. What we present here is an inverse electromagnetic solver for ultra high frequency (UHF) radar. This chapter presents a review of the sensing and tracking system proposed, a non-optimal formulation of aircraft plane measurement using the scattered electromagnetic fields captured by radar, an optimal Kalman estimator and tracNng algorithm for the electromagnetic field based system and a further application of the state estimator and tracking system for adaptive control of a vibrating body using a static electric field. The findings reported in this chapter can be summarized as follows: • The magnitude of the Doppler radar electromagnetic fields scattered by an aircraft, and its phase with respect to the transmitted electromagnetic pulse contain information on the aircraft wing position and its body position with respect to the radar plane. The scattered signal received by the radar is assumed to be corrupted by additive white Gaussian noise (AWGN). A maximum likelihood estimation (MLE) algorithm operating on the scattered

500

Chapter 13: Electromagnetic Signal Processing: A Case Study

electromagnetic field model is used to determine the position of the aircraft plane wi~ respect to the radar ce plane. • An algorithm which gives better convergence properties despite requiring more computer memo~, is the recursive least square (RLS) or the Kalman filter. This is an alternative to the MLE algorithm, where speed of computation is more important than the memory required. A further advantage of the Kalman algorithm reported here, is that it allows the radar system to track the flight path of an aircraft which is highly valuable data both in landing systems and in military applications. • In Section i3.4 is given a fur~er application of the electromagnetic sensor reported here for aircraft landing s2¢stems. Instead of using a flywheel type of mechanical arrangement to stabilize a vibrating body, the tracing system is used to tune adaptively an electrostatic force to stabilize the vibrating system. The technique could also be used to float micromotor rotors, and thereby eliminate frictional losses. In the Section 13.2 a description of the electromagnetic field signal processor for sensing and tracking system is given in the context of the instrument landing system(ILS) and microwave landing system(MLS). In the subsequent sections of this chapter a general form of the MLE applied to scattered magnetic fields is presented with simulation results. The Kalman algorithm for rapid measurement of the aircraft coordinate plane is develo~d for sensing and tracking. In Section 13.4 the application of the Kalman corrector predictor algorithm for adaptive vibration stabilization is given with results. The work is currently being extended to facilitate the recognition of aircraft using neural network signal processing principles, which for instance, could be quite important in preventing incidents like the downing of commercial aircraft mistakenly identified as military, aircraft by radar systems.

13.2 Overview The microwave landing system (MLS) is currently being proposed an an attractive alternative to the instrument landing system (ILS). An important feature to consider is the tilt and bank angles of an aircraft approaching the runway along a bent of highly compressed (area-wise) path. The important pitch auitude and bank angles axe defined in Figure 13.2.1.

(a) (b~ Figure 13.2.1: Pitch Attitude and Bank angle

T. Kirubarajan and P, R. P. Hoole

50I

In precision landing systems like ILS and MLS, the bank angle should be kept within 0 ° to 5 ° to avoid landing on the nose wheel or hitting the tail on the runway. The bank angle should be limited to 5 ° to avoid having a wing tip or engine pod hit the runway. In the ILS, similar situations are encountered, for instance, when an aircraft is to land on a nonstationary runway as on an aircraft carrier. The existing techniques for measuring bank angle and pitch attitude depend on the instruments placed inside the aircraft. This info~ation may be obtained by the controller through an interrogator signal. We have proposed a powerful signal processing method by which the bank angle and the pitch attitude may be extracted

(a)

MLS Motion

~S

I] ELE

SHIP

"~ Motion

SE a,

(b)

Figure 13.2.2: A Landing Aircraft a: On a Stationary Runway b: On a Moving Runway

5.0.2

Chapter 13: Electromagnetic Signal Processing: A Case Study

from the land-based (or runway-plane ba~ed)Doppler radar, or two single active microwave remote sensors placed close to a stationary or non-stationary runway. Fu~hermore, the i n f i ~ a t i o n extracted may be used to track both a fixed wing aircraft and a helicopter. Of the many advantages the system proposed here offers, one is the reduction of missed-landings on non-stationary runways, where now the values of the pitch attitude and bank angle are measured with respect to the runway, instead of the earth. The basic issue we address is, given a noisy magnetic field signal at the Doppler ~ e i v e r electronics, can we identify the plane of the aircraft with respect to the runway plane, the aircraft plane (defined by the bank angle and the pitch attitude which are the state variables of aircraft position)? The current state-of-the-an single element Doppler radar looks at the target as a moving object with a given reflection cross-section, and seeks to extract the following information of the target: distance, speed and elevation. Our argument is, if we process the electromagnetic signal turther, we could extract from it the information it contains on the bank angle and pitch attitude. Furthermore, it is also shown that info~ation on the yaw and roll (termed rotating in this chapter) could also be obtained using the signal processor reported here. Consider the two situations shown in Figure 13.2.2. In Figure 13.2.2(a) is shown the flight path of an aircraft coming in to land on a stationary runway with the aid of an MLS (or ILS). In Figure I3.2.2(b) is shown an aircraft landing on a non-stationary runway with the aid of an ILS (or MLS). In both situations, getting the right bank angle and pitch attitude is critical. In the MLS landing scenario, this is critical when the aircraft is taking a sha W turn before landing so that the manoeuvres a few seconds before landing involve sha~ tilts, In the ILS landing scenario depicted in Figure 13.2.2(b) the probability of missed landing on rough sea is higher due to difficulties in keeping the aircraft pitch attitude and bank angle within the specified limits with respect to the runway plane (X',Y', Z'). In both cases the system we describe here can effectively keep an accurate measure of these two angles by using two Doppler sensors, one mounted at the end of the runway close to the conventional distance measuring equipment (DME) and the other close to the elevation measuring equipment (EME) at the _~goise Target Helicopter

Kalman

/

Predictor to t÷At. . . .

Mathematical Model of the Sensor

Sensor Measurement ] State at Kalman Corrector i t

Figure 13.2.3: Microwave Remote Sensor and Tracker

T. Kimbarajan and P. R. P. Hoote

503

beginning of the runway. Indeed what is so attractive about the system is that it can be inco~orated into the existing DME and ELE, so that the hardware cost can be kept to a minimum. All that is required is the incorporation of the electromagnetic signal processor reported in this chapter so that it can operate on the signals scattered by the aircraft, picked up by the DME and ELE receiver electronics, conventionally to measure the distance at which the aircraft is from the runway (by DME) and the angle of elevation of the aircraft (by ELE). The system may be used for both f i x ~ wing mrcraft and for rotaD'-wing aircraft. The basic electromagnetic signal pr~essor is shown in Figure 13.2.3. The Doppler radar signals are reflected back to the radar with noise added to it. The noise is assumed to be AWGN. The signals scattered from the wings of a fixed wing aircraft could be used to determine the bank angle (Figure 13.2Aa) and the signal reflected from the body looking at it sideways (Figure 13.2.1b) could be used to obtain the pitch attitude. When the system is used for rotapy-wing aircraft (helicopters), the signal processor would have the additional t ~ k of filtering out the scattered fields by the rotor blades (which contain information on the flight path) by the fields scattered from the main body of the helicopter. The filtering could be done in the frequency domain, since the different Doppler shifts make it convenient for filtering, and the time-domain fields may be subsequently obtained using an inverse Fourier transform algorithm. For simulation purposes, we model the aircraft by a cylinder of similar dimensions. We are interested in extending this aspect of the work in an attempt to develop a model-independent neural network signal processor which could be trained to extract the required parameters from the scattered electromagnetic fields. The received signal (with AWGN superimposed on it) is compared with the model result, and the error is operated on by a Kalman corrector to give us the actual measure of the two state variables we are looking f o r - - the bank angle from the DME receiver antenna signals and the pitch attitude from the ELE receiver antenna signals. It was interesting to observe that when the MSE algorithm is used to obtain the results for tilts only (only changing the bank angle or changing the pitch attitude without any other manoeuvres like yawing), there is an analytical solution to the set of electromagnetic field signal processor equations. This can have limited use in functioning as a double check on the real-time measurements of the two angles. When the the Kalman predictor-corrector technique was used, we also get an estimate of the rate at which the aircraft bank angle and pitch attitude are changing at instant t(d0/dt = [0(t + At) - - 0(t - At)]/(2zXt), which in itself is a useful parameter to know in early warning systems, as well as being useful to track the aircraft.

13.3 Electromagnetic Signal Measurement We model the rotor blade of the helicopter as a homogeneous cylinder of radius r and its state is defined by a three dimensional coordinate system XYZ. The cases of rotation and tilt can be considered as a problem where the coordinate axes are

504

Chapter 13: Electromagnetic Signal Processing: A Case Study

Z

Oz

Yr Y

Xr

~,.

(a) (b) Figure 13.3.1: Rotation of the Axes allowed to rotate and thereby changing the state of the blade. Rotation of the blade is about the Z axis and tilt about the X axis. Figure I3.3.1(a) illustrates the rotation and Figure 13.3.1 (b) the tilt of the cylinder. When the cylinder is allowed to rotate, its new state is defined by the coordinate system X r, Yr, Zr, as illustrated in Figure 13.3. l(a), where Xr(SzX = 0 = YrOz Y and XrYrZ r is related to XYZ by (Collins, I968)

Zxl icos0s, 0 01Ixl

-sin 0 cos 0 0 Y (13.3.1) 0 0 1 Z Similarly for tilt about the X axis, the new state is defined by axes X t Yt Zt, which is related m XYZby =

[x,] 1' o 01E

Yt = cos 0 sin 0 0 (13.3.2) Zt _-sin 0 cos 0 0 When the cylinder is allowed to route, if the radius of the cylinder is r, and its surface angle, which is measured along the cylinder surface from the reference position, is q~, then the apparent or equivalent radius r r, by which the incident wave is scattered back, and the apparem surface angle ~r are found m be rr = r (I 3.3.3) ¢r = ¢" 0 (13.3.4) Similarly for tilt, the corresponding equivalent radius rt and surface angle ~t are given by r t = r ~4t I - sin 2 ¢ sin 2 p (13.3.5) ¢t = mn-I (tan ¢ cos 13) (13.3.6) "When the cylinder is allowed m rotate as well as to tilt, the equivalent radius rrt by which the incident field is scattered back and the equivalent surface angle ~rt axe given by rrt = r'~/1 - sin2¢ sin2p (cos Ocos~ sin~-sinOcos~"| ¢rt= tan'! k s i n O cos + c 0 s O cos ¢~']

~s-in-¢

(13.3.7) (13.3.8)

T. Kirubarajan and P. R. P. Hoole

505

Now an incident field, which is scattercd back by the rotating blade is used to derive the state estimator. We assume that the original surface angle of the rotating cylinder is known and constant. The incident field Ui, scattered electric field Use and the scattered magnetic field Ush are given by Ui = 1 -

1 m, -i~lreq

1-

~I req') 8- ) c o s Ceq

......... { -i(

-i U s e - In el.I

P exp 2alreq

Ush = i~I2

+ cos Ceq

al

req

exp

-

(13.3.9)

')}

~

{ ( -i

(13.3.10)

")

atreq - ~

(13.3.11)

where al = (co/c)rN, ~o is the signal frcquency, r the radius of the cylinder, N = (gvJ~0e0) !/2 the refractive index, and c the speed of light (Jones, 1964). Here req is given by equation (13.3.3), (I3.3.5) or (13.3.7) and Ceq by (13.3.4), (13.3.6) or (13.3.8). The receiver (Doppler radar) measures the magnitude and the phase of the scattered fields. We assume that the measurements are corrupted by Gaussian noise. For the case of rotation we assume that Ar the magnitude and p h ~ e me~urements are corrupted respectively by noise terms rtl and r12 whose probability distributions are given by f 2"1 PDF(nl)-/~

1

Cl exp ] ~ k

i exp PDF(nl) - .,,2U2~ c2

(13.3.12)

'1)

-rl2

(I3.3.13)

That is, nl and n2 are zero-mean Gaussian noise sequences whose known variances 2 2 are el and ~2 respectively. From (I3.3.9) and (13.3.11) the expression for magnitude and phase measurements can be derived. The measured values which are corrupted by their respective noise sequences are given by Ar = ~ ,2

+ cos (C-e)

3~ Br= ~ - a i r + 112

) 4 "~

+ "ql

(13.3.14) (13.3.15)

Similarly for the case of tilt, the magnitude measurement At and the phase measurement Bt are given by At

=

~(1

) + cos tan -1 (tan ¢ cos e) ...,~ 2~ir,,~ti_si n 2¢ sin a2 e + n l (13.3.I6)

506

,.......

Chapter !3: E,!ectromagneticSignal Processing: A Case Study Incident ~ l d magnitude

In,dent field phase

Su,-face Lngle in r a ~

(a) ScatteP'.,dlzza~rle~ ~

magnitude

Scattered magnetic ~ld phase

Tilt angle in ratl_iw

(b) F i g u r e 13.3.2: a. Incident Field vs Su~ace Angle b. Magnetic Field Measurements Versus Tilt Angle

T. Kirubarajan and P. R. P. H~le

507

Bt = T3re - air.V,'1-sin 2¢ sin a2 0 + n2 (13.3.17) where 1't1 and r12 are zero-mean Gaussian noise terrns.~e Corresponding equations for the case where the cylinder is allowed to tilt as well as to rotate are given by (1 ( c o s 0 cos ~ sin ~ - sin 0 cos ~ ) ) Ar t = 2 ~ ' + c o s tan'l cos 0 c o s ~ s i n ¢ sin 0 c o s + rll

'~

(I3.3.I8)

2alr"~/l-sin 2¢ sin 2 0

B ~ = ~3re - alr.~/1-sin 2¢ sin 20 + r12

(13.3.19)

Figure 13.3.2a gives the incident field on the cylinder surface for the case of rotation. Figure 13.3.2b shows a set of noisy magnitude and phase measurements for different values of tilt angle, B. The incident signal frequency is 1 GHz. Similar plots are obtained for the other cases. The actual surface angle is assumed to be constant at 0.523 radians. The magnitude and phase measurement noise t e ~ s are Gaussian with variances 0.00002 and 0.0005 rest~cfive!y.

Derivation of the Non-optimal Estimator Now from the measurements given by the above equations (13.3.14) and (13.3.15), (13.3.16) and (13.3.17), or (13.3.18) and (13.3.19), we can derive the state estimator based on the Maximum Likelihood Criterion (Stark and Woods, 1986). For the case of rotation, we want to find the values of r and o which will simultaneously maximize the probability distributions rli and n2 given by equations (13.3.12) and (13.3.13). Ttmt is, ~ e want to find ~ and ~r the values of r and 0 respectively, which will maximize [PDF(rll).PDF(rl2)]. Maximizing [PDF(ql).PDF(ri2)] is equivalent to maximizing log [PDF(rll).PDF(~2)], which is 2•2.

2 2

also equivalent to minimizing [(ql Ol) + (rl l/Oi)]. Then r and Or satisfy + k.~l 0 00

n =l)}r=

+

=0

(13.3.20)

-0

(13.3.21)

A

r

n

Therefore (" and O can be obtained by solving the following two equations simultaneously.

^)

A r - o~F-If~.~ ^ c

+cos (¢-0 r)

+

=0

2 2 _4+c°s(¢-0r) 2cu 1

(I3.3.22) ,,3 2raN r 2 c~ 2

508

Chapter 13: Electromagnetic Signal Processing: A Case Study

+ cos (, - 0

=0

(13.3.23)

Therefore the state estimator finds the approximate values of r, and 0 using the above two ~uafions. We assume that the statistical prope~ies of the noise t e ~ s are lcnown. The analytical solutions to these equations can also found to be r =~

(13.3.24)

e01',I

^ 2Amac] 2 I cos(, - Or) = ...."~x~n(3n-4Bm) " 4

(13.3.25)

Therefore we have derived a state estimator based on the Maximum Likelihood Criterion to find i~ and ~r, the estimated values of the radius of the rotating cylinder and its angle of rotation. Following similar arguments we find that for the case of tilt f" and ~t the estimated values of r and ~, satisfy (I3.3.26) and (13.3.27): - - 4 2 + cos tan "t tan ,_. cos _~

I

- ~ A t - k~ c1

( 1 - s i n 2 , s i n 2 ~'t) "1/4 ] x

1 + cos tan "1 tan , cbs I- sin 2 , sin 2 t ] ",~-i/

l~t7 I3

2~-J -sin2 * sin 2 ~t 2 B t . 3~ + k~ 2 4 e2 r^

4

[Bt " 3~+4 k~2

l

l'sin2 * sin2 ~t = 0(13.3.26)

l'sin2~sin2~tl

1-sin 2, sin 2^ ~U~[ 1

A t

"~ ~2

I +c°stan'ltan-~c°s~t l ....... ( 1 - s i n 2 , sin 2 ; t ) 1,4 ] x

ki:"% "

sintan "1 t a n , c o s

t t a n * sin ~t

( l + t a n 2 , cos 2 St) ( 1-sin2 * sin2 St) 1/4

(I +

+ c o s t a n "I t a n , c o s

s)t

sin 2 , s i n ~ t c"o s ~ t

2 ( t - s i n 2 , sin 2 ~t)I/4

^

=0

(13.3.27)

T. Kirubarajan and P. R. P. Hoole

509

where k = (o~N)/c. When the cylinder is rotating about the Z axis and tilting about the X axis, we A A get tbJ'ee equations which can be used to find ~, 0rt and 13rt, the estimated values of the radius and the angles of rotation and tilt respectively. ~ e three equations are

I,,

"1

l,ff' ~2 sin ~rt (1-sin2 * sin2 ¢' ~7 ~-x--"2 x • rt) ~, sin 0rt sin 13rt s i n , + cos 0rt cos

"J

01

(I3.3.28) ...... l'sin2 ~ sin2 I 3n 2~'kJr ~t Brt_ __ + kf. 2 "~ l_sin2 * sin2 ~t 2 4 cr2 1

I _/k~ Cl

^

~ cos *rt !- sin 2 ~ sin 2 ~rt) I/4

IA r t - k ~ ' "~

'

+ cos *rt

^~'rt)l/4J1 = 0

( 1 - s i n 2 , sin 2

1

A

1

i~ + cos ~rt ~ ~-~Art ,~i t -

- k~

(1-sin

(13.3.29)

, sin 2 13rt)1/4

^ I/4 ^ l'sin2 * sin2 ~rt sin ~rt cos 1+tan 2 *rt +

cOS~rtSio0 (~ 2

A ^ )2 sin 0rt cos ;rt sin ~ + cos 0rt cos

^

+ cos *rt

^

cos ~rt sin ~ + 2 a2 =0

3n ^2 Brt- 4 + kr

1

-

' 2 ¸¸¸~¸¸ -| sin 2~ sin 13rt] (13.3.30)

5 tO

Chapter 13: Electromagnetic Signal Processing: A Case Study

A

where Crt is given by ^ ,~-

tan -!

cos 0rt cos , r t sin ¢ - sin 0rt cos ~ ~ ~ ..sin 0rt cos ~rt sin ¢ cos 0rt cos

(13.3.31)

The above equations can be used to track the helicopter blade provided that scattered magnetic field measurements are available and the statistical properties of measurements noise are known. To illustrate the a0plication of the state estimator we use some real data to estimate the radius and the angles of rotation and tilt. The frequency of the signal is 1 GHz. We assume that the refractive index N is equal to one. Typical values for the magnitude of the back-scattered magnetic field are about

Error b e ~ n 0.015 [

............. ~- ...................... ; - . . . . . . . . . . . . . . . . . . . .

oo,,

,

,

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

~ ..........................

, .....

~

.

.

.

.

.

..........

II

" fii l,

l! ii!li

!

~0~ .o.ol F

actual and estimated v a l ~ s

~ ..................

!I, t ,

',~

|

it

I i

............"40

6,3

80

I00

120

1~

160

1811

measuring instants

Figure 13.3.3: Estimation Error of the State Estimator

200

T. Kirubarajan and P, R, P. HoNe

511

1 giXJm and the phase around rd3 radians. We also assume that ¢, the surface angle of the cylinder relative to the reference is known to be around n/4. For these typical values, when the cylinder is allowed to rotate we find that the estimated radius of the helicopter rotor is around .25m and the angle of rotation is about n/3 in the clockwise direction. These values agree with the results obtained using the analytical solution. The state estimator can be used on-line so that it will be possible to track the rotor blade as the measurements come in. The state estimator for all three cases has been implemented in software using MATLAB TM and its performance checked (See Appendix 13.A). To check the operation of the state estimator, the measurement sequence given in Figure 13.3.2b is fed into the state estimator and the estimated values for the angle of rotation are obtained. Figure 13.3.3 gives the error between the estimated values and the actual angles of rotation which produced the measurement sequence in Figure 13.3.2b. The results show that it is possible to get very accurate tracking using this estimator.

13.4 13.4.1 :

Application of the Estimator: Electromagnetic Vibration Controller Introduction

Low frequency mechanical vibrations are a source of major concern when mounting sensors on ground and space vehicles. In the Hubble Space Telescope, for instance, the 0.1 Hz jitter that vibrates the spacecraft for 3-6 minutes each time it passes into or out of the earth's shadow prevented the telescope from maintaining a fine lock on/hint guide stars for the highest-resolution imaging. Typically flywheellike reaction wheels are used to damp the jitter. However, it is not a reliable system and could create higher-frequency vibrations. In this paper, an electromagnetic field is used to produce countervailing forces to reduce mechanical jitter. The vibration frequency is computed using a sensor interfaced to an on-board microprocessor. This information is used to compute the required electric field to damp out the force causing the jitter. Two cases are considered: (i) a metallic cytinder (or plane) vibrates freely, remaining parallel to the ground plane and (ii) a cylinder (or plane), with one side bolted to a stationary body, vibrates about its fixed end. In this section we report the work documented in (Kirubarajan and Hoole, 1994). A further application of the technique reported here is t~or a closed control system to float the rotor of a micromotor or actuator (De Silva; I989; Kaplan, 1967; Kaplan and Regev, 1976). An important issue in the areas of industrial electronics and robotics is system integration, where the electromechanical subsystem is integrated with the electronic-system. In the design of a micro machine integrated with the control electronics, a major problem is the frictional losses due to the tiny motor rubbing against the mechanical mount. Using the system we have proposed here, we can float the rotor using an adaptively controlled voltage applied to the two stabilizing electrodes placed at the open ends of the

512

Chapter 13: Electromagnetic Signal Processing: A Case Study

Steady-state Balancing Force

Floating[ Displacemen L Body I

Stabilizing Electric Force

Figure 13.3.4: The Basic System stator. Some of the alternating current energy supplied to the stator may be tapped for use with the two stabilizing electrodes. The basic stabilizer system described is shown is Figure 13.3.4. For adaptive control of the stabilizing electric field, we have used the LMS algorithm. The LMS algorithm is attractive because it is computationally simple and in a continuous-time version it may be implemented in all-analog hardware. However the price paid for the simplicity is the slow convergence rate, since it has only a single adjustable parameter for controlling the convergence rate, namely the scale factor. For faster sta~ up and convergence, it may be necessary to resort to recursive-least-squares (RLS) algorithms, which however, are difficult to implement on existing digital hardware for high-frequency signals since they involve additional par~eters and computational complexity. We model the satellite sensor as a homogeneous cylinder of uniform radius r and length 21 freely hinged to the satellite at one end. The satellite exerts a deterministic force fs on the sensor at the hinged end. A random force, for example due to temperature variations in the surrounding, acts on the sensor and causes vibrations about the equilibrium position. The effect of the random force on the satellite, which is a heavy solid body, is assumed to be negligible and therefore, the whole system can be modelled as a beam freely hinged to a stationary body and vibrating about the hinged end. We apply the electric field of strength E which will exert an equating force on the sensor so that the sensor is brought to rest at the desired position. Figure 13.3.5a gives the geometrical setup of the system. Here 00 gives the equilibrium angle and H the current position of the sensor measured relative to the reference line OR. The motion of the sensor at time t is defined by ( I G + m12)'0 + eE 2 al cos 2 0 =0,

(13.3.33)

where I G is the moment of inertia of the sensor, m its mass and a cos 0 = 4Ir cos 0 the projected area on which the electric field exerts a force. The strength E of the field is varied depending on the deviation of the sensor from its equilibrium

T. Kimbarajan and P, R. P, H~ale

513

Pl

+ d ~E

O

R

P2

(a)

(b)

Figure 13.3.5: a. Vibrating Satellite Sensor b. Vibrating Electrode position. Therefore we need to devise an adaptive algorithm which will change the field strength continuously in order to minimize the deviation from the equilibrium point and to bring the sensor to rest at the equilibrium point. The error ~ t w e e n the current position and the equilibrium ~sition drives the adaptive algorithm. Another simpler possibility is a metallic plate which vibrates freely remaining parallel to the ground. We like to control the vibration again using an electric force. The geometrical setup for this problem is given in Figure 13.3.5b. In this case d o gives the equilibrium position and d the cur,ent position of the plate relative to the reference line OR. The following equation defines the motion of the vibrating plate: "d= f v ' e E 2 A m (13.3.34) where m is the mass of the vibrating sensor, A its area, and fv the driving force. The external circuit connected m the parallel plates PI and P2 can be conceived of as consisting of an inductor, a resistor and an alternating current source which produces E. The stability point of the whole system will be at the resonant frequency of the RJ_,C circuit, In this chapter we are concerned about the magnitude of the elec~ic field m be applied, based on an estimate of instantaneous positions of the vibrating plane. In other words, we deal with the more general proNem which assumes that the resonant frequency of the system cannot be determined or it is variable, as it often is in most practical systems. To drive the adaptive controller we need the state of the vibrating object. That is the current position and the angular velocity in the first case and the linear velocity in the second. In space objects, it is quite likely that these are not available for direct measurement. Therefore an alternative technique for finding out the current state is required. For this a scattered magnetic field based state estimator proposed by the authors can be used ( K i r u b ~ a n and Hoole I993a, I993b). In this case we do not have fiae exact state of the vibrating sensor, but the estimated state. The whole estimation and control system can be illustrated as in Figure 13.3.6.

514

Chapter 13: Electromagnetic Signal Processing: A Case Study

Here we assume that such a state estimator is available and that estimations of its value can be used for the control mechanism.

13.4.2: Derivation of the Controller

Non-optimal Case

Using Equation (I 3.3.33) we now proc~d to develop a linear state space model of the system. Let O and e be defined by two state variables X l and X 2 respectively and E by an input variable U. Then the state (Xl(t), X2(t)) of the vibrating sensor is given by

Xl(t) = X2(t) eU 2 al cos 2 Xl(t) X2(t) IG + ml 2

(I3.3.35)

It is also possible to derive a model, probably a nonlinear model, for the state estimator. For example in the case of a scattered magnetic field estimator proposed by Kirubarajan and Hoole (1993b), the measured quantity is Y(t), the magnitude and the phase of the magnetic field scattered back by the vibrating beam. Assume that the estimator model is given by Y(t) = £(X) (I 3.3.36) Equations (13.3.35) and (13.3.36) are nonlinear equations. We require linear equations to obtain an optimal state estimator. Therefore we linearize the system about its nominal equilibrium point (o0,0) and the nominal equilibrium input Eeq. If small perturbations in the angular position and velocity of the sensor and the electric field input t are 5Xi(t ) = x/, gX2(t ) = x 2 and 6U(t) = u respectively, then the linearized state space m ~ e l is given by

x2

,/sin 200 0

x

+

Process Noise

-

cos 2 o0 u

(13.3.37)

Measurement Noise I Scattered Field

1 Estimated[ Adaptive I 1 State Controller State Estimator

Figu~ 13.3.6: Estimation and Control Mechanism

T. Kirubarajan and P. R. P. Hoole

_•

515

Forward Time

u(k)

System 1 ~

.Calculation W

[I .....

x(k)

._ [

U(k)

e(k)

J Adaptive ! - - ~ Model [

Xl(k)

Figure 13.3.7: Adaptive Controller Mechanism 2 / where ~ = (EEeqar) (I G + mr2). Similarly we can derive a linearized model for the estimator in the form of y = Cx. The above equations satisfy the standard form of the state space model: x = Ax + Bu, y =Cx. For the floating body system illustrated in Figure I3.3.5b, a similar state space model can be derived. If d = x' I and d = x' 2, then the model is given by

I1

x, l = [ ~

L£,2j

1] i-x,,1] Lx 2J + -

0J

0 A u

(13.3.38)

where u = E, and other variables are as defined earlier. Now we derive the discrete-time equivalent of the above system so that it will be possible to derive the discrete-time adaptive controller. To derive the discretetime state space model we use Runge-Kutta numerical integration, which is based on the first order Taylor's series approximation (Reid, 1985). Here we find an approximate expression for the state at time kT using Taylor's series, where T is the sampling period. Then the discrete-time update equation is found to be XT(k) = [I + A(kT---T)]xT(k--I ) + B(kT- T)TUT(k--1 )T, (13.3.39) where xT and uT denote sampled values of state x and input u. Then the discretized state space model is defined by xq4~k)= I ~T sin1 200 T0 I XT(k'l)+ [ -

0 2 00 1 u~k-1) cos

(I3.3.40)

A similar discrete-time model YT(k) = H xT can be obtained. Now that we have a state space model for the vibrating satellite sensor, we proceed to develop the adaptive controller. For adaptation, which changes the electric field strength E so that equilibrium is achieved at the desired location, we use the adaptive model control (AMC) principle (Goodwin and Sin, I984; Widrow and Stearns, 1985). A model of the plant is used to dete~ine the control input, E in this case, to the system which will cause the desired output. The output of the

516

Chapter 13: Electromagnetic Signal Processing: A Case Study

^

system here is x(k). Once the required control input is found, we apply the same value of E to the actual system of state estimator and controller. This result causes the system output to match closely the desired output. The block diagram illustrating the operation of the adaptive controller is given in Figure 13.3.7. Since here we assume that we have access to a state estimator which can give the cu~ent state of the vibrating beam, the current state is taken as given in (I 3.3.40) for the adaptive process. In the combined system, where we have an estimator as well as a controller, we should use the estimated values from the estimator to drive the adaptive controller. As can be seen in Figure 13.3.7, we need an adaptive model of the system which has enough flexibility to match the dynamic response of the system containing unknown random components. In our case, the adaptive model is an adaptive linear combiner whose weights are adjusted to minimise the mean-square error. We can use an FIR filter of weight L + 1 as the adaptive model. The adaptive process automatically adjusts the weight vector W(k)

Differen~ between required and c u ~ n t positic*P~ 0.02

0.015 0.0I 0.005

-0.005 i

-0.01

0

-~0150 sampling instants

Figure 13.3.8: Simulation results for the Adaptive Controller

T. Kirubarajan and P. R. P. Hoole

517

so that for the given input-signal statistics, the model provides a best minimumrnean-square-error. We use the LMS algorithm to change the weight vector W(k). Updates of W(k) satisfy the following equation: W(k) = W(k--1) + 2 ~ ( k - - t ) U ( k - I ) (13.3.41) where I~ is the tuning parameter and U(k) gives the memory model of the system input. We want the estimated value of the current position and velocity of the rotating sensor to track the reference state of the system which is given by X~ef = (00, 0). The system control input, u(k) (the electric field strength E), is calculated using forward-time calculation (FTC). We generate u(k) from the reference state Xref and from the weight vector W(k) and the input vector U(k) which gives the past values of the system input. The adaptive process tries to make Xl(k) and Xref equal while maintaining A

E[e(k) 2] close to zero. When they are equal the system output x(k) wilt be close to Xref. The forward-time calculation tries to derive the appropriate value of u(k) which will satisfy the above condition. The weight vector W(k) is updated at each sampling instant using the LMS algorithm. The following equations give the expressions for Xl(k) and u(k) (Widrow and Stearns, 1985). L Xl(k ) = ZWn(k)u(k-n) (adaptive m ~ e l ) (13.3.42) n=0 1

(FTC) (13.3.43) Xre f - ~ W n ( k ) u(k-n) n=l Using the expected value of the control input u(k) as given in (13.3.43) we can vary the strength of the electric field. Therefore we have found ways of estimating the current state of the sensor and varying the electric field strength which will exea an electric force to bring the sensor to rest at the required position. Figure 13.3.8 illustrates the effect of the adaptive controller on the vibration of the rotating blade. By varying the electric field it is possible to dampen the magnitude of the vibration angle and bring the sensor to rest or reduce the vibration angle so that it lies within a certain minimal range. In most cases, for example in the HST, it is quite acceptable if the vibration angle is reduced below a certain limit and not necessarily made zero. We have used in our simulation the calculated values of the state. But in a real system, we will be using the estimated values from the state estimator. This will inevitably introduce errors in the controller due to estimation errors and the convergence rate will be slower. A similar state space model and an adaptive controller can be derived for the case where the object has a linear motion. This system is simulated using MATLAB TM and Figure 13.3.9 gives the deviation of the vibrating plate from the required equilibrium position with time. u(k) =

518

Chapter 13: Electromagnetic Signal Processing: A Case Study

13.4.3: State Space Model Optimal

Case

Using the results obtained in Equations (13.3.33), (13.3.5), (I 3.3.6) and (I 3.3.16), we now proceed m develop a linear state space model of the system. Let 0 and 0 be defined by two state variables X 1and X 2 respectively and E by an input variable U. Then the state (Xl(t), X2(t)) of the vibrating sensor is given by :'Kl(t) = X2(t) eU 2 al cos 2 Xl(t) J~z(t)

(13,3,44)

IG + m l 2

and the output measurement Y(t) = Am satisfies

~ a t i o n from the required position ......... ,...................................................

0.015 ~

I1. ! 0.005

-0.005 -0.01

-,3.015

50

. . . . . .I00 . . . . . . . . . . . . .150 ...

2~

sampling instants

Figure 13.3.9: Derivation from the Required Position

250

T. Kirubarajan and P. R. P. Hoole

519

1 r_l Y(t)= ~-'-~:~ | 4 + c o s t a n ' l ( t a n , cos 0 ) ] [ 1 - s i n 2 , s i n 2 0]-1/4

(13.3.45)

Equations (I3.3.44) and (13.3.45) are nonlinear equations. We require linear equations to obtain an optimal state estimator. Therefore we linearize the system about its nominal equilibrium point (e0, 0) and the nominal equilibrium input Eeq If small perturbations in the angular position and velocity of the sensor, the electric field input, and the output measurement are 8Xl(t) = x l, 8X2(t) = x 2, 8U(t) = u and BY(t) = y res~ctively, then the lineafized state space model is given by

x2

,i, sin 200 0

x2 + -

cos 2 00

u

y = [c 0Ix where V- (eEeq2ar)/(IG + mr 2) and c is given by ~ [ ' s i n tan -1 ( t a n , s i n 0 0 ) ( t a n c=~-~t,-2~L i + tan 2 0 c O s 200 [1- sin 2 , sin 2 00]-1/4 + x [~cos

(I3.3.46) (13.3.47)

,sin 00)lx

sin 2 , sin 200 [1- sin 2 , sin 2 00]'5/4

tan-l(tan, cos 00) ]

(13.3.48)

The above equations satisfy the standard f o ~ of state space model: x = Ax + Bu, y = Cx. Now we derive the discrete-time equivalent of the above system so that it will be possible to derive an optimal Kalman filter based state estimator. To derive the discrete-time state space model we use as before Runge-Kutta numerical integration, which is based on the first order Taylor's series approximation (Reid, I985). Here we find and approximate expression for the state at time kT using Taylor's series, where T is the sampling period. Then the discrete-time update equation is found to be XT(k) = [I + A(kT - T)] XT(k - 1 ) + B(kT - T)TUT(k - 1 )T, (13.3.49) where X T and uT denote sampled values of state x and input u. Then the discretized state space model is defined by XT(k)= TT sin 200 1 XT(k-1)+ ~ c o s

2 00 UT(.k-1)

(13.3.50)

L and y @ k ) - [c 01 xT(k) (13.3.51) Now, taking into account the random forces causing sensor vibration and random measurement noise, we derive the complete, standard discrete-time model of

520

Chapter 13: Electromagnetic Signal Processing: A Case Study

the vibration and measurement system. The following two equations define the m~el: x(k) = ~x(k-1) + ~ ( k - 1 ) + w(k-1), (!3.3.52) y(k) = Hx(k) + v(k) (!3.3.53) Here w and v are process and measurement noise temas respectively, w corresponds to the sudden (random) variations in environmental conditions and other random forces which make the sensor vibrate. Measurement error is due to the inherent limitations of measuring devices and to the atmospheric distortions. We assume the following regarding the statistics of the random processes w(k) and v(k): I. E[w(k)] = 0 for all k E[v(k)] = 0 for all k E

[w(j)T v(j)T] } _

[ Q(k)0 R(k)0 ] 5(k,j)

(!3.3.54)

where 5(kd) is the Kronecker delta. The matrix Q(k) is assumed to be nonnegative-definite, so it is possible that w(k) = O and R(k) is positive definite. 2. the initial state x(0) is a random variable of mean xo and cov~ance POThe above model will allow us to proceed with the derivation of the optimal filter to estimate the state of the rotating sensor. The optimal filter for estimating the state of the system (!3.3.52) and (13.3.53) is known to be (Goodwin and Sin, 1984). A A A x(k) = ,x(k) + K(k)[y(k) H x(k)] + Fu(k) (13.355) ~(O) = i o (the prior mean).

(133.56)

The optimal filter gain K(k)is given by K(k) = ,2(k)H T [HZ(k)H T + R] -1 (13.3.57) and ~(k) satisfies the following Riccati difference equation (PDE): Z(k) = ~ ( k ) , T + Q - K(k)[HZkH T + R]Kk T (133.58) Z(0) = Z 0 (13.3.59) The transient and steady state properties of the optimal filter have been studied extensively (Sorenson, 1985; Goodwin and Sin, 1984; Kwakernaak and Sivan, I972). Thus we have obtained an optimal state estimator which will enable us to estimate the angular position e and angular velocity i3 from the Doppler signature scattered back by the vibrating sensor. Now the estimator output can be used to vary the electric field strength such that the sensor is at rest at its original equilibrium point given by o = e0. To check the operation of the state estimator, the measurement sequence given in Figure 13.3.2b is fed into the state estimator and the estimated values for the angle of rotation are obtained. Figure 13.3.10 gives the error between the estimated values and the actual angles of rotation which produced the measurement sequence in Figure 13.3.2b. The results show that after some time, transients die away and in steady state the state estimator gives better tracking results.

T. Kirubarajan and P. R. P. Hoole

521

Difference ~t-,~een actual and estimated values

0.01 ~

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

,- . . . . . . . . . . . . . . . . . . . . . .

-

,

0.005

-0,005 -0.01 -0.015

.o.%

50

,00

.....

i5~

.................. 2 o o ~ - - - - - - ~ . ~ o

Sampling instants

Figure 13.3.10: Estimation Error of the Kalman Estimator

Angular position of the sensor

°.2I

0.15

0,05 |

-0~ -0, !

150

200

400

600

8~

1000

simulation time

Figure 13.3.11: Simulation Results for the Adaptive Controller

1200

522

Chapter 13: Electromagnetic Signal Processing: A Case Study

The authors have previously proposed a state estimator based on the Maximum Likelihood Criterion (Kirubarajan and Hoole, 1993a). To check the estimated values obtained using the Kalman estimator, that estimator is used. The same input sequence is used for the maximum likelihood estimator and the corresponding error sequence was given in Figure 13.3.3. It can be noted that, while the transient response of the maximum likelihood estimator is better than that of the Kalman estimator, in the steady state, the Kalman estimator gives better tracking. As earlier, we can use the LMS adaptive controller illustrated in Figure 13.3.7, since we have a state space model for the whole system. When new simulations are done for a set of input conditions using the Kalman estimator, the results obtained are as shown in Figure 13.3.1 I.

13.5 The electromagnetic signal processing work reported in this chapter has addressed the following three engineering problems: • How to determine the ~sition of the plane of an aircraft and its size from the scattered radar electromagnetic fields. ° How to stabilize optimally a vibrating sensor - - or for that matter, any cylindrical or plane body, hinged to a large structure vibrating at low frequency due to a disturbance force such as thermal stresses - - using an electric force adaptively controlled by the trackAng system. • How to float the tiny (e.g. 50 gm diameter, 1 gin thick silicon) rotor in micromotors used in robots, in order to eliminate completely the frictional losses, using a system similar to that described earlier. The scattered electromagnetic field based state estimator and tracker developed in this chapter may be used with the conventional Doppler radar system or the DME/ELE systems. It is applicable to both fixed wing aircraft and helicopters. Its practical significance lies in the fact that it may be an important component left unexplored in the MLS system now under experimentation. Furthermore, it could significantly reduce missed landings on non-stationary runways by adapting the runway plane as the reference plane with respect m which the aircraft plane is measured. The state estimator will have direct applications in many electromagnetic remote sensing applications, where the measurement of the tip plane orientation is useful for trajectory prediction and object identification. In the MLS it may be used to identify the approach of helicoptera, and military and commercial aircraft, and in addition, to guide automatically the aircraft from a landbased system, using the wing tilt angles or rotary-wing blade orientation. The method outlined in this chapter, applied to aircraft data and found to perform well, could also have important applications in signal processing applications in radio astronomy. We have also presented a novel application of electric fields to control low frequency mechanical vibrations. Such vibration problems are encountered in space vehicles, satellite sensors, aircraft and advanced manufacturing processes. Two

T. Kirubarajan and P. R. P, Hoo!e

523

instances which may ~ cited are the vibration problem faced by the Hubble Space Telescope and the manufacture and transport of silicon wafers in a clean environment where all mechanical contact is best minimized. The vibrating object is modelled by a homogeneous cylinder and tie state space model for the dynamic motion was reported, The optimal state estimator reported can be used to identify the current angular position of a vibrating body. The stabili~r is also intended for floating a micromotor rotor or microactuator moving arm. The state estimator formulation for a floating electrode was presented. Then an adaptive controller to vary the strength of the electric field was described. The electric field will exert a counteracting force and bring the sensor to rest at the required position. Computer simulations of the pro~sed methods have been found to perform very well. The work reported in this chapter highlights three important applications of a novel system which is an outcome of cross-fertilization of electromagnetic field computation and signal processing concepts.

524

Chapter 13: Electromagnetic Signal Processing: A Case Study

Appendix 13.A ii,

IHH

I,H!

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

The State Estimator Using MATLAB TM

% This is the main program file cI~ CC= ' X '; clc disp(' '); disp(' ') disp(' S C A T T A S (Matlab version) by T Kirubarajan') disp(' (Supervised by Dr P R P Hoole)') disp(' ') disp(' ') clcl= I; exi='n'; while exi=='n' I exi=='N' while cc ~=T & cc ~='F' & cc ~='s' & cc ~='S' if clc I ==2 clc ergJ disp(' ') disp('OPER.&~ON') cc=input('[F]ield calculation IS]rate estimation... : ','s'); clc1=2; end if cc=='f I c c ~ ' F jkl=l; and='y'; scatl CO= ' X ~ ; elseif cc=='s' t cc=='S' jk2=l; and2='y'; scat2 CC= ' X ' ; jkl=l;

T. Kirubarajan and P. R. P. Hoole

525

end antdd2='x'; while antdd2 ~='y' & antdd2 ~='Y' & antdd2 -='n' & antdd2 ='N' antdd2=input('Exit SCATTAS .... ? [YI?-I] ...', 's'); exi~ntdd2; er~J end disp(' ') disp('GOODBYE!') disp(' ') %subtile - scatl while and=='Y' I and=='y' if jkl==2 clc end disp(' ') disp('DATA') alpha=input('alpha ....... : ') r=input('radius... (meters)... : ') anm='y'; mo='x'; while anm=='Y' I anm=='y' disp(' ') dispCTYPE OF MOTION') while mo ~='N'& mo ~='R' & mo -='T' & mo --='B' ... & mo ~='n' & mo ~='r' & mo ~='t' & mo -='b' mo=input('[N]o motion JR]oration [T]ilt [B]oth ... ' , 's' ); e~cl phi=input('Range of phi... [initial step final] (radians) ... : '); phi_o=phi(1):phi(2):phi(3); phin=floor((phi(3)-phi(1))/phi(2))+ 1; r eq=zeros(pNn,l); phi_eq=zeros(phin,1); M a g U i =zeros (phin,l); phaseU_i=zeros(phin,l); MagU_e=zeros(phin,!); phaseU_e=zeros(pNn,l); MagU m=zeros(phin, !); phaseU_m=zeros(phin,I); if mo=='N' I mo=='n' cnt=l; xlab=?,lo motion'; for phii=phi(1):pN(2):phi(3) Leq(cnt)~; phi2~.q(cnt)~Ni;

526

Chapter 13: El~tmmagnetic Signal Processing: A Case Study

cnt-cnt+l; end mo='x'; elseif mo=='R' I mo=='r' theta=input('Angle of rotation(theta)... (radians) ... : '); cnt=l; xlab=['Rotation (theta=',num2str(theta),' radians)']; for phii=phi(1):phi(2):phi(3) r_eqCnt)=r; phi_eq(cnt)=phibtheta; cnt=cnt+l; mo='x'; elseif mo=='T I mo==T beta=input('Angle of tilt(beta) .... (radians)._ : '); cnt=l; xlab=['Tilt (beta=',num2str(beta),' radians)'l; for phii=phi(l):phi(2):phi(3) pb=sin(phii)*sin(beta); r_eq(cnt)=r*sqrt(1-(pN~2); pN_eq(cnt)=atan(tan(phii)*cos(beta)); cnt=cnt+l; erd mo='x'; elseif mo=='B' I mo=='b' theta=input(%ngle of rotation(theta) .... (radians)... : '); beta=input('Angle of tilt(beta) .... (radians) ... : '); cnt=l; xlab=['Rotation and Tilt (theta=',num2str(theta),' ... beta=',num2str(beta),' radiansy]; for phii=phi(|):phi(2):phi(3) pb=sin(phii)*sin(beta); r_eq(cn0~*sq~(1-(pb) 2); phi eq(cnt)=atan((cos(theta)*cos(beta)* sin(phii) sin(theta)*cos(phii)) /(sin(theta) *cos(beta)*sin(phii)+cos(theta)*cos(phii))); cnt-cnt+l; end mo=~x'; end for n=I:phin uil=(l-(alpha-2)*(r_eq(n) 2)/4); ui2=(alpha*r eq(n)*(l-(alpha 2)*(r;3q(n)~2)/8)* cos(pN_eq(n))); magU_i(n)=sqrt(uil~2+ui2 2); phaseU i(n)=atan(-ui2/ui!); magU, m(n)=(alpha)~2*(l N+cos(phi~(n)))* sqrt(pi/(2*alpha*r eq(n)));

T. Kirubarajan and P. R. P. Hoole

527

phaseU__m(n)=3*pit4 - alpha*r eq(n); magU_e(n)=(b'log(alpha))*sqrt(p~/(2*alpha*r_eq(n))); phaseU_e(n)~pil4-alpha*r_.~(n); end pla='y'; pI = ' X ' ; disp(' ') disp('FIELD PLOTS') while p 1 a== 'Y' I pla=='y' while pl -='I'& pl ~=T & pl ~='E' & pl ~ ='e' & ... pl ~='M' & pl -= 'm' pl=input('[I]ncident Scattered [E]lectric ... Scattered [M]agnetic: ','s'); end if pl==T f pl==T clg subplot(21 !) plot(phi_o,magU_i) ' ' Inclden " t field magm"t ude ' ); utle( xlabe!(x!ab); subplot(212) plot(phi_o,phaseU i ); title('Incident field phase'); xla~l('Surface angle in radians'); pl ' X " pause elseif p l = ' E ' I pl=='e' clg subplot(21 I) plot(phi_o,magU~) fitle('Scattered electric field magnitude'); xlabel(xlab); subplot(212) plot(phi_o,phaseU_e); title('Scattered electric field phas e ' ); xlabel('Surface angle in radians'); ' X ' ," =

p l

elseif pl=='M' I pI=='m' clg subplot(211) plot(phio,magUm) title('Scattered magnetic field magnitude'); xlabel(xlab); subplot(212) plot(phi o ,phaseU m); title('Scattered magnetic field phase'); xlabel('Surface angle in radians'); pl='x'; pause end

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Chapter 13: Electromagnetic Signal Processing: A Case Study

plaa='x'; while plaa ~='y' & plaa ~='Y' & plaa ~='n' & plaa ~='N' plaa=input('Another plot...? [YR,t] ...: ','s'); pla=plaa; end anmm='x'; while anmm ~='y' & anmm ~='Y' & anmm ~='n' & anmm -='N' anmm=input ( ' Change type of motion .... ? [Yfl'4] ...: ','s'); anm=anmm; era etvJ andd='x'; while andd ~='y' & andd ~='Y' & andd ~='n' & andd ~='N' andd=input('Ano~er set of data...? [Y/N] ...', 's'); endjkl=2; end

%subtile - scat2 while a n d 2 ~ ' y ' 1 and2=='Y' ifjk2=2 clc disp(' ') disp('DATA') am=input('Measured magnetic field magnitude... (A/m) ...:') bm=input('Measured magnetic field ph~e... (radians) ... " ') k=mput( alue of k... (m^(-1)) ') phi=input('Surface angle phi .... (radians) ... • ') sl=input('Cov~iance of noise in magnitude measurement... • ') s2=input( Covariance of noise in phase measurement .... • ') eI=input('Initial estimate of r ....... ' ') e2=input('Initial estimate of theta ...... " ') e3=input('Initial estimate of beta ...... " ' ) estim=[phi sl s2 am bm k]'; iniva=[e! e2]'; dtI=zeros(l 6,1 ); dtl(1)=l; [estl,code]=fsolve(rot,mlva,dtl,estlm) r=(I/2)*sqrt((3*pi-4*bm)/k); % for rotation only thetal=phi-acos((2* am~&)*~rt(2/(pi*(3*pi-4*bm)))-i I/4), "

%

disp(' ')

'V

.,.

"

T. l,'drubarajan and P. R. P. Hoole

529

disp('THE ESTIMA~S') disp(' ') fl=['radius (r) in meters ... : ',num2str(r)]; disp(rl) tl=['Angle of rotation (fihetal) in radians ...: ',num2str(thetal)]; disp(tl) t2=['Angle of rotation (theta2) in radians ._: ',num2str(theta2)]; disp(t2) antdd='x'; while antdd ~='y' & an~d ~='Y' & antdd ~='n' & antdd ~='N' antdd=input('Another set of data...? [Y/N] .. :','s'); end antdd='x'; jk2=2; end %subtile - rot function y=rot(x,z) r=x(1); t=x(2); p=Z (I) ; sl=z(2); s2=z(3); a=z(4); b~(5); k=z(6); y=zeros(2,1 ); y(1)=alr-k*sqrt(k*pi/2)*(1/4~os(p-t)); y(2)=((2/s2-2)*(b-3*pi/4)+(k~2*pi/(2*sl~2))*(I/4+cos(p-t)) 2)+... 2*k*r~~'s2.-2(a/(sl--2*r))*sqrt(k*pi/2)*(1/4~os(p-t)); %subtile- tiI function y=tit(x,z) ~-x(1); bt=x(2); p=z (1) ; sl=z(2); s2=z(3); a=z(4); b=z(5); k=z(6); y=zeros(2, I); l~&=sqrt(k*pi/2); pb=atan(tan(p)*cos(bt)); sp=(1-(sin(bt))-2*(sin(p))~2);

530

Chapter 13: Electromagnetic Signal Pr~essing: A Case Study

y(l)=(-!/sl 2)*(a-k*kk*(1/4~os(pb))*r/(sp~.25))* .... (kk*(l/4+cos(pb))/(sp~.25)+(I/s2~2)*(b-3*pi/4+k*r~2*sqrt(sp))*... (2*r*sqrt(sp)); y(2)=(r/s2-2)*(b-3*pi/4+k*r~2*sqrt(sp))*(1/sqrt(sp))-... (l/s2~2)*kk*((ak*kk*(t/4+cos(pb))*r/(sp-.25))*(sin(pb)*tan(p)* sin(bt))/... ((l+(tan(p)*cos(bt))~2)*(sp~.25))+.5*(1/4+cos(~o))*... (sin(p))~2*sin(bt)*cos(bt)/(sp~ 1.25)); %subtile- rtl function y=rtl(x,z) ~x(1); t=x(2); bt=x(3); p=Z (1) ; sl=z(2); s2=z(3); a=z(4); b=z(5); k=z(6); y=zeros(3,1); k2,c=sqrt(k*pi/2); pb=amn((cos(t)*cos(bt)*sin(p)-sin(O*cos(p))/... (sin(t)*cos(bt)*sin(p)~os(t)*cos(p))); sp=(l-(sin(bt))~2*(sin(p))~2); an=(sin(t)*cos(bt)*sin(p)+cos(t)*cos(p)); y(!)=(a/r-k*(I/4+cos(pb))*kk/(sp~.25)); y(2)=(2*sp.-.5/s2~2)*(b-3*pi/4+k*r~2*sqrt(sp))-... ([/s I ~2)*kk* (1/4~os(pb))/(sp .25 ) *(a/r-k*kk*( 1/4+cos(pb))/(sp. 25 )); y(3)=(1/sl-2)*(a/r-k*k&*(l/4+cos(pb))/sp .25)*kk*((sp~.25)*... sin(pb)*cos(p)/((1 +(tan(pb))~2)*(an 2))+5*(1/4~os(pb))* .... cos(bt)*sin(p))+(I/s2 2)*cos(bt)*sin(p)*(b-3*pi/(4*r) +k'r* sp'.5);

Chapter 14 Jlllll

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

P. Ratnamahilan P. Hoole and B. A. A. P. Balasuriya I

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I[

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

COMPUTATION OF PULSED ELECTROMATGNETIC FIELDS: WITH NEURAL NETWORK APPLICATIONS

14.1 Outline Transient and pulsed electromagnetic fields are frequently encountered in both natural phenomena - - like lightning and synchrotron (pulsed) radiation from stars as weI1 as in engineering systems like radar, digital communication and power systems (Stratton, 1941; Joos, 1934; Jordan and Balmain, 1968; Barlow, 1971; Jones, 1981; Cullen, 1981; Hoole and Hoole, 1986; Hoole and Hc~ole, 1987). The computation of pulsed electromagnetic pulses is essential in the analysis and design of a large number of devices and systems. The response of high temperature plasma to laser pulses is important in thermonuclear fusion experiments for bringing particles close together by implosion. A knowledge of lightning and nuclear electromagnetic pulses is critical in designing stable control systems and protection for fly-by-wire aircraft. The performance of semiconductor devices when subjected to nanosecond digital signals needs to be improved for semiconductor devices to takeover completely the role of high voltage gas discharge tubes as high power satellite communication amplifiers at microwave and millimeter wave frequencies. The stable operation of electric machines when controlled by power electronic devices or when subjected to power transients will require machines designed to give well distributed stator and rotor magnetic fields when subjected to transient currents. Effective radar antenna design involves electronically controlled changes in antenna current patterns to provide required scanning patterns and the prescribed response of the antenna system to incoming electromagnetic waves. All these and many more engineering systems necessitate powerful software tools to determine the relation between transient electromagnetic fields and their current source. In this chapter we shall set out an integral method and the finite element method for transient field computation, after first reviewing the basic

532

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

concepts involved and the simple transient current models for frequently encountered sources (Hoole, Hoole and Jayakumaran, 1986; Zienkiewicz, 1977; Bathe, 1982; Hoole, 1989; Watters, I980; Felsen, 1976). A detailed discussion of time-domain analysis using both the integration scheme and the mode representation scheme is given. The stability of both differential equations and the numerical methods most commonly used is analy~d. Some important features related to the development of general computer programs t\gr a variety of engineering electromagnetics problems are reported (Hoole and Hoole, 1984; Hoole, Jayakumaran, Anandarajah and Hoole, 1990; Hoo!e, Cendes and Hoole, 1986; Hoole, 1984; Hoole, I987).

14. 2 Basic Equations for Transient Electromagnetic Fields The two divergence equations of electrostatic and magnetostatic situations hold true for time-yawing transient electromagnetic fields. T,nese are V.BB= 0 (1.2.5) V.D= 9 (1.2.6) The other two Maxwell equations may be derived from simple experimental facts. The procedure is based on the conservation of energy, together with f o ~ u l a e for stored energy, power dissipated, and power flow. We shall go through this procedure in order to reinforce the important relation between energy radiated into space and the Maxwell equations for which we shall be seeking solutions. The energy stored in an inductor and capacitor are gH2 dv

(14.2.1)

eE 2 dv

(14.2.2)

W m = ~ LI 2 = V

W e = ~ L, 2 = V

If we now consider the current flow between two circular discs connected to the alternating current source, as shown in Figure 14.2.1, the ~ w e r flow between the plates P = 0I. But V = Ed, I = H 2ha; therefore P = jrE_.x H . ~

(14..2.3)

S

The power dissipated in a conductor

_,2 j z:dv

PL-R =

(I4.2.4)

P. R. P, Hoole and B. A. A. P. Balasuriya

533

t

E

dS

Figure 14..2.1" Current Flow between Two Plates Consider the finite volume V bounded by a surface shown in Figure 14.2.1. The rate of decrease of the energy stored in the volume must be equal to the sum of the rate at which energy is dissipated in the volume and the rate at which the energy leaves the volume across the boundary S. We therefore get - -d- f { dt

~1 ~H2 + ~1e E 2 } dv = .ferE2dv+ jrExH.ds s

(14.2.5)

V

Convening the surface integral into a volume integral using Gauss's theorem and rearranging, we obtain

f{ Hall -aT + EE ~a5+ c~.E + V.(ExH)

dv = 0

(14.2.6)

v Since this must be true for any volume, the integrand itself must be zero. Expanding the divergence term and equating the integrand to zero, we find

(0")

H. f a ~ + V x E

+_E. eff+er_E-VxH

)

=0

(14.2.7)

Since E and H are linearly independent, it follows that the multiplying factors must be separately zero. Thus we get the two curl equations of Max W ell ' s aH Vx__E= - ~ (14.2.8) OE VxH =ere + ~

(14.2.9)

In general, the reciprocity theorem holds true for Maxwell equations. For example, the mutual inductance between two coils is reciprocal, i.e., M12 = M2t whether the coils be identical or not. Similarly, for an antenna the radiation pattern when transmitting and receiving are identical, although the current distribution along the antenna may be different in both cases. The necessary condition that must be satisfied for the reciprocity theorem to hold is that the quantities er, e and g, should be symmetric tensors, of which, a scalar is a swcial and familiar case. There are, however, certain conditions under which there is an apparent

534

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

breakdown of reciprocity. Consider for example a pulsed electromagnetic signal (e.g., digital communication signal, pulsed radar signal, etc.) travelling over land and sea. It would be found that the transmission loss would depend on whether a land-based antenna is transmitting or one at sea. When a land based antenna is used as the transmitter, the field strength at ground level would begin to rise with increasing distance from the transmitter as the land/sea boundary is crossed. The explanation of the effect lies in the different vertical distribution of the field over land compared with that over sea. More field energy is f;ound high above the surface over land, whereas the energy is closer to the surface over sea. Thus as the land/sea boundary is crossed there is a redistribution of energy vertically in the vicinity of the boundary. Other practical examples of non-reciprocal transmission paths are ferfites at microwave frequencies and electric plasma immersed in a magnetic field (e.g., the ionosphere in the earth's magnetic field). The conjugate field theorem may be understood by considering Maxwell's equations for monochromatic fields in free space: (I4.2.10) VxE = -jo~0H (14.2.11) VxH = +J~0g Taking the complex conjugate of both sides (14.2.12) VxE* = ~ 0 H* (14.2.13) VxH* = - j ~ 0 E* which can be re-written in the f o ~ : (14.2.14) VxE* = -jo~go(-H*) (14,2.I5) Vx(-H*) = +j,~0 E* We may conclude that if (E,H) is a solution of Maxwell's equations, then (E*, -H*) must also be a solution. This theorem is useR~! when we need to ~.~verse the direction of travel of both guided and free waves. For example, a wave which has been distorted by passing through a dielectric sheet (e.g. glass) may be recovered by passing the wavefront back through the glass. One of the many important applications this opens up, is in image processing - - a fuzzy picture taken through a frosted glass may have all the distortions removed by reflecting the distorted wave back through the glass; there is a phase reversal, but the amplitude would be identical to the clean signal amplitude. Other applications of this theorem are in radio techniques, tracking laser beams, sonar and many more.

14.3 Some Common Types of Transient Pulses In this section we shall survey some of the most common types of electromagnetic, current and voltage pulses encountered in electrical power and telecommunication systems. These current pulses may be used with the computational techniques outlined in the next two section for a time-domain analysis of electromagnetic fields. Alternatively, the Fourier spectrum of the pulses may be used for monochromatic wave analysis - - where the currents may be taken to be sinusoidal - - and the inverse Fourier transform of the results may

P. R. P. Hoole and B. A. A. P. Balasuriya

535

be used to reconstruct the resultant electromagnetic fields. The transients f(t) are finite both in amplitude and in time, and may be continuous or discontinuous.

14.3.1: Double Exponential Function The transient is given by f(t) = A [ e ' a t ~ e "bt]

(14.3.1)

and is shown in Figure 14.3.1. In (14.3.1), constants a and b are selected such that b > a. The first derivative of the transient is df [ (14.3.2) = A t-ae'at + be "bt] dt from which the time to peak is obtained by setting df/dt = 0, to give tp =

a-b

(14.3.3)

The initial rate of rise (at t = O) is dt7dt = A (a -b) and the average rate of rise is A/tp. The fourier transform of (14.3. I) is A(b~a) F(m) = ab + jm(a+b) '0, 2 (14.3.4) The second derivative d2f= A[a2e "at- ~ e "bt] (14.3.5) dt t is the most important component of radiator current characteristics for radiation electromagnetic fields.

14.3.2: Parabolic Exponential Function The parabolic exponential transient differs from the double exponential transient, in that the wavefront is convex in the previous case and it is concave (parabolic) in the latter case, as shown in Figures 14.3.1 and 14.3.2. The parabolic A exponential

f(tJ A

AJ2

o

........

! |

I !

1 I

! !

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

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

i. t"

Figure 14.3.1: A Typical Transient from Double Exponentials

536

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

- -

.

.

.

.

.

.

.

.

.

.

.

.

f(t) A

0

t

t

Figure 14.3.2: A Transient from a Parabolic Expenential transient is given by fit)

[(at2)e "at]

(14.3.6)

and the time to ~ a k tp = 2/a. The first derivative is given by dfdt= Tlae 2 [2(at) - (at) 2] e "at and the fourier transfo~ e2I!p_ 1 F(w)

(14.3.7)

(14.3.8)

= .........i~---~

l+j 2w~ The second derivative is d2f a2e2 I [ 2 - 4at + (at) z] e -at dt 2 - 4

(14.3.9)

14.3.3: Infinite (Digital) Pulse Train The infinite pulse train resembles the digital data signal used in communication systems. The pulse train shown in Figure 14.3.3. is given by f(t) = V

1+2

sinn~(~/2) noh3(~/2)c o s n ~ 3

(14.3.10) n=l where ~ = 2~JT. and the bandwidth is 2~:. The envelope of the Fourier transform of the signal is sinno~0~t2 Fe(~ ) = VT n0~0~2 cos n~0t (14.3.11)

P. R. P. Hoote and B. A. A. P. Balasufiya

I

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537

if(t)

!

0

T

t

Figure 14.3.3: The Pulse Train

which has a well defined (sin x)/x shape. 14.3.4: Pulsed Radio - - Frequency Wave In Figure 14.3.4 is shown the pulsed radio-frequency wave representing the electromagnetic fields radiated by a pulsed radar or generated in the control circuit to turn on and off a transmitter. This pulsed wave train may be representedby Ii 0 -'d2 ~ t ~ -'d2 and "c/2 _
and the frequency spectrum or Fourier transform envelope is Isin (nohg-Wc)'C/2 Fc(~)=V~L

sin (no) 0 +O)c)~/21

(n~00:~

+

(14.3.I3)

(nw 0 + @

and the bandwidth is 47v'~.

14.3.5: Exponential Transition Transient The exponential transition transient, shown in Figure 14.3.5, frequently encountered in digital integrated logic circuits and power electronic circuits, may

'f(t) T !

|

',j :',,J

l,J t,2

!

-V

|

. . . . . . . . .

',,/,',,J

: )

,

i

........

!,

,

I

,-

|

....

!

,

,j,

...........

Figure 14.3.4: The Pulsed Radio Frequency Wave

538

Chapter 14: Computationof Pulsed ElectromagneticFields: Neural Netoworks .......

,,,,,,

f(t) A

0.9A t

I l I

0.1A, t

t

Figure 14.3.5: The Exponential Transition Transient be represented by t[_ Ae3'2189t/tr 2

i

f(t)=

A [ 1 - e'3"2189fftr] 2

t< 0 (14.3.14) t~0

the first and second derivatives of which a r e = ' d f+ A ~ ) 2 dt

e( + 3"2189t/tr)

d2~_+ A ~3.2189"-{e(+ 3.2189 t/tr) dt 2 - - 2 \ r ) and the Fourier transform 1 + F(m) = Atr jmtr 3.218~+ (o~tr)

for all t

(14.3.15) (14,3.16)

(14.3.17)

14.3.6: Damped Sinusoidal Transient When electric power circuits or electromechanical devices like motors are switched-on or interrupted, we have the sinusoidal transients shown in Figure 14.3.6 generated in the system; such pulses are also generated by power elex:tronic devices like the silicon- controlled rectifier during turn-on and turn-off. It is highly desirable that electric machines controlledby power electroniccircuitry be designed for magnetic fields generated by such sinusoidal transients. We may mathematically represent the transient shown in Figure 14.3.6 by f(t) = Ae"at sin (~t) (14.3.18) where a is the damping time constant and o30is the angular fi-equency of oscillation. The time for the pulse to decay to about 10% of initial peak is

P. R. P. Hoole and B. A. A. P. Balasuriya

I

539

!

Figure 14.3.6: Damped Sinusoidal Transient approximately 2.3 I/a seconds, and the peak of the energy specmam occurs at the frequency

. The Fourier transform of (14.3.18) is

A~ F(oJ) = (a2 + o>2 at~ + j2ao~)

(14.3.19)

and the first and second derivatives are dS- Ae'at [-asin ~ t + m0cos ~0 t] (14.3.20) drIn some communication systems like the LORAN navigation systems, the parabolic-exponential sinusoid is used, for which we have d2f-Ae'at [ - ( a 2 + ~(~)sin o,0t- 2ao~0o~cos~0 t ] 2 dt 2 f(t) = At2e'atsin ~ t

(14.3.21)

djf= Ae.at[(2t.at2)sin °~0t + t2e,0cos ~tJ dt

(14.3.23)

d _ Ae.at [sin ,~o0t+ (2t-at2)sin ~0 t + o,0t2cos ~t] dt 2 -

(14.3.24)

F(e,)= A

(I4.3.25)

[(a2+~2

~)+j2a~]3

(I4.3.22)

540

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

I

R I I I

f

i

Zi

Z2

Figure

14.4.1: A Linear Antenna Element

14.4 The Basic Linear Element for Integral Solution of Pulsed Electromagnetic Fields In the design and synthesis of wire antennas, travelling wave antennas, array antennas or aperture antennas, it is recognised that the linear current element antenna may ~ used as the elementary antenna building block. For example, according to Huygen's principle, by judiciously placing a number of elements at the aperture of a horn antenna we may obtain an expressive relationship between the radiated electromagnetic fields and the electromagnetic waves inside the waveguide. The latter links the microyJave source to the horn antenna. In this section we shall obtain a general exp~ssion for the electric and magnetic fields radiated by a pulsed cu~ent on the linear element of length (Z 2 - Z 1 ) shown in Figure t4.4.1. Consider the following equations governing the electromagnetic field 0A(R,t) E ( R , t ) - - bt - V,(R,t) (14.4.1) B(R,t) = -V X _A(R,t) (14.4.2) where, for an infinitesimal element cm-xying current, the vector potential is given by A(R,t) = Z k

h (14.4.3) 4~ R where vectors with subscript k indicate unit vectors and the square brackets indicate time retarded values, Le., [I] = I(t -P/c). ~ e static potential ~ is given by the Lorentz gauge. In spherical coordinates, we have = V,A(R,t) = 0 0t ,'- g0 1 A ~0 I A=rk ~hcose-0k ~hsin0 c2

Substituting (14.4.5) in (14.4.2) and using b/bR = -l/c 0/bt,

(I4.4.4) (14.4.5)

P. R. P. Hoole and B. A, A. P. Balasufiya

B0(R't)

541

=¢k(l ~ ~']~t0hsin0 ~.Rc dt + R 2 ;

(14.4.6)

before determining E(R,t) we find fiR,t) from (14.4.4) t

0

~(R,t) = -c 2 ~V.A(R,~)d~- c 2 JV.A(R,r) dr 0 .oo -{"l g 0 ~ h + ~ c 2 2 ° ) s 0 _ V.A(R,t) l, Rc dt 4rt Hence the first term on the right hand side of (I4.4.7) is given by t [Q~] .o h cos o IV.A(R,~)d~ = -

0 where

[I] + R2

4x

(14.4.7) (14.4.8)

(14.4.9)

r

[Q'] = I[I] dr 0 and the second term is given by 0 Qo go h cos 0 SV.A(R,~) d r R2 4n

(14.4.10)

(14.4.11),

ignoring any cu~ent on the radiating element prior to the application of the pulse current I. QO is the electric charge, if any, unifonni!y distributed along the element before applying the current. Setting [Q] = QO + [Q'], from (14.4.7) we have o(R,t) = , ~ / ~ hcos hcos 0 ~ e 0 4nR °[I] + 4980R2 ~ [Q] (14A.12). Substituting (14.4.5) and (14.4.12) into (14.4.1), and expressing (14.4.6) and (14.4.13) in cylindrical coordinates, ^ hcos 0 + ~(R,t) =R k 4n eR3

^

+ Ok

[O])

l.R

(14.4.13)

eR3

y

^ Dr0h r ~t0h r B__0(r,z,t)= PXk [,~1-- (r2 + Z 2 ) 3/2 [I] + 4nC (r2 + a 3h g ~ ( r Z x 2 ) E(r,z,t) = r k ~ r2

2 [I] + +Z

d[I] ) (14.4.14) 3/2 dt

"0 h rZx d[I] 4n ( r 2 + Z 2 ) 3 / 2 dt

542

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

3h

rZx

+4~O ( r 2 Z~) 5/2 [Q] ^

2Z~

+ Zx ~0h

r2

r

Z 2"~2" + x)

[I]

r2

4~ ( r 2 + Z ~ ) 3 / 2

at

Using (14.4.14) and (14.4.15)we may derive an expression for the field radiated from a finite length (z 2 - z I ) of thin conductor. By setting h = dz, ~ = zj - z, dz = dz x, and integrating, we obtain, using standrad integrals, for the finite length of cu~ent element

i lh

I1

i2

l ld

Perfect I1

12

Figure 14.4..2: Reflections of Vertical and Horizontal Current Elements

P. R. P. Hoole and B. A. A. P. Balasuriya

:0(r,z,t) = ~n0

r2 + (Zj-Z)2 tZ=Zl r

+~

Er(r,z,t) = ~

543

! [

tan -1..... r ~=zl dt

(14,4.16)

I

[r2+,~/~_Z.~213/q ? ] J

[Qlr

Z=Z 1

3

~0

!

.']z=z2

+ ~ "~4~°Jr2 + (zj-z)2J

[I]r

z=z 1 (14.4.17) + ~ Ez(r'z't) - -

(Zi_Z,}2~=zl a t

1 [[r2 "va Z - ' e _! ~ =z2 4roe0 + (a-Z"}2 ] 3/2

[Q]

z=z 1

1 ~0 - 8-~ ~ I r 2

3(Z-~) l tan. l (Z~Zj)jF=z 2 + (,_Z)2- r ............... [I] Z=Z

-~0[~4/4rt Z~_~ - --~¢=Z2d[I]dt r2 + ( a - Z " 1 2 | ~,

]

1

(14.4.t8)

_IZ=Z 1

Now the field contribution of each ( ~ -Zl) element - - which with other elements makes up the whole system - - to the field at a specified point (r,z) could be calculated from (14.4. !6), (14.4.17) and (I4.4.18) and Mded to give the total field. Tilted systems are allowed by resolving an element into horizontal and vertical components. A perfectly conducting ground plane could be replaced by the image of the element. Since electric charges reverse sign in their reflections, upward currents in the vertical elements have upward reflections, but for horizontal elements, left-moving currents have right-moving reflections and vice-versa, as shown in Figure I4.4.2. Thus to compute the field radiated by a long line of current, it may be broken up into little bits, and then each little bit into vertical and horizontal bits. Where the earth is involved, it may be replaced by reflections of the bits as shownin Fig. 14.4.2. More on this technique is given by Hoole and Hoole (1987)

544

Chapter 14: Computation of Puled Electroma~etic Fields: Neural Netoworks

Software Techniques for Linear

t Antennas

A Pascal program is given in Appendix 14.A for the evaluation of radiated electric and magnetic fields due to pulsed current on two vertically stacked linear elements of finite length. The proximity of the e ~ h is taken into account by including the two mirror images of the elements, as depicted in Figure I4.4.2. Figure 14.4.3 gives the source currents corresponding to a lightning stroke and the resulting radiated fields.

Kjthe M cloud ~S

Figure 14.4.3(a) Source Current Waveform.

~18xli Ol-7T m~4.5x/{~T I O-~T 1000m,~":~ 1.~I00 m)

-~(0~-. ,

5xlO-°mi -~[2x10 'T " (60.0)km :i(lO.O) km

~ ~ 2 4 T ~xlO-5T :111xl O-'6T I (50.O)m 27i (200.O)m""i 2 ,km Figure 14.4~b) Radiated Magnetic Field

P. R. P. Hoole and B. A. A. P. Balasudya

545

Eg " 500V/m t oi

(] O, 2) km

-

Er

J

(0,2,2

.

90kV/m

-11

L28

v/m

kV/m

"f / 1

-

0v,m :

14xlObv/m

Eg

75kVlm

- 3kV/m

E "

(2.0)km

O,Olm : w

F i b r e 14.4.3(c) Radiated Electric Fields(Er and Ez).

14.5 Finite Element Formulation for the Wave Equation In this section, we shall consider a more general solution of the wave equation, since such a f o ~ u l a t i o n would not only permit us to handle complex radiating

546

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

antenna elements, but also the effects of material media, including dielectric and conductor scatterers, which alter the radiation field magnitude and pattern. We intend solving for the electric and magnetic fields radiated by a plane symmetric or axisymmetric radiator in isotropic media. The magnetic flux density B is governed by Maxwelrs equation V x _ l B = j + aD at

(14.5.1)

where fa is the permeability, J is the current density, t is time and D is the electric Flux density. The electric flux density is g o v e m ~ by Maxwell's ~uation. aB Vx~e" 1D = - at (14.5.2) where e is permittivity. Considerable simplification is achieved by defining the magnetic vector potential A, B = VxA (1.3.7) and the electric scalar potential ~, OA D = - eV, - e at (14.5.3) using (1.3.7), (14.5. l) becomes OD Vx Vx A = gJ + taN

(14.5.4)

Eliminating D by using (14.5.3) results in V V.A - V.VA = gJ - ge V

+

(14.5.5)

which may be re-written as V . V A - ~ Eaa~ot 2 = - g.' + V ( V . A + lae ~t )

(14.5.6)

This equation can be simplified by specifying the divergence of A, by choosing the divergence of A to satisfy, the Lomntz condition. V.A = - ~ at (14.5.7) Thus (14.5.6) becomes the inhomogeneous wave equation. Given the current density J, the vector potential A may be calculated using (t4.5.8). The magnetic flux density B may now be calculated from (14.5.8). OA2 V.VA- p.e ~ = - ~ (14.5.8) The electric flux density E is calculated from (14.5.3) by first determining ~ from (14.5.7) 1 1

*='~6IV.A d~ + *0

(14.5.9)

where 00 is the initial potential on the radiator and any scatterer. The finite element method is used to solve (t4.5.8). Consider first a twodimensional (x,y) problem and re-casting (14.5.8) in terms of scalars A and J,

P. R. P. Hoole and B. A. A. P. Balasuriya

547

O2A + ~tJ = 0 (14.5. t0) at2 where (ge)-l/2 is the velocity of propagation. If (14.5.9) is to be satisfied, it follows that V . V A

f~(

-

ge

( V.VA - g ~O2A + ~tJ) dR=0

(14.5.11)

where -~ is a weighting function and R is the region of solution. Using integration by parts, f gl"

32A

dxdy j'J' xdy =

-

x2

+

y

dy

(14.5.I2)

Xl

At the boundaries x I and x2, either A = 0 or ~AxA= 0. Hence

ff["4'(V-VA)dR= f I (V~').(VA)dR

(14.5.13)

Therefore (14.5.11) may be re-written as - .,t" I (V,4t).(VA)dR.- ge J(" 1(]1" ' 4a2~" , ~ dR + g jt'~{"'4~JdR = 0 ff (14.5.14) The whole region, including the radiators and scatters are subdivided into triangles (Hoole and Hoole, 1984; Hoole, Jayakumaran, Anandarajah and Hoole, 1990); the vector pomntial A within each triangle is related to the vector potentials at the vertices of the triangles A 1, A 2, A 3 at time t, by 3 A(x,y,t) = ~fi(x,y)Ai(t) = [F][A] (14.5.15) i=l where fl, f2, f3 are area co-ordinates (a! +blX+elY ) 2A (a2 +bax +cgy ) f2 = 2A fl =

(14.5.I6)

548

Chapter l 4: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

f3 = in which

a + b3x + c3Y ) 2A

I x I Yl 1 1 x 2 Y2 1 x 3 Y3

= .Area of triangle

(t.5.30)

and a I = xlY 3 - x3Y2 (14.5.17a) bl = Y2 - Y3 (14.5.17b) c 1 = x 3 - x2 (14.5.17c) We set v such that V =~(x,y). The current density within each triangle is approximated by 3 J(x,y,t) = E f i ( x , y ) j ( t ) - [F][J] (14.5.18) i=l Assembling A , , and J for each element, gives the global equation (I4.5.I0) in terms of element values - [P][A] - tMT][A] + ~t[T][J] = 0 where the vector of unknowns [A] may be obtained from

(14.5.19)

[;£I FaaA] =L

J

PiJ= f It( Vfi'V~)dR

(14.5.20)

a"

(14.5.21)

so that gE[T][A] + [P][A] = ~t[W][J]

(14.5.22)

To solve (14.5.22), a futher approximation for A is required: At+At = (At)-2 [2A t+At - 5A t + 4At-A t . At-2At] which upon substitution gives { 2g~:(At)-2 [T] + [P] }[A] t+z~t = g[T][J] t+At + 5~e(At)-2 [T][A] t

(14.5.23)

P. R. P. Hoole and B. A. A. P. Balasuriya

549

- 4ge(At) -2 [T][A]At-At+ ge(At)-2 [TI[A] t-26t (14.5.24) The equation (14.5.24) is the final form of the equation solved to determine the potential A at each node in the triangular mesh at time t+6t, where 6t is the time increment. Expressions of the type (14.5.24) may be built up for all the triangles making up the region R and summed to correspond to the minimization of the full global functional

F(A) =

~

+ V2A - g/xJ dR

(14.5 25)

R

The resulting set of equations may be solved at each time step using (14.5.25) by either the preconditioned conjugate gradient method or frontal solvers (Hoole, Cendes and Hoole, 1986). For the case where A = Yk~, we shall show how the magnetic flux density B and electric flux density D are obtained. Once this is understood, we may formulate ~ and ~ when two components of A (e.g., A x and Ay) are present

B - VxA_ =Xk~'Yk~

+zk~ bX " ~ = )

= Zk ~ OX when & = ykAy, A x - A z = 0 ~=Zk{

+

(I4.5.26)

+

}

(14.5.27)

from which B within each triangle may be determined at each instant of time t, t t t since the values of potentials, A 1, A 2, A 3 at the nodes of each triangle, have already been determined from (I4.5.24). Now the scalar potential ~ is given by t L V ,t _ ~c!" .Adx + *0 t I cf(~A c2 c3 ) =" ~ 1 + ~ A2 + ~ A 3 at + 00 (14.5.28) 0 where the initial potential *0 in terms of initial potentials at the nodes of a triangle is given by q'0 = 001fl + *02f2 + *03f3 (14.5.29)

550

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

The electric flux density is readily obtained by assuming that the scalar potential rapidly decays to negligibly small values, except for very short distances from the radiator. Therefore, aA = xar e ~ + D O (I4.5.30) where Do is the initial value of flux density on which the radiator field is imposed. We get =-eL~l[aAlf aA2"~2 + ~ 3 ]

+ [D01f I + D0gf2 + D03f3]

(14.5.31)

The application of the technique described above is illustrated for a pulsed lightning current I = I0 e-z'! { e "at- e "bt} + Ip(t) + ~ (! 4.5.32) where Ip and Io are a sharp pulse current and a constant continuous current (Hoole and Hoole, 1986) respectively. The wave, as a typical example, travels along a 3 km long vertical lightning leader column. The radiated magnetic fields may be obtained from (I4.5.24) and (14.5.27).. It should be noted that for the example of a vertical column of cu~ent as in lightning, an axisymmetric cylindrical radiator ought to be considered. In this case we have a volume integral instead of an area integral in (14.5.11), to give (ai + bir + ciz) (14.5.33) fi= 2A a i = ~Zm- rmVj b i = zj - z m

(14.5.34)

ci = rm .

r i + rj + r m

r ~

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

3

_ ,

z --

+

3

zm

I Sij = 2n dfVfi.Vfj r dr dz = 2~ Vfi.(~,~) V~.(~,~) ~k =

(I4.5.35) (bibj+cicJ)2A (14.5.36)

Tij = 2~fi.(r,z))fi.(r,z) iA -- 2Art~(a i + bi r + ciz)( ~ + bjr + cjz)

(14.5.37)

where (rl ,Zl, (r2,z2) and (r3,z 3) are the nodal coordinates of a triangle, and the volume associated with each triangle is the area of the triangle multiplied by 2ft. The perfectly conducting ground plane may be, as already explained before, accounted for by a mirror image of the radiator. The computed results are presented in Figure 14.5.1 at increasing distance from the channel. Each figure, corresponding to a different radius, gives the fields at different altitudes.

P. R. P. Hoole and B. A. A. P. Balasuriya

551

xt O'gTe*~js

A: IK~MFrom Flash

~oo 90

70

6O

~0

Gto~d

e

5

~0

15

~

Z5 ~0

,~0 ~5 ~o liO~tQ~mlJl

45

50

KlO~OTemlsm

C: 25 KM From Flash 90

?0

50

~0 ?lu (~,.} 0

........ : ..............

$

..........

iO

:

......

rS

: ......

:

~

.... ~ .......

~

~

~

3S

O

5

~O

~5

20

~

38

35

Figure 14.5.1: The Magnetic Field Radiated from Lightning at Increasing Distances from Channel

552

Chapter t4: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

14.6

General Formulations for the Time-Domain Finite Element Scheme

A general finite element formulation of the wave equation will be developed, so that the various step-by-step procedures available to integrate the wave equation may be discussed simultaneously (Oden, 1969; Zienkiewicz and Lewis, 1975; Bathe and Wilson, 1973; Houbolt, 1950; Zienkiewicz, 1977; Newmark, I977). It should be noted that in this chapter the subject is the integration or recurrence methods applied to the wave equation and not the mode superposition technique. The latter is briefly touched upon at the end of this chapter (section 14.7). We proceed now to devise a finite element scheme in the time-domain tor equations containing the (O2~0t2) term. With time as the independent variable, we discretize just as in a spatial domain, A(t) = '~)"gi(t)Ai(t)

(14.6.1)

where ~ ( t ) is a set of A(t) at any given single node, at time t, t going through the steps t-At, t, t+At. Since the wave equation involves second derivatives, the function gj(t) should at least be a second order polynomial. Thus the interpolation functions gj(t) and their first and second derivatives can be written as (see Figure !4.6.1)

t

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

~ t

cJt*z~tj J J

Figure 14.6.1" The Function g at Sequential Time Steps

P. R. P. Hoole and B. A. A. P. Balasufiya

gt-At = " ~

1-

gt+At = ~

1+

gt-At = "

i

553

-1+

•, 2 gt = -(At) 2 -1 gt-At- (At) 2 If the wave equation holds true, we have already shown that

(14.6.2)

~tg [T][A] + [P][A] - g [T][J] = 0

(14.6.3)

which is satisfied by re-casting it as 1

+[p](gt.At[A]t-At + gt[a]t + gt+At[a] t+At) ~¢[T][J]~t } = 0

(14.6.4)

This initiN value problem may be solved. For [A~ +At given the sets of values lbr [A] t and [A] t-At. Substituting for gi and gi we get

At2 } [A]t

{ ~ae[T] + fiAt2[p] }[A] t+At + {-2~e[T] + ( I - 4fi + 2¥) + T [ P ] + {~[Tt + (1 + 2~- 27) ~ [ P ] }

[A]t-At + [J] At2=O

(14.6.5)

where 1

:vj~[Tl[Jl j = -I

dt (14.6.6)

1

554

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

Cfdt

7-

!

(14,6.7)

1

l"" ~ J t~ =

At]At l

(14,6.8)

AZ Using an identical time interpolation for the source term J, (1-4[~+2;~) ( I +2~-2V) J = ~u[T][J]t+kt + 2 ~[Wl[J]t+ 2 "[T][J]t'kt

(14.6.9)

A most frequently used three-point difference scheme is the central difference 1

scheme, with ~ = 0 and ~, = ~ for which we get .....l

[X~]t = At2 { [Alt+kt_ 2lAir + [A]t-At }

(14.6.10)

}

(14.6.11),

"At 2 ~ [ T ] ) [ a ] t + At ~ e [T][Alt-At

(14.6.12)

which on substitution gives ~[ [T][A]t+At P ] =A la[T][J]t t +(

To solve for [A] t+At we need to know [A~ and [A] t'At At the commencement of the computations, i. e. at time At, we need to specify lAP and [A]-At which may be obtained from At2 [A]-At = [A]0_ At[)k]0 + ~ [~10 (14.6.t3) where it is assumed that [A]0 at t=O is known, and [~,]0 is calculated from rtzlT][~d 0 = -[P][A] 0 + g[T][J] 0 (14.6.14) Although the central difference scheme is very widely used for stable solutions At must be limited to very, small values. In order to improve the numerical stability of the solution, we may use a four-point difference scheme instead of a three-point scheme. We have

P. R. P. Hoole and B. A. A. P. Balasufiya

555

(I4.6.15)

A(t) = 2gi(t)Ai(t)

where i = t-2At, t-At and t+At. After manipulations similar to that outlined above, we get gE(7-I)[T] + (6-2~ + ~)[PI[At 2 [A] t+At

2p-

+

~(4-3y~[T] + -~-

[P][at 2]

+

gE(37-5)[T] +

+

pe(2-7)[T]+ ( - 6 + ~ + 1 1 ~ + I ) [ P ] [ A t

(2-5~2+ 37)[P][At2I

[A] t [A] t'At

2 [A] t'2At

+ " ( 6 - 2 ~+ ~3 ) [T][J]t+At At2 + g ( 6 " 29 "3~2)[j]t At2 +p ~-

+37

(°

[Tl[J]t-AtAt2 + p -~+1~- 11 +1

[J]

At2=0(14.6.16)

where 3 3 2dt 3At =

3 " 2dr

(14.6.17)

3 2 2dt 3At

P=

(14.6.18)

3

jr,. 2dt 3At 3 2dt 3At ?'=

3 jr" 2dt ,v 3Xi

(I4.6.19)

556

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

Two very widely used four-point schemes are the Houbolt method, where ~ = 30, = 9 and -t = 3, which results in the equations [~]t+At_ ~ { 2[a]t+At 5[A~ + 4[A] t-At [A~"2At} At2 _

(14.6.20)

[j~ ]t+At = ~ 1 { 11 [Ajt+At - I8[A]t + 9[a] t'At - 2[a~ -2~t }

(14.6~21)

which we have used in the previous section. There am some difficulties in starting up the Houbolt scheme, which may be overcome by using the central difference method for the first two time steps, and then switching over to the four-point Houbolt method. The oth~ important scheme is the Wilson-® meth~, using o~=2 + 4 0 + 3 ® 2 + 0 3 = 4/3 + 2 0 + 202 ~'= 1 + O giving us the following approximations [~,]t+At=..... 6 ..... ([A] t+0At- [ A ~ ) - ~ 6 [A]t_ 2[~]t 02At2

(14.6.22)

oat [ ~ ] t + A t = . 3...... ([A] t+0~t - [ A ~ ) - 2[A] t- T [~]t (I4.6.23) 02At Very stable numerical solutions are obtained for O values in the range of 1.366 to 2.0. Both methods, the Houbolt and Wilson-O methods, give unconditionally stable solutions.

Eigenvalue Schemes To assess the stability of a numerical method we need to introduce the concept of mode superposition. To keep our discussion general, consider the equation T ~ + PA = J

(I4.6.24)

where we have set TA = ~[TI[A], PA = [P] [A] and J = g[T][Jl. Without the source term we get the fi'ee dynamic oscillation equation, TA + PA = 0 (14.6.25) which has a general solution of the form A = a exp~mt), which on substitution gives Pa = o~2Ta (14.6.26), which is the generalized eigenproblem, where for non-zero solution, the determinant I-o,2T + PI =0. (14.6.27) This determinantal equation may be solved to give us n-values of

0,2, i=l .... n for an

n node finite element mesh. We note that a, the orthonormalized eigenvectors, have the property

P, R. P. Hoole and B. A. A. P. Balasufiya

557

T 1 i =j (14.6.28) a Taj = 0 i<>j For each ith vector a i we have a corresponding natural frequency of the system ~i-

Defining a vector = [ a l , a 2 ..... a n ] and the diagonal matrix -

2 031

2 (14,6.29)

W= 2

03 n

using which we get 5 cc = Tee W Since aTTa= I we get c~Tso~ = W After solving (14.6.30) for the vector ~, we may solve (14.6,32) for the of the matrix W. With A(t) = ~ V(t) the n - uncoupled equilibrium equations are given by

(14.6.30. (14.6.31) (14.6.32) elements

;v~(t)+ ~,x,rV(t) = ocTJ(t) (14.6.33) The contrast of the above equation to (14.6.3) is that it is a set of scalar, uncoupled equations of the form

9i(t)+ 032 vi(t ) =ji(t) ji(t) = ~iF J(t)

for i = 1,2 ...... n.

t\or i = 1,2 ...... n.

(14.6.34) (14.6.35)

This set of equations may be individually solved for v(t) using either the centraldifference method, the Houbolt method or the Wilson-O method. The initial conditions to commence computation are 0 o T TA 0 vi =

"0 = ~ T T A O vi After solving the n equations, the resultant potential at each node may be obtained by superposition of the response at each node: n

A(t) = Z o q vi(t) i=l

558

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

14.7 Analysis of Stability and Accuracy A basic difference in the concepts dealt with in this chapter is the time-domain analysis involving second order terms. In eddy current and electric machine transient problems we are largely solving for first-order differential terms in the time-domain and second order in the space-domain. The numerical stability of the solution is an important issue in that the larger the time steps taken, although the computer memory-space required is less (the solution in the x-y frame, in a 2D problem, needs to be stored for each time step), the solution may well run unstable. Therefore in applying the difference method to the (b2/bt2) and (0/0t) terms, we must ensure the numerical stability of the computations. In other words, the inevitable numerical errors at one stage of the computation must not cause larger errors in the next stage of the computation. In a stable solution, the errors will gradually be damped out in the progress of computations. Instability might be due m the differential equation itself or due to the numerical method we have used to discretize the point differential terms. In this section we are mainly concerned with the judicial choice of a stable numerical method and suitable time steps. But for the sake of completeness, we shall first consider the stability of a differential equation. Let A be the exact solution of the differential equation (0A_/~t) = f(A,t), and U the approximate solution which varies with the round-off errors and the defects of the method of solution. Thus, with (0AJ0t) = A and (aUl0t) = fJ, we have ~ = f(A,t), [J = f(U,t). For the perturbed solution u(t), we write u(t) = A(t) + eP(t) where P(t) is a perturbation function and e is a small parameter, with e >> e2, ~ 3 . . Using the Taylor expansion (J = ~ + e/) = t;,'u,t) = J(A+eP,t) = f(A,t) + ePfA where fA is a given function of t. From this we get

U- A =EP-ePfA

(14.7.1)

or = fA p

(14.7.2)

the variation equation of the differential equation ~ = ffA,t), Integrating (14.7,2) results in

t"

P(t) = P0 exp(Jt~(t) dt where P0 is an arbitrary initial perturbation, which reduces to P = P0 exp a(t-t0)

(14.7.3),

P. R. P. Hoole and B. A. A. P. Balasuriya

559

when fA = a, a constant. Using t = b + jb, P = P0 exp (jt) where t - fA b = ab We may now make the toltowing remarks on the stability of the differential equation: When a is positive, for any small value of P0, the perturbation P will increase exponentially. Thus at each point in time t, the solution will contain an accumulation of perturbations carried over from previous computations. Such increasingly large errors as the computations are continued, also occur when fA is positive but not a constant. The differential equation is unstable if fA is predominantly positive. The inherent instability of a differential equation cannot be resolved by any method. To determine the stability of the numerical methods, consider the individual equation of the time dependent variable V, •" 2 v + 0~i v i =Ji Assuming a solution of the form t+At t vi = Xvi t t-At v, = :x.v. 1

(I4.7.4).

(I4.7.5) (14.7.6)

I

and using the three-point difference scheme we get the characteristic equation:

Solving the equation for ;~.will tell us whether we have chosen a value of At that will give a stable solution. For the numerical e~ors to be d a r n e d out, we require that the modulus of )~ ILl N 1 (14.7.8) Solving (14.7.7) for )~ XI'2 = where

(2-x) + N/)2-x) 2 - 4(l+y) 2

1 ( 1+2"~)m~At 2 x =2 l+~m2At2

(14.7.9)

(14.7.10)

1 (1-2~')°~2At2 y = ~ l+~o~?At 2 1

We get complex roots when

(14.7.11).

560

Chapmr 14: Computation of Pulsed Electromagnetic Fields: Neural Netowo~ks

(2-x)2 < 4 (l+y) Therefore to obtain N = "/i+i, ~ 1 we require that

(14.7.12)

1 (1"2~°~ At2 - l < y = ~ t+~ 2At2 <0

(14.7.13)

1

for unconditional numerical stability of the three-point difference scheme. I

For the central difference scheme, with B = 0 and "c= 2-- to achieve stability ~2At2 < 4 or At < m"i 2

(14.7.14)

Therefore the central difference scheme is only conditionally stable. For unconditional stability, we must have "c= (1/2) and B ~ (I/4). As we have already indicated, the four-point difference schemes of Houbolt's method and the Wilson® method are inherently stable approximations of first and second order derivatives.

14.8 Software Techniques for the One Dimensional Wave In the concluding section of this chapter we shall discuss the development of a Turbo-Pascal algorithm for the one dimensional wave equation in dissipative media. The potential wave along a transmission line is considered. The transmission line is used to describe a large number of plane electromagnetic wave propagation problems. These include impulse signals along telephone lines, electromagnetic pulses travelling through a multilayer material and digital signals propagating along computer data lines. For a transmission line with per unit length resistance, inductance and capacitance R, L and C respectively, placed along the z axis, the potential drop along the line is given by (14.8.1) = Ri + L ~ 3t az and the current variation is Ok= c~ (14.8.2) Oz From (I 4.8.1 ) and (I 4.8.2) we obtain the one dimensional wave equation

az2 -

(14.8.3) at 2 =

Using the central difference schen-m to discretize (14.8.3), we get I ~1 + ~

V(z,t)= I{ 2 ) 2 " 2L?Az)2~ V(z't'At)

P. R. P. Hoole and B. A. A. P. Balasufiya

561

Figure 14.8.1: Travelling Wave Surges in Communication Cable + 2L'~zXt)-(zXt)2 V(z,t-2~t) + LC(Az)2 [V(z+,~z,t-At) + V(z-,~z,t-zXt)](14.8.4) The program "Transmission" of Appendix 14.B solves the wave equation along a line with the given parameters. An obvious formulation would be to segment the transmission line into portions of length ( ~ ) and choose a time step of (zXT) and apply eqn. (I4.8.4). The time step is dependent on the rise time of the pulse. Hence pico-second pulses necessitate zXT to be in nano-seconds. For simulation time in the range of hundreds to a thousand pico-seconds the grid to iterate on will increase rapidly. With the limited amount of internal memory available in M s the problem becomes unmanageable. The program given below in Appendix 14.B maintains a queue of fixed length in internal memory and fine queue is periodically emptied on to auxiliary storage (hard disk). TbJs way pulses of pico-second ranges could be analyzed for fairly long intervals. The reader could implement a queue of v ~ a b l e length and explore the variation of the analysis time with queue length. Figures 14.8.1 and 14.8.2 give the computed surges on a communication cable and power line respectively.

Figure 14.8.2: Travelling Wave Surges in Overhead Power Lines.

562

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Nemworks

Noise T

Cockpit

Control circuit

Jet Engine

Figure 14.9.1: Block Diagram of Speed Control 14.9 Neural Network Filter for Electromagnetic Noise In the previous sections we have described the characterization of pulsed electromagnetic fields. Frequently, in critical systems like aerospace vehicles, electromagnetic pulses could cause unexpected control problems. A typical instance is where commercial and military aircraft experience engine surges when they fly into a thunderstorm environment. Lightning is an effective radiator of p u l s ~ electromagnetic fields. We briefly sketch a possible use of a neural network filter to handle such electromagnetic noise induced control problems. The spectrum of a typical nuclear or lightning electromagnetic pulse could span frequencies up to a few giga-Hertz. These easily find their way into the command, communication and control systems. In this section we introduce an elementary Hopfield algorithm (Eric and Naim, 1990) to keep a tag on the digital data pattern. If we should include the electromagnetic noise computed at different points in space, we could train the system even better to make use of its knowledge of the bit pattern as well as the radiated electromagnetic field pulses which induce the unwanted pulses. Hence we provide some hints of how a perceptron neural network could also be used to address the problem of aircraft control in a hostile electromagnetic environment. The block diagram for speed control system in an aircraft is shown in Fig. 14.9.1. As shown in this figure, noise can be induced in the data transmission line due to atmospheric (e.g. lightning) or nuclear electromagnetic pulses in the surroundings. The induced voltage pulses on the digital command line may corrupt the data transferred to the control circuit. This in turn will initiate a wrong engine control action. For simplification, the sets of commands can be minimized m the following eleven modes: Accelerate - five modes, Decelerate - five modes and Cruising speed - ! mode.

P. R. P. Hoole and B. A. A. P. Batasuriya

•I

Control Model [~ICircuit

Cockpit

Fig.

563

Jet Engine

14.9.2:. Training the Model in a Noise-free Environment

These eleven commands can be considered as eleven bit patterns that are transmitted m the control circuit. Therefore the problem is m filter out the noise in the bit pattern betbre it is fed into the control circuit. We include in the control circuit a Hopfield Net model m recognize a noisy pattern and to transfer only noise~free data into the engine control circuit. The trained examplers of the Hopfield model are the eleven modes of operation. The training of the system in a noise-free environment is shown in Fig. 14.9.2 and the real-time implementation in Fig. 14.9.3. The Hopfield algorithm is described below in Appendix I4.C

Noise 1 W

Cockpit

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

I~Control

Jet Engine

Fig.14.9.3: System With Filter- Real Time Implementation

5~

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

Appendix 14.A IIWIIJI!

III

[WI IIIIIII1!1111 !r!ll!l!ll!l

. . . . . . .

Field Radiated by Line Current

program element_antenna; (

~

~

~

program to evaluate Electric field c o m ~ n e n t s and Magnetic flux components at any given point due to linear antenna elements stacked vertically one on another ~ * ~ * ~ * ~ ~

)

uses Crt, Graph; const Pi = 3. i41592654; MuO = 1.2566370E-6; EpsO = 8.8541878E-12; C = 3.OE8; MaxSizeT = 1C~30; Omega = 2*Pi* 1E8; type BARR = array[O..MaxSizeT] of real; var

BPhi,Er,Ez, CVec K 1, K2, K3, K4, K5, K6, K7 T, Td DeltaT, MaxTime CH TScale, VScale Zj, r I,NEIm MaxBPhi,MinBPhi procedure InitialiseVar; var

: BARR;

: real; (* Constants for equations*) : real; (* Parameters ofthe pulse *) : real; : char; : real; : real; :integer; : real;

P. R. P. Hoole and B. A. A. P. Balasuriya

565

I : integer;

begin for I := 1 to M a x S i z e T do bgin BPhi[II := 0; Er[I] := 0; Ez[I] := 0; C V e c [ I ] := 0; end; end; function P o w e r T o ( N u , P : real) : real; begin P o w e r T o := e x p - P * In~Nu)); end; p r o c e d u r e C a l c C o n s m n t K s ( z 1 ,z2,zj,r : real); {This p r o c e d u r e C a l c u l a t e s the contants Ks stores it in Global variables } begin K1 : - M u O / ( 4 * P i * r ) * ((z2-zj)/sqrt(sqr(r) + sqr(zj-z2)) - ( z l - z j ) / s q ~ ( s q r ( r ) + sqr(zj-zI))); K2 := M u 0 / ( 4 * P i * C ) * ( a r c t a n ( z 2 - z j ) / r ) - arctan(zl-zj)/r)); K3 := r / ( 4 * P i * E p s O ) * ( P o w e r T o ( s q r ( r ) + sqr(zj-z2)), - 1.5) - P o w e r T o ( s q r ( r ) + sqr(zj-z I)), - t .5)); K4 := (3*r)/(8*Pi) * sqrt(MuOfl~psO) * (PowerTo(sqr(r) + sqr(zj-z2)) - 1.0) - PouerTo(sqr(r) + sqr(zj-z I)), - 1.0)); K5 : - ( M u O / ( 4 * P i ) ) * (PowerTo(sqrtr) + sqr(~-z2)), ~0.5) - P o w e r T o ( s q r ( r ) + s q r ( ~ - z l ) ) , -0.5)); K6 := 1.0/(4*Pi*EpsO) * ((z2 - zj)[PowerTo(sqr(r) + sqr(zj-z2)) - 1.5) - (z 1 - zj)/~owerTo(sqr(r) + sqr(zj-z 1)) - 1.5)); K7 := 1.0/(8*Pi) * s q r t ( M u O / E p s O ) * (3 * (z2 - zj))/(sqr(r) + sqr(zj - z2)) - l/r * arctan(z2 - zj)/r)) - (3 * (z2 - zj))/(sqr(r) + sqr(zj - z2)) ~ 1/r * arctan(z2 - zj)/r))); end; f u n c t i o n C a l c Q ( T b , T i m e , T e : real) : real; var

L o Q : real begin if ( T i m e < Tb) or ( T i m e > Te) then L o Q := O else L o Q := 0.5/0rnega * (1 ~ C o s ( O m e g a * ( T i m e -Tb))); C a l c Q := LoQ; end;

566

Chapter 14: Computation of Pulsed ELectromagnetic Fields: Neural Netoworks

function CalcI(Tb, Time,Te : real) : real; var LoQ : reaI; begin if (Time < Tb) or (Time > Te) then LoQ : = O else LoQ := 0.5 * Sin(Omega *(Time -Tb)); Calcl := LoQ; end;

function C a l c d I ( ~ , T i m e , T e : real) : real; vat

LoQ : real; begin if (Time < Tb) or (Time > Te) ~then LoQ := 0 else LoQ := 0.5 * Omega * Cos(Omega *(Time -Tb)); Calcd! := LoQ; end; procedure CalcCurrent(var CVec : BARR; Tb, Te, DeltaT,MaxTime: real); var

Time : real; Tindx : integer; begin Time := O; Tindx := O; while Time < MaxTime do begin CVec[Tindx] := CVec[Tindx] + CalcI(Tb, Time, Te); Time := Time + DeltaT; Tindx := Tindx + 1; end; end; procedure PlotGraph(Ar : BARR; Xscl,Yscl : real; Xoff, Yoff : integer; NE : integer; VLen, HLen : real; VBan : string); { Plots the Graph of Voltage Vs Time. } var Time : real; procedure Graphlnit; var

P, R. P. Hoole and B. A. A. P. Balasuriya

567

Gd, Gm : integer; begin Dete~tGraph(Gd,Gm); InitGraph(Gd,Gm,' ' ); if GraphResult <> grOk then Halt(l); end;

procedure DrawLine(X 1,Y I,X2,Y2 : real); { Draws a line from X1 ,Y1 m x2,r2, scale factors and Of'fsets are used appropriately as given } var

IX 1, IX2, IY 1, IY2 : integer; begin IX1 := round(X! * Xscl + Xoff); IX2 := round(X2 * Xscl + Xoff); IY 1 := round(GedvlaxY/2 - Y1 * Yscl + Yoff); IY2 := round(GetMaxY/2 - Y2 * Yscl + Yoff); Line(IX 1,1Y 1,1X2,IY2); end; procedure DrawAxis(VertLen, HoriLen : real; VertLabel : string); var VStep : real; begin DrawLine(O.O,O.O, HoriLen,O.O); SetTextStyle(2,VertDir,2); OutTextXY(30,70,VertLabel); SetTextStyle(2, HorizDir,2); DrawLine(O.O,O.O,O.O,VertLen); OutTextXY(4fN,225,'~ME/nano sec '); end; begin Graphlnit; DrawAxis(Vlen,Hlen,VBan); MoveTo(O,round(GetMaxW2-Yoff)); Time := O; for I := 1 to NE do begin DrawLine(Time,Ar[I],Time+DeltaT,Ar[ 1+ 1]); Time := Time + DeltaT; end; end; procedure Ca!cField(var Bphi, Er, Ez : BAIRN;

568

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

zl ,z2,zj,r, Tb,Te,De!taT, MaxTime : real); calculates electric/magnetic field components at a given point zj, r due the single element defined by z2, zl, Current function is given in I instantaneous values are calculated by CalcQ, Calcl, Calcdl var

Time :real; Tindx : integer; Q, 1, dl : real; begin CalcConstantKs(zl,z2,zj,r); Time := O; Tindx := O; repeat Q := CalcQ(Tb,Time,Te); I := CalcI(Tb,Time,Te); dl := Calcdl(Tb,Time,Te); BPhi[Tindx] := BPhi[Tindx] + KI*I + K2*dI; Er[Tindx] := Er[Tindx] + K3*Q + K4*l + K5*d!; Ez[Tindx] := Ez[Tindx] - K6*Q - KT*l - Kl*r*dl; Tindx := Tindx + 1; Time :=Time + DeltaT; until Time >= MaxTime; writeln(Tindx); end; procedure FindMaxMin(TA : BARN.; NE : integer; var MaxE, MinE : real); var I : integer; AMin, AMax : real; begin AMin := 5E35; AMax :=-5E35; for I := 1 to NE do begin if TA[I] < AMin then AMin := TA[I]; if TA[I] > AMax then AMax := TA[I]; end; MaxE := AMax; MinE := AMin; end;

begin { Main Program }

P. R. P. Hoole and B. A. A. P. Balasuriya

569

InitialiseVar; ClrScr; write('Enter Zj for point of evaluation write('Enter R for point of evaluation write('Enter time step DeltaT write('Enter Analysis time

: '); readln(zj); : '); readln(r); :'); readln(DeltaT); :'); readln(MaxTime); CalcField(BPhi,Er,Ez,O.0,O.O2,zj,r,O.O,3OE-9,DeltaT, MaxTime); CalcField(BPhi Er,Ez,-0.02,0.0 ~ r 0.0,30E-9,DeltaT, MaxTime); CalcFie!d(BPhi Er Ez,0.02,0.04 zj r 20E-9,50E-9,DeltaT, MaxTime); CalcField(BPhi,Er Ez,-0.04,-0.02,zj,r,2OE-9,50E-9,DeltaT, MaxTime); NEIm := round(MaxTime/DeltaT); TScale := 500~daxTime; CalcCurrent(CVec,0.0,30E-9,DeltaT,MaxTime); CalcCurrent(CVec,20E-9,50E-9,DeltaT,MaxTime); FindMaxMin(CVec, NElm, MaxBPhi, MinBPhi);

VScale := 100LMaxBPhi; PlotGraph(CVec,TScale, VScale,50,-35,NElm, 1.5"0.5, MaxTime, 'Source Current'); CH := ReadKeyNndMaxMin(BPhi, NEIm, MaxBPhi, MinBPhi); VScale := 100/MaxBPhi; PlotGraph(BPhi,TScale, VScale,50,-35,NEIm, 1.5 * MaxBPhi,MaxTime, 'Radiated Magnetic Ne!d'); CH := ReadKey; ClearDevice; NndMaxMin(Er NEIm, MaxBPhi,MinBPhi); VScale := 100jMaxBPhi; PlotGraph(Er,TScale, VScale,50,-35,NElm, 1.5 * MaxBPhi,MaxTime, 'Radiated Electric Field'); CH := ReadKey; ClearDevice; FindMaxMin(Ez N~lm MaxBPhi,MinBPhi); VScale := l~3jAbs(MinBPhi); NotGraph(Ez,TScale, VScale,50,-35,NElm, 1.5 * MaxBPhi, MaxTime, (Radiated Electric Neld'); CH := ReadKey; CloseGraph; end.

570

Chapter 14: Computation of Pulsed Etectroma~etic Fields: Neurat Netoworks

Appendix 14.B .

.

.

.

.

.

.

.

iiiiiiiiii

.

ii

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

ii!1!

i

•

iii

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

I

Program Transmission

program transmission ( ************** program to evaluate the response of a transmission line for double exponential pulse excitation - using Central Difference Scheme :¢ ~::# ~ ~ ~ : ~ : # ~ ~

~¢: ~ ~ ~

)

uses Crt, Graph; const MaxSizeT = I0; MaxSizeZ = IChg; type HARR = array[O..MaxSizeZ] of real; {Array to Section the line } VARR = array[- 1..MaxSi~T] of HARR; vat

VoltArray : VARR; (* Voltage Array *) K1, K2, K3, K4 : real; (* Constants for finite difference *) Vm, Vo, TI, T2, TAUI, TAU2 : real; (* Parameters of the pulse *) R, L, C : real; (* Parameters of the line *) DeltaT, D e l t ~ : real; MaxTime,MaxZlen: real; Back_Array : file of HARR; (* Disk backup of the array *) CH : char; TScale, VScale : real; procedure CalcConstantKs(R,L,C,OeltaT,DeltaZ : real); {This procedure Calculates the contants Ks, peak Vm, time constants and stores it in Global variables} begin K1 := I/((I/sqr(DeltaT)) +R/(2*L*DeltaT)); K2 := I/sqr(DeltaT)- RJ(2*L*DeltaT); K3 := 2/sqr(DeltaT) - 2/(L*C*sqr(DeltaZ));

P. R. P. Hoole and B. A. A. P. Balasuriya

571

K4 := 1/(L*C*sqr(Delt~)); end; procedure CalcVmandTau(V0,T1,T2 : real); begin Vm := V0 * exp(TI/(1.443*T2)); TAU1 := 0.2"TI; TAU2 := 1A43*T2; end;

procedure Inifialise(var V : VARR; SizeT,SizeZ : integer); vat

I,J : integer; begin for I := -I m SizeT do for J := O m SizeZ do V[I][JI := 0; end; function Vs(T : real) : real; { Returns the intan~neous value of the surge voltage at 'T' } begin Vs := Vm * (exp(-t[fAU2) - exp(-t/TAU1)); end; function VOLT(T,Z : integer) : real; { Returns the Voltage for given time and space slot } begin VOLT := Kl*(K3*VoltArray[T- 1]tZ] - K2*VoltArraytT-2]~] + K4*(VoltArraytT- 1] [Z+ 1] + VoltA~aytT- 1] [Z- 1])); end; procedure PlotGraph(XscI,Yscl : real; Xoff, Yoff : integer; ZStep : integer); { Plots the Graph of Voltage Vs Time. } var

CH : char; STime, CTime : real; It, TSlot, Jz : integer; procedure Graphlnit; var

Gd, Gm : integer;

572

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

begin DetectGraph(Gd,Gm); InitGraph(Gd,Gm,' ') if GraphResult <> grOk then Halt(l); end; procedure DrawLine(X1,YI,X2,Y2 : real); { Draws a line from X1,Y1 to X2,Y2, scale factors and Offsets are used appropriately as given} var

IXI, IX2, IY1, IY2 : integer; begin IX1 := round(Xl * Xscl + Xoff); IX2 := round(X2 * Xscl + Xoff'); IYI := round(Ge~,,taxY/2 - Y1 * Yscl + Yoff); IY2 := round(GetkdaxY/2 - Y2 * Yscl + Yoff); L i n e ( ~ 1,IY 1,IX2,IY2); end; procedure DrawAxis; var

VStep : real; begin SetTextStyle(!, HorizDir, 2); OuffexLXY(50,10, 'SURGE VARIATION IN O/H TRANSMISSION LINE '); DrawLine(0.0,0.0,MaxTime,0.()); DrawLine(0.0,0.0,0.0,1.5*Vo); VStep := 100.0; while VStep 1.5*Vo do begin DrawLine(-I ,VStep, I ,VStep) ; VStep := VStep + 100.0; end; VStep := 10.0; while VStep < MaxTime do ~ g i n DrawLine(Vstep,-5,VStep,5); VStep := VStep + 10.0; end; SetTextStyle(2,Ve~Di r,2); OutTextXY(30,7D, SURGE VOLTAGE kV ); SetTextStyle(2, HorizDir,2); OutTextXY(400,225, TIME/micro sec ); end;

P. R. P. Hoote and B. A. A. P. Balasufiya

573

begin Graphlnit; DrauAxis; ST i me : = - MaxS i zeT * De 1 t aT; reset(BackArray); It := 1; while STime < MaxTime do begin while not eof(BackArTay) and (It < (MaxSizeT + 2)) do begin read(BackArray, VoltA~aytlt-2] ); It := It + I; end; ST i me : = ST i me + MaxS i zeT*De 1 taT; Jz := 0; while Jz < MaxZlen do begin TSlot := 0; while TSlot < MaxSizeT do begin CTime := STime + TSlot * DeltaT; DrawLine(CTime-DeltaT,VoltArray [YSlot- 1] Oz], CTime,VoltArray[~lot] [Jz] ); TSlot := TSlot + 1; end; Jz := Jz + ZStep; end; It := 2; VoltA_tray[- 1] := VoltArray[9]; end; end; procedure Analyse(V0,T1 ,T2,DelmT,Delt~,MaxTime,MaxZlen: real ); ( analyse the transmission line and dump the results into the disk file 'c:\user'u'esu!t.out' } var Z, I : integer; TVolt : real; t : real; begi n Initialise(Volt,am'ay,MaxSizeT, MaxSizeZ); CalcVmandTau(V0,TI ~)CalcConstantKs0~_,LC,DeltaT,Delt~); t := DeltaT; reurite(BackArray); reFat Z := 1; VoltA~ay[1 ] [0] := Vs(t); while Z < M a x Z ! e n / D e l ~ do begin

574

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Nemworks

VoltA~ay[ 1] [Z] := Volt(I,Z); Z : = Z + 1; end; write(BackArray,VoltArray [- 1] ); for I : = O to 1 do VoltArray[I- 1] := VoltA~ay [1]; t := t + DeltaT; unti 1 t > MaxTime; Close(BackArray); end; begin { Main Program } Assign~ackArray, 'C:\USER'~uRESULT.OUT ); ClrScr; (** PARAMETER COUSTANTS FOR OVERHEAD LINE **) C := 0.01; R := 0.7; L := 10C~); write( Enter Surge Voltage : ); readln(Vo); write( Enter time T1 : ); readlnffl); write( Enter time T2 : ), readln(T2); write( Enter time step DeltaT : ); readln(DeltaT); write( Enter length step DeltaZ : ); readln(DeltaZ); write( Enter Analysis time : ); readln(MaxTime); write( Enter Max Length Z : ); readln(MaxZlen); write( Enter the Time axis scale: ); readtn(TScale); write( Enter the Volt axis scale: ); readln(VScale); write( Press A(nalyse P(Iot '); CH := ReadKey; if UpCase(CH) = A then Analyse(Vo,T 1,T2,DeltaT,DeltaZ,MaxTime,MaxZIen); PlotGraph(Tscale,Vscale,50,-35,5); end.

P. R. P. Hoole and B. A. A. P. Balasuriya

Append,ix,,,,, 14.C !!l

575

.....

!1

J

The Hopfield Algorithm for Recognizing Noise in EM Pulses

Number of recognizable patterns, M=I 1 Number of nodes, N= 100 Connection weight tij is between node i and node j. Examp!er bits - ~s 0 <= s <= M-I

0< = i <= N-1

On-line data - p~i(t) t=0 Updated data - ~tj(t) t>0 fh" hard limiting nonlinearity

The following assumptions are made: 1. Input to the Hopfield model is in binary form 2. Mode change occurs at a considerably low rate. The core of the Hopfield algorithm is given in the following four steps.

Step 1. Assign Connection Weights s=O tij

; s xj

i <>j

M-1 tij = 0

Step 2. Initialise with onqine data ~i (0) = x i

Step 3. Iteration

576

Chapter 14: Computation of Pulsed ElectromagneticFields: Neural Netoworks

i_-0 1

~j(t+l) = fh [ E tij/ai(t) [.N-1 ]

Step 4. Repeat by going to step 2

In the algorithm, step 1 is the training set and assignment of weigh~ is done in the noise free environment. The 'C' program listing for the above algorithm is given below. #include<stdio.h> #include<math.h> #include main()

{

int r,i,gd,k,d,s,f[ 11],x[ 11] [I00],u[10] [I00],t[100] [10~1,1[ 10][ 1001; clrscr0; printfCW',tThis Hopfield net has eleven patterns and a hundred nodes \n"); /* Training the network */ for(s=O;s< I ;s++)

{

printff"\n"); for(i=O;i
{

pfintf("\O,t Input pattern %d node %d ",s+l,i); scanff"%d",&x[s] [i]); printff"\n");

}

/* Calculation of wdghts */ for(s=O;s
{

for(i=O;i
for(j=O;j
1

P. R. P. Hoole and B. A. A. P. Balasuriya

}

else t[il[jl=t[ilUl+x[sl[il*x[s][jl;

} } while(l) { for(s=O;s
{

}

pfintfCElement %d ",i); scanf("%d',&u[r][il);

/* Iterafive loop */ printfC\n\n*************Iteration in progress***********"); d=O; do { for(i=O;i0) { u[r+l][i]=l; } else u[r+l] [i]=-l; } for(k=O;k
577

578

Chapter 14: Computation of Pulsed ElectromagneticFields: Neural Netoworks

iffu[r+l] [i]==X[k] [i])

{ }

f[kl=f[k]+ 1;

} iff~k]==lO0)

{

}

d=IC.3;

} for(i=O;i
{

if(u[r+ll[i]==u[r][i]) g=g+l;

}

r=r+l; if (r>IO)

{

for(i=O;i< 100;i++) u[O][i]=u[r][i]; r=O;

}

while ((d!=l(X)) && (g!=lO0)); if (d=l(h3)

{

printff'"\n\nMost closest pattern\n"); for(i=O;i<100;i++) printf("%d ",u[rl[i]); printf("\n");

} else

{

if(g=lO0)

{

}

printf("\n\nThe converged pattern after %d iterations\n", r-l); for(i=O;i< 100;i++) printf("%d ",u[r][i]); printf("\n");

} } The solution m the above control problem can also be tackled by implementing a single layer perceptron (Eric and Patric 1990). It could categorise the signals into two regions A and B. Categow A amy stand for a recognised signal pattern and B

P. R. P. Hoole and B. A. A. P. Balasufiya

579

for unrecognised signal pattern. Two sets of signals, one in its time domain and the other in the frequency domain are each divided into categories A and B. The control circuit responds in real time if and only if the frequency domain and time domain signatures of the incoming signal fall into the same category denoting a recognised pattern. The perceptron learns by means of a supervised learning process which proceeds by correcting errors. If the desired and the actual outputs are the same then it is not necessary to change the weights. Else the weights have to be updated until the desired and the actual outputs are the same, The perceptron algorithm is summarised in the following five steps:

Step I. Initialise Weights and Threshold set wi(O) and ® to small random values. 0 <= i <= N-I

Step 2. Present New input and Desired output continuous valued inputs x 0 .... XN-1 and desired output d(t).

Step 3. Calculate Actual Output

y(t) = fh

i=0 ") Z w i ( t ) xi(t) - ® \g-I

Step 4. Adapt Weigh~ wj(t+l) = wj(t) + g[d(t) - y(t)]xj(t) 0 <= i <= N-I +lif input in class B -I. d(t) -I if input in class B l g - positive gain fraction (0.0 <= g <= 1.0)

Step 5. Repeat by Going to step 2 The gain term mentioned in step 4 is tbr fast adaptation for real changes in the input distributions and averaging of past inputs to provide stable weight estimates. The lemming algorithm of the perceptron was described above by the s~uence: 1. x h, an example is presented to the network. 2.8 h = t h - o h is calculated. 3. Each weight wj is modified: wj = gShxih

580

Chapter 14: Computation of Pulsed Electromagnetic Fields: Neural Netoworks

The 'C' program for a single layer perceptron is given ~low. #include<stdio.h> #include<math.h> #include<stdlib,h> main()

{

int t,i,y[ 150],c,r, 1,d[ 150],x [ 150] [2],q [2]; float n,theeta,w [ 150] [2]; n=.05; i~; clrscrO; printff"\n'~a\n\ff\n\t\ffhis is a single layered perceptron algorithm\n"); printf("\f\twith 100 training data points"); pfintff,"\n\n\n\n\n\n\n\n\n\n\t\t\tPlease wait"); sleep(3); clrscr0; /* INITIALIZE WEIGHTS AND ~ R E S H O L D "/" while(i<2)

{

w[0][i] = rand0/1000; if((w[0][i]==0) It (w[0][i]>9))

{

}else i=i+l;

} printf("\nRandomly selected weight values %f %f~n",w[O] [O] ,w[O] [I] ); theeta=rand ( )/I0,~3; /* PRESENT NEW INPUT AND DESIRED OUTPUT*/ i=0;

for (t=0;t< 100;t++)

{

printf("\n~aPresent new continuous valued input %d\,n",t+l); for ( i=O; i<2; i++ ) scanf("%d",&x[t] [i]); pfintf ( "Enter the desired output ( I or - 1 )" ); scanf ("%d", &d[t] ); /* CALCULATE ACTUAL OUTPUT */ l=0 ;

P. R. P. Hoole and B. A. A. P. Balasuriya

for ( i=0; i<2; i++) l=l+w[t] [i]*x It] [i]; if ((l-theeta) >0)

{

y[t ] =1 ;

}

else y[t]=-I; /* A D A ~ WEIGHTS *t for ( i=0; i<2; i++)

{ c=d[t] -y[t]; wit+l] [i]=w[t] [i]+c*x[t] [i]*n;

} /* PRINTING OF UPDATED WEIGHTS */ prinff ( "Updated weights\n" ); for(i~3; i<2; i++) printf ("%f ",w[t+l] [i] );

} /* ENTERING AN UNRNOWN DATA */ while ( 1 )

{

clrscr ( ); printf ( "Enter one point to classif),\,n '' ); for(i=0;i<2;i++) scanff"%d",&q[i]); 14; for ( i ~ ; i<2; i++ ) l=l+w [t+l ] [ i ] *q[ i 1; if ((l-theeta) >0)

{

t=l; } else t=-l; printf ( "\n The input belongs m class %d", t); getch ( );

}

581

Chapter 15

Dhammika Kurumbalapitiya and S. Ratnajeevan H. Hoole IlllU

. . . . . . .

lllJl/ I1[//~ VIi!

III

II

IIIII

III ...........

lJJlJ/

IlJll I

I

OPEN BOUNDARY PROBLEMS VITH FINITE ELEMENTS

15.1

Boundary Problems

What are ~u.~enboundary problems? They are boundary value problems with a "far away" boundary modeling the unbounded nature of the geometry. The boundary condition at this far away boundary is that the fields are zero because it is too far away to feel the effect of the sources. But to solve the governing equations, we need to know what the limits are and where they exist. How do we obtain required limits for unbounded situations? The aim of this chapter is to look into this problem. There are numerous instances in electromagnetic systems where we encounter open boundaries. In fact the definition of one of the most primitive ideas, the potential, is expressed with r e s ~ c t to the potential at infinity which is assumed to be zero. There is no way to represent a domain with infinite dimensions within a scheme like "Finite Elements." This definition of potential will guide us to think of one important characteristic of open boundary problems: that is, in these problems we can always find a boundary with zero ~tential at infinity. This basic criterion, hand in hand with another vel2.¢attractive technique which we can find in c o n f o ~ a I mapping, will provide a powerful way to merge "Finite" and "Infinite" so that one could simply include the procedures in a classical finite element program. In secton I5.2 we present some material on a few methods used to analyze open boundary problems, and their advantages and disadvantages (Hoole, I989). At the end of section I5.2 a new concept will be described for open boundary problems introduced by Lowther and Freeman and Forghani (1989) and refined by

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

583

Imhoff, Meunier and Sobonnadiere (1990), and Brunotte, Meunier and Imhoff (1992). Then in section 15.3, since theirs is now the most powerful technique for such problems, we will take up the larger part of this chapter looking into the concept in greater detail. How it can be merged with the finite element routines will be discussed there. Finally, in section 15.4, two examples will be presented.

15.2 Existing Computational Schemes In finite element analysis, the problem region is divided into a finite number of element shapes. This cannot be done for an infinite problem region as in open boundary problems. It is common in practice that most engineering applications do not require a very precise answer. Therefore in some cases we might be able to impose the zero boundary condition not at infinity but at a finite distance away from the "Active Bodies." However, this is not acceptable where greater accuracy is mandated. Let us see how the difficulties which we have mentioned in the earlier section are overcome in some computational schemes. There are basically two ways to tackle the problem of open boundaries more exactly: 1. Using integral methods 2. Using ballooning The formulaton of the governing equations for an elctromagnetic field problem starts with Maxwell's equations. Maxwell's equations can be written in two ways. One way is to use the differential form of the equations. The other method is to use the integral form of the equations. The first method is straight~f:orward in the finite element context, but requires a finite bounded domain. When we have an integral equation, on the other hand, the boundary condition is implicit and the domain is infinite (see chapters 8 and 12). But the inhomogeneties invovled in the media, if any, will definitely complicate the discretized matrix f;~rm of the equation (Hoole 1989). Thus in the first class of using integral methods, the integral equation forms are used for the exterior region (in numerical or closed-form) and finite elements in the interior with some kind of matching at the interior/exterior boundary. We have already seen a form of this in chapter 8 where finite elements was used inside and boundary elements outside. In ballooning (Silvester, Lowther, Carpenter and Wyatt 1977; Hoole, 1989) it is possible to generate a graded mesh over the problem region, That is we have a fine mesh over the region where we need good accuracy and we get much larger elements geometrical graded up in size from the last layer of elements before the user defined boundary. That is, the last layer is repeated with geometrically increasing siT,e. Each layer is like an annulus. Because the same grading factor and mesh-shape are used for the annuli, the finite element equations for the annuli are known in terms of the matrices for the last layer and the geometric grading allows us to reach "practical infinity" with a few annuli. This looks similar to pumping air into "element balloons" such a way that they are small in the regions where high accuracy is required and vise versa. Hence this method carries the name

584

Chapter 15: Open Boundary Problems with Finite Elements

ballooning. As is described there must be a compromise between the accuracy and the size of the user defined problem region. The method found in Imhoff, Meunier and Sabonnadier (1990) is the one which we will deal with. Here we define a hy~thetical boundary in order to obtain two separate regions. The boundary is selected so that one of those regions is a closed one. The remaining geometry is open and homogeneous. Now it is possible at least to divide this closed region into finite elements. Next we must find a transformation which is capable of mapping all the points in our o ~ n region into this closed region. Such a transformation will preserve the shape of the hypothetical boundary and maps the points on this boundary on to themselves as well. Furthermore what happens to the points at infinity which have zero potential? They are now inside and we can "much" them. When we divide the exterior region into elements we must use all the points on theboundary with the interior region as nodes. These points will not be modified by the transformation, for that is how the transformation is designed. Now this is the important point. By these means we have managed to, first, reduce the infinite exterior to a finite region and, second, establish a link between the two regions with a set of common nodes for both regions on the boundary. Therefore now it is possible to use finite elements and assemble one finite element matrix common to both regions and solve it for the unknowns. The transformations which perform such mapping will be discussed in section 15.4.

15.3 Introduction Methods of this implementation are very similar to the methods of classical finite element programming. The aim of this document is to help the reader in a step by step manner to understand the authors' implementation of the method. There is no guarantee that this method is the best. The famous "Data Structures + Algorithms = Programs" rule has been followed during the program development. Therefore this is a discussion on the underlying data structures and procedural bodies. The mathematical formulation of the method will be converted into programming steps so that the reader can jump into writing the code. The reader is encouraged to develop his or her own data formats and associated routines.

Overview of Data Storage It is worthwhile to spend a few lines to discuss the storage mechanisms involved in general. The program must be able to provide enough memory to store the initial, intermediate and final data values. Disk storage is the obvious way to store data permanently for future applications. The amount of memory needed to hold the initial, intermediate and final data values depends on the size of the problem. The word "size" is used here to indicate the number of nodes and triangles that describe the problem,

D. Kururnbalapitiya and S. R. H. H~le

585

type Node_Pointer = ANode; Node= rc~rd Node, No 'integer; X,Y "real; Status "boolean; I/O " boolean; NextNode • NodePointer; end;

Figure 15.3.1: Type 15.3.1 - Type Definition of a Node Object. Linked lists are used here in the authors' implementation. The advantage of this approach is that the amount of memory required will be determined during run time. Therefore the memory allocated by the program is proportional to the size of the problem. Array implementation of the internal data structures is not considered in places where the amount of data depends on the problem size. Otherwise it is preferred. The difficulty with this approach is that the programmer has to provide procedures for allocating and deallocating memory as well as the access functions to read slots or fields in the data structure. Primary data such as the nodal and triangle information is stored in disks as text files and read into the memory during the execution.

Structures for Primary Data Storage The primary data consists of nodal and triangular information. The node data consist of N ~ a ! Data: The nodal data is stored in a text file according to the tollowing format. Node # X Y V a l u e Status 1/0 Node # - Is the node identifier. X,Y • X and Y coordinates of a node respectively. Value • The initial value of a node is written here,The initial values is known only at Dirichlet nodes. Other node values are ignored. Status " Character U is entered if the value is Unknown.Character K is entered if the value is Known. I/O Character I is entered if the node is Inside the finite domain. Character O is entered if it is Outside. Refer to the section 15.4, that describes the two domains in case of a problem. Type 15.3.1 (shown in Fig+ 15.3.1) is defined to create records to hold nodal data. And triangle data consist of:

586

Chapter 15: Open Boundary Problems with Finite Elements

type TrianglePointer = "Triangle; Triangle = record Triangle_No : integer; V : array [1..31 of integer; Prm 1, Prm2 : real; I/O: boolean; NextTriangle : Triangle_Pointer; end;

Figure 15.3.2: Type 15.3.2 - Type Definition of a Triangle Record. Triangle pata: The triangle data is stored in a text file according to the following format.

Triangle# Triangle # V1,V2,V3 Prm 1 Prm2 Status I/O

V1 V2 V3 P r m l Prm2 I/O : Is the triangle identifier. : Vertices identifiers. : Value of permitivity or permiability within the triangle. : Value of the charge density or current density within the triangle~ : Character U is entered if the value is Unknown. Character K is entered if the value is Known. : Character I is entered if the triangle is Inside the finite domain. Character O is entered if it is Outside.

The Type 15.3.2 (shown in Fig. 15.3.2) is defined to create records to hold the above triangle infomnation. First order triangular elements are used in the implementation. It is desirable type Triangle_Pointer = ^Triangle; Triangle = r~=ord Triangle3-,Io :integer; V : array [1..3] of integer; Prml, Prm2 : real; I/O :boolean; A_wa : real; b,c : array [ I..3, 1..3 ] of real; {Differentiation Matrices } NextTfiangle :Triangle_Pointer; end;

Figure 15.3.3: Type 15.3.3 - Type Definition of a Modified Triangle Record

D. Kummbalapitiya and S. R. H. H~le

587

to add three more very important fields which are common to any first order triangle into the above Triangle data type. One is the Area of the triangle. The other two are the two matrices that are known as the differentiation matrices that have been discussed in Algorithm 1.5.2, Therefore it is desirable to modify the triangle record as shown in Type I5,3.3 (shown in Fig. 15.3.3). The differentiation matrices are useful up to a certain stage of the analysis. After that they are not in use. Therefore one can implement them separately and there is an advantage to doing it separately. Once the program uses them it is possible to remove them from memory. This will definitely provide much free memory and it is always good for the coming steps in the program to ~ described. The values of these fields are evaluated just after reading both node and triangle information into the memory. To fill up the Area, b[l..31 and c[1..3] fields of a triangle, Algorithm 15.3. I. may be used. 1. Take the triangle where these slots m be evaluated. 2. Find out the coordinates of vertices by looking at the Point List. At this point it is possible to generate two arrays of local variables, X[ 1..3] and Y[ ! ..3] m store the vertex coordinates temporarily. 3. Now, Area := 0.5(X[2]xY[31 - X[3lxY[2] + X[3]xY[I] - X[I]xY[3] + X[ 1]xY[2] - X[2]xY[1 ]); f o r i : = l to3 begin if (i<>3) then j := i+l else j := I; k := (6 .~ (i + j)); b[i] := (Y~] - Y[k])/2¥Area; c[i] := (X[kl - XW)/2¥Area; end; A l g o r i t h m 15.3.1: Scheme to find Area and Differential Matrices.

Structures for Secondary Data Storage and Manipulation A large amount of memory is required tor secondary data storage. The secondary data consists of two objects. I. Set of Local Matrices attached to each triangular element. 2. The Global Matrix which is a square matrix with dimensions equal to (total no of nodes x total no of nodes).

~gcal Matrices: Each triangle has its own local matrix. The local matrix has the type definition given in Type 15.3.4 (shown in Fig. 15.3.4). In order to describe the reasons for the existence of the P M a t r i x and the Q_Matrix it is necessary to discuss some theory. It has been shown that the finite element approach to solving a partial differential equation starts with b3rming a suitable functional. The next step then is to minimize it over the problem region. As an example a magnetostatic problem is considered here. One of the reasons is

588

Chapter 15: Open Boundary Problems with Finite Elements

ty~ Local_Matrix_Pointer" ^Local_Matrix; Local_Matrix = ~ r d Matrix_No P_Matrix QMatrix Next_Local_Matrix end;

Figure 15.3.4:

: integer; :array [I..3,1..3] of real; : mxay[ I..3] of real; :Local_MatrixPointer;

Type 15,3.4 - Type Definition of a Local Matrix

that the example which will be considered at the end is a magnetostatic problem. Another reason is that most magnetostatic problems are really open boundary problems We have already seen that in magnetostatics Vx ± VxA - J = - G V~ g

(1.3.25),

which for two dimensional problems with a single-component vector potential -uV2A = J

(15.3.1)

I

where v = The functional corresponding to the above pa~ial differential equation is !

L[A] = g { ~,~ [VA]2- JA } dR

(15.3.2)

where the integration could be performed over a triangular element. That is what is exactly going to be done by us. The integral represents total energy over a triangular element if the integration is done over a triangle. The integration must be carried out over each and every triangle in order to find the total energy. Then it is possible to describe P_Matrix and Q_Matrix as: 1

P_Matrix • holds the ,fj"{ ~ ~ [VA] 2 } dR component of a triangle. Q M a t r i x " holds the f.[ { JA } dR component of a triangle. We have already seen in previous chapters how these integrations are performed (section 1.5). The potential at any point within a triangle is expressed as a function of individual potentials at each node and the triangular coordinates of at that point. This can be expressed in matrix form as A = [~] [a] where [a] = [ a I a 2 a3] and [~] = [ 4! ~2 ~3 ]t (15.3.3) In this treatment of open boundary problems there are two kinds of triangles triangles that are inside the hypothetical boundary with normal coordinates and triangles that are actually not inside the hypothetical boundary but are mapped inside the hypothetical boundary and therefore have different coordinates. It is not possible to treat them in the same way. This difference in coordinates will affect two places in the above integral. I. In the term VA.

D. Kurumbalapitiya and S. R. H. Hoole

589

2. In the integral operator fl In other words there are two different Local Matrix generation schemes. We begin by substituting the matrix form of the trial function, eq. (15.3.3), in place of A in the energy functional, eq. (I5.3.2), to obtain the total energy: 1

2

L[A] = Z fJr { ~v[V~a] - [Ra] J }dR.

(15.3.4)

In the above expression the "jfJ'" takes place over a triangle whereas the "E" takes place over the whole set of elements. The results of these integrations are put into the corresponding Local Matrices and then all the Local Matrices are put into the one "big matrix" called the Global Matrix as explained in section 1.5. Now it is the time to treat separately the cases of the natural triangles from the interior and the mapped triangles fi'om the exterior. Natural Triang&s :

There will not be any change from section 1.5. It is exactly similar to the classical finite element procedures. Expanding the term V leads to L[A]=

i

dill l,j[

3cca"| 3x I" 2 +

~3Ra"~27 , ~ j _ [c~a} J j d R

(15,3.5).

Terms inside the a matrix are independent of the space derivative. But terms inside the ~ matrix are not. Therefore 3oc ;9~ 3x = b and ~ = ¢. (15.3.6) Differentiation of a matrix produces the first order differentiation matrices b and e which have already been found at the beginning. Consider the integration in more detail. 1

L[A] = Z { ~.~ ([ba] 2 j'fdR + [ca] 2 J'J'dR) - [Ja IYa dRl }

(15.3.7)

j'J"dR = Area

(15.3.8)

SJ"ct. dR =

3 3 3

xArea = TxArea

(I 5.3.9)

where T is the matrix [ (1/3) (I/3) (1/3) ] and is known as the metric tensor. For simplicity at this stage, it is assumed that the term J remains constant over the triangle considered. Then I

L[A} =

~J Areaxat[btb + c t c ] a - A r e a x a ~ T J

and the following substitution hel~s to merge some of these long terms P M a t r i x = ,.~xAreax[ btb + e c ]; Q M a t r i x = AreaxT×J leadingto a more general tbrm like I

L[A] = Z { ~ a t [ P M a t r i x ] a - at[QMatrix] }

(15.3,10)

(15.3.tl) (15.3.12).

590

Chapter 15: Open Boundary Problems with Finite Elements

The steps in forming P_Matrix and Q_Matrix have already been presented in Algorithm 1.5.1.5.2.

Mapped Triangles In treating exterior triangles that are mapped to an area similar to the interior, the hange in both the operator V and the integral itself will complicate the integration. Consider the effect on VA first. In the x,y coordinate system V is defined as b b V= ux~+uyby (A1) Substituting the trial function in the latter expression will allow us to have a closer look at what happens in the transformed domain, i.e. on the X,Y plane. V(e~a ) -

b~a

Ux ~

b~a

+ Uy 5y

(I5.3.13).

on the x,y plane. It is possible at this point to use partial derivatives to transform the above expression into the X,Y domain: b~_~a0X baa bY V(aa ) = Ux OX b-x- + Uy bY by (15.3.14). Now, reducing this expression by using the substitution, 0X 0x _- VX and ~bY = VY: V(aa)=

baa baa ux~Vx+uy~VY

(15.3.15)

and [V(aa)]2=

+

2 (VY) 2

(15.3.16).

The last expression shows the effect on [V(aa)] 2. Two additional terms, (VX) 2 and (VY) 2 must be found. Now the effect on the integral ff { .. } dR must be found. The term dR can be described on x,y plane as dR = dxdy. But on the X,Y plane it becomes dR' = dXdY. It is required to perform the integration on the X,Y plane for all the mapped triangles. But the integral is defined over the original x,y plane of the physical problem. Therefore it must be transformed onto the X,Y plane, This is done according to the integral transformation rules under conformal mapping. According to these rules, a term called a Jacobian 0Y J = by] " is used At this stage, it is possible to rewrite the functional expression for the mapped triangles as follows. L[A] =

f

2(VY) 2] - [aa] J} JdR (15.3.18).

D. Kummbalapitiya and S. R. H. Hoole

591

This expression can be reduced to L[A] =E{2o([bal2g(VX)2jdR+[ca]2,fj'(Vy)2jdR)-- - atJffaJdR}

(15.3.19).

This is an integral expression which must be solved numerically. The terms f,[ dR and ~j" ( V y ) 2 j dR look similar. Therefore it is possible to apply the same method for them. But the t e ~ j;faJ dR needs different treatment. The numerical integration technique called Gaussian Quadrature is used in evaluating these integrals numerically. In this method, the function to be integrated (in this case (VX)2j.. etc ) is evaluated at set of selected points on the region over which the integration is done. Then these values are multiplied by certain factors called weights. Finally the summation of these point values gives the value of the integral. The final outcome of these integrals iss et of coefficients which can be defined as tbllows. K 1 = Jj" (VX)2a dR K 2 = j'.[ (Vy)2j dR K 3 = J'J~aJ dR (I5.3.20) K 1 and K 2 are real numbers and K 3 is a matrix with dimension lx3 containing real numbers and is very similar to the metric tensor, T. Thus the new form of the previous expression will be L[A] =

{~[

(b a)

2

Kt +(c

2xat[Ktbtb+

a

)2

K2

]

-a

K2ctc] a - a

t

J K3

K 3 J }

(15.3.21)

Now it looks very similar to the expression that we have obtained, eq. (I 5.3.10), for the natural triangle. Thus, assigning P_Matrix =,0 x _-[ K l b t b + K 2 c t e ] QMatrix = K 3 J (15.3.22) leads to, L[A] =

Ixat

[P_Matrix]a -at[Q

Matrix]

(15.3.23).

The procedure for finding P Matrix and Q_,Matrix in this case is now numerically very, similar to the previous one. But an additional piece of code must be designed to evaluate K1, K2 and K 3. This piece of code can ~ implemented as a procedure so that it evaluates them and passes them onto the main body which is similar to the previous one. Furthermore, this code segment depends on both, the transt\~ation used and the type of the problem at hand. Theretbre the best place to describe it along with a specific example.

592

Chapter 15: Open Boundary Problems with Finite Elements

ty~ ElementPointer = ^Element; Element = record Element_No "integer; P_Value • reaI; Nexl Pointer • Element_Pointer; end; Index_Pointer = AIndex Column; IndexColumn = rexcord RowNo'integer; Q , Value'real; Element_Start • ElementPointer; Next Index • IndexPointer; end;

Figure 15.3.5: Type 15-3-5 Type Definition for the Global Matrix

1. Arrange the node list with the unknown nodes first and knowns last. This step will make future operations easy, 2. Set Node_Address := Starting Address of the Node List Node_No := NodeAddress-> 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

Microflux Flux2D Flux3D Phi3D

Y Y Y

Y

CosmosM/Estar

Y

Ansys Ansys-Pc / Magnetic Ansys / ED

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

Chapter 16: On the Features of Electromagnetic FEA Software Packages

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

Ansys Ansys-Pc / Magnetic Ansys / ED

Y Y Y

Y Y Y

FS18

FS19 Y Y Y Y

"', I

Microflux Flux2D Flux3D Phi3D

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

Microflux Flux2D Flux3D Phi3D

Y Y Y Y

Y Y Y Y

Y Y Y Y

Y Y

CosmosM t~star

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 Y

PC Vector Fields, Inc.

Currents

Opera Opera 2d Tosca Electra

Y

lLu,,

Figure 16.3.1: Preprocessors

608

Chapter 16: On the Features of Electromagnetic FEA Software Packages

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.

Ansys Ansys-Pc / Magnetic Ansys / ED

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.

Structural Research & Analysis

.

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.

Structural Research & Analysis

Y

CosmosM/Estar

Y

Y

Y

Y

Ansys Ansys-Pc / Magnetic Ansys / ED

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

Figure 16.4.2: Postprocessors

610

Chapter 16: On the Features of Electromagnetic FEA Software Packages

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

Structural Research & Analysis

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

Figure 16.4.3: Postprocessors

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)

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628 -

.

.

.

.

.

.

.

.

.

,,,,

,,

,,

,,,

,.

.

.

.

.

.

.

.

.

.

.

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.

.

.

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