SCIENCE

Dual finite-element calculationsfor static electric and magnetic fields

Prof. P. Hammond, M.A., Sc.D., C.Eng., M.I.Mech.E., F.I.E.E., and T.D. Tsiboukis, Dipl. -Ing., Dr. -Ing.Mem. I.E.E.E.

Indexing terms: Electromagnetics, Magnetic fields, Modelling

Abstract: The finite-element method is widely used for the solution of field problems but the method, asgenerally applied, suffers from the fact that it is not known how close the solution is to the actual value.This uncertainty can be reduced by providing a dual finite-element method which is so arranged that bothmethods together provide upper and lower bounds to the correct solution. It is found that the double ap-proach also promises economies in the computation. The paper examines the physical basis of the dualmethod and applies it to Laplacian and Poissonian problems.

1 Introduction

The finite-element method is widely used for the solution offield problems, particularly if the geometry is complicatedand if there are subregions having different electric ormagnetic properties. As expected, large problems requireconsiderable computational effort and it is very desirablethat the computation should be handled in the most economicalmanner. In this paper we examine the physical basis of theproperty of duality which is inherent in vector fields andwhich can be used to achieve considerable economies in com-putation. When this paper was in draft our attention wasdrawn to a valuable study of the subject which had recentlybeen published by Penman and Fraser [ 1 ] . In that paper theauthors give a complete mathematical proof of the existenceof upper and lower bounds for magnetostatic problems. Ourown approach relies on physical description rather than onmathematics and we hope that this paper will be seen to comp-lement the previously published paper.

2 Variational principle in finite-element methods

The solution of a field problem by the finite-element methodis generally considered in terms of the solution of the under-lying differential or integral equation. For example an elec-trostatic problem may be described by Poisson's equation

V 2 0 = - p / e (1)

and may be considered solved when the potential function 0has been computed throughout the region of interest.

The finite-element method proceeds by setting up anenergy functional

(2)

The variation of this functional is given by

SW = <(p-V-Z)),50> (3)

and if the divergence of the flux density is equal to the chargedensity

5 W = 0 (4)

In terms of the calculus of variations eqn. 4 is the Euler-Lagrange equation of the functional of eqn. 2. Conversely

Paper 2407 A, first received 23rd August and in revised form 7thDecember 1982Prof. Hammond is with the Department of Electrical Engineering,University of Southampton., Southampton SO9 5NH, England, andDr. Tsiboukis is with the School of Electrical Engineering, AristotelianUniversity of Thessaloniki, Sfr

the use of eqn. 4 will ensure, for an arbitrary choice of 0and hence of E and D, that the divergence of the fieldcorresponds to the charge density in an average mannerthroughout the system.

Since

E = - \ 7

and

D = eE

(5)

(6)

by definition, the use of eqn. 4 ensures that eqn. 1 is satisfiedin an average manner also.

A convenient method of describing the potential is to put

= Z <*n (7)

This description is also used in the method of moments [2]and the resulting equations are the same, if in the method ofmoments the basis functions and weighting functions arechosen to be identical. This is Galerkin's method. It seems,therefore, that the variational method is merely a devicewhich provides a convenient alternative way of finding themoment equations in Galerkin's method, but this would,in the view of this paper, be a mistaken conclusion. Thevariational approach is much more than a mathematicaldevice. It offers insight into the physical behaviour of thesystem in a manner which is particularly helpful in an engin-eering context. Such insight also gives help in the choice ofappropriate and economical numerical methods.

We have already noted that the functional W describesthe energy of the system. At equilibrium this energy willbe stationary under arbitrary virtual displacements. Thevariational method uses this equilibrium property to findan estimate of the energy. The energy is given in terms ofthe quadratic functional of eqn. 2. Near equilibrium theenergy will be insentive to displacement and it is, therefore,more economical to calculate W than 0. If the energy of asystem is required, it is sensible to model 0 roughly andwithout great accuracy and thus to reduce the computationaleffort. The variational method encourages a view of theproblem which is quite different from the notion of solvingthe underlying equation. The problem is handled as a systemrather than as a set of particles and the method drawsattention to the equilibrium as a whole rather than to theequilibrium of each particle. In electric and magnetic fieldproblems the number of particles is enormous and a systemapproach has much to commend it. In the finite-elementmethod the analyst has the freedom to describe the systemwith the help of an arbitrary number of subsystems and

IEEPROC, Vol. 130, Pt. A, No. 3, MAY 1983 105

this number need only be large enough to ensure accuracywithin the appropriate engineering tolerances.

3 Duality in variational methods

The general principles of dual field calculations have beendiscussed by one of the authors in a book [3] . In thisSection we illustrate these principles by reference to theelectrostatic field problems discussed in Section 2. In thatSection the usual finite-element method was introduced byallowing a variation of 0, and therefore of E = — V4> and ofD = eE, and then by applying the equilibrium condition.

Suppose now that instead of 0 we vary D and thereforeE = D/e and also p — V • D. We can then use the equilibriumcondition

&W = < ( - V 0 - £ ' ) , 5 D > = 0

This means that the divergence condition

V D = p

(8)

(9)

is enforced by definition, but that the gradient relationshipof eqn. 5 is enforced only in an average, or integral, mannerthroughout the system. This is the dual of the previous pro-cedure. The energy can be obtained by integration of eqn. 8:

W = <0,p>2e

<\D\2) (10)

As expected the value of this functional at equilibrium is thesame as that of eqn. 2.

The distinction between the two variations is made clearerby writing

and

2e

(11)

(12)

where the notation p and 0 implies that these quantities arenot varied. To use the equations in this form we shouldrequire to know the charge in eqn. 11 and the potential at thecharge in eqn. 12. In general, it is likely that the potentialsof conductors will be specified, but that the charge will notbe known. In that case we can use the field energy by itselfand replace eqn. 11 by the expression

= ~<\E\2) (13)

where the specified values of 0 are implicit through therelationship of eqn. 5.

If the charge is known, but the potentials at the charges areunknown, eqn. 12 can similarly be replaced by

1

~2e<\D\2) (14)

The duality becomes clearer still by introducing the notionof capacitance. We can write Wl and W4 in terms of theassigned charge as O2/2C. Wx is a concave quadratic functionalwith a second variation of negative sign. This means thatCx is larger than the equilibrium value. W4 is convex and C4

is smaller than the correct value. Similarly W2 and W3 can bewritten as \ <p2C. Thus C2 is smaller and C3 larger than thecorrect value. We have

c> c4c> c2

(15)(16)

Thus, irrespective of the specification, the capacitance isincreased by varying the potential and reduced by varyingthe flux density. In terms of the sources of the field, theupper bound of the capacitance is produced by varying thedivergence sources and the lower bound by varying the curlsources. These bounds can be used as confidence limits for the •numerical value of C. Moreover, the average value of thebounds is closer to the true value than the less accurate of thetwo values.

4 The effect of discretisation

The energy functional of the form of eqns. 11 and 12 areuseful for open systems. To deal with closed systems andsubsystems it is convenient to write an additional term forthe surface energy. Thus we replace eqn. 2 by the expression

= [0,a] - ~ (\E\2)

where o is the surface density of charge and

90a = e dn

(17)

(18)

and where n is the outward normal at the surface and 90/dn is taken just inside the surface (Reference 3, pp. 76—79).In a system consisting of finite elements the surface energiesat internal boundaries depend on the discontinuities in 0 anda between adjacent elements. If there are no discontinuitiesthe internal boundaries have no net energy, because thedirection of the normal is of opposite sign from either sideof the boundary and the energies cancel except at theexternal surface bounding the system.

A discontinuity in a (or 90/8n) implies an additionalnet charge within the total system. A discontinuity in 0implies an additional double layer of sources. Such a doublelayer has the properties of a surface current and is associatedwith a step in the tangential component of field strength.

In terms of sources the discontinuity of the normal com-ponent of field provides a divergence source and thediscontinuity of the tangential component provides a curlsource. In finite-element discretisation care must be takenin modelling the field not only in each element but also atthe interfaces between elements. This will be illustrated bythe examples.

5 Dual finite-element solution of a system defined byLaplace's equation

To illustrate the method we choose an example which hasa known solution. Fig. 1 shows a tubular square capacitorof the dimensions shown. This has a capacitance of C ~ 10.2e0 F/m. Because of symmetry it is sufficient to examine

K;1

V

/K=0

106

Fig. 1 Tubular capacitor of square cross-option

IEEPROC, Vol. 130, Pt. A, No. 3, MAY 1983

Table 1 :

Number of elements

Approximate capacitancePercentage error

Upper bound of capacitance

3

12e0

+ 17.6

6 12

11e0 10.84 e0

+ 7.8 + 6.3

24

10.55 e0

+ 3.4

1/8 of the capacitor as indicated in the Figure. Fig. 2 showsvarious finite-element discretisations.

To obtain an upper bound of the capacitance we canchoose the scalar potential as the variational displacementas in the method of Section 2. It is convenient to specify theenergy in terms of assigned potentials, say, unit potentialon the inside conductor and zero potential on the outside.The energy functional is, therefore, of the form of eqn. 13.

In the first instance let us choose a linear spatial variationof 0 within each finite element e. Thus

= a? +af y (19)

In the usual finite-element method the coefficients a arerelated to the nodal values of the potential at the cornersof the elements. As these nodal values are shared by adjacentelements, the potential is continuous throughout the systemand the tangential components of the electric field at theinterfaces between elements are also continuous. In termsof sources this means that there are no curl sources on theinterfaces. The field within each element is constant as canbe seen by applying the gradient operator to eqn. 19. Unlessthe field is identical in adjacent elements, there will be astep in the normal component of the field and thus a surfacecharge a is created by the discretisation. Inside the elementsthere is no divergence as the potential is a linear function ofthe co-ordinates.

The variational method reduces the energy associated withthe surface charge a to a minimum. As explained in Section2 the curl sources are zero throughout the system and thedivergence sources are zero in an integral or average manner.Table 1 gives the values of the approximate capacitance andthe percentage error corresponding to the discretisationsof Fig. 2.

As expected we find that the increased number of elements,by providing an increase in the number of degrees of freedom,improves the accuracy of the solution.

We now seek to construct a dual method to give a lowerbound to the capacitance. We follow the method of Section3 by enforcing the divergence relationship everywhere andallowing the curl sources to be correct in an average manner.As this is a Laplacian problem, we can enforce the fact thatthe divergence is zero by putting

D = Vx K (20)

where K is an electric vector potential. For a two-dimensionalproblem only the z component of K is required. Inside eachelement we use a linear relationship as in eqn. 19:

Ke = a\ (21)

The field D is, therefore, constant in each element. On theinterfaces between elements we use the continuity of thenormal component of D to ensure that there are no divergencesources and we allow a discontinuity in the tangential com-ponents. Inside each element there are neither curl nordivergence sources because the field is constant. The variationalmethod, therefore, operates to reduce the energy associatedwith the curl sources on the interfaces between elements andon the boundary, which have been introduced by thediscretisation and by the use of eqn. 21. Table 2 gives theresults corresponding to Fig. 2.

Comparison between Tables 1 and 2 shows a very similarbehaviour for the lower and upper bounds.

Table 2: Lower bound of capacitance

Number of elements

Approximate capacitancePercentage error

Table 3

Number of elements

Average approximatecapacitancePercentage error

3

9.14e0

-10 .4

6

9.33 e0

- 8 . 5

: Average capacitance

3

10.57 e0

+ 3.6

6

10.17e0

- 0 . 3

12

9.76 e0

- 4 . 3

12

10.30e0

+ 1.0

24

9.88e0

- 3 . 1

24

10.215e0

+ 0.1

Fig. 2 Discretisation of problem of tubular capacitor

a—d 3, 6, 12 and 24 elements, respectively

In Table 3 the average of the bounds is given and also thecorresponding error.

It can be seen that with only three elements the averageerror is similar to the error of the normal finite-elementmethod using 24 elements. Confidence in the closeness ofthe solution is reinforced by noticing the oscillatory behaviourof the average value of the capacitance. It should be notedthat the two methods use the same discretisation, so that thecomputational labour is not greatly increased by the dualcalculation. There is, therefore, a very considerable savingin being able to reduce the number of elements quite apartfrom the important advantage of having confidence limitsfor the solution. The absence of such confidence limits ismentioned by several writers [4—6] as a problem of thenormal finite-element method.

The double use of scalar and vector potentials in aLaplacian problem leads to a further important change ofviewpoint. These potentials are now seen as auxiliary functionswhich can be chosen at will. They are convenient rather thanessential. The duality is inherent in the sources of the field.If we choose to use a scalar potential which is continuousthroughout a region we ensure that there are no curl sources.If on the other hand we choose a vector potential we ensurethat there are no divergence sources. The dual variational

IEEPROC, Vol. 130, Pt. A, No. 3, MA Y1983 107

processes vary either the divergence or the curl sources butnot both. By this means upper and lower bounds are obtained.Instead of using the potentials we can operate with the vectorfield themselves and with their sources. This becomesimportant if there are sources within the system which pre-clude the use of the potentials. We shall illustrate the methodin the Section dealing with Poissonian problems.

Before we leave the example of Fig. 1 we briefly look atsecond-order polynomials for the potentials. We can increasethe accuracy by putting

<t>e = a ? oc%x2 oce5y

2 + oce6xy (22)

The continuity of the tangential electric field on the interfacesbetween elements is now no longer implicit because thepotential on the interfaces does not depend only on the nodalpotentials at the ends of each element. Hence, the continuityexpression must be inserted explicitly to ensure that thereare no curl sources on the interfaces.

In the dual method a similar expression as in eqn. 22 canbe used for the vector potential K. The continuity of thenormal field at the interfaces must be inserted explicitly toensure that there are no divergence sources.

The capacitance has been calculated for the discretisationof Figs. 2a and b and is tabulated in Table 4.

Table 4: Capacitance calculation using second-order potentials

Number of elements

Approximate capacitance

Percentage errorAverage capacitanceAverage percentage error

3

10.67e0

+ 4.610.13e0

- 0 . 6

C

9.—

60 e0

5.9

6

c+

10.67 e0

+ 4.610.31 e0

+ 1.1

C

9.96 e0

— 2.4

The use of quadratic polynomials increases the com-putational labour. For this reason we did not calculate thecapacitance for a higher number of elements. A compromisehas to be reached between the complexity of the behaviourof the field in each element and the complexity imposed bythe discretisation. In this example comparison between Tables3 and 4 shows that there is little to choose in terms ofaccuracy. However, in a real problem the correct value willbe unknown, and so the sign of the average error will alsobe unknown. For this reason it may be desirable to establishthe oscillatory behaviour of the average value with increasein the number of elements in cases where great accuracy isessential.

6 Poissonian problems

In this Section we illustrate the dual finite-element methodby two examples described by Poisson's equation. Suchproblems arise in magnetic fields when it is required tocalculate the inductances of conductors near iron boundaries.The governing equation in terms of the magnetic vectorpotential is

V x V x / 1 = fiJ (23)

The use of the vector potential gives the magnetic field as

B = Vx A

and thereby enforces

V -B = 0

An energy functional is given by

W =2ju

(24)

(25)

(26)

and in the usual finite-element method the variation within thevolume is written

- V x H),bA) = 0 (27)

The divergence sources are therefore put equal to zero every-where, whereas the curl sources are put equal to the currentdensity in an average manner. Thus, the approximation allowsa faulty current distribution in modelling the system. Theprocedure gives a lower bound for the inductance. The upperbound is provided by fixing the curl sources in terms of theassigned current density by the relationship

V x / / = 7. (28)

and allowing the divergence sources to be correct in an averagemanner given by

8W = <V7 'B,8M = 0

This corresponds to the functional

W = -2

(\H\2)

(29)

(30)

where the tangential field Ht and therefore the surface current/ are specified on the bounding surface of the system. Therelation between B and H is given by

B = (31)

If the permeability varies it can be represented by additionalsources. In the usual finite-element method such sourcescan be treated as additional currents and in the dual methodthey would be magnetic poles.

Surface conditions are dealt with in a similar manner tothe electrostatic case discussed in Section 4. The surface layersare either surface currents or surface pole distributions. Theycorrespond to discontinuities in the tangential or normalcomponents of the magnetic field.

Fig. 3 Highly permeable conductor of square cross-section

Consider the problem of a highly permeable conductorillustrated by Figs. 3 and 4. If the permeability of the con-ductor is very much larger than that of the region outside it,it can be assumed that the boundary surface is a field line.This assumption permits the use of an analytical solutionfor the inductance per unit length given by [7]

T M a

L= Tib

tan/?

_2a192 a

175 b fe% (2Ar+l) : (32)

We are examining the case of a square conductor, so that a =b. The exact value of the inductance is 0.0351 /x.

108 IEE PROC, Vol. 130, Pt. A, No. 3, MA Y1983

Table 5: Lower bound of inductance of permeable conductors

Number of elements 16 32 64 800

Approximate inductancePercentage error

0.0278/u 0.0301 M 0.0327 0.0349— 20.96 -14.37 -7 .03 -0.81

If we use a linear relationship for the vector potential asin the normal finite-element method we obtain the resultsof Table 5.

It will be noted that the method operates with discontinuitiesof the tangential component of the field on the interfaces.Because of the linear expression for the vector potentialthe current sources in the system are confined to the interfaces.There are no divergence sources anywhere in the system.

B X

\ \ \

B B X

Fig. 4 Discretisation of problem of permeable conductor

a-d 16, 32, 64 and 800 elements, respectively

subject to the condition that

a s — a3 = / (37)

and of continuity of tangential components of H across the inter-faces. This will ensure that the variation is in terms of thedivergence of the field and that the curl sources are leftunchanged. It will be noticed that eqn. 37 cannot be satisfiedif both as and a3 are zero unless J is also zero. This impliesthat the dual variation is not possible if the field in eachelement is constant. We need at least a linear variation ofH to obtain an upper bound of the inductance. The lowerbound does not have this requirement. The results for Figs.4a and b are given in Table 6.

It will be noted that the percentage error of the averagevalue with a discretisation of 16 elements is of the same orderas the error obtained by the normal finite-element method for64 elements and that the error in the average value obtainedwith 32 elements is of the same order as that obtained with800 elements.

It will have been noticed that our treatment has avoidedthe use of potentials and that this has resulted in a con-siderably simpler form of the equations. Once again we seethat the method operates by controlling the divergence andcurl sources of the field.

Consider next the inductance of a nonmagnetic conductorpartially surrounded by highly permeable iron, as illustratedby Fig. 5. An accurate solution is available for this con-figuration and the inductance is given by L = 0.57 ju0 H/m.The discretisation used by us is shown in Fig. 6. The normalfinite-element method using a linear spatial relationship forthe vector potential gave the results shown in Table 7.

It will be seen that the convergence is slow. The reason mustbe that it is difficult to obtain a good estimate of the energy,if the actual current in the conductor is transferred to theinterfaces of the elements.

A better value for the lower bound of the inductance canbe obtained by using a linear spatial relationship for themagnetic field or a quadratic vector potential which allowsthe current to be distributed throughout the volume as wellas on the interfaces. An upper bound can be calculated byusing the correct value of current density and allowing the

There is, however, no need to confine the curl sourcesto the interfaces. We can relax the system by allowing a curldistribution within the elements. A linear variation of thefield components within the elements of the form

Bx = ocl+a2x + a3y (33)

By = a4 + a5x-a2y (34)

will ensure that there are no divergence sources in an element.If, in addition, the normal field is made continuous acrossinterfaces, we shall again obtain a lower bound for theinductance.

Similarly, we can obtain an upper bound for the inductanceby writing

(35)Hx = al +oc2x + a3y

Hy = + a s (36) Fig. 5 Conductor with iron boundaries

Number of elements

1632

Table 6

L_

0 . 0 3 2 6 M0 . 0 3 3 6 M

: Average

% error

- 7 . 3 8- 4 . 4 6

inductance of permeable conductors

< - •

0 . 0 4 1 7 M0 .0357 M

% error

+ 18.56+ 1.46

l-av

0.0371 M0.0346 M

% error

+ 5.59-1.50

IEEPROC, Vol. 130, Pt. A, No. 3, MA Y1983 109

Table 7: Lower bound of inductance of conductor with iron boundaries

Number of elements

Approximate inductancePercentage error

Number of elements

368

Table 8:

L_

0.420 Mo0.428 Mo0.486 Mo

3

0.405 Mo- 28.95

6

0.412 Mo- 2 7 . 7 0

8

0.412 Mo-27.70

12

0.412- 2 7 . 7 0

Average inductance of conductor with iron boundaries

% error

26.3224.9114.74

0.647 Mo0 . 6 1 5 M 0

0 . 6 1 5 M 0

% error

13.517.907.86

<-av

0.5340.5220.55

20

0.42- 26.32

% error

6.408.483.44

B X A

/

/

E

D

/

C

B X

X/

E

D (

/

B X A B X

B X

Fig. 6 Discretisation of problem of conductor with iron boundaries

a—e 3, 6, 8, 12 and 20 elements, respectively

divergence sources to vary both in the volume and on theinterfaces. The results are given in Table 8.

We notice that the average value of the inductance is anorder of magnitude better than the inductance obtained withthe normal finite-element method even when the number ofelements is halved. We also notice that the upper bound ismore accurate than the lower bound presumably because theenergy is markedly dependent on the correct distribution ofthe current. It is likely that in most Poissonian problems themore accurate solution will be obtained by the method whichmaintains the correct source distribution. This will always bethe dual method rather than the usual finite-element method.Thus, the estimate of the lower bound of capacitance andthe upper bound of inductance is likely to be the better one.In Laplacian problems it is likely that both bounds will showa similar behaviour.

7 Conclusions

Vector fields can be specified in terms of two independenttypes of sources defined by the divergence and curl of thefield. In electromagnetism such sources are typified bycharges and currents. A variational method has been devisedwhich varies one or other of these sources and by this meansobtains upper and lower bounds to parameters defining theenergy of a system. The subdivision of a system into finiteelements improves the accuracy of the bounds and at thesame time provides an estimate of energy density and, there-fore, of the field distribution itself.

The well known finite-element method approaches thesolution of field problems from one side only and does notprovide an estimate of the order of accuracy. Moreover,in that method it is usual to operate with a linear expressionfor the potential. This constrains the variation severely andmakes it necessary to use a large number of elements toobtain a good estimate.

The method of upper and lower bounds proposed in thispaper removes the uncertainty inherent in the finite-elementmethod by providing confidence limits. This enables theanalyst to restrict the complexity of the computation to alevel appropriate to the desired accuracy.

The method uses the same discretisation for both bounds,thus saving both memory space and computational effort.It can deal with 3-dimensional and nonlinear problems andcan be extended to deal with time-varying electromagneticfields.

It is expected that the method will enable many problemsto be solved with smaller computers and will make it possibleto solve larger problems which at present would requireexcessive computational effort. In both these ways the methodis likely to increase the scope of field solutions both in thedesign and analysis of electromagnetic systems and devices.

8 Acknowledgment

Thanks are due to the Universities of Thessaloniki (Greece)

110 IEEPROC, Vol. 130, Pt. A, No. 3, MAY 1983

and Southampton (England) for providing the opportunityfor one of the authors to work in the Department of ElectricalEngineering at Southampton during a period of study leave.

9 References

1 PENMAN, J., and FRASER, J.R.: 'Complementary and dual finiteelement principles in magnetostatics', IEEE Trans.. 1982, MAG-18,pp. 319-324

2 HARRINGTON, R.F.: 'Field computation by moment methods'(Macmillan, New York, 1968), pp. 6-7

3 HAMMOND, P.: 'Energy methods in electromagnetism' (ClarendonPress, Oxford, 1981)

4 ZIENKIEWICZ, O.C.: The finite-element method' (McGraw-Hill,Maidenhead, 1977), pp. 63, 232, 233

5 BROWN, M.L.: 'Calculation of 3-dimensional eddy currents atpower frequencies', IEE Proc. A, 1982, 129, (1), pp. 46-53

6 HOHL, J.H.: 'Variational principles for semi-conductor devicemodelling with finite elements', IBM J. Res. & Develop., 1978,22, pp. 159-167

7 TSIBOUKIS, T.D.: Doctorate thesis, University of Thessaloniki,1980, pp. 97-98

10 Appendix: Further details of the method of Section 6

To obtain an upper bound for the inductance we use thefunctional of eqn. 30 in the form

W = I ^(He,He)e = l 2

The approximating function is

He = Hxex + H*y

From eqns. 35—37 we obtain

H% = af + ote2 x + ae

3y

Hey =

(38)

(39)

(40)

(41)

On the interfaces the tangential components of H must becontinuous:

Het = He

t

If the equation of the interface is

y = Xi x + X2

we find

(42)

(43)

(at - a - Ǥ

Xl\2(cce5-oce

5) = 0 (44)

and

(a? - oce2) + 2 X, (a? - a\ ) + \j (a% - a?') = 0 (45)

Substituting eqns. 40 and 41 into eqn. 38, and making useof eqns. 44 and 45 we obtain

W = F{kx,k2,...,km) (46)

An estimate of the number m of the free coefficients canbe obtained as follows.

For n triangular elements and N boundary segments thereare \ (3n + N) sides and therefore 3n + N constraining con-ditions. The total number of coefficients describing the

field is 5«. Hence

m = 5n-(3n+N) = 2n~N

To find the ks we put

dW dW dW

dkr= 0

(47)

(48)

The values of k are then substituted into eqns. 46 and finallywe obtain

2W(49)

To obtain a lower bound for the inductance we use thefunctional of eqn. 26 in the form

W = f [<J,Ae)--^-(Be,Be)]+ f [A'xrf.no]e = i 2\i f%

(50)

where the last term signifies integration over the boundarysurface of the system. In that integration H is prescribedwhere A varies and fH • dl is prescribed where A is constant.The outward normal at the surface is defined by n0.

The linear variation of B described in eqns. 33 and 34 canconveniently be embodied in an expression for the vectorpotential

(51)

On the interfaces between elements the normal componentsof the field must be continuous, so that

If the equation of the interface is given by eqn. 43, then

and

+ X, X2 (a% - o f ' ) = 0

X]

The condition of continuity of A gives

Ce' = X2(af-af ')+y(a| -

(52)

(53)

(54)

(55)

If the equation of the interface is jv = yo, X2 is replaced byy0 in eqn. 56. If the equation of the interface is x = x0

Ce = ~x0 (at - a%) - | S (oce5

e5 - (56)

Substituting eqns. 33, 34 and 51 into eqn. 50, and takinginto account eqns. 53, 54 and 55 or 56, we obtain a functionalof the form of eqn. 46. The free coefficients km can beobtained by using eqn. 48 and finally we obtain

2WL~ " I2 (57)

IEE PROC, Vol. 130, Pt. A, No. 3, MAY 1983 111