SCIENCE

Whitney forms: a class of finite elements forthree-dimensional computations inelectromagnetism

Alain Bossavit

Indexing terms: Electromagnetic theory, Eddy currents, Mathematical techniques

Abstract: It has been recognised that numericalcomputations of magnetic fields by the finite-element method may require new types of ele-ments, whose degrees of freedom are not fieldvalues at mesh nodes, but other field-relatedquantities like e.g. circulations along edges of themesh. A rationale for the use of these special'mixed' elements can be obtained if one expressesbasic equations in terms of differential forms,instead of vector fields. The paper gives an ele-mentary introduction to this point of view, pre-sents Whitney forms (the mixed finite elementsalluded to), and sketches two numerical methods(dual, in some sense), for eddy-current studies,based on these elements.

1 Introduction

For those familiar with differential forms, Maxwell'sequations are best expressed in the language theyprovide: h and e are 1-forms, i.e. forms of degree one, band j are 2-forms. This, as we shall show in Section 2 ofthis paper, which consists in an elementary introductionto differential forms, means that the circulations of h ande along paths make sense from the physical pointof view, while the fluxes of b and j may be understoodthrough surfaces. Differential forms are, therefore, auseful tool, and some have argued that they should beused in electrodynamics, at least at the research level [2,10, 12, 14], and even for teaching [13]. But it is often feltthat, at least in the case of eddy-current studies, whichnever venture out of the nonrelativistic realm, the fullpower of this tool is not called for. Indeed, it can beobserved that most numerical analysts and engineersengaged in the study of electromagnetism prefer to thinkin terms of vector fields. Moreover, representations in co-ordinates are often preferred to vector expressions. It istherefore understandable that the bulk of the efforttowards finite-element modelling has consisted, up tonow, in adapting to vector-field methods which workedwell for scalar fields, like e.g. those used for solving theheat equation.

Paper 6282A (S8), first received 4th January and in revised form 7thJune 1988The author is with Electricite de France, 1 avenue du General deGaulle, 92141 Clamart, France

If such a trend were to continue, it would contrastwith the present tendency to geometrisation which can beseen to pervade all physics. According to this tendency,attention should focus on geometrical entities, and not ontheir representations as multiples of values in some co-ordinate system. If we accept this stand, it will not bedifficult to argue further than differential forms, and notvector fields, are the appropriate geometrical entities. Butthis will fail to convince the practising programmer whohas to deal with such objects as finite-element shape func-tions : these seem to require a co-ordinate system for theirmanipulation. The situation will be different if we candefine geometrical objects that are to differential formswhat finite-element interpolating functions are to scalarfields.

This is precisely the definition of Whitney forms.Briefly stated (Section 3 of the paper gives a more com-prehensive description), they are a family of differentialforms on a simplicial mesh (i.e. a network of tetrahedra,as used in finite-element studies), defined in such a waythat p-forms are determined by their integrals on p-simplices. One-forms (such as, in electricity, h and e) canthen be approximated by a suitable linear combination ofWhitney forms of degree 1, the coefficients being the cir-culations of the field along the edges of the mesh. Inother words, Whitney 1-forms play the role of finite ele-ments for h or e, but the so-called degrees of freedom areassociated with edges of the mesh, and are not the valuesof the components of the field at mesh nodes. This jus-tifies the nickname of 'edge elements' for Whitney 1-forms. Similarly, there are 'facet elements' whichaccommodate 2-forms, the degrees of freedom beingfluxes across facets. 'Node elements' are just piece wise-linear functions (they are commonly called P1), and'volume elements' are piecewise-constant functions(similarly known as P°).

Section 4 of the paper shows how these concepts canbe used to devise approximation methods for the eddy-current equations. The two methods we propose look, insome sense, symmetrical. This symmetry, or rather thisduality, is rooted in the duality properties of the mathe-matical structure that differential forms constitute.

All these arguments, we think, converge to suggestthat differential forms should be used as a working toolin numerical modelling of electromagnetic problems.

This does not imply such a radical change of thinkinghabits as we might fear. We have tried to present thetopic to simplify the transition from vector fields to dif-ferential forms, albeit at the risk of criticism from the side

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 8, NOVEMBER 1988 493

of mathematical tradition. (In particular, our ad-hoctreatment of the Hodge star operator certainly incurssuch a risk. We also ignore 'twisted forms', whose rele-vance to Maxwell equations is stressed in the excellentbook by Burke [8].)

Now, even if this plea for differential forms is not suc-cessful, Whitney forms as introduced here are interestingin their own right. Their value as finite elements for fieldcomputation has begun to be well documented [1].

Whitney forms were described in 1957, long before theuse of finite elements [20]. They were rediscovered in thefinite elements community since 1974, under the name of'mixed elements', which explains why our two numericalschemes fall in the category of 'mixed methods'. The rele-vance of Whitney elements to mixed methods, in general,have been exposed in Reference 6.

The rest of this Introduction will be devoted to nota-tions and to basic equations.

The three-dimensional (3D) eddy-current problem hasthe following mathematical description (Fig. 1). A region

Fig. 1 The situation: Q is a bounded region, F its boundary, C aconductor and C, an inducting coil

Q of space, with smooth boundary F, is divided up intothree regions: the conductor C, the inductor Cs (s forsource) and the air region. The permeability fi and theconductivity a {a = 0 and \i = n0 in air and in Cs) dependonly on position, not on the values of the fields.

It will be assumed in Section 4 that domains Q, C andCl — C are connected and simply-connected (no holes, noloops). Otherwise, the exposition would become tooinvolved, without immediate benefit. But it should bestressed that the approach outlined here works as well inthe general case. Indeed, the possibility of dealing withgeneral geometries is one of the advantages of thisapproach, although this will not be developed here.

A current density j s , divergence-free, null outside ofCs, is given as a function of space x and time t. We wishto compute the magnetic field h and the current density jwhich develop, starting from a given state of the electro-magnetic field at t = 0. The relevant equations are wellknown:

dtb + curl e = 0

curl h = j

j = ae+js

(1)

(2)

(3)

(4)

where bs is a given divergence-free field. bs, which is alsoa source of the electromagnetic field, corresponds tomagnets. It could be omitted (and will be droppedwithout further notice in Section 3), because magnets can

be replaced by equivalent current distributions, but wekeep it for the sake of symmetry. The following expectedequation:

div b = 0 (5)

is not necessary if b = 0 at t = 0, for it stems from eqn. 1.Similarly, div j = 0 is a consequence of eqn. 3.

We shall assume that one of the following sets ofboundary conditions holds:

nx h = and n • e = 0 on F

(the 'normal field' condition), or

n- b = 0 and n x e = 0 on F

(6)

(7)

(the 'tangential induction' condition), where n is a field ofoutward unit normals on F. As is well known, condition6 makes sense if the permeability n is infinite outside F('in the wall', in other words), and condition 7 corre-sponds to the case of a superconductive wall (or, morefrequently, to a symmetry plane). In the general case {p.and a finite), formulations 6 and 7 are approximationswith respect to physical reality, that are generally deemedacceptable if the boundary F is far enough from theregion of interest. The presence of such a wall at finitedistance is a simplifying assumption which we could dowithout. It allows us to consider a tetrahedral paving ofQ with a finite number of tetrahedra.

2 Eddy-current equations in the language ofdifferential forms

2.1 Differential formsDifferential forms are fields of alternating multilinearmappings from Up to U. If co is such a form, its value atpoint x is the mapping which, starting from p ordinarythree-dimensional vectors €lt..., £p (which can be seen asoriginating from x), yields a real number co( l̂5 . . . , £p).'Alternating' means that permuting two vectors invertsthe sign of the result (which, thus, is 0 if p > 3, so onlyforms of degree p = 0, 1, 2, or 3 will be considered). Wecall these 'p-forms', or 'forms of degree p\ In threedimensions, differential forms can be conceived as a dis-guise for vector-valued fields (if p = 1 or 2) or scalar-valued fields (p = 0 or 3). For, if u is a vector field in Q,the maps

£-«(*)•£ (8)

where ( , , ) denotes the mixed product, are alternatingones. So rule 8 yields a 1-form, which we shall note iu,and rule 9 yields a 2-form 2u. (Conversely, given a 1- or2-form co, there is a vector field u such that co = pu, forp = 1 or 2.) Similarly, to a function </> on fi correspondsthe 3-form 3(f> whose value at point x is the alternatingmap

The same function <f> generates a 0-form, which, by thegeneral definition, should be a field on fi whose value atx is a mapping from the empty set into U. Such amapping is a real-valued constant. This constant we takeas <j>(x), hence the 0-form °<f>. Now define the followingoperator d from the space of p-forms FP(Q) to Fp+1(fi) asfollows. If p = 0, d °<f> (the 0 is attached to <f>, not to d) isthe 1-form associated with the vector-field grad (f>, that is

d°<f> = '(grad (11)

494 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 8, NOVEMBER 1988

d is similarly defined for p = 1 or 2:

= 3(divu)

(12)

(13)

and, for p = 3, we set d 3(f> = 0.In a regular course on differential geometry (see e.g.

Reference 18), these definitions would work the otherway, as we would, first, define d in general, then introducegrad, curl and div as particular realisations of d. The fol-lowing formula, which is derived from eqns. 11-13, wouldthen be a consequence of the definition:

d o d = 0 (14)

The converse of eqn. 14, that is 'if a p-form <w is such thatdco = 0, then there is a (p — l)-form a such that co = da\which is true if Q is the whole space, is known as thePoincare lemma. When the space is simply-connected,this lemma is valid for p = 1. Spaces for which the lemmais valid for all p are said to be contractible.

2.2 The de Rham complexIn the general case, the situation would be depicted bythe following diagram (Fig. 2). (Such an algebraic struc-

od j (grad)

1

d j (curl)

2

d I (div)

3

Fig. 2 De Rham's complexSee text for graphic conventions

ture, where there is a family of sets of the same kind andan operator d between them which has the property ofeqn. 14, is known as a homological complex, and thestudy of such things is homological algebra. The presentstructure is de Rham's complex.)

Horizontal segments in Fig. 2 represent vector sub-spaces of Fp(il) for 0 ^ p ^ 3. (The 2 or 3 subspaces dis-played for a given p are, in fact, mutually orthogonalwith respect to the natural scalar product

- JJo

for p = 0 or 3, and

(pu, V) = \u • u'

(15)

(16)

for p = 1 or 2.) Now take two slanted arrows at somelevel of the diagram. Between the tails of the arrows is asubspace, between their heads is its image by d. So thediagram displays the kernels of d (i.e. the constants ifp = 0, curl-free fields if p = 1, div-free fields if p = 2), theimages by d (i.e. the constants if p = 0, curl-free fields ifp = 1, div-free fields if p = 2), the images by d (i.e. thespaces of gradients, of curls etc.) and the orthogonal com-plements of images with respect to kernels. These com-plements, here called Jf°, Jf1, J?2, are known as'cohomology spaces', and their dimensions (which arefinite in the case of non-pathological domains) measurehow short from being contractible Q falls. More precisely,

we have the following characterisations:

J f° = {0: <f> is constant on connected components of Q}

(17)

JV1 = I1!*: div u = 0, curl u = 0, n • u = 0 on T} (18)

Jff2 = {2u: div u = 0, curl u = 0, n x u = 0 on T} (19)

JV3 = {0} and

dim (Jf °) = number of connected components of il

(20)

(21)

(22)

dim (Jf1) = number of 'loops' in O

dim pf 2 ) = number of 'cavities' in Q

The purpose of the de Rham complex is precisely to givea handle (through an algebraic structure which can beinvestigated by algebraic techniques) on such globaltopological properties. In particular, a domain is simply-connected if Jf1 = {0}, contractible if 3tfp = {0} for all p.In the domain, we are not interested in topology for itsown sake, rather in the relations between de Rham's andWhitney's complexes, which will appear in Section 3. So,in order not to mix difficulties of independent origins, weshall assume, from now on, that 3V1 and 3V2 reduce to{0} (Q contractible).

Another general concept of differential geometrywhich can be readily introduced at this elementary levelis the Hodge operator • from Fp to F"~p (n = 3 here). Bydefinition,

*Cu) = 2u *(2U) = AM

(23)

(24)

2.3 Eddy-current equationsA first application of these considerations is Fig. 3, whichdisplays the eddy-current equations in the language of

1K_2;-- d'h=<j

3 0

Fig. 3 The structure of eddy-current equations

differential forms. The diagram (whose resemblance tothe one given in Reference 16 is not accidental) should beself-explanatory.

For instance, the relation b = nh + bs can be rewrittenas

2b =b = \i * Ch) + 2bs, which shows the relativelycomplex character of such a constitutive law. This relatesa 1-form (the field h) with a 2-form (the induction b) byfirst transforming lh into its Hodge dual 2h, now a2-form, multiplying by /i, then adding the (optional)source magnetisation bs. Clearly, the change of notationdoes not bring anything of interest by itself. What wegain, which is much more important, is an insight intothe structure of the equations, which is exposed by Fig. 3by the trick of locating b, h etc. at the right levels.

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 8, NOVEMBER 1988 495

3 Whitney forms

3.1 OrientationNow, as an attempt to provide a rationale for whatfollows, we shall outline the treatment of a particularsimple case of the stationary heat-flow problem (which isstructurally similar to the magnetostatics problem, as weshall see).

Let us suppose we want to find a vector field u in Q,satisfying

curl u = 0

div (KU) =f

(25)

(26)

for a given / and a constant K, and the boundary condi-tion n • u = 0 on P. We look for an approximation of uas

« = Z ui (27)

where w,- are P1 interpolants with respect to a tetrahedralpaving. So, if x, (a point in 3-space) is the position ofnode i, w,- is continuous, piecewise linear, with w^xj) = 0if j # i, and 1 if i = j . This way, eqn. 25 is satisfied. Buteqn. 26 cannot hold, because, owing to normal discontin-uities of grad w,- across facets, u in eqn. 27 and thus KUhave no divergence in the usual sense (their divergence isa distribution, not a function). As condition 26 cannot beenforced in this strong form, we express it in 'weak form',i.e. we look for some u of the form given in eqn. 27, whichin addition satisfies

— \ KU • g r a d <$>' = f(f>'Jn Jn

(28)

for all </>' which are linear combinations of the basis func-tions w,.

Note that eqns. 25 and 26 may be rewritten as

dYu = Q (29)

d2(Ku) = 3f (30)

two equations of similar form. Both cannot be exactlysatisfied simultaneously in discrete form. So, in the fam-iliar finite-element procedure described above, we chooseto take one of the equations at face value (here, eqn. 25),and to deal with the other one by a transposition of thedifferential operator d, which shifts from the left-handside (div in eqn. 26) to the right-hand side (grad in eqn.28) of a scalar product of forms.

It has been known for a while, among finite-elementspecialists, that the other choice is feasible: treat eqn. 26as is given and eqn. 25 in transposed form. This is knownas the 'mixed' approach [7, 11]. We purport to show thatthis idea works for eddy-current equations as well (andeven, although this will not be developed here, for a largeclass of equations in mathematical physics), thanks to theexistence, on a given mesh of tetrahedra, of a family ofpiecewise polynomial differential forms (Whitney'scomplex) which can be described as finite-element basesfor differential forms.

3.2 The Whitney complexLet us consider a tetrahedral paving of Q, with the usualconditions that two tetrahedra may have in common, afacet, an edge, a vertex, or they may be disjoint, to theexclusion of all other possibilities. Let V, E, F and T bethe sets of vertices, edges, facets and tetrahedra, respec-tively. They will be identified by the indices of their ver-tices: for instance edge {i,j}, facet {i,j, k} etc., and called

by their generic name of 'p-simplices' (p = 0 for vertices, 1for edges etc.).

In Reference 20, Whitney describes a family of formswith the following properties:

(i) they are polynomials of the first degree (at most) ontetrahedra

(ii) they 'match' on the facets, in a sense to be madeprecise below

(iii) they are uniquely determined from their integralson p-simplices.

Properties (ii) and (iii) need some explanation.We say that two p-forms match, or 'conform', on a

surface if they assign the same values to any given set ofp-vectors tangential to the surface at some point. Accord-ing to the definition of forms, 3-forms match uncondi-tionally (because 3 vectors in the same tangent plane aredependent), 2-forms 2u match if the normal componentsof u agree on both sides, and 1-forms lu if tangentialcomponents of u agree. Zero-forms 0<f> conform if 0 iscontinuous.

As for property (iii), it is natural, as p-forms act (at agiven point) on sets of p-vectors, to integrate the resultover manifolds of dimension p. So 1-forms have integralsover edges (that of lu is the circulation of the vector fieldu along the edge). The integral of 2u across a facet is theflux of M through this facet (the orientation of the normalis determined by the ordering of the vertices). The inte-gral of 3<p on a tetrahedron is that of <f>, and the integralof °<t> on 0-simplex i is simply 0(xf).

Whitney forms are constructed as follows: considervertex i and a point x belonging to one of the tetrahedrawhich share vertex i, let A,(x) be the barycentric weight ofx in its tetrahedron with respect to vertex i; with the con-vention that Xt{x) = 0 in other cases, we obtain a contin-uous, piecewise-linear function A, (which is w, above).Note that

ieN

where x, is the position of node i, and that

Remark: It may happen that the paving contains 'curved'tetrahedra, i.e. continuous images of straight tetrahedra.In this case, Xt{x) is defined as the barycentric co-ordinateof the preimage of x.

Now Whitney associates with any p-simplex i0, ilt ...,ip the differential form:

pwi0 ip — P - 2 - i \ 1 ' AUa Ah x x a Aij-i

x " Aij+i x • • • x a Aip yoi)

This rather awesome expression (which our notation doesnot make easier to read) results in much more manage-able formulas for low values of p, as we shall see.

3.3 Whitney elementsLet us begin with p = 0. We obtain 0W; = °X{, that is

w, = >H (32)

if we consider functions instead of the associated 0-forms.So Whitney elements (as we shall call the fields whichcorrespond to Whitney forms) are just the so-called Pl

finite-element interpolants for p = 0.If p = 1, and if i and j are the vertices of some edge,

expr. 31 similarly gives a 1-form, attached to this edge.

496 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 8, NOVEMBER 1988

The corresponding vector field is

wo. = AI.VAJ.-A7VAi (33)

(We denote grad by V for shortness.) Fig. 4 may help thereader to visualise it. As VA7- is orthogonal to facet {i, k, 1}

Fig. 4 Whitney edge element vvy

and VA, to facet {j, k, /}, the field turns around the axisk-l (its 'central axis'), is normal to planes containing k and/, and has (at points of such a plane) a magnitude pro-portional to the distance to the axis. The field is nonzeroover all tetrahedra which have {i,j} as one of their edges.

As the curl of wu, which is 2VA, x VAj, exists as afunction (and not only as a distribution), the tangentialpart of wtj is continuous across facets like {i, j , k}. It iseasy to check that its circulation is 1 along edge {i,j} and0 along all other edges. So properties (i), (ii) and (iii) aresatisfied. If

u= uiJwt (34)

is a linear combination of Whitney elements of degree 1,the degrees of freedom utj are the circulations of u alongthe edges, hence the nickname 'edge elements'.

If p = 2, we obtain Whitney elements of degree 2, or'facet elements':

w.Jk = 7Xj x VAk + Xj VAk x VA; + Ak VA,- x VA,) (35)

Now, instead of an axial field, we have a central field (thecentre is the fourth vertex) on each of the two tetrahedrawhich have facet {i, j , k) in common. We imagine thefield as coming from the 'source' /, growing, crossing thefacet and vanishing into the 'well' /', fourth vertex ofthe other tetrahedron. This field has normal continuity,its flux across facet {ijk} is 1, so such fluxes are thedegrees of freedom of the element. See Fig. 5.

(For reasons which cannot be exposed here, thedegrees of freedom should be localised at the centres ofcircles circumscribed to the facets. For edge elements,they are at mid edges.)

Whitney elements have been rediscovered in the finite-element literature since around 1974. Facet elements aredescribed in Reference 18, although in space-dimension 2,edge elements are described in Reference 15. There, theyare given in co-ordinate-dependent form:

w4x) = a x x 4- R nd\

where a and ft are three-dimensional vectors, constantover tetrahedra, for edge elements, and

= yx + S (37)

Fig. 5 Whitney facet element wijk

where y is scalar and d a vector, for facet elements. Eqns.33 and 35, derived from Whitney's general formula, aremuch more convenient, in particular when finite-elementassembly has to be performed.

Finally, for p = 3, we obtain functions wijkl, equal to aconstant on tetrahedron {i,j, k, /}, to zero elsewhere. (Thedegrees of freedom should be localised at the centres ofthe circumscribed spheres.)

We shall note by Wp the vector space generated byWhitney elements of order p. The similar notation PWwill serve, if necessary, to denote the corresponding spaceof differential forms. The most remarkable property ofWhitney forms is

= o, l, 2) (38)

Thanks to this, the complex of Whitney forms may servein place of de Rham's complex to deal with the same kindof topological problems in an economical way (as spacesof Whitney forms are yzmte-dimensional). But their inter-est to us is that, as we have seen, they are ready-madefinite elements for differential forms.

4 Two mixed methods

We shall now derive approximations for eqns. 1-4, byusing Whitney forms. To deal with boundary conditions,we define W% as the subspace of Wp of forms whichvanish on T (those whose degrees of freedom vanish onsimplices which lie entirely in T).

4.1 First mixed methodWe look for an approximation of h in Wl, therefore

(39)e e £

(recall that E is the set of edges, we is Whitney's elementof edge e). We note /i the vector of degrees of freedom:

h = {he :e€E) (40)

If we set j = curl h, eqn. 2 is exactly satisfied, and j e W2.From eqns. 2 and 4, b and e will be in Wl and W1,

respectively, so eqn. 1 cannot be satisfied exactly; thiswas to be expected. We deal with this mismatch by

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 8, NOVEMBER 1988 497

shifting to the transposed weak form. Hence, h and e willhave to satisfy

-^ I yh h'+ I ecm\h' = 0 W e W1 (41)dt Jn Jn

in addition to the weak form of eqn. 3, which is

I curl h e ' - \ ae • e' = \ j s • e' W e W2 (42)Jn Jn Jn

where

e = (43)

(F is the set of facets).This is a 'mixed', or 'two-fields' formulation [11]. It

results in a differential equation in h, e. To see this, let usintroduce the matrices A, B and C whose entries are

e e' = \Jn

we • we. e G E, e' e E

Bf.r = r feFJ'eF

Jncurl we • wr e e E,f e F

(44)

(45)

(46)

and call j s the vector of facet values such that js{f) =jj'Wf- Matrices A and B are square-symmetric and C isrectangular. Assuming a and ft constant for simplicity, wefind the following matrix form of eqns. 41 and 42:

- \LA C

C -oB(47)

Discretisation in time will lead to a series of linearsystems with a common rigidity matrix. Owing to thepeculiar form of this matrix (which is symmetric but notpositive definite), finding efficient resolution methods isstill a largely open problem.

In this method, the approximation of h is 'right', froma physical point of view: for h, being approximated as a1-form, will have tangential continuity. But approx-imating e as a 2-form is 'wrong', because e also shouldhave tangential continuity. Let us again stress that this isunavoidable with finite-element methods: already withthe simple case detailed in Section 3.1, we could notobtain tangential and normal continuity of u simulta-neously. But, if we cannot have the best of both worlds,we can minimise the effect of the mismatch on e by elimi-nating e from the equations (obtain e from eqn. 4 andsubstitute it into eqn. 41). This leads to the followingvariational formulation 'in h':

— \ nh W + a"1 curl h • curl h = 0 W e Wdt Jn Jc (48)

heW1 curl h = / (-A) w h e r e a = 0

We proposed this method in Reference 3. It has beenimplemented by J.C. Verite [4, 9], then developed into anindustrial code which is now in current use at Electricitede France. An example of application is given in Section4.3.

Remark: Boundary conditions 7 are implicit in eqns. 41and 42, in weak form. To enforce boundary conditions 6(now in exact form), we should replace Wl and W2 byWQ and W\, respectively.

4.2 Second mixed methodWe now look for an approximation of e in W1 and of binW2:

b =eeE feF

(49)

Faraday's law (eqn. 1) can then be satisfied exactly. Itsvariational form is

curl eh' = 0 W e W2 (50)

But there is a mismatch in Ampere's theorem (eqn. 2), as)must now lie in Wl and h in W2. We cure this by trans-position:

h . curl e' - \ae • e' = \ j s • e' W e Wl

Jn Jn Jn(51)

is imposed on h.The duality between the two mixed methods should

now be obvious.Eqns. 50 and 51 are a system of differential equations,

which can be recast, in the same notation as above, as

C -a A

where j s is now the vector of edge components:

Jn

(52)

(53)

Again, elimination of h (the wrong one in the pair h-e)suggests itself. Therefore,

- (ae+js)-e'+ n 'dt Jc u c, Jn

curl e • curl e' = 0

W € Wl (54)

This method was proposed in Reference 5. In that paper,the duality was shown, but not clearly understood. Thesimilarity with the proposal by Pillsbury [17] of a 'modi-fied vector potential method', based on the gauge

div {aa) = 0 (55)

was also mentioned there. The difference is in the choiceof Whitney elements to derive the discrete form of theequations.

No implementation of this method seems to have beendone. As the data structures (matrices A, B etc.) are thesame in both methods, it would probably be worthwhileto have them available in a single software system. For,in this way, we may hope to obtain bilateral bounds onsome global quantities like magnetic energy, Joule lossesetc., by solving the same problem with both methods, in asimilar manner as in Reference 12.

4.3 An exampleFor the time being, however, the software system we haveincorporates only the first method. The followingexample of application may help to show its possibilities.

This work has been done by Verite [9] to assess thesensitivity of a new design of eddy-current probe to thepresence of a 'transversal' crack (i.e. one which lies in aplane perpendicular to the axis) in a metallic tube (seeFig. 6). Such a probe would consist of two magneticcores, discoid, with six poles on each core and a coil on

498 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 8, NOVEMBER 1988

each probe. In the actual nondestructive testing oper-ation, it would move along the axis of the tube, and flawsin the metal would reveal themselves by a difference ofimpedance (as measured in a Wheatstone bridge) betweenthe two coils.

For such applications, it is absolutely necessary to

obtain a good insight into the magnetic field generated,and this requires a three-dimensional computation. Bysymmetry, this may be performed in a sector of 30° (Fig.7 shows the results). We can see half of one pole in therear and the corresponding part of the tube in the front.Shading indicates the density of the normal component(parallel to the axis) of the induced eddy currents in threeselected horizontal planes. Each run has been done usingabout 4000 tetrahedra; of course, with edge elements forh. The computation time was 4 min on a Cray IS, but webelieve it can be much reduced by improving on thelinear system solver (a biconjugate gradient solver, withrather crude preconditioning by the diagonal part of thematrix).

See also Reference 1 for information on the applicabil-ity of Whitney elements.

5 Conclusions

We have shown that, by expressing equations in the lan-guage of differential forms and by thinking of Whitneyforms as finite elements, we are led almost automaticallyto mixed methods. In the case of eddy currents, there aretwo main possibilities, which result in two 'dual' mixedmethods, one with emphasis on the magnetic field, theother one with emphasis on the electric field. Thisdemonstrates the usefulness of differential forms, even atthe level of finite-element discretisation.

6 Acknowledgments

I thank R. Kotiuga, of Harvard, USA, and L. Breen, ofEcole Polytechnique, France, for their help.

Fig. 6 Principle of eddy-current testing: two parallel six-pole probesinside a (flawed) tubeCoils around the poles are not shown

Fig. 7front)Eddy-current density

[Courtesy of Carre and Verite, Reference 9]

One-twelfth of one of the probes (in the rear) and of the tube (in

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