Global formulation for 3D magneto-static using flux and

gauged potential approaches

Maurizio Repetto, Francesco Trevisan1

Dep. of Ingegneria Elettrica, Industriale, Politecnico di Torino

C. Duca degli Abruzzi 24, 10129 Torino, Italy.

e-mail: repetto@polel1.polito.it, tel.: +39 0115647140, fax: +3901156471991Dep. of Ingegneria Elettrica, Gestionale e Meccanica, Universita di Udine

Via delle Scienze 208, 33100 Udine, Italy.

e-mail: trevisan@uniud.it, tel.: +39 0432558285, fax: +39 0432558251

Submitted to International Journal for Numerical Methods in Engineering, Wiley Interscience, on sept. the 24-th 2002

October 2, 2002

Abstract: The use of algebraic formulation for the solution of electromagnetic fields is becom-

ing more and more widespread. This paper presents the theoretical development of two algebraic

formulations of the magneto-static problem and their implementation in a three dimensional

computational procedure based on unstructured tetrahedral mesh. A complete description of the

variables used and of the solution algorithm is provided together with a discussion about the ad-

vantages of the method. The performances of the two procedures are tested and assessed versus

cases with known solution.

Index Terms: 3D magnetostatics, dual meshes, algebraic formulations.

1 Introduction

The numerical solution of electromagnetic field problems has been traditionally approached by meansof the differential formulation of Maxwell’s equations. These equations are often inserted in somevariational scheme, like for instance finite element method, to obtain a discrete numerical solution.

An alternative approach has been based on finite differences technique where a direct approx-imation of the differential operators is performed by using incremental ratios. This approach hasseen a remarkable success in high frequency electromagnetic field problems where the use of the Fi-nite Difference Time Domain scheme [1] is now widespread in many research works and engineeringapplications. A different line of reasoning has been frequently adopted in other research areas. Inthermal and fluid dynamic environment, a numerical solution of governing equations by means ofcontrol-volume technique is often performed. This technique does not take directly into accountscalar or vector fields as solution variables but instead their integral values over some space-timeregion. In this way, the fundamental laws of the physical phenomenon, for instance mass balance,are imposed directly by means of algebraic equations over a control-volume etc..

As it is well defined in Patankar book [2] introduction, this approach is a “step behind” withrespect to differential formulation of the fields, coming again to the “pre-calculus days” when balanceequations over control-volumes would have been the only way to write down a mathematical model.This approach is usually addressed as finite or global formulation of the physical problem. Someefficient applications of this scheme can be found also in electromagnetic area. Besides the already

1

mentioned FDTD method, the Finite Integration Technique proposed by Weiland is a well knownexample [3].

Finite formulation provides a theoretical frame which can be easily implemented in a compu-tational procedure defined on a structured regular grid. Furthermore, the definition of intertwinedgrids with related physical variables is very efficient, as for instance with the use of “staggered”grids as proposed in [2]. Notwithstanding the enormous advantages of this implementation, bothon the side of computational efficiency and of easy writing of code, its main drawback is related tothe “poor” geometrical discretization. A regular grid, in fact, is not well suited for the treatmentof curved boundaries or of uneaven geometrical dimensions inside the domain of the problem. Someattempts to overcome this limitation have been devised, but they often lead to very complex solu-tions, see for instance [4]. In this respect, the use of finite element method on non regular simplicialmesh is without doubt more efficient.

The theoretical work of Tonti in the definition of a Global Formulation of Electromagnetic Fields(GFEF), also referred as Finite Formulation, [5], allows to define, in a rigorous way, the main featuresof the basic geometrical entities over which global variables must be defined; global variables arethe starting point of Tonti’s formulation and they are equivalent to the integral variables commonlyused in the differential formulation of the electromagnetism. For this reason we refer here to theTonti’s work as to Global formulation instead to Finite formulation.

An other concept of the Global formulation is related to the introduction of a pair of orientedcell complexes, mainly based on simplexes, one dual of the other, where the duality present inMaxwell’s equation can be efficiently translated in a space-time discretization. In this way, Maxwell’sequations become topological constraints applied on the global variables and they can be enforcedalso on irregular space-time cells. While the topological constraints are formally identical on bothregular and irregular space-time cells, the enforcement of material constitutive equations becomesless straightforward then in the case of orthogonal cell complexes but, as it will be shown later inthis paper, this difficulty can be overcome by the use of interpolating functions over the cell.

Because at the base of GFEF are the ideas of algebraic topology, the implementation of a solutionalgorithm for the field problems is for many aspects resembling the approaches to the solution ofelectric networks. This theoretical treatment can be applied to the solution of the full Maxwell’sscheme involving space and time discretization. In this work, GFEF has been applied to the solutionof magneto-static problems, limiting thus the to the space cellulisation of the domain.

Two formulations of the magneto-static problem are presented and implemented: one usingmagnetic flux as solution variable and the other using the circulation of magnetic vector potential.In both the cases, the solution algorithm requires to define a tree along some mesh edges in orderto find out the set of independent variables and equations.

In the case of magnetic vector potential formulation, the use of the theoretical scheme allows totackle the problem of uniqueness of solution. The problem of uniqueness and gauging of magneticvector potential has been addressed by many researchers, see for instance [6],[7], [8] and [9] in theframe of variational formulation.

The problem can be as well faced on the basis of physical definition of solution variables andlinking the minimal set of independent unknowns to the minimal set of variables needed to definethe solution. In this way, gauge conditions are no more related to the space where the solution islooked for, but to the topological structure of the discretization. Similar approaches for a “physical”definition of gauge conditions can be found in [10].

The paper presents the subject with the following structure: section 2 recalls the main aspects ofthe global formulation particularised for the case of magneto-static, section 3 considers the magneto-static laws recast in global form, and 4 presents the approaches to the magnetic constitutive equation.In section 5 respectively flux and magnetic vector potential formulation are discussed while section6 reports some results on test cases and then conclusions are drawn.

2

2 Global Formulation of magneto-static

According to the Global Formulation, it is possible to deduce a set of algebraic equations directlyfrom physics instead of obtaining them from a discretisation process applied to differential or integralequations written in terms of the field quantities. Global variables are the starting point of the globalformulation; using the global variables, like currents or magnetic fluxes, a direct discrete formulationof physical laws can be obtained ready for the numerical implementation.

Global variables are referred not only to points or instants, like the field variables, but alsoto oriented space and time elements like points P, lines L, surfaces S, volumes V, time instantsI and time intervals T (the bold face evidence that the elements are oriented) and are thereforedomain functions instead of point functions. Moreover global variables are continuous in presence ofdifferent materials and do not require any restriction, like field functions do, in terms of derivabilityconditions on the material media parameters.

The global variables, used in static magnetic field analysis are:

• the magnetic flux Φ;

• the circulation p of the magnetic vector potential;

• the electric current I;

• the magnetic voltage F .

The Global Formulation is also based on a further classification of the physical variables, inconfiguration, source and energy variables.

The configuration variables describe the configuration of the field or of the system without theintervention of the material parameters; also those variables linked to them by algebraic operationslike sum or division by a length, an area or a volume are configuration variables. Examples of theconfiguration variables are: the magnetic flux Φ or the circulation p of the magnetic vector potential,but also the induction field B, and the magnetic vector potential A.

The source variables describe the sources of the field without involving the material parameters;also those variables linked to them by algebraic operations like sum or division by a length, an areaor a volume are configuration variables. Examples of the source variables are: the electric currentI, the magnetic voltage F , but also the electric current density J or the magnetic field H.

The energy variables are obtained as the product of a configuration variable by a source variable;the magnetic energy density is an example of energy variable.

The global variables involved in magneto-static analysis are reported in Table 1. Their depen-dence on the oriented geometrical elements (in bold face) is evidenced within a square bracket;moreover, to distinguish one of the two possible orientations (inner and outer), a tilde is used tospecify the outer orientation respect to the inner orientation, as it will explained later. The globalvariables are related to the field functions by means of an integration performed on oriented lines,surfaces, volumes; therefore they are equivalent to the commonly used integral variables.

Table 1: Global variables in magneto-static and their relation with the field functions.These variables are also divided into configuration and source variables.

configurationglobal variables (Wb)

Φ =∫

S

B · dS p =∫

L

A · dL

sourceglobal variables (A)

I =∫

S

J · dS F =∫

L

H · dL

3

2.1 Space discretisation

The implementation of the Global Formulation requires the use of a pair of oriented cell complexes.The oriented geometrical elements, into which the space is discretized, can be thought as parts of aprimal cell complex that fills the whole region where the physical phenomena are considered. Verticesph, edges li, faces sj and cells vk are representative of points P, lines L, surfaces S and volumesV of the primal cell complex K; the subscripts are required in computational electromagnetism tonumber the geometrical elements.

From the primal complex , we can introduce the dual complex K made of geometrical elementsdenoted by vh, si, lj , pk with a tilde to distinguish them from the corresponding geometrical elementsof the primal cell complex K. The duality between the two complexes K and K assures that thegeometrical elements correspond as follows: ph ↔ vh, li ↔ si, sj ↔ lj ,vk ↔ pk.

The primal cell complex K considered here is based on tetrahedral cells vk, while the dualcomplex K is built starting from K according to the barycentric subdivision. A dual node pk is thebarycentre of the tetrahedron vk; a dual edge lj , crossing a primal face sj , is built in two segmentseach one going from the barycentre of the face sj to the barycentres of the two tetrahedra shearingthat face. A dual face si, crossed by a primal edge li, is the union of a number of quadrilateralfacets; each of them is individuated by the following four nodes: the barycentre of the edge li, thebarycentre of the tetrahedron individuated by the two faces sj having the edge li in common, thebarycentres of those two faces sj . A dual cell vh forms a volume enclosing a primal node ph; itsboundary faces are a number of dual faces sj .

In the paper the domain of interest is discretised in Nv tetrahedra as primal cells vk, in Ns

primal faces sj , in Nl primal lines li and Np primal points ph. Due to the duality between K andK it can be stated that Nv is the number of dual points pk, Ns is the number of dual lines lj , Nl isthe number of dual feces si and Np is the number of dual cells vh.

2.2 Orientation of cell complexes

Points, lines, surfaces and volumes of the two cell complexes K, K are endowed respectively withinner and outer orientation. A point ph can be oriented as a sink, this gives the inner orientationof it, the inner orientation of a line li is a verse chosen on it, the inner orientation of a face sj

is a direction to go along its boundary and the inner orientation of a cell vk is a congruent innerorientation of all its bounding faces (e. g. all faces taken with counter-clock wise verse).

Considering now the dual complex K, automatically all geometrical elements of K are endowedwith outer orientation. Precisely the outer orientation of a volume vh corresponds to an outwarddirection across its boundary faces and it is the inner orientation of the point ph, the outer orientationof a face si is the inner orientation of the primal line crossing it, the outer orientation of a dual edgelj is the inner orientation of the primal face sj crossing the it, the outer orientation of a dual pointpk is the inner orientation of the primal cell vk containing the point.

2.3 Global variables and cell complexes

The classification of global variables in configuration and source variables has a paramount impactin the numerical implementation of the Global Formulation. It has been shown that configurationvariables are naturally associated with geometrical elements endowed with inner orientation andtherefore are referred to the primal complex K; source variables are associated with geometricalelements with outer orientation and are therefore referred to the dual complex K.

This unique association of the global variables to the respective primal or dual cell complex, isone of the key points of the Global Formulation.

Therefore the global variables of magneto-static are associated to the oriented geometrical ele-ments of the cell complexes K, K as follows:

4

• Φ is the vector, of dimension Ns, of the fluxes relative to the primal faces sj ;

• p is the vector, of dimension Nl, of the circulations of A relative to the primal edges li;

• I is the vector, of dimension Nl, of the electric currents across the dual faces si;

• F is the vector, of dimension Ns, of the magnetic voltages relative to the dual edges lj .

3 Global form of magneto-static laws

The physical laws of magneto-static rewritten in global form, are expressed as topological constraintson global variables of the same kind. These constraints tie a global variable associated to a geomet-rical element to be equal to another global variable associated to its boundary. With reference to thepair of primal-dual cell complexes K, K above defined, the physical laws governing the magneto-static analysis are:

• the magnetic Gauss law:DΦ = 0 (1)

with D the incidence matrix, of dimension Nv ×Ns, between the inner orientations of a primalcell vk and a primal face sj ; Φ is the vector of fluxes;

• the Ampere law:CF = I (2)

with C the incidence matrix, of dimension Nl×Ns, between the outer orientations of a dual facesi and a dual edge lj ; F and I are the vectors of magnetic voltages and currents respectively;

• the charge conservation law:DI = 0 (3)

with D the incidence matrix, of dimension Np × Nl, between the outer orientations of a dualcell vh and a dual face si; I is the vector of currents.

4 Magnetic constitutive equation

The link between the configuration and the source variables are the constitutive equations thatcontain the material properties and the metrical notions such as lengths, areas and volumes. Inthe case of magneto-static analysis, the magnetic constitutive equation can be derived according totwo equivalent approaches described in the following. In both the approaches the media inside aprimal cell vk is assumed as uniform with permeability matrix µk of dimension 3; moreover it isconvenient to introduce the local vectors of fluxes Φk = [Φ1, ...,Φ4]T and of the magnetic voltagesFk = [F1, ..., F4]T along the four portions of dual edges l1, ..., l4 inside the tetrahedon and with localnumbering. It should be noted that, due to the barycentric subdivision, the primal face sj is notorthogonal to the portion of dual edge lj , Fig. 2.

4.1 Facet elements interpolation

In the first approach an interpolation is performed to link locally the two vectors Fk, Φk. Thisinterpolation is built using Whitney facet element shape functions inside each tetrahedron, [12].Using these interpolating functions, the magnetic flux density in each primal cell can be written as:

Bk =4∑

j=1

dkjwjΦj (4)

5

primal edge connected to

node

dual edge bounding a dual face

dual node(barycenter of tetrahedron)

Figure 1: A primal cell complex based on tetrahedra is shown together with primal nodes, edges,and faces; moreover some dual nodes, dual edges, and dual faces are shown.

ab

c

d

~~

~ ~

s1s2

s3

s4

l1

l2

l3

l4 l5

l6l1

l3

l2

l4

si~ pk

~

vk

Figure 2: A detail of a primal cell vk (a tetrahedron) is displayed together with the four primalnodes a, b, c, d, the six primal edges l1, ..., l6 and the four primal faces s1, ..., s4 with local number-ing. Moreover the dual node pk located at the barycentre of the tetrahedron is shown togetherwith the four portions of dual edges l1, ..., l4, with local numbering, and a portion of a dual facesi tailored inside the primal cell.

6

being dkj the incidence numbers between the inner orientations of the primal cell vk and of itsprimal faces sj , wj are the facet elements shape function vectors for the j − th face having nodes(l, m, n) as vertices; it is defined as:

wk = 2 [Nl(P )∇Nm ×∇Nn + Nm(P )∇Nn ×∇Nl + Nn(P )∇Nl ×∇Nm] (5)

where Nl(P ) is the affine nodal shape function for the node l. From (4) the resulting induction Bk

results uniform and therefore a uniform field Hk can be deduced from the constitutive law in termsof the fields quantities:

Hk = (µk)−1 Bk (6)

Being Hk uniform inside vk, the magnetic voltage vector Fk can be derived as:

Fk = Lk Hk (7)

where Lk is the 4×3 matrix whose rows are the four row vectors l1, ..., l4 of the four portions of dualedges internal to vk.

Substituting in (7), the expressions (6) and (4), the final matrix form of the magnetic constitutiveequation can be derived as:

Fk = MkΦk (8)

being Mk the square non symmetric elemental matrix of dimension 4 relative to the cell vk.

4.2 Uniform field approach

A second approach to derive the local constitutive equation, starts from the assumption of uniformityof the fields Bk and Hk within the cell vk. With reference to Fig. 2, the following four vectors ofthe magnetic voltages along three of the four portions of dual edges can be introduced:

Fa = [F2, F3, F4]T,Fb = [F1, F3, F4]T,Fc = [F1, F2, F4]T,Fd = [F1, F2, F3]T (9)

the field Hk can be expressed as:

Hk = L−1a Fa = L−1

b Fb = L−1c Fc = L−1

d Fd (10)

where La, Lb, Lc, Ld are the 3×3 matrices storing by rows the corresponding row vectors of the dualedges. In a compact form (10) can be rewritten in terms of the vector Fk as:

Hk = Aa Fk = Ab Fk = Ac Fk = Ad Fk (11)

being:

• Aa a 3 × 4 matrix whose elements are those of L−1a and in addition the 1-st column of zeros;

• Ab a 3 × 4 matrix whose elements are those of L−1b and in addition the 2-nd column of zeros;

• Ac a 3 × 4 matrix whose elements are those of L−1c and in addition the 3-rd column of zeros;

• Ad a 3 × 4 matrix whose elements are those of L−1d and in addition the 4-th column of zeros.

From (11) and the inverse of the constitutive law (6), the uniform induction Bk can be derived byaveraging in turn as:

Bk =13µk(Ab + Ac + Ad)F =

13µk(Aa + Ac + Ad)F =

13µk(Aa + Ab + Ad)F =

13µk(Aa + Ab + Ac)F

(12)

7

The magnetic constitutive equation expressing Φk as a function of Fk is then derived as:

Φk =13

S1 µk(Ab + Ac + Ad)S2 µk(Aa + Ac + Ad)S3 µk(Aa + Ab + Ad)S4 µk(Aa + Ab + Ac)

Fk = Nk Fk (13)

with S1, ...,S4 are the four row area vectors of the four oriented primal faces of the cell vk. Finally,by inversion of the 4× 4 matrix Nk, exactly the magnetic constitutive matrix Mk in (8) is obtained.

By assembling element by element the local constitutive equation (8), the global constitutiveequation can be derived as:

F = MΦ (14)

being M the Ns × Ns global constitutive matrix.

5 Formulations for magneto-static

The algebraic system of equations that allows to solve the magneto-static problem for given externalcurrent and media permeability distributions, can be cast according to two formulations:

• the flux formulation, where the flux vector Φ = [Φ1, ..., ΦNs]T is considered;

• the p−formulation, where the vector p = [p1, ..., pNl]T of the circulations of A is considered.

5.1 Flux formulation

In this case the Ns fluxes relative to the primal faces sj can be split in Nse fluxes relative to theexternal boundary faces and Nsi fluxes relative to the internal faces such that Ns = Nse + Nsi. Weconsider in this subsection only the case of a closed boundary surface S, enclosing the volume V, onwhich the external fluxes Φej , with j = 1, ..., Nse, are known being assigned as boundary conditionsthat must comply with the magnetic Gauss law as follows:

Nse∑j=1

djΦj = 0 (15)

dj are the incidence numbers between the inner orientation of the external boundary faces andthe inner orientation of the external volume V containing the primal cell complex K. The case ofsymmetry planes on which symmetry conditions are imposed will be treated later.

The relation (15) must be considered in addition to the Nv equations from (1); therefore Nv + 1equations can be written according to the Gauss law but only Nv of them are linearly independent.An independent set of Nv equations based on the Gauss law can be written by adding to (15), Nv−1equations obtained by eliminating from (1) a row in the matrix D; the corresponding reduced systembecomes:

DrΦ = 0 (16)

with Dr the reduced incidence matrix of dimension (Nv − 1) × Ns.By substituting in the Ampere law (2) the constitutive equation (14) the following Nl equations

can be derived:CM Φ = I (17)

These equations are linearly dependent due to the Np − 1 independent constraints on the currentsimposed by the charge conservation law (3). From the duality of the cell complexes K, K thefollowing relation holds:

D = −GT (18)

8

being G the incidence matrix, of dimension Nl ×Np, between the inner orientations of a primal lineli and of a primal point ph; the rank of G is Np − 1 being null the sum of its columns.

The total number of independent equations becomes:

Neq = [Nv]Gauss law + [Nl − (Np − 1)]Ampere+constitutive laws (19)

Substituting in (19) for Nv the Euler ’s formula:

Nv = Np − Nl + Ns − 1 (20)

we obtain that the number of independent equations Neq = Ns equals the total number of fluxes.Of course, from a computational point of view only Nsi fluxes relative to the internal primal

faces are the actual unknowns. The corresponding set of Nsi independent equations can be derivedby considering:

• Nv − 1 independent equations from (16), assuming (15) satisfied by the boundary conditionson the external fluxes;

• a number of closed paths can be defined, as in basic circuit theory, made of dual edges layinginside the domain. The set of internal dual edges lj connecting dual nodes pk can be treatedas a connected graph G of Nsi dual edges and Nv dual points. From the graph G a tree canbe derived of Nv − 1 branches and the corresponding co-tree individuates the Nsi − Nv + 1fundamental loops, respect to which Nsi −Nv + 1 independent equations of the kind (17) canbe written.

5.2 p−formulation

In this case it is convenient to introduce the vector p of circulations of the magnetic vector potential,relative to the primal edges li, such that:

Cp = Φ (21)

being C the incidence matrix, of dimension Ns ×Nl, between the inner orientations of a primal facesj and of a primal line li. Expression (21) identically satisfies the Gauss law (1), because of theidentity:

DC ≡ 0 (22)

Therefore by substituting (21) in (17), the following set of equations can be derived:

CMC p = I (23)

where from the duality of the cell complexes C = CT. It can be numerically verified that the Nl ×Nl

matrix CTMC of the system (23) is symmetric even if the constitutive matrix M is not.These Nl equations are not independent because of charge conservation equation (3) that imposes

Np − 1 independent constraints on the currents. Therefore the resulting number of independentequations is:

Neq = Nl − Np + 1 (24)

These Neq independent equations constrain Nl−Np+1 circulations pi; this means that the remainingNp − 1 are free.

From a computational point of view it is important to individuate which are the actual unknownspi considering that some of them are imposed as boundary conditions on the boundary edges andsome of them are irrelevant in the numerical solution of (23).

By distinguishing the primal edges Nl = Nle + Nli in Nle external boundary edges and Nli

internal edges and the primal nodes Np = Npe + Npi in Npe external boundary nodes and Npi

internal nodes, the number of Nl − Np + 1 constrained circulations can be rewritten as:

Nl − Np + 1 = [Nle − (Npe − 1)] + [Nli − Npi] (25)

9

that can be interpreted as follows:

• considering the connected graph Ge based on Npe boundary primal nodes and on Nle boundaryprimal edges located on the external boundary, (Npe−1) corresponds to the number of branchesof a tree Te of the graph Ge;

• Nle − (Npe − 1) is the number of branches of the co-tree Ce of the graph Ge corresponding tothe selected tree Te.

• Npi − 1 is the number of the branches of a tree Ti of the graph formed by the Nli internalprimal nodes and by a number of internal primal edges connecting those nodes; by adding tothe branches of Ti a further primal edge, linking an internal primal point to a primal point onthe external boundary, a new graph T ′

i is obtained with Npi branches;

• the complementary graph C′i of T ′

i respect to set of the Nli internal primal edges can beintroduced; C′

i has Nli − Npi branches.

Therefore the actual unknowns of the magneto-static problem are the Nli − Npi values of thecirculations pi associated to the branches of the graph C′

i, while the values of pi associated to thebranches of the graph T ′

i can be eliminated.Observing that the union of the graphs T ′

i and Te is a tree T of the whole graph G basedon Np primal nodes and Nl primal edges and that the union of the graphs C′

i and Ce is the co-tree corresponding to the tree T of G, this process to individuate the independent circulations p isanalogous the tree co-tree decomposition of the vector potential described in [6].

5.3 Comparison of the two formulations

In the p−formulation the Nv − 1 independent equations from (16), assuming (15) satisfied by theboundary conditions on the external fluxes, are automatically satisfied thanks to (21).

The Nli − Npi independent equations/ unknowns in the system (23) are strictly related to thenumber of Nsi −Nv + 1 independent equations of the kind (17) in the flux formulation. The Euler’sformula (20) can be rewritten as:

Npe + Npi − Nle − Nli + Nse + Nsi − Nv = 1 (26)

Observing that from Euler’s formula (20), written for the whole volume Ve enclosing the domain ofthe magneto-static problem, gives:

Npe − Nle + Nse = 2, (27)

(26) becomes:Nsi − Nv + 1 = Nli − Npi (28)

that expresses the equivalence between the two formulations.Moreover a further correspondence between flux and p−formulations with the gauge on the

magnetic vector potential suggested by Albanese-Rubinacci can be derived.Considering the whole graph G, from its internal co-tree C′

i, which branches are Nli−Npi internalprimal edges, Nli −Npi fundamental loops Lm can be constructed. Nli −Npi independent fluxes Φf

m

can be associated to the fundamental loops Lm, in order to form a base to express the Nsi unknowninternal fluxes Φj associated to the corresponding internal primal faces sj , Fig. 3. Therefore theNli −Npi fundamental fluxes Φf

m univocally correspond to the circulations pcm relative to the primal

edges of the internal co-tree C′i; if the circulations relative to the internal tree T ′

i are set to zerothe following equivalence holds:

Φfm ≡ pc

m (29)

10

Φ1 Φ2f f

primal tree

fondamental loopsand

associated fluxes

boundary edges

primal co-tree

Figure 3: With reference to a plane primal cellulisation, a tree, the co-tree and the some of thecorresponding fundamental loops are shown together with the associated independent fluxes Φf

j .

assuming that each fundamental loop Lm is oriented internally as the corresponding co-tree edge.The zeroing of the circulations relative to the internal tree T ′

i corresponds to the gauge onthe vector potential of Albanese-Rubinacci. However if the flux formulation has a direct linkwith physics the resulting system obtained from (16) and (17) leads to a non symmetric matrix.Moreover, the corresponding computational scheme requires to store for each dual co-tree branchthe fundamental loop topology, this fact in large three dimensional problems can be very demandingin terms of memory requirements. In additions, with usual tetrahedral mesh generators the obtainedalgebraic system is larger for the flux formulation respect to the p−formulation. A further advantageof the p−formulation is that it can be easily extended to the case of magneto quasi-static problems.

5.4 Symmetry conditions

In order to reduce the computational effort the symmetry conditions can be profitably used when ap-plicable to the magnetosatic problem. Two types of symmetry conditions can be imposed dependingon the kind of the problem:

• the flux is zero relative to surfaces Σt, internal to the domain of interest V; this condition isequivalent to (B · n)Σt

= 0 that expresses the tangent field condition to Σt;

• the flux is maximum relative to surfaces Σn, internal to the domain of interest V; this conditionis equivalent to (H · t)Σn

= 0 that expresses the normal field condition to Σn.

5.4.1 Tangent field condition

In this case the fluxes relative to the primal faces laying on the symmetry surface Σt are null,together with the circulations p along the primal edges bounding that faces. In this way only onefraction of the domain V needs to be analyzed, bounded by a part of the external boundary S andby the symmetry surface Σt. From a computational point of view this condition is treated in thesame way as the boundary conditions on the external boundary S.

11

5.4.2 Normal field condition

In this case the field is orthogonal to the symmetry surface Σn but the fluxes relative to the primalfaces on Σn are unknowns as well as the circulations p along the primal edges bounding that faces.In this way only one half of the domain V needs to be analyzed, bounded by half of the externalboundary surface S and by the symmetry surface Σn. From a computational point of view theprimal faces or the primal edges on Σn are treated in the same way as the internal faces or theinternal edges with unknown flux or circulation p.

The equivalence of the flux and p formulations even in this case can be shown starting from (26)written as:

Npe + NpΣn+ Npi − Nle − NlΣn

− Nli + Nse + NsΣn+ Nsi − Nv = 1 (30)

where the geometrical entities on Σn are accounted for explicitly. Because the portion of S, wherethe fluxes or the circulations p are assigned as boundary conditions, is an open surface and does notbound a volume anymore, the Euler’s formula gives:

Npe − Nle + Nse = 1 (31)

and therefore (30) simplifies as:

(Nsi + NsΣn) − Nv = (Nli + NlΣn

) − (Npi + NpΣn) (32)

that is the corresponding relation to (28).

6 Numerical results

6.1 Implementation

The theoretical methods presented in the previous sections, have been implemented in two numericalprocedures for magneto-static computations. One procedure is implemented in the Matlab environ-ment and translates in a computer code the matrix formulation of the problem. An interface versussome commercial three-dimensional mesh generators is defined and then, starting from the primaltetrahedral mesh, all the topological and material dependent matrices are computed. Another pro-cedure has been implemented in a standard programming language (Fortran). In this case, theprimal mesh is generated by some external pre-processor both commercial and in-house developed.Again, starting from the tetrahedral mesh, all other geometrical entities are defined and all theirtopological links are stored in tables.

In the solver phase the system matrix is not assembled by means of the topological matricesbut instead working on a element-by-element procedure. For each tetrahedron the local reluctivitymatrix is built and then the contribute of all single entities is summed in the solution matrix. Inthis way memory requirement and computational effort are minimized.

6.1.1 Cube with uniform applied field

A cubic domain with a unit length edge is used to test the accuracy of the method by applying auniform field along z direction. A primal mesh with distorted tetrahedra made of 384 volumes, 864faces, 604 edges and 125 nodes is used as test bench. The domain of the problem together with meshis shown in Fig. 4. Boundary conditions requiring a magnetic flux density of 1 T along z directionis applied. The solution obtained with both flux and p formulations gives an internal flux densityequal to 1 T uniform at most of 10−11 T. The same domain has been used to assess the stability ofthe solution to the choice of the tree. Two trees along the mesh have been used and they have beengenerated by means of a depth first and breadth first algorithm [13]. The solution has not shown anappreciable dependence on tree choice.

12

Figure 4: Flux density plot produced by imposing as boundary conditions a flux density of 1 Ton the upper and lower faces of the cube.

6.1.2 Ferromagnetic sphere in uniform field

This problem with two materials has an analytical solution which can be used as reference. Thedomain of the problem is shown in Fig. 5 together with the tetrahedral primal mesh. The primalmesh is made of 3240 tetrahedra, 6676 faces, 4088 edges and 653 nodes. A uniform flux is imposedon the two bases of the external cylinder corresponding to a uniform magnetic flux density of 1 T,while the sphere has a value of relative permeabilty of 1000. The vector plot of the magnetic fluxdensity around the sphere is reported in Fig. 6. The analytical solution for the magnetic flux densityinside the sphere is given by:

B = 3µr − 1µr + 2

B0 = 2.9917 (33)

The results obtained by the two schemes are reported in Table 2. In this case, the pattern of thesolution is following the expected behaviour leading to a local increase of magnetic flux density valuesaround the sphere and to a uniform flux density inside it. The error with respect to the analyticalsolution is small and can be attributed both to a discretization error of the spherical surface and toa uniform flux condition imposed at a finite distance from the sphere and not to infinity.

Table 2: Comparisons for the ferromagnetic sphere case.

flux formulation p formulation

B in the sphere centre [T] 2.853 2.894

error respect toanalytical solution [%] 4.6 3.2

13

Figure 5: Domain geometry enclosing a magnetic sphere with constant relative permeability ofµr = 1000; as boundary conditions a flux density of 1 T on the upper and lower faces of the cubehas been imposed.

Figure 6: Detail of the flux density plot around the spherical magnetic core.

14

6.1.3 Coil around a ferromagnetic cylinder

In this case a ferromagnetic cylinder with relative permeability µr = 1000 is partially inserted withina coil feeded with a uniform current of 1.2 A. The number of tetrahedra of the primal mesh is 1981.In Table 3 the amplitude of the computed flux density is reported along a number of points closeto the axis of the cylindrical core compared with those obtained from a 3D finite elements analysisperformed with the ANSYS code.

coil

magnetic core

Figure 7: Flux density plot produced by a coil with uniform current density, around a magneticcore with constant relative permeability of µr = 1000; cylinder radius rc = 5 mm and heighthc = 10 mm, coil inner radius Ri = 6 mm, outer radius Ro = 9 mm. The core is 4.5 mm outsidethe coil, while the uniform coil current is 1.2 A.

Table 3: Comparisons of the induction amplitude in the magnetic core.

Point coordinates (mm) B (mT) p formulation B (mT) Finite Elements

(0.63, -0.48, 1.1) 0.115 0.122

(1.2, -0.04, 2.5) 0.183 0.193

(1.2, 0.09, 4.3) 0.201 0.259

(1.2, 0.07, 7.5) 0.220 0.271

(0.66, 0.07, 8.7) 0.197 0.217

(1.3, 0.09, 10) 0.161 0.144

7 Conclusions

The Global formulation of electromagnetic fields proposed by Tonti has been the starting point forthe definition of two algorithms aiming to the solution of three-dimensional magneto-static problems.The work presented has allowed to define the theoretical details of these algorithms and to assesstheir performances in a computational procedure. It can be said that:

15

• the work has shown the feasibility of a solution based on an algebraic formulation of theelectromagnetic field on an unstructured grid. This approach overcomes the difficulties relatedto the orthogonal and structured grids usually employed in other “finite” formulations;

• the phi-formulation has been based on an algorithm which is very much related to the solutionof a circuit approach showing the analogies between global formulation and electric circuit;

• the p-formulation has the advantage of enforcing directly some physical properties of the fluxdensity but introduces some problems related to the uniqueness of the solution. In this case,uniqueness is enforced using a tree-co-tree scheme and this result, which is formally identicalto the one obtained by other Authors on variational basis, is obtained by enforcing physicallaws;

• the study performed has highlighted a substantial independence of the solution accuracy ontree selection;

• the formulations proposed can have, in Authors’ view, a notable importance for educationalpurposes, where a complete theoretical consistent solution scheme can be defined withoutrequiring a deep knowledge of variational calculus.

While this study has allowed to show the feasibility of the approach, the work will have to go onto compare its computational efficiency with other well established numerical procedures based onfinite elements.

8 Acknowlegments

The authors are greatful to Dr. A. Bossavit and to Prof. E. Tonti for the helpful discussions andsuggestions about the fundamentals of the Global Formulation.

References

[1] K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations inisotropic media”, IEEE Trans. on Antennas and Propagation, Vol. AP-14, pp. 302-307, 1966.

[2] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere Publishing Co., New York, USA,1980.

[3] T. Weiland, “Time Domain Electromagnetic Field Computation with Finite Difference Method”, Int.J. of Num. Modelling, 9, pp. 295-319, 1996.

[4] M. Clemens, T. Weiland, “Magnetic field simulation using conformal FIT formulations”, IEEE Trans.Mag. Vol. 38, No. 2, March 2002, pp. 389-392.

[5] E. Tonti, Finite Formulation of Electromagnetic Field, IEEE Trans. Mag. Vol. 38, No. 2, March 2002,pp. 333-336.

[6] R. Albanese, G. Rubinacci, “Finite Element Method for the Solution of 3D Eddy Current Problems”,Advances in Imaging and Electron Physics, vol.102, pp.1–86, April 1998.

[7] T. Morisue, “The gauge and topology problem in using the magnetic vector potential”, COMPEL, Vol.9(A), pp. 1-6, 1990.

[8] B. Menges, Z. Czendes, “A generalized tree-cotree gauge for magnetic field computation”, IEEE Trans.on Mag, Vol. 31, pp. 1342-1347, 1995.

[9] O. Biro, K. Preis, K. Richter, “On the use of the magnetic vector potential in the nodal and edge finiteelement analysis of 3D magnetostatic problems”, IEEE Trans. on Mag, Vol. 32, pp. 651-654, 1996.

[10] L. Kettunen, L. Turner, “A volume integral formulation for nonlinear magnetostatics and eddy currentsusing edge elements”, IEEE Trans Magn, Vol. 28, pp. 1539-1642, 1992.

16

[11] M. Clemens, T. Weiland, “Transient eddy-current calculation with FI-method”, IEEE Trans. on Mag,Vol. 35, pp. 1163-1166, 1999.

[12] A. Bossavit, “Computational Electromagnetism”, Academic Press, 1998.

[13] E. Kreyszig, “Advanced engineering mathematics”, John Wiley & Sons, Inc., 1993.

17