Virtual work approach to the computation of magnetic force distribution from finite element field solutions

A.Benhama, A.C.Williamson and A.B.J.Reece

Abstract: A method for the computation of magnetic force distribution from finite element (FE) field solutions is proposed, based on the principle of virtual work. 111 this method, a force is derived for each nodc by direct differentiation of the stored magnctic energy or co-energy or the finite clcmcnts surrounding the node with respect to tlic virtual displaccmcnt while keeping the magnetic potential conslant. The force distribution around the node is then obtained by spreading the ‘nodal’ forcc over an appropriate ‘touching’ area or volume of the clcmcnts surrounding the node. This method can be easily implemcnled in 2D and 3D FE field calculation programs. Numerical cvaluation for both linear and nonlinear ficld problems in two dimensions and (by implication from an axisymmctric treatment) in three dimensions shows that the results computed by the proposed heuristic method are in good agreement with analytical solutions and experiincntal results.

1 Introduction

Thcre are two forcc quantities of particular interest to dcsigncrs of electromcclianical devices: net forcc and force distribution. Thc net force acting on a rigid body can be computed from field solutions cithcr by integration of the Maxwcll stress or by the virtual work principlc and both methods can be used to compute the net forcc on ferro- magnetic bodies as well as current-carrying structures. Experience with such application to tinitc clement formula- tions is described in [l].

Local surl:ace force iiitctisities arc usually calculated by the Maxwell stress tensor (MST) method, but tlierc are problems in the case of solutions obtained using the finilc element (FE) method or field calculation. The langcnlial and the normal Maxwell strcsscs on a surrice both depend on the tangential component of the magnetic ficld H, and the normal flux density B,, at this surracc. A problcm asso- ciated with the discretisation necessary in FE methods is the inevitable introduction of discontinuitics in field quanti- tics which should he continuous. If the magnetic scalar potential formulation is tiscd, the normal llux density, ,!It,,, will not he continuous across the surfacc between elements and iT the magnetic vector potential is used, the tangential field, H,, will not be continuous. When determining the rorce density distribution on an air-iron interkice, one has to evaluate the Maxwcll stresses on thc common borders of two finite elements and the user is faced with the problcm of which value of B,, or If, to choose. Wignall et U/ . [2] suggested an approach using the magnetic vector potential, which involved a weighted average of the air and iron tangential components of the magnetic field intensity where

0 IRE, 2000 IEI; P I Y J ~ ~ / ~ I I ~ . s online 110. 2CUW724 DOL IO. I M9/ipcpa:2oooO724 l’apcr first received I6th I’chruaiy and in revised form 4111 .luly 2000 A. Bcnh;oii;i is will1 Brook Croiiipton, SI Thoinas’ Road, Huddcrsficld, Wcsl Yol-kshirc. HDI 3U, UI< A.C. Williainson and A.B.J. Reccc are wilh the Dcpartiiient ol‘ Elcc1l.ical Ungi- necring and Electronics, UMIST, PO Box 88, Manchester, M60 IQD, UK

the reelalive permeabilitics of air and iron were used as the weighting factor. Although thcrc is some evidence that this works satisfactorily for 2D problems its application to 3D problems does not appear to have been succcssfully imple- mented.

A ncw method of local rorce coinputation was recently proposed by Bossavits [3] who applied the virtual work principle lo edge elemcnl rormulations of the ckxlromag- netic equations. The application and numcrical valiclation of this method have been reporlcd [4]. However, the vast majority of available FE clcctromagnetic packages are based on convcntional finite elements, and Bossavits’ con- tribution, though important, is not helpful to many currcnt FE users. The present paper describes a method of calculat- ing s u r f ~ c force intensities which is applicable to conven- tional FE formulations. The method was derived somewhat intuitivcly from earlier work by the authors aimed at producing algorithms for the Coulomb virtual woi-IC (CVW) method of calculating total force, originally proposed by Coulomb [5] and dcscribed and validated in [l]. Unlike the CVW method where a set of nodes sur- rounding an object arc simultaneously displaced en bloc, the proposed method displaces a single node at a time. A local forcc, associated with the displaced node, is obtained a s tlic analytical differential of the energy or co-encrgy of the elements surrounding tlic nodc, with respect to the vir- tual displaccmcnt while keeping the magnetic potential con- stant. A nodal forcc implies infinite stress, but in the heuristic approach proposed in this paper rcalistic local stresses arc oblained by distributing the nodal force uni- formly over an appropriate adjacent surface. The method is general and easy to implemcnt in F E electromagnetic pro- grams. It can be applied to 2D or 3D linear or nonlinear magnetic problcnis. It will be callcd the local virtual work (LVW) method.

2 Local virtual work theory

The interface between two different media such as air and iron is shown for a 2D system in Fig. I, together with the F E discretisation. Considcr the portion of the interface

437

defined by nodcs FKD. To determine the local force FFt/ associated with node K, the node is virtually displaccd in the g-direction by an increment &I, as illustlatcd in Fig. 1, while the neighbouring nodes remain fixed. When the inag- netic vector potcntial formulation is used, the force FK,/ associated with node K can be calculated as thc derivative of the stored magnetic cncrgy with respect to tlie virtual displacement at constant flux linkage, or the dcrivative of the magnetic co-encrgy with rcspcct to thc virtual displace- ment at constant currcnt. The energy method is chosen hcre, and tlie local force FKT, is derived by differciitiating the niagiictic stored energy with respect to the direction q. with flux linltagcs held constant, i.e.

Fig. 1 Displwed node 111 on I I ; ~ - ; ~ ; i iwf i iw

The stored magnetic energy W is given by [5]

w = / [!..dB] d V (2) V

where I.‘ is thc volumc of the field region, B is the flux density and fl is the niagnctic ficld intensity. I n the FE formulation the domain V is divided into a set of finite elements, and the total cncrgy W is obtained by adding the energy contribution of each element. Therefore the total energy W becomes

where M is the total number of finitc clcinciits in tlic field region and v, is the volume of eleiiient ‘e’. Whcn using first-order elements, eqn. 3 simplifies to

(4)

whcrc vc thc reluctivity of element e is expressed in terms of the flux density squared, i.c.

I / , = U ? ( P ) (5) Substituting eqn. 4 into eqn. 1 and dirferentiating with respect to the virtual displaccmeuts yiclds the cxpression for the local forcc FKq associatcd with node K

N V, 2 a u , aB2 I / , + --R ~ _ _ Fzc, = - 5 [:% 2 d B 2 dq

It should bc noted that in this equation the summation is only over the N elements directly conncctcd to node K,

438

since tlie energy associated with the remaining elcmcnts is unaffected by the displacement.

The teim dv,/dB* is easily obtainable for cach nonlinear fiinilc element, aiid is already available when the Newton- Raphson algorithm is used to solve the nonlinear system of equations rcsulting from tlic FE formulation. The deriva- tives dB2/dq and dV,/dq rcquirc the knowledge of the co- ordinate derivatives of thc nodcs of the clcincnts attachcd to the displaced nodc. Section 8 details the FE implcmcnta- tioii of the local virtual mcthod for 2D Cartesian and axisynimetric problems. Thc formulation for thc 3D cases is obtaincd with exactly the Same philosophy but with a complexity which, it is bclicvcd, would serve only to cloucl tlie issue at this stage.

3

Thc proposed LVW mcthod can be used to calculate not only the local forces but also the global forces by summing the nodal forccs over an appropriate region. Another potentially attractive feature of thc LVW iiicthod is that the nodal forces can be input dircctly into a mcclianical FE package to compute the resulting displacements. T o make the nodal forces more nicaiiingliil to designers of clectro- inechanical devices, they can be translated into stresses. For inslance, the force dciisity or stress 011 the interface FKD in Fig. 1 inay be obtained by dividing the nodal force FKc/ by a sensibly chosen area surrounding the node.

For 2D problems, it is proposed that the force dcnsity ociated with the area around a node is obtained by

assuming the nodal force per unit length normal to the cross-section to be distributed uniformly over a length along the surface equal to (FK + KD)/2.0.

For 2D axisymnietric problems thc force dcnsity associ- ated with the area around a nodc is obtained by distribut- ing thc nodal force computed for the full circlc Lmiforinly over a peripheral surrace equal to mK(F1< + KD), whcrc rK is the radius at node K.

For 3D problems it is proposed that a nodal force is translated into a stress by assuming it is uniforinly distrib- uted over one-third or tlic total area of the surfkes of the cleinents attached to the node.

4

To validate the results of the proposed (LVW) method, three numerical examples were considered. Thc first is a lin- ear case described in [6]. It consists of a hollow magnetic sphere of specified permeability placcd in the magnetic field of a currcnt in a circular coil. The normal surface force density on thc spherc is known analytically, so peimitting a check 011 the results of the LVW mcthod. Thc second is tlie severely nonliiicar casc of an axisyinmetrical actuator on which the net magnetic forcc acting on its moving part has been measured. The LVW incthod was applied to deter- mine both force distribution and, by summation, thc net force exerled on the moving part of the actuator. The force distribution is compared to thc results obtained by applica- tion of the MST. Thc third cxample is a switched reluc- tance motor, where the LVW and MST methods arc used to calculate the normal and tangential force densities acting on the stator pole of thc cxcitcd phase.

4. I Problem 1: Magnetic levitation of a hollow sphere This problem is described in [6] and consists of a hollow ferromagnetic sphere placed in a magnetic field created by a current flowing through a circular coil. The gcoinetry and

Global force and force density calculation

Validation of the LVW method

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the dimensions or this problem arc shown in Fig. 2. Fig. 3 shows tlic computcd field distribution. The objcct is to deteimine tlie normal force density on the surfxe of tlie sphere by the proposed LVW method and to comparc thcsc results with those calculated analytically.

1

I F coil M-l

bution is compared with the analytical rcsulls in Fig. 4. Thc authors consider this close agreement to be a convincing demonstration of thc potential of the proposed method.

[ I , = 50.0 I = 200A

Fig. 3 %K d~roil~uiiuii

Reference [6] determines analytically the normal force dcnsity on the sphere by the MST method using the follow- ing equation:

where and p arc the permeabilities of air and iron, respectively. The results obtained from h i s equation arc shown in Fig. 4.

In the FE model, axisyninietry allowed the problcm to he discretised in the (TZ) plane and the magnetic vector potential formulation was used. The adequacy of thc number of nodes in thc FE mesh was judgcd by using the criterion of action and reaction balance, i s . the nct force exertcd on the sphere must he equal and opposite to thc nct force exertcd on the coil. Hence, lhe FE mesh was refined until the action and reaction forces werc closely equal in magnitude. The nct force excrtcd on the coil was calculatcd by both the Lorcntz and the LVW methods. Having deter- mined satisfactory discretisation, the local force calculation program was used to determine the distribution of tlie nor- mal force dcnsity on the surface of the sphere. This dis1l.i-

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nyioii support fw MS moving

z axis

I /cylinder

4.2 Problem 2: Experimental actuator This actuator was used in [l] which describes the calcula- tion of forccs by the CVW method. It has axisymnictric geometry with the radial cross-scction shown in Fig. 5. The moving part consists of a cylindrical liner of nyloil which carries a mild steel cylinder wliich is acted upon by electro- magnetic forces. The ncl rorce acting on the cylinder was measured by applying a known mass to the cylinder and then increasing thc coil currcnt slowly until the balance between thc magnetic rorce and the gravitational force was obtained. Mapct ic vector potentials wcrc calculated using a 2D axisymmetric model. It was found [I] that the use of the layer of air elements closest to the stecl surface in the CVW method gave the least accurate net force compared to more reniotc layers. Thercfore, to improve the accuracy of force intensity calculations, thc net force was computed from the LVW results, and if it differcd from the net forcc computed by tlic CVW nicthod with a rcmote distorted layer of air elements, the mesh was rcfined until the agrce- ment was adequate.

Fig. 6 gives a flux plot and Fig. 7 compares the net axial forces computed by summing the nodal forces given by the LVW method and their measured values in tcrnis of the applicd current for a given position of thc moving cylinder.

430

10.0 ~

7.5 ~

z m 2 +

5.0 -

2.5 ~

I-

0.000 0.005 0.010 0015 0.020 0.025 -0.5

path ABCDA

Fig. 8 ~~ ~ from axisyminetric MSI'

from axisymmetric LVW

Norinul ,suifiice,{brcr ilwsiiy 012 uutw ,sui:/Uce u /~y~ l i nde r

Fig. 8 shows the normal force density distribution deter- mined from the axisymmetric LVW method on the outer surface of the moving cylinder. The results were compared with the those given by the MST niethod using the normal and tmgential flux densities associated with air elements. Fig. 8 indicates that the results of the LVW and MST methods are in a very good agreement in regions of high permeability. In these regions the MST method is accuratc because the inaccurate and troublesome tangential compo- nent of the flux density is vcry small comparcd to its nor- mal component.

440

4.3 Problem 3: Switched reluctance motor This motor, which also was used as a tesl vehicle in [l], is a switchcd reluctance motor with six poles on the stator and b u r poles on the rotor. After carrying out a 2D FE analy- sis using the magnetic vector potential Formulation, the LVW method was applied lo a stator pole of thc excited phase to dctcrminc both thc normal and tangential force dcnsitics on tlic surfticc of the pole. One relative position of stator and rotor is illustrated in Fig. 9, which also givcs thc cornputcd flux distribution. As in the previous problem, the results obtaincd wcrc compared to those given by the MST. Figs. 10 and 11 compare alternatively dcrivcd values of thc normal and the kmgential force densities, respectively. Fig. I O indicates that relatively good agreement exists

1.5

between the norinal force densities obtained by the LVW and the MST methods. As in thc previous case, agrccment is better in high pcrmeability rcgions. It should hc noted that the values computed from the Maxwell stress, espe- cially the tangential stress, can be expected to be in error because the clement adjacent to the iron ail- interfxe is tlie most critical [I] .

5 Conclusions

A incthod of calculating the electromagnetic force distribu- tion based on the virtual work principle and the use of con- vcntioiial finite clcmcnts has bcen presented. This method can be applied to 2D and 3D linear and nonlincar field problems, and is easy to implement in FE electromagnetic programs.

Although the method is heuristic, it avoids problems aris- ing with the use of Maxwell stress on iron boundaries, where the Tailure of the FE method to obey both continuity conditions can lead to some uncertainty in the result.

Validation of a forcc distribution can only be indircct, but thc results prcsented in this paper, for three very differ- ent problems, show encouragingly good agreement with valucs obtained by the MST method (here using the air val- ues of I f i ) and with total measured force for the case where test results are availablc.

6 Acknowledgments

The authors would liltc to thank the ALSTOM Research and Technology Centre and EPSRC for supporting the work.

References

BENHAMA, A., WILLIAMSON, A.C., ;ind REECC, A.B.J.: ‘I:orcc and torque conipuliilion from 2 - 0 ;ind 3-D finite element lield solti-

RT, A H . , and YANC, S.I.: ‘Calculation of forces on magnetised ferrous cores using l l i e Miixwcll stress

, MAG24,(I ) , pp.459462 magneioslatics and their computalion’, .I

Appi !’/ijx, 1990, 67, (9). pp. 5812-5814 REN, Z., iind RAZEK, A.: ‘I forcc compulation in dclbrmablc hodics using edge clcincnts’. Twm, 1992. MAG-ZX, (2) , pp. 1212-1215 COULOMB, J.L.: ‘A incthodology for dclcrmination 01’ global elcc- tromcchanical quantiiies from ii finite clemcnt analysis and iis applica- tion to ihe cvaluaiion of maenctic forces. Li>raucs. and slifliiesr’. IEEE

I ’W 6!!J/l/., 1999, 146, (I), p13. 25-31

7iwn.s.. 1984, MAG-30, ( S ) , Fp 2514 2519 ’ ’

RDN, Z.: ‘Compwison if different forcc calculiltion mcihods in 3D f ini te clement modelling’, JEKC Tr~iirs., 1994, MAG-30, (3, pp. 3471- 1474 . , . I I

BRAUCK, J.R.: ‘Finite elcinciil xnalysis of clcclromiignctic induction in transforincrs’. Proccedings o1‘lEEH P I 3 inccting, 1977, pp. 122-125

Appendix

8. I 20 Cartesian FE implementation using the magnetic vector potential In 2D FE analysis the domain of tlie problem is dividcd LIP into triangular clcments, and an approximate solution for the magnetic potential is assumcd in each element such that its valuc is a linear interpolatc of its values at the clement nodcs. This is concisely expressed by

%=I

whcrc q(x, y ) arc first-orcicr functions given by

wlierc n l = r;:21/3 -y2:c3 a:( = sl?/z-y[ ,cz

CI = .z:j - 2 2 c 2 = 2 1 -sL’g c 3 = x2-21

U1 = z 3 ? / 1 -y:j21

01 = ?/a-?/,? 0 2 = 113 -?/I h3 = y1-?/2

n = -(OtC2 - h l C , ) 1 2

(10) the co-ordinates (xl, vl), (x2, yz), (x3, y3) represent the nodal co-ordinatcs of each triangular finite element. The volume V,. of element ‘e’ is given by the following detenni- nant:

(11)

v, = -

where L is the axial length of the field problem under con- sideration.

Differentiation of thc volume V , with respect to the virtual displacement q yields

The flux density vector oblained from the equation B = curlA, is given by

from which the flux density displacement derivatives can be easily calculated

Now the s and y components of the nodal force are given by

+-vu,- ( E n ) c32 B21 2

8.2 Axisymmetric FE implementation using the magnetic vector potential In this case, the domain ofthc problem is broken up into a set oT rings, each ring having a triangular cross-section area QC that has been revolved around the z-axis. The magnetic potential in the (U) planc is approximated in the same way as Tor thc 2D case, with r and z rcplacing x and y , respec- tively.

44 I

n A(r, 2) = q ( r , z )A , (16)

2=1

the flux density vector B is given by

hi the FE analysis, thc triangular elements are sufficiently sinall so that the axial component B2 of the flux density a t the centroid can be taken as representative for the triangle as a whole [7].

where the centroidal values A, and I^, are given by

thc dcrivativcs of B2 with rcspcct to I’ and z are given by

The axial and the radial nodal forces associated with node k arc given by

(2%)

442