IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 5, MAY 2014 7400204

Two-Dimensional Magnetostatic Finite-Element Simulationfor Devices With a Radial Symmetry

Dries Vanoost1,2, Herbert De Gersem2,3, Joan Peuteman1,4, Georges Gielen5, and Davy Pissoort1,5

1ReMI Research Group, Katholieke Universitet Leuven Kulab, Oostende 8400, Belgium2Wave Propagation and Signal Processing Research Group, Katholieke Universitet Leuven Kulak, Kortrijk 8500, Belgium

3Department of Physics and Astronomy, Solid-State Physics and Magnetism,Katholieke Universitet Leuven, Heverlee 3001, Belgium

4Department of Electrical Engineering, Electrical Energy and Computer Architecture,Katholieke Universitet Leuven, Heverlee 3001, Belgium

5Department of Electrical Engineering, Microelectronics and Sensors, Katholieke Universitet Leuven, Heverlee 3001, Belgium

This paper proposes a 2-D magnetostatic finite-element solver for radially symmetric devices, complementary to the standardCartesian and axisymmetric solvers, which are typically used for translatory and cylindrically symmetric configurations. In contrastto the Cartesian and axisymmetric cases, a specific difficulty is encountered due to the particular dependence of the magnetic vectorpotential on the radial coordinate caused by the requirement for radial symmetry. Dedicated finite-element shape functions aredeveloped such that the partition-of-unity property, consistency, and convergence of the formulation are guaranteed. Implementationaspects and modeling pecularities are discussed. The new solver is validated for models in which analytical solutions exist. Themodeling accuracy of the new 2-D solver is compared with a 3-D model for the calculation of the electromotive force and theelectromagnetic torque of a twin-rotor axial-flux permanent magnet synchronous machine.

Index Terms— Convergence of numerical methods, finite-element methods, magnetostatics, partial differential equations, permanentmagnet machines.

I. INTRODUCTION

WHENEVER possible, 3-D magnetic field simulation isavoided in favor of 2-D simulation. This is possible

when the geometry and the boundary conditions feature atranslatory or cylindrical symmetry and the excitation currentsare perpendicular to the symmetry plane, such that the mag-netic flux lines lie in the plane of symmetry. Considering themagnetostatic formulation

∇ × (ν∇ × A) = Js − ∇ × Hs (1)

with A the magnetic vector potential, ν the reluctivity,Js the applied current density and Hs the source magneticfield strength of permanent magnet (PM) material, in theCartesian and axisymmetric cases, Js and A have only z- orθ -components, respectively, whereas Hs and the magnetic fluxdensity B = ∇ × A are confined to the perpendicular xy- orr z-plane. For the Cartesian case, degrees of freedom (DoFs)are defined for the z-component of A. The choice of DoFsfor the cylindrical case is less obvious and was an item ofdiscussion during the early 1990s [14]. Eventually, a con-sensus arose on defining DoFs for 2πr Aθ , where Aθ is theθ -component of A [10]. It is advantageous to solve for fluxlinkages �a = �z Az or �a = 2πr Aθ in unit Wb, because thena generic implementation of the solver and the field-circuitcoupling becomes possible [7], [9].

Two-dimensional Cartesian and axisymmetric solversare standard, both in commercial software and freeware.Reductions to 2-D for arbitrary symmetries have been

Manuscript received February 22, 2013; revised September 5, 2013;accepted November 15, 2013. Date of publication November 25, 2013; dateof current version May 1, 2014. Corresponding author: D. Vanoost (e-mail:dries.vanoost@kuleuven-kulak.be).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2013.2292672

Fig. 1. Reduction of a 3-D model of an axial-flux machine to a number ofcylindrical shells, and further to a single stratified 2-D model plane at whichthe 2-D simulation is carried out.

proposed in [1], [11] and [18]. Reductions for helicalcoordinates are reported in [1], [15] and [18]. A 2-Dreduction of (1) for radial symmetry has not been proposedso far. This is remarkable because applications with a more orless radial symmetry exist (disk motors, cylindrical magneticbrakes, see Fig. 1) and an appropriate 2-D reduction seemsstraightforward [16]. In [2] and [20], axial-flux PM machineswere represented by stacks of thin cylindrical shells (Fig. 1),each represented by a 2-D Cartesian model. This techniquewas organized as a workaround for dealing with radial insteadof Cartesian symmetry.

In case of a symmetric geometry but nonsymmetric excita-tions, the magnetic field is no longer symmetric. Nevertheless,

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7400204 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 5, MAY 2014

when a smooth variation of the field along the direction ofgeometrical symmetry may be assumed, a quasi-2-D solvermay be the method of choice. The variation of the fields alongthe symmetry direction is represented by a small number oforthogonal functions, e.g., by harmonic functions in a studyof machine end windings [3] or by Legendre polynomials ina study of a superconducting magnet system [12]. Couplingsbetween 2-D and 3-D model parts have been proposed in [8].In this paper, for the sake of conciseness, we restrict ourselvesto a single model part featuring radial symmetry and toa linear magnetostatic formulation with isotropic materials.These restrictions can be alleviated easily but would needthe introduction of a heavier notation, which would hide themessage of this paper.

Devices such as axial flux machines and cylindrical mag-netic brakes have a more or less radially symmetric symmetryand carry magnetic flux lines confined in the θz-plane. Then,a promising reduction from 3-D to 2-D simulation consistsof calculating A = (Ar (r, θ, z), 0, 0) in a θz-plane. Fromthis paper, it will become clear that this goal is not easilyobtained. In contrast to the Cartesian and axisymmetric cases,the unknown component Ar (r, θ, z) depends on all threespatial coordinates. Moreover, in analogy with the axisym-metric case, one should be careful when choosing FE shapefunctions. The remainder of this paper is organized as follows.Section II introduces the reduction to a 2-D model, Section IIIdevelops an appropriate discretization, Section IV validates theapproach, Section V address the peculiar behavior for bendingflux, and Section VI discusses an application.

II. REDUCTION TO RADIAL SYMMETRY

A cylindrical coordinate system (r, θ, z) with r the radial,θ the peripheral, and z the axial coordinate is considered(Fig. 1). The model domain V reaches between r1(θ, z) andr2(θ, z) and has an arbitrary shape in the θz-plane. The modeldomain has a reluctivity ν, a current density Js = (h j/r, 0, 0)and a source magnetic field strength Hs = (0, hθ /r, hz), whereν, h j , hθ , and hz only depend on θ and z. The specificdependencies of Js and Hs on r ensure the divergence-freenessof Js and the curl-freeness of Hs . A reference plane Sref atreference radius rref is considered. For modeling conve-nience and visualization, the shell coordinates (rrefθ, z) areused (Fig. 1).

The assumptions allow us to express (1) by

−1

r

∂

∂θ

(ν

r

∂ Ar

∂θ

)− ∂

∂z

(ν∂ Ar

∂z

)= 1

r

(h j − ∂hz

∂θ+ ∂hθ

∂z

)(2)

∂

∂r

(ν

1

r

∂ Ar

∂θ

)= 0 (3)

1

r

∂

∂r

(rν

∂ Ar

∂z

)= 0. (4)

The additional (3) and (4) force the solution Ar (r, θ, z) to havea particular dependence on the radial coordinate r

Ar = r f (θ) + 1

rg(z) (5)

where f (θ) only depends on θ and g(z) only depends on z.The magnetic flux density is then

B =(

0,1

r

∂g

∂z,−∂ f

∂θ

).

Fig. 2. (a) Edge shape function w(e)j (r, θ, z) for element Ve constructed by

extruding the bended triangle Se from r1 to r2. (b) Axial flux turning intoazimuthal flux.

The azimuthal and axial components of B have differentdependencies on r . In a cylindrical setting, flux redistributingbetween θ - and z-direction requires a radial flux component,which is contradictory to the assumption of radial symme-try. Therefore, dealing with a bending flux in a 2-D solverwith radial symmetry necessarily comes together with anapproximation.

III. FINITE-ELEMENT DISCRETIZATION

Formulation (2) needs to be discretized by edge shapefunctions that feature the particular dependence (5) on theradial coordinate r . An edge shape function w j (r, θ, z) isassociated with a radial line through node (θ j , z j ) [Fig. 2(a)].Se is a triangle with straight edges in the θz-plane. An elementVe = Se × [r1, r2] is constructed by extruding the triangle Sefrom r1 to r2 while keeping the θ - and z-coordinate unchanged.In element Ve, w j is expressed by

w(e)j (r, θ, z) = N (e)

j (r, θ, z)

�rer (6)

N (e)j (r, θ, z) = a(e)

j1r + b(e)

j rθ + c(e)j

1r z

2De. (7)

Here, �r = r2 − r1 is the extent of element Ve in radialdirection, N (e)

j is a scalar function associated with node jand linearly decaying toward the other nodes of the elementand a(e)

j , b(e)j , and c(e)

j and De are the coefficients such

that N (e)j (rref , θi , zi ) = δi j . The constructed shape functions

fulfill the following properties.1) Partition-of-unity: The integration of w j along a radial

line through the associated node j yields 1, whereasintegration along a radial line through any other nodeyields 0 [4].

2) Consistency: The set of edge shape functions are ableto represent a homogeneous axial field and an azimuthalfield with dependence 1/r exactly.

3) Convergence: The set of edge shape functions approx-imates a field distribution with a discretization error oforder O(h2), where h denotes the mesh size.

Formulation (2) is discretized by the Ritz–Galerkinapproach, using w j (r, θ, z) both as test and trial functions,

VANOOST et al.: TWO-DIMENSIONAL MAGNETOSTATIC FINITE-ELEMENT SIMULATION 7400204

leading to the system of equations

Kν�a = ��

j s + ��j pm (8)

where

Kν,i j =∫

V(ν∇ × wi ) · (∇ × w j

)dV (9)

��j s,i =

∫V

Js · wi dV (10)

��j pm,i = −

∫V

Hs · (∇ × wi ) dV . (11)

Substituting (6) and (7) and integrating for a single elementVe leads to elementary 3 × 3 matrices K(e)

ν and elementary

3 × 1 vectors��j(e)s and

��j(e)pm to be assembled into Kν ,

��j s and

��j pm by the standard FE procedure

K(e)ν,i j = c(e)

i νc(e)j χ + b(e)

i νb(e)j γ

4De�r(12)

��j(e)s,i = De

3rrefh j (13)

��j(e)pm,i = −hθc(e)

i

2χ + hzb(e)

i

2γ (14)

where χ = (ln r2 − ln r1)/(�rrref), γ = rm/rref , and rm =(r1 + r2)/2 are the element-wise adaptation factors calculatedfor the average inner and outer radii of the considered element.The choice rref = rm substantially simplifies the expressions

for a(e)j , b(e)

j , c(e)j , De, K(e)

ν,i j , and��j(e)pm,i , and is applied in the

implementation. After introducing the boundary conditions,the algebraic system (8) is sparse, symmetric, and positivedefinite and can be solved by a sparse direct solver or by apreconditioned conjugate gradient method [19].

IV. CONSISTENCY AND CONVERGENCE CHECK

The consistency and convergence of the discretizationis verified for four analytical models that have an extent[r1, r2]×[0, θ2]×[0, z2], are filled with homogeneous materialand are submitted to four different excitations (Fig. 3). Theyhave an excitation as follows:

1) a homogeneous axial flux φax exerted by the boundaryconditions �a(0, z) = 0 and �a(θ2, z) = φax;

2) an azimuthal flux φaz exerted by the boundary conditions�a(θ, 0) = 0 and �a(θ, z2) = φaz;

3) an axial flux exerted by a radial current density Jr =(1/r)(Iapp/θ2z2) with Iapp the applied current;

4) a flux exerted by a PM with remanence Br at thereference plane and a magnetization angle ϕ with respectto the azimuthal direction.

The discretization error converges to machine precision forthe models with homogeneous axial and curl-free azimuthalfluxes (Fig. 4). This shows the consistency of the FE shapefunctions [5]. For the other analytic test cases, the convergenceof the discretization error is order O(h2) = O(n−1) withh the mesh size and n the number of DoFs, which is theexpected convergence order for linear FE shape functions [5].These checks validate the particular choice of the FE shapefunctions. Other choices, e.g., with other dependencies on r ,will typically fail in these tests.

Fig. 3. Analytical models for consistency and convergence checks and theirsolutions with (a) impressed axial flux (Ar = −(φax/Sax)rθ ), (b) impressedazimuthal flux (Ar = −((φaz)/(r ln(r2/r1)))(z/z2)), (c) axial flux generatedby a radial current (Ar = ((μIapp)/(θ2z2))(rθ/2)(θ2 − θ)), and (d) PMmaterial (Ar = (rref /r)Br z cos ϕ − Br rθ sin ϕ + α(rθ2 − (z2/r))).

Fig. 4. Convergence of the discretization error of formulation (1),discretization by the radially symmetry edge shape functions (6).

V. BENDING FLUX

Because the azimuthal and axial components of B havedifferent dependences on r , the reduced formulation is notcapable of representing a bended magnetic field. In Fig. 2(b),kz(r) denotes the line flux density (in Wb/m) entering theconsidered volume at a radius r in the axial direction, whereaskθ (r) denotes the line flux density leaving the volume inazimuthal direction. These line flux densities depend on r askz ∼ r and kθ ∼ (1/r). Therefore, when the line flux densityis conserved at a shell at radius ra , i.e., kz(ra) = kθ (ra), thenthis is no longer true for the shell at another radius rb. Thisinconsistency of the continuous formulation (2) is, however,alleviated by the discretization procedure. The DoFs �a attributefluxes to the radial lines perpendicular to the nodes of the2-D model. This guarantees that all flux entering an element Vethrough one of its faces leaves the element through one ofits other faces. Any necessary redistribution of axial fluxcomponents into azimuthal flux components or vice versais possible by radial components inside Ve experiencing azero reluctivity in the radial direction. The fourth analyticalexample deals with a bending flux and shows a second-orderconvergence for the discretization error.

VI. APPLICATION

The 2-D FE solver for radially symmetric models is appliedto calculate the performance of a three-phase twin-rotor

7400204 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 5, MAY 2014

Fig. 5. Electromotive force and electromagnetic torque generated by thetwin-rotor axial-flux PM synchronous machine at 3000 r/m.

Fig. 6. Magnetic flux lines at a θ z-shell at r = rref .

axial-flux PM synchronous machine (Fig. 1) [17]. The sta-tor consists of 12 coils wound around six iron yoke partsattached to a back-iron disk. The rotors, one at the frontand the other at the back of the stator, each consist oftwo PMs attached to each other one in the middle of themachine with a nonmagnetic material. The two rotors areπ radians shifted such that the opposite magnetic polesare across each other. A 2-D model with 6800 DoFs isconstructed using finite element method magnetics [13] andsimulated by the newly developed FE solver implemented inMATLAB. A reference 3-D model with 200 000 DoFs is con-structed and simulated using CST EMStudio [6]. The no-loadelectromotive force is postprocessed from the solution for �a.The electromagnetic torque is derived from the magnetic coen-ergy. The 2-D reduction introduces severe approximations:1) flux fringing at the inner and outer ends is neglected, as it isthe case for any 2-D model; 2) PM cubes are approximated byring segments (with the same remanent flux; and 3) bendingmagnetic fluxes are assumed to redistribute along radial pathswith zero reluctivity inside the FEs. Nevertheless, the 2-D solver for radially symmetric models achieves acceptableresults (Fig. 5). The 2-D results for the electromotive forceand the electromagnetic torque differ by 5.50% and 8.50%,respectively, from their 3-D counterparts. These differencesare fully attributed to the 2-D modeling assumptions. The2-D solver is valuable for fast determination of major machineparameters during initial design steps. The magnetic flux linesare visualized on the θz-shell at r = rref (Fig. 6).

VII. CONCLUSION

The components of a radially symmetric magnetic flux den-sity have specific dependencies on the radial coordinate. A 2-Dformulation in terms of the radial component of the magneticvector potential needs to be discretized using edge shapefunctions that support these dependencies. At the continuouslevel, the 2-D reduction does not allow exchange of axial andazimuthal flux components. In the discrete setting, however,bending fluxes becomes possible because a radial field com-ponent may exist at the inner element level. Such particular

behavior is not encountered for Cartesian and axisymmetric2-D reductions. Two-dimensional simulations for a three-phasetwin-rotor axial-flux PM synchronous machine give accurateresults.

ACKNOWLEDGMENT

This work was supported in part by the Agency forInnovation by Science and Technology in Flanders and inpart by the Electromagnetic field simulation for future particleaccelerators under Grant KUL_3E100118.

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