Some two-dimensional multiscale nite element formulations

ASC Report No. 21/2017

Some two-dimensional multiscale finiteelement formulations for the eddy currentproblem in iron laminates

K. Hollaus, and J. Schoberl

Institute for Analysis and Scientific Computing

Vienna University of Technology — TU Wien

www.asc.tuwien.ac.at ISBN 978-3-902627-00-1

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Some Two-Dimensional Multiscale Finite ElementFormulations for the Eddy Current Problem in Iron

LaminatesK. Hollaus, and J. Schoberl

Institute for Analysis and Scientific Computing, Vienna University of Technology,Wiedner Hauptstraße 8-10, A-1040 Wien, Austria, [email protected]

Abstract—The aim of this work is to introduce and to study theperformance of some multiscale finite element formulations forthe eddy current problem in laminated iron in two dimensions.The case of the main magnetic flux parallel to the laminates andperpendicular to the plane of projection is considered. Multiscaleapproaches based on the magnetic vector potential (MVP), thesingle component current vector potential (SCCVP) and on amixed formulation are presented. An approach for a multiscaleformulation (MSF) with the MVP is constructed at the bestby examining and representing the eddy current distribution inlaminated iron of a reference solution. The associated weak formof the multiscale finite element method (MSFEM) is presented.Similarly to the MVP the SCCVP and a mixed formulation withthe MVP and the current density are studied.This work does not present a mathematical analysis of MSFEMfor eddy currents in laminates. The performance of MSFEMis studied by numerous numerical experiments of differentexamples. The paper covers the topics edge effect, averaging ofcoefficients and p-refinement of multiscale shape-functions andof standard finite element polynomials (SFEPs) and stability ofMSFEM in a particular case.Simulations show the capability of MSFEM to efficiently andaccurately approximate eddy currents in iron laminates. Thesimulations shall also provide a collection of numerical examplesto evaluate other multiscale or homogenization methods.

Index Terms—Eddy currents in 2D, edge effect, generalizedfinite element method GFEM, laminated iron cores, micro-shapefunction, multiscale formulation MSF, multiscale finite elementmethod MSFEM, p-refinement, linear dependence.

I. Introduction

The discretization of the micro-structure of laminated ironcores i.e. of each laminate by finite elements (FEs) would leadto prohibitively large systems of equations, whose solution isfar away from being a routine task for engineers in the designof electrical devices with modern computer power or evenimpossible. To overcome this unpleasant fact homogenizationand multiscale methods have been developed, which requirebasically the solution of the large scale problem.

A homogenization method has been proposed in [1] withaveraging the material properties for static magnetic fields inlaminates in two dimensions.Homogenization methods based on a truncated asymptoticexpansion as an approach for the solution of the eddy currentproblem (ECP) lead to an average value and corrector terms,see for instance [2] or [3]. An asymptotic expansion with theMVP A was used in [4]. To improve the local approximation

first and second order corrector terms were determined in [5]solving associated cell problems for the SCCVP T in twodimensions.A multiscale method has been presented in [6]. The mag-netic flux density parallel to the lamination is expandedinto orthogonal even polynomials, so-called orthogonal skineffect sub-basis functions, to improve the local approximation.The arising coefficients are averaged and the edge effect isneglected.

The present paper deals with multiscale finite elementmethods (MSFEMs) and possibly applies averaging of the co-efficients. Some two-dimensional multiscale finite element for-mulations are introduced and analyzed. Simulations of smalland simple numerical examples and of a more demanding largeand nonlinear one are carried out to study the performance ofthe proposed MSFEMs with respect to different parameters,for example, the penetration depth. Solutions with MSFEMare compared with reference solutions (RS) [7] where eachlaminate is modeled by FEs.

Section II gives an introduction to MSFEM in the contextof eddy currents in laminated media. The boundary valueproblem of the eddy current problem to be solved is describedin Sec. III. Details of the small numerical examples aresummarized in Sec. IV. All topics mentioned in the followingare supplemented by numerical experiments. The multiscalefinite element formulation with A is presented in Sec. V. Aphysically motivated explanation of the approach for a MSFis given by the eddy current distribution in laminates. Howto construct a MSF by means of a RS for A is discussedin detail. Coefficients haven’t been averaged for the sake ofaccuracy, except in Sec. V-E and V-F where the influence ofaveraging is investigated and the possibility to use triangularmeshes is demonstrated, respectively. The behavior of MS-FEM concerning p-refinement of the micro-shape functions(MSFs), the power of higher order MSFEM (HMSFEM), isstudied in sections V-G. Stability of MSFEM with respectto the degree of SFEPs in a particular case is addressed inSec. V-H. A large and nonlinear problem has been studied todemonstrate the ability of the MSFEM to cope with nonlinearproblems and show the benefit compared with the standardfinite element method (SFEM) in Sec. VI. MSFEM for theSCCVP T is shown in Sec. VII. The behavior of MSFEMconcerning p-refinement of the SFEP bases [8] is studied in

Figure 1: Transformer core with large scale dimensions L etc.

Figure 2: Small scale dimensions, d+d0 represents one periode,micro-shape functions φi.

VII-E. The improvement of the accuracy increasing the degreeof the SFEPs is investigated. Attention is paid to the edgeeffect in VII-F. Finally, a mixed MSF using the A and thecurrent density J is introduced in Sec. VIII.

II. Multiscale Finite ElementMethodMSFEM

The problem we are confronted with exhibits two differentscales. A large scale characterized by large scale dimensions,for instance the length L, the height H, the width W and thedimensions of the windows, E and F, of a transformer coreshown in Fig. 1. The thickness d of the laminates, the widthof the air gap d0 in between the laminates, see Fig. 2, andthe penetration depth δ are dimensions at the small scale or atthe micro-scale. The ratio of the different scales is extremelylarge, about 105 up to 106.

Mapped polynomials are used by the standard finite elementmethod (SFEM) to approximate the unknown solution. TheSFEM performs well as long as the solution or the coefficientsin the equations are smooth. However, to obtain an accurateapproximation also in case of equations with rough coeffi-cients, for instance materials with a micro-structure (laminatediron core) or for problems with singularities like boundarylayers, extremely fine meshes are required [9]. This is thereason for the prohibitively large equation systems which mayrequire exorbitant amounts of computer resources to obtain anaccurate solution [10]. This work addresses solely the eddycurrent problem in 2D with rough coefficients because of alaminated iron core.

To avoid large algebraic equation systems the generalizedfinite element method (GFEM) as a general framework forequations with rough coefficients or for problems with singu-larities seems to be a very promising option [9], [10]. Apartfrom the finite dimensions laminated iron cores represent aproblem with a periodic micro-structure. The SFEP basis isaugmented by special functions including a priori informationinto the ansatz space

uh(x) =

n∑i=1

m∑j=1

ui jϕi(x)φ j(x) =

n∑i=1

m∑j=1

ui jψi j(x), (1)

where n is the number of the SFEPs ϕi, m is the numberof special functions φ j and ui j are the coefficients of theapproximate solution uh. The special functions, which arecustom tailored ansatz functions, may stem from an analyticsolution or, for example, from a FE solution of a basicproblem, i. e. these functions are known. The local basis ofthe special functions approximates well the solution locally.Micro-shape functions φ j represent a local space and standardpolynomials ϕi(x) a global space. Multiplication of standardpolynomials ϕi with special functions φ j yields the new basisfunctions ψi j.In the present context of micro- and macro-scale we call themethod MSFEM and the special functions φ j the micro-shapefunctions, see Fig. 2.

III. Eddy Current ProblemThe simplified eddy current problem (ECP) in two dimen-

sions to be solved in the present work is shown in Fig. 3. Theentire domain Ω = Ωm∪Ω0 consists of the laminated mediumΩm and air Ω0, Ωm comprises laminates Ωc, i.e. the conductingdomain, and air gaps in between. Eddy currents represented bythe current density J occur in Ωc. The interface Γm0 separatesΩm from Ω0. The outer boundary of Ω is denoted by Γ and nis the unit normal vector pointing out of Ω. The magneticflux density B is assumed to be parallel to the laminatesand perpendicular to the plane of projection. The materialparameters are the magnetic permeability µ(B), which can benonlinear, and the electric conductivity σ. Relations (2) to (6)are valid in Ωc, whereas (7) to (9) belong to air Ω0 includingthe air gaps in Ωm and (10, 11) are possible

curl H = J in Ωm (2)curl E = − ∂

∂t B (3)div B = 0 (4)

J = σE (5)B = µH (6)

curl H = J0 in Ω0 (7)div B = 0 (8)

B = µH (9)

H × n = K on ΓH (10)B · n = b on ΓB (11)

2

boundary conditions on Γ. The boundary value problem BVP(2) to (11) considers Dirichlet boundary conditions only. TheBVP of the ECP in the frequency domain is straightforwardand thus not shown here.

Figure 3: Problem in 2D.

IV. Numerical Simulations

A. Relative errors in percentage

To evaluate the performance of MSFEM with respect to forinstance the penetration depth δ relative errors of the potentialu, the flux density curl u and the eddy current losses P,

‖uMS FEM−uRS ‖L2‖uRS ‖L2

· 100%,‖curl (uMS FEM )−curl (uRS )‖L2

‖curl (uRS )‖L2· 100%

and PMS FEM−PRSPRS

· 100%, (12)

respectively, are computed by means of a RS and that ofMSFEM. Single laminates are modeled by FEs for the RS.Errors (12) are evaluated in Ωc, i.e. iron laminates, only. Thefrequency was changed to get different δ.

B. Finite element models

A small example has been deliberately chosen to demon-strate also errors which would not be visible in large problemsby means of (12). All data are given so that the simulationresults can be verified. The problem is symmetric with respectto both axes, x and y, assuming the origin in the center of theproblem domain, compare with Fig. 3. The dimensions of thedomains are |Ωm| = 20 × 20 mm2 and |Ω| = 40 × 40 mm2.The FE models shown in Figs. 4 for the RS and for MSFEMconsider 10 laminates, d = 1.8mm, and air gaps in between,d0 = 0.2mm. For meaningful errors (12) the discretization iny-direction is the same for both FE models. The dimensionsof the FE models in Figs. 4 are summarized in Table I.The fill factor f f = d

d+d0·100 of 90% is valid for all examples.

In case of averaging d + d0 = 0.5 mm was selected. Most ofthe simulations use the models with dimensions summarizedin Tab. I.

V. Multiscale Finite Element Formulation with A

In brief, the time domain is only presented here. In caseof linear material properties the problem is formulated in thefrequency domain with the phasor convention e jωt.

Figure 4: FE models for a) RS (left) and b) MSFEM (right).

Table I: Dimensions of the FE Models in mm a

M x, RS 9 1 0.1 1.8 0.2 ... b

M x, MSFEM 9 1 2 8

M y 9 1 0.5 1.5 8

a Due to the symmetry only one half of the dimensions in x- and y-directionare presented.b After 10 laminates with air gaps in between the dimensions repeat accord-ingly to obtain a symmetric problem.

A. Boundary value problem with AThe magnetic vector potential (MVP) A is introduced by

B = curl A with

A : Ω ⊂ R2 → K2, (x, y) 7→ (A1(x, y), A2(x, y))T (13)

curl A : Ω ⊂ R2 → K, (x, y) 7→(∂A2

∂x−∂A1

∂y

)(x, y) (14)

whereK stands either for real numbersR or complex numbersC. Mapping to R or C belongs to the time domain orto the frequency domain, respectively. The valid domain isindicated at the given case. Considering the relevant Maxwell’sequations leads to the BVP of the ECP in the time domain

curl1

µ(A)curl A + σ

∂

∂tA = 0 in Ω ⊂ R2, (15)

A × n = α on Γ. (16)

B. Weak form with AThe weak form of the SFEM belonging to the time domain

reads:Find Ah ∈ Vh,α := Ah ∈ Vh : Ah × n = αh on Γ, such that∫

Ω

1µ(Ah)

curl Ah · curl vh dΩ +∂

∂t

∫Ω

σAh · vh dΩ = 0 (17)

for all vh ∈ Vh,0, where Vh ⊂ H(curl,Ω). Index h indicates FEdiscretization. The solution of (17) serves as RS to evaluatethe accuracy of MSFEMs.Since σ = 0 in air a regularization by a penalty term [11]was applied to get a unique solution of (17).

C. An approach for the MSF with AThere is no rigorous mathematical rule to construct an

approach for a MSF. Therefore, in the context of a MVP Athe eddy current distribution of a RS in a laminated medium is

3

studied to devise an approach for the MSF. Figure 5 shows adetail of the RS with eddy currents consisting of a laminar part,currents are flowing parallel to the laminates, up and down iny-direction, and a second part at the end of the laminates,where the eddy currents turn around flowing in x-direction tobuilt closed loops. The later part or this fact is frequently calledthe ”edge effect” in the literature and sometimes neglected[12]. Note, that nL points into the x-direction as shown inFig. 3.

Figure 5: Eddy currents in laminates, detail.

Particular attention is paid to an accurate representation ofthe edge effect by the MSF in the present work. A separateterm considers the edge effect in MSF (18). The first term A0in (18) takes into account the smooth variations in the solutiondue to the large scale dimensions and the second and third termconsider the solution due to the micro-structure. The secondterm is a vector with the component A1 in the y-directionmodeling the laminar part of the eddy currents and the thirdterm with (w1φ1) considers the edge effect. The tilde in (18)indicates MSF. The zig-zag function φ1 is the micro-shapefunction which fits to the periodic micro-structure as depictedin Fig. 2. Maximum and minimum are selected arbitrarily, thechoice 1 and −1 is convenient. The micro-shape function isperiodic, continuous across the lamination and known. Theselection of φ1 stems from the linear approximation of theanalytic solution of eddy currents in an infinite slab where Bis parallel to the slab with the MVP. Without the third term in(18) for the edge effect the associated MSFEM does not workat all.

A = A0 + φ1

(0A1

)+ ∇(φ1w1) (18)

The support of the MSF (18) is depicted in Fig. 6, A0 is validin the entire domain Ω, whereas A1 and w1 are restricted tothe laminated domain Ωm.

Figure 6: Support, boundary and interface conditions.

D. Weak form of MSFEM with ASimply speaking replacing A in the weak form (17) of the

primary formulation by MSA (18) yields the weak form ofMSFEM with A:Find (A0h, A1h, w1h) ∈ Vh,α := (A0h, A1h, w1h) : A0h ∈ Uh,A1h ∈ Vh , w1h ∈ Wh and A0h × n = αh on Γ, such that∫

Ω

1µ

curl Ah · curl vh dΩ +∂

∂t

∫Ω

σAh · vh dΩ = 0 (19)

for all (v0h, v1h, q1h) ∈ Vh,0, where Uh is a FE subspace ofH(curl,Ω), Vh of L2(Ωm), Wh of H1(Ωm) and φ1 is in thespace of periodic and continuous functions H1

per(Ωm). TheMSFEM with A1 ∈ H1(Ωm) is analyzed in Sec. V-F.Dirichlet boundary conditions are prescribed for A0h on Γ andhomogenous natural boundary conditions hold for w1 and forA1 in Γm0.The FE approximation of the MSF (18) is consistent with theH(curl) space. A FE subspace of H(curl,Ω) is selected forthe first term A0h. The x-dimension of the rectangular FEs isassumed to be an integer multiple of the period p = d +d0 andφ1 = 0 along the relevant FE boundaries. This means that thesecond term is tangentially continuous across the laminates.The third term is a gradient of continuous functions. Thus,the latter two belong also to FE subspaces of H(curl,Ωm).Solutions of A0h, A1h and w1h are very smooth as can be seenin Figs. 43 to 44, in Appendix B. Therefore, a rather coarseFE mesh and thus much less unknowns suffice for an accurateapproximation of the solutions of MSFEM, see Figs. 4 b) or23 b). This fact explains the big advantage of MSFEM overSFEM and the RS.

E. Exact integration versus averaging of the coefficients

The ECP in the frequency domain with the complex unitj and the angular frequency ω is considered in this section.The bilinear forms in (19), the curl of gradients vanishes, arewritten in detail in (20) in Appendix A. Two different nu-merical techniques to compute the FE matrices are compared,averaging and exact integration, see also [6], [12], [7], [13]and [14].

Averaging of coefficients λ in (19) over the period p = d+d0

λ =1p

∫ p

0λ(x)dx, (21)

where λ stands for example for 1µ, jωσ, jωσ∇φ1, jωσφ1,

etc., with φ1x := ∇φ1, in (20) in Appendix A, leads to (22) inAppendix A. The averaged coefficients are constant throughoutΩm. Bilinear forms with averaged coefficients and in turn theweak form are modified. Therefore, all quantities are markedwith a bar. Then, the usual Gauss integration of the modifiedbilinear forms (22) is carried out. The numerical effort isreduced significantly. Quantities of the new weak form aremarked with a bar to indicate averaging. This technique clearlyimplies an error. The solution of the modified weak form with(22) is hopefully as accurate as the original one with (19).

For the exact integration it is assumed that the x-dimensionof the rectangular FEs is an integer multiple of p because nl

4

points into the x-direction, see Fig. 3. Exact integration meansthat the FEs are split into rectangular subdomains according tod and d0, respectively, and Gauss integration on the differentsubdomains is made considering the relevant standard poly-nomials, micro-shape function and material parameters. Thecontributions of the integrals over the subdomains are addedup for the respective FE matrix.

The errors defined in (12) can be seen in Figs. 7 to 9. Thereis a significant difference between the RS Ah according to (17)and the MSFEM solution Ah obtained by (19) of the potentialsand eddy current losses depending on whether the exact or theaveraging integration technique is used. This is clearly visibleat small errors due to the logarithmic scaling. The error inA is essentially higher than that of P over the investigatedrange. The magnetic flux density curl A is hardly affect bythe selected integration technique, see Fig. 8. The polynomialdegrees of the SFE basis of MSFEM (18) were one for A0and A1 and two for w1. In general errors grow strongly onceδ becomes smaller than d + d0 regardless of the integrationtechnique. The HMSFEM in Sec. V-G can solve this problem.

Figure 7: Magnetic vector potential A.

Figure 8: Magnetic flux density curl A.

Figure 9: Eddy current losses P.

F. Triangular mesh

MSFEMs with averaged coefficients facilitate an easy useof triangular FE meshes. Two examples are presented inFigs. 10. The computational domain Ωm ∪ Ω0 is uniformlysubdivided into triangular FEs. The side length of the trianglesis approximately s. In case of s = 20mm Ωm consists of onlytwo triangular FEs. All simulations are based on the MSFEM(19) and relation (21).

Figure 10: Triangular FE models of Ωm∪Ω0 in red and green,respectively, with a) s = 10mm (left) and b) s = 4mm (right).

Eddy current distributions are shown in Figs. 11.

Figure 11: Magnitude of the eddy current density in Ωm fora) s = 10mm (left) and b) s = 4mm (right).

The error in A with respect to s is large as can be seenin Fig. 12, the error in curl A is almost independent froms and clearly grows with the frequency (Fig. 13). The limitof the losses P with decreasing s is too small because the

5

local space is insufficient, see Fig. 14. On the other handthe error grows with s on and on, the insufficiency of theglobal space becomes significant. The penetration depths are




ρ25 = 2.25mm, ρ50 = 1.59mm and ρ100 = 1.13mm for thefrequencies f = 25Hz, f = 50Hz and f = 100Hz, respectively.The width of the laminates is d = 1.8mm.

The degree of the SFEPs is one for A0h and A1h and twofor w1h in Figs. 12 to 14.

G. Higher order MSF with A, p-refinement

To cope with problems, where δ is essentially smaller thand, approach (18) is extended. Higher order terms for thelaminar part as well as for the edge effect of eddy currents areadded [14]. The dimension of the local basis B is increased,for instance, from one B1 = φ1 in (18) to two or threeB3 = φ1, φ3, φ5 leading to the approach of a higher orderMSF

A = A0 + φ1

(0A1

)+ ∇(φ1w1) + φ3

(0A3

)+ ∇(φ3w3)

+ φ5

(0A5

)+ ∇(φ5w5). (23)

Since the magnetic flux density B is an even function acrossthe laminates, only odd higher order terms for the MSF with Aare considered in (23). The higher order multiscale functionsφ3 and φ5, see Fig. 15, are living in Ωc, i.e. on iron subin-tervalls. The multiscale functions φ3 and φ5 are equal to zeroin air and, thus, represent bubble functions. Therefore, theypreserve the required continuity of the tangential componentof A.

The associated weak form for the higher order multiscalefinite element method (HMSFEM) of (23) is a straightforwardextension of (19) and, therefore, not shown here. The choice ofthe FE subspaces A0h ∈ Uh ⊂ H(curl,Ω), A1h, A3h and A5h ∈

Vh ⊂ L2(Ωm), w1h, w3h and w5h ∈ Wh ⊂ H1(Ωm) andφ1, φ3 and φ5 ∈ Hper(Ωm) is obvious.

Figure 15: Odd micro-shape functions.

Figures 16 to 18 show results for different dimensions of thelocal space. Adding of 3rd order terms improves the accuracyenormously. The 5th order approach performs clearly betterthan the 3rd order one for small δ. The degree of the SFEPswas one for A0h, one for A1h, A3h and A5h and two for w1h,w3h and w5h. The computational costs are compared in TableII. A fairly good improvement has been achieved increasingthe dimension of the local basis for this small problem.

H. Stability of MSFEM

This section deals with the stability of MSFEM with respectto the degree of SFEPs mainly with A in a particular case.The MSFEM for A in Sec. V with A1 ∈ L2(Ωm) offers theopportunity to apply static condensation to the degrees of

6




freedom belonging to L2 with which the number of unknownscan essentially be reduced. The FE-mesh described in TableI and Fig. 4 has been used for all numerical investigations.One period p = d + d0 equals to 2 mm and fits exactly to oneFE-layer on the left and on the right side of Ωm.Linear dependence is an important issue of multiscale methodsbut it is rather seldom addressed [15], [16]. Strictly speaking,the basis of functions ψi j(x) of the MSFEM (19) are linear

Table II: Number of Degrees of Freedom

Total No. H(curl,Ω) L2(Ωm) H1(Ωm)

RS 24,745 a) 24,745 - -MSFEM 1,675 b) 676 152 c) 181 c)

a) For 6th order H(curl) - elements for the smallest δ.b) For the 5th higher order MSF, static condensation eliminates all degrees offreedom of L2(Ω) and possible higher order ones of H(curl,Ω) and H1(Ωm).c) Holds for one term in approach (23)

independent but not well-conditioned. Contrary to A1 ∈ L2partial derivatives of A1 are considered in case of A1 ∈ H1introducing some degree of smoothing, compare Figs. 19.Analysis of the FE system matrices showed very similarcondition numbers and eigenvalue spectra of both MSFEMsand with averaged coefficients. Surprisingly, the MSFEM withA1 ∈ H1 is stable with respect to the degree of SFEPs as canbe seen in Figs. 20 to 22. The abscissa shows the degree ofthe SFEPs of A0 and A1, that of w1 is one greater than those.The error for A1 ∈ H1(Ωm) decreases slightly with degree ofSFEPs whereas that for A1 ∈ L2(Ωm) grow rapidly and for thelosses in an uncontrolled manner. The method with averagedcoefficients indicated with AC is robust against instability.The same instability can be observed for HMSFEM withA1, A3 and A5 ∈ L2. Since Dirichlet boundary conditions haveto be prescribed for T2 only T2 ∈ H1 is possible. Investigationsof MSFEM with T showed no instability.An effective and simple remedy for the instability is to resolvethe outermost laminates by the SFEM while the MSFEM withexact integration is retained for the rest of the laminates. Twolaminates (p = 1.0 mm) in each outermost layer is stable upto a polynomial degree of the SFE basis with four for A0 andA1 and five for w1.

Figure 19: Real part of A1 in Ωm for a) A1 ∈ L2(Ωm) (right)and b) A1 ∈ H1(Ωm) (left).

VI. Large Nonlinear Problem

To demonstrate the ability of MSFEM to cope with large andnonlinear problems and to clearly point out the benefit of theMSFEM compared with the SFEM a large nonlinear problemwith FE models in Figs. 23 have been studied. The nonlinearproblem with the magnetization curve in Fig. 24 consists of1,000 laminates. MSFEM of 1st order of (19) has been studied.The implicit Euler method was used for the time discretizationand Newton’s method to solve the nonlinear problem. The

7




agreement of the eddy current losses with respect to time inFig. 25 is very satisfactory. The reduction of computationalcosts by means of MSFEM compared with SFEM and the RSis impressive as can be seen in Table III. The computationalrequirements of MSFEM for this large problem and the linearcase are almost the same as those for the small problem inTable II, Sec. V-G, for the 1st order MSFEM with only 10laminates.

Figure 23: FE models for a) RS (left) and b) MSFEM (right).

Figure 24: Magnetization curve.

Figure 25: Evolution over time of eddy current losses.

Table III: Number of Degrees of Freedom

Total No. H(curl,Ω) L2(Ωm) H1(Ωm)

RS 164, 430 164, 430 - -MSFEM 1, 289 a) 840 208 241

a) For the 1st order MSF in (23)

VII. Higher OrderMultiscale Finite ElementMethod withSingle Component Current Vector Potential T

A. Boundary value problem with T

The current density J can be represented with a currentvector potential T by J = curl T. This section deals with the

8

single component current vector potential T , e.g., pointing inz-direction T = Tez in the frequency domain:

T : Ω ⊂ R2 → C, (x, y) 7→ T (x, y) (24)

curl T : Ω ⊂ R2 → C2, (x, y) 7→

( ∂T∂y− ∂T∂x

)(x, y) (25)

A simple BVP of the ECP in the frequency domain reads, seeFig. 3:

curl1σ

curl T + jµωT = 0 in Ω ⊂ R2 (26)

T = T0 on Γ (27)

B. Weak form with T

The weak form in the frequency domain reads:Find Th ∈ Vh,T0 := Th ∈ Uh : Th = T0 on Γ, such that

∫Ω

1σ

curl Th · curl th dΩ + jω∫

Ω

µThth dΩ = 0 (28)

for all th ∈ Vh,0, where Uh ⊂ H1(Ω).The solution of (28) serves as RS to evaluate the MSFEMwith T .

C. Higher order multiscale finite element method with T

The MSF up to the order 4 for the single component currentvector potential

T (x, y) = T0(x, y) + φ2(x)T2(x, y) + φ4(x)T4(x, y) (29)

with even micro-shape functions φ2 and φ4 shown in Fig. 26yields the multiscale current density

J = curl T =

(T0y

−T0x

)+

(φ2T2y

−φ2xT2 − φ2T2x

)+

(φ4T4y

−φ4xT4 − φ4T4x

).

(30)

Simply speaking T corresponds to the magnetic field strengthH which is an even function, therefore the multiscale functionsφ2 and φ4 are used.

Figure 26: Even micro-shape functions.

D. Weak form of HMSFEM with T

Find (T0h,T2h,T4h) ∈ Vh,T0 := (T0h,T2h,T4h) : T0h ∈

Uh,T2h and T4h ∈ Vh,T0h = T0 on Γ and T2h = 0 andT4h = 0 on Γm0,1 ⊂ Γm0, such that∫

Ω

1σ

curl Th · curl th dΩ + jω∫

Ω

µTh th dΩ = 0 (31)

for all (t0h, t2h, t4h) ∈ Vh,0, where Uh is a subspace of H1(Ω),Vh of H1(Ωm) and φ2 and φ4 ∈ H1

per(Ωm).Homogenous essential boundary conditions are prescribed forT2 = 0 and T4 = 0 on the interface Γm0,1 of Γm0 = Γm0,1∪Γm0,2,see Fig. 27, to ensure the edge effect that can easily be seenby the y-component in (30).

E. p-refinement of the standard FE basis

This time attention is paid to the accuracy with respect tothe degree of the SFEPs, see Figs. 28 to 36. The degree ofthe SFEPs is one for T0h, T2h and T4h in Figs. 28 to 30, onefor T0h and two for T2h and T4h in Figs. 31 to 33 and one forT0h and three for T2h and T4h in Figs. 34 to 36. The errors arevisibly reduced by increasing the degree of the SFEPs. The4th order MSFEM performs clearly better than the 2nd orderone. This holds in general for small δ

d (Figs. 28 to 36) and inparticular for the 3rd degree of SFEPs (Figs. 34 to 36).

Figure 27: Support, boundary and interface conditions.

Figure 28: Current vector potential T .

The computational costs are summarized in Table IV. A fairly

9

Figure 29: Current density curl T .



good improvement has been achieved for this small problem.It is worth mentioning that both the degree of SFEPs

and the order of the MSA along with the micro-shapefunctions can not be increased arbitrarily because of thelinear dependence due to the tensor product of MSA (18) andthe use of quadrilateral FEs.




F. Edge effect

The difference between considering and neglecting the edgeeffect is shown by means of eddy currents in Figs. 37. Incontrast to MSFEM with A in Sec. V MSFEM with T worksalso without prescribing homogenous essential boundary con-ditions for T2 and T4 on Γm0,1, i.e. without considering theedge effect, reasonably and offers therefore the opportunity to

10



Table IV: Number of Degrees of Freedom

Total No. H1(Ω) H1(Ωm) H1(Ωm)

RS 9,211 9,211 a) - -MSFEM 721 99 b) 311 c) 311 d)

a) T of 6th order H1(Ω)-elements for the smallest δb) T0 of 1st order H1(Ωm)c) T2 of 3rd order H1(Ωm)d) T4 of 3rd order H1(Ωm)

study the influence of the edge effect. Dashed lines in Figs. 38to 40 stand for edge effect neglected and solid lines for edgeeffect considered. Note, that the abscissa in all Figs. 38 to 40represents the thickness d of the laminates and δ was about0.5mm. The influence of the edge effect can easily be seen.

Figure 37: Eddy current distribution a) considering (left) andb) neglecting (right) the edge effect.

The error of curl Tez neglecting the edge effect is relatively

large throughout the range of d. A substantial improvementcan be observed considering the edge effect for this smallexample. In general errors decrease due to the edge effect forsmaller d. The 2nd order MSA shows clearly faster growingerrors with large d compared to the 4th order one.The degree of the SFEPs is one for T0h and two for T2h andT4h in Figs. 38 to 40.




11

VIII. Multiscale Finite ElementMethod with aMixedFormulation using (A, J)

Introducing the current density

Jh = − jωσAh (32)

also as an unknown yields the additional relation

−

∫Ω

Ah · gh dΩ +jω

∫Ω

1σ

Jh · gh dΩ = 0 (33)

for the weak form of the mixed formulation:

A. Weak form of the standard mixed formulation with (A, J)

Find (Ah, Jh) ∈ Vh,α := (Ah, Jh) : Ah ∈ Uh, Jh ∈ Mh andAh × n = αh on Γ, such that∫

Ω

1µ

curl Ah · curl vh dΩ −

∫Ω

Jh · vh dΩ = 0 (34)

−

∫Ω

Ah · gh dΩ +jω

∫Ω

σJh · gh dΩ = 0 (35)

for all (vh, gh) ∈ Vh,0 with Uh ⊂ H(curl,Ω) and Mh ⊂

H(div,Ω).

B. Weak form of the mixed multiscale finite element method(MMSFEM) with (Ah, Jh)

The approach of the current densityJ = J0 + curl(φ2T2ez) (36)

for the MSF has been constructed with the single componentcurrent vector potential T2. Together with (18) for A the weakform of the mixed MSF is obtained:Find (A0h, A1h, w1h, J0h, T2h) ∈ Vh,α := (A0h, A1h, w1h, J0h,T2h) : A0h ∈ Uh, A1h ∈ Vh, w1h and T2h ∈ Wh, J0h ∈ Mh

and A0h × n = αh on Γ, such that∫Ω

1µ

curl Ah · curl vh dΩ −

∫Ω

Jh · vh dΩ = 0 (37)

−

∫Ω

Ah · gh dΩ +jω

∫Ω

1σ

Jh · gh dΩ = 0 (38)

p∫

Ω

div J0h div g0h dΩ = 0 (39)

for all (v0h, v1h, q1h, g0h, t2h) ∈ Vh,0 with Uh ⊂ H(curl,Ω),Vh ⊂ L2(Ωm), Wh ⊂ H1(Ωm), Mh ⊂ H(div,Ω) and φ1 andφ2 ∈ H1

per(Ωm) and for a sufficiently large p ∈ R. The micro-scale currents in (36) are divergence free per se. To get also adivergence free macro-scale current density J0h and a uniquesolution the penalty term in (39) is additionally required.The computational costs of the mixed formulation are certainlyhigher than that of the corresponding standard formulationwith A. However, the mixed formulation performs much betterin reproducing the edge effect as can easily be seen in Fig. 41.A comparison of the losses obtained by different methods areshown in Fig. 42. The error of the MMSFEM and that ofthe mixed SFEM are practically identical and are enclosed bythose of the MSFEM of 1st with A order as upper bound andthose of the MMSFEM with ( Ah, Jh) of 2nd order as lowerbound. The degrees of the SFEPs are one for A0h, A1h andJ0h and two for w1h and T2h.

Figure 41: Edge effect by means of the current density: a)MSFEM with Ah (left) and b) MMSFEM with (Ah, Jh) (right).

Figure 42: Eddy current losses P of mixed formulations with(Ah, Jh) and of the MSFEM with A.

Acknowledgment

This work was supported by the Austrian Science Fund(FWF) under Project P 27028-N15.

References

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[6] P. Dular, J. Gyselinck, and L. Krahenbuhl, “A time-domain finite elementhomogenization technique for lamination stacks using skin effect sub-basis functions,” COMPEL, vol. 25, no. 1, pp. 6–16, 2006.

[7] K. Hollaus and J. Schoberl, “Homogenization of the eddy currentproblem in 2D,” ser. 14th Int. IGTE Symp., Graz, Austria, Sep. 2010,pp. 154–159.

[8] J. Schoberl and S. Zaglmayr, “High order Nedelec elements with localcomplete sequence properties,” COMPEL, vol. 24, no. 2, pp. 374–384,2005.

[9] I. Babuska and J. M. Melenk, “The partition of unity method,” Int. J.Numer. Meth. Engng., vol. 40, pp. 727–758, 1997.

[10] T. Strouboulis, K. Copps, and I. Babuska, “The generalized finiteelement method: an example of its implementation and illustration ofits performance,” Int. J. Numer. Meth. Engng., vol. 47, pp. 1401–1417,2000.

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[12] P. Dular, “A time-domain homogenization technique for laminationstacks in dual finite element formulations,” J. Comput. Appl. Math., vol.215, no. 2, pp. 390–399, 2008.

[13] K. Hollaus, A. Hannukainen, and J. Schoberl, “Two-scale homogeniza-tion of the nonlinear eddy current problem with FEM,” IEEE Trans.Magn., vol. 50, no. 2, pp. 413–416, Feb 2014.

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13

Appendix ASome Equations

∫Ω

1µ

curl(A0h + φ1

(0

A1h

))· curl

(v0h + φ1

(0

v1h

))dΩ + jω

∫Ω

σ

(A0h + φ1

(0

A1h

)+ ∇(φ1w1h)

)·

(v0h + φ1

(0

v1h

)+ ∇(φ1q1h)

)dΩ = 0

(20)

∫Ω

(curl A0h

A1h

)T ν νφ1x

νφ1x νφ21x

(curl v0h

v1h

)dΩ + jω

∫Ω

(A0h)x

(A0h)y

A1h

w1h

w1hx

w1hy

T

σ 0 0 σφ1x σφ1 00 σ σφ1 0 0 σφ1

0 σφ1 σφ21 0 0 σφ2

1

σφ1x 0 0 σφ21x σφ1xφ1 0

σφ1 0 0 σφ1xφ1 σφ21 0

0 σφ1 σφ21 0 0 σφ2

1

(v0h)x

(v0h)y

v1h

q1hq1hxq1hy

dΩ = 0

(22)

Appendix BMultiscale Finite ElementMethod Solutions

Typical solutions are selected to make the advantage of MSFEM over standard FEM quite clear. In this context it is worthmentioning that the corresponding solutions of the 3rd order approach are almost the same. Results with the 5th order MSFare shown in Figs. 43 to 44.

Figure 43: MSFEM solution: Re(A0h(1)) (left), Re(A0h(2)) (right).

Figure 44: MSFEM solution: Re(A1h) (left), Re(w1h) (right).

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Documents

Some two-dimensional multiscale nite element formulations