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Microsoft Word 00 a loinoidau(moi thang12 2016)(tienganh) docx ISSN 1859 1531 THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 12(109) 2016 29 A SUBPROBLEM METHOD FOR ACCURATE THIN SHEL[.]

ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 12(109).2016 29

A SUBPROBLEM METHOD FOR ACCURATE THIN SHELL MODELS

BETWEEN CONDUCTING AND NONCONDUCTING REGIONS

Dang Quoc Vuong

Hanoi University of Science and Technology; vuong.dangquoc@hust.edu.vn

Abstract - A subproblem method with a finite element

magnetodynamic formulations are developed for correcting the

inaccuracies near edges and corners arising from using thin shell

models that replace thin volume regions by surfaces The

surface-to-volume correction problem is defined as one of the multiple

subproblems (SPs) applied to a complete problem, considering

sucessive additions of inductors and magnetic or conducting regions,

some of these being thin regions Each SP is solved on its own

separate domain and mesh, which facilitates meshing and solving

while controlling the importance and usefulness of each correction

Key words - Eddy current; finite element method (FEM);

magnetodynamics; subproblem method (SPM); thin shell (TS);

conducting regions; non-conducting regions

1 Introduction

The thin shell (TS) model asumes that the fields in the

thin regions are approximated by a priori 1-D analytical

distribution along the shell thickness [1],[2] Their interior

is not meshed and rather extracted from the studied

domain, which is reduced to a zero-thickness double layer

with interface conditions (ICs) linked to 1-D analytical

distribution that, however, generally neglect end and

curvature effects [1],[2] This leads to inaccuracies near

edges and corners that increase with the thickness In order

to overcome these drawbacks, the authors have recently

proposed a SPM for a thin region located between

non-conducting regions [3] The magnetic field h is herein

defined in non-conducting regions by means of a magnetic

scalar potential φ, i.e., h= -gradφ, with discontinuities of

φthrough the TS

In this paper, the subproblem method (SPM) is

extended to account for thin regions located between

conducting regions (CRs) or between conducting and

nonconducting regions (NCRs) (Figure 1) In these regions,

the potential φ is not defined anymore on both sides of the

TS and the problem has to be expressed in terms of the

discontinuities of h, possibly involving φon one side only, to

be strongly defined via an IC through the TS

Figure 1 Geometry of SPs with a thin region located between

CRs and NCRs: The TS model (left) and volume correction

(right)

In the proposed SP strategy, a reduced problem with

only inductors is first solved on a simplified mesh without

thin and volume regions Its solution gives surface sources

(SSs) as ICs for added TS regions, and volume sources

(VSs) for possible added volume regions The TS solution

is further improved by a volume correction via SSs and

VSs that overcome at the TS assumptions, respectively

suppressing the TS model and adding the volume model

2 Subproblem coupling with TS models

2.1 Generalities

In the frame SPM, two importance SPs can be defined: for “adding a TS” in a configuration with an already calculated solution other sources and for “correcting a TS” via its actual volume extension

2.2 Canonical magnetodynamic problem

A canonical magnetodynamic problem i, to be solved

at step i of the SPM (i≡ u, p or k), is defined in a domain

Ωi, with boundary∂Ω = Γ = Γ ∪Γi i h i, b i, The eddy current conducting part of Ωi is denoted Ωc,i and the

non-conducting one Ωc , i C, with , C,

Ω =Ω ∪Ω Stranded (multifilamentary) inductors belong to Ωc , i C, whereas

massive inductors belong to Ωc,i The equations, material relations and boundary conditions (BCs) of SP iare

(2a - b) (3a - b) (3c)

e ,i b ,i

i i i i t i

f ,i f ,i

f ,i

i

h = b + h , j = e + j

Γ ⊂Γ

×

where h i is the magnetic field, b i is the magnetic flux

density, e i is the electric field, j i is the electric current density, μi is the magnetic permeability, σi is the electric

conductivity and n is the unit normal exterior to Ωι The

field h s,i and j s,iand in (2a) and (2b) are VSs that can be used for expressing changes of permeability or conductivity from previous SPs [3], [4]

The fields j f,i and k f,iin (3a) and (3b) are SSs and generally equal zero for classical homogeneous BCs ICs can define their discontinuities through any interface γi(γi +

and γi –) in Ω i, with the notation[ ⋅ ]γ

i = ⋅ |γ

i + – ⋅ |γ

i – If

nonzero, they define possible SSs that account for particular phenomena occurring in the idealized thin region betweenγi + and γ i – [3] A typical case appears when some

field traces in a previous problem are forced to be discontinuous (e.g in TS model), whereas their continuity must be recovered via a correction problem: the SSs in SPi

are then fixed as the opposite of the trace discontinuities accumulated from the previous problems

2.3 From SP u to SP p – inductor alone to TS model

The solution of an SPuis first known for an inductor

30 Dang Quoc Vuong

alone (Figure 2, left).The next SP p consists of adding a

TS to SPu (Figure 2, right) SP p is constrained via a SS

that is related to the BCs and ICS given by the TS model

[2], to be combined with contributions from SPu The

b-formulation uses a magnetic vector potential ai (such

that curl a i = b i ), split as a = a c,i + a d,i [2] The

h-formulation uses a similar splitting for the magnetic

field, h = h c,i + h d,i

Figure 2 Regions defining SPu and SP p

The fields a c,i , h c,i , and a d,i , h d,i , are continuous and

discontinuous respectively through the TS The traces

discontinuities in SPp [n × h p]γp and [n × e p]γp (with n t = -n)

in both formulations can be expressed as paper [2]

p

p

1

p p t

p p t

μ β β ( d / ), ( j ) /

−

∂

∂

σ

because there are no discontinuities in SP u (before

the frequency and j is an imaginary unit Also, the traces of

e p andh pon the positive side γp + are expressed as [2]

2

2

p p

p p

p p

p p

β

β β

β

+ +

+ +

γ γ

γ γ

σ

σ μ

σ

The TS solution obtained in an SPpis then corrected by

means of SPk that overcomes the TS assumptions The SPM

offers tools to perform such a model refinement, thanks to

simultaneous SSs and VSs A volume mesh of the shell is

now required and extended to its neighborhood without

including the other regions of previous SPs This allows for

focus on the fineness of the mesh only in the shell and its

neighborhood SPk has to suppress the TS representation

via SSs opposed to TS discontinuities, in parallel to VSs in

the added actual volume [3] that account for changes of

material properties in the added volume region from μpand

σpin SPp from μkand σkin SPk(withμp = μ0, μk = μ volume, σk=

0 and σk= σvolume) This correction can be limited to the

neighborhood of the shell, which permits benefit from a

reduction of the extension of the associated mesh [3] The

VSs for SPk are paper [3], [4]

k p p p u p

3 Finite Element Weak Formulations

The weak b i -formulation (in terms ofa i ) of SP i (i≡ u, p

or k) is obtained from the weak form of the Ampère

equation (1a), i.e [3], [4]

1

1 ,

i

where Fi1(Ωi) is a curl-conform function space defined in

Ωi, gauged in Ωc,iC, and containing the basis functions for

this space is defined via edge FEs; the gauge is based on a tree-co-tree technique); (·, ·)Ω and < ·, · >Γ respectively denote a volume integral in Ω and a surface integral on Γ

of the product of their vector field arguments The surface integral term on Γh,i accounts for natural BCs of type (3a),

usually zero At the discrete level, the required meshes for

each SP i differ

3.1.1 Inductor alone – SP u

The weak form of an SP u with the inductor alone is first solved via the first and last volume integrals in (11)

(i≡ u) where j i is the fixed current density in on Ωs

3.1.2 Thin shell FE model- SP p

The TS model is defined via the term

, [ p] ,γp d p' γp

< ×n h a > in (11)(i≡ p) The test function a i ' is

split into continuous and discontinuous parts a' c,p and a' d,p

(with a' d,p zero on γ−p) [2] One thus has

, [ p] , 'γp p γp [ p] ,γp c p' γp

< ×n h a > =< ×n h a > +

,

| , '

p γ + d p γ +

< ×n h a > (12) The terms of the RHS of (12) are developed using (4) and (7) respectively, i.e

[ p] ,γp c p' γp [ ] ,γp c p' γp

< ×n h a > = < ×n h a >

p p t c p d p c p γ

=<σ β ∂ a +a a > (13)

< ×n h a > = −< ×n h a > +

σ β

The last surface integral term in (14) is related to a SS that can be naturally expressed via the weak formulation of

SP u (11), i.e

1

At the discrete level, the volume integral in (15) is thus limited to a single layer of FEs on the side Ω+

p touching

γp+, because it involves only the associated trace

ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 12(109).2016 31

u, has to be projected on the mesh of SP p, using a

projection method [5], [6]

3.1.3 Volume correction replacing the TS representation-

SP k

The TS SP p solution is then corrected by SP k via the

p

p

(11) The VSs j s,k and h s,k are given in (9) and (10)

respectively

Simultaneously, to the VSs in (11), SSs have to

suppress the TS discontinuities, with ICs to be defined as

k p k

k

k γ

×

< ×n h a > = −< ×n h a > (16)

and can be weakly evaluated from a volume integral from

SP p similarly to (15) However, direct use of the explicit

k

p γ

×

is thus preferred

3.2.1 h-Formulation with source and reaction magnetic

fields

The h i− φi formulation of SP i (i≡u; p or k) is obtained

from the weak form of Faraday’s law (1 c) [6] The field h i

is split into two parts, h i =h s,i + h r,i where h s,iis a source

fielddefined by curl h s,i= j s,i and h r,iis unknown One has

1

,

1

whereFi1(Ωi) is a curl-conform function space defined in

Ωiand contains basic functions for h i as well as for the test

function h i ' The surface integral term on Γe,i ,which

accounts for natural BCs of type (3 b), is usually zero

3.2.2 Inductor model SP u

The model SP u with only the inductor is first solved

with (17) (i≡ p) The source field h s,u is defined via a

projection method of a known distribution j s,u [5], i.e

1

3.2.3 Thin shell FE model–SP p

The TS model is defined via the term

,

< ×n h > in (11)(i≡ p) The test function h i ' is

split into continuous and discontinuous parts h' c,p and h' d,p

(with h' d,p zero on γ−

p) [2] One thus has

, [ e p] , 'γp p γp [ e p] ,γp c p' γp

< ×n h > =< ×n h > +

,

< ×n h > (19) The terms of the right-hand side of (19) are developed using (5) and (8) respectively, i.e

[ e p] ,γp c p' γp [ e] ,γp c p' γp

< ×n h > = < ×n h >

p p t c p d p c p γ

=<μ β ∂ h +h h > (20)

< ×n h > = −< ×n h > +

μ β

The last surface integral term in (21) is related to a SS that can be naturally expressed via the weak formulation of

SP u (17), i.e

The sources h s,i and h r,i, initially in the mesh of SP u, have to be projected on the mesh of SP p using a using a

projection method [5]

3.2.4 Volume correction replacing the TS representation

SP k

Once obtained, the TS solution in SP p is corrected by

SP k via the volume integrals ∂t s p(b, , ')h Ωpand

, ( , curl ')

k

and (10) respectively The VS e s,kin (10) is to be obtained

from the still unknown electric fields e u ande p and their determination needs to solve an electric problem [6]

In parallel with the VSs in (17), ICs compensate the TS discontinuities to suppress the TS representation via SSs

opposed to previous TS ICs, i.e., h d,k = -h d,k to be strongly

k p

e γ

×

[ e k] , 'γk k γp [ e p] , 'γk k γk

< ×n h > = −< ×n h > (23) and can be weakly evaluated from a volume integral from

SP p similarly to (22)

4 ApplicationExamples

The first test problem considers a thin region located between CRs and between CRs (Figure 3)

A thin region is located between CRs (Figure 4, top

left); it is first considered with the h-formulation via an

SPu with the inductor alone (Figure 4, φu , top right),

discontinuity Δφp is defined to zero on both sides of the

solution is indicated as well (Figure 3, φk = φu +φp +φk , topleft)

32 Dang Quoc Vuong

Figure 3.Geometry of SPs with a thin region located between

CRs: Complete problem (top left), the TS model (right) and

volume correction (bottom), with μ1 =μ2 =μ3 =μ4 =100,

σ1 =σ2 =σ3 =σ4 = 59 MS/m, σ1 =1/σ4 , d 1 =d 1 = d 3 = 5mm and

d 2 = d 4 = 2mm

Figure 4 Distribution of magnetic scalar potential for a

reduced model SP u (φu , left) with the inductors alone, added thin

shell SP p (φp , middle) and volume correction SP k (φk , right)

Figure 5 Eddy current density along the y-axis for a thin

region located between CRs with h-formulation, for affects of

μ1 , μ2 ,μ3 , μ4 and σ1 ,σ2 , σ3 , σ4 (given in Figure 3)

inaccuracy on the eddy current density of TS SPp along

y-axis is shown via the importance of the volume correction

SPk(Figure 5) This is presented via a superposition of

the SP solutions (i.e TS + volume) which is checked to be

closed to the complete/reference solution with

h-formulation The results are also illustrated and validated

with b-formulation (Figure 6) Significant errors on the

eddy current density descrease with a smaller thicnkness,

e.g d3 = 2mm

Figure 6 Eddy current density along the y-axis for a thin

region located between CRs with b-formulation, for affects of

μ1 , μ2 ,μ3 , μ4 and σ1 ,σ2 , σ3 , σ4 (given in Figure 3)

Figure 7 Geometry of SPs with a thin region located between

CRs and NCRs: Complete problem (top left), the TS model (right) and volume correction (bottom), with μ1 =μ2 =μ3 =μ4

=100, σ1 =σ2 =σ3 =σ4 = 59 MS/m, σ1 =1/σ4 , d 1 =d 1 = d 3 = 5mm

and d 2 = d 4 = 2mm

Figure 8 Eddy current density along the y-axis for a thin region

located between CRs and NCRs with h-formulation (top) and b-formulation (bottom), for affects of μ1 , μ2 ,μ3 , μ4 and σ1 ,σ2 ,

σ3 , σ4 (given in Figure 7)

The second test problem considers a thin region located between CRs and between CRs and NCRs (Figure7) The error on the eddy current density of TS SPp is also pointed

solution is then checked to be closed to the complete

solution (Figure 8, bottom)

ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 12(109).2016 33

5 Conclusions

The SPM has been developed for correcting the local

quantities inherent to the TS finite elements models for

the simply connected TS region Accurate eddy current

densities are obtained, especially along the edges and

corners of the thin regions, also for significant

thicknesses The method has been successfully applied to

thin regions located between CRs and NCRs All the steps

of the method have been illustrated and validated via

practical tests with the b – and h– formulations As an

example of a practical case, the proposed approach can be

directly applied to the model of the lamination stack of

a transformer

Acknowledgment

of Hanoi University of Science and Technology (HUST)

REFERENCES

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1450-1455, 1993

[2] C Geuzaine, P Dular, and W Legros, “Dual formulations for the modeling of thin electromagnetic shells using edge elements,” IEEE Trans Magn., vol 36, no 4, pp 799–802, 2000

[3] Vuong Q Dang, P Dular, R.V Sabariego, L Krähenbühl, C

Geuzaine, “Subproblem approach for Thin Shell Dual Finite

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410, 2012

[4] P Dular, R V Sabariego, M V Ferreira da Luz, P Kuo-Peng and

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[5] C Geuzaine, B Meys, F Henrotte, P Dular and W Legros, “A Galerkin projection method for mixed finite elements," IEEE Trans Magn., Vol 35, No 3, pp 1438-1441, 1999

[6] P Dular and R V Sabariego, “A perturbation method for computing field distortions due to conductive regions with h-conform magnetodynamic finite element formulations," IEEE Trans Magn., vol 43, no 4, pp 1293-1296, 2007

(The Board of Editors received the paper on 17/8/2016, its review was completed on 02/10/2016)