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-tovolume 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 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 nonconducting 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 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 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 ∂Ωi = Γi = Γh,i ∪Γb,i The eddy current conducting part of Ωi is denoted Ωc,i and the nonC conducting one Ωc,iC, with Ωi =Ωc,i ∪Ω c,i Stranded (multifilamentary) inductors belong to Ωc,iC, whereas massive inductors belong to Ωc,i The equations, material relations and boundary conditions (BCs) of SP iare curl hi = ji , div bi = 0, curl ei = −∂ t bi hi = μi -1 bi + hs,i , ji = σ i ei + js,i n × hi Γ = j f ,i , n ⋅ bi Γ = f f ,i n × ei h ,i Γe ,i ⊂Γb ,i b ,i = k f ,i (1a - b - c) (2a - b) (3a - b) (3c) where hi is the magnetic field, bi is the magnetic flux density, ei is the electric field, ji 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 hs,iand js,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 jf,iand kf,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 Ωi, with the notation[ ⋅ ]γ = ⋅ |γ + – ⋅ |γ – If i i i nonzero, they define possible SSs that account for particular phenomena occurring in the idealized thin region betweenγi+ and γi– [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 SPu to SPp – 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 SPp consists of adding a TS to SP u (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 bformulation uses a magnetic vector potential (such that curl = bi), split as a = ac,i + ad,i[2] The hformulation uses a similar splitting for the magnetic field, h = hc,i + hd,i Finite Element Weak Formulations 3.1 Magnetic Vector Potential Formulation The weak bi -formulation (in terms ofai) of SP i (i≡ u, p or k) is obtained from the weak form of the Ampère equation (1a), i.e [3], [4] (μi −1 curl , curl ')Ωi + (hs ,i , curl ')Ωi + ( js,i , ')Ωi + (σi ∂t , ')Ωi +< n× hi , ' >Γh,i −Γt,i +< [n× hi ]Γt,i , ' >Γt,i = ( js ,i , ')Ωi , ∀ ' ∈ Fi1 (Ωi ) Figure Regions defining SPu and SPp The fields ac,i, hc,i, and ad,i, hd,i, are continuous and discontinuous respectively through the TS The traces discontinuities in SPp [n × hp]γ and [n × ep]γ (with nt = -n) p p in both formulations can be expressed as paper [2] [n × (hu × hp )]γ = [n × hp ]γp = μ p β p ∂ t (2ac, p + ad, p ) [n × (eu × e p )]γ = [n × e p ]γp = σ p β p ∂ t (2hc, p + hd, p ) (4) (5) β p = θ p −1 tanh( θ p d p / ), θ p = ( + j ) / δ p (6) because there are no discontinuities in SP u (before addingγp), where dp is the local TS thickness, δ p = /(ωσ p μ p ) is the skin depth in the TS, ω = 2πf,fis the frequency and j is an imaginary unit Also, the traces of epandhpon the positive side γp+ are expressed as [2] n× hp n× e p γ +p γ +p 1 a ] -n× hu γ + = [σ p βp∂t (2ac, p + ad, p ) + p σ p βp d, p 1 a ] -n× eu γ + = [μp βp∂t (2hc, p + hd, p ) + p σ p βp d, p (7) bs,k = (μk - μp )(hu + hp ), js,k = (σ p -σ p )(eu + ep ) hs,k = ( μ−1k - μ−1p )(bu + bp ), es,k = -(eu + ep ) where Fi1(Ωi) is a curl-conform function space defined in Ωi, gauged in Ωc,iC, and containing the basis functions for a as well as for the test function ai' (at the discrete level, 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 – SPu 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 ji is the fixed current density in on Ωs 3.1.2 Thin shell FE model- SPp The TS model is defined via the term < [n×hp ]γ p , ad, p ' >γ p in (11)(i≡ p) The test function ai' is split into continuous and discontinuous parts a'c,pand a'd,p (with a'd,p zero on γ−p) [2] One thus has < [n×hp ]γ p , ap ' >γ p =< [n×hp ]γ p , ac, p ' >γ p + (8) 2.4 From SPp to SPk –TS model to volume correction 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 = μvolume, σk= 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] (9a - b) (10a -b) (11) < n×hp |γ+ , ad, p ' >γ+ p (12) p The terms of the RHS of (12) are developed using (4) and (7) respectively, i.e < [n×hp ]γ p , ac, p ' >γ p = < [n×h]γ p , ac, p ' >γ p =γ p = −< n×hu |γ+ , ad, p ' >γ+ + p p p 1 , ad , p ' >γ p < σ pβ p∂t (2ac, p + ad, p ) + σ pβ p (14) 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 − < n × hu |γ + , a d , p ' > γ + = (μ u −1 curl au , curl ad , p ')Ω + p p p 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 n × ad,p'|γ+p Also, the source au, initially in the mesh of SP 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 representationSP k The TS SP p solution is then corrected by SP k via the volume integrals (hs, p ,curl a ')Ω p and ( js , p , a ')Ω in p (11) The VSs js,k and hs,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 [n × hk ]γ k = −[ n × h p ]γ k and [n × a]γ k = −[ n × a p ]γ k The trace discontinuity [ n × hk ]γ k occurs in (11) via < [n×hk ]γk , ak ' >γ p = −< [n×hp ]γk , ak ' >γk (16) and can be weakly evaluated from a volume integral from SP p similarly to (15) However, direct use of the explicit form (4) for [ n × h p ]γ k gives the same contribution, which is thus preferred 3.2 Magnetic Field Formulation 3.2.1 h-Formulation with source and reaction magnetic fields The hi − 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 hi is split into two parts, hi =hs,i + hr,iwhere hs,iis a source fielddefined by curl hs,i= js,iand hr,iis unknown One has ∂t (μi (hr ,i + hs,i ), hi ')Ωi + (σi −1 curl hi ,curl hi ')Ωi + ∂t (bs,i , hi ,')Ωi + (es,i ,curl hi ')Ωi + < n × ei , hi ' >Γe,i + < [n × ei ]γi , hi ' >γi = 0, ∀ hi ' ∈ Fi1 (Ωi ) whereFi1(Ωi) is a curl-conform function space defined in Ωiand contains basic functions for hi as well as for the test function hi' 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 hs,u is defined via a projection method of a known distribution js,u [5], i.e (curl hs,u , curl hs ,u ')Ωu = ( js,u , curl hs ,u ')Ωu , ∀ hs,u ' ∈ Fu1 (Ωu ) (18) 3.2.3 Thin shell FE model–SPp The TS model is defined via the term < [n×ep ]γ p , hd, p ' >γ p in (11)(i≡ p) The test function hi' is split into continuous and discontinuous parts h'c,pand h'd,p (with h'd,p zero on γ−p) [2] One thus has < [n×ep ]γ p , hp ' >γ p =< [n×ep ]γ p , hc, p ' >γ p + 31 < n×ep |γ+ , hd, p ' >γ+ p (19) p The terms of the right-hand side of (19) are developed using (5) and (8) respectively, i.e < [n×ep ]γ p , hc, p ' >γ p = < [n×e]γ p , hc, p ' >γ p =γ p = − < n×eu |γ+ , hd, p ' >γ+ + p p p 1 , hd, p ' >γ p < μ pβ p∂t (2hc, p + hd, p ) + μ pβ p (21) 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 − < n × eu |γ + , hd , p ' >γ + = (μu ∂t (hs,u + hs,u ), hd , p ')Ω+ (22) p p p The sources hs,iand hr,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 (bs, p , h ')Ω p and (es , p , curl hk ')Ω k The VSs bs,k and es,kare also given in (9) and (10) respectively The VS es,kin (10) is to be obtained from the still unknown electric fields eu andep 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., hd,k= -hd,k to be strongly defined, and [n × e p ]γ k = −[n × e p ]γ k The trace discontinuity [n × e p ]γk occurs in (17) via < [n×ek ]γk , hk ' >γ p = −< [n×ep ]γk , hk ' >γk (23) and can be weakly evaluated from a volume integral from SP p similarly to (22) 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), followed by the addition of a TS model SPp without the inductors anymore (Figure 4, φp, top left); the discontinuity Δφp is defined to zero on both sides of the TS A volume correction SPk eventually corrects the TS model (Figure 4, φk, bottomright) The complete solution is indicated as well (Figure 3, φk = φu +φp+φk, topleft) 32 Dang Quoc Vuong Figure 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 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, d1 =d1 = d3 = 5mm and d2 = d4 = 2mm Figure 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, d1 =d1 = d3 = 5mm and d2 = d4 = 2mm Figure Distribution of magnetic scalar potential for a reduced model SPu (φu, left) with the inductors alone, added thin shell SPp (φp, middle) and volume correction SPk (φk, right) Figure 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) For d3 = 5mm (thickness of the thin region), the inaccuracy on the eddy current density of TS SPp along yaxis 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 hformulation 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 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 out via SPk (Figure 8, top) A superposition of the SP 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 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 This work was supported by the fund (T2016-PC-085) of Hanoi University of Science and Technology (HUST) 33 REFERENCES [1] L Krähenbühl and D Muller, “Thin layers in electrical engineering Examples of shell models in analyzing eddy-currents by boundary and finite element methods,” IEEE Trans Magn., Vol 29, No 2, pp 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 Element Formulations,” IEEE Trans Magn., vol 48, no 2, pp 407– 410, 2012 [4] P Dular, R V Sabariego, M V Ferreira da Luz, P Kuo-Peng and L Krähenbühl, “Perturbation Finite Element Method for Magnetic Model Refinement of Air Gaps and Leakage Fluxes," IEEE Trans Magn., vol.45, no 3, pp 1400-1403, 2009 [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) ... +p γ +p 1 a ] -n× hu γ + = [σ p βp∂t (2ac, p + ad, p ) + p σ p βp d, p 1 a ] -n× eu γ + = [μp βp∂t (2hc, p + hd, p ) + p σ p βp d, p (7) bs,k = (μk - μp )(hu + hp ), js,k = (σ p -? ? p )(eu + ep... (μk - μp )(hu + hp ), js,k = (σ p -? ? p )(eu + ep ) hs,k = ( μ−1k - μ−1p )(bu + bp ), es,k = -( eu + ep ) where Fi1(Ωi) is a curl-conform function space defined in Ωi, gauged in Ωc,iC, and containing... (T2016-PC-085) of Hanoi University of Science and Technology (HUST) 33 REFERENCES [1] L Krähenbühl and D Muller, “Thin layers in electrical engineering Examples of shell models in analyzing eddy-currents