VIEWS: 59 PAGES: 4 CATEGORY: Technology POSTED ON: 11/12/2009 Public Domain
Miklós KUCZMANN Széchenyi István University Nodal and Edge Finite Element Analysis of Eddy Current Field Problems Abstract. Eddy current field problems can be solved by different potential formulations based on the “quasi-static” Maxwell’s equations. The potential formulations are obtained using a vector potential and a scalar potential, the most widely used techniques are the A,V-A, the T,Φ-Φ formulations and their combinations. Vector potentials can be approximated by nodal or edge finite elements, scalar potentials are approximated by scalar elements. The paper presents and compares these formulations through TEAM Problem No. 7 containing a multiply connected region. Streszczenie. Problemy związane w prądami wirowymi mogą być rozwiązywane przy wykorzystaniu różnych sformułowań i różnych potencjałów, bazujących na quasistatycznych równaniach Maxwella. Wykorzystywany jet w tych sformułowaniach potencjał wektorowy i potencjał skalarny, najczęściej są to sformułowania A,V-A, oraz T,Φ-Φ, a także ich kombinacje. Potencjał wektorowy jest aproksymowany przez elementy węzłowe i krawędziowe, a potencjał skalarny przez elementy skalarne. Artykuł przedstawia różne sformułowania i prowadzi ich analizę komparatystyczną na bazie problemu No. 7 z katalogu TEAM Workshop. (Węzłowe i krawędziowe elementy w analizie problemów wiroprądowych) Keywords: eddy current field, potential formulations, finite element method Słowa kluczowe: prądy wirowe, sformułowania potencjałów, metoda elementów skończonych Introduction There are well-known potential formulations [1-5], here a A linear eddy current field problem contains a source of short summary is presented. the electromagnetic field, which can be a current driven or a The static magnetic field in Ω n can be described by the voltage driven coil, and it generates a time varying magnetic reduced magnetic scalar potential Φ or by the magnetic field in the vicinity of the coil. This field induces eddy vector potential A. The combination of these formulations is currents inside a conducting material (e.g. inside an also valid. aluminum plate), which currents generate a magnetic field, The source current density can be represented by the and this field modifies the magnetic field supplied by the so-called impressed current vector potential T0 by sources. In this case, not only the magnetic field, but the eddy J 0 = ∇ × T0 , since ∇ ⋅ J 0 = 0 , and the magnetic field current field must be simulated. intensity can be represented by Eddy current field problems can be solved by using vector potentials combined with scalar potentials in the (2) H = T0 − ∇Φ frame of the Finite Element Method (FEM). from the first equation in (1). Using the magnetic Gauss law Governing equations results in the following partial differential equation: The following linear static and “quasi-static” Maxwell’s equations can be used to simulate linear eddy current field (3) − ∇ ⋅ [μ∇Φ ] = − μ∇ ⋅ T0 in Ωn . problems: ⎧∇ × H = J, Here, the impressed current vector potential T0 can be ⎪ calculated by the Biot-Savart law. ⎧∇ × H = J 0 , ⎪∇ × E = − B, The tangential component of magnetic field intensity can ⎪ ⎪ (1) ⎨∇ ⋅ B = 0 , in Ω n , and ⎨∇ ⋅ B = 0 , in Ωc . be prescribed by the Dirichlet boundary condition Φ = Φ 0 ⎪ B = μH, ⎪ B = μH, ⎩ ⎪ on ΓH n , where Φ 0 is constant [2,5], and a Neumann ⎪J = σE ⎩ boundary condition μ [T0 − ∇Φ ]⋅ n = 0 on ΓB prescribes B⋅n = 0 . Here H, J0, J, E, B, σ and μ are the magnetic field Magnetic vector potential is introduced by the magnetic intensity, the source current density, the eddy current Gauss law, i.e. density, the electric field intensity, the magnetic flux density, the conductivity, and the permeability, respectively. (4) B = ∇ × A , The domain Ω n is the air region where source current and from the first equation in (1) the partial differential is placed and μ = μ 0 , furthermore Ωc is the region where equation eddy currents are present. The two regions are coupled by interface conditions, i.e. the tangential component of the (5) ∇ × [ ∇ × A] = J 0 ν in Ωn magnetic field intensity and the normal component of the magnetic flux density are continuous on Γnc . On the can be obtained. Here ν = 1 / μ is called the magnetic boundary ∂Ω n = ΓH n ∪ ΓB , and on ∂Ω c = ΓH c ∪ ΓE reluctivity. boundary conditions are prescribed as H × n = 0 on ΓH n , The tangential component of magnetic field intensity is prescribed by a Neumann boundary condition and B ⋅ n = 0 on ΓB , and H × n = 0 on ΓH c , and [ν∇ × A]× n = 0 on ΓH n , and a Dirichlet boundary condition E × n = 0 on ΓE . n × A = 0 on ΓB prescribes B ⋅ n = 0 . 194 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 12/2008 Two potentials can be introduced in the eddy current and region, either a current vector potential T or a magnetic vector potential A to simulate either the eddy current field or ∇ × [ ∇ × A] − ∇[ ∇ ⋅ A] + σA + σ∇V = 0, and ν ν [ ] the magnetic field. The first one must be coupled with a (11) reduced magnetic scalar potential Φ, while the magnetic − ∇ ⋅ σA + σ∇V = 0. vector potential must be coupled with an electric scalar Some extra boundary conditions have to be satisfied in potential V. these situations, too, A ⋅ n = 0 on ΓH n , and ν∇ ⋅ A = 0 on Eddy currents are solenoidal, i.e. ∇ ⋅ J = 0 , from which the current vector potential can be introduced as ΓB in case of (9), ρ∇ ⋅ T = 0 on ΓH c , and T ⋅ n = 0 on ΓE J = ∇ × T . The magnetic field intensity can be represented in case of (10), A ⋅ n = 0 on ΓH c , and ν∇ ⋅ Α = 0 on ΓE in by two unknown potentials as case of (11). (6) H = T0 + T − ∇Φ Coulomb gauge is satisfied automatically when applying edge element based FEM to represent vector potentials, from Ampere’s law. however the right hand side of the partial differential The following partial differential equations can be written equation (5) must be in the same function space, i.e. T0 from Maxwell’s equations: must be used represented by edge elements [4], ∇ × [ρ∇ × T ] + μT − μ∇Φ = − μT0 , and (12) ∇ × [ ∇ × A] = ∇ × T0 . ν (7) ∇ ⋅ [μT − μ∇Φ ] = −∇ ⋅ [μT0 ] in Ω c . The partial differential equations of the vector element Here ρ = 1 / σ is the resistance of the material. The representation of T,Φ-formulation, and A,V-formulation are the same as in (4) and in (5). These formulations are called Dirichlet boundary conditions T × n = 0 , and Φ = Φ 0 on ungauged version of the potential formulations, however ΓH c , and the Neumann boundary conditions (9), (10) and (11) result in the gauged version, when vector potentials are approximated by nodal elements. [ρ∇ × T ]× n = 0 , and μ [T0 + T − ∇Φ ]⋅ n = 0 on ΓE prescribe the boundary conditions of the eddy current field Numerical simulation by FEM problem. Benchmark problem No. 7 of the TEAM Workshop The following partial differential equations are valid in consists of an asymmetrical conductor made of aluminum 7 the case of eddy current field problems when applying ( σ = 3.526·10 S/m ) with a hole (Fig. 1) [6]. magnetic vector potential: The source of magnetic field is the sinusoidal current (maximum ampere-turn is 2742AT, and f = 50Hz, ∇ × [ ∇ × A] + σA + σ∇V = 0 , and ν f = 200Hz) flowing in the coil placed above the plate. The [ ] (8) − ∇ ⋅ σA + σ∇V = 0 in Ω c . problem is a linear steady state eddy current field problem with multiply connected eddy current region. It has been The Neumann boundary conditions [ ∇ × A]× n = 0 and ν solved in the frequency domain. [ ] − σ A − ∇V ⋅ n = 0 on ΓH c , and the Dirichlet boundary conditions n × A = 0 , and V = V0 on ΓE prescribe the boundary conditions of the eddy current field problem. Eddy current field problem must be coupled to static magnetic field problem when a conducting material is placed into a non-conducting one. The coupling of the above potentials results in the following formulations: T,Φ−Φ, A,V−A, T,Φ−A, T,Φ−A−Φ, A,V−Φ, A,V−A−Φ. In this case, the interface conditions have to be implemented by the introduced potentials [1-5]. Scalar potentials can be approximated by nodal shape functions and the unknowns are associated to the nodes of the FEM mesh. Nodal shape functions as well as vector shape functions can be used to represent vector potentials. In the first case, three unknowns are associated to the nodes, and in the second case, unknowns are associated to the edges. It is well known from the literature that these formulations are very sensitive to gauging when representing vector potentials by nodal shape functions. It can be worked out by the implicit enforcement of Coulomb gauge on the left hand side of partial differential equations in (5), (7) and (8), i.e. the following modifications must be applied [2]: (9) ∇ × [ ∇ × A] − ∇[ ∇ ⋅ A] = J 0 , ν ν [ ] Fig.1. Geometry and FEM mesh of TEAM Problem No. 7 ∇ × [ρ∇ × T ] − ∇[ρ∇ ⋅ T ] + μ T − ∇Φ = − μT0 , and (10) The reference solution is obtained by the A,V−A- ∇ ⋅ [μT − μ∇Φ ] = − μ∇ ⋅ T0 , formulation. The eddy currents have a path around the hole filled with air, as it can be seen in Fig. 2. The use of the PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 12/2008 195 reduced magnetic scalar potential Φ in the eddy current free well the A,V-A-Φ is faster than the nodal version of these region results in a false solution, as it is illustrated in Fig. 3. formulations. The nodal version of the scalar potential This is the situation when applying the A,V−Φ-formulation or based formulation is faster than the edge based ones. the T,Φ−Φ-formulation. It is noted that the aim of applying reduced scalar potential in the air region is to decrease the degree of freedom however the solution is incorrect because the aluminum plate is a multiply connected region. Fig.2. Correct distribution of eddy currents in the aluminum Fig.3. False distribution of eddy currents in the aluminum Applying some tricks, this problem can be eliminated. The problem of the A,V−Φ-formulation can be solved by using the so-called A,V−A−Φ-formulation, when the magnetic vector potential A is introduced in the hole, too, i.e. the multiply connected region can be transformed into a simple connected one. The incorrect solution of the T,Φ−Φ- formulation can be corrected by employing the T,Φ−A- formulation or the more economic T,Φ−A−Φ-formulation. The magnetic vector potential A is introduced in the whole air region, or only in the hole, respectively. The other possibility is filling the hole with a conducting material with very low conductivity, i.e. the T,Φ−Φ-formulation can be used without any modification. Fig. 4 and Fig. 5 show some comparisons between measured [6] and simulated results. Here, the z component of the magnetic flux density and the y component of the eddy current density have been shown along the line x=0,…,288mm, y=72mm, and y=144mm, z=34mm, and x=0,…,288mm, y=72mm, z=0mm, and z=19mm, respectively. The solutions of the different potential formulations are practically identical. Degree of freedom (DOF), number of iteration of the linear system solver, and Fig. 4. Bz component, f = 50Hz the CPU time (AMD Athlon 64 X2 Dual Core Processor 4600+ 2.41GHz, 4GByte RAM) can also be studied in Table 1-Table 4. The edge element based A,V-A formulation as 196 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 12/2008 Table 2. Data of computations A,V-A-Φ DOF Iteration Time 55554/nodal 980 1450 55554/nodal 9995 9919 58377/vector 185 259 58377/vector 180 255 Table 3. Data of computations T,Φ-Φ DOF Iteration Time 53204/nodal 50 319 53204/nodal 60 345 55949/vector 216 278 55949/vector 277 355 Table 4. Data of computations T,Φ-A-Φ DOF Iteration Time 53422/nodal 47 340 53422/nodal 44 325 55999/vector 2353 2409 55999/vector 1092 1171 Conclusions The paper is a summary of some potential formulations and their FEM representation. Detailed presentation is shown in the book [5]. The advantages and disadvantages of the presented methods have been shown by a problem containing a multiply connected region. The aim of further research is to solve nonlinear eddy current field problems. Acknowledgments: This paper was supported by the János Bolyai Research. Scholarship of the Hungarian Academy of Sciences (BO/00064/06), by the Hungarian Scientific Research Fund (OTKA PD 73242), and by the Széchenyi István University (15-3210-02). REFERENCES [1] Jackson J. D., Classical electrodynamics, J. Wiley, New York, 1962. [2] Bíró O., Richter K. R., CAD in electromagnetism, Adv. in Elect. and Elect. Physics, Vol. 82, 1991. [3] Jin J., The finite element method in electromagnetics, John Wiley and Sons, New York, 2002. [4] Bíró O., Edge element formulations of eddy current problems. Comput. Meth. Appl. Mech. Engrg., Vol. 169, 1999, pp. 391– 405. [5] Kuczmann M., Iványi A., The finite element method in magnetism, Akadémiai Kiadó, Budapest, to be published in 2008. [6] Fujiwara K., Nakata T., Results for benchmark problem 7. COMPEL, Vol. 11, pp. 335–344, 1990. Author: Dr. Miklós Kuczmann, PhD, Széchenyi István University, Department of Telecommunications, Laboratory of Electromagnetic Fields, Egyetem tér 1, H-9026 Győr, Hungary, E-mail: kuczmann@sze.hu. Fig. 5. Jy component, f = 200Hz Table 1. Data of computations A,V-A DOF Iteration Time 122020/nodal 532 1924 122020/nodal 857 2454 132344/vector 132 616 132344/vector 132 616 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 12/2008 197