NONLINEAR FINITE ELEMENT METHOD IN MAGNETISM

POLLACK PERIODICA An International Journal for Engineering and Information Sciences DOI: 10.1556/Pollack.4.2009.2.2 Vol. 4, No. 2, pp. 13 24 (2009) www.akademiai.com NONLINEAR FINITE ELEMENT METHOD IN MAGNETISM Miklos KUCZMANN Laboratory of Electromagnetic Fields, Department of Telecommunication Széchenyi István University, Egyetem tér 1, H-9026 Győr, Hungary e-mail: kuczmann@sze.hu Received 9 February 2009; accepted 26 May 2009 Abstract: The paper presents and compares three potential formulations to solve nonlinear static magnetic field problems by applying the fixed-point technique and the Newton-Raphson scheme. Nonlinear characteristics have been handled by the polarization method in the two algorithms. The proposed combination of Newton-Raphson scheme and the polarization formulation result in a very effective nonlinear solver, because only the derivate of the characteristics, i.e. only the permeability or the reluctivity has to be used. That is why, this method can be prosperous to solve nonlinear problems with hysteresis, and it is faster than the classical fixed-point method. Keywords: Finite element method, Fixed-point method, Newton-Raphson method, Nonlinear magnetics 1. Introduction Nonlinear electromagnetic field problems can only be solved by iterative algorithms. There are two widely used techniques, the fixed-point method, and the Newton- Raphson scheme. The first one is known as a stable, but very slow algorithm, while the second one has quadratic convergence speed close to the solution of nonlinear equations, however Newton-Raphson scheme is sometimes divergent [1]-[7]. The nonlinear characteristics of ferromagnetic materials can be handled by the polarization formulation [1]-[5]. This formulation is used to prescribe the nonlinear constitutive relations between the magnetic field intensity H and the magnetic flux density B as HU ISSN 1788 1994 2009 Akadémiai Kiadó, Budapest

14 M. KUCZMANN and B = µ H + R, (1) H = ν B + I, (2) where R and I are the nonlinear residual terms determined iteratively, moreover µ and ν are permeability-like and reluctivity-like quantities. In other words, the nonlinear output of the hysteresis models can be split into a linear part and a nonlinear part. In the case of fixed-point method, µ and ν are usually constant [1]-[5], and µ max + µ µ = min, (3) 2 ν max + ν ν = min. (4) 2 Here µ max, µ min, ν max, and ν min are the maximum and the minimum slope of the nonlinear characteristics. This selection results in global convergence of the fixed-point technique. The other possibility for representation of nonlinear behavior is the application of nonlinear permeability or nonlinear reluctivity [6]-[8], i.e. B = [ µ ( H )]H or H = [ ν ( B) ]B. The main disadvantage of this method is that the derivatives of the elements of tensors [ µ () ] or [ ν () ] have to be determined, and these functions must be monotonous. In general, [ µ () ] and [ ν ( ) ] is not monotonous. This is the reason why, in the case of hysteresis, the classical Newton-Raphson technique can be difficult to use. This paper presents a combination of the polarization formulation (1) or (2) and the Newton-Raphson method, which results in an algorithm as simple as the fixed-point technique, and only the characteristics have to be monotonous, neither the permeability nor the reluctivity. 2. Governing nonlinear equations Let suppose the domain of interest = 0 m, ( 0 and m are filled with air and with magnetic material, respectively) is surrounded by the boundary Γ, which can be decomposed into two parts (refer to Fig. 1), i.e. Γ = Γ H ΓB, where boundary conditions are prescribed [9], [10]. Nonlinear static magnetic field problems can be described by the following Maxwell s equations [9], [10]

NONLINEAR FINITE ELEMENT METHOD IN MAGNETISM 15 H = J, B = 0, in, (5) where H, B and J are the magnetic field intensity, the magnetic flux density and the source current density. Source current density is placed only in the air region. The following boundary conditions are prescribed on the boundary [9], [10] H n = K, B n = b, on on ΓH, ΓB, (6) where K and b are the surface current density, and the charge density of fictitious magnetic surface charges, moreover n is the outer normal unit vector of the domain. The constitutive relation between the magnetic field intensity and the magnetic flux density is linear in air, B = µ 0H or H = ν 0B, where µ 0 and ν 0 = 1 µ 0 are the permeability and the reluctivity of vacuum. In the nonlinear region filled with ferromagnetic material, the constitutive laws in the form of (1) or (2) can be used. Fig. 1. Structure of a static magnetic field problem 2.1. The magnetic vector potential by nodal FEM The magnetic vector potential A can be introduced from the second equation in (5), i.e. [9], [10] B = A. (7) Substituting it to (2) and (5), and enforcing Coulomb gauge implicitly, result in the following partial differential equations [9], [10] ( ν 0 A ) ( ν 0 A) = J, in 0, (8)

16 M. KUCZMANN ( ν A ) ( ν A) = I, in m. (9) The following boundary conditions are appended to these equations [9], [10], ( ν A + I ) ( ν A) n = K, 0 n = K, on Γ H, (10) A n = 0, on Γ H, (11) n A = α, on Γ B, (12) ν A = 0, ν0 A = 0, on Γ B. (13) Here the term α is a constant representing b in (6) [9]. In this case, the magnetic vector potential is approximated by nodal shape functions. 2.2. The magnetic vector potential by edge FEM The other possibility is using the ungauged formulation of magnetic vector potential, which is approximated by the edge shape functions [9], [10], [11], i.e. the unknowns are associated to the edges of the mesh. The source current density must be approximated by the impressed current vector potential T as T = J, in, (14) since J = 0, i.e. the following partial differential equations are valid [9], [10], [11], ( ν 0 A ) = T, in 0, (15) ( ν A ) = T I, in m, (16) with the boundary conditions (10) and (12), and K = 0 according to the first equation in (23). 2.3. The reduced magnetic scalar potential The source current density is approximated by using the impressed current vector potential T as in (14), and from the first Maxwell s equation in (5) [9], [10], [11] H = T Φ (17)

NONLINEAR FINITE ELEMENT METHOD IN MAGNETISM 17 can be obtained, where Φ is the so called reduced magnetic scalar potential, and it is approximated by nodal FEM. Substituting it back to (1) and to (5) results in the partial differential equations, ( µ 0 Φ ) = ( µ 0T ), in 0, (18) ( µ Φ ) = ( µ T ) R, in m, (19) and in the following boundary conditions are valid Φ = 0, on Γ H, (20) [ µ ( T Φ ) + R] ( T Φ ) n = b, µ 0 n = b, on ΓB. (21) 2.4. The impressed current vector potential The impressed current vector potential can be obtained by the solution of the partial differential equation [10] T = J, in, (22) with the following boundary conditions T n = K, T n = 0, on on ΓH, ΓB. (23) This ungauged formulation can be solved by the usual finite element procedures (see in [10]). 3. The Newton-Raphson method in the polarization formulation The magnetic vector potential as well as the reduced magnetic scalar potential has to be updated by A and by Φ in every nonlinear iteration steps [5], [6], [7]. The weak form of the potential formulations can be built up by using the weighted residual method and the Galerkin scheme [9], [10]. The nonlinear terms I and R are linearized as and I ( A + A) I( A) + ([ ] [ 1] ν ) ( A) ν d, (24) ( Φ Φ ) R( Φ ) ([ µ ] [ 1] µ ) ( Φ ) R + d, (25)

18 M. KUCZMANN where [ ν d ] and [ µ d ] are the differential reluctivity tensor and the differential permeability tensor of the nonlinear characteristics, which are updated in every nonlinear steps. Here µ and ν are selected as in (1) and in (2). Applying (24) and (25) in the weighted residual method, the weak form of the potential formulations can be obtained. 3.1. The magnetic vector potential by nodal FEM After applying the weighted residual method to the partial differential equations and boundary conditions presented in 2.1, the following weak formulation can be obtained, ν 0 W ( A + A) + ν W A 0 + W [ ν d ] m ( A) + ν W ( A + A) + W I m W J W K dγ = 0. Γ 0 H + ν 0 W ( A + A) m m 0 (26) Here W is the vector weighting function [9], [10]. The magnetic vector potential is updated by A, which is the solution of (26) in every iteration steps. After applying the weighted residual method to the partial differential equations and boundary conditions presented in 2.2, a weak formulation similar to (23) can be obtained. 3.2. The magnetic vector potential by edge FEM After applying the weighted residual method to the partial differential equations and boundary conditions presented in 2.2, the following weak formulation can be obtained ν 0 W ( A + A) + ν W A 0 m + W [ ν d ] m ( A) + W I W T = 0, m 0 m (27) where W is the vector weighting function. The magnetic vector potential is updated by A, i.e. the solution of (27). 3.3. The reduced magnetic scalar potential After applying the weighted residual method to the partial differential equations and boundary conditions presented in 2.3, the following weak formulation can be obtained,

NONLINEAR FINITE ELEMENT METHOD IN MAGNETISM 19 µ 0 N ( Φ + Φ ) + µ N Φ + N [ µ d ] 0 m m N R µ N T Nb dγ = 0. m 0 m ΓB ( Φ ) (28) Here N is the scalar weighting function [9], [10]. The magnetic scalar potential is updated by Φ, which is the solution of (28) in every iteration steps. 4. Simulation results Three problems (a modified version of TEAM 10, TEAM 13, and a modified version of TEAM 24 defined by the COMPUMAG Society) have been solved by the above-introduced Newton-Raphson algorithm and by the classical fixed-point technique. The same nonlinear characteristics presented in Fig. 2 have been used, and the curve has been approached by a simple piecewise linear approximation. Originally, this nonlinearity is measured using the material from TEAM 13. 4.1. Modified version of TEAM problem No. 10 The problem can be seen in Fig. 3 [12]. Steel plates have been placed around a racetrack shaped coil. Only the eighth, of the problem can be analyzed, because of symmetry. Fig. 2. The nonlinear characteristics of the simulated problems Fig. 3. The arrangement of TEAM problem No. 10 Two types of mesh have been analyzed. The first one consists of 8413 tetrahedra, the second one has been built up by 44828 finite elements. The first mesh results in 12850, 57278, and 43539 unknowns for Φ, vector A, and nodal A formulation, respectively. The second discretization results in 63807, 293814, and 218100 unknowns for the same formulation.

20 M. KUCZMANN Fig. 4 shows the finer mesh of the problem. The magnetic flux density is driven by the steel plates around the coil as it can be seen in Fig. 5. Fig. 4. The FEM mesh of TEAM 10 Fig. 5. Magnetic flux is driven by the plates Fig. 6 shows the magnetic flux density along a line [12] placed inside the plates (the source current density is given in the figures) and along a line just below the horizontal plate. The three formulations give practically the same results. It is noted that, the nodal A formulation is more sensitive to the density of the finite element mesh. Fig. 6. Distribution of magnetic flux density inside the plates and in air under the horizontal plate The Newton-Raphson method is much faster than the classical fixed-point method as it can be seen in Fig. 7. The measured magnetic flux density in the center of the central plate is 1.67 T [12]. The simulated data are 1.7065 T, 1.6968 T, 1.6060 T by the reduced magnetic scalar potential formulation, by the edge element represented magnetic vector potential and by

NONLINEAR FINITE ELEMENT METHOD IN MAGNETISM 21 the magnetic vector potential approximated by nodal finite elements. It can be concluded that the edge element based A formulation gives the best result, moreover B simulated by the reduced magnetic scalar potential is a little larger than the results obtained from the magnetic vector potential, moreover the nodal vector potential formulation is more sensitive to the density of the mesh 4.2. TEAM problem No. 13 Fig. 7. Comparison of the number of iterations by Newton-Raphson method and the fixed-point method This problem is a modified version of TEAM 10 [13], [14]. The U-shaped yokes have been translated as it can be seen in Fig. 8. Only the half of the geometry has been analyzed. Fig. 8. The arrangement of TEAM problem No. 13 The coarse mesh consists of 28777 tetrahedra, which results in 40180, 186626, and 134022 unknowns for Φ, edge element based A, and nodal A, respectively. The

22 M. KUCZMANN dense mesh contains 147231 finite elements. It means 198913, 938694, and 695382 unknowns for the same potentials. Fig. 9 shows some comparisons between the results simulated by the three, presented potential formulations. The magnetic flux density simulated by the reduced magnetic scalar potential is a little larger than the results obtained from the magnetic vector potential formulations, moreover the nodal vector potential formulation is more sensitive to the density of the mesh, as it was experienced in the last example, too. Comparison between measured and simulated data shows that the vector A formulation is the closest to the measured data. Fig. 9. Distribution of magnetic flux density inside the plates 4.3. Modified version of TEAM problem No. 24 This is a modified version of the problem TEAM 24 (Fig. 10) [15]. The problem contains a motor which rotor has been locked. The source current of the coils is constant, i.e. a static magnetic field problem has been analyzed, and the characteristics of the stator and the rotor are given in Fig. 2. Fig. 10. Model of TEAM problem 24

NONLINEAR FINITE ELEMENT METHOD IN MAGNETISM 23 The number of finite elements is 39060, which results in 55825 and 256530 unknowns for the Φ -formulation and for the edge element based A -formulation, respectively. The magnetic flux is mainly driven by the stator and the rotor steels as it can be seen in Fig. 11, and a comparison between the simulated results is shown in Fig. 12. Fig. 11. The magnetic flux density inside the motor Fig. 12. Magnetic flux density along the path shown in Fig. 11 5. Conclusion It can be concluded that the mentioned Newton-Raphson method combined with the polarization formulation is much faster than the fixed-point technique, however the weak formulations and the applied algorithms are almost the same. Only the permeability or the reluctivity of the material has to be simulated.

24 M. KUCZMANN The aim of further research is to find the way to solve nonlinear eddy current field problems, and taking the nonlinearity into account by the vector Preisach model of hysteresis. Acknowledgements This paper was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences (BO/00064/06), by Széchenyi István University (15-3210-02), by the Hungarian Scientific Research Fund (OTKA PD 73242), and by Hungarian Science and Technology Foundation (OMFB-00725/2008). References [1] Dlala E., Belachen A., Arkkio A. A fast fixed-point method for solving magnetic field problems in media with hysteresis, IEEE Trans. on Magn, Vol. 44, 2008, pp. 1214 1217. [2] Dlala E., Arkkio A. Analysis of the convergence of the fixed-point method used for solving nonlinear rotational magnetic field problems, IEEE Trans. on Magn, Vol. 44, 2008, pp. 473 478. [3] Auserhofer S., Bíró O., Preis K. A strategy to improve the convergence of the fixed-point method for nonlinear eddy current problems, IEEE Trans. on Magn, Vol. 44, 2008, pp. 1282 1285. [4] Saitz J. Newton-Raphson method and fixed-point technique in finite element computation of magnetic field problems in media with hysteresis, IEEE Trans. on Magn, Vol. 35, 1999, pp. 1398 1401. [5] Chiampi M., Repetto M., Chiarabaglio D. An improved technique for nonlinear magnetic problems, IEEE Trans. on Magn, Vol. 30, 1994, pp. 4332 4334. [6] O Dwyer J., O Donnell T. Choosing the relaxation factor for the solution of nonlinear magnetic field problems by the Newton-Raphson method, IEEE Trans. Magn, Vol. 31, 1995, pp. 1484 1487. [7] Koh C. S., Ryu J. S., Fujiwara K. Convergence acceleration of the Newton-Raphson method using successive quadratic function approximation of residual, IEEE Trans. Magn, Vol. 42, 2006, pp. 611 614. [8] Fonteyn K., Belahcen A., Arkkio A. Properties of electrical steel sheets under strong mechanical stress, Pollack Periodica, Vol. 1, 2006, pp. 93 104. [9] Bíró O. CAD in electromagnetism, Advances in electronics and electron physics, Vol. 82, 1991, pp. 1 96. [10] Kuczmann M., Iványi A. The finite element method in magnetism. Akadémiai Kiadó, Budapest, 2008. [11] Bíró O. Edge element formulations of eddy current problems, Computer Methods in Applied Mechanics and Engineering, Vol. 160, 1999 pp. 391 405. [12] Preis K., Bárdi I., Bíró O., Magele C., Renhart W., Richter K.R., Vrisk G. Numerical analysis of 3D magnetostatic fields, IEEE Trans. Magn, Vol. 27, 1991, pp. 3798 3803. [13] Nakata T., Fujiwara K. Summary of results for benchmark problem 13 (3-D nonlinear magnetostatic model), COMPEL, Vol. 10. 1992, pp. 231 252. [14] Preis K., Bárdi I., Bíró O., Magele C., Vrisk G., Richter K. R. Different finite element formulations of 3D magnetostatic fields, IEEE Trans. Magn, Vol. 28, 1992, pp. 1056 1059. [15] Allen N., Rodger D. Description of TEAM workshop problem 24, nonlinear time-transient rotational test rig, http://www.compumag.co.uk/problems/problem24.pdf.