Handling Nonlinearity by the Polarization Method and the Newton-Raphson Technique

Transcription

1 Handling Nonlinearity by the Polarization Method and the Newton-Raphson Technique M. Kuczmann Laboratory of Electromagnetic Fields, Department of Telecommunications, Széchenyi István University, Egyetem tér 1, Györ, H-96, Hungary 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. The nonlinear characteristics have been handled by the polarization method. The combination of Newton-Raphson scheme and the polarization formulation results in a very effective nonlinear solver, because only the derivate of the characteristics, i.e. only the permeability or the reluctivity must be used. That is why, the proposed Newton-Raphson scheme can be used to solve nonlinear problems with hysteresis, which is much faster than the fixed point method. Keywords: Nonlinear static magnetic field, polarization method, fixed point technique, Newton-Raphson technique. I. INTRODUCTION Nonlinear electromagnetic field problems can only be solved by iterative techniques. There are two widely used techniques, the fixed point method, and the Newton- Raphson method. The first one is known as a stable, but very slow algorithm, the second one has quadratic convergence speed close to the solution of nonlinear equations, however it is sometimes divergent [1,,3,4,5,6]. The nonlinear characteristics of ferromagnetic materials can be handled by the polarization method [1,,3]. This formulation is used to reformulate the nonlinear constitutive relations between the magnetic field intensity H and the magnetic flux density B as or B = H { H} = µ opt H + R, 1 H = B{ B} = ν opt B + I, where R and I are the nonlinear residual terms determined iteratively, moreover µ opt and ν opt are permeability-like and reluctivity-like quantities. In other words, the nonlinear output of a hysteresis model H { }, or B{ } can be split into a linear and a nonlinear part. In the case of fixed point method, µ opt and ν opt are usually constant [1], µ opt = µ max + µ min, and ν opt = ν max + ν min. 3 Here µ max, µ min, ν max, and ν min are the maximum and the minimum slope of the nonlinear characteristics represented by H { } and B{ }. This selection results in global convergence of the fixed point technique. The other possibility of representation of nonlinear behavior is applying nonlinear permeability or reluctivity [4,5,6], i.e. B = [µ H] H, or H = [ν B] B. It has the main disadvantage that the derivatives of the tensor [µ ], or [ν ] has to be determined, and these functions must be monotonous. In general [µ ] and [ν ] are not monotonous. In this case, usual Newton-Raphson technique can not be used. This paper presents a combination of the polarization formulation and the Newton-Raphson method, which results in an algorithm as simple as the fixed point technique, and only the characteristics must be monotonous, neither the permeability nor the reluctivity. II. GOVERNING NONLINEAR EQUATIONS Let us suppose that the domain of interest Ω = Ω Ω m Ω and Ω m are filled with air and with magnetic material, respectively is surrounded by the boundary Ω = Γ, which can be decomposed into two parts refer to Fig.1, i.e. Γ = Γ H, where boundary conditions are prescribed. Nonlinear static magnetic field problems can be described by the following Maxwell s equations [7,8], H = J, in Ω, H =, in Ω m, 4 and boundary conditions B =, in Ω Ω m, 5 H n = K, on Γ H, 6 B n = b, on, 7 where J, K, b are the source current density, 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 H, or H = ν B, where µ and ν = 1/µ are the permeability and the reluctivity of vacuum. In the nonlinear region filled with ferromagnetic material, the constitutive relations 1 and can be used. n Γ H Γ H air µ magnetic material Γ m Ω m µ r, ν r or B{ }, H { } Γ m Ω n m H, B Figure 1. General scheme of nonlinear static magnetic field problems. n i

2 A. The magnetic vector potential by nodal FEM The magnetic vector potential A can be introduced from 5, i.e. B = A [7,8]. Substituting it back to and 4, and enforcing Coulomb gauge implicitly result in the linear and nonlinear partial differential equations ν A ν A = J, 8 ν opt A ν opt A = I, 9 in Ω and in Ω m, respectively. The following boundary conditions are appended to these equations: ν opt A + I n = K, on Γ H, 1 A n =, on Γ H, 11 n A = α, on, 1 ν opt A =, on. 13 Equations 1 and 13 are valid on the segment of Γ H and, which are the boundary of the nonlinear material, moreover ν opt = ν, and I = must be used on the rest. In this case, the magnetic vector potential is approximated by nodal shape functions, i.e. three unknowns are associated to every node of the finite element mesh. B. The magnetic vector potential by edge FEM The other possibility is using the ungauged formulation of the magnetic vector potential, which is approximated by the edge shape functions, i.e. the unknowns are associated to the edges of the finite element mesh. The source current density must be approximated by using the impressed current vector potential T, where [7,8,9] T = J, 14 since J =, i.e. the following partial differential equations and boundary conditions are valid: ν A = T, in Ω, 15 ν opt A = T I, in Ω m, 16 ν opt A + I n = K, on Γ H, 17 n A = α, on. 18 In 17, ν opt = ν and I = are assumed if Γ H bounds the air region. C. 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 4 [7,8] H = T 19 can be obtained, where is called reduced current vector potential. Substituting it back to 1 and to 5 results in the partial differential equations µ = µ T, in Ω, µ opt = µ optt R, in Ω m, 1 and boundary conditions =, on Γ H, µ T opt n + R n = b, on. 3 In 3, µ opt = µ and R = are assumed if bounds the air region. It is evident that the reduced magnetic scalar potential is approximated by nodal finite elements. D. The impressed current vector potential The impressed current vector potential has been obtained by the solution of the partial differential equation [8] T = J, in Ω Ω m 4 with the following boundary conditions T n = K, on Γ H, 5 T n =, on. 6 This ungauged formulation can be solved by the usual finite element procedures assuming edge finite elements to approximate T. III. THE NEWTON-RAPHSON METHOD WITH THE POLARIZATION FORMULATION The magnetic vector potential as well as the reduced magnetic scalar potential have to be updated by A and by in every nonlinear iteration steps [4,5,6], A A + A, and +, 7 and the modification of the potentials is the unknown. The weak form of the potential formulations can be built up in the usual way using the updated values in 7. Only the nonlinear terms I, and R are linearized, and I A + A = I A + [ν d ] [1]ν opt A, 8 R + = R [µ d ] [1]µ opt, 9 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. Applying 8 and 9 in the weighted residual method, the weak form of the potential formulations can be obtained. A. The magnetic vector potential by nodal FEM After applying the weighted residual method to the nonlinear partial differential equation and boundary conditions presented in II.A, the following weak formulation can be obtained: ν W AdΩ + ν W AdΩ Ω Ω m Ω Ω m + ν W AdΩ + ν W A dω Ω Ω Ω m + W [ν d ] A dω + W I dω Ω m Ω m W J dω W K dγ =. Ω Γ H 3

3 Here W is the vector weighting function. In the first, second and fourth integrals ν = ν and ν = ν opt in Ω and in Ω m, respectively. The magnetic vector potential is updated by A, which is the solution of 3 in every iteration steps. B. The magnetic vector potential by edge FEM After applying the weighted residual method to the partial differential equations and boundary conditions presented in II.B, the following weak formulation can be obtained: ν W AdΩ + ν W AdΩ Ω Ω m Ω + W [ν d ] AdΩ + W I dω Ω m Ω m W T dω =, Ω Ω m 31 where W is the vector weighting function. In the first integral ν = ν and ν = ν opt in Ω and in Ω m, respectively. The magnetic vector potential is updated by A, which is the solution of 31 in every iteration steps. C. The reduced magnetic scalar potential After applying the weighted residual method to the partial differential equations and boundary conditions presented in II.C, the following weak formulation can be obtained: µ N dω + µ N dω Ω Ω m Ω + N [µ d ] dω N R dω Ω m Ω m µ N T dω Nb dγ =. Ω Ω m 3 Here N is the scalar weighting function. In the first and last volume integrals µ = µ and µ = µ opt in Ω and in Ω m, respectively. The magnetic scalar potential is updated by, which is the solution of 3 in every iteration steps. D. The under relaxation technique Sometimes, the updating term A or can be too large, which results in a divergent process. That is why, an under relaxation technique must be realized while applying the Newton-Raphson technique. From the magnetic energy W = w dω = 1 H B dω, 33 Ω Ω m Ω Ω m the following energy balance equation can be obtained when using the reduced scalar potential formulation: 1 [ µ ] T µ dω 1 Ω Ω m R dω =. Ω m 34 Here µ = µ and µ = µ opt in Ω and in Ω m, respectively. The aim of under relaxation is to find an optimal value of the parameter α, which results in a decreasing sequence of the expression in 34 while updating the scalar potential by + α. 35 In this case, the left hand side of 34 can be denoted by the function fα, which should reach zero. The under relaxation algorithm is as follows, 1. Initialization: α 1 =, α = 1, α 3 =.5, where fα i, and f α i = dfα/dα are calculated, and i = 1,, 3.. A fourth order polinomial fα is fitted applying five parameters and fα i i = 1,, 3, and f α i i = 1,. 3. Find the real roots of fα, and select α [α 1, α ], if any. 4. If all the roots are complex, then find α, which minimize the polinomial fα by applying f α. 5. The solution from point 3. or 4. is denoted by x. Calculate fx and fx, then check the difference between them, i.e. fx fx < ε. 36 If the error is small enough then α = x can be used as an under relaxation parameter, otherwise find a new triplet bracketing the solution, and step back to. The under relaxation algorithm according to the magnetic vector potential can be realized in a similar way. IV. THE NEWTON-RAPHSON METHOD AND THE FIXED POINT ALGORITHM It is interesting to note that, the classical fixed point method can be formulated by using the proposed Newton- Raphson algorithm by using the following substitutions: [ν d ] = [1]ν opt, and [µ d ] = [1]µ opt. V. SIMULATION RESULTS Three problems a modified version of TEAM 1, TEAM 13, and a modified version of TEAM 4 given by the Compumag Society have been solved by the above mentioned Newton-Raphson algorithm and by the classical fixed point technique. The results and the computational costs have been compared. The same nonlinear characteristics presented in Fig. have been used, and the curve has been approached by a piecewise linear approximation. Originally, this nonlinearity is measured using the material from TEAM 13. The permeability can be seen in Fig. 3. A. Modified version of TEAM Problem No. 1 The problem can be seen in Fig. 4 [1]. Steel plates have been placed around a racetrack shaped coil. Only the eigth of the problem can be analyzed, because of symmetry. Two types of mesh have been tried out and analyzed. The first one consists of 8413 tetrahedra, the second one has been built up by 4488 finite elements. The first mesh results in 185, 5778, and unknowns for, vector A, and nodal A formulations, respectively. The second discretization results in 6387, 93814, and 181 unknowns for the same formulations.

4 B [T] 1 z [mm] H [A/m] y [mm] Figure : The nonlinear characteristics of the problems. Figure 5: Magnetic flux density inside the central plate. 8 x AT µ [Vs/Am] AT H [A/m].8.4 A B C D Figure 3. The permeability as a function of the magnetic field. Figure 6. Distribution of magnetic flux density inside the plates AT. Figure 4: Steel plates around a coil, Problem TEAM AT The distribution of magnetic flux density vectors inside the central plate can be seen in Fig. 5. Fig. 6 and Fig. 7 show the magnetic flux density along a line placed inside the plates, and below the horizontal plate the source current is given in the figures. 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. Table I presents the number of nonlinear steps and the computation time in the case of Newton-Raphson method and the fixed point method, respectively in the form NR/FP for the coarse and the fine mesh. It can be seen that the Newton-Raphson method is much faster. Some measured data is known from [1]. The comparison between mea- Figure 7. plate. E Distribution of magnetic flux density under the horizontal sured and simulated results can be studied in Table II. The edge element based A formulation gives the best results. B. TEAM Problem No. 13 This problem is a modified version of TEAM 1 [11,1]. The U-shaped yokes have been translated as it can be seen in Fig. 8. The coarse mesh consists of 8777 tetrahedra, which results in 418, 18666, and 134 unknowns for, edge F

5 TABLE I: SIMULATION RESULTS, TEAM 1 Formulation No. of steps Time[sec] 11/88 39/7587 A-vector 1/ /5869 A-nodal 7/77 98/ / /1445 A-vector 11/ /5576 A-nodal 1/ /1975 TABLE II: SIMULATION RESULTS I, TEAM 1 Meas / / / A-v / / /1.184 A-n. 1.66/ / / z [mm] y [mm].5 3 Figure 1: Magnetic flux density inside the central plate Measured.8.4 Figure 8: Steel plates around a coil, Problem TEAM 13. element based A, and nodal A, respectively. The dense mesh contains finite elements. It means , , and unknowns for the same potentials. The fine mesh is shown in Fig. 9. Only the half of the geometry has been analyzed, periodic boundary condition has not taken into consideration. The magnetic flux density vectors inside the central I shaped plate can be seen in Fig. 1, where the effect of translated plates can be seen easily. Fig. 11 and Fig. 1 show 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 Figure 11. A B C D Distribution of magnetic flux density inside the plates E F Figure 1. plate. Distribution of magnetic flux density under the horizontal Figure 9: The FEM mesh of the half of the problem. in the last example, too. Comparison between measured and simulated data shows that the vector A formulation is the closest to the measured data. Table III presents the number of nonlinear steps and the computation time in the case of Newton-Raphson method and the fixed point method, respectively in the form NR/FP for the coarse and the fine mesh, which shows that the Newton-Raphson method is much faster. Here, the fixed point method has not used in the case of fine mesh.

6 TABLE III: SIMULATION RESULTS, TEAM 13 Formulation No. of steps Time[sec] 7/ /46 A-vector 5/ /4663 A-nodal 6/ /63 8/- 1753/- A-vector 7/- 587/- A-nodal 7/- 489/- B [T] C. TEAM Problem No. 4 This is a modified version of the problem TEAM 4 Fig. 13 [13]. 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 is given by Fig.. The number of finite elements is 396, which results in 5585 and 5653 unknowns for the -formulation and for the edge element based A-formulation, respectively see Table IV for the comparisons between the Newton- Raphson method and the fixed point technique. The magnetic flux is mainly driven by the stator and the rotor steels as it can be seen in Fig. 14, and a comparison between simulated results is shown in Fig. 15. Figure 13: Model of TEAM Problem 4. Figure 14: Magnetic flux density vectors inside the motor. TABLE IV: SIMULATION RESULTS, TEAM 4 Formulation No. of steps Time[sec] 1/ /3979 A-vector 7/- 445/ Position Figure 15. Magnetic flux density along the path presented in Fig. 14. VI. CONCLUSION The combination of the polarization method and the Newton-Raphson technique results in a powerful and fast nonlinear solver to solve nonlinear magnetic field problems as it can be read in this paper. 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 Preisach model of hysteresis. This paper was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences BO/64/6, by Széchenyi István University , and by the Hungarian Scientific Research Fund OTKA PD 734. REFERENCES [1] I. F. Hantila, Mathematical model of the relation between B and H for non-linear media, Revue Roumaine Des Sciences Techniques, Electrotechnique et Energetique, Bucarest, vol. 19, pp , [] W. Peterson, Fixed-Point Technique in Computing Nonlinear Eddy Current Problems, COMPEL, vol., no., pp. 31-5, 3. [3] J. Saitz, 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, no. 3, pp , [4] K. Fujiwara, T. Nakata, N. Okamoto, and K. Muramatsu, Method for Determining Relaxation Factor for Modified Newton-Raphson Method, IEEE Trans. on Magn., vol. 9, no., pp , [5] J. O Dwyer, and T. O Donnell, Choosing the Relaxation Factor for the Solution of Nonlinear Magnetic Field Problems by the Newton- Raphson Method, IEEE Trans. on Magn., vol. 31, no. 3, pp , [6] C. S. Koh, J. S. Ryu, and K. Fujiwara, Convergence Acceleration of the Newton-Raphson Method Using Successive Quadratic Function Approximation of Residual, IEEE Trans. on Magn., vol. 4, no. 4, pp , 6. [7] O. Biro, CAD in Electromagnetism, Advances in electronics and electron physics, vol. 8, pp. 1-96, [8] M. Kuczmann, A. Ivanyi, The Finite Element Method in Magnetism, Akadmiai Kiad, Budapest, 8, submitted for publication. [9] O. Biro, Edge Element Formulations of Eddy Current Problems, Computer Methods in Applied Mechanics and Engineering, vol. 16, pp , [1] K. Preis, I. Brdi, O. Biro, C. Magele, W. Renhart, K. R. Richter, and G. Vrisk, Numerical Analysis of 3D Magnetostatic Fields, IEEE Trans. on Magn., vol. 7, no. 5, pp , [11] T. Nakata, and K. Fujiwara, Summary of Results for Benchmark Problem 13 3-D Nonlinear Magnetostatic Model, COMPEL, 199. [1] K. Preis, I. Bardi, O. Biro, C. Magele, G. Vrisk, and K. R. Richter, Different Finite Element Formulations of 3D Magnetostatic Fields, IEEE Trans. on Magn., vol. 8, no., pp , 199. [13] N. Allen, and D. Rodger, Description of TEAM Workshop Problem 4: Nonlinear time-transient rotational test rig, 1996.