PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 153 > pp. 33-55

THEORETICAL FORMULATION OF A TIME-DOMAIN FINITE ELEMENT METHOD FOR NONLINEAR MAGNETIC PROBLEMS IN THREE DIMENSIONS (Invited Paper)

By S. Yan and J.-M. Jin

Full Article PDF (560 KB)

Abstract:
In this work, a numerical solution of nonlinear ferromagnetic problems is formulated using the three-dimensional time-domain finite element method (TDFEM) combined with the inverse Jiles-Atherton (J-A) vector hysteresis model. After a brief introduction of the J-A constitutive model, the second-order nonlinear partial differential equation (PDE) is constructed through the magnetic vector potential in the time domain, which is then discretized by employing the Newmark-β scheme, and solved by applying the Newton-Raphson method. Different Newton-Raphson schemes are constructed and compared. The capability of the proposed methods is demonstrated by several numerical examples including the simulation of the physical demagnetization process, the prediction of the magnetic remanence in the ferromagnetic material, and the generation of higher-order harmonics.

Citation:
S. Yan and J.-M. Jin, "Theoretical Formulation of a Time-Domain Finite Element Method for Nonlinear Magnetic Problems in Three Dimensions (Invited Paper)," Progress In Electromagnetics Research, Vol. 153, 33-55, 2015.
doi:10.2528/PIER15091005
http://www.jpier.org/PIER/pier.php?paper=15091005

References:
1. Mayergoyz, I. D., Mathematical Models of Hysteresis, Springer-Verlag, New York, 1991.
doi:10.2172/6911694

2. Smith, R. C., Smart Material Systems: Model Development, SIAM, Philadelphia, 2005.
doi:10.1137/1.9780898717471

3. Dupré, L. and J. Melkebeek, "Electromagnetic hysteresis modelling: From material science to finite element analysis of devices," International Compumag Society Newsletter, Vol. 10, No. 3, 4-15, 2003.

4. Preisach, F., "Über die magnetische nachwirkung," Zeitschrift für Physik, Vol. 94, 277-302, 1935.
doi:10.1007/BF01349418

5. Jiles, D. C. and D. L. Atherton, "Theory of the magnetisation process in ferromagnetics and its application to the magnetomechanical effect," J. Phys. D: Appl. Phys., Vol. 17, No. 6, 1265-1281, Jun. 1984.
doi:10.1088/0022-3727/17/6/023

6. Jiles, D. C. and D. L. Atherton, "Theory of the magnetisation process in ferromagnetics and its application to the magnetomechanical effect," J. Phys. D: Appl. Phys., Vol. 17, No. 6, 1265-1281, Jun. 1984.
doi:10.1088/0022-3727/17/6/023

7. Mayergoyz, I. D., "Dynamic preisach models of hysteresis," IEEE Trans. Magn., Vol. 24, No. 6, 2925-2927, Nov. 1988.
doi:10.1109/20.92290

8. Bertotti, G., "Dynamic generalization of the scalar Preisach model of hysteresis," IEEE Trans. Magn., Vol. 28, No. 5, 2599-2601, Sep. 1992.
doi:10.1109/20.179569

9. Bergqvist, A. J., "A simple vector generalization of the Jiles-Atherton model of hysteresis," IEEE Trans. Magn., Vol. 32, No. 5, 4213-4215, Sep. 1996.
doi:10.1109/20.539337

10. Leite, J. V., N. Sadowski, P. Kuo-Peng, N. J. Batistela, J. P. A. Bastos, and A. A. de Espíndola, "Inverse Jiles-Atherton vector hysteresis model," IEEE Trans. Magn., Vol. 40, No. 4, 1769-1775, Jul. 2004.
doi:10.1109/TMAG.2004.830998

11. Vecchio, R. D., "An efficient procedure for modelling complex hysteresis processes in ferromagnetic materials," IEEE Trans. Magn., Vol. 16, 809-811, 1980.
doi:10.1109/TMAG.1980.1060680

12. Miano, G., C. Serpico, L. Verolino, and C. Visone, "Comparison of different hysteresis models in FE analysis of magnetic field diffusion," IEEE Trans. Magn., Vol. 31, 1789-1792, 1995.
doi:10.1109/20.376383

13. Dupré, L., O. Bottauscio, M. Chiampi, M. Repetto, and J. Melkebeek, "Modelling of electromagnetic phenomena in soft magnetic materials under unidirectional time periodic flux excitations," IEEE Trans. Magn., Vol. 35, 4147-4184, 1999.

14. Takahashi, N., S. Miyabara, and K. Fujiwara, "Problems in practical finite element analysis using Preisach model," IEEE Trans. Magn., Vol. 35, 1243-1246, 1999.
doi:10.1109/20.767175

15. Park, G., S. Hahn, S. Lee, and H. Jung, "Implementation of hysteresis characteristics using the Preisach model with MB-variables," IEEE Trans. Magn., Vol. 29, 1542-1545, 1993.
doi:10.1109/20.250697

16. Dupré, L., R. V. Keer, and J. Melkebeek, "Complementary 2D finite element procedures for the magnetic field analysis using a vector hysteresis model," Intern. J. for Num. Meth. in Eng., Vol. 42, 1005-1023, 1998.
doi:10.1002/(SICI)1097-0207(19980730)42:6<1005::AID-NME396>3.0.CO;2-N

17. Matsuo, T., Y. Osaka, and M. Shimasaki, "Eddy current analysis using vector hysteresis models with play and stop hysterons," IEEE Trans. Magn., Vol. 36, 1172-1177, 2000.
doi:10.1109/20.877649

18. Jin, J.-M., The Finite Element Method in Electromagnetics, 3rd Ed., Wiley, Hoboken, NJ, 2014.

19. Kuczmann, M., "Using the Newton-Raphson method in the polarization technique to solve nonlinear static magnetic field problems," IEEE Trans. Magn., Vol. 46, No. 3, 875-879, 2010.
doi:10.1109/TMAG.2009.2034260

20. Fujiwara, K., T. Nakata, N. Takahashi, and K. Muramatsu, "Method for determining relaxation factor for modified Newton-Raphson method," IEEE Trans. Magn., Vol. 29, 1962-1965, Mar. 1993.
doi:10.1109/20.250793

21. Li, Y. and J.-M. Jin, "A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems," IEEE Trans. Antennas Propag., Vol. 54, No. 10, 3000-3009, Oct. 2006.
doi:10.1109/TAP.2006.882191

22. Li, Y.-J. and J.-M. Jin, "Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations," J. Comput. Phys., Vol. 228, No. 9, 3255-3267, 2009.
doi:10.1016/j.jcp.2009.01.029

23. Li, Y.-J. and J.-M. Jin, "A new dual-primal domain decomposition approach for finite element simulation of 3-D large-scale electromagnetic problems," IEEE Trans. Antennas Propag., Vol. 55, No. 10, 2803-2810, Oct. 2007.
doi:10.1109/TAP.2007.905954

24. Yao, W., J.-M. Jin, and P. T. Krein, "An efficient domain decomposition method for 3-D finite element analysis of nonlinear electric machine problems," 2013 IEEE International Electric Machines & Drives Conference (IEMDC), 709-715, May 2013.
doi:10.1109/IEMDC.2013.6556171

25. Yan, S. and J.-M. Jin, "Analysis of nonlinear electromagnetic problems using time-domain finite element method," Proc. IEEE Antennas Propag. Symp., Orlando, FL, Jul. 2013.

26. Ising, E., "Beitrag zur theorie des ferromagnetismus," Zeitschrift für Physik, Vol. 31, No. 1, 253-258, 1925.
doi:10.1007/BF02980577

27. Jiles, D. C., Introduction to Magnetism and Magnetic Materials, Chapman and Hall, New York, 1991.
doi:10.1007/978-1-4615-3868-4

28. Chikazumi, S., Physics of Ferromagnetism, 2nd Ed., Clarendon Press, Oxford, 1997, English edition prepared with the assistance of C. D. Graham, Jr.

29. Jiles, D. C., J. B. Thoelke, and M. K. Devine, "Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis," IEEE Trans. Magn., Vol. 28, No. 1, 27-35, Jan. 1992.
doi:10.1109/20.119813

30. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd Ed., Cambridge University Press, New York, 2007.

31. Ren, Z., "Influence of the R.H.S. on the convergence behaviour of the curl-curl equation," IEEE Trans. Magn., Vol. 32, No. 3, 655-658, May 1996.
doi:10.1109/20.497323

32. Whitney, H., Geometric Integration Theory, Princeton University Press, Princeton, NJ, 1957.

33. Nédélec, J. C., "Mixed finite elements in R3," Numer. Meth., Vol. 35, 315-341, 1980.
doi:10.1007/BF01396415

34. Webb, J. P., "Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements," IEEE Trans. Antennas Propag., Vol. 47, No. 8, 1244-1253, Aug. 1999.
doi:10.1109/8.791939

35. Peterson, A. F., "Absorbing boundary conditions for the vector wave equation," Microw. Opt. Tech. Lett., Vol. 1, No. 2, 62-64, 1988.
doi:10.1002/mop.4650010206

36. Webb, J. P. and V. N. Kanellopoulos, "Absorbing boundary conditions for the finite element solution of the vector wave equation," Microw. Opt. Tech. Lett., Vol. 2, No. 10, 370-372, 1989.
doi:10.1002/mop.4650021010

37. Newmark, N. M., "A method of computation for structural dynamics," J. Engineering Mechanics Division. ASCE, Vol. 85, 67-94, Jul. 1959.

38. Zienkiewicz, O. C., "A new look at the Newmark, Houboult and other time stepping formulas: A weighted residual approach," Earthquake Engineering and Structural Dynamics, Vol. 5, 413-418, 1977.
doi:10.1002/eqe.4290050407

39. Gedney, S. D. and U. Navsariwala, "An unconditionally stable finite element time-domain solution of the vector wave equation," IEEE Microw. Guided Wave Lett., Vol. 5, No. 10, 332-334, Oct. 1995.
doi:10.1109/75.465046

40. Saad, Y. and M. H. Schultz, "GMRes: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 7, No. 3, 856-869, Jul. 1986.
doi:10.1137/0907058

41. Sleijpen, G. L. G. and D. R. Fokkema, "BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum," Electronic Trans. Numer. Anal., Vol. 1, 11-32, Sep. 1993.

42. Carpentieri, B., I. S. Duff, and L. Giraud, "Experiments with sparse preconditioning of dense problems from electromagnetic applications," CERFACS, Tech. Rep. TR/PA/00/04, Toulouse, France, 2000.

43. Alléon, G. M. Benzi, and L. Giraud, "Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics," Numer. Algorithms, Vol. 16, 1-15, 1997.
doi:10.1023/A:1019170609950

44. Hantila, F. I., G. Preda, and M. Vasiliu, "Polarization method for static fields," IEEE Trans. Magn., Vol. 36, No. 4, 672-675, Jul. 2000.
doi:10.1109/20.877538

45. International Compumag Society, Testing electromagnetic analysis methods (T.E.A.M.), http://www.compumag.org/jsite/team.

46. Nakata, T., N. Takahashi, and K. Fujiwara, "Summary of results for benchmark problem 10 (steel plates around a coil)," Compel, Vol. 14, No. 2/3, 103-112, Sep. 1995.
doi:10.1108/eb010141

47. Bottauscio, O., M. Chiampi, C. Ragusa, L. Rege, and M. Repetto, "A test-case for validation of magnetic field analysis with vector hysteresis," IEEE Trans. Magn., Vol. 38, No. 2, 893-896, Mar. 2002.
doi:10.1109/20.996230

48. Yamada, S., K. Bessho, and J. Lu, "Harmonic balance finite element method applied to nonlinear AC magnetic analysis," IEEE Trans. Magn., Vol. 24, No. 4, 2971-2973, Jul. 1989.


© Copyright 2014 EMW Publishing. All Rights Reserved