volume | PIER Journals

Abstract

In this work, numerical analysis of nonlinear ferromagnetic problems is presented using the three-dimensional time-domain finite element method (TDFEM). Formulated with the secondorder nonlinear partial differential equation (PDE) combined with the inverse Jiles-Atherton (J-A) vector hysteresis model, the nonlinear problems are solved in the time domain with the Newton-Raphson method. To solve the ordinary differential equation (ODE) representing the magnetic hysteresis accurately and efficiently, several ODE solvers are specifically designed and investigated. To improve the computational efficiency of the Newton-Raphson method, the multi-dimensional secant methods, aka Broyden's methods, are incorporated in the nonlinear TDFEM solver. A nonuniform time-stepping scheme is also developed using the weighted residual approach to remove the requirement of a uniform time-step size during the simulation. The capability and the performance of the proposed methods are demonstrated by various numerical examples.

1. Yan, S. and J.-M. Jin, "Theoretical formulation of a time-domain finite element method for nonlinear magnetic problems in three dimensions (Invited Paper)," in the Commemorative Collection on the 150-Year Anniversary of Maxwell's Equations, Progress In Electromagnetics Research, Vol. 153, 33-55, 2015.

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

3. 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

4. Jiles, D. C. and D. L. Atherton, "Theory of ferromagnetic hysteresis," Journal of Magnetism and Magnetic Materials, Vol. 61, 48-60, Sep. 1986.
doi:10.1016/0304-8853(86)90066-1

5. 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

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

7. Broyden, C. G., "A class of methods for solving nonlinear simultaneous equations," Math. Comp., Vol. 19, 577-593, 1965.
doi:10.1090/S0025-5718-1965-0198670-6

8. 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

9. 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

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

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

12. 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

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

14. 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

15. 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

16. 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

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

18. Albanese, R. and G. Rubinacci, "Solution of three dimensional eddy current problems by integral and differential methods," IEEE Trans. Magn., Vol. 24, 98-101, Jan. 1998.

19. Lee, S. H., "Efficient finite element electromagnetic analysis for high-frequency/high-speed circuits and multiconductor transmission line,", Ph.D. dissertation, University of Illinois at Urbana-Champaign, Urbana, IL, USA, 2009.

20. Jorgensen, E., J. L. Volakis, P. Meincke, and O. Breinbjerg, "Higher order hierarchical Legendre basis functions for electromagnetic modeling," IEEE Trans. Antennas Propag., Vol. 52, No. 11, 2985-2995, Nov. 2004.
doi:10.1109/TAP.2004.835279

21. Nakata, T., T. Takahashi, K. Fujiwara, and P. Olszewski, "Analysis of magnetic fields of 3-D nonlinear magnetostatic model (problem 13)," Proc. of the European TEAM Workshop and Int. Sem. on Elecmagn. Field Anal., Oxford, England, Apr. 1990.

22. 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

23. 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

24. 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.

Dr. Su Yan

Professor Jian-Ming Jin

Dr. Chao-Fu Wang

Dr. Joseph D. Kotulski