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.
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.
4. Jiles, D. C. and D. L. Atherton, "Theory of ferromagnetic hysteresis," Journal of Magnetism and Magnetic Materials, Vol. 61, 48-60, Sep. 1986.
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.
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.
7. Broyden, C. G., "A class of methods for solving nonlinear simultaneous equations," Math. Comp., Vol. 19, 577-593, 1965.
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.
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.
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.
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.
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.
15. Peterson, A. F., "Absorbing boundary conditions for the vector wave equation," Microw. Opt. Tech. Lett., Vol. 1, No. 2, 62-64, 1988.
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.
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.
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.
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.
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.