Solvability analysis of a magneto-heat coupling magnetization model for ferromagnetic materials. (English) Zbl 1499.35364
Summary: This paper studies the magneto-heat coupling magnetization problem with ferromagnetic materials, which consists of quasi-static Maxwell’s equations coupled with a heat diffusion equation. The ferromagnetic properties are described by a power material law and the electric conductivity is dependent on temperature. The temperature is governed by the heat equation with a nonlinear radiation boundary condition and the heat source term is approximated by the truncated quadratic Joule heating term. We suggest a formulation to solve the nonlinear magneto-heat coupling system and discuss the solvability of the problem. The theoretical analysis is given by means of the theory of monotone operator and Rothe’s method. Finally, the finite element method is employed to solve it approximately and some numerical results are shown here.
MSC:
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 35Q61 | Maxwell equations |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 78A25 | Electromagnetic theory (general) |
Keywords:
magnetization model; magneto-heat coupling; nonlinear degenerate eddy current equation; radiation boundary condition; solvabilityReferences:
| [1] | Adams, RA; Fournier, JJF, Sobolev spaces. Pure and applied mathematics (2003), New York: Academic Press, New York · Zbl 1098.46001 |
| [2] | Ainslie, MD; Fujishiro, H., Modelling of bulk superconductor magnetization, Supercond Sci Technol, 28, 5, 053002 (2015) · doi:10.1088/0953-2048/28/5/053002 |
| [3] | Akrivis, G.; Larsson, S., Linearly implicit finite element methods for the time-dependent Joule heating problem, BIT Numer Math, 45, 3, 429-442 (2005) · Zbl 1113.65091 · doi:10.1007/s10543-005-0008-1 |
| [4] | Barglik, J.; Doležel, I.; Karban, P.; Ulrych, B., Modelling of continual induction hardening in quasi-coupled formulation, COMPEL Int J Comput Math Electr Electron Eng, 24, 1, 251-260 (2005) · Zbl 1082.78001 · doi:10.1108/03321640510571273 |
| [5] | Bermúdez, A.; Gómez, D.; Muñiz, MC; Salgado, P., Transient numerical simulation of a thermoelectrical problem in cylindrical induction heating furnaces, Adv Comput Math, 26, 1-3, 39-62 (2006) · Zbl 1115.78010 |
| [6] | Brenner, SC; Ridgway Scott, L., The mathematical theory of finite element methods (2008), New York: Springer, New York · Zbl 1135.65042 · doi:10.1007/978-0-387-75934-0 |
| [7] | Chovan, J.; Geuzaine, C.; Slodička, M., \({{\varvec{A}}}-\phi\) formulation of a mathematical model for the induction hardening process with a nonlinear law for the magnetic field, Comput Methods Appl Mech Eng, 321, 294-315 (2017) · Zbl 1439.35459 · doi:10.1016/j.cma.2017.03.045 |
| [8] | DiBenedetto, E., Degenerate parabolic equations (1993), New York: Springer, New York · Zbl 0794.35090 · doi:10.1007/978-1-4612-0895-2 |
| [9] | Druet, P-É, Existence of weak solutions to the time-dependent MHD equations coupled to the heat equation with nonlocal radiation boundary conditions, Nonlinear Anal Real World Appl, 10, 5, 2914-2936 (2009) · Zbl 1173.80002 · doi:10.1016/j.nonrwa.2008.09.015 |
| [10] | Dunford, N., Linear operators (1988), New York: Interscience Publishers, New York · Zbl 0635.47003 |
| [11] | Elliott, CM; Larsson, S., A finite element model for the time-dependent Joule heating problem, Math Comput, 64, 212, 1433-1453 (1995) · Zbl 0846.65047 · doi:10.1090/S0025-5718-1995-1308451-4 |
| [12] | Evans LC (1990) Weak convergence methods for nonlinear partial differential equations. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence · Zbl 0698.35004 |
| [13] | Evans, LC, Partial differential equations. Graduate studies in mathematics (2010), New York: American Mathematical Society, New York · Zbl 1194.35001 |
| [14] | Hömberg, D., A mathematical model for induction hardening including mechanical effects, Nonlinear Analy Real World Appl, 5, 1, 55-90 (2004) · Zbl 1330.74044 · doi:10.1016/S1468-1218(03)00017-8 |
| [15] | Huang, Q.; Jia, S.; Xu, F.; Xu, Z.; Yao, C., Solvability of wave propagation with Debye polarization in nonlinear dielectric materials and its finite element methods approximation, Appl Numer Math, 146, 145-164 (2019) · Zbl 1477.78002 · doi:10.1016/j.apnum.2019.07.002 |
| [16] | Jiang, X.; Zhang, D.; Zhang, L.; Zheng, W., Finite element analysis for nonstationary magneto-heat coupling problem, Commun Comput Phys, 26, 5, 1471-1489 (2019) · Zbl 1473.65306 · doi:10.4208/cicp.2019.js60.08 |
| [17] | Kačur, J., Method of Rothe in evolution equations, volume 80 of Teubner-Texte zur Mathematik (1985), Leipzig: B.G. Teubner, Leipzig · Zbl 0582.65084 |
| [18] | Kang, T.; Wang, R.; Zhang, H., Potential field formulation based on decomposition of the electric field for a nonlinear induction hardening model, Commun Appl Math Comput Sci, 14, 2, 175-205 (2019) · Zbl 1434.35207 · doi:10.2140/camcos.2019.14.175 |
| [19] | Kang, T.; Wang, R.; Zhang, H., Fully discrete \({\varvec{{T}}}-\psi\) finite element method to solve a nonlinear induction hardening problem, Numer Methods Partial Differ Equ, 37, 1, 1-37 (2021) · Zbl 1535.65195 · doi:10.1002/num.22540 |
| [20] | Kufner, A.; John, O.; Fučik, S., Function spaces (1977), London: Springer, London · Zbl 0364.46022 |
| [21] | Landau, LD; Lifshitz, EM, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Perspectives in theoretical physics, 51-65 (1992), New York: Elsevier, New York · doi:10.1016/B978-0-08-036364-6.50008-9 |
| [22] | Landau, LD; Lifshitz, EM; Sykes, JB; Bell, JS; Dill, EH, Electrodynamics of continuous media, Phys Today, 14, 10, 48-50 (1961) · doi:10.1063/1.3057154 |
| [23] | Li, X.; Mao, S.; Yang, K.; Zheng, W., On the magneto-heat coupling model for large power transformers, Commun Comput Phys, 22, 3, 683-711 (2017) · Zbl 1488.35299 · doi:10.4208/cicp.OA-2016-0194 |
| [24] | Monk, P., Finite element methods for Maxwell’s equations (2003), Oxford: Clarendon Press, Oxford · Zbl 1024.78009 · doi:10.1093/acprof:oso/9780198508885.001.0001 |
| [25] | Nečas, J., Introduction to the theory of nonlinear elliptic equations (1986), Chichester: Wiley, Chichester · Zbl 0643.35001 |
| [26] | Slodička, M., A time discretization scheme for a non-linear degenerate eddy current model for ferromagnetic materials, IMA J Numer Anal, 26, 1, 173-187 (2006) · Zbl 1089.78026 · doi:10.1093/imanum/dri030 |
| [27] | Slodička, M.; Chovan, J., Solvability for induction hardening including nonlinear magnetic field and controlled Joule heating, Appl Anal, 96, 16, 2780-2799 (2017) · Zbl 1386.35230 · doi:10.1080/00036811.2016.1243661 |
| [28] | Slodička, M.; Zemanová, V., Time-discretization scheme for quasi-static Maxwell’s equations with a non-linear boundary condition, J Comput Appl Math, 216, 2, 514-522 (2008) · Zbl 1151.78004 · doi:10.1016/j.cam.2007.06.004 |
| [29] | Vaǐnberg, MM, Variational method and method of monotone operators in the theory of nonlinear equations (1973), Jerusalem: Wiley Israel Program for Scientific Translations, Jerusalem · Zbl 0279.47022 |
| [30] | Vrábel’, V.; Slodička, M., An eddy current problem with a nonlinear evolution boundary condition, J Math Anal Appl, 387, 1, 267-283 (2012) · Zbl 1234.35262 · doi:10.1016/j.jmaa.2011.08.067 |
| [31] | Yao, C., The solvability of coupling the thermal effect and magnetohydrodynamics field with turbulent convection zone and the flow field, J Math Anal Appl, 476, 2, 495-521 (2019) · Zbl 1448.76199 · doi:10.1016/j.jmaa.2019.03.061 |
| [32] | Yin, H-M, Regularity of weak solution to Maxwell’s equations and applications to microwave heating, J Differ Equ, 200, 1, 137-161 (2004) · Zbl 1068.35169 · doi:10.1016/j.jde.2004.01.010 |
| [33] | Yin, H-M; Wei, W., Regularity of weak solution for a coupled system arising from a microwave heating model, Eur J Appl Math, 25, 1, 117-131 (2013) · Zbl 1314.35183 · doi:10.1017/S0956792513000326 |
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