On the double critical-state model for type-II superconductivity in 3D
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 3, pp. 333-374.

In this paper we mathematically analyse an evolution variational inequality which formulates the double critical-state model for type-II superconductivity in 3D space and propose a finite element method to discretize the formulation. The double critical-state model originally proposed by Clem and Perez-Gonzalez is formulated as a model in 3D space which characterizes the nonlinear relation between the electric field, the electric current, the perpendicular component of the electric current to the magnetic flux, and the parallel component of the current to the magnetic flux in bulk type-II superconductor. The existence of a solution to the variational inequality formulation is proved and the representation theorem of subdifferential for a class of energy functionals including our energy is established. The variational inequality formulation is discretized in time by a semi-implicit scheme and in space by the edge finite element of lowest order on a tetrahedral mesh. The fully discrete formulation is an unconstrained optimisation problem. The subsequence convergence property of the fully discrete solution is proved. Some numerical results computed under a rotating applied magnetic field are presented.

DOI : https://doi.org/10.1051/m2an:2008010
Classification : 65M60,  65M12,  47J20,  49J40
Mots clés : the double critical-state model for superconductivity, evolution variational inequality, Maxwell's equations, edge finite element, convergence, computational electromagnetism
@article{M2AN_2008__42_3_333_0,
author = {Kashima, Yohei},
title = {On the double critical-state model for type-II superconductivity in 3D},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {333--374},
publisher = {EDP-Sciences},
volume = {42},
number = {3},
year = {2008},
doi = {10.1051/m2an:2008010},
zbl = {pre05288663},
mrnumber = {2423790},
language = {en},
url = {www.numdam.org/item/M2AN_2008__42_3_333_0/}
}
Kashima, Yohei. On the double critical-state model for type-II superconductivity in 3D. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 3, pp. 333-374. doi : 10.1051/m2an:2008010. http://www.numdam.org/item/M2AN_2008__42_3_333_0/

[1] H. Attouch, Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, Boston-London-Melbourne (1984). | MR 773850 | Zbl 0561.49012

[2] H. Attouch and A. Damlamian, Application des méthodes de convexité et monotonie à l'étude de certaines équations quasi-linéaires. Proc. Roy. Soc. Edinburgh 79A (1977) 107-129. | MR 477473 | Zbl 0374.35022

[3] A. Badía and C. López, Critical state theory for nonparallel flux line lattices in type-II superconductors. Phys. Rev. Lett. 87 (2001) 127004.

[4] A. Badía and C. López, Vector magnetic hysteresis of hard superconductors. Phys. Rev. B 65 (2002) 104514.

[5] A. Badía and C. López, The critical state in type-II superconductors with cross-flow effects. J. Low. Temp. Phys. 130 (2003) 129-153.

[6] J.W. Barrett and L. Prigozhin, Dual formulations in critical state problems. Interfaces Free Boundaries 8 (2006) 349-370. | MR 2273233 | Zbl 1108.35098

[7] C.P. Bean, Magnetization of high-field superconductors. Rev. Mod. Phys. 36 (1964) 31-39.

[8] A. Bermúdez, R. Rodríguez and P. Salgado, A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations. SIAM J. Num. Anal. 40 (2002) 1823-1849. | MR 1950624 | Zbl 1033.78009

[9] A. Bossavit, Numerical modelling of superconductors in three dimensions: a model and a finite element method. IEEE Trans. Magn. 30 (1994) 3363-3366.

[10] H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis, E. Zarantonello Ed., Academic Press, Madison, WI (1971) 101-156. | MR 394323 | Zbl 0278.47033

[11] S.J. Chapman, A hierarchy of models for type-II superconductors. SIAM Rev. 42 (2000) 555-598. | MR 1814048 | Zbl 0967.82014

[12] J.R. Clem and A. Perez-Gonzalez, Flux-line-cutting and flux-pinning losses in type-II superconductors in rotating magnetic fields. Phys. Rev. B 30 (1984) 5041-5047.

[16] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland Publishing Co., Amsterdam (1976). | MR 463994 | Zbl 0322.90046

[13] C.M. Elliott and Y. Kashima, A finite-element analysis of critical-state models for type-II superconductivity in 3D. IMA J. Num. Anal. 27 (2007) 293-331. | MR 2317006 | Zbl 1119.82046

[14] C.M. Elliott, D. Kay and V. Styles, A finite element approximation of a variational formulation of Bean's model for superconductivity. SIAM J. Num. Anal. 42 (2004) 1324-1341. | MR 2113687 | Zbl 1071.82063

[15] C.M. Elliott, D. Kay and V. Styles, Finite element analysis of a current density - electric field formulation of Bean's model for superconductivity. IMA J. Num. Anal. 25 (2005) 182-204. | MR 2110240 | Zbl 1100.78018

[17] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Theory and algorithms. Springer, Berlin (1986). | MR 851383 | Zbl 0585.65077

[18] S. Guillaume and A. Syam, On a time-dependent subdifferential evolution inclusion with a nonconvex upper-semicontinuous perturbation. E. J. Qualitative Theory Diff. Equ. 11 (2005) 1-22. | MR 2151540 | Zbl 1093.35018

[19] Y. Kashima, Numerical analysis of macroscopic critical state models for type-II superconductivity in 3D. Ph.D. thesis, University of Sussex, Brighton, UK (2006).

[20] N. Kenmochi, Solvability of Nonlinear Evolution Equations with Time-Dependent Constraints and Applications, The Bulletin of The Faculty of Education 30. Chiba University, Chiba, Japan (1981). | Zbl 0662.35054

[21] P. Monk, Finite element methods for Maxwell's equations. Oxford University Press, Oxford (2003). | MR 2059447 | Zbl 1024.78009

[22] J.C. Nédélec, Mixed finite elements in ${ℝ}^{3}$. Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[23] A. Perez-Gonzalez and J.R. Clem, Response of type-II superconductors subjected to parallel rotating magnetic fields. Phys. Rev. B 31 (1985) 7048-7058.

[24] A. Perez-Gonzalez and J.R. Clem, Magnetic response of type-II superconductors subjected to large-amplitude parallel magnetic fields varying in both magnitude and direction. J. Appl. Phys. 58 (1985) 4326-4335.

[25] A. Perez-Gonzalez and J.R. Clem, ac losses in type-II superconductors in parallel magnetic fields. Phys. Rev. B 32 (1985) 2909-2914.

[26] L. Prigozhin, On the Bean critical-state model in superconductivity. Eur. J. Appl. Math. 7 (1996) 237-247. | MR 1401169 | Zbl 0873.49007

[27] L. Prigozhin, The Bean model in superconductivity: variational formulation and numerical solution. J. Comput. Phys. 129 (1996) 190-200. | MR 1419742 | Zbl 0866.65081

[28] L. Prigozhin, Solution of thin film magnetization problems in type-II superconductivity. J. Comput. Phys. 144 (1998) 180-193. | MR 1633053

[29] J. Rhyner, Magnetic properties and AC-losses of superconductors with power law current-voltage characteristics. Physica C 212 (1993) 292-300.

[30] R.T. Rockafellar, Integrals which are convex functionals. Pacific J. Math. 24 (1968) 525-539. | MR 236689 | Zbl 0159.43804

[31] R.T. Rockafellar and R.J.-B. Wets, Variational analysis. Springer, Berlin-Heidelberg-New York (1998). | MR 1491362 | Zbl 0888.49001

[32] R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (2003) 395-431. | Numdam | MR 2005609 | Zbl 1150.46014

[33] W. Rudin, Functional analysis. McGraw-Hill, New York-Tokyo (1991). | MR 1157815 | Zbl 0867.46001

[34] A. Schmidt and K.G. Siebert, Design of adaptive finite element software, the finite element toolbox ALBERTA, Lect. Notes Comput. Sci. Engrg. 42. Springer, Berlin-Heidelberg (2005). | MR 2127659 | Zbl 1068.65138

[35] H. Si, TetGen: A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangular. Version 1.4.1 (http://tetgen.berlios.de), Berlin (2006).

[36] J. Simon, Compact sets in the space ${L}^{p}\left(0,T;B\right)$. Ann. Math. Pure. Appl. 146 (1987) 65-96. | MR 916688 | Zbl 0629.46031

[37] V. Thomée, Galerkin finite element methods for parabolic problems. Springer, Berlin (1997). | MR 1479170 | Zbl 0884.65097

[38] S. Yotsutani, Evolution equations associated with the subdifferentials. J. Math. Soc. Japan 31 (1978) 623-646. | MR 544681 | Zbl 0405.35043