On the double critical-state model for type-II superconductivity in 3D
ESAIM: 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 : 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: 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},
     mrnumber = {2423790},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2008010/}
}
TY  - JOUR
AU  - Kashima, Yohei
TI  - On the double critical-state model for type-II superconductivity in 3D
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 333
EP  - 374
VL  - 42
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2008010/
DO  - 10.1051/m2an:2008010
LA  - en
ID  - M2AN_2008__42_3_333_0
ER  - 
%0 Journal Article
%A Kashima, Yohei
%T On the double critical-state model for type-II superconductivity in 3D
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 333-374
%V 42
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2008010/
%R 10.1051/m2an:2008010
%G en
%F M2AN_2008__42_3_333_0
Kashima, Yohei. On the double critical-state model for type-II superconductivity in 3D. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 3, pp. 333-374. doi : 10.1051/m2an:2008010. http://www.numdam.org/articles/10.1051/m2an:2008010/

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 (0,T;B). Ann. Math. Pure. Appl. 146 (1987) 65-96. | MR | Zbl

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

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

Cité par Sources :