Hostname: page-component-6d856f89d9-72csx Total loading time: 0 Render date: 2024-07-16T06:15:04.075Z Has data issue: false hasContentIssue false
  • English
  • Français

On the double critical-state model for type-II superconductivity in 3D

Published online by Cambridge University Press:  03 April 2008

Yohei Kashima
Yohei Kashima*
Affiliation:
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan. y.kashima@mail.sci.hokudai.ac.jp
Get access

Abstract

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.

Keywords

The double critical-state model for superconductivityevolution variational inequalityMaxwell's equationsedge finite elementconvergencecomputational electromagnetism.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 3 , May 2008 , pp. 333 - 374
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H. Attouch, Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, Boston-London-Melbourne (1984).
Attouch, H. and Damlamian, A., Application des méthodes de convexité et monotonie à l'étude de certaines équations quasi-linéaires. Proc. Roy. Soc. Edinburgh 79A (1977) 107–129. CrossRef
Badía, A. and López, C., Critical state theory for nonparallel flux line lattices in type-II superconductors. Phys. Rev. Lett. 87 (2001) 127004. CrossRef
Badía, A. and López, C., Vector magnetic hysteresis of hard superconductors. Phys. Rev. B 65 (2002) 104514. CrossRef
A. Badía and C. López, The critical state in type-II superconductors with cross-flow effects. J. Low. Temp. Phys. 130 (2003) 129–153.
Barrett, J.W. and Prigozhin, L., Dual formulations in critical state problems. Interfaces Free Boundaries 8 (2006) 349–370. CrossRef
Bean, C.P., Magnetization of high-field superconductors. Rev. Mod. Phys. 36 (1964) 31–39. CrossRef
Bermúdez, A., Rodríguez, R. and Salgado, P., A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations. SIAM J. Num. Anal. 40 (2002) 1823–1849. CrossRef
Bossavit, A., Numerical modelling of superconductors in three dimensions: a model and a finite element method. IEEE Trans. Magn. 30 (1994) 3363–3366. CrossRef
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.
Chapman, S.J., A hierarchy of models for type-II superconductors. SIAM Rev. 42 (2000) 555–598. CrossRef
Clem, J.R. and Perez-Gonzalez, A., Flux-line-cutting and flux-pinning losses in type-II superconductors in rotating magnetic fields. Phys. Rev. B 30 (1984) 5041–5047. CrossRef
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland Publishing Co., Amsterdam (1976).
Elliott, C.M. and Kashima, Y., A finite-element analysis of critical-state models for type-II superconductivity in 3D. IMA J. Num. Anal. 27 (2007) 293–331. CrossRef
Elliott, C.M., Kay, D. and Styles, V., A finite element approximation of a variational formulation of Bean's model for superconductivity. SIAM J. Num. Anal. 42 (2004) 1324–1341. CrossRef
Elliott, C.M., Kay, D. and Styles, V., Finite element analysis of a current density – electric field formulation of Bean's model for superconductivity. IMA J. Num. Anal. 25 (2005) 182–204. CrossRef
V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Theory and algorithms. Springer, Berlin (1986).
Guillaume, S. and Syam, A., On a time-dependent subdifferential evolution inclusion with a nonconvex upper-semicontinuous perturbation. E. J. Qualitative Theory Diff. Equ. 11 (2005) 1–22.
Y. Kashima, Numerical analysis of macroscopic critical state models for type-II superconductivity in 3D. Ph.D. thesis, University of Sussex, Brighton, UK (2006).
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).
P. Monk, Finite element methods for Maxwell's equations. Oxford University Press, Oxford (2003).
Nédélec, J.C., Mixed finite elements in ${\mathbb{R}}^3$ . Numer. Math. 35 (1980) 315–341. CrossRef
Perez-Gonzalez, A. and Clem, J.R., Response of type-II superconductors subjected to parallel rotating magnetic fields. Phys. Rev. B 31 (1985) 7048–7058. CrossRef
Perez-Gonzalez, A. and Clem, J.R., 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. CrossRef
Perez-Gonzalez, A. and Clem, J.R., ac losses in type-II superconductors in parallel magnetic fields. Phys. Rev. B 32 (1985) 2909–2914. CrossRef
Prigozhin, L., On the Bean critical-state model in superconductivity. Eur. J. Appl. Math. 7 (1996) 237–247.
Prigozhin, L., The Bean model in superconductivity: variational formulation and numerical solution. J. Comput. Phys. 129 (1996) 190–200. CrossRef
Prigozhin, L., Solution of thin film magnetization problems in type-II superconductivity. J. Comput. Phys. 144 (1998) 180–193. CrossRef
Rhyner, J., Magnetic properties and AC-losses of superconductors with power law current-voltage characteristics. Physica C 212 (1993) 292–300. CrossRef
Rockafellar, R.T., Integrals which are convex functionals. Pacific J. Math. 24 (1968) 525–539. CrossRef
R.T. Rockafellar and R.J.-B. Wets, Variational analysis. Springer, Berlin-Heidelberg-New York (1998).
Rossi, R. and Savaré, G., Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (2003) 395–431.
W. Rudin, Functional analysis. McGraw-Hill, New York-Tokyo (1991).
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).
H. Si, TetGen: A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangular. Version 1.4.1 (http://tetgen.berlios.de), Berlin (2006).
J. Simon, Compact sets in the space Lp(0,T;B). Ann. Math. Pure. Appl. 146 (1987) 65–96. CrossRef
V. Thomée, Galerkin finite element methods for parabolic problems. Springer, Berlin (1997).
Yotsutani, S., Evolution equations associated with the subdifferentials. J. Math. Soc. Japan 31 (1978) 623–646. CrossRef