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

• Volume: 42, Issue: 3, page 333-374
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Kashima, Yohei. "On the double critical-state model for type-II superconductivity in 3D." ESAIM: Mathematical Modelling and Numerical Analysis 42.3 (2008): 333-374. <http://eudml.org/doc/250415>.

@article{Kashima2008,
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. },
author = {Kashima, Yohei},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {The double critical-state model for superconductivity; evolution variational inequality; Maxwell's equations; edge finite element; convergence; computational electromagnetism.; double critical-state model; type-II superconductor; energy functional; evolution variational equality; solution existence; representation theorem; discretization; finite-element method},
language = {eng},
month = {4},
number = {3},
pages = {333-374},
publisher = {EDP Sciences},
title = {On the double critical-state model for type-II superconductivity in 3D},
url = {http://eudml.org/doc/250415},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Kashima, Yohei
TI - On the double critical-state model for type-II superconductivity in 3D
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/4//
PB - EDP Sciences
VL - 42
IS - 3
SP - 333
EP - 374
AB - 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.
LA - eng
KW - The double critical-state model for superconductivity; evolution variational inequality; Maxwell's equations; edge finite element; convergence; computational electromagnetism.; double critical-state model; type-II superconductor; energy functional; evolution variational equality; solution existence; representation theorem; discretization; finite-element method
UR - http://eudml.org/doc/250415
ER -

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