The Bean Model in Superconductivity: Variational Formulation and Numerical Solution

Prigozhin, Leonid

doi:10.1006/jcph.1996.0243

Cited by 126 publications

(133 citation statements)

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“…This model and its equivalent variational formulations have been studied extensively by Prigozhin [51,53,52].…”

Section: Infinite Mobility In the Long-wave Limitmentioning

confidence: 99%

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A Hierarchy of Models for Type-II Superconductors

Chapman¹

2000

SIAM Rev.

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“…This model and its equivalent variational formulations have been studied extensively by Prigozhin [51,53,52].…”

Section: Infinite Mobility In the Long-wave Limitmentioning

confidence: 99%

“…When J c is constant it is known as the Bean model [7,8], and when J c is a function of H it is known as the Kim-Anderson model [43,44,2]. As in the finite λ case, there is an equivalent variational formulation [51], which is to find H ∈ K such that…”

Section: Infinite Mobility In the Long-wave Limitmentioning

confidence: 99%

A Hierarchy of Models for Type-II Superconductors

Chapman¹

2000

SIAM Rev.

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“…In our numerical simulations in Section 5, however, we will confirm that the resistivity ρ 0 always vanishes and the term ρ 0 J does not affect the nature of the model. The numerical analysis of the variational inequality formulation of macroscopic critical-state models for type-II superconductivity was initiated by Prigozhin [26,27]. Prigozhin proposed the subdifferential formulation of the Bean critical-state model [7], proved the well-posedness of the formulation in [26] and intensively studied the numerical simulations in 2D in [27].…”

Section: Introductionmentioning

confidence: 99%

“…The numerical analysis of the variational inequality formulation of macroscopic critical-state models for type-II superconductivity was initiated by Prigozhin [26,27]. Prigozhin proposed the subdifferential formulation of the Bean critical-state model [7], proved the well-posedness of the formulation in [26] and intensively studied the numerical simulations in 2D in [27]. Elliott et al [14,15] established error estimates for their finite element approximation of the variational inequality formulation of the Bean model in 2D.…”

Section: Introductionmentioning

confidence: 99%

“…Though this condition holds true in some geometric configurations where the component J is predicted to vanish such as in axially symmetric superconductors (see [27]) or superconducting thin films (see [28]) under perpendicular applied field, the redistribution of the pinned magnetic flux driven by Lorentz force may induce the electric field which is not parallel to the current in general 3D geometry. See [11] or [27] for the limitation of the Bean model argued from a modelling perspective.…”

Section: Introductionmentioning

confidence: 99%

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

Kashima¹

2008

ESAIM: M2AN

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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.Mathematics Subject Classification. 65M60, 65M12, 47J20, 49J40.

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