Hyperbolic Maxwell Variational Inequalities for Bean's Critical-State Model in Type-II Superconductivity

Abstract

This paper focuses on the numerical analysis for three-dimensional Bean's critical-state model in type-II superconductivity. We derive hyperbolic mixed variational inequalities of the second kind for the evolution Maxwell equations with Bean's constitutive law between the electric field and the current density. On the basis of the variational inequality in the magnetic induction formulation, a semidiscrete Ritz-Galerkin approximation problem is rigorously analyzed, and a strong convergence result is proven. Thereafter, we propose a concrete realization of the Ritz-Galerkin approximation through a mixed finite element method based on edge elements of Nédélec's first family, Raviart-Thomas face elements, divergence-free Raviart-Thomas face elements, and piecewise constant elements. As a final result, we prove error estimates for the proposed mixed finite element method.We note that Bean made a simplifying assumption of a constant critical current density j c ∈ R + , which is physically reasonable in the case of a not so strong magnetic field. According to experiments, however, the critical current density can depend on the magnetic field j c = j c (|H|) in the case of strong external fields. This physical phenomenon was observed by Kim, Hempstead, and Strnad [21]. We refer the reader to [10] for a comprehensive review on the derivation of the Bean critical-state constitutive relation from different mathematical models, including Ginzburg-Landau and London equations. See also [11,12] for mathematical and numerical results on Ginzburg-Landau equations.Let Ω ⊂ R 3 be a bounded Lipschitz domain and let Ω sc be an open set satisfying Ω sc ⊂ Ω. Here, the subset Ω sc represents a type-II superconductor. Assuming that the temperature of the superconductor Ω sc is below the critical one, the evolution of the electromagnetic waves in Ω is described by the Maxwell equations

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Fully discrete solution for Bean's critical‐state model in type‐II superconductivity

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On an optimal control problem for a class of elliptic VIs of the second kind involving unbounded operators

“…where g ∈ L ∞ (Ω; R n ) satisfies g ≥ 0 a.e. in Ω and g i are the components of g, i = 1, ..., n. We note that the variationalinequality-constraint in (P) arises from applications such as the time-discretized Bean critical-state model in type-II superconductivity (see [5,6]). In this case, n = 6 with g 1 = g 2 = g 3 = j c , where j c : Ω → R is the critical current density and g 4 = g 5 = g 6 = 0 a.e.…”