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

Abstract

We consider a hyperbolic Maxwell-type variational inequality arising in Bean's critical state model for type-II superconductivity. Combining the implicit Euler method in time with a mixed Ritz-Galerkin approximation in space results in a fully discrete scheme for the concerned variational inequality. We present some numerical experiments, which visualize the Meissner-Ochsenfeld effect in type-II superconductivity and verify that the proposed approach is physically reasonable. Fully discrete schemeFor a bounded, polyhedral and simply connected domain Ω ⊂ R 3 with a connected Lipschitz boundary we consider the following hyperbolic mixed variational inequality of the second kind for the evolutionary Maxwell's equations derived from Bean's critical-state model for type-II superconductivity [1,2]:where , µ ∈ L ∞ (Ω) denote the material parameters, f ∈ W 1,∞ ([0, T ], L 2 (Ω)) stands for the applied current source and (E 0 , B 0 ) is some initial electromagnetic field. Furthermore, E denotes the electric field and B the magnetic induction. The non-linear function ϕ is defined by ϕ(v) = Ω g(x)|v(x)| dx for v ∈ L 1 (Ω). In the context of Bean's model, the scalar function g ∈ L ∞ (Ω) is given by g(x) = j c χ Ωsc (x), where j c ∈ R + is the critical current density and Ω sc ⊂ Ω the domain of the type-II superconductor. Moreover, χ Ωsc denotes the indicator function of Ω sc . In order to obtain a time discretization for (VI) we introduce the time step ∆t = T N with N ∈ N and consider an equidistant partition of [0, T ] by t n = n∆t for all n = 0, . . . , N . With this partition at hand, we use Euler's implicit method to replace the time derivatives in (VI) with difference quotients. Now, the fully discrete scheme of (VI) is obtained by a mixed Ritz-Galerkin approximation in space based on Nédélec's edge elements for E and piecewise constant elements for B: