A finite-element analysis of critical-state models for type-II superconductivity in 3D

Abstract

We consider the numerical analysis of evolution variational inequalities which are derived from Maxwell's equations coupled with a nonlinear constitutive relation between the electric field and the current density and governing the magnetic field around a type-II bulk superconductor located in three dimensional space. The nonlinear Ohm's law is formulated using the sub-differential of a convex energy so the theory is applied to the Bean critical state model, a power law model and an extended Bean critical state model. The magnetic field in the nonconductive region is expressed as a gradient of a magnetic scalar potential in order to handle the curl free constraint. The variational inequalities are discretized in time implicitly and in space by Nédélec's curl conforming finite element of lowest order. The non-smooth energies are smoothed with a regularization parameter so that the fully discrete problem is a system of nonlinear algebraic equations at each time step. We prove various convergence results. Some numerical simulations under a uniform external magnetic field are presented.Keywords: macroscopic models for superconductivity, variational inequality, Maxwell's equations, edge finite element, convergence, computational electromagnetism.

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

“…Accordingly, the energy functional depends not only on the electric current but also on the magnetic flux and is not convex with respect to the unknown magnetic field. The existence theorem by Brezis [10] does not apply in this case and the analysis for the existence of a solution differs from that of the preceding articles [13,26]. We show the solvability by applying the Schauder fixed point theorem coupled with the unique existence theorem for nonlinear evolution system driven by time dependent subdifferentials (see [18,20,38]).…”

“…theorem of nonlinear evolution system by Brezis [10] (see also [13,26] for the proof of the well-posedness of these formulations). In our formulation of the model (1.2), however, the current density needs to be decomposed into parallel and perpendicular components to the magnetic flux.…”

“…Barrett and Prigozhin [6] derived the dual formulation of the Bean model in terms of the electric field as the conjugate variable to the magnetic field and proved the convergence property of a practical finite element approximation of their dual formulation. Recently Elliott and Kashima [13] reported a 3D finite element analysis of the variational inequality formulation of the critical-state models governing the magnetic field and the current density around a bulk type-II superconductor. For more on the preceding mathematical work concerning the critical-state models see the introduction of [13] and the references therein.…”

“…Recently Elliott and Kashima [13] reported a 3D finite element analysis of the variational inequality formulation of the critical-state models governing the magnetic field and the current density around a bulk type-II superconductor. For more on the preceding mathematical work concerning the critical-state models see the introduction of [13] and the references therein.…”

A macroscopic model for an intermediate state between type-I and type-II superconductivity

“…If n → ∞, the solution to the power law formulation converges to the solution to Beans critical-state formulation [16,17]. This relation in combination with Maxwell's equations is investigated in [22][23][24][25][26]. Using (1) and taking the curl of (6) leads to the following equation for the magnetic field:…”