In this paper, the author analyzes an evolution variational inequality to being a formulation of the double critical-state model for type-II superconductivity proposed by Clem and Perez-Gonzalez who developed the general critical-state theory by postulating that the electric field should be decomposed into two parts. One of them corresponds to the perpendicular component of the electric current density to the magnetic flux density and the other corresponds to the parallel component.
In this paper, this double critical state model is modified by adding the term with the resistivity which vanishes if the magnitude of the current density is smaller than a certain critical value. This additional term makes the energy density deriving the corresponding constitutive relation coercive with respect to current so that a solution to the variational inequality formulation of the problem exists and the convex optimization problem is derived as the fully discrete formulation admits the existence of its unique minimizer. In order to investigate the macroscopic behavior of the electromagnetic fields around a bulk type-II superconductor in general 3D configurations, it is necessary to formulate the double critical-static model in a 3D configuration and propose a finite-element method to carry out its numerical simulations.
The author shows solvability of the problem by applying the Schauder fixed point theorem coupled with the unique existence theorem for nonlinear evolution system driven by time dependent subdifferentials. It is given the characterization theorem of a subdifferential for a class of energy functionals including the energy deriving the author’s formulation and it is observed that Faraday’s law and the nonlinear Ohm’s law can be recovered in the superconductor in the sense of almost everywhere. The space discretization is carried out by means of the lowest order edge finite element on a tetrahedral mesh. In the time discretization, it is employed a semi-implicit time-stepping scheme so the fully discrete formulation is an unconstrained minimization problem. In order to handle the curl-free constraint imposed on the magnetic field outside the superconductor, it is introduced a scalar magnetic potential and proposed the magnetic fields-scalar potential hybrid formulation, which is equivalent to the original formulation.
Finally, it is solved the discrete optimization problem in the hybrid space numerically by means of Newton’s method coupled with the conjugate gradient method. The numerical results are obtained under a rotating applied magnetic field.