Minimal Model for the Topology of the Critical State in Hard Superconductors

The critical state problem in type-II superconductivity is described theoretically by a direct optimization method, which allows a straightforward treatment for non idealized geometries. Based on Faraday's law and the principle of minimum entropy production, the magnetic history is built up just by a constrained minimization of the field changes along the process. Constraints are in the form $\vec{J}\in\Delta$, with $\vec{J}$ the electric current density and $\Delta$ some bounded set. This incorporates the vortex pinning and interaction phenomena and may be used for the modelling of anisotropy, inhomogeneities and flux cutting interactions. In this work, our variational statement is posed on the finite element discretization and provides a minimal tool for investigating the effects of the sample's topology on the field penetration patterns. Simulations of (i) the contraction and splitting of boundaries between the flux free and penetrated regions, (ii) the effect of granularity on the superconducting properties, (iii) the influence of defects, and (iv) surface curvature phenomena are presented.