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 3D space. The nonlinear Ohm's law is formulated using the subdifferential 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 nonconducting region is expressed as a gradient of a magnetic Scalar potential in order to handle the curl-free constraint. The variational inequalities ate discretized in time implicitly and in space by Nedelec's curl-conforming finite element of lowest order. The honsmooth 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.