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

Abstract

Abstract. In this paper we mathematically analyse an evolution variational inequality which formulates the double critical-state model for type-II superconductivity in 3D space and propose a finite element method to discretize the formulation. The double critical-state model originally proposed by Clem and Perez-Gonzalez is formulated as a model in 3D space which characterizes the nonlinear relation between the electric field, the electric current, the perpendicular component of the electric current to the magnetic flux, and the parallel component of the current to the magnetic flux in bulk type-II superconductor. The existence of a solution to the variational inequality formulation is proved and the representation theorem of subdifferential for a class of energy functionals including our energy is established. The variational inequality formulation is discretized in time by a semi-implicit scheme and in space by the edge finite element of lowest order on a tetrahedral mesh. The fully discrete formulation is an unconstrained optimisation problem. The subsequence convergence property of the fully discrete solution is proved. Some numerical results computed under a rotating applied magnetic field are presented.Mathematics Subject Classification. 65M60, 65M12, 47J20, 49J40.

Fast Fourier transform-based solution of 2D and 3D magnetization problems in type-II superconductivity

“…It was ran with the default values of parameters (the relative tolerance 3 10 ,  the absolute tolerance 6 10  ) . Randomly chosen one-sided finite differences were used in [1,2] to approximate the derivatives in (9). Here we differentiated in the Fourier space and used the Gaussian filter to smooth the results.…”

A Quasi-Variational Inequality Problem in Superconductivity

“…The latter phenomenon, often called "the fishtail effect", can be described by the eddy current model with a non-monotonic J c (|b|) dependance; see, e.g., Johansen et al 19 . We mention here also the primal variational formulation for a generalized double critical-state model, see Badía and Lopez 2 and Kashima 20 , in which Φ is the characteristic function of the set of admissible currents satisfying j(x, t) ∈ Δ(b(x, t)) and Δ(b) ⊂ R 3 being a given family of closed convex sets. Below we consider a simple geometric configuration of an infinite superconducting cylinder having a cross section Ω ⊂ R 2 and placed into a parallel non-stationary uniform external magnetic field b e (t).…”

3D simulation of superconducting magnetic shields and lenses using the fast Fourier transform

“…Our first example is a hollow, closed at one end superconducting cylinder (Fig 1 ); its sizes are taken from [2] and scaled in accordance with (8). The cylinder was centrally positioned in a cube with sides 3.2 (the computation domain); in this example we set out 20   .…”

“…Here we solve such a 3D problem for a hollow superconducting cylinder sealed at one end (the configuration considered in [2]) using the Fast Fourier Transform (FFT) based method [6]. This method is an easily implemented alternative to the finite element methods for 3D magnetization problems (see, e.g., [7][8][9][10][11][12]) and we employ it to model also field concentration by a magnetic lens. The configuration of a bulk superconductor in the latter problem is that of the magnetic lenses in [13].…”