Maxwell quasi-variational inequalities in superconductivity

Irwin Yousept

doi:10.1051/m2an/2021028

Abstract

This paper is devoted to the mathematical modeling and analysis of a hyperbolic Maxwell quasi-variational inequality (QVI) for the Bean-Kim superconductivity model with temperature and magnetic field dependence in the critical current. Our analysis relies on local (resp. global) boundedness and local (resp. global) Lipschitz continuity assumptions on the critical current with respect to the temperature (resp. magnetic field). Emerging from the Euler time discretization, we analyze the corresponding H(curl)-elliptic QVI and prove its existence using a fixed-point argument in combination with techniques from variational inequalities and Maxwell’s equations. Based on the existence result for the H(curl)-elliptic QVI, we examine the stability and convergence of the Euler scheme, which serve as our fundament for the global well-posedness of the governing hyperbolic Maxwell QVI.

Highlights

Ever since the discovery of superconductivity by Heike Kamerlingh Onnes in 1911, various modern applications and key technologies have been developed
A prominent critical-state model describing the irreversible magnetization process in high-temperature superconductivity was proposed by Bean [5, 6]
His model postulates a nonlinear and non-smooth constitutive relation between the current density and the electric field through the so-called critical current as follows: (B1) the current density strength |J | cannot exceed the critical current jc (B2) the electric field E vanishes if |J | is strictly less than jc (B3) the electric field E is parallel to the current density J

Summary

Introduction

Ever since the discovery of superconductivity by Heike Kamerlingh Onnes in 1911, various modern applications and key technologies have been developed. Among many other profound applications, we mention magnetic resonance imaging, magnetic confinement fusion, and magnetic levitation Such technological advances are made possible by superconductors due to their fundamental properties of vanishing electrical resistance and expulsion of applied magnetic fields (Meissner effect) occurring when the temperature is cooled below the critical temperature. A prominent critical-state model describing the irreversible magnetization process in high-temperature superconductivity was proposed by Bean [5, 6]. His model postulates a nonlinear and non-smooth constitutive relation between the current density and the electric field through the so-called critical current as follows:.

YOUSEPT

Preliminaries

Time-discrete problem

Further discussions