Variational Inequalities in Critical-State Problems

Similar evolutionary variational inequalities appear as convenient formulations for continuous quasistationary models for sandpile growth, formation of a network of lakes and rivers, magnetization of type-II superconductors, and elastoplastic deformations. We outline the main steps of such models derivation and try to clarify the origin of this similarity. New dual variational formulations, analogous to mixed variational inequalities in plasticity, are derived for sandpiles and superconductors.Comment: Submitted for publicatio

Electric field formulation for thin film magnetization problems

We derive a variational formulation for thin film magnetization problems in type-II superconductors written in terms of two variables, the electric field and the magnetization function. A numerical method, based on this formulation, makes it possible to accurately compute all variables of interest, including the electric field, for any value of the power in the power law current-voltage relation characterizing the superconducting material. For high power values we obtain a good approximation to the critical state model solution. Numerical simulation results are presented for simply and multiply connected films, and also for an inhomogeneous film.Comment: 15 p., submitte

AC losses in type-II superconductors induced by nonuniform fluctuations of external magnetic field

Magnetic field fluctuations are inevitable in practical applications of superconductors and it is often necessary to estimate the AC losses these fluctuations induce. If the fluctuation wavelength is greater than the size of a superconductor, known estimates for an alternating uniform external magnetic field can be employed. Here we consider the opposite case and analyze, using a model critical-state problem, penetration of spatially nonuniform fluctuations into type-II superconductors. Numerical simulation is based on a variational formulation of the Bean model. The analytical solutions, found in a weak penetration limit, are used to evaluate AC losses for two types of fluctuations: the running and standing waves. It is shown that for spatially nonuniform fluctuations the losses are better characterized by the fluctuation penetration depth than by the fluctuation amplitude. The results can be used to estimate the AC losses in flywheels, electric motors, magnetic shields, etc.Comment: 18 pages, 5 fugure

3D Simulation of Superconducting Magnetic Shields and Lenses using the Fast Fourier Transform

Shielding sensitive scientific and medical devices from the magnetic field environment is one of the promising applications of superconductors. Magnetic field concentration by superconducting magnetic lenses is the opposite phenomenon based, however, on the same properties of superconductors: their ideal conductivity and ability to expel the magnetic field. Full-dimensional numerical simulations are necessary for designing magnetic lenses and for estimating the quality of magnetic shielding under arbitrary varying external fields. Using the recently proposed Fast Fourier Transform based three-dimensional numerical method [Prigozhin and Sokolovsky, ArXiv 1801.04869] we model performance of two such devices made of a bulk type-II superconductor: a magnetic shield and a magnetic lens. The method is efficient and can be easier to implement than the alternative approaches based on the finite element methods.Comment: 7 pages, 6 figure

A Quasi-Variational Inequality Problem Arising in the Modeling of Growing Sandpiles

Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized mixed formulation involving both the primal (sand surface) and dual (sand flux) variables. We derive, analyse and compare two methods for the approximation, and numerical solution, of this mixed problem. We prove subsequence convergence of both approximations, as the mesh discretization parameters tend to zero; and hence prove existence of a solution to this mixed model and the associated regularized quasi-variational inequality problem. One of these numerical approximations, in which the flux is approximated by the divergence-conforming lowest order Raviart-Thomas element, leads to an efficient algorithm to compute not only the evolving pile surface, but also the flux of pouring sand. Results of our numerical experiments confirm the validity of the regularization employed.Comment: 51 p., low resolution fig