460 research outputs found
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
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
Partial L-1 Monge-Kantorovich problem: variational formulation and numerical approximation
Published versio
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