Leonid Prigozhin

Analysis of critical-state problems in type-II superconductivity

An efficient numerical scheme is proposed for modeling the hysteretic magnetization of type-II superconductors. Numerical examples are presented for the Bean and Kim critical state models. It is shown that Beans model is a Hele-Shaw type problem.

Variational model of sandpile growth

A model describing the evolving shape of a growing pile is considered, and is shown to be equivalent to an evolutionary quasi-variational inequality. If the support surface has no steep slopes, the inequality becomes a variational one. For this case existence and uniqueness of the solution are proved.

On the Bean critical-state model in superconductivity

We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

Computing AC losses in stacks of high-temperature superconducting tapes

Superconducting tape coils and Roebel cables are often modeled as stacks of parallel superconducting tapes carrying the same transport current. We solved, in the infinitely thin approximation, the transport current and magnetization problems for such stacks using an efficient numerical scheme based on a variational formulation of the Kim critical-state model. We also refined the anisotropic bulk approximation, introduced by Clem et al in order to simplify AC loss estimates for densely packed stacks of many tapes; this was achieved by removing the simplifying a priori assumptions on the current sheet density in the subcritical zone and the shape of this zone boundary. Finally, we studied the convergence of stack problem solutions to the solution of the modified bulk problem. It was shown that, due to the fast convergence to the anisotropic bulk limit, accurate AC loss estimates for stacks of hundreds of tapes can usually be obtained also using a properly rescaled model of a stack containing only ten to twenty tapes.

Nonlinear dynamics of Aeolian sand ripples

We study the initial instability of flat sand surface and further nonlinear dynamics of wind ripples. The proposed continuous model of ripple formation allowed us to simulate the development of a typical asymmetric ripple shape and the evolution of a sand ripple pattern. We suggest that this evolution occurs via ripple merger preceded by several soliton-like interaction of ripples.

Formation of aeolian ripples and sand sorting.

We present a continuous model capable of demonstrating some salient features of aeolian sand ripples: the realistic asymmetric ripple shape, coarsening of the ripple field at the nonlinear stage of ripple growth, saturation of ripple growth for homogeneous sand, typical size segregation of sand, and formation of armoring layers of coarse particles on ripple crests and windward slopes if the sand is inhomogeneous.

A QUASI-VARIATIONAL INEQUALITY PROBLEM IN SUPERCONDUCTIVITY

We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical finite element method based on the lowest order Raviart–Thomas element, which yields approximations to both the primal and dual variables (the magnetic and electric fields). We prove the subsequence convergence of this approximation, and hence prove the existence of a solution to both the dual and primal formulations, for strictly star-shaped domains. The effectiveness of the approximation is illustrated by numerical examples with and without this domain restriction.

Dual formulations in critical state problems

Similar evolutionary variational inequalities appear as convenient formulations for continuous models for sandpile growth, magnetization of type-II superconductors, and evolution of some other dissipative systems characterized by the multiplicity of metastable states, long-range interactions, avalanches, and hysteresis. The origin of this similarity is that these are quasistationary models in which the multiplicity of metastable states is a consequence of a unilateral condition of equilibrium (critical-state constraint). Existing variational formulations for critical-state models of sandpiles and superconductors are convenient for modeling only the “primary” variables (evolving pile shape and magnetic field, respectively). The conjugate variables (the surface sand flux and the electric field) are also of interest in various applications. Here we derive dual variational formulations, which have some similarities to mixed variational inequalities in plasticity, for the sandpile and superconductor models. We then approximate them by fully practical finite element methods based on the lowest order Raviart‐Thomas element. We prove convergence of these approximations, and hence existence of a solution, to these dual formulations. Finally, we present some numerical experiments.

A variational problem of bulk solids mechanics and free-surface segregation

Abstract A proposed mathematical model of filling formation describes the evolution of the free surface of a filling and transport of cohesionless polydisperse granular material during some typical operations of bulk solids handling. The form of free surface is determined as a solution of an evolutionary variational inequality. Equations of material transport allow us to find the distribution of different materials inside the granular body and to take into account the free-surface segregation of polydisperse bulk material.

Electric field formulation for thin film magnetization problems

We derive a variational formulation for thin film magnetization problems for 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.