Leonid Prigozhin

John W. Barrett, Leonid Prigozhin

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.

Leonid Prigozhin, Vladimir Sokolovsky

Numerical methods for modeling thin-film magnetization are primarily focused on computing the current density distribution. The highly nonlinear current-voltage characteristic of type-II superconductors significantly complicates the accurate computation of the electric field. The T-E formulation-based mixed finite element method, previously derived for flat superconducting films, enables the simultaneous, accurate determination of both variables. Another advantage of this method is that the computational domain is limited to the film itself: no meshing of the surrounding space is required. The thin-shell approximation reduces the problem to a two-dimensional one. This work extends the T-E formulation and numerical method to non-flat superconducting shells with a metal substrate. We validate the method with several test examples, including modeling the magnetization of a sphere. The method is then applied to a realistic model of a cylindrical magnetic dynamo pump, and the generated open-circuit voltage is computed.

Leonid Prigozhin

Solutions to Monge-Kantorovich equations, expressing optimality condition in mass transportation problem with cost equal to distance, are stationary points of a critical-slope model for sand surface evolution. Using a dual variational formulation of sand model, we compute both the optimal transport density and Kantorovich potential as a stationary limit of evolving sand flux and sand surface, respectively.