William G. Litvinov

A Modified TV-Stokes Model for Image Processing

We introduce and investigate the modified total variation (TV)-Stokes model for two classical image processing tasks, i.e., image restoration and image inpainting. The modified TV-Stokes model is a two-step model based on a TV minimization in each step and the use of geometric information of the image. In the first step, a smoothed and divergence-free tangential field of the given image is recovered, and in the second step, the image is reconstructed from the corresponding normals. The existence and the uniqueness of the solution to the minimization problems are established for both steps of the model. Numerical examples and comparisons are presented to illustrate the effectiveness of the model.

Flow of electrorheological fluid under conditions of slip on the boundary

We study a mathematical model describing flows of electrorheological fluids. A theorem of existence of a weak solution is proved. For this purpose the approximating-topological method is used.

Problems of Stationary Flow of Electrorheological Fluids in a Cylindrical Coordinate System

We consider the general problem on stationary flow of the electrorheological fluid with the constitutive equation developed in [R. H. W. Hoppe and W. G. Litvinov, Comm. Pure. Appl. Anal., 3 (2004), pp. 809--848] in the cylindrical coordinate system. The problem is studied under mixed boundary conditions wherein velocities are specified on one part of the boundary and surface forces are given on the other part. The existence of a solution to this problem and the convergence of Galerkin approximations are established. Then, we consider the occasion where the flow is axially symmetric and study a problem on an electrorheological clutch. This problem is solved numerically, and the results of calculations of the electric field and velocities are presented.

Modeling, Simulation and Optimization of Electrorheological Fluids

Publisher Summary This chapter describes the modeling, simulation, and optimization of electrorheological fluids. It is concerned with the balance equations and constitutive laws for isothermal and nonisothermal electrorheological fluid flows and with the existence and/or uniqueness of solutions. It also discusses numerical methods both for steady and time-dependent fluid flows. It presents numerical simulation results for some selected electrorheological devices and briefly addresses optimal design issues. Electrorheological fluids are the concentrated suspensions of the electrically polarizable particles of small size in the range of micrometers in nonconducting or semiconducting liquids such as silicone oils. Under the influence of an outer electric field, the particles form chains along the field lines followed by a coalescence of the chains into columns in the plane orthogonal to the field because of short-ranged potentials arising from charge–density fluctuations. The formation of the chains is a process that happens in milliseconds, whereas the aggregation to columns occurs on a timescale that is larger by an order of magnitude. On a macroscopic scale, the chainlike and columnar structures have a significant impact on the rheological properties of the suspensions. In particular, the viscosity increases rapidly with increasing electric field strength in the direction perpendicular to the field. The fluid experiences a phase transition to a viscoplastic state, and the flow shows a pronounced anisotropic behavior. Under the influence of large stresses, the columns break into continuously fragmenting and aggregating volatile structures, which tilt away from strict field alignment. As a result, the viscosity decreases and the fluid flow behave less anisotropically.

Modelling and computation of axially symmetric flows of electrorheological fluids

In this article, we discuss the extended Bingham fluid model introduced in the paper [2] for electrorheological fluids, and formulate the problem in the axially symmetric cyllindrical coordinates system. As an application we choose the ER Shockabsorber, and present some numerical simulation of its behaviour.

Basic Definitions and Auxiliary Statements

We suppose the reader to be familiar with notions connected with sets and functions; nevertheless, we state the terminology and introduce notations that will be used throughout the book.

Optimization of Deformable Solids

Plates and shells are main elements of many advanced structures. One of the most important characteristics of a construction is its weight, which determines the consumption of material needed for production of the construction as well as some operating features of the latter. For example, the increase of weight of an aircraft causes growth of the fuel rate in flight and degradation of some flight characteristics.

Control by the Right-hand Sides in Elliptic Problems

Let U be a reflexive Banach space, and suppose

Direct Problems for Plates and Shells

\left. {\begin{array}{*{20}{c}} {u,v \to \pi (u,v)\,is\,a\,bilinear,\,symmetric,\,continuous,\; \,\,\quad \quad } \\ {positive\,form\;on\;U \times U,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ \end{array} } \right\}