Anca Capatina

Algorithms and convergence results for an inverse problem in heat propagation

The aim of this paper is to study from the theoretical and numerical points of view an optimal control problem in heat propagation theory. An iteration algorithm is defined and its convergence is proved. For the discretization of this problem an internal approximation in the space variables and a backward Euler scheme in the time variable are used. Stability and convergence results are also established. Some numerical examples are given to illustrate the accuracy of the proposed algorithms in the cases of traction and torsion tests.

Optimal control of a signorini contact problem

We are interested in finding the coefficient of friction which leads us to a given displacement on the contact surface between an elastic solid body and a rigid foundation. The mathematical formulation of the problem is an optimal control problem governed by a quasivariational inequality. We obtain an approximative caracterization, by using two families of penalized and regularized problems, for a given optimal control.

Optimal control of a non isothermal Navier-Stokes flow

Abstract This paper deals with a boundary optimal control problem associated with the stationary Navier-Stokes equations coupled with the heat equation. The most general type of boundary condition for the temperature is considered. The existence of a solution of this problem and, for some values of the viscosity coefficient, the uniqueness are proved. The control problem consists in finding a temperature of the surrounding medium which leads to a desired configuration of the temperature of the fluid. The existence of an optimal control is proved and necessary conditions of optimality are derived by introducing a family of regularized optimal problems.

Spaces of Vector-Valued Functions

In this chapter we will introduce additional tools which are fundamentals for the study of evolutionary problems studied later in this book. We consider here spaces of functions defined on a time interval \(I \subset \mathbb{R}\) with values into a Banach or Hilbert space X. The results are presented without proofs and for details we refer to the bibliography.

Homogenization results for elliptic problems in periodically perforated domains with mixed-type boundary conditions

The asymptotic behaviour of a class of elliptic equations with highly oscillating coefficients, in a perforated periodic domain, is analyzed. We consider, in each period, two types of holes and we impose, on their boundaries, a Signorini and, respectively, a Neumann condition. Using the periodic unfolding method, we prove that the limit problem contains two additional terms, a right-hand side term and a “strange” one.

Numerical analysis of a control problem in heat conducting navier-stokes fluid

Abstract An optimal control problem for the coupled system Navier-Stokes and heat equations is examined from a numerical point of view. The control considered is the temperature of the surrounding medium. The functional minimizes the L 2 -distance of the candidate temperature to some desired temperature. Finite element approximation of the optimality system is defined. The convergence of the proposed algorithms for solving the discrete problem is proved. The analysis of the numerical results and their physical meaning are discussed.

A quasistatic frictional contact problem with normal compliance and unilateral constraint

A mathematical model describing the quasistatic process of frictional contact between a nonlinearly elastic body and an elastic-rigid foundation is considered. The contact is modeled by normal compliance with unilateral constraint, and the friction by a slip-dependent version of Coulomb’s law. A weak formulation of the problem is derived and, under a smallness assumption on contact and friction functions, an existence result is proved by using incremental techniques, Kakutani’s fixed point theorem, and compactness, monotonicity, and lower semicontinuity arguments.

Approximations of Variational Inequalities

This chapter is devoted to the discrete approximation of abstract elliptic and implicit evolutionary quasi-variational inequalities. Convergence results for internal approximations in space of elliptic quasi-variational inequalities together with a backward difference scheme in time of implicit evolutionary quasi-variational inequalities are proved. The results obtained in this chapter, representing generalizations of the approximations of variational inequalities of the first and second kinds, can be applied to a large variety of static and quasistatic contact problems, including unilateral and bilateral contact or normal compliance conditions with friction. In particular, static and quasistatic unilateral contact problems with nonlocal Coulomb friction in linear elasticity will be considered in last part of this book.

Spaces of Real-Valued Functions

This chapter is a brief background on spaces of continuous functions and some Sobolev spaces including basic properties, embedding theorems and trace theorems. We recall some classical definitions and theorems of functional analysis which will be used throughout this work. These results are standard and so they are stated without proofs; details and proofs can be found in to many references. We only deal with real valued functions. We assume that the reader is familiar with the basic concepts of general topology and functional analysis.

Existence and Uniqueness Results

This chapter deals with existence and uniqueness results for variational and quasi-variational inequalities. With the intention of focusing the differences among the proofs of results, we first consider elliptic variational inequalities of the first and second kind with linear and continuous operators in Hilbert space or monotone and hemicontinuous operators in Banach space. Next, we deal with elliptic quasi-variational inequalities involving monotone and hemicontinuous or potential operators. The last section concerns the study of a class of evolutionary quasi-variational inequalities. The results presented here will be applied, in the last part of the book, in the study of frictional contact problems.