# Analysis of optimal control problems for the incompressible MHD equations and implementation in a finite element multiphysics code

Bornia, Giorgio (2012) Analysis of optimal control problems for the incompressible MHD equations and implementation in a finite element multiphysics code, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Ingegneria energetica, nucleare e del controllo ambientale, 24 Ciclo. DOI 10.6092/unibo/amsdottorato/4868.
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## Abstract

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

Abstract
Tipologia del documento
Tesi di dottorato
Autore
Bornia, Giorgio
Supervisore
Dottorato di ricerca
Scuola di dottorato
Ingegneria industriale
Ciclo
24
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
Optimal control, MHD equations, finite element method
URN:NBN
DOI
10.6092/unibo/amsdottorato/4868
Data di discussione
11 Maggio 2012
URI