|Abstract:|| In the last decades Shape Optimization problems have raised an ever-increasing attention, mainly thanks to the several applications in the fields of Physics and Engineering. Their aim is to find the best shape for an object in order to minimize (or maximize) a certain quantity, depending on the characteristics of the object itself and of its use. In this thesis we introduce a reference-domain approach to solve such problems, with the application to two-dimensional, incompressible, low-Reynolds flows, modelled by Stokes equations.
The main idea of this approach is to map the physical domain, on which a certain differential problem is defined, to a reference domain which depends no more on the physical shape. In this way, no handling with moving domains is needed, and the difficulty of the problem moves to the presence of variable coefficients in the equations describing the physical problem.
The rst part of this work is devoted to the theoretical analysis of the problem of minimizing the energy dissipation of a Stokes flow in a two-dimensional domain. After the denition of the state problem, the set of admissible shapes and the functional to minimize, some properties of the state solution operator are provided. Then, an existence result for the optimization problem and the first order optimality conditions are given. Finally, some a priori estimates for the discretization error of the control and the state variables are derived.
In the second part, we introduce the algorithm implemented for the solution of the optimization problem: the Python code implemented for this purpose makes use FEniCS library, and it is documented in the Appendix. Afterwards, the solver is applied to some test cases, in order to verify the ecacy of the theoretical model and of the code itself.|