Ottimizzazione di forma per problemi di fluidodinamica: analisi teorica e metodi numerici

Manzoni, Andrea
Ottimizzazione di forma per problemi di fluidodinamica: analisi teorica e metodi numerici
Thursday 23rd October 2008
Salsa, S.
Advisor II:
Parolini, N.
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Optimal Control of systems governed by Partial Di erential Equations (PDE) is one of the most advanced elds of the whole Mathematical Modeling and allows us to face the study of many engineering problems. This is done by mixing theoretical methods of Analysis and Calculus of Variations with numerical tools for solving PDEs and optimization problems. Optimal control problems can be divided in two groups, depending on the nature of the object under control: variables (such as data, coeffcients or boundary values) or shape of the domain itself. Shape optimization and Optimal design belong to the second group; they arise from problems of structural mechanics and fluid dynamics, where they have assumed an increasing role in the last decades, thanks to the better performances of computing and numerical techniques. One of the most appealing applications deals with the shape optimization of a body in relative motion in a fluid - for example, wing pro les in aerospace engineering or boats - or optimal design of biomedical devices such as aorto-coronaric by-passes. In this Master Thesis we consider the problem of drag minimization for a body embedded in a viscous incompressible fluid, discussing both theoretical aspects and numerical tools for the calculus of solutions. At a deeper level, this problem can be inserted in a larger research eld, aiming at the performances improvement of racing boats, by shape optimization of the appendages of the hull. With this aim, we consider optimal control approach applied to the steady Navier-Stokes equations, using an adjoint-based method for the formulation of the optimality conditions system. We illustrate some useful techniques from Shape analysis for computing the shape gradient of a cost functional and a formulation of Lagrange multipliers method for Shape optimization problems. Moreover, we develop some numerical optimization procedures based on Computational Fluid Dynamics methods such as Finite Element approximation for solving PDEs, using steepest descent methods and two di erent approaches for shape deformation and mesh treatment. In particular, we focus on the problem of drag reduction for the bulb of a Class America boat in the Chapter 1, discussing some important features of Shape Optimization in Fluid Dynamics. In Chapter 2 we consider an abstract formulation of Optimal design problems and we present some general results about admissible shapes. We introduce a lot of theoretical tools for the study of Optimal design problems in Chapter 3, dealing with domain deformations, functional shape gradients and shape/material derivatives of linear elliptic PDEs solutions. Moreover, Function shape parametrization and Space embedding techniques are introduced, aiming to develop a Lagrangian approach for the analysis of optimality conditions. In the Chapter 4 we discuss the formulation of dissipated energy and drag minimization problems, we explore their well posedness and we obtain the related systems of optimality conditions. In Chapter 5 we introduce some techniques for solving numerically optimization problems and we briefly recall Finite Element methods and some stabilization techniques for High Reynolds numbers flows; we also present two di erent strategies for domain description and transformation in Optimal design problems. Moreover, some features about the structure of the FreeFem++ library used for code implementation are summarized. Finally, we present in Chapter 6 some results for the dissipated energy and drag reduction problems obtained by the numerical simulations - both at low and high Reynolds numbers - underlining positive and critical aspects of various approaches.