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

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.

System innovation and life cycle thinking in packaging value chain: the circularity of plastics.

Compared to other, plastic materials have registered a strong acceleration in production and consumption during the last years. Despite the existence of waste management systems, plastic_based materials are still a pervasive presence in the environment, with negative consequences on marine ecosystem and human health. The recycling is still challenging due to the growing complexity of product design, the so-called overpackaging, the insufficient and inadequate recycling infrastructure, the weak market of recycled plastics and the high cost of waste treatment and disposal. The Circular economy package, the European Strategy for plastics in a circular economy and the recent European Green Deal include very ambitious programmes to rethink the entire plastic value chain. As regards packaging, all plastic packaging will have to be 100% recyclable (or reusable) and 55% recycled by 2030. Regions are consequently called upon to set up a robust plan able to fit the European objectives. It takes on greater importance in Emilia Romagna where the Packaging valley is located. This thesis supports the definition of a strategy aimed to establish an after-use plastics economy in the region. The PhD work has set the basis and the instruments to establish the so-called Circularity Strategy with the aim to turn about 92.000t of plastic waste into profitable secondary resources. System innovation, life cycle thinking and participative backcasting method have allowed to deeply analyse the current system, orientate the problem and explore sustainable solutions through a broad stakeholder participation. A material flow analysis, accompanied by a barrier analysis, has supported the identification of the gaps between the present situation and the 2030 scenario. Eco-design for and from recycling (and a mass _based recycling rate (based on the effective amount of plastic wastes turned into secondary plastics), valorized by a value_based indicator, are the key-points of the action plan.

Incompressible limit and well-posedness of PDE models of tissue growth

Both compressible and incompressible porous medium models are used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. A coupled system of equations describes the cell density and the nutrient concentration and the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state.