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.

Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics

Over the years the Differential Quadrature (DQ) method has distinguished because of its high accuracy, straightforward implementation and general ap- plication to a variety of problems. There has been an increase in this topic by several researchers who experienced significant development in the last years. DQ is essentially a generalization of the popular Gaussian Quadrature (GQ) used for numerical integration functions. GQ approximates a finite in- tegral as a weighted sum of integrand values at selected points in a problem domain whereas DQ approximate the derivatives of a smooth function at a point as a weighted sum of function values at selected nodes. A direct appli- cation of this elegant methodology is to solve ordinary and partial differential equations. Furthermore in recent years the DQ formulation has been gener- alized in the weighting coefficients computations to let the approach to be more flexible and accurate. As a result it has been indicated as Generalized Differential Quadrature (GDQ) method. However the applicability of GDQ in its original form is still limited. It has been proven to fail for problems with strong material discontinuities as well as problems involving singularities and irregularities. On the other hand the very well-known Finite Element (FE) method could overcome these issues because it subdivides the computational domain into a certain number of elements in which the solution is calculated. Recently, some researchers have been studying a numerical technique which could use the advantages of the GDQ method and the advantages of FE method. This methodology has got different names among each research group, it will be indicated here as Generalized Differential Quadrature Finite Element Method (GDQFEM).

Fluid-structure interaction by a coupled lattice Boltzmann-finite element approach

In this thesis, a strategy to model the behavior of fluids and their interaction with deformable bodies is proposed. The fluid domain is modeled by using the lattice Boltzmann method, thus analyzing the fluid dynamics by a mesoscopic point of view. It has been proved that the solution provided by this method is equivalent to solve the Navier-Stokes equations for an incompressible flow with a second-order accuracy. Slender elastic structures idealized through beam finite elements are used. Large displacements are accounted for by using the corotational formulation. Structural dynamics is computed by using the Time Discontinuous Galerkin method. Therefore, two different solution procedures are used, one for the fluid domain and the other for the structural part, respectively. These two solvers need to communicate and to transfer each other several information, i.e. stresses, velocities, displacements. In order to guarantee a continuous, effective, and mutual exchange of information, a coupling strategy, consisting of three different algorithms, has been developed and numerically tested. In particular, the effectiveness of the three algorithms is shown in terms of interface energy artificially produced by the approximate fulfilling of compatibility and equilibrium conditions at the fluid-structure interface. The proposed coupled approach is used in order to solve different fluid-structure interaction problems, i.e. cantilever beams immersed in a viscous fluid, the impact of the hull of the ship on the marine free-surface, blood flow in a deformable vessels, and even flapping wings simulating the take-off of a butterfly. The good results achieved in each application highlight the effectiveness of the proposed methodology and of the C++ developed software to successfully approach several two-dimensional fluid-structure interaction problems.

Finite element models of coseismic deformation due to the 2009 L'Aquila (Italy) and 2008 Wenchuan(China) earthquakes

The topic of my Ph.D. thesis is the finite element modeling of coseismic deformation imaged by DInSAR and GPS data. I developed a method to calculate synthetic Green functions with finite element models (FEMs) and then use linear inversion methods to determine the slip distribution on the fault plane. The method is applied to the 2009 L’Aquila Earthquake (Italy) and to the 2008 Wenchuan earthquake (China). I focus on the influence of rheological features of the earth's crust by implementing seismic tomographic data and the influence of topography by implementing Digital Elevation Models (DEM) layers on the FEMs. Results for the L’Aquila earthquake highlight the non-negligible influence of the medium structure: homogeneous and heterogeneous models show discrepancies up to 20% in the fault slip distribution values. Furthermore, in the heterogeneous models a new area of slip appears above the hypocenter. Regarding the 2008 Wenchuan earthquake, the very steep topographic relief of Longmen Shan Range is implemented in my FE model. A large number of DEM layers corresponding to East China is used to achieve the complete coverage of the FE model. My objective was to explore the influence of the topography on the retrieved coseismic slip distribution. The inversion results reveals significant differences between the flat and topographic model. Thus, the flat models frequently adopted are inappropriate to represent the earth surface topographic features and especially in the case of the 2008 Wenchuan earthquake.

Active fibre-reinforced composites with embedded shape memory alloys

This dissertation concerns active fibre-reinforced composites with embedded shape memory alloy wires. The structural application of active materials allows to develop adaptive structures which actively respond to changes in the environment, such as morphing structures, self-healing structures and power harvesting devices. In particular, shape memory alloy actuators integrated within a composite actively control the structural shape or stiffness, thus influencing the composite static and dynamic properties. Envisaged applications include, among others, the prevention of thermal buckling of the outer skin of air vehicles, shape changes in panels for improved aerodynamic characteristics and the deployment of large space structures. The study and design of active composites is a complex and multidisciplinary topic, requiring in-depth understanding of both the coupled behaviour of active materials and the interaction between the different composite constituents. Both fibre-reinforced composites and shape memory alloys are extremely active research topics, whose modelling and experimental characterisation still present a number of open problems. Thus, while this dissertation focuses on active composites, some of the research results presented here can be usefully applied to traditional fibre-reinforced composites or other shape memory alloy applications. The dissertation is composed of four chapters. In the first chapter, active fibre-reinforced composites are introduced by giving an overview of the most common choices available for the reinforcement, matrix and production process, together with a brief introduction and classification of active materials. The second chapter presents a number of original contributions regarding the modelling of fibre-reinforced composites. Different two-dimensional laminate theories are derived from a parent three-dimensional theory, introducing a procedure for the a posteriori reconstruction of transverse stresses along the laminate thickness. Accurate through the thickness stresses are crucial for the composite modelling as they are responsible for some common failure mechanisms. A new finite element based on the First-order Shear Deformation Theory and a hybrid stress approach is proposed for the numerical solution of the two-dimensional laminate problem. The element is simple and computationally efficient. The transverse stresses through the laminate thickness are reconstructed starting from a general finite element solution. A two stages procedure is devised, based on Recovery by Compatibility in Patches and three-dimensional equilibrium. Finally, the determination of the elastic parameters of laminated structures via numerical-experimental Bayesian techniques is investigated. Two different estimators are analysed and compared, leading to the definition of an alternative procedure to improve convergence of the estimation process. The third chapter focuses on shape memory alloys, describing their properties and applications. A number of constitutive models proposed in the literature, both one-dimensional and three-dimensional, are critically discussed and compared, underlining their potential and limitations, which are mainly related to the definition of the phase diagram and the choice of internal variables. Some new experimental results on shape memory alloy material characterisation are also presented. These experimental observations display some features of the shape memory alloy behaviour which are generally not included in the current models, thus some ideas are proposed for the development of a new constitutive model. The fourth chapter, finally, focuses on active composite plates with embedded shape memory alloy wires. A number of di®erent approaches can be used to predict the behaviour of such structures, each model presenting different advantages and drawbacks related to complexity and versatility. A simple model able to describe both shape and stiffness control configurations within the same context is proposed and implemented. The model is then validated considering the shape control configuration, which is the most sensitive to model parameters. The experimental work is divided in two parts. In the first part, an active composite is built by gluing prestrained shape memory alloy wires on a carbon fibre laminate strip. This structure is relatively simple to build, however it is useful in order to experimentally demonstrate the feasibility of the concept proposed in the first part of the chapter. In the second part, the making of a fibre-reinforced composite with embedded shape memory alloy wires is investigated, considering different possible choices of materials and manufacturing processes. Although a number of technological issues still need to be faced, the experimental results allow to demonstrate the mechanism of shape control via embedded shape memory alloy wires, while showing a good agreement with the proposed model predictions.