Hydrodynamic vortex separators (HDVSs) are used in wastewater treatment to separate solids from wastewater. The aim of this research is to devise a CFD-based methodology that optimises their performance through modification of their design. A validation study is performed to assess whether OpenFOAM can be used to reliably model the flow of water in an HDVS. The results of the simulations are compared with experimental readings, showing a good fit when the appropriate boundary layer height and turbulence model are used. The continuous adjoint method is employed to derive the adjoint equations, associated with the drift flux equations used to model the flow of wastewater. They are specialised to the typical boundary conditions of ducted flows and are coded using OpenFOAM. An optimal design is found for boundary conditions, corresponding to typical values used in practice, and is shown to improve the performance of a simplified initial design by 40%. This optimal design is subsequently subjected to a different hydraulic loading rate and dispersed-phase volume fraction at the inlet, to assess the performance variation in these circumstances. Though the optimal design removes all the solids when the dispersed-phase fraction is reduced at the inlet, initial results suggest that the design is sensitive to hydraulic loading rate and further tests are recommended before drawing more explicit conclusions. This is the first time the adjoint drift flux equations have been derived. It is also the first time they have been coded and applied to an HDVS to optimise its performance. The methodology developed in this thesis could be applied to any device that separates solids from liquid or two immiscible liquids, in order to optimise its performance.

Design optimization is a method that can be used to automate the design process to obtain better results. When applied to aeroelastic structures, design optimization often leads to the creation of highly flexible aeroelastic structures. There are, however, a number of conventional design procedures that must be modified when dealing with highly flexible aeroelastic structures. First, the deformed geometry must be the baseline for weight, structural, and stability analyses. Second, potential couplings between aeroelasticity and rigid-body dynamics must be considered. Third, dynamic analyses must be modified to handle large nonlinear displacements. These modifications to the conventional design process significantly increase the difficulty of developing an optimization framework appropriate for highly flexible aeroelastic structures. As a result, when designing these structures, often either gradient-free optimization is performed (which limits the optimization to relatively few design variables) or optimization is simply omitted from the design process. Both options significantly decrease the design exploration capabilities of a designer compared to a scenario in which gradient-based optimization is used. This dissertation therefore presents various contributions that allow gradient-based optimization to be more easily used to optimize highly flexible aeroelastic structures. One of our primary motivations for developing these capabilities is to accurately capture the design constraints of solar-regenerative high-altitude long-endurance (SR-HALE) aircraft. In this dissertation, we therefore present a SR-HALE aircraft optimization framework which accounts for the peculiarities of structurally flexible aircraft while remaining suitable for use with gradient-based optimization. These aircraft tend to be extremely large and light, which often leads to significant amounts of structural flexibility. Using this optimization framework, we design an aircraft that is capable of flying year-round at \SI{35}{\degree} latitude at \SI{18}{\kilo\meter} above sea level. We subject this aircraft to a number of constraints including energy capture, energy storage, material failure, local buckling, stall, static stability, and dynamic stability constraints. Critically, these constraints were designed to accurately model the actual design requirements of SR-HALE aircraft, rather than to provide a rough approximation of them. To demonstrate the design exploration capabilities of this framework, we also performed several parameters sweeps to determine optimal design sensitivities to altitude, latitude, battery specific energy, solar efficiency, avionics and payload power requirements, and minimum design velocity. Through this optimization framework, we demonstrate both the potential of gradient-based optimization applied to highly flexible aeroelastic structures and the challenges associated with it. One challenge associated with the gradient-based optimization of highly flexible aeroelastic structures, is the ability to accurately, efficiently, and reliably model the large deflections of these structures in gradient-based optimization frameworks. To enable large-scale optimization involving structural models with large deflections to be performed more easily, we present a finite-element implementation of geometrically exact beam theory which is designed specifically for gradient-based optimization. A key feature of this implementation of geometrically exact beam theory is its compatibility with forward and reverse-mode automatic differentiation, which allows accurate design sensitivities to be obtained with minimal development effort. Another key feature is its native support for unsteady adjoint sensitivity analysis, which allows design sensitivities to be obtained efficiently from time-marching simulations. Other features are also presented that build upon previous implementations of geometrically exact beam theory, including a singularity-free rotation parameterization based on Wiener-Milenkovi\'c parameters, an implementation of stiffness-proportional structural damping using a discretized form of the compatibility equations, and a reformulation of the equations of motion for geometrically exact beam theory from a fully implicit index-1 differential algebraic equation to a semi-explicit index-1 differential algebraic equation. Several examples are presented which verify the utility and validity of each of these features. Another challenge associated with the gradient-based optimization of highly flexible aeroelastic structures is the ability to reliably track and constrain individual dynamic stability modes across the design iterations of an optimization framework. To facilitate the development of mode-specific dynamic stability constraints in gradient-based optimization frameworks we develop a mode tracking method that uses an adaptive step size in order to maintain an arbitrarily high degree of confidence in mode correlations. This mode tracking method is then applied to track the modes of a linear two-dimensional aeroelastic system and a nonlinear three-dimensional aeroelastic system as velocity is increased. When used in a gradient-based optimization framework, this mode tracking method has the potential to allow continuous dynamic stability constraints to be constructed without constraint aggregation. It also has the potential to allow the stability and shape of specific modes to be constrained independently. Finally, to facilitate the development and use of highly flexible aeroelastic systems for use in gradient-based optimization frameworks, we introduce a general methodology for coupling aerodynamic and structural models together to form modular monolithic aeroelastic systems. We also propose efficient methods for computing the Jacobians of these coupled systems without significantly increasing the amount of time necessary to construct these systems. For completeness we also discuss how to ensure that the resulting system of equations constitutes a set of first-order index-1 differential algebraic equations. We then derive direct and adjoint sensitivities for these systems which are compatible with automatic differentiation so that derivatives for gradient-based optimization can be obtained with minimal development effort.

Le progrès technologique nécessite des techniques de plus en plus sophistiquées et précises de traitement de matériaux. Nous étudions le traitement de matériaux par faisceaux de haute énergie : un jet d’eau abrasif, une sonde ionique focalisée, un laser. L’évolution de la surface du matériau sous l’action du faisceau de haute énergie est modélisée par une EDP. Cette équation contient l’ensemble des coefficients inconnus - les paramètres de calibration de mo- dèle. Les paramètres inconnus peuvent être calibrés par minimisation de la fonction coût, c’est-à-dire, la fonction qui décrit la différence entre le résultat de la modélisation et les données expérimentales. Comme la surface modélisée est une solution du problème d’EDP, cela rentre dans le cadre de l’optimisation sous contrainte d’EDP. L’identification a été rendue bien posée par la régularisation du type Tikhonov. Le gradient de la fonction coût a été obtenu en utilisant les deux méthodes : l’approche adjointe et la différen- ciation automatique. Une fois la fonction coût et son gradient obtenus, nous avons utilisé un minimiseur L-BFGS pour réaliser la minimisation.Le problème de la non-unicité de la solution a été résolu pour le problème de traitement par le jet d’eau abrasif. Des effets secondaires ne sont pas inclus dans le modèle. Leur impact sur le procédé de calibration a été évité. Ensuite, le procédé de calibration a été validé pour les données synthétiques et expérimentales. Enfin, nous avons proposé un critère pour distinguer facilement entre le régime thermique et non- thermique d’ablation par laser. / The technological progress demands more and more sophisticated and precise techniques of the treatment of materials. We study the machining of the material with the high energy beams: the abrasive waterjet, the focused ion beam and the laser. Although the physics governing the energy beam interaction with material is very different for different application, we can use the same approach to the mathematical modeling of these processes.The evolution of the material surface under the energy beam impact is modeled by PDE equation. This equation contains a set of unknown parameters - the calibration parameters of the model. The unknown parameters can be identified by minimization of the cost function, i.e., function that describes the differ- ence between the result of modeling and the corresponding experimental data. As the modeled surface is a solution of the PDE problem, this minimization is an example of PDE-constrained optimization problem. The identification problem was regularized using Tikhonov regularization. The gradient of the cost function was obtained both by using the variational approach and by means of the automatic differentiation. Once the cost function and its gradient calculated, the minimization was performed using L-BFGS minimizer.For the abrasive waterjet application the problem of non-uniqueness of numerical solution is solved. The impact of the secondary effects non included into the model is avoided as well. The calibration procedure is validated on both synthetic and experimental data.For the laser application, we presented a simple criterion that allows to distinguish between the thermal and non-thermal laser ablation regimes.