Variational principles for circle patterns

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow cone-like singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variational principle. The functionals (one for the euclidean, one for the hyperbolic case) are convex functions of the radii of the circles. The analogous functional for the spherical case is not convex, hence this case is treated by stereographic projection to the plane. From the existence and uniqueness of circle patterns in the sphere, we derive a strengthened version of Steinitz' theorem on the geometric realizability of abstract polyhedra. We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and Rivin for circle packings and circle patterns from our variational principles. In the case of Br\"agger's and Rivin's functionals. Leibon's functional for hyperbolic circle patterns cannot be derived directly from our functionals. But we construct yet another functional from which both Leibon's and our functionals can be derived. We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics

Dynamic Modeling of Autorotation for Simultaneous Lift and Wind Energy Extraction

The goal of this thesis is to develop a multi-body dynamics model of autorotation with the objective of studying its application in energy harvesting. A rotor undergoing autorotation is termed an Autogyro. In the autorotation mode, the rotor is unpowered and its interaction with the wind causes an upward thrust force. The theory of an autorotating rotorcraft was originally studied for achieving safe flight at low speeds and later used for safe descent of helicopters under engine failure. The concept can potentially be used as a means to collect high-altitude wind energy. Autorotation is inherently a dynamic process and requires detailed models for characterization. Existing models of autorotation assume steady operating conditions with constant angular velocity of the rotor. The models provide spatially averaged aerodynamic forces and torques. While these steady-autorotation models are used to create a basis for the dynamic model developed in this thesis, the latter uses a Lagrangian formulation to determine the equations of motion. The aerodynamic effects on the blades that produce thrust forces, in-plane torques, and out-of-plane torques, are modeled as non-conservative forces within the Lagrangian framework. To incorporate the instantaneous aerodynamic forces, the above-mentioned spatial averaging is removed. The resulting model is causal and consists of a system of differential equations. To investigate the dynamics under energy-harvesting operation, an additional in-plane regenerative torque is added to simulate the effect of a generator. The aerodynamic effects of this regenerative braking is incorporated into the model. In addition, the dynamic model relaxes assumptions of small flapping angles, and the periodic flapping behavior of the blades are naturally generated by the dynamics instead of assuming Fourier expansions. The dynamic model enables the study of transients due to change in operating conditions or external influences such as wind speeds. It also helps gain insight into force and torque fluctuations. Model verification is conducted to ensure that the dynamic model produces similar steady-operating conditions as those reported in prior works. In addition, the behavior of autorotation under energy harvesting is evaluated. The thesis also explores the viability of achieving sufficient lift while extracting energy from prevailing winds. A range of regenerative torques are applied to determine the optimal energy state. Finally, a complete high-altitude energy harvesting system is modeled by incorporating a tether utilizing a catenary model. Overall, the thesis lends support to the hypothesis that a tethered autogyro can support its weight while harvesting energy from strong wind-fields, when augmented with appropriate control systems

Pattern-Equivariant Homology of Finite Local Complexity Patterns

This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a "pattern" is introduced, which may be seen as an analogue of the space group of isometries preserving a tiling but where, instead, one considers partial isometries preserving portions of it. These inverse semigroups of partial transformations are the suitable analogue of the space group for patterns with FLC but few global symmetries. In a similar vein we introduce the notion of a \emph{collage}, a system of equivalence relations on the ambient space of a pattern, which we show is capable of generalising many constructions applicable to the study of FLC tilings and Delone sets, such as the expression of the tiling space as an inverse limit of approximants. An invariant is constructed for our abstract patterns, the so called pattern-equivariant (PE) homology. These homology groups are defined using infinite singular chains on the ambient space of the pattern, although we show that one may define cellular versions which are isomorphic under suitable conditions. For FLC tilings these cellular PE chains are analogous to the PE cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are shown to be related through Poincar\'{e} duality. An efficient and highly geometric method for the computation of the PE homology groups for hierarchical tilings is presented. The rotationally invariant PE homology groups are shown not to be a topological invariant for the associated tiling space and seem to retain extra information about global symmetries of tilings in the tiling space. We show how the PE homology groups may be incorporated into a spectral sequence converging to the \v{C}ech cohomology of the rigid hull of a tiling. These methods allow for a simple computation of the \v{C}ech cohomology of the rigid hull of the Penrose tilings.Comment: 159 pages, 8 figures, PhD thesi

Direct Nonlinear Trajectory Optimization and State Estimation for a Tethered Underwater Energy Harvesting Kite

This dissertation addresses the coupled challenges of state estimation and trajectory optimization for a marine hydro-kinetic energy harvesting kite. The optimization objective is to maximize the kite's average mechanical power output. This work is motivated by the potential of ``pumping-mode" tethered kites to provide attractive levelized costs of electricity, especially when cross-current motion is exploited to maximize energy harvesting. In ``pumping-mode" kites, the kite is tethered to platform carrying a motor/generator, and electricity generation is achieved by reeling the kite out and in at high and low tether tension levels, respectively. Marine hydro-kinetic (MHK) systems are heavily influenced by wind energy systems. In both contexts, for instance, tethered kites can be used for electricity generation instead of stationary turbines. Similar to airborne wind energy (AWE) systems, the power production capacities of MHK kites are heavily influenced by their flight trajectories. While trajectory optimization is a well-established research area for AWE systems, it is a nascent but growing field for MHK kites. Moreover, although both AWE and MHK kites have the potential to benefit from trajectory optimization, the lessons learned from AWE systems might not be directly applicable to MHK kites, since MHK systems are often close to neutral buoyancy whereas AWE systems are not. Finally, there is little work in the literature that co-optimizes the spooling and cross-current trajectories of a pumping-mode MHK kite. The first contribution of this dissertation is to explore the simultaneous optimization of the cross-current trajectory and the spooling motion of a pumping-mode kite using direct transcription. While the results highlight the degree to which simultaneous optimization can be beneficial for these systems, they also motivate the need for a solution approach that satisfies the constraints imposed by the kite dynamics exactly, as opposed to approximately. This leads to the second contribution of this dissertation, namely, finding an analytic solution to the inverse dynamics of the MHK kite, i.e., mapping a desired combination of kite position, velocity, and acceleration onto the corresponding actuation inputs. The dissertation then proceeds to its third contribution, namely, solving the kite trajectory optimization problem based on the above exact solution of the kite's inverse dynamics. The resulting simulation provides more realistic optimization results. However, all of the above work focuses on the special case where the free-stream fluid velocity is known and spatio-temporally constant. This motivates the fourth and final contribution of this dissertation, namely, the development of an unscented Kalman filter for simultaneously estimating both the kite's state and the free-stream fluid velocity. One interesting outcome of the estimation study is the finding that simple unscented Kalman filtering is not able to estimate the fluid velocity accurately without the direct measurement of the attitude of the kite