Functional data usually consist of a sample of functions, with each function observed on a discrete grid. The key idea of functional data analysis is to consider each function as a single, structured object rather than a collection of data points. To represent and investigate functional data, curve registration and functional regression are two important techniques. Curve registration is used to align random curves that display time variations. This procedure, known as functional convex averaging, leads to phase-variance adjusted mean functions. Therefore, compared to a simple averaged mean function, phase-variance adjusted mean function by functional convex averaging is a more accurate representation of the inherent function from which the functional data arise. Several curve registration methods are reviewed in this work, including landmark, self-warping and Bayesian hierarchical curve registration (BHCR). For BHCR, when the number of random curves is large or the sampling grid is intensive, the computational cost increases dramatically. To solve this problem, we introduce an accelerated BHCR algorithm via a predictive process model (PPM), known as PPM-BHCR. Tested by a simulation study and real data, this new method is demonstrated to save large amounts of computing time, without a large sacrifice of accuracy.

Functional regression is used to explore the relationship between the outcome and the predictor, where either or both of them are functional. In this work, several functional regression methods are reviewed according to the function-on-scalar, scalar-on-function and function-on-function categories. Registration is traditionally performed as a data preprocessing step before regression. In this work, we introduce a new method called warped functional regression (WFR), which integrates curve registration and functional regression into one joint model. Therefore, we are able to provide prediction based on an unwarped predictor using this new model. The proposed method is evaluated by simulation studies and demonstrates high accuracy. Several case studies illustrate the key contributions of the proposed method in addressing complex scientific questions.

A high-performance Discrete-Vortex Method (DVM) is successfully developed and implemented to directly simulate two-dimensional low to medium-Reynolds number flows around multiple arbitrarily shaped bodies undergoing either prescribed or free motions. The deterministic Viscous-Vortex-Domain (VVD) formulation is adopted to simulate vorticity diffusion. Through the use of CPU and Graphics-Processing-Unit (GPU) parallel computing, significant speedup of the simulation compared to a serial implementation on a CPU is achieved. The validity of the present DVM simulation is confirmed by comparing the present results with published ones for a variety of test cases. The current implementation of DVM has been used to study two novel flow problems of practical interest and has led to significant findings.

First, the full and partial ground effects on the lift generation of a flapping (air)foil in normal hovering mode are investigated. To achieve full ground effect, the foil of chord c is made to hover above the center of a finite-sized platform of length 10c. The computed force-enhancement, force-reduction, and force-recovery regimes at low, medium, and high ground clearances are observed to be in line with existing literature. This research puts special focus on partial ground effects when the foil is hovering near the edge of the platform. Lift-modifying mechanisms not previously observed under full ground effect have been discovered. When stroke reversal of the flapping occurs near the edge of the platform, a relatively stationary strong vortex may form above the platform edge. This strong vortex can either increase or decrease the instantaneous lift force on the foil depending on the position of the foil relative to the platform edge. Further, the platform edge may lead to the formation of an additional vortex pair which increases the instantaneous lift force as the foil sweeps past the edge under certain suitable conditions. Lastly, the platform edge can lead to the formation of a reverse von Kármán vortex street that extends well below the stroke plane under suitable geometric arrangements.

Second, the flow past a Bach-type vertical-axis wind or current turbine is simulated using the DVM at a Reynolds number of 1,500. The main purpose of the study is to evaluate the suitability of Bach-type turbines for use as micro-scale energy harvesters that can be applied to power, for example, sensor nodes of a Wireless Sensor Network. Through simulations, the maximum power coefficient of the turbine operating at a prescribed constant tip-speed ratio is found to be 0.18, which is comparable to the performance of a turbine of the same geometry at much higher Reynolds numbers. This indicates that there is only minimal performance penalty for miniaturization. The angular velocity of the turbine has a strong influence on the evolution of vortical flow structures. A new wake-capturing mechanism that boosts the performance of the turbine is discovered from the simulations for a certain range of tip-speed ratios where the vortex shed by the advancing blade helps drive the returning blade. In addition to the condition of prescribed rotation, free rotation of a steel Bach-type turbine under a steady current in water is also investigated. Significant fluctuation in angular velocity over one period of rotation is observed. This speed fluctuation is found to be detrimental to energy extraction, reducing the maximum power coefficient to approximately 0.16. This level of power-generation capability implies that such micro-scale turbines can significantly extend the life expectancy of a wireless sensor node or even maintain the node in a low-power state indefinitely.