Browsing by Subject "Mechanical engineering"

This dissertation presents a novel distributed optimal control (DOC) problem formulation that is applicable to multiscale dynamical systems comprised of numerous interacting systems, or agents, that together give rise to coherent macroscopic behaviors, or coarse dynamics, that can be modeled by partial differential equations (PDEs) on larger spatial and time scales. The DOC methodology seeks to obtain optimal agent state and control trajectories by representing the system's performance as an integral cost function of the macroscopic state, which is optimized subject to the agents' dynamics. The macroscopic state is identified as a time-varying probability density function to which the states of the individual agents can be mapped via a restriction operator. Optimality conditions for the DOC problem are derived analytically, and the optimal trajectories of the macroscopic state and control are computed using direct and indirect optimization algorithms. Feedback microscopic control laws are then derived from the optimal macroscopic description using a potential function approach.

The DOC approach is demonstrated numerically through benchmark multi-agent trajectory optimization problems, where large systems of agents were given the objectives of traveling to goal state distributions, avoiding obstacles, maintaining formations, and minimizing energy consumption through control. Comparisons are provided between the direct and indirect optimization techniques, as well as existing methods from the literature, and a computational complexity analysis is presented. The methodology is also applied to a track coverage optimization problem for the control of distributed networks of mobile omnidirectional sensors, where the sensors move to maximize the probability of track detection of a known distribution of mobile targets traversing a region of interest (ROI). Through extensive simulations, DOC is shown to outperform several existing sensor deployment and control strategies. Furthermore, the computation required by the DOC algorithm is proven to be far reduced compared to that of classical, direct optimal control algorithms.

In this thesis, a hybrid numerical approach is proposed for modeling dynamic fracture in brittle materials. This method is based on a combination of embedded finite element methods and extrinsic cohesive zone models. The effect of different methods to enforce the kinematics at the embedded interface for crack initiation and propagation are investigated and numerically compared. Finally, Nitsche's method is suggested within the hybrid numerical schemes to simulate dynamic fracture. In the pre-failure stage, terms for consistency and stabilization are introduced into the finite element framework with Nitsche's method. When the fracture criterion is met, the extrinsic cohesive law governs the behavior of the opening surfaces by a simple change of framework without modifications of the mesh. This traction and separation law is directly implemented at the interface through an interface approach. Upon closure of the crack surfaces in compression, Nitsche's method is suggested to weakly enforce contact conditions at crack surfaces.

The applicability of the proposed hybrid method is investigated in numerical examples. By using Nitsche's method, the main advantage of the hybrid method for modeling dynamic crack propagation is to avoid unphysical initial slopes in the numerical implementation of extrinsic cohesive laws, which affords us more accurate crack initiation than with the penalty method. Another advantage is that the consistency and stability at unfractured interfaces during crack propagation are maintained and hence the issues caused by the penalty method in explicit dynamic schemes are avoided. Importantly, Nitsche's method performs better than the penalty method conventionally used to prevent interpenetration under compressive loadings.

It is currently an exciting time to be doing research at the intersection of sports and engineering. Advances in wearable sensor technology now enable large quantities of physiological and biomechanical data to be collected from athletes with minimal obstruction and cost. These technological advances, combined with an increased public awareness of the relationship between exercise, fitness, and health, has created an environment where engineering principles can be integrated with biomechanics, exercise physiology, and sports science to dramatically improve methods for physiological assessment, injury prevention, and athletic performance.

The first part of this dissertation develops a new method for analyzing heart rate (HR) and oxygen uptake (VO2) dynamics. A dynamical system model was derived based on the equilibria and stability of the HR and VO2 responses. The model accounts for nonlinear phenomena and person-specific physiological characteristics. A heuristic parameter estimation algorithm was developed to determine model parameters from experimental data. An artificial neural network (ANN) was developed to predict VO2 from HR and exercise intensity data. A series of experiments was performed to validate: 1) the ability of the dynamical system model to make accurate time series predictions for HR and VO2; 2) the ability of the dynamical system model to make accurate submaximal predictions for maximum heart rate (HRmax) and maximal oxygen uptake (VO2max); 3) the ability of the ANN to predict VO2 from HR and exercise intensity data; and 4) the ability of a system comprising an ANN, dynamical system model, and heuristic parameter estimation algorithm to make submaximal predictions for VO2max without requiring VO2 data collection. The dynamical system model was successfully validated through comparisons with experimental data. The model produced accurate time series predictions for HR and VO2 and, more importantly, the model was able to accurately predict HRmax and VO2max using data collected during submaximal exercise. The ANN was successfully able to predict VO2 responses using HR and exercise intensity as system inputs. The system comprising an ANN, dynamical system model, and heuristic parameter estimation algorithm was able to make accurate submaximal predictions for VO2max without requiring VO2 data collection.

The second part of this dissertation applies a support vector machine (SVM) to classify lower extremity movement patterns that are associated with increased lower extremity injury risk. Participants for this study each performed a jump-landing task, and experimental data was collected using two video cameras, two force plates and a chest-mounted single-axis accelerometer. The video data was evaluated to classify the lower extremity movement patterns of the participants as either excellent or poor using the Landing Error Scoring System (LESS) assessment method. Two separate linear SVM classifiers were trained using the accelerometer data and the force plate data, respectively, with the LESS assessment providing the classification labels during training and evaluation. The same participants from this study also performed several bouts of treadmill running, and an additional set of linear SVM classifiers were trained using accelerometer data and gyroscope data to classify movement patterns, with the LESS assessment again providing the classification labels during training and evaluation. Both sets of SVM's performed with a high level of accuracy, and the objective and autonomous nature of the SVM screening methodology eliminates the subjective limitations associated with many current clinical assessment tools.

Wind energy has emerged as one of the most promising and rapidly growing renewable energy technologies in the United States and over the world. The offshore wind energy is of special interest because it has more consistent and faster wind speed, and is usually close to large population areas that are along the coast. However, wake shielding on offshore wind farms substantially reduces the efficiency of downstream wind turbines due to the interaction with the energy-depleted wakes from upwind turbines. This research considers a method to mitigate the wake shielding effect by tilting the turbine axes upward, which causes streamwise vorticity in the near wake so that the energy depleted wakes transport upward alleviating shielding, and pumping more energetic fluid into downstream turbines.

The wake simulations in this research employ a specially developed hybrid free-wake method for wind turbine wakes, that utilizes Vortex Lattice Method (VLM) for near wake representation with appropriate stall and unsteady models, and Constant Circulation Contours Method (CCCM) for turbine far wake representation with a large degree of downwind vorticity diffusion. This approach has been implemented to capture the natural behavior of multi-filament multi-blade complex turbine wakes in relatively short computation time, with the capability to simulate wake interaction with downstream turbines. It is validated through comparison to two wind tunnel tests, NREL/NASA-Ames Wind Tunnel Test and MEXICO, and two turbine wake numerical models, BEM and QBlade.

The wake steering effect for tilted turbines is verified and the degree of effectiveness is assessed. Detailed turbine wake structure is studied to obtain insights into how to strengthen the steering effect and decrease wake velocity deficit. Inline two turbine simulations where one turbine operates in the wake of the other have been performed to assess the advantage of wake steering in power generation of a system of turbines. Beyond the single rotor tilted turbine, an intermeshed rotor wind turbine configuration, consisting of two partially overlapping counter-rotating rotors, has been studied to assess its potential to strength wake steering effect and to intensify wake deficit recovery. These two turbine configurations are compared along with a discussion of potential advantages and challenges. Several model refinements for more robust turbine wake simulation are under development or considered as future research goals.

A National Aeronautics and Space Administration computational fluid dynamics code, OVERFLOW 2, was modified to utilize a harmonic balance solution method. This modification allows for the direct calculation of the nonlinear frequency-domain solution of a periodic, unsteady flow while avoiding the time consuming calculation of long physical transients that arise in aeroelastic applications.

With the usual implementation of this harmonic balance method, converting an implicit flow solver from a time marching solution method to a harmonic balance solution method results in an unstable numerical scheme. However, a relatively simple and computationally inexpensive stabilization technique has been developed and is utilized. With this stabilization technique, it is possible to convert an existing implicit time-domain solver to a nonlinear frequency-domain method with minimal modifications to the existing code.

This new frequency-domain version of OVERFLOW 2 utilizes the many features of the original code, such as various discretization methods and several turbulence models. The use of Chimera overset grids in OVERFLOW 2 requires care when implemented in the frequency-domain. This research presents a harmonic balance version of OVERFLOW 2 that is capable of solving on overset grids for sufficiently small unsteady amplitudes.

The treasure hunt problem to determine how a computational agent can maximize its ability to detect and/or classify multiple targets located in a region of interest (ROI) populated with multiple obstacles. One particular instance of this problem involves optimizing the performance of a sensor mounted on an unmanned aerial vehicle (UAV) flying over a littoral region in order to detect mines buried underground.

Buried objects (including non-metallic ones) have an effect on the thermal conductivity and heat retention of the soil in which they reside. Because of this, objects that are not very deep below the surface often create measurable thermal anomalies on the surface soil. Because of this, infrared (IR) sensors have the potential to find mines and minelike objects (referred to in this thesis as clutters).

As the sensor flies over the ROI, sensor data is obtained. The sensor receives the data as pixellated infrared light signatures. Using this, ground temperature measurements are recorded and used to generate a two-dimensional thermal profile of the field of view (FOV) and map that profile onto the geography of the ROI.

The input stream of thermal data is then passed to an image processor that estimates the size and shape of the detected target. Then a Bayesian Network (BN) trained from a database of known mines and clutters is used to provide the posterior probability that the evidence obtained by the IR sensor for each detected target was the result of a mine or a clutter. The output is a confidence level (CL), and each target is classified as a mine or a clutter according to the most likely explanation (MLE) for the sensor evidence. Though the sensor may produce incomplete, noisy data, inferences from the BN attenuate the problem.

Since sensor performance depends on altitude and environmental conditions, the value of the IR information can be further improved by choosing the flight path intelligently. This thesis assumes that the UAV is flying through an environmentally homogeneous ROI and addresses the question of how the optimal altitude can be determined for any given multi-dimensional environmental state.

In general, high altitudes result in poor resolution, whereas low altitudes result in very limited FOVs. The problem of weighing these tradeoffs can be addressed by creating a scoring function that is directly dependent on a comparison between sensor outputs and ground truth. The scoring function provides a flexible framework through which multiple mission objectives can be addressed by assigning different weights to correct detections, correct non-detections, false detections, and false non-detections.

The scoring function provides a metric of sensor performance that can be used as feedback to optimize the sensor altitude as a function of the environmental conditions. In turn, the scoring function can be empirically evaluated over a number of different altitudes and then converted to empirical Q scores that also weigh future rewards against immediate ones. These values can be used to train a neural network (NN). The NN filters the data and interpolates between discrete Q-values to provide information about the optimal sensor altitude.

The research described in this thesis can be used to determine the optimal control policy for an aircraft in two different situations. The global maximum of the Q-function can be used to determine the altitude at which a UAV should cruise over an ROI for which the environmental conditions are known a priori. Alternatively, the local maxima of the Q-function can be used to determine the altitude to which a UAV should move if the environmental variables change during flight.

This thesis includes the results of computer simulations of a sensor flying over an ROI. The ROI is populated with targets whose characteristics are based on actual mines and minelike objects. The IR sensor itself is modeled by using a BN to create a stochastic simulation of the sensor performance. The results demonstrate how Q-learning can be applied to signals from a UAV-mounted IR sensor whose data stream is preprocessed by a BN classifier in order to determine an optimal flight policy for a given set of environmental conditions.

The dynamics of impacting components is of particular interest to engineers due to concerns about noise and wear, but is particularly difficult to study due to impact's non-linear nature. To begin transferring concepts studied purely analytically to the world of physical mechanisms, four experiments are outlined, and important non-linear concepts highlighted with these systems. A linear oscillator with a kicked impact, an impacting forced pendulum, two impacting forced pendulums, and a cam follower pair are studied experimentally, with complementary numerical results.

Some important ideas highlighted are limit cycles, basins of attraction with many wells, grazing, various forms of coexistence, super-persistent chaotic transients, and liftoff. These concepts are explored using a variety of non-linear tools such as time lag embedding and stochastic interrogation, and discussions of their intricacies when used in non-smooth systems yield important observations for the experimentalist studying impacting systems.

The focus is on experimental results with numerical validation, and spends much time discussing identification of these concepts from an experiment-first mindset, rather than the more traditional analytical-first approach. As such a large volume of experimentally important information on topics such as transducers and forcing mechanism construction are included in the appendices.

Autonomous systems are emerging across many industries. From unmanned aircraft to self-driving cars to closed-loop medical devices, these systems offer great benefits but also pose new risks. Regulators must grapple with how to manage these risks, challenged to keep pace with technological developments and exhibit appropriate precaution without stifling innovation. Seeking inspiration for a viable approach to the regulation of autonomous systems, this thesis draws from the practices of systems engineering, an interdisciplinary field of engineering aimed at managing the risks of complex projects. By comparing systems engineering practices to regulatory options, current regulations, and the inherent challenges of regulating emerging technologies, this thesis concludes that a systems engineering-based approach to regulating autonomous systems offers great potential for managing the risks of autonomous systems while also driving innovation.

Fracture is a common phenomenon in engineering applications. Many types of fracture exist, including, but not limited to, brittle fracture, quasi-brittle fracture, cohesive fracture, and ductile fracture. Predicting fracture has been one of the most challenging research topics in computational mechanics. The variational treatment of fracture and its associated phase-field regularization have been employed with great success for modeling fracture in brittle materials. Extending the variational statement to describe other types of fracture and coupled field phenomena has proven less straightforward. Main challenges that remain include how to best construct a total potential that is both mathematically sound and physically admissible, and how to properly describe the coupling between fracture and other phenomena.

The research presented in this dissertation aims at addressing the aforementioned challenges. A variational framework is proposed to describe fracture in general dissipative solids. In essence, the variational statement is extended to account for large deformation kinematics, inelastic deformation, dissipation mechanisms, dynamic effects, and thermal effects. The proposed variational framework is shown to be consistent with conservations and laws of thermodynamics, and it provides guidance and imposes restrictions on the construction of models for coupled field problems. Within the proposed variational framework, several models are instantiated to address practical engineering problems. A brittle and quasi-brittle fracture model is used to investigate fracture evolution in polycrystalline materials; a cohesive fracture model is applied to revisit soil desiccation; a novel ductile fracture model is proposed and successfully applied to simulate some challenging benchmark problems; and a creep fracture model is developed to simulate the spallation of oxide scale on high temperature heat exchangers.

Cancer comprises a collection of diseases, all of which begin with abnormal tissue growth from various stimuli, including (but not limited to): heredity, genetic mutation, exposure to harmful substances, radiation as well as poor dieting and lack of exercise. The early detection of cancer is vital to providing life-saving, therapeutic intervention. However, current methods for detection (e.g., tissue biopsy, endoscopy and medical imaging) often suffer from low patient compliance and an elevated risk of complications in elderly patients. As such, many are looking to “liquid biopsies” for clues into presence and status of cancer due to its minimal invasiveness and ability to provide rich information about the native tumor. In such liquid biopsies, peripheral blood is drawn from patients and is screened for key biomarkers, chiefly circulating tumor cells (CTCs). Capturing, enumerating and analyzing the genetic and metabolomic characteristics of these CTCs may hold the key for guiding doctors to better understand the source of cancer at an earlier stage for more efficacious disease management.

The isolation of CTCs from whole blood, however, remains a significant challenge due to their (i) low abundance, (ii) lack of a universal surface marker and (iii) epithelial-mesenchymal transition that down-regulates common surface markers (e.g., EpCAM), reducing their likelihood of detection via positive selection assays. These factors potentiate the need for an improved cell isolation strategy that can collect CTCs via both positive and negative selection modalities as to avoid the reliance on a single marker, or set of markers, for more accurate enumeration and diagnosis.

The technologies proposed herein offer a unique set of strategies to focus, sort and template cells in three independent microfluidic modules. The first module exploits ultrasonic standing waves and a class of elastomeric particles for the rapid and discriminate sequestration of cells. This type of cell handling holds promise not only in sorting, but also in the isolation of soluble markers from biofluids. The second module contains components to focus (i.e., arrange) cells via forces from acoustic standing waves and separate cells in a high throughput fashion via free-flow magnetophoresis. The third module uses a printed array of micromagnets to capture magnetically labeled cells into well-defined compartments, enabling on-chip staining and single cell analysis. These technologies can operate in standalone formats, or can be adapted to operate with established analytical technologies, such as flow cytometry. A key advantage of these innovations is their ability to process erythrocyte-lysed blood in a rapid (and thus high throughput) fashion. They can process fluids at a variety of concentrations and flow rates, target cells with various immunophenotypes and sort cells via positive (and potentially negative) selection. These technologies are chip-based, fabricated using standard clean room equipment, towards a disposable clinical tool. With further optimization in design and performance, these technologies might aid in the early detection, and potentially treatment, of cancer and various other physical ailments.

Biosensing is a critical area of research that involves detecting and measuring biological molecules. Among the various types of biosensors, acoustic biosensors are attractive for their simplicity, robustness, and low cost, particularly in point-of-care (POC) applications. However, the quality factor (Q-factor) of acoustic biosensors is often low, limiting their sensitivity and accuracy in terms of in-liquid gravimetric detection for biosensing applications. In this dissertation, we present a novel approach that eliminates nearly all dissipation and damping from sample liquids, rendering a significant improvement in Q-factor for in-liquid gravimetric detection. We constructed rigid microfluidic channels to confine liquids and the associated acoustic energy, thereby eliminating acoustic radiation damping. We also used the channels' side walls to create pressure waves, confining the liquids within and suppressing acoustic damping due to the viscous layer. The quartz crystal microbalance (QCM) was selected as the model system for implementing the new paradigm due to its widespread usage in various applications, simplicity, cost-effectiveness, and relevance of its principles to other types of acoustic biosensors. We hypothesized that the ratio of the wavelength of the pressure wave to the width of the channels is a crucial determining factor for optimal performance. We then tested the hypothesis by building the microfluidic QCM (the µ-QCM) to improve the Q-factor of conventional QCM. The combination of experiments, simulations, and theoretical studies demonstrated a 10-fold improvement in the Q-factor. The new system offers many other advantages, including direct data interpretation, minimized sample volume requirement, and easier temperature control for in-liquid gravimetric detection. Additionally, the same principles can be applied to other acoustic biosensors, benefiting the entire field.

In this dissertation work, the aim was to garner better mechanistic understanding of how shock wave lithotripsy (SWL) breaks stones in order to guide design improvements to a modern electromagnetic (EM) shock wave lithotripter. To accomplish this goal, experimental studies were carefully designed to isolate mechanisms of fragmentation, and models for wave propagation, fragmentation, and stone motion were developed. In the initial study, a representative EM lithotripter was characterized and tested for in vitro stone comminution efficiency at a variety of field positions and doses using phantom kidney stones of variable hardness, and in different fluid mediums to isolate the contribution of cavitation. Through parametric analysis of the acoustic field measurements alongside comminution results, a logarithmic correlation was determined between average peak pressure incident on the stone surface and comminution efficiency. It was also noted that for a given stone type, the correlations converged to an average peak pressure threshold for fragmentation, independent of fluid medium in use. The correlation of average peak pressure to efficacy supports the rationale for the acoustic lens modifications, which were pursued to simultaneously enhance beam width and optimize the pulse profile of the lithotripter shock wave (LSW) via in situ pulse superposition for improved stone fragmentation by stress waves and cavitation, respectively. In parallel, a numerical model for wave propagation was used to investigate the variations of critical parameters with changes in lens geometry. A consensus was reached on a new lens design based on high-speed imaging and stone comminution experiments against the original lens at a fixed acoustic energy setting. The results have demonstrated that the new lens has improved efficacy away from the focus, where stones may move due to respiration, fragmentation, acoustic radiation forces, or voluntary patient movements. Using traditional theory of brittle fragmentation and newfound understanding of average peak pressure correlation to stone comminution, the entire set of stone comminution data for lens comparison was modeled using a Weibull-style distribution function. This model linked both the average peak pressure and shock wave dose to efficacy, including their respective threshold parameters, and demonstrated correlation of coefficients to cavitation activity. Subsequently, this model was used in prediction of stone comminution efficiency from mimicked respiratory motions in vitro, which compared favorably to actual simulated motion studies using both the new and original lenses. Under a variety of mimicked respiratory motions, the new lens produced statistically higher stone comminution efficiency than the original lens. These results were confirmed in vivo in a swine model, where the new lens produced statistically higher stone comminution after 1,000 and 2,000 shocks. Finally, a mechanistic investigation into the effects of cavitation with the original lens was conducted using an integrated, self-focusing annular ring transducer specially designed for tandem pulse lithotripsy. It was found that cavitation and stone comminution efficiency are progressively enhanced by tandem pulsing as source energies of both the primary LSW and trailing pressure pulse increase, which suggests cavitation and stress waves act synergistically enhance the efficacy in kidney stone fragmentation.

Laser scalpels are utilized across a variety of invasive and non-invasive surgical procedures due to their precision and non-contact nature. Meanwhile, robotic and robotic-assisted surgeries are becoming more prevalent with the promise of improving surgical outcomes through increased precision, reduced operating times, and minimally invasive procedures. This dissertation presents methods and devices developed to enable assistive robotic laser surgery, with the goal of realizing the surgical benefits of both and ultimately improving surgical outcomes for patients.

The device is first used to demonstrate targeted soft tissue resection in porcine brain in an open-loop fashion. This device, coined the "TumorCNC" combines 3D scanning capabilities with a steerable surgical laser. Results show high variance around target cut depths which motivats the need for a closed-loop feedback and control as well as characterization of laser-tissue interactions for predictive modeling.

To begin to address the technical difficulties of closed-loop ablation, a model-based approach is taken. A soft tissue ablation simulator is developed and used in conjunction with an optimization routine to select parameters which maximize the total resection of target tissue while minimizing the damage to surrounding tissue. The optimization is performed using genetic algorithms. The simulator predicts the ablative properties of tissue from an interrogation cut for tuning and simulates the removal of a tumorous tissue embedded on the surface of healthy tissue using a laser scalpel. This demonstrates the ability to control depth and smoothness of cut using genetic algorithms to optimize the ablation parameters and cutting path. The laser power level, cutting rate and spacing between cuts are optimized over multiple surface cuts to achieve the desired resection volumes.

Noting that the modeling approached developed is applicable to other laser treatments requiring uniformity of laser energy deposition, a study of superficial region ablation is performed for applications in dermatology. The TumorCNC is now outfitted with an RGB-D camera. To accurately ablate targets chosen from the color image, a 3D extrinsic calibration method between the RGB-D camera frame and the laser coordinate system is implemented. The accuracy of the calibration method is tested on phantoms with planar and cylindrical surfaces. Positive error and negative error, as defined as undershooting and overshooting over the target area, are reported for each test. For 60 total test cases, the root-mean-square of the positive and negative error in both planar and cylindrical phantoms is less than 1.0mm, with a maximum absolute error less than 2.0mm. This work demonstrates the feasibility of automated laser therapy with surgeon oversight via our sensor system.

As a demonstration of the culmination of these techniques, a closed-loop, adaptive online estimation of ablative properties for soft tissue laser resection of tumors is demonstrated. First, a laser photoablation feature is created in an agarose based tissue phantom using a robotic laser photoablation device equipped with a carbon dioxide laser. Second, the device measures the surface profile of the ablated feature for analysis. Genetic algorithms in conjunction with the photoablation simulator based on the steady-state photoablation model are used to estimate the photoablation enthalpy, density, and ablative radiant threshold of the tissue phantom. The parameters and model are validated through comparison of predicted and measured surface ablations at varying depths. This approach proved effective for predicting the resulting surface profiles for small cut depths (<= 2mm) and generating laser cut paths to reach a desired depth of cut for a large surface area. This work is enabling of closed-loop resection of tissue in robotic laser surgery.

Polynomial chaos expansions provide an efficient and robust framework to analyze and quantify uncertainty in computational models. This dissertation explores the use of adaptive sparse grids to reduce the computational cost of determining a polynomial model surrogate while examining and implementing new adaptive techniques.

Determination of chaos coefficients using traditional tensor product quadrature suffers the so-called curse of dimensionality, where the number of model evaluations scales exponentially with dimension. Previous work used a sparse Smolyak quadrature to temper this dimensional scaling, and was applied successfully to an expensive Ocean General Circulation Model, HYCOM during the September 2004 passing of Hurricane Ivan through the Gulf of Mexico. Results from this investigation suggested that adaptivity could yield great gains in efficiency. However, efforts at adaptivity are hampered by quadrature accuracy requirements.

We explore the implementation of a novel adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed adaptive pseudo-spectral projection (aPSP) algorithm that is based on a direct application of Smolyak's sparse grid formula, and that allows for the use of arbitrary admissible sparse grids. Such a construction ameliorates the severe restrictions posed by insufficient quadrature accuracy. The adaptive algorithm is tested using an existing simulation database of the HYCOM model during Hurricane Ivan. The {\it a priori} tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling.

In order to provide a finer degree of resolution control along two distinct subsets of model parameters, we investigate two methods to build polynomial approximations. The two approaches are based with pseudo-spectral projection (PSP) methods on adaptively constructed sparse grids. The control of the error along different subsets of parameters may be needed in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid pseudo-spectral projection is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the projection error along individual dimensions. Specifically, we use a Sobol decomposition of the projection surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube ignition model involving 22 uncertain parameters and 3 design parameters. The numerical experiments indicate that whereas both methods provide effective means for tuning the quality of the representation along distinct subsets of parameters, adaptive PSP in the global parameter space generally requires fewer model evaluations than the nested approach to achieve similar projection error.

In order to increase efficiency even further, a subsampling technique is developed to allow for local adaptivity within the aPSP algorithm. The local refinement is achieved by exploiting the hierarchical nature of nested quadrature grids to determine regions of estimated convergence. In order to achieve global representations with local refinement, synthesized model data from a lower order projection is used for the final projection. The final subsampled grid was also tested with two more robust, sparse projection techniques including compressed sensing and hybrid least-angle-regression. These methods are evaluated on two sample test functions and then as an {\it a priori} analysis of the HYCOM simulations and the shock-tube ignition model investigated earlier. Small but non-trivial efficiency gains were found in some cases and in others, a large reduction in model evaluations with only a small loss of model fidelity was realized. Further extensions and capabilities are recommended for future investigations.

Interfaces play a dominant role in governing the response of many biological systems and they pose many challenges to traditional finite element. For sharp-interface model, traditional finite element methods necessitate the finite element mesh to align with surfaces of discontinuities. Diffuse-interface model replaces the sharp interface with continuous variations of an order parameter resulting in significant computational effort. To overcome these difficulties, we focus on developing a computationally efficient spline-based finite element method for interface problems.

A key challenge while employing B-spline basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. We begin by examining weak enforcement of such conditions for B-spline basis functions, with application to both second- and fourth-order problems based on Nitsche's approach. The use of spline-based finite elements is further examined along with a Nitsche technique for enforcing constraints on an embedded interface. We show that how the choice of weights and stabilization parameters in the Nitsche consistency terms has a great influence on the accuracy and robustness of the method. In the presence of curved interface, to obtain optimal rates of convergence we employ a hierarchical local refinement approach to improve the geometrical representation of interface.

In multiple dimensions, a spline basis is obtained as a tensor product of the one-dimensional basis. This necessitates a rectangular grid that cannot be refined locally in regions of embedded interfaces. To address this issue, we develop an adaptive spline-based finite element method that employs hierarchical refinement and coarsening techniques. The process of refinement and coarsening guarantees linear independence and remains the regularity of the basis functions. We further propose an efficient data transfer algorithm during both refinement and coarsening which yields to accurate results.

The adaptive approach is applied to vesicle modeling which allows three-dimensional simulation to proceed efficiently. In this work, we employ a continuum approach to model the evolution of microdomains on the surface of Giant Unilamellar Vesicles. The chemical energy is described by a Cahn-Hilliard type density functional that characterizes the line energy between domains of different species. The generalized Canham-Helfrich-Evans model provides a description of the mechanical energy of the vesicle membrane. This coupled model is cast in a diffuse-interface form using the phase-field framework. The effect of coupling is seen through several numerical examples of domain formation coupled to vesicle shape changes.

Numerical simulation of hyperbolic systems remains a challenge, particularly in the case of complex geometries. In particular, the need to construct meshes for complicated geometries is a bottleneck in many cases. This is especially evident when doing rapid prototyping and design optimization, where generating a new mesh for every trial geometry is prohibitive. These difficulties can be obviated by employing an embedded/immersed boundary method, in which boundary conditions are enforced weakly.

In my Ph.D. work, a new Nitsche-type approach is proposed for the weak enforcement of Dirichlet and Neumann boundary conditions in the context of time-domain wave propagation problems in mixed form. A peculiar feature of the proposed method is that, due to the hyperbolic structure of the problem considered, two penalty parameters are introduced, corresponding to Dirichlet and Neumann conditions, respectively. A stability and convergence estimate is also provided, in the case of a discontinuous-in-time Galerkin space–time integrator. The spatial discretization used is based on a stabilized method with equal order interpolation for all solution components. In principle, however, the proposed methodology is not confined to stabilized methods. An extensive set of tests are provided to validate the robustness and accuracy of the proposed approach.

The proposed Nitsche method is then extended to embedded domain computations of hyperbolic systems, using as models the equations of acoustic wave propagation and shallow water flows, through Shifted Boundary Method (SBM). The SBM belongs to the class of surrogate/approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforced weakly, are appropriately modified to preserve optimal error convergence rates. Accuracy, stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations. Comparisons with standard weak boundary conditions imposed on grids that conform to the geometry of the computational domain boundaries are also presented.

This dissertation explores the aeroelastic stability of a folding wing using both theoretical and experimental methods. The theoretical model is based on the existing clamped-wing aeroelastic model that uses beam theory structural dynamics and strip theory aerodynamics. A higher-fidelity theoretical model was created by adding several improvements to the existing model, namely a structural model that uses ANSYS for individual wing segment modes and an unsteady vortex lattice aerodynamic model. The comparison with the lower-fidelity model shows that the higher-fidelity model typical provides better agreement between theory and experiment, but the predicted system behavior in general does not change, reinforcing the effectiveness of the low-fidelity model for preliminary design of folding wings. The present work also conducted more detailed aeroelastic analyses of three-segment folding wings, and in particular considers the Lockheed-type configurations to understand the existence of sudden changes in predicted aeroelastic behavior with varying fold angle for certain configurations. These phenomena were observed in carefully conducted experiments, and nonlinearities - structural and geometry - were shown to suppress the phenomena. Next, new experimental models with better manufacturing tolerances are designed to be tested in the Duke University Wind Tunnel. The testing focused on various configurations of three-segment folding wings in order to obtain higher quality data. Next, the theoretical model was further improved by adding aircraft longitudinal degrees of freedom such that the aeroelastic model may predict the instabilities for the entire aircraft and not just a clamped wing. The theoretical results show that the flutter instabilities typically occur at a higher air speed due to greater frequency separation between modes for the aircraft system than a clamped wing system, but the divergence instabilities occur at a lower air speed. Lastly, additional experimental models were designed such that the wing segments may be rotated while the system is in the wind tunnel. The fold angles were changed during wind tunnel testing, and new test data on wing response during those transients were collected during these experiments.

One of the grand challenges faced by industry is the accurate prediction of unsteady aerodynamics events, including frequency lock-in and forced response. These aeromechanical incidents occurring in airplane engines and gas turbines can cause high-amplitude blade vibration and potential failure of the engine or turbine. During the last decades, the development of computational fluid dynamics has allowed the design and optimization of complex components while reducing the need for expensive engine testing. However, the validation of frequency lock-in and forced response numerical results with experimental data is very incomplete. Despite tremendous advances in computational capabilities, industry is still looking to validate design tools and guidelines to avoid these potentially costly aeroelastic events early in the design process.

The research efforts presented in this dissertation investigate the aeroelastic phenomena of frequency lock-in and forced response in turbomachinery. First, frequency lock-in is predicted for two structures, namely a two-dimensional cylinder and a single three-dimensional airfoil, and the results are compared to experimental data so that the methods can be extended to more complex structures. For these two simpler structures, a frequency domain harmonic balance code is used to estimate the natural shedding frequency and the corresponding lock-in region. Both the shedding frequencies and the lock-in regions obtained by an enforced motion method agree with experimental data from previous literature and wind tunnel tests. Moreover, the aerodynamic model of the vibrating cylinder is coupled with the structural equations of motion to form a fluid-structure interaction model and to compute the limit-cycle oscillation amplitude of the cylinder. The extent of the lock-in region matches the experimental data very well, yet the peak amplitude is underestimated in the numerical model. We demonstrate that the inclusion of the cylinder second degree of freedom has a significant impact on the cylinder first degree of freedom amplitude. Moreover, it is observed that two harmonics need to be kept in the equations of motion for accurate prediction of the unsteady forces on the cylinder.

The second important topic covered is a comprehensive forced response analysis conducted on a multi-stage axial compressor and compared with the initial data of the largest forced response experimental data set ever obtained in the field. Both a frequency domain and a time domain codes are used. The steady-state and time-averaged aerodynamic performance results compare well with experimental data, although losses are underestimated due to the lack of secondary flow paths and fillets in the model. The use of mixing planes in the steady simulations underpredicts the wakes by neglecting the important interactions between rows. Therefore, for similar cases with significant flow separation, the use of a decoupled method for forced response predictions cannot yield accurate results. A full multi-row transient analysis must be conducted for accurate prediction of the wakes and surface unsteady pressures. Finally, for the first time, predicted mistuned blade amplitudes are compared to mistuned experimental data. The downstream stator is found to be necessary for the accurate prediction of the modal forces and vibration amplitudes. The mistuned rotor is shown to be extremely sensitive to perturbations in blade frequency mistuning, aerodynamic asymmetry, and excitation traveling wave content. Since this dissertation presents the initial results of a five-year research program, more research will be conducted on this compressor to draw guidelines that can be used by aeromechanical engineers to safely avoid forced response events in the design of jet engines and gas turbines.

This dissertation concerns the optimization of the aeroelastic performance of conventional

helicopter rotors, considering various design variables such cyclic and higher

harmonic controls. A nite element model is introduced to model the structural

eects of the blade, and a coupled induced velocity/projected force model is used

to couple this structural model to the aerodynamic model constructed in previous

works. The system is then optimized using two separate objective functions: minimum

power and minimum vibrational loading at the hub. The model is validated

against several theoretical and experimental models, and good agreement is demonstrated

in each case. Results of the rotor in forward

ight demonstrate for realistic

advance ratios the original lifting surface model is sucient for modeling normalized

induced power. Through use of the dynamics model the vibrational loading minimization

is shown to be extremely signicant, especially when using more higher

harmonic control. However, this decrease comes at an extreme cost to performance

in the form of the normalized induced power nearly doubling. More realistic scenarios

can be created using multi-objective optimization, where it is shown that vibrational

loading can be decreased around 60% for a 5% increase in power.