In this thesis, we mainly investigate the application of a nodal discontinuous Galerkin spectral element method (DGSEM) for simulating processes in column liquid chromatography. Additionally, we investigate the effectiveness of a total variation diminishing in the mean (TVDM) limiter in controlling spurious oscillations related to the Gibbs phenomenon. With an order-of-accuracy test, we demonstrated that our nodal DGSEM achieved and, in multiple instances, even exceeded theoretical convergence rates, especially with an increased number of elements, validating the use of high-order basis functions for achieving high-order accuracy. We also demonstrated how setup parameters could affect process outcomes, which suggests that numerical simulations can help guide the development of experimental methods since they can explore the solution space of an optimization problem much faster than experimental procedures by leveraging computational speed. Finally, we showed that the TVDM limiter successfully eliminated severe oscillations and negative concentrations near shock regions but introduced significant smearing of the shocks. These findings validate the nodal DGSEM as a highly accurate and reliable tool for detailed modeling of column liquid chromatography, which is essential for improving efficiency, yield, and product quality in biopharmaceutical manufacturing.

This thesis deals with Large Eddy Simulations (LES) of the Atmospheric Boundary Layer (ABL), focusing on studying the resolution dependence of turbulent passive scalar transport within the layer. The ABL is the lowest part of the atmosphere, where humans live and conduct most of their daily activities. Here, a scalar was injected at four different heights in a mixed shear- and convective-driven ABL, which was simulated using the Spectral Element Method (SEM) code Nek5000. The statistics of the four scalars were analysed and their resolution dependence was studied and compared to that of non-scalar quantities. No significant resolution dependence was found with regards to non-scalar quantities, while scalar quantities show a rather strong dependence on resolution especially in the first quarter of the simulation. Negative concentration values are found within the layer and some approaches to solve the problem are proposed. Statistics alone provide an accurate description of the general ABL behaviour, but are found to be insufficient to capture the dynamics of the scalar injection, which ought to be analysed with more advanced methods (e.g. modal decomposition). The structures arising within the layer are also analysed, and further work regarding the study of scalar fronts is suggested.

The vulnerability of a ship to the impact of underwater explosions (UNDEX) and how to incorporate this factor into early-stage ship design is an important aspect in the ship survivability study. In this dissertation, attention is focused on the cost-efficient simulation of the ship response to a far-field UNDEX which involves fluid shock waves, cavitation, and fluid-structural interaction. Traditional fluid numerical simulation approaches using the Finite Element Method to track wave propagation and cavitation requires a high-level of mesh refinement to prevent numerical dispersion from discontinuities. Computation also becomes quite expensive for full ship-related problems due to the large fluid domain necessary to envelop the ship. The burden is aggravated by the need to generate a fluid mesh around the irregular ship hull geometry, which typically requires significant manual intervention. To accelerate the design process and enable the consideration of far-field UNDEX vulnerability, several contributions are made in this dissertation to make the simulation more efficient. First, a Cavitating Acoustic Spectral Element approach which has shown computational advantages in UNDEX problems, but not systematically assessed in total ship application, is used to model the fluid. The use of spectral elements shows greater structural response accuracy and lower computational cost than the traditional FEM. Second, a novel fully automatic all-hexahedral mesh generation scheme is applied to generate the fluid mesh. Along with the spectral element, the all-hex mesh shows greater accuracy than the all-tetrahedral finite element mesh which is typically used. This new meshing approach significantly saves time for mesh generation and allows the spectral element, which is confined to the hexahedral element, to be applied in practical ship problems. A further contribution of this dissertation is the development of a surrogate non-numerical approach to predict structural peak responses based on the shock factor concept. The regression analysis reveals a reasonably strong linear relationship between the structural peak response and the shock factor. The shock factor can be conveniently employed in the design aspects where the peak response is sufficient, using much less computational resources than numerical solvers. / Doctor of Philosophy / The vulnerability of a ship to the impact of underwater explosions (UNDEX) and how to incorporate this factor into early-stage ship design is an important aspect in the ship survivability study. In this dissertation, attention is focused on the cost-efficient simulation of the ship response to a far-field UNDEX which involves fluid shock waves, cavitation, and fluid-structural interaction. Traditional fluid numerical simulation approaches using the Finite Element Method to track wave propagation and cavitation requires a highly refined mesh to deal with large numerical errors. Computation also becomes quite expensive for full ship-related problems due to the large fluid domain necessary to envelop the ship. The burden is aggravated by the need to generate a fluid mesh around the irregular ship hull geometry, which typically requires significant manual intervention. To accelerate the design process and enable the consideration of far-field UNDEX vulnerability, several contributions are made in this dissertation to make the simulation more efficient. First, a Cavitating Acoustic Spectral Element approach, which has shown computational advantages in UNDEX problems but not systematically assessed in total ship application, is used to model the fluid. The use of spectral elements shows greater structural response accuracy and lower computational cost than the traditional FEM. Second, a novel fully automatic all-hexahedral mesh generation scheme is applied to generate the fluid mesh. Along with the spectral element, the all-hex mesh shows greater accuracy than the all-tetrahedral finite element mesh which is typically used. A further contribution of this dissertation is the development of a non-numerical approach which can approximate peak structural responses comparable to the numerical solution with far less computational effort.

To obtain predictive modeling of a spacecraft, the author investigates the effects of adding cables to a simple structure with the goal of developing an understanding of how cables interacting with a structure. In this research, the author presents predictive and accurate modeling of a cable-harnessed structure by means of the Spectral Element Method (SEM). A double beam model is used to emulate a cable-harnessed structure. SEM modeling can define the location and the number of connections between the two beams in a convenient fashion. The proposed modeling is applied and compared with the conventional FEM. The modeling approach was compared to and validated by measurement data. The validated modeling was implemented to investigate the effect of the number of connections, of the spring stiffness of interconnections, and of mass portion of an attached cable. Damping has an important role in structural design because it reduces the dynamic response, thereby avoiding excessive deflection or stress, fatigue loads, and settling times. Experimental results with some specimens indicate a clear change of damping on the main structure with the inclusion of cable dynamics. The author investigated the modification of the damping of the host structure induced by various attached cables. The identification of a damping matrix is performed using measured data. The effect of the flexibility of a cable harness on damping is observed through experiments with various types of cables. The effect of the number of connections on damping is also investigated by changing the number of connections. Moreover, to overcome the sensitivity to noise in measured data of damping matrix identification approach, various methods are compared with a simulated lumped model and real test results. An improved damping matrix identification approach is proposed and can generate the unique damping matrix over the full frequency range of interest. / Ph. D.

Detection, characterization and prognosis of damage in civil, aerospace and mechanical structures, known as structural health monitoring (SHM), have been a growing area of research over the last few decades. As several in-service civil, mechanical and aerospace structures are approaching or even exceeding their design life, the implementation of SHM systems is becoming a necessity. SHM is the key for transforming schedule-driven inspection and maintenance into condition-based maintenance, which promises enhanced safety and overall life-cycle cost reduction. While damage detection and characterization can be achieved, among other techniques, by analyzing the dynamic response of the structure under test, damage prognosis requires the additional knowledge of loading patterns acting on the structure. Accurate, nondestructive, and reference-free measurement of the state-of-stress in structural components has been a long standing challenge without a fully-satisfactory outcome. In light of this, the main goal of this research effort is to advance the current state of the art of structural health and loading monitoring, with focus being cast on impedance-based SHM and acoustoelastic-based stress measurement techniques. While impedance-based SHM has been successfully implemented as a damage detection technique, the utilization of electromechanical impedance measurements for damage characterization imposes several challenges. These challenges are mainly stemming from the high-frequency nature of impedance measurements. Current acoustoelastic-based practices, on the other hand, are hindered by their poor sensitivity and the need for calibration at a known state of stress. Addressing these challenges by developing and integrating theoretical models, numerical algorithms and experimental techniques defines the main objectives of this work. A key enabler for both health and loading monitoring techniques is the utilization of piezoelectric transducers to excite the structure and measure its response. For this purpose, a new three-layer spectral element for piezoelectric-structure interaction has been developed in this work, where the adhesive bonding layer has been explicitly modeled. Using this model, the dynamic response of piezoelectric-augmented structures has been investigated. A thorough parametric study has been conducted to provide a better understanding of bonding layer impact on the response of the coupled structure. A procedure for piezoelectric material characterization utilizing its free electromechanical impedance signature has been also developed. Furthermore, impedance-based damage characterization has been investigated, where a novel optimization-based damage identification approach has been developed. This approach exploits the capabilities of spectral element method, along with the periodic nature of impedance peaks shifts with respect to damage location, to solve the ill-posed damage identification problem in a computationally efficient manner. The second part of this work investigates acoustoelastic-based stress measurements, where model-based technique that is capable of analyzing dispersive waves to calculate the state of stress has been developed. A criterion for optimal selection of excitation waveforms has been proposed in this work, taking into consideration the sensitivity to the state of stress, the robustness against material and geometric uncertainties, and the ability to obtain a reflections-free response at desired measurement locations. The impact of material- and geometry-related uncertainties on the performance of the stress measurement algorithm has also been investigated through a comprehensive sensitivity analysis. The developed technique has been experimentally validated, where true reference-free, uncalibrated, acoustoelastic-based stress measurements have been successfully conducted. Finally, the applicability of the aforementioned health and loading monitoring techniques to railroad track components has been investigated. Extensive in-lab experiments have been carried out to evaluate the performance of these techniques on lab-scale and full-scale rail joints. Furthermore, in-field experiments have been conducted, in collaboration with Norfolk Southern and the Transportation Technology Center Inc., to further investigate the performance of these techniques under real life operating and environmental conditions. / Ph. D.

This thesis presents a GPU accelerated implementation of a high order splitting scheme with a spectral element discretization for the incompressible Navier Stokes (INS) equations. While others have implemented this scheme on clusters of processors using the Nek5000 code, to my knowledge this thesis is the first to explore its performance on the GPU. This work implements several of the Nek5000 algorithms using OpenCL kernels that efficiently utilize the GPU memory architecture, and achieve massively parallel on chip computations. These rapid computations have the potential to significantly enhance computational fluid dynamics (CFD) simulations that arise in areas such as weather modeling or aircraft design procedures. I present convergence results for several test cases including channel, shear, Kovasznay, and lid-driven cavity flow problems, which achieve the proven convergence results.

<p>In this study we propose a fast hybrid spectral-element time-domain (SETD) / finite-element time-domain (FETD) method for transient analysis of multiscale electromagnetic problems, where electrically fine structures with details much smaller than a typical wavelength and electrically coarse structures comparable to or larger than a typical wavelength coexist.</p><p>Simulations of multiscale electromagnetic problems, such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and electronic packaging, can be very challenging for conventional numerical methods. In terms of spatial discretization, conventional methods use a single mesh for the whole structure, thus a high discretization density required to capture the geometric characteristics of electrically fine structures will inevitably lead to a large number of wasted unknowns in the electrically coarse parts. This issue will become especially severe for orthogonal grids used by the popular finite-difference time-domain (FDTD) method. In terms of temporal integration, dense meshes in electrically fine domains will make the time step size extremely small for numerical methods with explicit time-stepping schemes. Implicit schemes can surpass stability criterion limited by the Courant-Friedrichs-Levy (CFL) condition. However, due to the large system matrices generated by conventional methods, it is almost impossible to employ implicit schemes to the whole structure for time-stepping.</p><p>To address these challenges, we propose an efficient hybrid SETD/FETD method for transient electromagnetic simulations by taking advantages of the strengths of these two methods while avoiding their weaknesses in multiscale problems. More specifically, a multiscale structure is divided into several subdomains based on the electrical size of each part, and a hybrid spectral-element / finite-element scheme is proposed for spatial discretization. The hexahedron-based spectral elements with higher interpolation degrees are efficient in modeling electrically coarse structures, and the tetrahedron-based finite elements with lower interpolation degrees are flexible in discretizing electrically fine structures with complex shapes. A non-spurious finite element method (FEM) as well as a non-spurious spectral element method (SEM) is proposed to make the hybrid SEM/FEM discretization work. For time integration we employ hybrid implicit / explicit (IMEX) time-stepping schemes, where explicit schemes are used for electrically coarse subdomains discretized by coarse spectral element meshes, and implicit schemes are used to overcome the CFL limit for electrically fine subdomains discretized by dense finite element meshes. Numerical examples show that the proposed hybrid SETD/FETD method is free of spurious modes, is flexible in discretizing sophisticated structure, and is more efficient than conventional methods for multiscale electromagnetic simulations.</p> / Dissertation

Supercritical fluid extraction has been used to recover biomaterials from natural matrices. Mathematical modeling of the extraction is required for process design and scale up. Existing models in literature are correlative and dependent upon the experimental data. Construction of predictive models giving reliable results in the lack of experimental data is precious. The long term objective of this study was to construct a predictive mass transfer model, representing supercritical fluid extraction of biomaterials in packed beds by the method of volume averaging. In order to develop mass transfer equations in terms of volume averaged variables, velocity and velocity deviation fields, closure variables were solved for a specific case and the coefficients of volume averaged mass transfer equation for the specific case were computed using one and two-dimensional geometries via analytical and numerical solutions, respectively. Spectral Element method with Domain Decomposition technique, Preconditioned Conjugate Gradient algorithm and Uzawa method were used for the numerical solution. The coefficients of convective term with additional terms of volume averaged mass transfer equation were similar to superficial velocity. The coefficients of dispersion term were close to diffusivity of oil in supercritical carbon dioxide. The coefficients of interphase mass transfer term were overestimated in both geometries. Modifications in boundary conditions, change in geometry of particles and use of three-dimensional computations would improve the value of the coefficient of interphase mass transfer term.

<p>Numerical methods such as Monte Carlo method (MCM), finite difference method (FDM) and finite element method (FEM) have been successfully implemented to solve financial partial differential equations (PDEs). Sophisticated computational algorithms are strongly desired to further improve accuracy and efficiency.</p><p>The relatively new spectral element method (SEM) combines the exponential convergence of spectral method and the geometric flexibility of FEM. This dissertation carefully investigates SEM on the pricing of European options and their Greeks (Delta, Gamma and Theta). The essential techniques, Gauss quadrature rules, are thoroughly discussed and developed. The spectral element method and its error analysis are briefly introduced first and expanded in details afterwards.</p><p>Multi-element spectral element method (ME-SEM) for the Black-Scholes PDE is derived on European put options with and without dividend and on a condor option with a more complicated payoff. Under the same Crank-Nicolson approach for the time integration, the SEM shows significant accuracy increase and time cost reduction over the FDM. A novel discontinuous payoff spectral element method (DP-SEM) is invented and numerically validated on a European binary put option. The SEM is also applied to the constant elasticity of variance (CEV) model and verified with the MCM and the valuation formula. The Stochastic Alpha Beta Rho (SABR) model is solved with multi-dimensional spectral element method (MD-SEM) on a European put option. Error convergence for option prices and Greeks with respect to the number of grid points and the time step is analyzed and illustrated.</p> / Dissertation

A Direct Numerical Simulation of an iced Natural Laminar Flow NLF-0414 airfoil is carried out using a high-order spectral element method for low chord Reynolds numbers (O(10^5)). This study aims to advance the state-of-the-art for accurate computational modeling of transition, iced airfoil aerodynamics, and irregular surface spectral element method Direct Numerical Simulation. Ice accretion over an aircraft, ranging from light to severe, changes the aerodynamic profile of the airfoil and alters the overall performance. The literature presents simulations that have been carried out with a range of turbulence models which fail to accurately capture the complex physics of these flows. The iced profiles being studied, Run 606 and 622-2D, were obtained from a Technical Publication by NASA on iced airfoils including the NLF-0414, and were selected as they are relatively lightly iced profiles of the NLF-0414. The largest bottleneck with the current advancement in High Performance Computing is the computation time required for Direct Numerical Simulation. Results such as lift, drag, pressure, and skin friction coefficients, for a clean NLF-0414 and two lightly iced NLF-0414 airfoils at chord Reynolds numbers of Rec = 1 x 10^5 and Rec = 2 x 10^5 are visualized and discussed, showing the degradation of the natural laminar flow due to ice accretion. Turbulence statistics are calculated to study the effective contributions of turbulent fluctuations in the flow to further understand the flow physics near transition. The detailed study of these six cases has led us to 1) further understand the complexities of the transition process on iced airfoils, 2) observe and explain the sometimes unexpected changes in aerodynamic performance due to varying iced geometries, and 3) establish a methodology for spectral element method Direct Numerical Simulations.