The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.

This thesis focuses on the effects of slip-slip, slip-twin, and slip-precipitates interactions on strain hardening, with the intent of developing comprehensive modelling capabilities enabling to investigate unit processes and their collective effects up to the macroscopic response. To this end, the modelling strategy adopted in this work relies on a two-way exchange of information between predictions obtained by discrete dislocation dynamics (DDD) simulations and crystal plasticity laws informed by DDD. At the scale of lattice defects, a DDD tool enabling simulations on any crystalline structure is developed to model dislocation-dislocation, dislocation-twin and dislocation-particles interactions. The tool is first used to quantify the collective effect and strength of dislocation-dislocation interactions on latent-hardening, especially in the case of pure Mg. With regards to slip-twin interactions, a transmission mechanism is implemented in the DDD framework so as to investigate the collective effects of dislocation transmission across a twin-boundary. With respect to slip-particles interactions, an efficient novel DDD approach based on a Fast Fourier Transform (FFT) technique is developed to include the field fluctuations related to elastic heterogeneities giving rise to image forces on dislocation lines. In addition, the DDD-FFT approach allows for the efficient treatment of anisotropic elasticity, thereby paving the way towards performing DDD simulations in low-symmetry polycrystals. The information extracted from the collective dislocation interactions are then passed to a series of constitutive models, and later used to quantify their effects at the scale of the polycrystal. For such purpose, a constitutive framework capable of receiving information from lower scales and establishing a direct connection with DDD simulations is notably developed.

<div>Incompressible single-phase and two-phase flows are widely encountered in and underlie many engineering applications. In this thesis, we aim to develop efficient methods and algorithms for numerical simulations of these classes of problems. Specically, we present two schemes: (1) a modied consistent splitting scheme for incompressible single-phase flows with open/out flow boundaries; (2) a three-dimensional hybrid spectral element-Fourier spectral method for wall-bounded two-phase flows.</div><div><br></div><div><div>In the first part of this thesis, we present a modied consistent splitting type scheme together with a family of energy stable outflow boundary conditions for incompressible single-phase outflow simulations. The key distinction of this scheme lies</div><div>in the algorithmic reformulation of the viscous term, which enables the simulation of outflow problems on severely-truncated domains at moderate to high Reynolds numbers. In contrast, the standard consistent splitting scheme is observed to exhibit a numerical instability even at relatively low Reynolds numbers, and this numerical instability is in addition to the backflow instability commonly known to be associated with strong vortices or backflows at the outflow boundary. Extensive numerical experiments are presented for a range of Reynolds numbers to demonstrate the effectiveness and accuracy of the proposed algorithm for this class of flows.</div></div><div><br></div><div><div>In the second part of this thesis, we present a numerical algorithm within the phase-field framework for simulating three-dimensional (3D) incompressible two-phase flows in flow domains with one homogeneous direction. In this numerical method, we represent the flow variables using Fourier spectral expansions along the homogeneous direction and C0 spectral element expansions in the other directions. This is followed by using fast Fourier transforms so that the solution to the 3D problem is obtained by solving a set of decoupled equations about the Fourier modes for each flow variable. The computations for solving these decoupled equations are performed in parallel to effciently simulate the 3D two-phase</div><div>ows. Extensive numerical experiments are presented to demonstrate the performance and the capabilities of the scheme in simulating this class of flows.</div></div>

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In this dissertation, we study the evolution and properties of the relic (or cosmic) neutrino distribution from neutrino freeze-out at T=O(1) MeV through the free-streaming era up to today, focusing on the deviation of the neutrino spectrum from equilibrium and in particular we demonstrate the presence of chemical non-equilibrium that continues to the present day. The work naturally separates into two parts. The first focuses on aspects of the relic neutrinos that can be explored using conservation laws. The second part studies the neutrino distribution using the full general relativistic Boltzmann equation. Part one begins with an overview of the history of the Universe, from just prior to neutrino freeze-out up through the present day, placing the history of cosmic neutrino evolution in its proper context. Motivated by the Planck CMB measurements of the effective number of neutrinos, we derive those properties of neutrino freeze-out that depend only on conservation laws and are independent of the details of the scattering processes. Part one ends with a characterization of the present day neutrino spectrum as seen from Earth. The second part of this dissertation focuses on the properties of cosmic neutrinos that depend on the details of the neutrino reactions, as is necessary for modeling the non-thermal distortions from equilibrium and computing freeze-out temperatures. We first develop some geometry background concerning volume forms and integration on submanifolds that is helpful in computations. We then detail a new spectral method for solving the Boltzmann equation, based on a dynamical basis of orthogonal polynomials. Next, we detail an improved procedure for analytically simplifying the corresponding scattering integrals for subsequent numerical computation. Using this, along with the spectral method mentioned above, we solve the Boltzmann equation through the neutrino freeze-out period. Finally, we conclude by using our novel solution methods to perform parametric studies of the dependence of the neutrino freeze-out standard model parameters. This exploration is performed with the aim of recognizing mechanisms in the neutrino freeze-out process that are capable of leading to the measured value of the effective number of neutrinos.

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A time spectral and coupled adjoint based sensitivity analysis of rotor blade is carried out in this study. The time spectral method is an efficient technique to solve unsteady periodic problems by transforming unsteady equation of motion to a steady state one. Due to the availability of the governing equations in the steady form, the steady form of the adjoint equations can be applied for the sensitivity analysis of the coupled fluid-structure system. An expensive computational time and memory requirement for the unsteady adjoint sensitivity analysis is thus avoided. A coupled analysis of fluid, structural, and flight dynamics is carried out through a CFD/CSD/CA coupling procedure that combines FSI analysis with enforced trim condition. Coupled sensitivity analysis results and their validations are presented and compared with aerodynamics only sensitivity analysis results. The fluid-structure coupled adjoint based sensitivity analysis will be applied to the shape optimization of a rotor blade in the future work. Minimization of required power is the objective of the optimization problem with constraints on thrust and drag of the rotor. The bump functions are considered as the design variables. Rotor blade shape changes are obtained by using the bump function on the surface of the airfoil sections along the span. / Doctor of Philosophy / The work in this dissertation is motivated by the reducing the computational cost at the early design stage with guaranteed accuracy. In the research, the author proposes that the goal can be achieve through coupled adjoint based sensitivity analysis using a fluid structure interaction in time spectral form. Adjoint based sensitivity analysis is very efficient for solving design problems with a large number of design variables. The time spectral approach is used to overcome inefficient calculation of rotor flows by expressing flow and structural state variables as Fourier series with small number of harmonics. The accuracy and the efficiency of flow solver are examined by simulating UH-60A forward flight condition. A significant reduction in the computational cost is achieved by its Fourier series form of the periodic time response and the assumption of periodic steady state. A good agreement between time accurate and time spectral analysis is noted for the high speed forward flight condition of UH-60A configuration. Prediction from both methods also agree quite well with the experimental data. The adjoint based sensitivity analysis results are compared with the finite difference sensitivity analysis results. Even with presence of small discrepancies, these two results show a good agreement to each other. Coupled sensitivity analysis includes not only the effect of fluid state changes but also the contribution of structural deformation.

Yes / The development of the patchwise Partial Di erential Equation (PDE) framework a few years a- go has paved the way for the PDE method to be used in mesh signal processing. In this paper we, for the rst time, extend the use of the PDE method to progressive mesh compression and mesh denoising. We, meanwhile, upgrade the existing patchwise PDE method in patch merging, mesh partitioning, and boundary extraction to accommodate mesh signal processing. In our new method an arbitrary mesh model is partitioned into patches, each of which can be represented by a small set of coe cients of its PDE spectral solution. Since low- frequency components contribute more to the recon- structed mesh than high-frequency ones, we can achieve progressive mesh compression and mesh denoising by manipulating the frequency terms of the PDE solution. Experimental results demonstrate the feasibility of our method in both progressive mesh compression and mesh denoising.

Métodos de resolução numérica de equações diferenciais parciais que utilizam ondaletas como base vêm sendo desenvolvidos nas últimas décadas, mas existe uma carência de estudos mais profundos das características computacionais dos mesmos. Neste estudo analisou-se detalhadamente um método espectral de Galerkin com base de ondaletas harmônicas. Revisou-se a teoria matemática referente às ondaletas harmônicas, que mostrou ter grande similaridade com a teoria referente à base trigonométrica de Fourier. Diversos testes numéricos foram realizados. Ao analisarmos a resolução da equação do transporte linear, e também de transporte não linear (equação de Burgers), obtivemos boas aproximações da solução esperada. O custo computacional obtido foi similar ao método com base de Fourier, mas com ondaletas harmônicas foi possível usar a localidade das ondaletas para detectar características de localidade do sinal. Analisamos ainda uma abordagem pseudo-espectral para os casos não lineares, que resultaram em um expressivo aumento de eficiência. Tendo em vista o uso das propriedades de localidade das ondaletas, usamos o método de Galerkin com base de ondaletas harmônicas para resolver um sistema de equações referente a um modelo de propagação de frentes de precipitação. O método mostrou boas aproximações das soluções esperadas, custo computacional ótimo e ainda a possibilidade de se obter espectralmente informações sobre a localização da frente de precipitação. / Numerical methods to solve partial differential equations based on wavelets have been developed in the last two decades, but there is a lack of studies on their computational characteristics. In this study a Galerkin spectral method using harmonic wavelets base has been thoroughly analyzed. We performed a review on the mathematics of harmonic wavelets, that showed a great similarity with Fourier basis. Several numerical experiments were made. Analyzing the use of the Galerkin method, with harmonic wavelets, on linear and non linear transport equations, we achieved good approximations in respect to the expected solution. The computational cost resulted to be similar to the same method with Fourier basis. On the other hand, employing harmonic wavelets we were able to obtain local information of the solution by simple inspection of the spectral coeffcients. We also analyzed a pseudo-spectral method based on harmonic wavelets for the non linear equations, resulting in a great improvement in efficiency. Looking towards using the locality propriety of harmonic wavelets, we tested the Galerkin method on a precipitation front propagation model. The method resulted in good approximations to the expected solution, optimal computational cost and the possibility of obtaining information on the locality of the precipitation fronts spectrally.