Doctor of Philosophy

dissertationPlatelet aggregation, an important part of the development of blood clots, is a complex process involving both mechanical interaction between platelets and blood, and chemical transport on and o the surfaces of those platelets. Radial Basis Function (RBF) interpolation is a meshfree method for the interpolation of multidimensional scattered data, and therefore well-suited for the development of meshfree numerical methods. This dissertation explores the use of RBF interpolation for the simulation of both the chemistry and mechanics of platelet aggregation. We rst develop a parametric RBF representation for closed platelet surfaces represented by scattered nodes in both two and three dimensions. We compare this new RBF model to Fourier models in terms of computational cost and errors in shape representation. We then augment the Immersed Boundary (IB) method, a method for uid-structure interaction, with our RBF geometric model. We apply the resultant method to a simulation of platelet aggregation, and present comparisons against the traditional IB method. We next consider a two-dimensional problem where platelets are suspended in a stationary fluid, with chemical diusion in the fluid and chemical reaction-diusion on platelet surfaces. To tackle the latter, we propose a new method based on RBF-generated nite dierences (RBF-FD) for solving partial dierential equations (PDEs) on surfaces embedded in 2D domains. To robustly tackle the former, we remove a limitation of the Augmented Forcing method (AFM), a method for solving PDEs on domains containing curved objects, using RBF-based symmetric Hermite interpolation. Next, we extend our RBF-FD method to the numerical solution of PDEs on surfaces embedded in 3D domains, proposing a new method of stabilizing RBF-FD discretizations on surfaces. We perform convergence studies and present applications motivated by biology. We conclude with a summary of the thesis research and present an overview of future research directions, including spectrally-accurate projection methods, an extension of the Regularized Stokeslet method, RBF-FD for variable-coecient diusion, and boundary conditions for RBF-FD

Spectral methods applied to axisymmetric thin film flows

We employ numerical techniques to investigate the influence of slot injec- tion/suction on the thin film axisymmetric flow of a Newtonian fluid subject to centrifugal and Coriolis forces, gravity and rotation. Surface tension ef- fects are neglected. We obtain a nonlinear diffusion equation when modeling the spreading of the free surface of a thin film under gravity with blowing or suction at the base. When we model the spreading of the free surface of a thin film under both gravity and rotation with blowing or suction we obtain a nonlinear second order partial differential equation. A first order quasi-linear partial differential equation is obtained when modeling the thickness of the thin film under the effects of rotation only with blowing or suction at the base. We compare and contrast spectral methods with MATLAB built-in functions as well as finite differences. We also examine the effect that the slot has on the wave breaking process

Symmetry-based stability theory in fluid mechanics

The present work deals with the stability theory of fluid flows. The central subject is the question under which circumstances a flow becomes unstable. Instabilities are a frequent trigger of laminar-turbulent transitions. Stability theory helps to explain the emergence of structures, e.g. wave-like perturbation patterns. In this context, the use of Lie symmetries allows the classification of existing and the construction of new solutions within the framework of linear stability theory. In addition, a new nonlinear eigenvalue problem (NEVP) is presented, whose derivation is completely based on Lie symmetries. In classical linear stability theory, a normal ansatz is used for perturbations. Another ansatz that has been shown in early work is the Kelvin mode ansatz. In the work of Nold and Oberlack (2013) and Nold et al. (2015) it was shown that these ansätze can be traced back to the Lie symmetries of the linearized perturbation equations. Interestingly, knowledge of the symmetries also allows for the construction of new ansatz functions that go beyond the known ansätze. For a plane rotational shear flow, in addition to the normal mode ansatz, an algebraic mode ansatz with algebraic behavior in time t^s (eigenvalue s) can be constructed. The flow is stable according to Rayleigh's inflection point criterion, which is also confirmed by the algebraic mode ansatz. Furthermore, exact solutions of the eigenfunctions can be found and new stable modes can be determined by asymptotic methods. Thereby, spiral-like structures of the vorticity can be recognized, which propagate in the region with time. Another key result of this work is the formulation and solution of an NEVP based on the Lie symmetries of the Euler equation. It can is shown that an NEVP can be formulated for a class of flows with a constant velocity gradient. These include, for example, linear shear flows, strained flows, and rotating flows. The NEVP for linear shear flows shows a relation to experimental data from turbulent shear flows. It can be theoretically shown that the turbulent kinetic energy scales exponentially with the eigenvalue of the NEVP. The eigenvalue is determined numerically using a parallel spectral solver. Initially, nonlinear terms are neglected. The determined eigenvalues are in the range of known literature values for turbulent shear flows. Furthermore, the NEVPs for plane flows with pure rotation and pure strain are solved. It is shown that the flow is invariant to rotation, while oscillatory eigenfunctions are found in the case of strain. In addition, an algorithm to solve the NEVP including the nonlinear terms is presented. The results allow an exciting insight into a new stability theory and form the basis for further investigation and understanding of the full nonlinear dynamics of the fluid flows based on the NEVP

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period April l, 1988 through September 30, 1988

Computational and numerical analysis of differential equations using spectral based collocation method.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally efficient spectral collocation-based methods, both modified and new, and apply them to solve differential equations. Spectral collocation-based methods are the most commonly used methods for approximating smooth solutions of differential equations defined over simple geometries. Procedurally, these methods entail transforming the gov erning differential equation(s) into a system of linear algebraic equations that can be solved directly. Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported in the literature, researchers often transform their models to reduce the number of variables or narrow them down to problems with fewer dimensions. Such a process is accomplished by making a series of assumptions that limit the scope of the study. To address this deficiency, the present study explores the development of numerical algorithms for solving ordinary and partial differential equations defined over simple geometries. The solutions of the differential equations considered are approximated using interpolating polynomials that satisfy the given differential equation at se lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the computational domain is particularly emphasized as it plays a key role in determining the number of grid points that are used; a feature that dictates the accuracy and the computational expense of the spectral method. To solve differential equations defined on large computational domains much effort is devoted to the development and application of new multidomain approaches, based on decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con firms the superiority of these multiple domain techniques in terms of accuracy and computational efficiency over the single domain approach when applied to problems defined over large domains. The structure of the thesis indicates a smooth sequence of constructing spectral collocation method algorithms for problems across different dimensions. The process of switching between dimensions is explained by presenting the work in chronological order from a simple one-dimensional problem to more complex higher-dimensional problems. The preliminary chapter explores solutions of or dinary differential equations. Subsequent chapters then build on solutions to partial differential i equations in order of increasing computational complexity. The transition between intermediate dimensions is demonstrated and reinforced while highlighting the computational complexities in volved. Discussions of the numerical methods terminate with development and application of a new method namely; the trivariate spectral collocation method for solving two-dimensional initial boundary value problems. Finally, the new error bound theorems on polynomial interpolation are presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical algorithms. The numerical results of the study confirm that incorporating domain decomposition techniques in spectral collocation methods work effectively for all dimensions, as we report highly accurate results obtained in a computationally efficient manner for problems defined on large do mains. The findings of this study thus lay a solid foundation to overcome major challenges that numerical analysts might encounter

A Wiener Chaos Based Approach to Stability Analysis of Stochastic Shear Flows

As the aviation industry expands, consuming oil reserves, generating carbon dioxide gas and adding to environmental concerns, there is an increasing need for drag reduction technology. The ability to maintain a laminar flow promises significant reductions in drag, with economic and environmental benefits. Whilst development of flow control technology has gained interest, few studies investigate the impacts that uncertainty, in flow properties, can have on flow stability. Inclusion of uncertainty, inherent in all physical systems, facilitates a more realistic analysis, and is therefore central to this research. To this end, we study the stability of stochastic shear flows, and adopt a framework based upon the Wiener Chaos expansion for efficient numerical computations. We explore the stability of stochastic Poiseuille, Couette and Blasius boundary layer type base flows, presenting stochastic results for both the modal and non modal problem, contrasting with the deterministic case and identifying the responsible flow characteristics. From a numerical perspective we show that the Wiener Chaos expansion offers a highly efficient framework for the study of relatively low dimensional stochastic flow problems, whilst Monte Carlo methods remain superior in higher dimensions. Further, we demonstrate that a Gaussian auto-covariance provides a suitable model for the stochasticity present in typical wind tunnel tests, at least in the case of a Blasius boundary layer. From a physical perspective we demonstrate that it is neither the number of inflection points in a defect, nor the input variance attributed to a defect, that influences the variance in stability characteristics for Poiseuille flow, but the shape/symmetry of the defect. Conversely, we show the symmetry of defects to be less important in the case of the Blasius boundary layer, where we find that defects which increase curvature in the vicinity of the critical point generally reduce stability. In addition, we show that defects which enhance gradients in the outer regions of a boundary layer can excite centre modes with the potential to significantly impact neutral curves. Such effects can lead to the development of an additional lobe at lower wave-numbers, can be related to jet flows, and can significantly reduce the critical Reynolds number.EPSR

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions