This study is concerned with defining the mathematical framework in which the finite element procedure can most advantageously be considered. It is established that the finite element method generates an approximate solution to a given equation which is defined in terms of assumed co-ordinate functions and unknown parameters. The advantages of determining the parameters by Galerkin's method are discussed and the convergence characteristics of this method are reviewed using functional analysis principles. Comparisons are made between the Galerkin and Rayleigh-Ritz procedures and the connection between virtual work and Galerkin's method is illustrated. The convergence results presented for the Galerkin procedure are used to provide sufficient conditions that ensure the convergence of a finite element solution of a general system of time independent linear differential equations. Application of the principles developed is illustrated with a convergence proof for a finite element solution of a non-symmetric eigenvalue problem and by developing a computer program for the finite element analysis of the two-dimensional steady state flow of an incompressible viscous fluid. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

"May 2003." Bibliography: leaves 125-130. This study considers use of an interior source method for modelling supercavitation in 2-dimensional (planar) and 3-dimensional axisymmetric flows. Aspects considered include the determination of where the fluid separates from the body, the shapes of the cavities formed, the pressure distribution on the body and the pressure on the cavity surface, and the resulting drag and lift forces on the body.

"May 2003." / Bibliography: leaves 125-130. / vii, 130 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This study considers use of an interior source method for modelling supercavitation in 2-dimensional (planar) and 3-dimensional axisymmetric flows. Aspects considered include the determination of where the fluid separates from the body, the shapes of the cavities formed, the pressure distribution on the body and the pressure on the cavity surface, and the resulting drag and lift forces on the body. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 2003

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Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 2003. / "May 2003." Bibliography: leaves 125-130.

Thesis (M.S.)--Florida State University, 2004. / Advisor: Dr. Robert van Engelen, Florida State University, College of Arts and Sciences, Dept. of Computer Science. Title and description from dissertation home page (viewed Sept. 30, 2004). Includes bibliographical references.

<p> This dissertation presents studies on several statistical-mechanical problems, many of which involve exotic many-particle systems. In Chapter 2, we present an algorithm to generate Random Sequential Addition (RSA) packings of hard hyperspheres at the infinite-time saturation limit, and investigate this limit with unprecedented precision. In Chapter 3, we study the problem of devising smooth, short-ranged isotropic pair potentials such that their ground state is an unusual targeted crystalline structure. We present a new algorithm to do so, and demonstrate its capability by targeting several singular structures that were not known to be achievable as ground states with isotropic interactions. </p><p> A substantial portion of this dissertation examines exotic many-particle systems with so-called &ldquo;collective-coordinate&rdquo; interactions. They include &ldquo;stealthy&rdquo; potentials, which are isotropic pair potentials with disordered and infinitely degenerate ground states as well as &ldquo;perfect-glass&rdquo; interactions, which have up to four-body contributions, and possess disordered and <i>unique</i> ground states, up to trivial symmetry operations. Chapters 4-7 study the classical ground states of &ldquo;stealthy&rdquo; potentials. We establish a numerical means to sample these infinitely-degenerate ground states in Chapter 4 and study exotic &ldquo;stacked-slider&rdquo; phases that arise at suitable low densities in Chapter 5. In Chapters 6 and 7, we investigate several geometrical and physical properties of stealthy systems. Chapter 8 studies lattice-gas systems with the same stealthy potentials. Chapter 9 is concerned with the introduction and study of the perfect-glass paradigm. Chapter 10 demonstrates that perfect-glass interactions indeed possess disordered and unique classical ground states&mdash;a highly counterintuitive proposition. </p><p> In Chapter 11, we use statistical-mechanical methods to characterize the spatial distribution of the prime numbers. We show that the primes are much more ordered than anyone previously thought via the structure factor. Indeed, they are characterized by infinitely many Bragg peaks in any non-zero interval of wave vectors, yet unlike quasicrystals, the ratio between the heights or locations of any two Bragg peaks is always rational. We analytically explain the locations and heights of all such peaks.</p><p>

Meshless methods have been developed to avoid the numerical burden imposed by meshing in the Finite Element Method. Such methods are especially attrac- tive in problems that require repeated updates to the mesh, such as problems with discontinuities or large geometrical deformations. Although meshing is not required for solving problems with meshless methods, the use of meshless methods gives rise to different challenges. One of the main challenges associated with meshless methods is imposition of essential boundary conditions. If exact interpolants are used as shape functions in a meshless method, imposing essen- tial boundary conditions can be done in the same way as the Finite Element Method. Another attractive feature of meshless methods is that their use involves compu- tations that are largely independent from one another. This makes them suitable for implementation to run on highly parallel computing systems. Highly par- allel computing has become widely available with the introduction of software development tools that enable developing general-purpose programs that run on Graphics Processing Units. In the current work, the Moving Regularized Interpolation method has been de- veloped, which is a novel method of constructing meshless shape functions that achieve exact interpolation. The method is demonstrated in data interpolation and in partial differential equations. In addition, an implementation of the Element-Free Galerkin method has been written to run on a Graphics Processing Unit. The implementation is described and its performance is compared to that of a similar implementation that does not make use of the Graphics Processing Unit.

This dissertation describes several new connections between a polynomial's coefficients and its zeros. The most important of these, the Finite Nyquist Theorem (FNT), states that one can prove a polynomial has all its roots in an open, bounded or unbounded, convex region ${\cal D}$ of the complex plane given only the polynomial's degree and its phase at finitely many points along ${\cal D}$'s boundary. An immediate and very useful corollary to FNT is the Finite Inclusions Theorem (FIT), with which one can prove a class of polynomials has all its zeros in ${\cal D}$ given only the polynomials' degree and approximate knowledge of the class's value set at finitely many points along ${\cal D}$'s boundary. From FIT a procedure we call FIT synthesis is developed for synthesizing robustly ${\cal D}$-stabilizing controllers for parametrically uncertain systems (note, all the systems considered here are assumed to be linear time-invariant and finite dimensional). This procedure uses FIT to directly search for robust controllers by way of solving a sequence of systems of linear inequalities. Two numerical examples of this procedure are given to show its effectiveness. In these examples the systems of inequalities are solved via the projection method which is an elegantly simple technique for solving (finite or infinite) systems of convex inequalities in an arbitrary Hilbert space. Since this method has yet to appear in standard textbooks on numerical methods, it is covered here in detail with the aim of better popularizing the method and, where possible, extending the known theory concerning its convergence.