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This thesis is concerned with the fast iterative solution of linear systems of equations of saddle point form. Saddle point problems are a ubiquitous class of matrices that arise in a host of computational science and engineering applications. The focus here is on improving the convergence of iterative methods for these problems by preconditioning. Preconditioning is a way to transform a given linear system into a different problem for which iterative methods converge faster. Saddle point matrices have a very specific block structure and many preconditioning strategies for these problems exploit this structure. The preconditioners considered in this thesis are constraint preconditioners. This class of preconditioner mimics the structure of the original saddle point problem. In this thesis, we prove norm- and field-of-values-equivalence for constraint preconditioners associated to saddle point matrices with a particular structure. As a result of these equivalences, the number of iterations needed for convergence of a constraint preconditioned minimal residual Krylov subspace method is bounded, independent of the size of the matrix. In particular, for saddle point systems that arise from the finite element discretization of partial differential equations (p.d.e.s), the number of iterations it takes for GMRES to converge for theses constraint preconditioned systems is bounded (asymptotically), independent of the size of the mesh width. Moreover, we extend these results when appropriate inexact versions of the constraint preconditioner are used. We illustrate this theory by presenting numerical experiments on saddle point matrices that arise from the finite element solution of coupled Stokes-Darcy flow. This is a system of p.d.e.s that models the coupling of a free flow to a porous media flow by conditions across the interface of the two flow regions. We present experiments in both two and three dimensions, using different types of elements (triangular, quadrilateral), different finite element schemes (continuous, discontinuous Galerkin methods), and different geometries. In all cases, the effectiveness of the constraint preconditioner is demonstrated. / Mathematics

The Euler and Navier-Stokes-Fourier equations will be directly expressed as distribution evolution equations, where a new and proper continuum prescription will be derived. These equations of motion will be numerically solved with the development of a new and unique finite element formulation. Out of this framework, the 7D phasetime element has been born. To provide optimal stability, a new quantization procedure is established based on the principles of quantum theory. The entirety of this framework has been coined the "quantized velocity finite element method" (QVFEM). The work performed herein lays the foundational development of what is hoped to become a new paradigm shift in computational fluid dynamics. / Doctor of Philosophy / To model any of the four fundamental states of matter, for practical engineering applications, we must first recognize the complexity of such states. In consequence, a new and novel approach is presented on how to numerically simulate the dynamics of a gas using both the Euler and Navier-Stokes-Fourier equations of continuum mechanics and thermodynamics. In contrast to direct numerical simulation, a statistical mechanical prescription will be given where the equations of motion will be quantized using methods taken from the study of quantum mechanics. This newly developed discretization of the phase space and time, or phasetime, provides optimal stability for compressible flow simulations. From the newly proposed framework, the 7D phasetime element has been born.

Diese Dissertation verallgemeinert die nichtkonformen Finite-Elemente-Methoden (FEMn) nach Morley und Crouzeix und Raviart durch neue gemischte Formulierungen für das Poisson-Problem, die Stokes-Gleichungen, die Navier-Lamé-Gleichungen der linearen Elastizität und m-Laplace-Gleichungen der Form $(-1)^m\Delta^m u=f$ für beliebiges m=1,2,3,... Diese Formulierungen beruhen auf Helmholtz-Zerlegungen. Die neuen Formulierungen gestatten die Verwendung von Ansatzräumen beliebigen Polynomgrades und ihre Diskretisierungen stimmen für den niedrigsten Polynomgrad mit den genannten nicht-konformen FEMn überein. Auch für höhere Polynomgrade ergeben sich robuste Diskretisierungen für fast-inkompressible Materialien und Approximationen für die Lösungen der Stokes-Gleichungen, die punktweise die Masse erhalten. Dieser Ansatz erlaubt außerdem eine Verallgemeinerung der nichtkonformen FEMn von der Poisson- und der biharmonischen Gleichung auf m-Laplace-Gleichungen für beliebiges m>2. Ermöglicht wird dies durch eine neue Helmholtz-Zerlegung für tensorwertige Funktionen. Die neuen Diskretisierungen lassen sich nicht nur für beliebiges m einheitlich implementieren, sondern sie erlauben auch Ansatzräume niedrigster Ordnung, z.B. stückweise affine Polynome für beliebiges m. Hat eine Lösung der betrachteten Probleme Singularitäten, so beeinträchtigt dies in der Regel die Konvergenz so stark, dass höhere Polynomgrade in den Ansatzräumen auf uniformen Gittern dieselbe Konvergenzrate zeigen wie niedrigere Polynomgrade. Deshalb sind gerade für höhere Polynomgrade in den Ansatzräumen adaptiv generierte Gitter unabdingbar. Neben der A-priori- und der A-posteriori-Analysis werden in dieser Dissertation optimale Konvergenzraten für adaptive Algorithmen für die neuen Diskretisierungen des Poisson-Problems, der Stokes-Gleichungen und der m-Laplace-Gleichung bewiesen. Diese werden auch in den numerischen Beispielen dieser Dissertation empirisch nachgewiesen. / This thesis generalizes the non-conforming finite element methods (FEMs) of Morley and Crouzeix and Raviart by novel mixed formulations for the Poisson problem, the Stokes equations, the Navier-Lamé equations of linear elasticity, and mth-Laplace equations of the form $(-1)^m\Delta^m u=f$ for arbitrary m=1,2,3,... These formulations are based on Helmholtz decompositions. The new formulations allow for ansatz spaces of arbitrary polynomial degree and its discretizations coincide with the mentioned non-conforming FEMs for the lowest polynomial degree. Also for higher polynomial degrees, this results in robust discretizations for almost incompressible materials and approximations of the solution of the Stokes equations with pointwise mass conservation. Furthermore this approach also allows for a generalization of the non-conforming FEMs for the Poisson problem and the biharmonic equation to mth-Laplace equations for arbitrary m>2. A new Helmholtz decomposition for tensor-valued functions enables this. The new discretizations allow not only for a uniform implementation for arbitrary m, but they also allow for lowest-order ansatz spaces, e.g., piecewise affine polynomials for arbitrary m. The presence of singularities usually affects the convergence such that higher polynomial degrees in the ansatz spaces show the same convergence rate on uniform meshes as lower polynomial degrees. Therefore adaptive mesh-generation is indispensable especially for ansatz spaces of higher polynomial degree. Besides the a priori and a posteriori analysis, this thesis proves optimal convergence rates for adaptive algorithms for the new discretizations of the Poisson problem, the Stokes equations, and mth-Laplace equations. This is also demonstrated in the numerical experiments of this thesis.

Thesis (MScEng (Mechanical and Mechatronic Engineering))--University of Stellenbosch, 2011. / Please refer to full text to view abstract. / Imported from http://etd.sun.ac.za. / np2011

The work presented in this thesis is concerned with the propagation of acoustic waves through phononic crystal systems and their ability to attenuate sound in the low frequency regime. The plane wave expansion method and finite element method are utilised to investigate the properties of conventional phononic crystal systems. The acoustic band structure and transmission measurements of such systems are computed and verified experimentally. Good agreement between band gap locations for the investigative methods detailed is found. The well known link between the frequency range a phononic crystal can attenuate sound over and its lattice parameter is confirmed. This leads to a reduction in its usefulness as a viable noise barrier technology, due to the necessary increase in overall crystal size. To overcome this restriction the concept of an acoustically resonant phononic crystal system is proposed, which utilises acoustic resonances, similar to Helmholtz resonance, to form additional band gaps that are decoupled from the lattice periodicity of the phononic crystal system. An acoustically resonant phononic crystal system is constructed and experimental transmission measurements carried out to verify the existence of separate attenuation mechanisms. Experimental attenuation levels achieved by Bragg formation and resonance reach 25dB. The two separate attenuation mechanisms present in the acoustically resonant phononic crystal, increase the efficiency of its performance in the low frequency regime, whilst maintaining a reduced crystal size for viable noise barrier technology. Methods to optimise acoustically resonant phononic crystal systems and to increase their performance in the lower frequency regime are discussed, namely by introducing the Matryoshka acoustically resonant phononic crystal system, where each scattering unit is composed of multiple concentric C-shape inclusions.

In this thesis, three parts, each with several chapters, are respectively devoted to hydrostatic, viscous and inertial fluids theories and applications. In the hydrostatics part, the classical Maclaurin spheroids theory is generalized, for the first time, to a more realistic multi-layer model, which enables the studies of some gravity problems and direct numerical simulations of flows in fast rotating spheroidal cavities. As an application of the figure theory, the zonal flow in the deep atmosphere of Jupiter is investigated for a better understanding of the Jovian gravity field. High viscosity flows, for example Stokes flows, occur in a lot of processes involving low-speed motions in fluids. Microorganism swimming is such typical a case. A fully three dimensional analytic solution of incompressible Stokes equation is derived in the exterior domain of an arbitrarily translating and rotating prolate spheroid, which models a large family of microorganisms such as cocci bacteria. The solution is then applied to the magnetotactic bacteria swimming problem and good consistency has been found between theoretical predictions and laboratory observations of the moving patterns of such bacteria under magnetic fields. In the analysis of dynamics of planetary fluid systems, which are featured by fast rotation and very small viscosity effects, three dimensional fully nonlinear numerical simulations of Navier-Stokes equations play important roles. A precession driven flow in a rotating channel is studied by the combination of asymptotic analyses and fully numerical simulations. Various results of laminar and turbulent flows are thereby presented. Computational fluid dynamics requires massive computing capability. To make full use of the power of modern high performance computing facilities, a C++ finite-element analysis code is under development based on PETSc platform. The code and data structures will be elaborated, along with the presentations of some preliminary results.

Master of Science / Department of Architectural Engineering and Construction Science / Kimberly Waggle Kramer / This report examines the effects of openings located at the base of reinforced concrete slender wall panels (tilt-up panels) designed in accordance with the American Concrete Institute (ACI) Committee 318-11 Building Code Requirements for Structural Concrete Section 14.8 Alternative Design of Slender Walls. The parametric study calculates the reinforcement (longitudinal) required for specific panels in accordance with ACI 318-11 Section 14.8 and compares the designs to a finite element analysis conducted with SAP 2000 version 14 to determine the appropriateness of the assumptions made in Section 14.8. Furthermore, this report compares the design of a tilt-up panel designed by Section 14.8 Alternative Design of Slender Walls and designed by Section 10.10 Slenderness Effects in Compression Members.

Submitted by Maria Cristina (library@lncc.br) on 2017-08-10T15:25:59Z No. of bitstreams: 1 Tese_Geraldo_Belmonte_2016.pdf: 2695418 bytes, checksum: 2a9690a5f6ddd075770a578b20f23383 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-08-10T15:26:12Z (GMT) No. of bitstreams: 1 Tese_Geraldo_Belmonte_2016.pdf: 2695418 bytes, checksum: 2a9690a5f6ddd075770a578b20f23383 (MD5) / Made available in DSpace on 2017-08-10T15:26:21Z (GMT). No. of bitstreams: 1 Tese_Geraldo_Belmonte_2016.pdf: 2695418 bytes, checksum: 2a9690a5f6ddd075770a578b20f23383 (MD5) Previous issue date: 2016-10-03 / A new finite element method is proposed for mixed variational formulation by use of the hybridization technique and of the adding of several stabilization mechanisms to the classical Galerkin methods applied to the elasticity problems. The method is designed by hibridization technique of the classical dual mixed formulation applied to the element level, adding various least-squares residual terms of the locally governing equations and of the interelement continuity conditions of the fields. The residual terms are added without violating the consistency condition of the methods and include mesh-parameter dependent coefficients. The method is designed to enhance stability in a better norm, adding features such as: flexibility in the choice of the approximation spaces, including equal-order interpolation, by adding residual stabilization terms; improvement of the convergence rate of the dual variables, using mixed formulation; very efficient solver with global system assembled with Lagrange multiplier (hybridized variable) only via static condensation; robustness to solver problems with non smooth fields and internal limits, including discontinuous fields (e.g. cracks), typical features of Galerkin discontinuous; possibility of easily handling local enrichment with polynomial (p-adaptivity) and non polynomial functions in different elements; and under certain conditions we have local conservation. The stability of the methods is proved and various numerical experiments are provided to show the features listed above, including convergence rates, stability and accuracy. The method is applied the several problems of plane stress, plane strain and axisymmetric solid, including the cases of compressible elasticity, Girkmann problem and linear elastic fracture. / Um método de elementos finitos baseado na formulação mista hibridizada e estabilizada aplicado a problemas de elasticidade é proposto. O método é construído pela hibridização, no nível do elemento, da formulação mista dual clássica de Galerkin com a adição de vários termos de resíduos de mínimos quadrados das equações que governam o problema localmente e resíduos de mínimos quadrados das equações de continuidade interelemento. Os termos de resíduos são adicionados de forma a não violar a consistência do método e incluem coeficientes dependentes do parâmetro de malha h. O método é projetado para melhorar a estabilidade em normas mais fortes, adicionando características, tais como: flexibilidade nas escolhas dos espaços de aproximação, incluindo igual ordem, através dos termos de estabilização; melhora da taxa de convergência das variáveis duais, usando métodos mistos; uma estratégia de solver mais eficiente (menor custo computacional) com a montagem do sistema global apenas no multiplicador de Lagrange (variável hibridizada) via condensação estática; robustez do método na solução de problemas com campos não suaves e problemas limites, incluindo campos descontínuos, característica típica de métodos de Galerkin descontínuo; facilidade para implementar processos de enriquecimento local com funções polinomiais (p-adaptatividade) e não polinomiais em diferentes elementos; e sob determinadas condições, obtenção de conservação local. A estabilidade dos métodos é provada e experimentos numéricos são realizados para demonstrar as características elencadas anteriormente, incluindo taxas de convergência, estabilidade e exatidão. Os métodos são aplicados a diversas classes de problemas em estado plano de tensão e deformação e sólido axissimétrico, incluindo elasticidade compressível, problema de Girkmann e fratura elástica linear.

O emprego das estruturas de membrana é cada dia mais frequente em edificações de relevância civil e arquitetônica, em especial para a cobertura de grandes vãos. Sua aplicabilidade, contudo, vai além da construção civil, sendo igualmente importante nas indústrias das engenharias mecânica, naval, oceânica, aeroespacial e biomédica: aeronaves, satélites, paraquedas, airbags, velas de embarcações, moinhos de vento e até aplicações biomecânicas com tecidos humanos ou artificiais utilizam a tecnologia das estruturas de membrana. O comportamento mecânico de grande parte das membranas estruturais pode ser idealizado como uma casca isótropa de pequena espessura reforçada por uma membrana ortótropa. O objetivo desta pesquisa de mestrado é dar continuidade aos estudos referentes às teorias de casca geometricamente exatas desenvolvidas por [1], [2], [3] e [4], e sua generalização para o âmbito das membranas iniciada em [5]. Pretende-se contribuir, principalmente, para o desenvolvimento de equações constitutivas ortótropas consistentes para grandes deformações, apresentar uma metodologia para calibração dos parâmetros materiais destas equações constitutivas e para a análise de estabilidade com vistas ao estudo do fenômeno do enrugamento. / The use of membrane structures is becoming increasingly common in buildings of architectural and civil engineering importance, especially to cover large spans. Its applicability, however, goes beyond the construction industry being equally important in the industries of mechanical, naval, ocean, aerospace and biomedical engineering: aircraft, satellites, parachutes, airbags, sails of boats, windmills and even biomechanical applications with human and artificial tissues use the technology of membrane structures. The mechanical behavior of most structural membranes can be idealized as an isotropic thin shell reinforced by an orthotropic membrane. The objective of this work is to continue the studies on the geometrically exact shell theories developed by [1], [2] , [3] and [4] and its generalization to the scope of the membranes initially studied by [5]. It aims intended to contribute mainly to the development of consistent orthotropic constitutive equations for large deformations, to present a methodology for the calibration of the material parameters in those constitutive relations and for the stability analysis in order to study the phenomenon of wrinkling.