Growing interest in the simulation of first order transient systems, typical of those encountered in transient heat conduction, flow transport, and fluid dynamics, has prompted the development of a variety of time integration methods for solving these systems numerically. The primary contribution of this thesis is the design and development of a new time integration/discretization framework, under the class of single step single solve algorithms which are the most popular, for use in such first order transient systems with computationally attractive features. These include second order accuracy, unconditional stability, zero-order overshoot, and controllable numerical dissipation with a new selective control feature which overcomes the restrictions in the existing and current state-of-the-art methods. Throughout the thesis, we demonstrate the capability and advantage of the newly developed framework, termed GS4-1, in comparison to existing methods using various types of numerical examples (both linear and nonlinear). The numerical results consistently demonstrate the roles played by the new feature in improving the numerical solutions of both the primary variable and its time derivative which is important to correctly capture the dynamics of the problems, in contrast to the existing methods without such a feature. Additionally, a breakthrough contribution presented in this thesis is the development of an isochronous integration framework (iIntegrator), stemming from the novel relations between the newly developed GS4-1 framework and the existing GS4-2 framework (for second order dynamic systems). Such a development enables the use of the same computational framework to solve both first and second order dynamic systems without having to resort to the individual GS4-1 and GS4-2 frameworks; hence the practicality in the computational and implementation aspects. Finally, the application of the new GS4-1 framework to silica particle deposition, which is a practical problem of interest, is presented with the focus primarily on the physics of the problem. In this part of the thesis, a numerical model of the problem is presented and employed to investigate the effects of the flow and physicochemical parameters on the rate of deposition. The results of the parametric studies undertaken based on the employed numerical model enable some recommendations for the mitigation of the problem, and therefore serve as additional valuable contribution of the thesis.

[pt] Mais de três décadas atrás, Przemieniecki introduziu uma formulação para análise de elementos de barra e treliça baseada em uma expansão em série de freqüências. Recentemente esta formulação foi generalizada para análise de sistemas elásticos submetidos a carregamento qualquer e deslocamentos iniciais. Baseado no método da superposição modal, um sistema acoplado, com equações diferenciais de movimento de alta ordem, é transformado em um sistema desacoplado com equações diferenciais de segunda ordem, que pode ser resolvido por qualquer método conhecido na literatura. A motivação para este desenvolvimento é o Método Híbrido dos Elementos de Contorno, que tem sido desenvolvido para problemas dependentes do tempo e problemas dependentes da freqüência. Esta formulação, assim como a introduzida por Pian para o Método dos Elementos Finitos, obtém uma matriz de rigidez utilizando apenas integrais de contorno, para um domínio de forma qualquer contendo vários graus de liberdade. O uso de termos com freqüências de alta ordem melhora muito a precisão numérica. A análise modal de um problema dinâmico, conforme se apresenta, é aplicável a qualquer formulação de elementos finitos, em geral, desde que a matriz de rigidez generalizada possa ser obtida. Este trabalho é uma tentativa de consolidação da formulação teórica proposta, em que se faz uso de integrais exclusivamente no contorno, com a discussão de diversos casos particulares e a conseqüente avaliação numérica: estruturas restringidas ou não; consideração de deslocamentos e velocidades iniciais, tanto em termos de valores nodais quanto de campos prescritos no domínio (incluindo deslocamentos de corpo rígido); deslocamentos forçados dependentes do tempo; forças de massa dependentes do tempo; cálculo de resultados em pontos internos. Vários exemplos acadêmicos para problemas de potencial bidimensionais ilustram este trabalho. / [en] More than three decades ago, Przemieniecki introduced a formulation for the free vibration analysis of bar and beam elements based on a power series of frequencies. Recently, this formulation was generalized for the analysis of the dynamic response of elastic systems submitted to arbitrary nodal loads as well as initial displacements. Based on the mode-superposition method, a set of coupled, higher-order differential equations of motion is transformed into a set of uncoupled second order differential equations, which may be integrated by means of standard procedures. Motivation for this theoretical achievement is the hybrid boundary element method, which has been developed for time-dependent as well as frequency-dependent problems. This formulation, as a generalization of Pian`s previous achievements for finite elements, yields a stiffness matrix for which only boundary integrals are required, for arbitrary domain shapes and any number of degrees of freedom. The use of higher-order frequency terms drastically improves numerical accuracy. The introduced modal assessment of the dynamic problem is applicable to any kind of finite element for which a generalized stiffness matrix is available. The present work is an attempt of consolidating this boundary- only theoretical formulation, in which a series of particular cases are conceptually outlined and numerically assessed: Constrained and unconstrained structures; initial displacements and velocities as nodal values as well as prescribed domain fields (including rigid body movement); forced time-dependent displacements; time-dependent body forces; evaluation of results at internal points. Several academic examples for 2D problems of potential illustrate the formulation.