Abstract Several numerical methods and algorithms, for solving the mathematical model of a continuous casting process, are presented, and theoretically studied, in this work. The numerical algorithms can be divided in to three different groups: the Schwarz type overlapping methods, the nonoverlapping Splitting iterative methods, and the Predictor-Corrector type nonoverlapping methods. These algorithms are all so-called parallel algorithms i.e., they are highly suitable for parallel computers. Multiplicative, additive Schwarz alternating method and two asynchronous domain decomposition methods, which appear to be a two-stage Schwarz alternating algorithms, are theoretically and numerically studied. Unique solvability of the fully implicit and semi-implicit finite difference schemes as well as monotone dependence of the solution on the right-hand side are proved. Geometric rate of convergence for the iterative methods is investigated. Splitting iterative methods for the sum of maximal monotone and single-valued monotone operators in a finite-dimensional space are studied. Convergence, rate of convergence and optimal iterative parameters are derived. A two-stage iterative method with inner iterations is analyzed in the case when both operators are linear, self-adjoint and positive definite. Several new finite-difference schemes for a nonlinear convection-diffusion problem are constructed and numerically studied. These schemes are constructed on the basis of non-overlapping domain decomposition and predictor-corrector approach. Different non-overlapping decompositions of a domain, with cross-points and angles, schemes with grid refinement in time in some subdomains, are used. All proposed algorithms are extensively numerically tested and are founded stable and accurate under natural assumptions for time and space grid steps. The advantages and disadvantages of the numerical methods are clearly seen in the numerical examples. All of the algorithms presented are quite easy and straight forward, from an implementation point of view. The speedups show that splitting iterative method can be parallelized better than multiplicative or additive Schwarz alternating method. The numerical examples show that the multidecomposition method is a very effective numerical method for solving the continuous casting problem. The idea of dividing the subdomains to smaller subdomains seems to be very beneficial and profitable. The advantages of multidecomposition methods over other methods is obvious. Multidecomposition methods are extremely quick, while being just as accurate as other methods. The numerical results for one processor seem to be very promising.

Neste projeto, é apresentado um método numérico com uma abordagem do tipoMAC para a simulação de escoamentos viscoelásticos incompressíveis tridimensionais com superfície livre governados pelo modelo de fluido SXPP. A formulação apresentada nesse trabalho é uma extensão dos resultados obtidos por Oishi et al. (2011), sobre o estudo de métodos numéricos para a simulação de escoamentos incompressíveis viscoelásticos com superfície livre a baixos números de Reynolds, para o caso bidimensional. No contexto de problemas transientes, metodologias explícitas para solução numérica das equações governantes apresentam restrições de estabilidade muito severas para a definição do passo temporal, acarretando em um custo computacional relativamente alto. Sendo assim, utilizamos um método implícito para resolver a equação de conservação da quantidade de movimento, eliminando assim, a restrição de estabilidade parabólica e diminuindo significativamente o custo computacional. Mas tal estratégia acopla os campos de velocidade e pressão. Dessa forma, para desacoplar esses campos, foi utilizado uma abordagem que combina método de projeção com uma técnica implícita para o tratamento da pressão na superfície livre. A equação constitutiva foi resolvida pelo método de Runge-Kutta de segunda-ordem. A validação do método numérico foi realizada utilizando refinamento da malha no escoamento em um canal. Como aplicação, apresentamos resultados numéricos sobre o problema do jato oscilante e do inchamento do extrudado / In this work, we present a numerical method with a MAC type approach to simulate tridimensional incompressible viscoelastic free surface flows governed by a SXPP (Single eXtended Pom-Pom) model. The formulation presented in this work is an extension to the work of Oishi et al. (2011). They have studied numerical methods for solving incompressible viscoelastic free surface flows with low Reynolds number, for the bidimensional case. In the context of transient problems, explicitmethodologies for the numerical solution of the governing equations present severe stability constraints for defining the time step, what highly increases the computational cost. Due to this fact, an implicit method is used to solve the momentum equation, eliminating the parabolic stability constraint and decreasing significantly the computational cost. However, this strategy couples velocity and pressure fields. To decouples this fields, it was used an approach that combines a projection method and an implicit technique for the treatment of the pressure at the free surface. The constitutive equation is solved by a second-order Runge-Kutta method. The numerical method validation was achieved by a mesh refinement for a flow in a channel. As applications, numerical results of the die-swell problem and the jet buckling phenomenon are presented

>Magister Scientiae - MSc / This thesis deals with analysis and implementation of a positivity preserving numerical method for a vaccination model for the transmission dynamics of two HIVsubtypes in a given community. The continuous model is analyzed for stability and equilibria. The qualitative information thus obtained is used while designing numerical method(s). Three numerical methods, namely, Implicit Finite Difference Method (IFDM), Non-standard Finite Difference Method (NSFDM) and the Runge-Kutta method of order four (RK4), are designed and implemented. Extensive numerical simulation are carried out to justify theoretical outcomes.

The steady flow method (SFM), most commonly used for permeability measurement in the laboratory, is not applicable for tight rocks, higher rank coals and coals under highly stressed condition because of the difficulty in measuring steady-state gas flowrates resulting from the tight rock structure of. However, accurate estimation of permeability of highly stressed coals is pivotal in coalbed methane (CBM) operations in order to precisely and effectively model and project long-term gas production. A fast and accurate permeability measurement technique is, therefore, required to investigate gas flow behavior of CBM reservoirs. The pulse-decay method (PDM) of permeability measurement is believed to be better suited for low-permeability rocks. In this study, application of the currently used pulse-decay laboratory permeability measurement techniques for highly stressed coals were evaluated. Considering the limitations of these techniques in permeability measurement of unconventional gas reservoirs, such as coal and gas shales, the conventional PDM was optimized by adjusting the experimental apparatus and procedures. Furthermore, the applicability of an optimized PDM was verified numerically and experimentally. This dissertation is composed of five chapters. To complete the research objectives as discussed above, it is necessary to have a profound understanding of the basic theories, such as, gas storage mechanism, gas migration, and permeability evolution during gas depletion in coalbed reservoirs. In Chapter 1, a brief discussion regarding the basic knowledge of reservoir properties and transport mechanisms is presented. The chapter also provides the appropriate background and rationale for the theoretical and experimental work conducted in this study. Chapter 2 presents the transient pressure-decay technique in permeability measurement of highly stressed coals and verifies the validity of Brace et al.’s solution (1968) by comparing it with Dicker and Smits’s solution (1988) and Cui et al.’s solution. The differences between these three solutions are discussed in detail. Based on the established permeability trends from these different solutions, a persuasive suggestion is presented for selection of the best alternative when testing coal permeability. Furthermore, permeability is regarded as a coupled parameter, resulting from the combined effects of mechanical compression and “matrix shrinkage” caused by desorption of gas. To isolate the role of gas desorption from the coupled result, a series of experiments were carried out under constant effective stress condition and a stress-dependent permeability trend was established. Chapter 3 proposes an optimized experimental design in order to improve the accuracy of the calculated permeability for sorptive rocks. In order to verify the optimized design theoretically, a modified mathematical model is presented and describes the one-dimensional fluid flow in porous media by a partial differential equation. The numerical solutions of the model are presented graphically to evaluate the fluid flow behavior in porous media. Finally, the validity of Brace et al.’s solution when testing sorptive rocks, without the need of consideration on the compressive storage and sorption effect, is elucidated. Chapter 4 demonstrates the efficiency and applicability of the optimized PDM through its direct application to experimental work designed to establish the permeability trend under best replicated in situ conditions. In this chapter, CO2 was used as the test fluid to profile and characterize the pulse decay plots due to its higher affinity towards coal than methane, and then establish the stress-dependent-permeability trend for highly-stressed CBM reservoirs. In this chapter, Brace et al.’s solution was also verified by comparing the laboratory data and computer simulated results obtained from the optimized mathematical model proposed in Chapter 3. The experimental work demonstrates that the optimized technique can be used for permeability tests of sorptive rocks without the need to carry out additional experimental work required to measure rock porosities and sorption isotherms. Finally, a summary and future research perspectives are presented in Chapter 5.

No abstract is provided. / Thesis / Master of Engineering (MEngr) / Scope and contents: The aim of this thesis is to present a finite difference method, for analysing coupled shear walls with constant or variable cross-section, resting on rigid or elastic foundations and with elastic or inelastic connecting beams. It is also intended to compare the finite difference method with the continuous connection method, which can be developed using Rosman's approach or Newmark's concept for analysing composite beams or the energy approach, and with the finite element method. An analysis of coupled shear walls with multiple piers is presented.

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A one-dimensional model of a heterogeneous material consisting of a matrix with embedded separated particles is considered, and the melting or solidification of the particles is investigated. The matrix is in imperfect contact with the particles, and the lumped capacity approximation applies to each individual particle. Heat is generated inside the particles or is transferred from the matrix to the particles coupled through a contact conductance. The matrix is not allowed to change phase and energy is either generated inside the matrix or transferred from the boundaries, which is initially conducted through the matrix material. The physical model of this coupled, two-step heat transfer process is solved using the energy method. The investigation is conducted in several phases using a building block approach. First, a lumped capacity system during phase transition is studied, then a one-dimensional homogeneous material during phase change is investigated, and finally the one-dimensional heterogeneous material is analyzed. A numerical solution based on the finite difference method is used to solve the model equations. This method allows for any kind of boundary conditions, any combination of material properties, particle sizes and contact conductance. In addition, computer programs, using Mathematica, are developed for the lumped capacity system, homogeneous material, and heterogeneous material. Results show the effects of control volume thickness, time step, contact conductance, material properties, internal sources, and external sources. / Master of Science