Browsing by Subject "Numerical Analysis"
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Item Approximate Solutions to the Allen-Cahn Equation Using the Finite Difference Method(Texas A&M International University, 2016-06-13) Villarreal, Jamil Malik; Lin, RunchangSeeking a deeper understanding of the world has been a driving factor in Applied Mathematics. From counting and measuring physical objects to developing equations and ratios that resemble patterns in nature, mathematics is used to interpret and explain the intricate structures that we observe everyday. The field of Applied Mathematics almost always involves setting up and then solving, or approximating solutions to, at least one partial differential equation that takes the physical and mathematical properties into consideration. This is the process of creating mathematical models. For this thesis, we will investigate approximate solutions to the Allen-Cahn equation whose analytic solution is still unknown due to the nonlinearities of the problem as well as its sensitivity to certain constants as we shall see. The numerical schemes involved in these approximations are obtained from the finite difference method.Item Approximation Techniques for Incompressible Flows with Heterogeneous Properties(2011-10-21) Salgado Gonzalez, Abner JonatanWe study approximation techniques for incompressible flows with heterogeneous properties. Speci cally, we study two types of phenomena. The first is the flow of a viscous incompressible fluid through a rigid porous medium, where the permeability of the medium depends on the pressure. The second is the ow of a viscous incompressible fluid with variable density. The heterogeneity is the permeability and the density, respectively. For the first problem, we propose a finite element discretization and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence is exponential, we propose a splitting scheme which involves solving only two linear systems. For the second problem, we introduce a fractional time-stepping scheme which, as opposed to other existing techniques, requires only the solution of a Poisson equation for the determination of the pressure. This simpli cation greatly reduces the computational cost. We prove the stability of first and second order schemes, and provide error estimates for first order schemes. For all the introduced discretization schemes we present numerical experiments, which illustrate their performance on model problems, as well as on realistic ones.Item Multiscale Methods for Fluid-Structure Interaction with Applications to Deformable Porous Media(2012-10-19) Brown, DonaldIn this dissertation we study multiscale methods for slowly varying porous media, fluid and solid coupling, and application to geomechanics. The thesis consists of three closely connected results. We outline them and their relation. First, we derive a homogenization result for Stokes flow in slowly varying porous media. These results are important for homogenization in deformable porous media. Traditionally, these techniques are applied to periodic media, however, in the case of Fluid-Structure Interaction (FSI) slowly varying domains occur naturally. We then develop a computational methodology to compute effective quantities to construct homogenized equations for such media. Next, to extend traditional geomechanics models based primarily on the Biot equations, we use formal two-scale asymptotic techniques to homogenize the fully coupled FSI model. Prior models have assumed trivial pore scale deformation. Using the FSI model as a fine-scale model, we are able to incorporate non-trivial pore scale deformation into the macroscopic equations. The primary challenge here being the fluid and solid equations are represented in different coordinate frames. We reformulate the fluid equation in the fixed undeformed frame. This unified domain formulation is known as the Arbitrary Lagrange-Eulerian (ALE). Finally, we utilize the ALE formulation of the Stokes equations to develop an efficient multiscale finite element method. We use this method to compute the permeability tensor with much less computational cost. We build a dense hierarchy of macro-grids and a corresponding collection of nested approximation spaces. We solve local cell problems at dense macro-grids with low accuracy and use neighboring high accuracy solves to correct. With this method we obtain the same order of accuracy as we would if we computed all the local problems with highest accuracy.Item Natural Convection and Radiation in Small Enclosures with a Non-Attached Obstruction(Texas A&M University, 2004-09-30) Lloyd, Jimmy LynnNumerical simulations were used to investigate natural convection and radiation interactions in small enclosures of both two and three-dimensional geometries. The objectives of the research were to (1) determine the relative importance of natural convection and radiation, and to (2) estimate the natural convection heat transfer coefficients. Models are generated using Gambit, while numerical computations were conducted using the CFD code FLUENT. Dimensions for the two-dimensional enclosure were a height of 2.54 cm (1 inch), and a width that varied between 5.08 cm and 10.16 cm (2 inches and 4 inches). The three-dimensional model had a depth of 5.08 cm (2 inches) with the same height and widths as the two-dimensional model. The obstruction is located at the centroid of the enclosure and is represented as a circle in the two-dimensional geometry and a cylinder in the three-dimensional geometry. Obstruction diameters varied between .51 cm and 1.52 cm (0.2 inches and 0.6 inches). Model parameters used in the investigation were average surface temperatures, net total heat flux, and net radiation heat flux. These parameters were used to define percent temperature differences, percent heat flux contributions, convective heat transfer coefficients, Nusselt numbers, and Rayleigh numbers. The Rayleigh numbers varied between 0.005 and 300, and the convective heat transfer coefficients ranged between 2 and 25 W/m2K depending on the point in the simulation. The simulations were conducted with temperatures ranging between 310 K and 1275 K on the right boundary. For right boundary temperatures above 800 K, the estimated error on the obstruction temperature is less than 6.1% for neglecting natural convection and conduction from the heat transfer analysis. Lower right boundary temperatures such as 310 K had significant contributions, over 50%, from heat transfer modes other than radiation. For lower right boundary temperatures, a means of including natural convection should be included. When a bulk fluid temperature and average surface temperature values are available, a time average heat transfer coefficient of 6.73 W/m2K is proposed for simplifying the numerical calculations. In the transient right boundary temperature analysis, all modes of heat transfer other than radiation can be neglected to have an error below 8.1%.