The Peaceman–Rachford ADI-dG method for linear wave-type problems

In this thesis the discretization in space and time of a broad class of linear first order wave-type problems is investigated. The full discretization is achieved by using a method of lines approach utilizing a central-fluxes discontinuous Galerkin (dG) method in space and the Peaceman–Rachford scheme in time. Rigorous error bounds are derived, which show full order of convergence in space and time if the exact solution is sufficiently regular. Additionally, despite the fact that the Peaceman–Rachford scheme is an implicit method exhibiting unconditional stability, it is shown that by combining it with an alternating direction implicit (ADI) approach, it can be applied to certain problems at roughly the cost of an explicit time integration scheme. Problems for which this is possible are characterized precisely, and the implementation used to achieve this efficiency is described in detail. This class of problems comprises, e.g., the 2D advection and wave equations and the 3D Maxwell\u27s equations, for which the ADI method was previously proposed in literature

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp

Analysis of Heat Partitioning During Sliding Contact At High Speed and Pressure

This research develops a mathematical formulation and an analytical solution to frictional heat partitioning in a high speed sliding system. Frictional heating at the interface of sliding materials impacts temperature and the wear mechanisms. The heat partition fraction for a sliding system is an important parameter in calculating the distribution of frictional heat flux between the contacting surfaces. The solution presented in this dissertation considers the characteristics of the slipper\u27s frictional heat partition values along with the experimental loading data. With a physics based, rather than a phenomenological approach, this solution improves the estimate for the slipper\u27s heat partition function. Moreover, this analytical solution is practical in calculating the average surface temperature and estimating the total melt wear volume. The heat partition function compares favorably with existing experimental and analytical data. Using the Strang\u27s Splitting and ADI methods, a numerical method for surface temperature and corresponding wear percentage under dynamic bounce conditions was extensively developed

ADI schemes for heat equations with irregular boundaries and interfaces in 3D with applications

In this paper, efficient alternating direction implicit (ADI) schemes are proposed to solve three-dimensional heat equations with irregular boundaries and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a modified ADI scheme is constructed to mitigate the issue of accuracy loss in solving problems with time-dependent boundary conditions. The unconditional stability of the new ADI scheme is also rigorously proven with the Fourier analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes, the KFBI discretization takes advantage of the Cartesian grid and preserves the structure of the coefficient matrix so that the fast Thomas algorithm can be applied to solve the linear system efficiently. Second-order accuracy and unconditional stability of the KFBI-ADI schemes are verified through several numerical tests for both the heat equation and a reaction-diffusion equation. For the Stefan problem, which is a free boundary problem of the heat equation, a level set method is incorporated into the ADI method to capture the time-dependent interface. Numerical examples for simulating 3D dendritic solidification phenomenons are also presented

An investigation of alternating-direction implicit finite-difference time-domain (ADI-FDTD) method in numerical electromagnetics

In this thesis, the alternating-direction implicit method (ADI) is investigated in conjunction with the finite difference time-domain method (FDTD) to allow crossing of the Courant-Friedrich-Levy (CFL) stability criterion while maintaining stability in the FDTD algorithm. The main reason for this is to be able to use a larger numerical time step than that governed by the CFL criterion. The desired effect is a significant reduction in numerical run-times. Although the ADI-FDTD method has been used in the literature, most analysis and application have been performed on simple three-dimensional cavities.This work makes original contribution in two aspects. Firstly, a new modified alternating-direction implicit method for a three-dimensional FDTD algorithm has been successfully developed and implemented in this research. This new method allows correct modelling of a realistic physical structure such as a microstrip patch with the ADI scheme without causing instability even when the CFL criterion is not observed. However, due to the inherent property of this modified ADI-FDTD method, a decreasing reflection coefficient is observed using this scheme.The second and more important contribution this research makes in the field of numerical electromagnetics is the development of a new method of simulating realistic complex structures such as geometries comprising copper patch antennas on a dielectric substrate. With this new method, for the first time, the ADl-FDTD algorithm remains stable while still in violation of the CFL criterion, even when complex structures are being modelled.However, there is a trade-off between accuracy and computational speed in ADI-FDTD and modified ADI-FDTD methods. The larger the numerical time step, the shorter is the simulation run-time but an increase in numerical time step causes a degradation in accuracy of numerical results. Comparison between speed and accuracy is shown in this thesis and it has to be mentioned here that these values are very much dependent on the structure being modelled

Numerical simulation of nanopulse penetration of biological matter using the ADI-FDTD method

Nanopulses are ultra-wide-band (UWB) electromagnetic pulses with pulse duration of only a few nanoseconds and electric field amplitudes greater than 105 V/m. They have been widely used in the development of new technologies in the field of medicine. Therefore, the study of the nanopulse bioeffects is important to ensure the appropriate application with nanopulses in biomedical and biotechnological settings. The conventional finite-difference time-domain (FDTD) method for solving Maxwell\u27s equations has been proven to be an effective method to solve the problems related to electromagnetism. However, its application is restricted by the Courant, Friedrichs, and Lewy (CFL) stability condition that confines the time increment and mesh size in the computation in order to prevent the solution from being divergent. This dissertation develops a new finite difference scheme coupled with the Cole-Cole expression for dielectric coefficients of biological tissues to simulate the electromagnetic fields inside biological tissues when exposed to nanopulses. The scheme is formulated based on the Yee\u27s cell and alternating direction implicit (ADI) technique. The basic idea behind the ADI technique is to break up every time step into two half-time steps. At the first half-step, the finite difference operator on the right-hand side of the Maxwell\u27s equation is implicit only along one coordinate axis direction. At the second half-step, the finite difference operator on the right-hand side of the Maxwell\u27s equation is implicit only along the other coordinate axis direction. As such, only tridiagonal linear systems are solved. In this numerical method, the Cole-Cole expression is approximated by a second-order Taylor series based on the z-transform method. In addition, the perfectly matched layer is employed for the boundary condition, and the total/scattered field technique is employed to generate the plane wave in order to prevent the wave reflection. The scheme is tested by numerical examples with two different biological tissues. For the purpose of comparison, both the proposed ADI-FDTD scheme and the conventional FDTD scheme are employed to the numerical examples. The results show that the proposed ADI-FDTD scheme breaks through the CFL stability condition and provides a stable solution with a larger time step, where the conventional FDTD scheme fails. Results also indicate that the computational time can be reduced with a larger time step

Miscible flow through porous media

This thesis is concerned with the modelling of miscible fluid flow through porous media, with the intended application being the displacement of oil from a reservoir by a solvent with which the oil is miscible. The primary difficulty that we encounter with such modelling is the existence of a fingering instability that arises from the viscosity and the density differences between the oil and solvent.\ud \ud We take as our basic model the Peaceman model, which we derive from first principles as the combination of Darcy’s law with the mass transport of solvent by advection and hydrodynamic dispersion. In the oil industry, advection is usually dominant, so that the Peclet number, Pe, is large. \ud \ud We begin by neglecting the effect of density differences between the two fluids and concentrate only on the viscous fingering instability. A stability analysis and numerical simulations are used to show that the wavelength of the instability is proportional to $\mathrm{Pe}^{−1/2}$, and hence that a large number of fingers will be formed. We next apply homogenisation theory to investigate the evolution of the average concentration of solvent when the mean flow is one-dimensional, and discuss the rationale behind the Koval model. We then attempt to explain why the mixing zone in which fingering is present grows at the observed rate, which is different from that predicted by a naıve version of the Koval model. We associate the shocks that appear in our homogenised model with the tips and roots of the fingers, the tip-regions being modelled by Saffman-Taylor finger solutions.\ud \ud We then extend our model to consider flow through porous media that are heterogeneous at the macroscopic scale, and where the mean flow is not onedimensional. We compare our model with that of Todd & Longstaff and also models for immiscible flow through porous media.\ud \ud Finally, we extend our work to consider miscible displacements in which both density and viscosity differences between the two fluids are relevant

Computational Fluid Dynamics Studies in Heat and Mass Transfer Phenomena in Packed Bed Extraction and Reaction Equipment: Special Attention to Supercritical Fluids Technology

El entendimiento de los fenómenos de transferencia de calor y de masa en medios porosos implica el estudio de modelos de transporte de fluidos en la fracción vacía del medio; este hecho es de fundamental importancia en muchos sistemas de Ingeniería Química, tal como en procesos de extracción o en reactores catalíticos. Los estudios de flujo realizados hasta ahora (teóricos y experimentales) usualmente tratan al medio poroso como un medio efectivo y homogéneo, y toman como válidas las propiedades medias del fluido. Este tipo de aproximación no tiene en cuenta la complejidad del flujo a través del espacio vacío del medio poroso, reduciendo la descripción del problema a promedios macroscópicos y propiedades efectivas. Sin embargo, estos detalles de los procesos locales de flujo pueden llegar a ser factores importantes que influencien el comportamiento de un proceso físico determinado que ocurre dentro del sistema, y son cruciales para entender el mecanismo detallado de, por ejemplo, fenómenos como la dispersión de calor, la dispersión de masa o el transporte entre interfaces.La Dinámica de Fluidos Computacional (CFD) como herramienta de modelado numérico permite obtener una visión mas aproximada y realista de los fenómenos de flujo de fluidos y los mecanismos de transferencia de calor y masa en lechos empacados, a través de la resolución de las ecuaciones de Navier - Stokes acopladas con los balances de materia y energía y con un modelo de turbulencia si es necesario. De esta forma, esta herramienta permite obtener los valores medios y/o fluctuantes de variables como la velocidad del fluido, la temperatura o la concentración de una especie en cualquier punto de la geometría del lecho empacado.El objetivo de este proyecto es el de utilizar programas comerciales de simulación CFD para resolver el flujo de fluidos y la transferencia de calor y de masa en modelos bi/tri dimensionales de lechos empacados, desarrollando una estrategia de modelado aplicable al diseño de equipos para procesos de extracción o de reacción catalítica. Como referencia se tomaran procesos de tecnología supercrítica debido a la complejidad de los fenómenos de transporte involucrados en estas condiciones, así como a la disponibilidad de datos experimentales obtenidos previamente en nuestro grupo de investigación. Estos datos experimentales se utilizan como herramienta de validación de los modelos numéricos generados, y de las estrategias de simulación adoptadas y realizadas durante el desarrollo de este proyecto.An understanding of the heat and mass transfer phenomena in a porous media implies the study of the fluid transport model within the void space; this fact is of fundamental importance to many chemical engineering systems such as packed bed extraction or catalytic reaction equipment. Experimental and theoretical studies of flow through such systems often treat the porous medium as an effectively homogeneous system and concentrate on the bulk properties of the flow. Such an approach neglects completely the complexities of the flow within the void space of the porous medium, reducing the description of the problem to macroscopic average or effective quantities. The details of this local flow process may, however, be the most important factor influencing the behavior of a given physical process occurring within the system, and are crucial to understanding the detailed mechanisms of, for example, heat and mass dispersion and interface transport.Computational Fluid Dynamics as a simulation tool allows obtaining a more approached view of the fluid flow and heat and mass transfer mechanisms in fixed bed equipment, through the resolution of 3D Reynolds averaged transport equations, together with a turbulence model when needed. In this way, this tool permit to obtain mean and fluctuating flow and temperature values in any point of the bed. The goal of this project is to use commercial available CFD codes for solving fluid flow and heat and mass transfer phenomena in two and three dimensional models of packed beds, developing a modeling strategy applicable to the design of packed bed chemical reaction and extraction equipment. Supercritical extraction and supercritical catalytic reaction processes will be taken as reference processes due to the complexity of the transport phenomena involved within this processes, and to the availability of experimental data in this field, obtained in the supercritical fluids research group of this university. The experimental data priory obtained by our research group will be used as validation data for the numerical models and strategies dopted and followed during the developing of the project

Performance of the Horizontal Wells in a Naturally Fractured Carbonate Reservoir

About 60 percent of the world\u27s proven oil reserves and 40 percent of gas reserves are trapped in carbonate reservoirs. Recovery rates are relatively low in carbonate reservoirs, and it is extremely challenging to predict due to the heterogeneous nature of these reservoirs. The majority of carbonate reservoirs contain fractures which may vary in size from millimeters to kilometers. A typical example of this type of reservoirs is oil and gas fields in the northern Iraq. These fields are almost all developed by vertical wells. This study will investigate the use of horizontal wells to enhance the productivity in one of these reservoirs.;CMG software is used to simulate the Upper Qamchuqa reservoir of the Khabbaz oil field northeastern of Iraq. The Upper Qamchuqa reservoir is a subsurface anticline with a major normal fault on the eastern flank. Moreover, it mainly consists of dolomite, dolomitic limestone, limestone and marly limestone. The simulation model was validated by the history matching the production rates. Subsequently, horizontal wells were added to the model, and the optimum placement and lateral length investigated. Furthermore, the reservoir parameters that have a significant influence on the history matching and the predicted horizontal wells\u27 performance were identified for future development of the field. The results of this study can provide a guideline for reducing the operating costs and increasing the productivity of similar naturally fractured reservoirs in the area