Browsing by Subject "Numerical methods"
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Item A Boundary Element Method for the strongly nonlinear analysis of ventilating water-entry and wave-body interaction problems(2009-08) Vinayan, Vimal; Kinnas, Spyros A.A two-dimensional Boundary Element Method (BEM) is developed to study the strongly nonlinear interaction between a surface-piercing body and the free-surface. The scheme is applied to problems with and without the possibility of ventilation resulting from the motion and geometric configuration of the surface-piercing body. The main emphasis of this research work is on the development of numerical methods to improve the performance prediction of surface-piercing propellers by including the whole range of free-surface nonlinearities. The scheme is applied to predict the ventilated cavity shapes resulting from the vertical and rotational motion of a blade-section with fully nonlinear free-surface boundary conditions. The current method is able to predict the ventilated cavity shapes for a wide range of angles of attack and Froude numbers, and is in good agreement with existing experimental results. Through a comparison with a linearized free-surface method, the current method highlights the shortcomings of the negative image approach used commonly in two-dimensional and three-dimensional numerical methods for surface-piercing hydrofoils or propellers. The current method with all its capabilities makes it a unique contribution to improving numerical tools for the performance prediction of surface-piercing propellers. The scheme is also applied to predict the roll and heave dynamics of two-dimensional Floating Production Storage and Offloading (FPSO) vessel hull sections within a potential flow framework. The development of the potential flow model is aimed at validating the free-surface dynamics of an independently developed Navier Stokes Solver for predicting the roll characteristics of two-dimensional hull sections with bilge keels.Item Connecting the dots : tracking galaxy evolution using constant cumulative number density at 3(2015-12) Jaacks, Jason Dale; Finkelstein, Steven L.; Bromm, VolkerUsing the cosmological smoothed particle hydrodynamical code GADGET-3 we make a realistic assessment of the technique of using constant cumulative number density as a tracer of galaxy evolution. We find that over a redshift range of 3Item Discontinuous Galerkin methods for resolving non linear and dispersive near shore waves(2014-05) Panda, Nishant; Dawson, Clinton N.Near shore hydrodynamics has been an important research area dealing with coastal processes. The nearshore coastal region is the region between the shoreline and a fictive offshore limit which usually is defined as the limit where the depth becomes so large that it no longer influences the waves. This spatially limited but highly energetic zone is where water waves shoal, break and transmit energy to the shoreline and are governed by highly dispersive and non-linear effects. An accurate understanding of this phenomena is extremely useful, especially in emergency situations during hurricanes and storms. While the shallow water assumption is valid in regions where the characteristic wavelength exceeds a typical depth by orders of magnitude, Boussinesq-type equations have been used to model near-shore wave motion. Unfortunately these equations are complex system of coupled non-linear and dispersive differential equations that have made the developement of numerical approximations extremely challenging. In this dissertation, a local discontinuous Galerkin method for Boussinesq-Green Naghdi Equations is presented and validated against experimental results. Currently Green-Naghdi equations have many variants. We develop a numerical method in one horizontal dimension for the Green-Naghdi equations based on rotational characteristics in the velocity field. Stability criterion is also established for the linearized Green-Naghdi equations and a careful proof of linear stability of the numerical method is carried out. Verification is done against a linearized standing wave problem in flat bathymetry and h,p (denoted by K in this thesis) error rates are plotted. The numerical method is validated with experimental data from dispersive and non-linear test cases.Item An efficient solution procedure for simulating phonon transport in multiscale multimaterial systems(2013-05) Loy, James Madigan; Murthy, JayathiOver the last two decades, advanced fabrication techniques have enabled the fabrication of materials and devices at sub-micron length scales. For heat conduction, the conventional Fourier model for predicting energy transport has been shown to yield erroneous results on such length scales. In semiconductors and dielectrics, energy transport occurs through phonons, which are quanta of lattice vibrations. When phase coherence effects can be ignored, phonon transport may be modeled using the semi-classical phonon Boltzmann transport equation (BTE). The objective of this thesis is to develop an efficient computational method to solve the BTE, both for single-material and multi-material systems, where transport across heterogeneous interfaces is expected to play a critical role. The resulting solver will find application in the design of microelectronic circuits and thermoelectric devices. The primary source of computational difficulties in solving the phonon BTE lies in the scattering term, which redistributes phonon energies in wave-vector space. In its complete form, the scattering term is non-linear, and is non-zero only when energy and momentum conservation rules are satisfied. To reduce complexity, scattering interactions are often approximated by the single mode relaxation time (SMRT) approximation, which couples different phonon groups to each other through a thermal bath at the equilibrium temperature. The most common methods for solving the BTE in the SMRT approximation employ sequential solution techniques which solve for the spatial distribution of the phonon energy of each phonon group one after another. Coupling between phonons is treated explicitly and updated after all phonon groups have been solved individually. When the domain length is small compared to the phonon mean free path, corresponding to a high Knudsen number ([mathematical equation]), this sequential procedure works well. At low Knudsen number, however, this procedure suffers long convergence times because the coupling between phonon groups is very strong for an explicit treatment of coupling to suffice. In problems of practical interest, such as silicon-based microelectronics, for example, phonon groups have a very large spread in mean free paths, resulting in a combination of high and low Knudsen number; in these problems, it is virtually impossible to obtain solutions using sequential solution techniques. In this thesis, a new computational procedure for solving the non-gray phonon BTE under the SMRT approximation is developed. This procedure, called the coupled ordinates method (COMET), is shown to achieve significant solution acceleration over the sequential solution technique for a wide range of Knudsen numbers. Its success lies in treating phonon-phonon coupling implicitly through a direct solution of all equations in wave vector space at a particular spatial location. To increase coupling in the spatial domain, this procedure is embedded as a relaxation sweep in a geometric multigrid. Due to the heavy computational load at each spatial location, COMET exhibits excellent scaling on parallel platforms using domain decomposition. On serial platforms, COMET is shown to achieve accelerations of 60 times over the sequential procedure for Kn<1.0 for gray phonon transport problems, and accelerations of 233 times for non-gray problems. COMET is then extended to include phonon transport across heterogeneous material interfaces using the diffuse mismatch model (DMM). Here, coupling between phonon groups occurs because of reflection and transmission. Efficient algorithms, based on heuristics, are developed for interface agglomeration in creating coarse multigrid levels. COMET is tested for phonon transport problems with multiple interfaces and shown to outperform the sequential technique. Finally, the utility of COMET is demonstrated by simulating phonon transport in a nanoparticle composite of silicon and germanium. A realistic geometry constructed from x-ray CT scans is employed. This composite is typical of those which are used to reduce lattice thermal conductivity in thermoelectric materials. The effective thermal conductivity of the composite is computed for two different domain sizes over a range of temperatures. It is found that for low temperatures, the thermal conductivity increases with temperature because interface scattering dominates, and is insensitive to temperature; the increase of thermal conductivity is primarily a result of the increase in phonon population with temperature consistent with Bose-Einstein statistics. At higher temperatures, Umklapp scattering begins to take over, causing a peak in thermal conductivity and a subsequent decrease with temperature. However, unlike bulk materials, the peak is shallow, consistent with the strong role of interface scattering. The interaction of phonon mean free path with the particulate length scale is examined. The results also suggest that materials with very dissimilar cutoff frequencies would yield a thermal conductivity which is closest to the lowest possible value for the given geometry.Item Reservoir simulation and optimization of CO₂ huff-and-puff operations in the Bakken Shale(2014-08) Sanchez Rivera, Daniel; Balhoff, Matthew T.; Mohanty, Kishore KumarA numerical reservoir model was created to optimize CO₂ Huff-and-Puff operations in the Bakken Shale. Huff-and-Puff is an enhanced oil recovery treatment in which a well alternates between injection, soaking, and production. Injecting CO₂ into the formation and allowing it to “soak” re-pressurizes the reservoir and improves oil mobility, boosting production from the well. A compositional reservoir simulator was used to study the various design components of the Huff-and-Puff process in order to identify the parameters with the largest impact on recovery and understand the reservoir’s response to cyclical CO₂ injection. It was found that starting Huff-and-Puff too early in the life of the well diminishes its effectiveness, and that shorter soaking periods are preferable over longer waiting times. Huff-and-Puff works best in reservoirs with highly-conductive natural fracture networks, which allow CO₂ to migrate deep into the formation and mix with the reservoir fluids. The discretization of the computational domain has a large impact on the simulation results, with coarser gridding corresponding to larger projected recoveries. Doubling the number of hydraulic fractures per stage results in considerably greater CO₂ injection requirements without proportionally larger incremental recovery factors. Incremental recovery from CO₂ Huff-and-Puff appears to be insufficient to make the process commercially feasible under current economic conditions. However, re-injecting mixtures of CO₂ and produced hydrocarbon gases was proven to be technically and economically viable, which could significantly improve profit margins of Huff-and-Puff operations. A substantial portion of this project involved studying alternative numerical methods for modeling hydraulically-fractured reservoir models. A domain decomposition technique known as mortar coupling was used to model the reservoir system as two individually-solved subdomains: fracture and matrix. A mortar-based numerical reservoir simulator was developed and its results compared to a tradition full-domain finite difference model for the Cinco-Ley et al. (1978) finite-conductivity vertical fracture problem. Despite some numerical issues, mortar coupling closely matched Cinco-Ley et al.'s (1978) solution and has potential applications in complex problems where decoupling the fracture-matrix system might be advantageous.Item Statistical analysis of three fourth-order ordinary differential equation solvers(2007-12) He, Bo; Martin, Clyde F.; Surles, James; Neusel, Mara D.We develop an autoregressive integrated moving average model (ARIMA) to study the statistical behavior of the numerical error generated from three fourth-order ODE Solvers: Milne's method, Adams-Bashforth method and a new method which randomly switches between Milne and Adams-Bashforth methods. With the actual error data based on three differential equations we desire to identify an ARIMA model to each data series. Results show that some of data series can be described by ARIMA models but others can not. Based on the mathematical form of the error data, other statistical models should be applied in the future. Finally we assess the multivariate normality of sample mean vectors which are generated by the switching method as an application of the multivariate central limit theorem.Item Steady-state spherical accretion using smoothed particle hydrodynamics(2011-12) Baumann, Mark Chapple; Matzner, Richard A. (Richard Alfred), 1942-; Dicus, Duane; Klein, Josh; Kopp, Sacha; Marder, MichaelDue to its adaptable nature in a broad range of problem domains, Smoothed Particle Hydrodynamics (SPH) is a popular numerical technique for computing solutions in astrophysics. This dissertation discusses the SPH technique and assesses its capabilities for reproducing steady-state spherically-symmetric accretion flow. The accretion scenario is of great interest for its applicability in a diverse array of astrophysical phenomena and, under certain assumptions, it also provides an accepted analytical solution against which the numerical method can be validated. After deriving the necessary equations from astrophysical fluid dynamics, giving a detailed review of solving the steady-state spherical accretion problem, and developing the SPH methodology, this work suggests solutions to the issues that must be overcome in order to successfully employ the SPH methodology to reproduce steady-state spherical accretion flow. Several techniques for setting initial data are addressed, resolution requirements are illustrated, inner and outer boundary conditions are discussed, and artificial dissipation parameters and methodologies are explored.