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Item A survey of discontinuous Galerkin methods for solving the time domain Maxwell's equations(2006) Binford, Tommy L., Jr.; Warburton, TimShow more The discontinuous Galerkin (DG) method with different numerical fluxes is applied to the square wave guide problem to avoid spurious modes that arise from the application of standard finite element methods. These numerical fluxes are the central, upwind and Lax-Friedrichs found in the literature. A new scheme, called penalty DG, is presented. Each scheme is tested with and without a locally divergence-free basis for the magnetic field. The spectral properties of the DG spatial discretization matrix for each flux are surmised by considering three different meshes and example eigenvalue plots. The convergence rate of the first ten eigenvalues is observed for h - and p -refinements. The central flux scheme is determined to be a poor choice for problems involving Maxwell's equations. It is proved that the kernel is empty for the DG spatial discretization matrix corresponding to the Lax-Friedrichs divergence-free scheme.Show more Item Accelerated Discontinuous Galerkin Solvers with the Chebyshev Iterative Method on the Graphics Processing Unit(2011) Tullius, Toni Kathleen; Riviere, Beatrice M.; Warburton, TimShow more This work demonstrates implementations of the discontinuous Galerkin (DG) method on graphics processing units (GPU), which deliver improved computational time compared to the conventional central processing unit (CPU). The linear system developed when applying the DG method to an elliptic problem is solved using the GPU. The conjugate gradient (CG) method and the Chebyshev iterative method are the linear system solvers that are compared, to see which is more efficient when computing with the CPU's parallel architecture. When applying both methods, computational times decreased for large problems executed on the GPU compared to CPU; however, CG is the more efficient method compared to the Chebyshev iterative method. In addition, a constant-free upper bound for the DC spectrum applied to the elliptic problem is developed. Few previous works combine the DG method and the GPU. This thesis will provide useful guidelines for the numerical solution of elliptic problems using DG on the GPU.Show more Item Hydrodynamic Modeling of Heating Processes in Solar Flares(2014-10-09) Reep, Jeffrey; Bradshaw, Stephen J; Alexander, David; Warburton, TimShow more This thesis examines the heating of the solar atmosphere due to energy release in solar flares. A one-dimensional hydrodynamic model, which solves the equations of conservation of mass, momentum, and energy along a magnetic flux tube, is described in detail and employed to study the dynamic response of the solar atmosphere to large amounts of energy release from the magnetic field. A brief introduction to the solar atmosphere and solar flares, from both observational and theoretical perspectives, is given. Then, the hydrodynamic model is described, along with derivations of energy deposition due to a beam of highly energetic electrons colliding with the ambient atmosphere (and their implementation in the model is explained). Using this model of heating along with RHESSI-derived parameters of observed flares, the sensitivity of the GOES flare classification to the parameters of the electron beam (the non-thermal energy, the power-law index of the electron distribution, and the low-energy cut-off) are examined, and clear correlations are determined. Next, the response of the atmosphere to heating due to isoenergetic beams of electrons are studied, to elucidate the importance of electrons at different energy. It is found that at high total energy fluxes, the energy of individual electrons are unimportant, but that at lower fluxes, lower energy electrons are significantly more efficient at heating the atmosphere and driving chromospheric evaporation than high energy electrons. It is also found that the threshold for explosive evaporation is strongly dependent on the cut-off energy, as well as the beam flux. A case study of a well-observed flare is performed. The flare, a C-class flare that occurred on 28 November 2002, was modeled for various cases of heating due to a beam of electrons, in situ coronal heating, and a hybrid model that combines both forms of heating. It is found that the observation of X-ray source heights seen with RHESSI are most consistent with a hybrid model. The results indicate that the energy must be partitioned between thermal and kinetic energy, and the implications are discussed. This work is then summarized, and future avenues of research are discussed. Improvements that can be made to the model, the forward modeling of emission, and comparisons to observations are discussed.Show more Item Instabilities in a Crystal Growth Melt Subjected to Alternating Magnetic Fields(2013-09-16) Davis, Kenny; Houchens, Brent C.; Akin, John Edward.; Warburton, TimShow more In confined bulk crystal growth techniques such as the traveling heater method, base materials in an ampoule are melted and resolidified as a single crystal. During this process, flow control is desired so that the resulting alloy semiconductors are uniform in composition and have minimal defects. Such control allows for tuned lattice parameters and bandgap energy, properties necessary to produce custom materials for specific electro-optical applications. For ternary alloys, bulk crystal growth methods suffer from slow diffusion rates between elements, severely limiting growth rates and reducing uniformity. Exposing the electrically conducting melt to an external alternating magnetic field can accelerate the mixing. A rotating magnetic field (RMF) can be used to stir the melt in the azimuthal direction, which reduces temperature variations and controls the shape at the solidification front. A traveling magnetic field (TMF) imposes large body forces in the radial and axial directions, which helps reduce the settling of denser components and return them to the growth front. In either case, mixing is desired, but turbulence is not. At large magnetic Taylor numbers the flow becomes unstable to first laminar and then turbulent transitions. It is imperative that crystal growers know when these transitions will occur and how the flow physics is affected. Here, the melt driven by electromagnetic forces is analyzed through the use of 3D numerical simulations of the flow field up to and beyond the point of laminar instability. The analysis aims to emulate laboratory conditions for generating electromagnetic forces for both types of alternating magnetic fields and highlights the differences between laboratory forces and the analytical approximations that are often assumed. Comparisons are made between the resulting forces, flow fields, and points of instability as the frequency of the alternating field varies. Critical Taylor numbers and the resulting unstable flow fields are compared to the results from linear stability theory.Show more Item Locally Mass-Conservative Method With Discontinuous Galerkin In Time For Solving Miscible Displacement Equations Under Low Regularity(2013-09-16) Li, Jizhou; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.; Warburton, TimShow more The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.Show more Item Moment Matching and Modal Truncation for Linear Systems(2013-07-24) Hergenroeder, AJ; Embree, Mark; Sorensen, Danny C.; Warburton, TimShow more While moment matching can effectively reduce the dimension of a linear, time-invariant system, it can simultaneously fail to improve the stable time-step for the forward Euler scheme. In the context of a semi-discrete heat equation with spatially smooth forcing, the high frequency modes are virtually insignificant. Eliminating such modes dramatically improves the stable time-step without sacrificing output accuracy. This is accomplished by modal filtration, whose computational cost is relatively palatable when applied following an initial reduction stage by moment matching. A bound on the norm of the difference between the transfer functions of the moment-matched system and its modally-filtered counterpart yields an intelligent choice for the mode of truncation. The dual-stage algorithm disappoints in the context of highly nonnormal semi-discrete convection-diffusion equations. There, moment matching can be ineffective in dimension reduction, precluding a cost-effective modal filtering step.Show more Item On the Constants in Inverse Inequalities in L2(2010-05) Ozisik, Sevtap; Riviere, Beatrice; Warburton, TimShow more In this paper we determine the constants in multivariate Markov inequalities in the L2-norm on an interval, a triangle and a tetrahedron. Using orthonormal polynomials, we derive explicit expression for the constants on that given 1-simplex, 2-simplex and 3-simplex. Accurate values for the constants are crucial for the correct derivation of a priori and a posteriori error estimations in adaptive computation.Show more Item On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows(Frontiers Media S.A., 2022) Chan, Jesse; Ranocha, Hendrik; Rueda-Ramírez, Andrés M.; Gassner, Gregor; Warburton, TimShow more High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.Show more Item Radial MILO: A 4D Image Registration Algorithm Based on Filtering Block Match Data via l1-minimization(2015-04-21) Vargas, Arturo; Zhang, Yin; Castillo, Edward; Tapia, Richard; Warburton, TimShow more Minimal l1 Perturbation to Block Match Data (MILO) is a spatially accurate image registration algorithm developed for thoracic CT inhale/exhale images. The MILO algorithm consists of three components: (1) creating an initial estimate for voxel displacement via a Mutual Minimizing Block Matching Algorithm (MMBM), (2) a filtering step based on l1 minimization and a uniform B-spline parameterization, and (3) recovering a full displacement field based on the filtered estimates. This thesis presents a variation of MILO for 4DCT images. In practice, the use of uniform B-splines has led to rank deficient linear systems due to the spline's inability to conform to non-structured MMBM estimates. In order to adaptively conform to the data an octree is paired with radial functions. The l1 minimization problem had previously been addressed by employing QR factorization, which required substantial storage. As an alternative a block coordinate descent algorithm is employed, relieving the need for QR factorization. Furthermore, by modeling voxel trajectories as quadratic functions in time, the proposed method is able to register multiple images.Show more Item Searching For FFLO States in Ultracold Polarized Fermi Gases: A Numerical Approach(2013-07-24) Lu, Hong; Pu, Han; Hulet, Randall G.; Warburton, TimShow more Ultracold atomic gases have emerged as an ideal laboratory system to emulate many-body physics in an unprecedentedly controllable manner. Numerous many-body quantum states and phases have been experimentally explored and characterized using the ultracold atomic gases, offering new insights into many exciting physics ranging from condensed matters to cosmology. In this thesis, we will present a systematic numerical study of a novel experimental system, population imbalanced two-component ultracold Fermi gases. We explore the phase diagram of this system in both 3D and 1D especially focusing on the exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, which is characterized by a spatially oscillating order parameter. In 3D, we solve for the stationary states of trapped imbalanced Fermi gases in a wide range of parameter space with a home-made parallel eigen-solver for Bogoliubov-de Gennes (BdG) equations. Our results show that there exists a metastable state with a FFLO type oscillating order parameter. In 1D, we simulate the dynamical expansion of the population imbalanced Fermi gases from the trap. A numerically quasi-exact scheme, time-evolving block decimation (TEBD), is introduced for the comparative studies with the solution of the time-dependent BdG equation. Our results predict that the existence of FFLO states will leave conspicuous signatures in the density profiles during the expansion. For further understanding of the interplay between the population imbalance and two-body pairing interaction between two spin components, we also study the spin transport properties through trapped ultracold Fermi gases. The preliminary results will be discussed.Show more Item Transfer-of-approximation Approaches for Subgrid Modeling(2013-07-24) Wang, Xin; Symes, William W.; Warburton, Tim; Riviere, Beatrice M.; Zelt, Colin A.Show more I propose two Galerkin methods based on the transfer-of-approximation property for static and dynamic acoustic boundary value problems in seismic applications. For problems with heterogeneous coefficients, the polynomial finite element spaces are no longer optimal unless special meshing techniques are employed. The transfer-of-approximation property provides a general framework to construct the optimal approximation subspace on regular grids. The transfer-of-approximation finite element method is theoretically attractive for that it works for both scalar and vectorial elliptic problems. However the numerical cost is prohibitive. To compute each transfer-of-approximation finite element basis, a problem as hard as the original one has to be solved. Furthermore due to the difficulty of basis localization, the resulting stiffness and mass matrices are dense. The 2D harmonic coordinate finite element method (HCFEM) achieves optimal second-order convergence for static and dynamic acoustic boundary value problems with variable coefficients at the cost of solving two auxiliary elliptic boundary value problems. Unlike the conventional FEM, no special domain partitions, adapted to discontinuity surfaces in coe cients, are required in HCFEM to obtain the optimal convergence rate. The resulting sti ness and mass matrices are constructed in a systematic procedure, and have the same sparsity pattern as those in the standard finite element method. Mass-lumping in HCFEM maintains the optimal order of convergence, due to the smoothness property of acoustic solutions in harmonic coordinates, and overcomes the numerical obstacle of inverting the mass matrix every time update, results in an efficient, explicit time step.Show more Item Transparency Property of One Dimensional Acoustic Wave Equations(2013-07-24) Huang, Yin; Symes, William W.; Riviere, Beatrice M.; Warburton, TimShow more This thesis proposes a new proof of the acoustic transparency theorem for material with a bounded variation. The theorem states that if the material properties (density, bulk modulus) is of bounded variation, the net power transmitted through the point z = 0 over a time interval [−T,T] is greater than some constant times the energy at the time zero over a spatial interval [0,Z], provided that T equals the time of travel of a wave from 0 to Z. This means the reflected energy of an input into the earth will be received. Otherwise, the reflections may not arrive at the surface. A proof gives a lower bound for material properties (density, bulk modulus) with bounded variation using sideways energy estimate. A different lower bound that works only for piecewise constant coefficients is also given. It gives a lower bound by analyzing reflections and transmissions of the waves at the jumps of the material properties. This thesis also gives an example to illustrate that the bounded variation assumption may not be necessary for the medium to be transparent. This thesis also discusses relations between the transparency property and the data of an inverse problem.Show more