This work deals with numerical modeling of electromagnetic induction in 3D environment with heterogeneous conductivity. We develop a program to solve Maxwell's equations in quasistatic approximation by using Continuous and Discontinuous Finite Elements. Their implementation in the numerical library deal.ii is discussed. The obtained numerical results are compared with each other and also with a quasianalytic solution for an environment with 1D heterogeneous conductivity. We discuss different numerical methods, limits of our code for practical use and possible future enhancements.

The subject of this thesis is the development of an accurate, general and robust numerical method capable of predicting the flow behavior of two-phase immiscible fluids, separated by a well defined interface. In the quest of an accurate and robust numerical method for the modeling of two-phase flows, one has to keep in mind the intrinsic properties and difficulties associated with the problem: (i) those flows are mostly three-dimensional, (ii) some flows are steady, others unsteady, (iii) the interface might encounter a lot of topology changes (like merger or break-up), (iv) large jumps of density and viscosity might exist across the interface (e.g. ratio of density of 1/1000 for water and air), (v) surface tension forces may play a very important role in the interface dynamics. Hence, the influence of this force should be accurately evaluated and incorporated into the model, (vi) mass conservation is of primary importance. All these issues are addressed in this thesis, and special techniques are proposed for their treatment, which enables to construct the desired computational method. The chosen computational method combines a pressure stabilized finite element method for the Navier Stokes equations with a discontinuous Galerkin (DG) method for the level set equation. Such a combination of those two numerical methods results in a simple and effective algorithm that allows to simulate diverse flow regimes presenting large density and viscosity ratios (ratio up to 1/1000).

In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Pade approximation of exp(-z) when the p-th degree basis functions are used. Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids. Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cutting an embedded geometry out of the Cartesian grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.

Coastal regions are the most densely populated regions of the world. The populations of these regions continue to grow which has created a high demand for water that stresses existing water resources. Coastal aquifers provide a source of water for coastal populations and are generally part of a larger system where freshwater aquifers are hydraulically connected with a saline surface-water body. They are characterized by salinity variations in space and time, sharp freshwater/saltwater interfaces which can lead to dramatic density differences, and complex groundwater chemistry. Mismanagement of coastal aquifers can lead to saltwater intrusion, the displacement of fresh water by saline water in the freshwater regions of the aquifers, making them unusable as a freshwater source. Saltwater intrusion is of significant interest to water resource managers and efficient simulators are needed to assist them. Numerical simulation of saltwater intrusion requires solving a system of flow and transport equations coupled through a density equation of state. The scale of the problem domain, irregular geometry and heterogeneity can require significant computational resources. Also, modeling sharp transition zones and accurate flow velocities pose numerical challenges. Discontinuous Galerkin (DG) finite element methods (FEM) have been shown to be well suited for modeling flow and transport in porous media but a fully coupled DG formulation has not been applied to the variable density flow and transport model. DG methods have many desirable characteristics in the areas of numerical stability, mesh and polynomial approximation adaptivity and the use of non-conforming meshes. These properties are especially desirable when working with complex geometries over large scales and when coupling multi-physics models (e.g. surface water and groundwater flow models). In this dissertation, we investigate a new combined local discontinuous Galerkin (LDG) and non-symmetric, interior penalty Galerkin (NIPG) formulation for the non-linear coupled flow and solute transport equations that model saltwater intrusion. Our main goal is the formulation and numerical implementation of a robust, efficient, tightly-coupled combined LDG/NIPG formulation within the Department of Defense (DoD) Proteus Computational Mechanics Toolkit modeling framework. We conduct an extensive and systematic code and model verification (using established benchmark problems and proven convergence rates) and model validation (using experimental data) to verify accomplishment of this goal. Lastly, we analyze the accuracy and conservation properties of the numerical model. More specifically, we derive an a priori error estimate for the coupled system and conduct a flow/transport model compatibility analysis to prove conservation properties. / text

In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Pade approximation of exp(-z) when the p-th degree basis functions are used. Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids. Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cutting an embedded geometry out of the Cartesian grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.

A look into an implementation of turbulence model into the ADGFEM code for viscous flow. Discretization, theory background and development of the method will be carried during this thesis. Also some numerical examples of the application of the code will be provided. 1

abstract: This dissertation describes a process for interface capturing via an arbitrary-order, nearly quadrature free, discontinuous Galerkin (DG) scheme for the conservative level set method (Olsson et al., 2005, 2008). The DG numerical method is utilized to solve both advection and reinitialization, and executed on a refined level set grid (Herrmann, 2008) for effective use of processing power. Computation is executed in parallel utilizing both CPU and GPU architectures to make the method feasible at high order. Finally, a sparse data structure is implemented to take full advantage of parallelism on the GPU, where performance relies on well-managed memory operations. With solution variables projected into a kth order polynomial basis, a k+1 order convergence rate is found for both advection and reinitialization tests using the method of manufactured solutions. Other standard test cases, such as Zalesak's disk and deformation of columns and spheres in periodic vortices are also performed, showing several orders of magnitude improvement over traditional WENO level set methods. These tests also show the impact of reinitialization, which often increases shape and volume errors as a result of level set scalar trapping by normal vectors calculated from the local level set field. Accelerating advection via GPU hardware is found to provide a 30x speedup factor comparing a 2.0GHz Intel Xeon E5-2620 CPU in serial vs. a Nvidia Tesla K20 GPU, with speedup factors increasing with polynomial degree until shared memory is filled. A similar algorithm is implemented for reinitialization, which relies on heavier use of shared and global memory and as a result fills them more quickly and produces smaller speedups of 18x. / Dissertation/Thesis / Doctoral Dissertation Aerospace Engineering 2015

This thesis studies the numerical solution of non-linear convection-diffusion problems using the space- time discontinuous Galerkin method, which perfectly suits the space as well as time local adaptation. We aim to develop a posteriori error estimates reflecting the spatial, temporal, and algebraic errors. These estimates are based on the measurement of the residuals in dual norms. We derive these estimates and numerically verify their properties. Finally, we derive an adaptive algorithm and apply it to the numerical simulation of non-stationary viscous compressible flows. Powered by TCPDF (www.tcpdf.org)

In this dissertation we prove the superconvergence of the minimal-dissipation local discontinuous Galerkin method for elliptic problems and construct optimal immersed finite element approximations and discontinuous immersed finite element methods for the Stokes interface problem. In the first part we present an error analysis for the minimal dissipation local discontinuous Galerkin method applied to a model elliptic problem on Cartesian meshes when polynomials of degree at most <i>k</i> and an appropriate approximation of the boundary condition are used. This special approximation allows us to achieve <i>k</i> + 1 order of convergence for both the potential and its gradient in the L² norm. Here we improve on existing estimates for the solution gradient by a factor &#8730;h. In the second part we present discontinuous immersed finite element (IFE) methods for the Stokes interface problem on Cartesian meshes that does not require the mesh to be aligned with the interface. As such, we allow unfitted meshes that are cut by the interface. Thus, elements may contain more than one fluid. On these unfitted meshes we construct an immersed Q₁/Q₀ finite element approximation that depends on the location of the interface. We discuss the basic features of the proposed Q₁/Q₀ IFE basis functions such as the unisolvent property. We present several numerical examples to demonstrate that the proposed IFE approximations applied to solve interface Stokes problems maintain the optimal approximation capability of their standard counterpart applied to solve the homogeneous Stokes problem. Similarly, we also show that discontinuous Galerkin IFE solutions of the Stokes interface problem maintain the optimal convergence rates in both L² and broken H¹ norms. Furthermore, we extend our method to solve the axisymmetric Stokes interface problem with a moving interface and test the proposed method by solving several benchmark problems from the literature. / Ph. D.

As traditional communication technology we use in our day-to-day life reaches its limitations, the international community searches for new methods to communicate information. One such novel approach is the so-called molecular communication system. During the last few decades, molecular communication systems become more and more popular. The main difference between traditional communication and molecular communication systems is that in the latter, information transfer occurs through chemical means, most often between microorganisms. This process already happens all around us naturally, for example, in the human body. Even though the molecular communication topic is attractive to researchers, and a lot of theoretical results are available - one cannot claim the same about the practical use of molecular communication. As for experimental results, a few studies have been done on the macroscale, but investigations at the micro- and nanoscale ranges are still lacking because they are a challenging task. In this work, a self-contained introduction of the underlying theory of molecular communication is provided, which includes knowledge from different areas such as biology, chemistry, communication theory, and applied mathematics. Two numerical methods are implemented for three well-studied partial differential equations of the MC field where advection, diffusion, and the reaction are taken into account. Numerical results for test cases in one and three dimensions are presented and discussed in detail. Conclusions and essential analytical and numerical future directions are then drawn.