In this thesis, four different numerical techniques are implemented for the purpose of simulating resonant tunnelling diodes (RTDs). The chosen methods were: piecewise constant transfer matrix (TMM-C), piecewise linear transfer matrix (TMM-L), quantum transmitting boundary method (QTBM), and the Crank-Nicolson method (CN). The numerical methods converged compared with the known analytic simulation for plane waves tunnelling through a single barrier. To better represent the semiconductor-based RTDs, the effective mass approximation was adopted with accompanying modifications to the Hamiltonian operators to ensure the continuity of the wave function and its derivative. Using the Tsu-Esaki formula, the current density was calculated as a function of bias voltage for two different RTD devices. The numerically obtained current density was of the same order of magnitude as referenced experimental values but differed significantly enough to require better models if engineering applications are decided. The models in this thesis were able to display resonant tunnelling and a negative differential resistance (NDR), giving them plausible educational value.

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Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons are made between the coarse grid with refinement case and a fine grid without refinement case. Also, a non-negativity condition for two dimensions is derived to give a bound on the three major parameters: the diffusion coefficient and the spatial and time discretizations.

An unsteady, finite-rate, one-dimensional model has been developed for the analysis for gas-turbine combustors. The basis of the model is the one-dimensional, integral form of the conservation equations for multi-species, non-equilibrium, reacting mixtures. Special procedures were devised for the flow-division of the inlet flow into primary- and annular-flows, for both straight- and reverse-flow combustors. This allows the model to handle complete combustor configurations, which at present are beyond the reach of more sophisticated CFD tools. The model was validated with a steady-state analytical solution for a basic problem, and with steady-state results from a production code applied to a production combustor. Additional calculations show the ability of the code to predict blow-out due to rich and lean mixtures, and to predict the response of a combustor to perturbations in operating and boundary conditions. / Ph. D.

Many complexities arise when writing software for computational physics. The choice of underlying data structures, physics model representation, and numerical methods used for the solver all add to the overall complexity of a code and significantly affect the simulation speed and accuracy of the solution. This work has integrated multiple recently developed software tools into a unified framework called ELEMENTS. ELEMENTS contains tools to address the complexities of data representation and numerical methods implementation for computational physics applications. ELEMENTS consists of multiple software packages: Elements, MATAR, Swage, Geometry, and SLAM. MATAR is a performance portability and productivity implementation of data-oriented design that leverages KOKKOS for multi-architecture portability. MATAR's data-oriented design allows for highly efficient memory use through the use of contiguous memory allocation and access for optimal performance. The elements library contains the requisite mathematical functions for a wide range of numerical methods and high order field representation, including the Serendipity basis set that allows for a higher-order solution with fewer degrees of freedom than the more standard tensor product elements. Swage is a novel mesh class capable of representing all of the geometric entities required to implement low and high-order continuous and discontinuous Galerkin methods on unstructured hexahedral meshes as well as connectivity structures between the disparate index spaces. SLAM is a library for linear algebra solvers and tools for linking to external solver packages. Combining these tools allows for the research and development of novel methods for solving problems in computational physics. This work discusses the ELEMENTS package and reviews multiple numerical methods built using ELEMENTS.

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ABSTRACT The performance of aircraft engines can be significantly affected by the variety of foreign particles that are mixed into the air while operating under miscellaneous conditions. In particular, aircraft engines that operate in sandy or dusty conditions may fail within minutes of exposure to particle-laden flow due to foreign particle deposition on hot section components or erosion occurring on the compressor and turbine blades. For these reasons, the effect of sand ingestion on erosion, which may occur in the turbine and compressor blades, was studied in this master's thesis. In this master's thesis, the effect of sand ingestion on erosion on the M250 turboshaft engine's compressor blades will be investigated with the aid of numerical methods. In this study, we used the OpenFOAM software to solve the multiphase flow problem from the standpoint of finite control methods and the Eulerian-Lagrangian framework. The initial sand distribution conditions were taken from the Ph.D. thesis written by Olshefski, K. T. (2023) [1]. The compressor blade was modeled as 2D, which has a NACA 6510 profile shape, with a chord length of 63 mm. The results show that the leading edge and the suction side of the compressor, i.e. the upper half of the compressor, eroded more compared to the trailing edge, and the pressure side. Results also show that as the sand particle distribution becomes non-uniform the most eroded region shifts toward the trailing edge. In addition, for varying angles of attack, the region where the erosion occurs alters periodically. We observed that as the angle of attack increases, the eroded region shifts toward the trailing edge, but when the angle of attack is kept increasing the eroded region shifts back to the leading edge again. In conclusion, the non-uniformity of sand particle loading has a strong effect on the determination of the eroded regions. Furthermore, the variation of the angle of attack has a huge role in both the determination of eroded regions and the amount of eroded material. / Master of Science / GENERAL AUDIENCE ABSTRACT In this master's thesis, the effect of sand ingestion on compressor blade erosion was investigated with the help of numerical methods. The compressor is one of the vital parts of air-breathing engines such as turboshaft, turbofan, turbojet, and turboprop engines. Therefore, the erosion on the compressor blades may cause pressure surges, which could cause severe problems in the operation of aircraft or airplanes operating under dusty conditions. Historically, it is reported that a TransAmerican aircraft propelled by Alison T-56 engines lost two of its four engines after 3 to 4 minutes of exposure to volcanic ash while flying over Mt. St. Helens in 1980. Another example of the effects of sand ingestion is an MV-22 Osprey crash that happened during a training exercise in Hawaii, claiming the lives of two US Marines and injuring twenty other personnel in 2015. It was attributed that the cause of the fatal accident was the ingestion of dust that caused engine failure. Therefore, our intention in studying this field is to have an understanding of the regions of compressor blades that are vulnerable to erosion. In this master's thesis, numerical methods based on the finite volume method were used to obtain numerical solutions to estimate erosion on the compressor blade by utilizing OpenFOAM. We would like to recommend a nice OpenFOAM tutorial for those who are interested in applying numerical methods using OpenFOAM, taught by Jozsef Nagy accessible on YouTube, https://www.youtube.com/@OpenFOAMJozsefNagy. Also, for creating geometry and mesh generation of an airfoil for the use of OpenFOAM, we would like to recommend the tutorial presented by Ali Ikhsanul, accessible on YouTube via this link https://www.youtube.com/@aliikhsanul7982. These tutorial videos could help those who are interested in Openfoam but do not have much experience with Openfoam. The work in this master's thesis indicates that the leading edge of the compressor blade is more prone to be eroded than the trailing edge. In addition, it is shown that the eroded region distribution is highly dependent on the angle of attack of sand particles.

In this thesis we present TREVR (Tree-based Reverse Ray Tracing), a general algo- rithm for computing the radiation field, including absorption, in astrophysical sim- ulations. TREVR is designed to handle large numbers of sources and absorbers; it is based on a tree data structure and is thus suited to codes that use trees for their gravity or hydrodynamics solvers (e.g. Adaptive Mesh Refinement). It achieves com- putational speed while maintaining a specified accuracy via controlled lowering of resolution of both sources and rays from each source. TREVR computes the radiation field in O(N log(N)) time without absorption and O (Nlog(N)log(N)) time with absorption. These claims are substantiated by mathematically predicting and testing the algorithm’s general scaling. The scalings arise from merging sources of radiation according to an opening angle criterion and walking the tree structure to trace a ray to a depth that gives the chosen accuracy for absorption. The absorption-depth refinement criterion is unique to TREVR and is presented here for the first time. We provide a suite of tests demonstrating the algorithm’s ability to accurately compute fluxes, ionization fronts and shadows. Two novel test cases are presented here for the first time as part of this suite. / Thesis / Master of Science (MSc) / In this thesis we present TREVR (Tree-based Reverse Ray Tracing), a general method for computing the effects of of radiation in astrophysical simulations.

The problem of rough surface scattering and propagation over rough terrain in ducting environments has been receiving considerable attention in the literature. One popular method of modeling this problem is the parabolic wave equation (PWE) method. In this method, the Helmholtz wave equation is replaced by a PWE under the assumption of predominant forward propagation and scattering. The resulting PWE subjected to the appropriate boundary condition(s) is then solved, given an initial field distribution, using marching techniques such as the split-step Fourier algorithm. As is obvious from the assumption on which it is based, the accuracy of the PWE approximation deteriorates in situations involving appreciable scattering away from the near-forward direction, i.e. when the terrain under consideration is considerably rough. The backscattered field is neglected in all PWE-based models. An alternative and more rigorous method for modeling the problem under consideration is the boundary integral equation (BIE) method, which is formulated in two steps. The first step involves setting up an integral equation (the magnetic field integral equation, MFIE, or the electric field integral equation EFIE) governing currents induced on the rough surface by the incident field and solving for these currents numerically. The resulting currents are then used in the appropriate radiation integrals to calculate the field scattered by the surface everywhere in space. The BIE method accounts for all orders of multiple scattering on the rough surface and predicts the scattered field in all directions in space (including the backscattering direction) in an exact manner. In homogeneous media, the implementation of the BIE approach is straightforward since the kernel (Green's function or its normal derivative) which appears in the integral equation and the radiation integrals is well known. This is not the case, however, in inhomogeneous media (ducting environments) where the Green's function is not readily known. Due to this fact, there has been no attempt, up to our knowledge, at using the BIE (except under the parabolic approximation) to model the problem under consideration prior to the work presented in this thesis. In this thesis, a closed-form approximation of the Green's function for a two-dimensional ducting environment formed by the presence of a linear-square refractivity profile is derived using the asymptotic methods of stationary phase and steepest descents. This Green's function is then modified to more closely model the one associated with a physical ducting medium, in which the refractivity profile decreases up to a certain height, beyond which it becomes constant. This modified Green's function is then used in the BIE approach to study low grazing angle (LGA) propagation over rough surfaces in the aforementioned ducting environment. The numerical method used to solve the MFIE governing the surface currents is MOMI, which is a very robust and efficient method that does not require matrix storage or inversion. The proposed method is meant as a benchmark for people studying forward propagation over rough surfaces using the parabolic wave equation (PWE). Rough surface scattering results obtained via the PWE/split-step approach are compared to those obtained via the BIE/MOMI approach in ducting environments. These comparisons clearly show the shortcomings of the PWE/split-step approach. / Ph. D.