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Browsing by Author "Warburton, Timothy"

Now showing 1 - 9 of 9
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    The Automation of Numerical Models of Coseismic Tsunamis
    Wiersma, Codi Allen (Virginia Tech, 2019-08-26)
    The use of tsunami models for applications of 'now-casting', which is the prediction of the present and near future behavior, has limited exploration, and could potentially be of significant usefulness. Tsunamis are most often caused by earthquakes in subduction zones, which generates coupled uplift and subsidence, and displaces the water column. The behavior of the fault failure is difficult to describe in the short term, often requiring seismic waveform inversion, which takes a length of time on the order of weeks to months to properly model, and is much too late for any use in a now-casting sense. To expedite this length of time, a series of source models are created with variable fault geometry behaviors, using fault parameters from Northern Oceanic and Atmospheric Administration's Short-term Inundation and Forecasting of Tsunamis (SIFT) database, in order to model a series of potential tsunami behaviors using the numerical modelling package, GeoClaw. The implementation of modeling could identify areas of interest for further study that are sensitive to fault failure geometry. Initial results show that by varying the geometry of sub-faults of a given earthquake, the resulting tsunami models behave fairly differently with different wave dispersion behavior, both in pattern and magnitude. While there are shortcomings of the potential geometries the code created in this study, and there are significant improvements that can be made, this study provides a good starting point into now-casting of tsunami models, with future iterations likely involving statistical probability in the fault failure geometries.
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    Continuum Kinetic Simulations of Plasma Sheaths and Instabilities
    Cagas, Petr (Virginia Tech, 2018-09-07)
    A careful study of plasma-material interactions is essential to understand and improve the operation of devices where plasma contacts a wall such as plasma thrusters, fusion devices, spacecraft-environment interactions, to name a few. This work aims to advance our understanding of fundamental plasma processes pertaining to plasma-material interactions, sheath physics, and kinetic instabilities through theory and novel numerical simulations. Key contributions of this work include (i) novel continuum kinetic algorithms with novel boundary conditions that directly discretize the Vlasov/Boltzmann equation using the discontinuous Galerkin method, (ii) fundamental studies of plasma sheath physics with collisions, ionization, and physics-based wall emission, and (iii) theoretical and numerical studies of the linear growth and nonlinear saturation of the kinetic Weibel instability, including its role in plasma sheaths. The continuum kinetic algorithm has been shown to compare well with theoretical predictions of Landau damping of Langmuir waves and the two-stream instability. Benchmarks are also performed using the electromagnetic Weibel instability and excellent agreement is found between theory and simulation. The role of the electric field is significant during nonlinear saturation of the Weibel instability, something that was not noted in previous studies of the Weibel instability. For some plasma parameters, the electric field energy can approach magnitudes of the magnetic field energy during the nonlinear phase of the Weibel instability. A significant focus is put on understanding plasma sheath physics which is essential for studying plasma-material interactions. Initial simulations are performed using a baseline collisionless kinetic model to match classical sheath theory and the Bohm criterion. Following this, a collision operator and volumetric physics-based source terms are introduced and effects of heat flux are briefly discussed. Novel boundary conditions are developed and included in a general manner with the continuum kinetic algorithm for bounded plasma simulations. A physics-based wall emission model based on first principles from quantum mechanics is self-consistently implemented and demonstrated to significantly impact sheath physics. These are the first continuum kinetic simulations using self-consistent, wall emission boundary conditions with broad applicability across a variety of regimes.
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    Design, Analysis, and Application of Immersed Finite Element Methods
    Guo, Ruchi (Virginia Tech, 2019-06-19)
    This dissertation consists of three studies of immersed finite element (IFE) methods for inter- face problems related to partial differential equations (PDEs) with discontinuous coefficients. These three topics together form a continuation of the research in IFE method including the extension to elasticity systems, new breakthroughs to higher degree IFE methods, and its application to inverse problems. First, we extend the current construction and analysis approach of IFE methods in the literature for scalar elliptic equations to elasticity systems in the vector format. In particular, we construct a group of low-degree IFE functions formed by linear, bilinear, and rotated Q1 polynomials to weakly satisfy the jump conditions of elasticity interface problems. Then we analyze the trace inequalities of these IFE functions and the approximation capabilities of the resulted IFE spaces. Based on these preparations, we develop a partially penalized IFE (PPIFE) scheme and prove its optimal convergence rates. Secondly, we discuss the limitations of the current approaches of IFE methods when we try to extend them to higher degree IFE methods. Then we develop a new framework to construct and analyze arbitrary p-th degree IFE methods. In this framework, each IFE function is the extension of a p-th degree polynomial from one subelement to the whole interface element by solving a local Cauchy problem on interface elements in which the jump conditions across the interface are employed as the boundary conditions. All the components in the analysis, including existence of IFE functions, the optimal approximation capabilities and the trace inequalities, are all reduced to key properties of the related discrete extension operator. We employ these results to show the optimal convergence of a discontinuous Galerkin IFE (DGIFE) method. In the last part, we apply the linear IFE methods in the literature together with the shape optimization technique to solve a group of interface inverse problems. In this algorithm, both the governing PDEs and the objective functional for interface inverse problems are discretized optimally by the IFE method regardless of the location of the interface in a chosen mesh. We derive the formulas for the gradients of the objective function in the optimization problem which can be implemented efficiently in the IFE framework through a discrete adjoint method. We demonstrate the properties of the proposed algorithm by applying it to three representative applications.
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    Finite-element simulations of interfacial flows with moving contact lines
    Zhang, Jiaqi (Virginia Tech, 2020-06-19)
    In this work, we develop an interface-preserving level-set method in the finite-element framework for interfacial flows with moving contact lines. In our method, the contact line is advected naturally by the flow field. Contact angle hysteresis can be easily implemented without explicit calculation of the contact angle or the contact line velocity, and meshindependent results can be obtained following a simple computational strategy. We have implemented the method in three dimensions and provide numerical studies that compare well with analytical solutions to verify our algorithm. We first develop a high-order numerical method for interface-preserving level-set reinitialization. Within the interface cells, the gradient of the level set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved. For the non-interface cells, we compute the gradient of the level set function by solving a Hamilton-Jacobi equation as a conservation law system using the discontinuous Galerkin method. This follows the work by Hu and Shu [SIAM J. Sci. Comput. 21 (1999) 660-690]. The missing constant for these cells is recovered using the continuity of the level set function while taking into account the characteristics. To treat highly distorted initial conditions, we develop a hybrid numerical flux that combines the Lax-Friedrichs flux and a penalty flux. Our method is accurate for non-trivial test cases and handles singularities away from the interface very well. When derivative singularities are present on the interface, a second-derivative limiter is designed to suppress the oscillations. At least (N + 1)th order accuracy in the interface cells and Nth order accuracy in the whole domain are observed for smooth solutions when Nth degree polynomials are used. Two dimensional test cases are presented to demonstrate superior properties such as accuracy, long-term stability, interface-preserving capability, and easy treatment of contact lines. We then develop a level-set method in the finite-element framework. The contact line singularity is removed by the slip boundary condition proposed by Ren and E [Phys. Fluids, vol. 19, p. 022101, 2007], which has two friction coefficients: βN that controls the slip between the bulk fluids and the solid wall and βCL that controls the deviation of the microscopic dynamic contact angle from the static one. The predicted contact line dynamics from our method matches the Cox theory very well. We further find that the same slip length in the Cox theory can be reproduced by different combinations of (βN; βCL). This combination leads to a computational strategy for mesh-independent results that can match the experiments. There is no need to impose the contact angle condition geometrically, and the dynamic contact angle automatically emerges as part of the numerical solution. With a little modification, our method can also be used to compute contact angle hysteresis, where the tendency of contact line motion is readily available from the level-set function. Different test cases, including code validation and mesh-convergence study, are provided to demonstrate the efficiency and capability of our method. Lastly, we extend our method to three-dimensional simulations, where an extension equation is solved on the wall boundary to obtain the boundary condition for level-set reinitializaiton with contact lines. Reinitialization of ellipsoidal interfaces is presented to show the accuracy and stability of our method. In addition, simulations of a drop on an inclined wall are presented that are in agreement with theoretical results.
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    Higher Order Immersed Finite Element Methods for Interface Problems
    Meghaichi, Haroun (Virginia Tech, 2024-05-17)
    In this dissertation, we provide a unified framework for analyzing immersed finite element methods in one spatial dimension, and we design a new geometry conforming IFE space in two dimensions with optimal approximation capabilities, alongside with applications to the elliptic interface problem and the hyperbolic interface problem. In the first part, we discuss a general m-th degree IFE space for one dimensional interface problems with many polynomial-like properties, then we develop a general framework for obtaining error estimates for the IFE spaces developed for solving a variety of interface problems, including but not limited to, the elliptic interface problem, the Euler-Bernoulli beam interface problem, the parabolic interface problem, the transport interface problem, and the acoustic interface problem. In the second part, we develop a new m-th degree finite element space based on the differential geometry of the interface to solve interface problems in two spatial dimensions. The proposed IFE space has optimal approximation capabilities, easy to construct, and the IFE functions satisfy the interface conditions exactly. We provide several numerical examples to demonstrate that the IFE space yields optimally converging solutions when applied to the elliptic interface problem and the hyperbolic interface problem with a symmetric interior penalty discontinuous Galerkin formulation.
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    Model Development and Monte-Carlo Methods for the Simulation and Analysis of Coastal Impacts of Barrier Island Breach During Hurricanes
    Jeffries, Catherine Renae (Virginia Tech, 2024-05-07)
    Barrier islands can protect the mainland from flooding during storms through reduction of storm surge and dissipation of storm generated wave energy. However, the protective capability is reduced when barrier islands breach and a direct hydrodynamic connection between the water bodies on both sides of the barrier island is established. Breaching of barrier islands during large storm events is complicated, involving nonlinear processes that connect water, sediment transport, dune height, and island width among other factors. In order to assess the impacts barrier island breaching has on flooding on the mainland, we modified a storm surge model, GeoClaw, to impose a Gaussian bell-curve on the barrier island that opens during a hurricane simulation and deepened over time. We added a new method of generating storm surge with storm forcing inputs in the form of wind and pressure fields to expand GeoClaw's current utilization of best track information so that storm forcing from planetary boundary layer models can also be utilized in simulations. We created a statistical method to assess the sensitivity of mainland storm-surge to barrier island breaching by randomizing the location, time, and extent of a breach event across the barrier island at Moriches, NY. My results show that total mainland inundation is affected by the changes in location, size, timing and numbers of breaches. Total inundation has a logarithmic relationship with total breach area and breach location is an important predictor of inundation and bay surge. The insights from this study can help prepare shoreline communities for the differing ways that breaching affects the mainland coastline. The model updates created can also allow others to use this framework to study differing regions. Understanding which mainland locations are vulnerable to breaching, planners and coastal engineers can design interventions to reduce the likelihood of a breach occurring in areas adjacent to high flood risk.
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    Multimethods for the Efficient Solution of Multiscale Differential Equations
    Roberts, Steven Byram (Virginia Tech, 2021-08-30)
    Mathematical models involving ordinary differential equations (ODEs) play a critical role in scientific and engineering applications. Advances in computing hardware and numerical methods have allowed these models to become larger and more sophisticated. Increasingly, problems can be described as multiphysics and multiscale as they combine several different physical processes with different characteristics. If just one part of an ODE is stiff, nonlinear, chaotic, or rapidly-evolving, this can force an expensive method or a small timestep to be used. A method which applies a discretization and timestep uniformly across a multiphysics problem poorly utilizes computational resources and can be prohibitively expensive. The focus of this dissertation is on "multimethods" which apply different methods to different partitions of an ODE. Well-designed multimethods can drastically reduce the computation costs by matching methods to the individual characteristics of each partition while making minimal concessions to stability and accuracy. However, they are not without their limitations. High order methods are difficult to derive and may suffer from order reduction. Also, the stability of multimethods is difficult to characterize and analyze. The goals of this work are to develop new, practical multimethods and to address these issues. First, new implicit multirate Runge–Kutta methods are analyzed with a special focus on stability. This is extended into implicit multirate infinitesimal methods. We introduce approaches for constructing implicit-explicit methods based on Runge–Kutta and general linear methods. Finally, some unique applications of multimethods are considered including using surrogate models to accelerate Runge–Kutta methods and eliminating order reduction on linear ODEs with time-dependent forcing.
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    A Parallel Aggregation Algorithm for Inter-Grid Transfer Operators in Algebraic Multigrid
    Garcia Hilares, Nilton Alan (Virginia Tech, 2019-09-13)
    As finite element discretizations ever grow in size to address real-world problems, there is an increasing need for fast algorithms. Nowadays there are many GPU/CPU parallel approaches to solve such problems. Multigrid methods can be used to solve large-scale problems, or even better they can be used to precondition the conjugate gradient method, yielding better results in general. Capabilities of multigrid algorithms rely on the effectiveness of the inter-grid transfer operators. In this thesis we focus on the aggregation approach, discussing how different aggregation strategies affect the convergence rate. Based on these discussions, we propose an alternative parallel aggregation algorithm to improve convergence. We also provide numerous experimental results that compare different aggregation approaches, multigrid methods, and conjugate gradient iteration counts, showing that our proposed algorithm performs better in serial and parallel.
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    Time Integration Methods for Large-scale Scientific Simulations
    Glandon Jr, Steven Ross (Virginia Tech, 2020-06-26)
    The solution of initial value problems is a fundamental component of many scientific simulations of physical phenomena. In many cases these initial value problems arise from a method of lines approach to solving partial differential equations, resulting in very large systems of equations that require the use of numerical time integration methods to solve. Many problems of scientific interest exhibit stiff behavior for which implicit methods are favorable, however standard implicit methods are computationally expensive. They require the solution of one or more large nonlinear systems at each timestep, which can be impractical to solve exactly and can behave poorly when solved approximately. The recently introduced ``lightly-implicit'' K-methods seek to avoid this issue by directly coupling the time integration methods with a Krylov based approximation of linear system solutions, treating a portion of the problem implicitly and the remainder explicitly. This work seeks to further two primary objectives: evaluation of these K-methods in large-scale parallel applications, and development of new linearly implicit methods for contexts where improvements can be made. To this end, Rosenbrock-Krylov methods, the first K-methods, are examined in a scalability study, and two new families of time integration methods are introduced: biorthogonal Rosenbrock-Krylov methods, and linearly implicit multistep methods. For the scalability evaluation of Rosenbrock-Krylov methods, two parallel contexts are considered: a GPU accelerated model and a distributed MPI parallel model. In both cases, the most significant performance bottleneck is the need for many vector dot products, which require costly parallel reduce operations. Biorthogonal Rosenbrock-Krylov methods are an extension of the original Rosenbrock-Krylov methods which replace the Arnoldi iteration used to produce the Krylov approximation with Lanczos biorthogonalization, which requires fewer vector dot products, leading to lower overall cost for stiff problems. Linearly implicit multistep methods are a new family of implicit multistep methods that require only a single linear solve per timestep; the family includes W- and K-method variants, which admit arbitrary or Krylov based approximations of the problem Jacobian while maintaining the order of accuracy. This property allows for a wide range of implementation optimizations. Finally, all the new methods proposed herein are implemented efficiently in the MATLODE package, a Matlab ODE solver and sensitivity analysis toolbox, to make them available to the community at large.