Current Search: Burkardt, John V. (x)
Search results
 Title
 Approximating Nonlocal Diffusion Problems Using Quadrature Rules Generated by Radial Basis Functions.
 Creator

Lyngaas, Isaac Ron, Peterson, Janet S., Gunzburger, Max D., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Nonlocal models differ from traditional partial differential equation (PDE) models because they contain no spatial derivatives; instead an appropriate integral is used. Nonlocal models are especially useful in the case where there are issues calculating the spatial derivatives of a PDE model. In many applications (e.g., biological systems, flow through porous media) the observed rate of diffusion is not accurately modeled by the standard diffusion differential operator but rather exhibits so...
Show moreNonlocal models differ from traditional partial differential equation (PDE) models because they contain no spatial derivatives; instead an appropriate integral is used. Nonlocal models are especially useful in the case where there are issues calculating the spatial derivatives of a PDE model. In many applications (e.g., biological systems, flow through porous media) the observed rate of diffusion is not accurately modeled by the standard diffusion differential operator but rather exhibits socalled anomalous diffusion. Anomalous diffusion can be represented in a PDE model by using a fractional Laplacian operator in space whereas the nonlocal approach only needs to slightly modify its integral formulation to model anomalous diffusion. Anomalous diffusion is one such case where approximating the spatial derivative operator is a difficult problem. In this work, an approach for approximating standard and anomalous nonlocal diffusion problems using a new technique that utilizes radial basis functions (RBFs) is introduced and numerically tested. The typical approach for approximating nonlocal diffusion problems is to use a Galerkin formulation. However, the Galerkin formulation for nonlocal diffusion problems can often be difficult to compute efficiently and accurately especially for problems in multiple dimensions. Thus, we investigate the alternate approach of using quadrature rules generated by RBFs to approximate the nonlocal diffusion problem. This work will be split into three major parts. The first will introduce RBFs and give some examples of how they are used. This part will motivate our approach for using RBFs on the nonlocal diffusion problem. In the second part, we will derive RBFgenerated quadrature rules in one dimension and show they can be used to approximate nonlocal diffusion problems. The final part will address how the RBF quadrature approach can be extended to higher dimensional problems. Numerical test cases are shown for both the standard and anomalous nonlocal diffusion problems and compared with standard finite element approximations. Preliminary results show that the method introduced is viable for approximating nonlocal diffusion problems and that highly accurate approximations are possible using this approach.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Lyngaas_fsu_0071N_13512
 Format
 Thesis
 Title
 Modeling and Simulating Vortex Pinning and Transport Currents for High Temperature Superconductors.
 Creator

Sockwell, K. Chadwick (Kenneth Chadwick), Gunzburger, Max D., Peterson, Janet S., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of...
Show moreSockwell, K. Chadwick (Kenneth Chadwick), Gunzburger, Max D., Peterson, Janet S., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Superconductivity is a phenomenon characterized by two hallmark properties, zero electrical resistance and the Meissner effect. These properties give great promise to a new generation of resistance free electronics and powerful superconducting magnets. However this possibility is limited by the extremely low critical temperature the superconductors must operate under, typically close to 0K. The recent discovery of high temperature superconductors has brought the critical temperature closer to...
Show moreSuperconductivity is a phenomenon characterized by two hallmark properties, zero electrical resistance and the Meissner effect. These properties give great promise to a new generation of resistance free electronics and powerful superconducting magnets. However this possibility is limited by the extremely low critical temperature the superconductors must operate under, typically close to 0K. The recent discovery of high temperature superconductors has brought the critical temperature closer to room temperature than ever before, making the realization of room temperature superconductivity a possibility. Simulations of superconducting technology and materials will be necessary to usher in the new wave of superconducting electronics. Unfortunately these new materials come with new properties such as effects from multiple electron bands, as is the case for magnesium diboride. Moreover, we must consider that all high temperature superconductors are of a Type II variety, which possess magnetic tubes of flux, known as vortices. These vortices interact with transport currents, creating an electrical resistance through a process known as flux flow. Thankfully this process can be prevented by placing impurities in the superconductor, pinning the vortices, making vortex pinning a necessary aspect of our model. At this time there are no other models or simulations that are aimed at modeling vortex pinning, using impurities, in twoband materials. In this work we modify an existing GinzburgLandau model for twoband superconductors and add the ability to model normal inclusions (impurities) with a new approach which is unique to the twoband model. Simulations in an attempt to model the material magnesium diboride are also presented. In particular simulations of vortex pinning and transport currents are shown using the modified model. The qualitative properties of magnesium diboride are used to validate the model and its simulations. One main goal from the computational end of the simulations is to enlarge the domain size to produce more realistic simulations that avoid boundary pinning effects. In this work we also implement the numerical software library Trilinos in order to parallelize the simulation to enlarge the domain size. Decoupling methods are also investigated with a goal of enlarging the domain size as well. The OneBand GinzburgLandau model serves as a prototypical problem in this endeavor and the methods shown that enlarge the domain size can be easily implemented in the twoband model.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Sockwell_fsu_0071N_13577
 Format
 Thesis
 Title
 Numerical Analysis of Nonlocal Problems.
 Creator

Guan, Qingguang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Burkardt, John V., Wang, Xiaoqiang, Florida State University, College of Arts and Sciences, Department...
Show moreGuan, Qingguang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Burkardt, John V., Wang, Xiaoqiang, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

In this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the timedependent nonlocal diffusion and wave equations, formulated in the peridynamics setting. Initial and boundary data are given. For nonlocal diffusion equation, the time derivative is approximated using either an explicit Forward Euler, or implicit Backward Euler scheme. For nonlocal wave equation, we get the dispersion relations and use the Newmark...
Show moreIn this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the timedependent nonlocal diffusion and wave equations, formulated in the peridynamics setting. Initial and boundary data are given. For nonlocal diffusion equation, the time derivative is approximated using either an explicit Forward Euler, or implicit Backward Euler scheme. For nonlocal wave equation, we get the dispersion relations and use the Newmark method to discretize the equation. We have reformulated the standard timestep stability conditions, in light of the peridynamics formulation. Also we have obtained convergence results. Secondly, we consider the spacetime fractional diffusion equation which is used to model anomalous diffusion in physics. Finite difference, finite element and other methods are used to solve it. For finite difference method, the stability of the numerical schemes is well studied. However, for finite element method, we have not found the results for the stability of the θ schemes, especially for the explicit scheme. Here we get the stability and convergence results for all schemes with 0 ≤ θ ≤ 1. Thirdly, an obstacle problem for a nonlocal operator equation is considered; the operator is a nonlocal integral analogue of the Laplacian operator and, as a special case, reduces to the fractional Laplacian. In the analysis of classical obstacle problems for the Laplacian, the obstacle is taken to be a smooth function. For the nonlocal obstacle problem, obstacles are allowed to have jump discontinuities. We cast the nonlocal obstacle problem as a minimization problem wherein the solution is constrained to lie above the obstacle. We prove the existence and uniqueness of a solution in an appropriate function space. Then, the well posedness and convergence of finite element approximations are demonstrated. The results of numerical experiments are provided that illustrate the theoretical results and the differences between solutions of the nonlocal and local obstacle problems. Then we use sparse grid collocation, reduced basis and simplified reduced basis methods to solve nonlocal diffusion equation with random input data. Regularity of the solution and the convergence results for numerical methods are proved. The efficiency of these methods for solving the problem is investigated. As the radius of the spatial interaction zone changes, the computation cost varies due to the density of the stiffness matrix. This is quite different from local problems. Finally, the 1d nonlocal diffusion equation is solved by a continuous piecewiselinear collocation method using a uniform mesh. The time derivative is approximated using any of forward Euler, backward Euler, or CrankNicolson scheme. By developing a technique to deal with the singular integral, we are able to extend the method so that its validity is extended to include the case 1/2 ≤ s [less than] 1. We also derive stability conditions and convergence rates.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Guan_fsu_0071E_13425
 Format
 Thesis
 Title
 Improvement of a Tracer Correlation Problem with a NonIterative Limiter.
 Creator

Lopez, Nicolas A., Gunzburger, Max D., Burkardt, John V., Peterson, Janet C., Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

A functional relation between two chemical species puts observational constraints on attempts to model the atmosphere. For example, adequate representation of these relations is important when modeling the depletion of stratospheric ozone by nitrous oxide. Previous work has shown a case where a linear functional relation is not preserved in the tracer transport scheme of the Higher Order Methods Modeling Environment (HOMME), which is the spectral element dynamics core used by the Community...
Show moreA functional relation between two chemical species puts observational constraints on attempts to model the atmosphere. For example, adequate representation of these relations is important when modeling the depletion of stratospheric ozone by nitrous oxide. Previous work has shown a case where a linear functional relation is not preserved in the tracer transport scheme of the Higher Order Methods Modeling Environment (HOMME), which is the spectral element dynamics core used by the Community Atmosphere Model (CAM). Application of a certain simple tracer chemistry reaction before each model time step can test whether the scheme actually preserves linear tracer correlations (LCs) to machine precision. Using this method, we confirm previous results that, the implementation of the default shapepreserving filter of HOMME used in the transport scheme does not preserve LCs. However, since we prove that this limiter along with a few other limiter algorithms do in fact preserve LCs in exact arithmetic, we suggest that these limiter algorithms exacerbate the growth of roundoff error in elements where tracers have very different magnitudes. Nevertheless, we manage to put forth a limiting scheme that improves the tracer correlation problem. We also derive another new limiter that relies on multiplicative rescaling of nodal values within a given element. This algorithm does not rely on iterations for convergence and thus has the advantage of being more computationally efficient than the current default CAMSE limiter. Results also show that the default limiter does not always introduce the lowest amount of L₂ error, which contradicts its purpose, since it was derived to minimize error in the L₂ norm.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SU_Lopez_fsu_0071N_13470
 Format
 Thesis
 Title
 A Multiscale Implementation of Finite Element Methods for Nonlocal Models of Mechanics and Diffusion.
 Creator

Xu, Feifei, Gunzburger, Max D., Wang, Xiaoming, Burkardt, John V., Wang, Xiaoqiang, Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

The nonlocal models considered are free of spatial derivatives and thus are suitable for modeling problems with solutions exhibiting defects such as fractures in solids. Those models feature a horizon parameter that specifies the maximum extent of nonlocal interactions. A multiscale finite element implementation in one dimension and two dimensions of the nonlocal models is developed by taking advantage of the proven fact that, for smooth solutions, the nonlocal models reduce, as the horizon...
Show moreThe nonlocal models considered are free of spatial derivatives and thus are suitable for modeling problems with solutions exhibiting defects such as fractures in solids. Those models feature a horizon parameter that specifies the maximum extent of nonlocal interactions. A multiscale finite element implementation in one dimension and two dimensions of the nonlocal models is developed by taking advantage of the proven fact that, for smooth solutions, the nonlocal models reduce, as the horizon parameter tends to zero, to wellknown local partial differential equations models. The implementation features adaptive abrupt mesh refinement based on the detection of defects and resulting in an abrupt transition between refined elements that contain defects and unrefined elements that do not do so. Additional difficulties encountered in the implementation that are overcome are the design of accurate quadrature rules for stiffness matrix construction that are valid for any combination of the grid size and horizon parameter. As a result, the methodology developed can attain optimal accuracy at very modest additional costs relative to situations for which the solution is smooth. Portions of the methodology can also be used for the optimal approximation, by piecewise linear polynomials, of given functions containing discontinuities. Several numerical examples are provided to illustrate the efficacy of the multiscale methodology.
Show less  Date Issued
 2015
 Identifier
 FSU_2016SP_Xu_fsu_0071E_12974
 Format
 Thesis
 Title
 Reduced Order Modeling for a Nonlocal Approach to Anomalous Diffusion Problems.
 Creator

Witman, David, Gunzburger, Max D., Peterson, Janet C., Stagg, Scott, Shanbhag, Sachin, Burkardt, John V., Florida State University, College of Arts and Sciences, Department of...
Show moreWitman, David, Gunzburger, Max D., Peterson, Janet C., Stagg, Scott, Shanbhag, Sachin, Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

With the recent advances in using nonlocal approaches to approximate traditional partial differential equations(PDEs), a number of new research avenues have been opened that warrant further study. One such path, that has yet to be explored, is using reduced order techniques to solve nonlocal problems. Due to the interactions between the discretized nodes or particles inherent to a nonlocal model, the system sparsity is often significantly less than its PDE counterpart. Coupling a reduced...
Show moreWith the recent advances in using nonlocal approaches to approximate traditional partial differential equations(PDEs), a number of new research avenues have been opened that warrant further study. One such path, that has yet to be explored, is using reduced order techniques to solve nonlocal problems. Due to the interactions between the discretized nodes or particles inherent to a nonlocal model, the system sparsity is often significantly less than its PDE counterpart. Coupling a reduced order approach to a nonlocal problem would ideally reduce the computational cost without sacrificing accuracy. This would allow for the use of a nonlocal approach in large parameter studies or uncertainty quantification. Additionally, because nonlocal problems inherently have no spatial derivatives, solutions with jump discontinuities are permitted. This work seeks to apply reduced order nonlocal concepts to a variety of problem situations including anomalous diffusion, advection, the advectiondiffusion equation and solutions with spatial discontinuities. The goal is to show that one can use an accurate reduced order approximation to formulate a solution at a fraction of the cost of traditional techniques.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Witman_fsu_0071E_13130
 Format
 Thesis
 Title
 Using Deal.II to Solve Problems in Computational Fluid Dynamics.
 Creator

Bystricky, Lukas, Peterson, Janet C., Shanbhag, Sachin, Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

Finite element methods are a common tool to solve problems in computational fluid dynamics (CFD). This thesis explores the finite element package deal.ii and specific applications to incompressible CFD. Some notation and results from finite element theory are summarised, and a brief overview of some of the features of deal.ii is given. Following this, several CFD applications are presented, including the Stokes equations, the NavierStokes equations and the equations for Darcy flow in porous...
Show moreFinite element methods are a common tool to solve problems in computational fluid dynamics (CFD). This thesis explores the finite element package deal.ii and specific applications to incompressible CFD. Some notation and results from finite element theory are summarised, and a brief overview of some of the features of deal.ii is given. Following this, several CFD applications are presented, including the Stokes equations, the NavierStokes equations and the equations for Darcy flow in porous media. Comparison with benchmark problems are provided for the Stokes and NavierStokes equations and a case study looking at foam deformation is provided for Darcy flow. Code is provided where applicable.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Bystricky_fsu_0071N_13237
 Format
 Thesis