Current Search: Gunzburger, Max D. (x)
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 Title
 Centroidal Voronoi Tessellations for Mesh Generation: from Uniform to Anisotropic Adaptive Triangulations.
 Creator

Nguyen, Hoa V., Gunzburger, Max D., ElAzab, Anter, Peterson, Janet, Wang, Xiaoming, Wang, Xiaoqiang, Department of Mathematics, Florida State University
 Abstract/Description

Mesh generation in regions in Euclidean space is a central task in computational science, especially for commonly used numerical methods for the solution of partial differential equations (PDEs), e.g., finite element and finite volume methods. Mesh generation can be classified into several categories depending on the element sizes (uniform or nonuniform) and shapes (isotropic or anisotropic). Uniform meshes have been well studied and still find application in a wide variety of problems....
Show moreMesh generation in regions in Euclidean space is a central task in computational science, especially for commonly used numerical methods for the solution of partial differential equations (PDEs), e.g., finite element and finite volume methods. Mesh generation can be classified into several categories depending on the element sizes (uniform or nonuniform) and shapes (isotropic or anisotropic). Uniform meshes have been well studied and still find application in a wide variety of problems. However, when solving certain types of partial differential equations for which the solution variations are large in some regions of the domain, nonuniform meshes result in more efficient calculations. If the solution changes more rapidly in one direction than in others, nonuniform anisotropic meshes are preferred. In this work, first we present an algorithm to construct uniform isotropic meshes and discuss several mesh quality measures. Secondly we construct an adaptive method which produces nonuniform anisotropic meshes that are well suited for numerically solving PDEs such as the convection diffusion equation. For the uniform Delaunay triangulation of planar regions, we focus on how one selects the positions of the vertices of the triangulation. We discuss a recently developed method, based on the centroidal Voronoi tessellation (CVT) concept, for effecting such triangulations and present two algorithms, including one new one, for CVTbased grid generation. We also compare several methods, including CVTbased methods, for triangulating planar domains. Furthermore, we define several quantitative measures of the quality of uniform grids. We then generate triangulations of several planar regions, including some having complexities that are representative of what one may encounter in practice. We subject the resulting grids to visual and quantitative comparisons and conclude that all the methods considered produce highquality uniform isotropic grids and that the CVTbased grids are at least as good as any of the others. For more general grid generation settings, e.g., nonuniform and/or anistropic grids, such quantitative comparisons are much more difficult, if not impossible, to either make or interpret. This motivates us to develop CVTbased adaptive nonuniform anisotropic mesh refinement in the context of solving the convectiondiffusion equation with emphasis on convectiondominated problems. The challenge in the numerical approximation of this equation is due to large variations in the solution over small regions of the physical domain. Our method not only refines the underlying grid at these regions but also stretches the elements according to the solution variation. Three main ingredients are incorporated to improve the accuracy of numerical solutions and increase the algorithm's robustness and efficiency. First, a streamline upwind Petrov Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor. Our algorithm has been tested on a variety of 2dimensional examples. It is robust in detecting layers and efficient in resolving nonphysical oscillations in the numerical approximation.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd2616
 Format
 Thesis
 Title
 A Study of Shock Formation and Propagation in the ColdIon Model.
 Creator

Cheung, James, Gunzburger, Max D., Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

The central purpose of this thesis is to explore the behavior of the numerical solution of the Cold Ion model with shock forming conditions in one and two dimensions. In the one dimensional case, a comparison between the numerical solution of the Vlasov equation is made. It is observed that the ColdIon model is no longer representative of the coldion limit of the VlasovPoisson equation when a spike forms in the solution. It was found that the lack of a spike in the solution of the Cold...
Show moreThe central purpose of this thesis is to explore the behavior of the numerical solution of the Cold Ion model with shock forming conditions in one and two dimensions. In the one dimensional case, a comparison between the numerical solution of the Vlasov equation is made. It is observed that the ColdIon model is no longer representative of the coldion limit of the VlasovPoisson equation when a spike forms in the solution. It was found that the lack of a spike in the solution of the ColdIon model does not necessarily mean that a bifurcation has not formed in the solution of the VlasovPoisson equation. It was also determined that the spike present in the solution of the one dimensional problem appears again in the two dimensional simulation. The findings presented in this thesis opens up the question of determining which initial and boundary conditions of the ColdIon model causes a shock to form in the solution.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9158
 Format
 Thesis
 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
 Efficient and Accurate Numerical Schemes for Long Time Statistical Properties of the Infinite Prandtl Number Model for Convection.
 Creator

Woodruff, Celestine, Wang, Xiaoming, Sang, QingXiang Amy, Case, Bettye Anne, Ewald, Brian D., Gunzburger, Max D., Florida State University, College of Arts and Sciences,...
Show moreWoodruff, Celestine, Wang, Xiaoming, Sang, QingXiang Amy, Case, Bettye Anne, Ewald, Brian D., Gunzburger, Max D., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show...
Show moreIn our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show numerically that our schemes align with current knowledge of the model's characteristics at low Rayleigh numbers.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Woodruff_fsu_0071E_12813
 Format
 Thesis
 Title
 High Order LongTime Accurate Methods for the StokesDarcy System and Uncertainty Quantification of Contaminant Transport.
 Creator

Sun, Dong, Wang, Xiaoming, Gunzburger, Max D., Wang, Xiaoqiang, Ewald, Brian D., Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of...
Show moreSun, Dong, Wang, Xiaoming, Gunzburger, Max D., Wang, Xiaoqiang, Ewald, Brian D., Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The dissertation includes two parts. The first part consists of designing and analyzing high order longtime accurate numerical methods for StokesDarcy system. We propose second and thirdorder efficient and longtime accurate numerical methods, called IMplicitEXplicit methods (IMEX) for the coupled StokesDarcy system. Although the original continuum StokesDarcy PDE system is fully coupled, our algorithm is capable of decoupling the system into two subsystems so that a single Stokes and...
Show moreThe dissertation includes two parts. The first part consists of designing and analyzing high order longtime accurate numerical methods for StokesDarcy system. We propose second and thirdorder efficient and longtime accurate numerical methods, called IMplicitEXplicit methods (IMEX) for the coupled StokesDarcy system. Although the original continuum StokesDarcy PDE system is fully coupled, our algorithm is capable of decoupling the system into two subsystems so that a single Stokes and a single Darcy system can be computed in a parallel fashion without iteration. All the schemes we proposed are proven to be unconditionally stable and longtime stable. The bound on the error is uniformintime, which is among the first of this kind for second and thirdorder methods of StokesDarcy system. Error estimates for the second order BackwardDifferentiation scheme are proved. The second part concerns the Uncertainty of Quantification (UQ) of the contaminant transport. We compute the convectiondiffusion equation with Streamline Upwind PetrovGalerkin (SUPG) method. The quantity of interest is acquired using Monte Carlo and Sparse Grid methods in order to study the sensitivity with respect to the random parameters.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9692
 Format
 Thesis
 Title
 ReducedOrder Modeling of Reactive Solute Transport for AdvectionDominated Problems with Nonlinear Kinetic Reactions.
 Creator

McLaughlin, Benjamin R. S., Peterson, Janet S., Ye, Ming, Duke, D. W. (Dennis W.), Gunzburger, Max D., Shanbhag, Sachin, Florida State University, College of Arts and Sciences,...
Show moreMcLaughlin, Benjamin R. S., Peterson, Janet S., Ye, Ming, Duke, D. W. (Dennis W.), Gunzburger, Max D., Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Groundwater is a vital natural resource, and our ability to protect and manage this resource efficiently and effectively relies heavily on our ability to perform reliable and accurate computer modeling and simulation of subsurface systems. This frequently raises research questions involving parameter estimation and uncertainty quantification, which are often prohibitively expensive to answer using standard highdimensional computational models. We have previously demonstrated the ability to...
Show moreGroundwater is a vital natural resource, and our ability to protect and manage this resource efficiently and effectively relies heavily on our ability to perform reliable and accurate computer modeling and simulation of subsurface systems. This frequently raises research questions involving parameter estimation and uncertainty quantification, which are often prohibitively expensive to answer using standard highdimensional computational models. We have previously demonstrated the ability to replace the highdimensional models used to solve linear, uncoupled, diffusiondominated multispecies reactive transport systems with lowdimension approximations using reduced order modeling (ROM) based on proper orthogonal decomposition (POD). In this work, we seek to apply ROM to more general reactive transport systems, where the reaction terms may be nonlinear, mathematical models may be coupled, and the transport may be advectiondominated. We discuss the use of operator splitting, which is prevalent in the reactive transport field, to simplify the computation of complex systems of reactions in the transport model. We also discuss the use of some stabilization methods which have been developed in the computational science community to treat advectiondominated transport problems. We present a method by which we are able to incorporate stabilization and operator splitting together in the finite element setting. We go on to develop methods for implementing both operator splitting and stabilization in the ROM setting, as well as for incorporating both of them together within the ROM framework. We present numerical results which establish the ability of this new approach to produce accurate approximations with a significant reduction in computational cost, and we demonstrate the application of this method to a more realistic reactive transport problem involving bioremediation.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9649
 Format
 Thesis
 Title
 Investigating Vesicle Adhesions Using Multiple Phase Field Functions.
 Creator

Gu, Rui, Wang, Xiaoqiang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Ye, Ming, Florida State University, College of Arts and Sciences, Department of Scientific...
Show moreGu, Rui, Wang, Xiaoqiang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Ye, Ming, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

We construct a phase field model for simulating the adhesion of a cell membrane to a substrate. The model features two phase field functions which are used to simulate the membrane and the substrate. An energy model is defined which accounts for the elastic bending energy and the contact potential energy as well as, through a penalty method, vesicle volume and surface area constraints. Numerical results are provided to verify our model and to provide visual illustrations of the interactions...
Show moreWe construct a phase field model for simulating the adhesion of a cell membrane to a substrate. The model features two phase field functions which are used to simulate the membrane and the substrate. An energy model is defined which accounts for the elastic bending energy and the contact potential energy as well as, through a penalty method, vesicle volume and surface area constraints. Numerical results are provided to verify our model and to provide visual illustrations of the interactions between a lipid vesicle and substrates having complex shapes. Examples are also provided for the adhesion process in the presence of gravitational and point pulling forces. A comparison with experimental results demonstrates the effectiveness of the two phase field approach. Similarly to simulating vesiclesubstrate adhesion, we construct a multiphasefield model for simulating the adhesion between two vesicles. Two phase field functions are introduced to simulate each of the two vesicles. An energy model is defined which accounts for the elastic bending energy of each vesicle and the contact potential energy between the two vesicles; the vesicle volume and surface area constraints are imposed using a penalty method. Numerical results are provided to verify the efficacy of our model and to provide visual illustrations of the different types of contact. The method can be adjusted to solve endocytosis problems by modifying the bending rigidity coefficients of the two elastic bending energies. The method can also be extended to simulate multicell adhesions, one example of which is erythrocyte rouleaux. A comparison with laboratory observations demonstrates the effectiveness of the multiphase field approach. Coupled with fluid, we construct a phase field model for simulating vesiclevessel adhesion in a flow. Two phase field functions are introduced to simulate the vesicle and vessel respectively. The fluid is modeled and confined inside the tube by a phase field coupled NavierStokes equation. Both vesicle and vessel are transported by fluid flow inside our computational domain. An energy model regarding the comprehensive behavior of vesiclefluid interaction, vesselfluid interaction, vesiclevessel adhesion is defined. The vesicle volume and surface area constraints are imposed using a penalty method, while the vessel elasticity is modeled under Hooke's Law. Numerical results are provided to verify the efficacy of our model and to demonstrate the effectiveness of our fluidcoupled vesicle vessel adhesion phase field approach by comparison with laboratory observations.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Gu_fsu_0071E_12873
 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
 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
 Peridynamic Multiscale Models for the Mechanics of Materials: Constitutive Relations, Upscaling from Atomistic Systems, and Interface Problems.
 Creator

Seleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, ElAzab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific...
Show moreSeleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, ElAzab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific Computing, Florida State University
Show less  Abstract/Description

This dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the statebased peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models,...
Show moreThis dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the statebased peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models, establishing a hierarchy that reveals that some models are special cases of others. Furthermore, this allows for the modulation of the strength of non local interactions, even for a fixed radius of interactions between material points in the peridynamics model. The multiscale aspect of peridynamics is demonstrated through its connections to molecular dynamics. Using higherorder gradient models, it is shown that peridynamics can be viewed as an upscaling of molecular dynamics, preserving the relevant dynamics under appropriate choices of length scales. The statebased peridynamic model is shown to be appropriate for the description of multiscale and multimodel systems. A formulation for nonlocal interface problems involving scalar fields is presented, and derivations of non local transmission conditions are derived. Specializations that describe local, non local, and local/non local transmission conditions are considered. Moreover, the convergence of the non local transmission conditions to their classical local counterparts is shown. In all cases, results are illustrated by numerical experiments.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd0273
 Format
 Thesis
 Title
 Comparison of Different Noise Forcings, Regularization of Noise, and Optimal Control for the Stochastic NavierStokes Equations.
 Creator

Zhao, Wenju, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Quaife, Bryan, Huang, Chen (Professor of Scientific Computing), Florida State University, College of Arts and...
Show moreZhao, Wenju, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Quaife, Bryan, Huang, Chen (Professor of Scientific Computing), Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Stochastic NavierStokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic NavierStokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic NavierStokes equations perturbed by a large range of random noises in time and...
Show moreStochastic NavierStokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic NavierStokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic NavierStokes equations perturbed by a large range of random noises in time and space; effective Martingale regularized methods for the stochastic NavierStokes equations with additive noises; and the stochastic NavierStokes equations constrained stochastic boundary optimal control problems. We systemically provide numerical simulation methods for the stochastic NavierStokes equations with different types of noises. The noises are classified as colored or white based on their autocovariance functions. For each type of noise, we construct a representation and a simulation method. Numerical examples are provided to illustrate our schemes. Comparisons of the influence of different noises on the solution of the NavierStokes system are presented. To improve the simulation accuracy, we impose a Martingale correction regularized method for the stochastic NavierStokes equations with additive noise. The original systems are split into two parts, a linear stochastic Stokes equations with Martingale solution and a stochastic modified NavierStokes equations with smoother noise. In addition, a negative fractional Laplace operator is introduced to regularize the noise term. Stability and convergence of the pathwise modified NavierStokes equations are proved. Numerical simulations are provided to illustrate our scheme. Comparisons of nonregularized and regularized noises for the NavierStokes system are presented to further demonstrate the efficiency of our numerical scheme. As a consequence of the above work, we consider a stochastic optimal control problem constrained by the NavierStokes equations with stochastic Dirichlet boundary conditions. Control is applied only on the boundary and is associated with reduced regularity, compared to interior controls. To ensure the existence of a solution and the efficiency of numerical simulations, the stochastic boundary conditions are required to belong almost surely to H¹(∂D). To simulate the system, state solutions are approximated using the stochastic collocation finite element approach, and sparse grid techniques are applied to the boundary random field. Oneshot optimality systems are derived from Lagrangian functionals. Numerical simulations are then made, using a combination of Monte Carlo methods and sparse grid methods, which demonstrate the efficiency of the algorithm.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Zhao_fsu_0071E_14002
 Format
 Thesis
 Title
 Ensemble Proper Orthogonal Decomposition Algorithms for the Incompressible NavierStokes Equations.
 Creator

Schneier, Michael, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Erlebacher, Gordon, Huang, Chen, Florida State University, College of Arts and Sciences, Department of...
Show moreSchneier, Michael, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Erlebacher, Gordon, Huang, Chen, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the timedependent NavierStokes equations for which a recently developed ensemblebased method allows for...
Show moreThe definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the timedependent NavierStokes equations for which a recently developed ensemblebased method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. In this work we incorporate a proper orthogonal decomposition (POD) reducedorder model into the ensemblebased method to further reduce the computational cost; in total, three algorithms are developed. Initially a first order accurate in time scheme for low Reynolds number flows is considered. Next a second order algorithm useful for applications that require longterm time integration, such as climate and ocean forecasting is developed. Lastly, in order to extend this approach to convection dominated flows a model incorporating a POD spatial filter is presented. For all these schemes stability and convergence results for the ensemblebased methods are extended to the ensemblePOD schemes. Numerical results are provided to illustrate the theoretical stability and convergence results which have been proven.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Schneier_fsu_0071E_14687
 Format
 Thesis
 Title
 Overcoming Geometric Limitations in the Finite Element Method by Means of Polynomial Extension: Application to Second Order Elliptic Boundary Value and Interface Problems.
 Creator

Cheung, James, Gunzburger, Max D., Steinbock, Oliver, Bochev, Pavel B., Perego, Mauro, Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and...
Show moreCheung, James, Gunzburger, Max D., Steinbock, Oliver, Bochev, Pavel B., Perego, Mauro, Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

In this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using geometric manipulations to fit the discrete domain to the curved domain given by the continuous problem, we use polynomial extensions to enforce that a suitably constructed extension of the numerical solution matches the boundary condition given by the...
Show moreIn this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using geometric manipulations to fit the discrete domain to the curved domain given by the continuous problem, we use polynomial extensions to enforce that a suitably constructed extension of the numerical solution matches the boundary condition given by the continuous problem in the weak sense. This method is thus aptly named the Polynomial Extension Finite Element Method (PEFEM). Using this approach, we may approximate the solution of elliptic interface problems by enforcing that the extension of the solution on their respective subdomains matches weakly the continuity conditions prescribed by the continuous problem on a curved interface. This method is then called the Method of Virtual Interfaces (MVI), since, while the continuous interface exists in the context of the continuous problem, it is virtual in the context of its numerical approximation. The key benefits of this polynomial extension approach is that it is simple to implement and that it is optimally convergent with respect to the best approximation results given by interpolation. Theoretical analysis and computational results are presented.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Cheung_fsu_0071E_14328
 Format
 Thesis
 Title
 Mass Conserving HamiltonianStructurePreserving Reduced Order Modeling for the Rotating Shallow Water Equations Discretized by a Mimetic Spatial Scheme.
 Creator

Sockwell, K. Chad (Kenneth Chad), Gunzburger, Max D., Wahl, Horst, Peterson, Janet S., Quaife, Bryan, Huang, Chen, Florida State University, College of Arts and Sciences,...
Show moreSockwell, K. Chad (Kenneth Chad), Gunzburger, Max D., Wahl, Horst, Peterson, Janet S., Quaife, Bryan, Huang, Chen, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Ocean modeling, in a climatemodeling context, requires long timehorizons over global scales, which when combined with accurate resolution in time and space makes simulations very timeconsuming. While highresolution oceanmodeling simulations are still feasible on large HPC machines, performing uncertainty quantification or other many query applications at these resolutions is no longer feasible. Developing a more efficient model would allow for efficient uncertainty quantification, data...
Show moreOcean modeling, in a climatemodeling context, requires long timehorizons over global scales, which when combined with accurate resolution in time and space makes simulations very timeconsuming. While highresolution oceanmodeling simulations are still feasible on large HPC machines, performing uncertainty quantification or other many query applications at these resolutions is no longer feasible. Developing a more efficient model would allow for efficient uncertainty quantification, data assimilation, and spinup initializations. For these techniques to be feasible in practice, a faster model must be designed which can still attain sufficient accuracy. Techniques such as reduced order modeling produce an efficient reduced model based on existing highresolution simulation data. Models produced by these techniques provide a tremendous speedup at the cost of reduced accuracy. To offset this tradeoff, novel strategies are developed to retain as much accuracy as possible while still achieving tremendous speedups. Some of these methods improve accuracy by incorporating physical properties into the reduced model, leading to better solution quality. In this dissertation, a novel reduced order modeling method, the Hamiltonianstructurepreserving reduced order modeling method, will be derived and analyzed. The Hamiltonian structure is possessed by many physical systems and is directly related to energy conservation. This method produces a reduced model which retains the Hamiltonian structure of noncanonical Hamiltonian systems, which are the category of systems that many ocean models fall into. Error estimates are proven for the new method. The model is also be made to preserve linear invariants in the reduced model which are Casimirs. Casimirs are a class of special conserved quantities in the Hamiltonian Framework. For oceanmodeling, the Casimirs we consider are mass and potential vorticity. The new reduced model is proven to conserve both of these quantities. The model is also implemented in a special inner product derived from the Hamiltonian Framework, the approximate energy inner product. This special inner product not only improves the accuracy of the new method but also improves the accuracy of the traditional reduced order modeling method and leads to favorable analytical properties for problems with quadratic Hamiltonian functionals. The new method will be applied to the rotating shallow water equations, which act as a proxy to real ocean models, and compared to the traditional reduced order modeling method. Both energy conserving and forced testcases are considered where energy conservation, accuracy, and stability are investigated. Special techniques are also implemented to ensure that the new method is as efficient as possible.
Show less  Date Issued
 2019
 Identifier
 2019_Summer_Sockwell_fsu_0071E_15277
 Format
 Thesis
 Title
 Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data.
 Creator

Webster, Clayton G. (Clayton Garrett), Gunzburger, Max D., Gallivan, Kyle, Peterson, Janet, Tempone, Raul, Department of Mathematics, Florida State University
 Abstract/Description

The objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either ClenshawCurtis or Gaussian abscissas. Even in the presence of...
Show moreThe objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either ClenshawCurtis or Gaussian abscissas. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. The full tensor product spaces suffer from the curse of dimensionality since the dimension of the approximating space grows exponentially in the number of random variables. When this number is moderately large, we combine the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh equally in the solution; the latter approach is ideal when solving highly anisotropic problems depending on a relatively small number of random variables. We also include a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each problem. These procedures are very effective for the problems under study. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. Numerical examples illustrate the theoretical results and compare this approach with several others, including the standard Monte Carlo. For moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy is very efficient and superior to all examined methods. Due to the high cost of effecting each realization of the PDE this work also proposes the use of reducedorder models (ROMs) that assist in minimizing the cost of determining accurate statistical information about outputs from ensembles of realizations. We explore the use of ROMs, that greatly reduce the cost of determining approximate solutions, for determining outputs that depend on solutions of stochastic PDEs. One is then able to cheaply determine much larger ensembles, but this increase in sample size is countered by the lower fidelity of the ROM used to approximate the state. In the contexts of proper orthogonal decompositionbased ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd1223
 Format
 Thesis
 Title
 Analysis of Two Partial Differential Equation Models in Fluid Mechanics: Nonlinear Spectral EddyViscosity Model of Turbulence and InfinitePrandtlNumber Model of Mantle Convection.
 Creator

Saka, Yuki, Gunzburger, Max D., Wang, Xiaoming, ElAzab, Anter, Peterson, Janet, Wang, Xiaoqiang, Department of Mathematics, Florida State University
 Abstract/Description

This thesis presents two problems in the mathematical and numerical analysis of partial differential equations modeling fluids. The first is related to modeling of turbulence phenomena. One of the objectives in simulating turbulence is to capture the large scale structures in the flow without explicitly resolving the small scales numerically. This is generally accomplished by adding regularization terms to the NavierStokes equations. In this thesis, we examine the spectral viscosity models...
Show moreThis thesis presents two problems in the mathematical and numerical analysis of partial differential equations modeling fluids. The first is related to modeling of turbulence phenomena. One of the objectives in simulating turbulence is to capture the large scale structures in the flow without explicitly resolving the small scales numerically. This is generally accomplished by adding regularization terms to the NavierStokes equations. In this thesis, we examine the spectral viscosity models in which only the highfrequency spectral modes are regularized. The objective is to retain the largescale dynamics while modeling the turbulent fluctuations accurately. The spectral regularization introduces a host of parameters to the model. In this thesis, we rigorously justify effective choices of parameters. The other problem is related to modeling of the mantle flow in the Earth's interior. We study a model equation derived from the Boussinesq equation where the Prandtl number is taken to infinity. This essentially models the flow under the assumption of a large viscosity limit. The novelty in our problem formulation is that the viscosity depends on the temperature field, which makes the mathematical analysis nontrivial. Compared to the constant viscosity case, variable viscosity introduces a secondorder nonlinearity which makes the mathematical question of wellposedness more challenging. Here, we prove this using tools from the regularity theory of parabolic partial differential equations.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd2108
 Format
 Thesis
 Title
 SparseGrid Methods for Several Types of Stochastic Differential Equations.
 Creator

Zhang, Guannan, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet, Wang, Xiaoqiang, Ye, Ming, Webster, Clayton, Burkardt, John, Department of Scientific Computing, Florida...
Show moreZhang, Guannan, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet, Wang, Xiaoqiang, Ye, Ming, Webster, Clayton, Burkardt, John, Department of Scientific Computing, Florida State University
Show less  Abstract/Description

This work focuses on developing and analyzing novel, efficient sparsegrid algorithms for solving several types of stochastic ordinary/partial differential equations and corresponding inverse problem, such as parameter identification. First, we consider linear parabolic partial differential equations with random diffusion coefficients, forcing term and initial condition. Error analysis for a stochastic collocation method is carried out in a wider range of situations than previous literatures,...
Show moreThis work focuses on developing and analyzing novel, efficient sparsegrid algorithms for solving several types of stochastic ordinary/partial differential equations and corresponding inverse problem, such as parameter identification. First, we consider linear parabolic partial differential equations with random diffusion coefficients, forcing term and initial condition. Error analysis for a stochastic collocation method is carried out in a wider range of situations than previous literatures, including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both semidiscrete and fullydiscrete solutions. Second, we consider multidimensional backward stochastic differential equations driven by a vector of white noise. A sparsegrid scheme are proposed to discretize the target equation in the multidimensional timespace domain. In our scheme, the time discretization is conducted by the multistep scheme. In the multidimensional spatial domain, the conditional mathematical expectations derived from the original equation are approximated using sparsegrid GaussHermite quadrature rule and adaptive hierarchical sparsegrid interpolation. Error estimates are rigorously proved for the proposed fullydiscrete scheme for multidimensional BSDEs with certain types of simplified generator functions. Third, we investigate the propagation of input uncertainty through nonlocal diffusion models. Since the stochastic local diffusion equations, e.g. heat equations, have already been well studied, we are interested in extending the existing numerical methods to solve nonlocal diffusion problems. In this work, we use sparsegrid stochastic collocation method to solve nonlocal diffusion equations with colored noise and MonteCarlo method to solve the ones with white noise. Our numerical experiments show that the existing methods can achieve the desired accuracy in the nonlocal setting. Moreover, in the white noise case, the nonlocal diffusion operator can reduce the variance of the solution because the nonlocal diffusion operator has "smoothing" effect on the random field. At last, stochastic inverse problem is investigated. We propose sparsegrid Bayesian algorithm to improve the efficiency of the classic Bayesian methods. Using sparsegrid interpolation and integration, we construct a surrogate posterior probability density function and determine an appropriate alternative density which can capture the main features of the true PPDF to improve the simulation efficiency in the framework of indirect sampling. By applying this method to a groundwater flow model, we demonstrate its better accuracy when compared to bruteforce MCMC simulation results.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5298
 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
 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
 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
 The Impact of Microstructure on an Accurate Snow Scattering Parameterization at Microwave Wavelengths.
 Creator

Honeyager, Ryan Erick, Liu, Guosheng, Gunzburger, Max D., Ahlquist, Jon E., Ellingson, R. G., Wu, Zhaohua, Florida State University, College of Arts and Sciences, Department of...
Show moreHoneyager, Ryan Erick, Liu, Guosheng, Gunzburger, Max D., Ahlquist, Jon E., Ellingson, R. G., Wu, Zhaohua, Florida State University, College of Arts and Sciences, Department of Earth, Ocean and Atmospheric Science
Show less  Abstract/Description

High frequency microwave instruments are increasingly used to observe ice clouds and snow. These instruments are significantly more sensitive than conventional precipitation radar. This is ideal for analyzing icebearing clouds, for ice particles are tenuously distributed and have effective densities that are far less than liquid water. However, at shorter wavelengths, the electromagnetic response of ice particles is no longer solely dependent on particle mass. The shape of the ice particles...
Show moreHigh frequency microwave instruments are increasingly used to observe ice clouds and snow. These instruments are significantly more sensitive than conventional precipitation radar. This is ideal for analyzing icebearing clouds, for ice particles are tenuously distributed and have effective densities that are far less than liquid water. However, at shorter wavelengths, the electromagnetic response of ice particles is no longer solely dependent on particle mass. The shape of the ice particles also plays a significant role. Thus, in order to understand the observations of high frequency microwave radars and radiometers, it is essential to model the scattering properties of snowflakes correctly. Several research groups have proposed detailed models of snow aggregation. These particle models are coupled with computer codes that determine the particles' electromagnetic properties. However, there is a discrepancy between the particle model outputs and the requirements of the electromagnetic models. Snowflakes have countless variations in structure, but we also know that physically similar snowflakes scatter light in much the same manner. Structurally exact electromagnetic models, such as the discrete dipole approximation (DDA), require a high degree of structural resolution. Such methods are slow, spending considerable time processing redundant (i.e. useless) information. Conversely, when using techniques that incorporate too little structural information, the resultant radiative properties are not physically realistic. Then, we ask the question, what features are most important in determining scattering? This dissertation develops a general technique that can quickly parameterize the important structural aspects that determine the scattering of many diverse snowflake morphologies. A Voronoi bounding neighbor algorithm is first employed to decompose aggregates into welldefined interior and surface regions. The sensitivity of scattering to interior randomization is then examined. The loss of interior structure is found to have a negligible impact on scattering cross sections, and backscatter is lowered by approximately five percent. This establishes that detailed knowledge of interior structure is not necessary when modeling scattering behavior, and it also provides support for using an effective medium approximation to describe the interiors of snow aggregates. The Voronoi diagrambased technique enables the almost trivial determination of the effective density of this medium. A bounding neighbor algorithm is then used to establish a greatly improved approximation of scattering by equivalent spheroids. This algorithm is then used to posit a Voronoi diagrambased definition of effective density approach, which is used in concert with the Tmatrix method to determine singlescattering cross sections. The resulting backscatters are found to reasonably match those of the DDA over frequencies from 10.65 to 183.31 GHz and particle sizes from a few hundred micrometers to nine millimeters in length. Integrated error in backscatter versus DDA is found to be within 25% at 94 GHz. Errors in scattering crosssections and asymmetry parameters are likewise small. The observed crosssectional errors are much smaller than the differences observed among different particle models. This represents a significant improvement over established techniques, and it demonstrates that the radiative properties of dense aggregate snowflakes may be adequately represented by equalmass homogeneous spheroids. The present results can be used to supplement retrieval algorithms used by CloudSat, EarthCARE, Galileo, GPM and SWACR radars. The ability to predict the full range of scattering properties is potentially also useful for other particle regimes where a compact particle approximation is applicable.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Honeyager_fsu_0071E_13726
 Format
 Thesis
 Title
 An Optimal Control Problem for a TimeDependent GinzburgLandau Model of Superconductivity.
 Creator

Lin, Haomin, Peterson, Janet, Gunzburger, Max, Schwartz, Justin, Wang, Xiaoming, Horne, Rudy, Trenchea, Catalin, Department of Mathematics, Florida State University
 Abstract/Description

The motion of vortices in a Type II superconductor destroys the material's superconductivity because it dissipates energy and causes resistance. When a transport current is applied to a clean TypeII superconductor in the mixed state, the vortices will go into motion due to the induced Lorentz force and thus the superconductivity of the material is lost. However, various pinning mechanisms, such as normal inclusions, can inhibit vortex motion and pin the vortices to specific sites. We...
Show moreThe motion of vortices in a Type II superconductor destroys the material's superconductivity because it dissipates energy and causes resistance. When a transport current is applied to a clean TypeII superconductor in the mixed state, the vortices will go into motion due to the induced Lorentz force and thus the superconductivity of the material is lost. However, various pinning mechanisms, such as normal inclusions, can inhibit vortex motion and pin the vortices to specific sites. We demonstrate that the placement of the normal inclusion sites has an important effect on the largest electrical current that can be applied to the superconducting material while all vortices remain stationary. Here, an optimal control problem using a time dependent GinzburgLandau model is proposed to seek numerically the optimal locations of the normal inclusion sites. An analysis of this optimal control problem is performed, the existence of an optimal control solution is proved and a sensitivity system is given. We then derive a gradient method to solve this optimal control problem. Numerical simulations are performed and the results are presented and discussed.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd1334
 Format
 Thesis
 Title
 Using RBFGenerated Quadrature Rules to Solve Nonlocal Continuum Models.
 Creator

Lyngaas, Isaac R., Peterson, Janet S., Musslimani, Ziad H., Gunzburger, Max D., Quaife, Bryan, Shanbhag, Sachin, Florida State University, College of Arts and Sciences,...
Show moreLyngaas, Isaac R., Peterson, Janet S., Musslimani, Ziad H., Gunzburger, Max D., Quaife, Bryan, Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Recently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating timedependent nonlocal integrodifferential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due...
Show moreRecently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating timedependent nonlocal integrodifferential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due to the approximation of the nonlocal integral. In this work, we propose a novel method based on using radial basis functions to generate accurate quadrature rules for the nonlocal integral appearing in the model and then coupling this with a finite difference approximation to the timedependent terms. The viability of our method is demonstrated through various numerical tests on time dependent nonlocal diffusion, nonlocal anomalous diffusion, and nonlocal advection problems in one and two dimensions. In addition to nonlocal problems with continuous solutions, we modify our approach to handle problems with discontinuous solutions. We compare some numerical results with analogous finite element results and demonstrate that for an equivalent amount of computational work we obtain much higher rates of convergence.
Show less  Date Issued
 2018
 Identifier
 2018_Fall_Lyngaas_fsu_0071E_14886
 Format
 Thesis