A general finite element method is presented to solve the Euler equations in a Lagrangian reference frame. This FEM framework allows for separate arbitrarily high order representation of kinematic and thermodynamic variables. An accompanying hydrodynamics code written in Matlab is presented as a test-bed to experiment with various basis function choices. A wide range of basis function pairs are postulated and a few choices are developed further, including the bi-quadratic Q2-Q1d and Q2-Q2d elements. These are compared with a corresponding pair of low order bi-linear elements, traditional Q1-Q0 and sub-zonal pressure Q1-Q1d. Several test problems are considered including static convergence tests, the acoustic wave hourglass test, the Sod shocktube, the Noh implosion problem, the Saltzman piston, and the Sedov explosion problem. High order methods are found to offer faster convergence properties, the ability to represent curved zones, sharper shock capturing, and reduced shock-mesh interaction. They also allow for the straightforward calculation of thermodynamic gradients (for multi-physics calculations) and second derivatives of velocity (for monotonic slope limiters), and are more computationally efficient. The issue of shock ringing remains unresolved, but the method of hyperviscosity has been identified as a promising means of addressing this. Overall, the curvilinear finite elements presented in this thesis show promise for integration in a full hydrodynamics code and warrant further consideration.

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The solution of a single optimization problem often requires computationally-demanding evaluations; this is especially true in optimal design of engineering components and systems described by partial differential equations. We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of partial differential equations with affine parameter dependence. The critical ingredients of the method are: (i) reduced-basis techniques for dimension reduction in computational requirements; (ii) an "off-line/on-line" computational decomposition for the rapid calculation of outputs of interest and respective sensitivities in the limit of many queries; (iii) a posteriori error bounds for rigorous uncertainty and feasibility control; (iv) Interior Point Methods (IPMs) for efficient solution of the optimization problem; and (v) a trust-region Sequential Quadratic Programming (SQP) interpretation of IPMs for treatment of possibly non-convex costs and constraints. / Singapore-MIT Alliance (SMA)

This thesis studies the behavior of the Eulerian scheme, with "Wave Confinement" (WC), when propagating periodic waves. WC is a recently developed method that was derived from the scheme "vorticity confinement" used in fluid mechanics, and it efficiently solves the linear wave equation. This new method is applicable for numerous simulations such as radio wave propagation, target detection, cell phone and satellite communications. The WC scheme adds a nonlinear term to the discrete wave equation that adds stability with negative and positive diffusion, conserves integral quantities such as total amplitude and wave speed, and it allows wave propagation over long distances with minimal numerical diffusion, which contrasts to other numerical methods where wave propagation is affected by numerical dissipation. Previous studies have shown that WC propagates short pulses/surfaces as thin nonlinear solitary waves. In this thesis, a one-dimensional (1D) periodic wave is propagated by WC using the advection and wave equations. For the advection equation, the parameters and the initial condition (IC) used in WC are analyzed to establish for which conditions the method can be implemented. When the IC is a positive periodic wave, the converged solution consists of a series of hyperbolic secants where the number of cycles of the IC represents the number of hyperbolic secants. Waves with varying signs are analyzed by changing the wave confinement term. For this case, the converged solution is a series of positive and negative hyperbolic secants where each hyperbolic secant is represented by half cycle of the IC. For the wave equation, parameters and different IC's are studied to determine when WC is feasible. For positive periodic waves, the converged solution retains its sinusoidal shape and does not converge to a series of hyperbolic secants. The waves with varying signs, however, converge to a series of hyperbolic secants as seen for the advection equation. WC is stable for various periodic waves for both advection and wave equations, which shows WC is useful for numerically propagating periodic waveforms. Convergence depends on the wave number of the IC and on the parameters (convection speed, positive diffusion, negative diffusion) used in WC.

A unique mathematical tool is developed to deal with global optimization of a set of engineering problems. These include image processing, mechanical topology optimization, and optimal path planning in a variational framework, as well as some benchmark problems in parameter optimization. The optimization tool in these applications is based on the level set theory by which an evolving contour converges toward the optimum solution. Depending upon the application, the objective function is defined, and then the level set theory is used for optimization. Level set theory, as a member of active contour methods, is an extension of the steepest descent method in conventional parameter optimization to the variational framework. It intrinsically suffers from trapping in local solutions, a common drawback of gradient based optimization methods. In this thesis, methods are developed to deal with this drawbacks of the level set approach. By investigating the current global optimization methods, one can conclude that these methods usually cannot be extended to the variational framework; or if they can, the computational costs become drastically expensive. To cope with this complexity, a global optimization algorithm is first developed in parameter space and compared with the existing methods. This method is called "Spiral Bacterial Foraging Optimization" (SBFO) method because it is inspired by the aggregation process of a particular bacterium called, Dictyostelium Discoideum. Regardless of the real phenomenon behind the SBFO, it leads to new ideas in developing global optimization methods. According to these ideas, an effective global optimization method should have i) a stochastic operator, and/or ii) a multi-agent structure. These two properties are very common in the existing global optimization methods. To improve the computational time and costs, the algorithm may include gradient-based approaches to increase the convergence speed. This property is particularly available in SBFO and it is the basis on which SBFO can be extended to variational framework. To mitigate the computational costs of the algorithm, use of the gradient based approaches can be helpful. Therefore, SBFO as a multi-agent stochastic gradient based structure can be extended to multi-agent stochastic level set method. In three steps, the variational set up is formulated: i) A single stochastic level set method, called "Active Contours with Stochastic Fronts" (ACSF), ii) Multi-agent stochastic level set method (MSLSM), and iii) Stochastic level set method without gradient such as E-ARC algorithm. For image processing applications, the first two steps have been implemented and show significant improvement in the results. As expected, a multi agent structure is more accurate in terms of ability to find the global solution but it is much more computationally expensive. According to the results, if one uses an initial level set with enough holes in its topology, a single stochastic level set method can achieve almost the same level of accuracy as a multi-agent structure can obtain. Therefore, for a topology optimization problem for which a high level of calculations (at each iteration a finite element model should be solved) is required, only ACSF with initial guess with multiple holes is implemented. In some applications, such as optimal path planning, objective functions are usually very complicated; finding a closed-form equation for the objective function and its gradient is therefore impossible or sometimes very computationally expensive. In these situations, the level set theory and its extensions cannot be directly employed. As a result, the Evolving Arc algorithm that is inspired by "Electric Arc" in nature, is proposed. The results show that it can be a good solution for either unconstrained or constrained problems. Finally, a rigorous convergence analysis for SBFO and ACSF is presented that is new amongst global optimization methods in both parameter and variational framework.

This thesis is devoted to studies of initial-boundary value problems (IBVPs) for systems of partial differential equations (PDEs) arising from fluid mechanics modeling, especially for the compressible Euler equations with frictional damping, the Boussinesq equations, the Cahn-Hilliard equations and the incompressible density-dependent Navier-Stokes equations. The emphasis of this thesis is to understand the influences to the qualitative behavior of solutions caused by boundary effects and various dissipative mechanisms including damping, viscosity and heat diffusion.

Thesis (M.S.)--University of North Carolina Wilmington, 2009. / Title from PDF title page (February 16, 2010) Includes bibliographical references (p. 30)

Thesis (M.S.)--University of North Carolina Wilmington, 2009. / Title from PDF title page (February 16, 2010) Includes bibliographical references (p. 34-35)

In [2], Beals, Gaveau and Greiner find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This fundamental solution is based on the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces [4] and the Heisenberg group [6]. In this thesis, we look to generalize the work in [2] for a p-Laplace-type equation. After discovering that the "natural" generalization fails, we find two generalizations whose solutions are based on the fundamental solution to the p-Laplace equation.

The aim of this work is to use the Pfaffian technique, along with the Hirota bilinear method to construct different classes of exact solutions to various of generalized integrable nonlinear partial differential equations. Solitons are among the most beneficial solutions for science and technology, from ocean waves to transmission of information through optical fibers or energy transport along protein molecules. The existence of multi-solitons, especially three-soliton solutions, is essential for information technology: it makes possible undisturbed simultaneous propagation of many pulses in both directions. The derivation and solutions of integrable nonlinear partial differential equations in two spatial dimensions have been the holy grail in the field of nonlinear science since the late 1960s. The prestigious Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, as well as the ,Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable nonlinear partial differential equations in (1+1) and (2+1) dimensions, respectively. Do there exist Pfaffian and soliton solutions to generalized integrable nonlinear partial differential equations in (3+1) dimensions? In this dissertation, I obtained a set of explicit exact Wronskian, Grammian, Pfaffian and N-soliton solutions to the (3+1)-dimensional generalized integrable nonlinear partial differential equations, including a generalized KP equation, a generalized B-type KP equation, a generalized modified B-type KP equation, soliton equations of Jimbo-Miwa type, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. A set of sufficient conditions consisting of systems of linear partial differential equations involving free parameters and continuous functions is generated to guarantee that the Wronskian determinant or the Pfaffian solves these generalized equations. On the other hand, as part of this dissertation, bilinear Bäcklund transformations are formally derived for the (3+1)-dimensional generalized integrable nonlinear partial differential equations: a generalized B-type KP equation, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. As an application of the obtained Bäcklund transformations, a few classes of traveling wave solutions, rational solutions and Pfaffian solutions to the corresponding equations are explicitly computed. Also, as part of this dissertation, I would like to apply the Pfaffianization mechanism of Hirota and Ohta to extend the (3+1)-dimensional variable-coefficient soliton equation of Jimbo-Miwa type to coupled systems of nonlinear soliton equations, called Pfaffianized systems. Examples of the Wronskian, Grammian, Pfaffian and soliton solutions are explicitly computed. The numerical simulations of the obtained solutions are illustrated and plotted for different parameters involved in the solutions.