Numerical Analysis of First and Second Order Unconditional Energy Stable Schemes for Nonlocal Cahn-Hilliard and Allen-Cahn Equations

This PhD dissertation concentrates on the numerical analysis of a family of fully discrete, energy stable schemes for nonlocal Cahn-Hilliard and Allen-Cahn type equations, which are integro-partial differential equations (IPDEs). These two IPDEs -- along with the evolution equation from dynamical density functional theory (DDFT), which is a generalization of the nonlocal Cahn-Hilliard equation -- are used to model a variety of physical and biological processes such as crystallization, phase transformations, and tumor growth. This dissertation advances the computational state-of-the-art related to this field in the following main contributions: (I) We propose and analyze a family of two-dimensional unconditionally energy stable schemes for these IPDEs. Specifically, we prove that the schemes are (a) uniquely solvable, independent of time and space step sizes; (b) energy stable, independent of time and space step sizes; and (c) convergent, provided the time step sizes are sufficiently small. (II) We develop a highly efficient solver for schemes we propose. These schemes are semi-implicit and contain nonlinear implicit terms, which makes numerical solutions challenging. To overcome this difficulty, a nearly-optimally efficient nonlinear multigrid method is employed. (III) Via our numerical methods, we are able to simulate crystal nucleation and growth phenomena, with arbitrary crystalline anisotropy, with properly chosen parameters for nonlocal Cahn-Hilliard equation, in a very efficient and straightforward way. To our knowledge these contributions do not exist in any form in any of the previous works in the literature

Designing isoelectronic counterparts to layered group V semiconductors

In analogy to III-V compounds, which have significantly broadened the scope of group IV semiconductors, we propose IV-VI compounds as isoelectronic counterparts to layered group V semiconductors. Using {\em ab initio} density functional theory, we study yet unrealized structural phases of silicon mono-sulfide (SiS). We find the black-phosphorus-like $\alpha$-SiS to be almost equally stable as the blue-phosphorus-like $\beta$-SiS. Both $\alpha$-SiS and $\beta$-SiS monolayers display a significant, indirect band gap that depends sensitively on the in-layer strain. Unlike 2D semiconductors of group V elements with the corresponding nonplanar structure, different SiS allotropes show a strong polarization either within or normal to the layers. We find that SiS may form both lateral and vertical heterostructures with phosphorene at a very small energy penalty, offering an unprecedented tunability in structural and electronic properties of SiS-P compounds.Comment: 7 pages, 5 figure

Qualitative and Quantitative Comparisons between Spar and Semi-Submersible Platforms

Spars and semisubmersibles are two floating platforms commonly used in offshore deepwater exploration and production. However, currently lack of studies are available to comparatively evaluate the two platforms under similar cost range. This study compares the hydrodynamic motions of a truss spar and a semisubmersible under similar cost and wave environment. The platforms were numerically modelled using the radiation/diffraction software HydroSTAR. The response amplitude operators (RAO) were obtained for surge, heave and pitch respectively. The findings indicate the responses for the truss spar are generally lower than the semisubmersible, with the exception for the peak heave and pitch RAO. At 1000 m, the percentage difference of the peak surge RAO for spar and semisubmersible is 0.005%; heave RAO for semisubmersible is 34% of the spar’s value; and pitch RAO for semisubmersible is 79% of the spar’s value. Additionally, the semisubmersible achieved its peak RAO at a higher frequency for the heave and pitch. The findings proved that overall the spar’s dynamic responses are better for the majority of wave frequencies

The Nature of the Interlayer Interaction in Bulk and Few-Layer Phosphorus

An outstanding challenge of theoretical electronic structure is the description of van der Waals (vdW) interactions in molecules and solids. Renewed interest in resolving this is in part motivated by the technological promise of layered systems including graphite, transition metal dichalcogenides, and more recently, black phosphorus, in which the interlayer interaction is widely believed to be dominated by these types of forces. We report a series of quantum Monte Carlo (QMC) calculations for bulk black phosphorus and related few-layer phosphorene, which elucidate the nature of the forces that bind these systems and provide benchmark data for the energetics of these systems. We find a significant charge redistribution due to the interaction between electrons on adjacent layers. Comparison to density functional theory (DFT) calculations indicate not only wide variability even among different vdW corrected functionals, but the failure of these functionals to capture the trend of reorganization predicted by QMC. The delicate interplay of steric and dispersive forces between layers indicate that few-layer phosphorene presents an unexpected challenge for the development of vdW corrected DFT.Comment: 8 pages, 6 figure