Covering Partial Cubes with Zones

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results

Discrete Fourier Analysis and Chebyshev Polynomials with G2G_2 Group

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type

Investigating the Effects of Topology on the Fracture and Failure Mechanisms of Low Density Metamaterials

Advances in additive manufacturing have enabled the creation of low density metamaterials with fine features and complex topographies. These new metamaterial topologies and size scales not previously possible broaden the spectrum of lightweight materials with unique properties that are advantageous in a variety of applications. There however is a lack of understanding of metamaterial failure and fracture behaviors. Studies tend to report only a few material properties rather than a comprehensive description of behavior. Due to this, there is a hesitancy to incorporate metamaterials into engineering designs despite proven remarkable properties. This work seeks to investigate in three parts the fracture and failure mechanisms controlling the deformation behavior of three different types of low-density metamaterials. The first part of the study explored increasing the fracture toughness of sheet-based metamaterials using designed porosity to redirect crack growth away from its original crack path to a less damaging direction. The crack was diverted into features in the metamaterial base topology, which served to toughen the material. It was identified that base material plays a role in the crack arrest mechanism activated. The added porosity was able to increase the fracture toughness of the metamaterial by a factor of three. The second part of the study calculated yield surfaces for common cellular material topologies that incorporates the anisotropy of tension, compression, and shear of cellular materials between different loading orientations. The shear component was the weakest of the topologies, atypical of monolithic material behavior. The third part of this study is currently on-going work to analyze the deformation of lattice metamaterials in compressive creep and compare the creep exponent and activation energy of the lattice to the base material as well as identify the mechanisms controlling the deformation of the lattice unit cell

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains

DEVELOPING ELECTROMAGNETIC AND PHOTONIC DEVICES BY USING ARTIFICIAL DIELECTRIC MATERIALS

Transformation-Optics (TO) is a new theoretical tool that allows for designing advanced electromagnetic and photonic devices. TO theory often prescribes material parameters for transformed media that cannot be found in nature. Metamaterials (MMs) were initially used for realization of TO-based devices. However, conventional MMs possess noticeable losses caused by their metallic parts that prevents their utilization in optical range. Alternatively, photonic crystals (PhCs) formed from arrays of low-loss all-dielectric elements can be good substitutes for building TO-prescribed devices. Metasurfaces (MSs) comprised from 2D arrays of dielectric resonators (DRs) have been found as other promising candidates for realizing flat and efficient devices. In our work, we explored incorporation of all-dielectric artificial media in invisibility cloaks, representing the most exciting TO application, wave collimators, and MSs. We studied associated electromagnetic and photonic phenomena and solved engineering problems met at the development of device prototypes. We designed and used anisotropic PhCs composed of rectangular lattice dielectric rod arrays to build up a cylindrical cloak medium realizing prescriptions of TO (Chapter 2). We also formed another cylindrical invisibility cloak by utilizing the self-collimation phenomenon in PhCs without considering TO prescriptions for turning the wave in the cloak medium (Chapter 3). Furthermore, we designed a wave collimator by employing high-anisotropic rectangular lattice dielectric rod arrays with unidirectional near-zero refractive indices (Chapter 4). Then, we studied the resonance and scattering responses of MSs composed of dielectric disks, while altering the periodicity of MSs. Our results demonstrated that periodicity of arrays has significant influence on defining the responses of MSs. (Chapter 5). Increasing lattice constants of dielectric MSs provided us with an opportunity to investigate interactions between lattice resonances (LRs) and dipolar electric and magnetic resonances that affected characteristics of MSs (Chapter 6). We analyzed the formation of Fano responses and wave interference processes in dense MSs to reveal the nature of electromagnetically induced transparency (EIT) that was detected at the frequency of electric dipolar resonance. (Chapter 7)

Negative definite spin filling and branched double covers

We investigate the negative definite spin fillings of branched double covers of alternating knots. We derive some obstructions for the existence of such fillings and find a characterization of special alternating knots based on them.Comment: 22 pages, 16 figure