An algorithm for two-dimensional mesh generation based on the pinwheel tiling

We propose a new two-dimensional meshing algorithm called PINW able to generate meshes that accurately approximate the distance between any two domain points by paths composed only of cell edges. This technique is based on an extension of pinwheel tilings proposed by Radin and Conway. We prove that the algorithm produces triangles of bounded aspect ratio. This kind of mesh would be useful in cohesive interface finite element modeling when the crack propagation pathis an outcome of a simulation process.Comment: Short version appears in Proceedings of 2004 International Meshing Roundtable at http://www.imr.sandia.go

Nondifferentiable energy minimization for cohesive fracture in a discontinuous Galerkin finite element framework

Until recently, most works on the computational modelling of fracture relied on a Newtonian mechanics approach, i.e., momentum balance equations describing the motion of the body along with fracture criteria describing the evolution of fractures. Robustness issues associated with this approach have been identified in the previous literature, several of which, as this thesis shows, due to the discontinuous dependence of stress field on the deformation field at the time of insertion of displacement discontinuities. Lack of continuity limits applicability of the models and undermines reliability of the numerical solutions. In particular, solutions often show non-convergent behaviour with time step refinement and exhibit nonphysical velocity fields and crack activation patterns. In addition, implicit time-stepping schemes, which are favoured in quasi-static and low-velocity problems, are challenging in such models. This is not a coincidence but a manifestation of algorithmic pitfalls of such methods. Continuity of stresses is in general hard to achieve in a computational model that employs a crack initiation criterion. Energy (variational) approaches to fracture have gained increased popularity in recent years. An energy approach has been shown to avoid introduction of a crack initiation criterion. The central idea of this model is the minimization of a mechanical energy functional, whose term representing the energy due to the cracks is a nondifferentiable function of the interface openings at zero opening displacement. A consequence of this formulation is that crack initiation happens automatically as a by-product of energy minimization. This avoids the complexities arising from the introduction of an extrinsic activation criterion but entails minimization of a nondifferentiable potential. The aim of this research is to develop robust and efficient computational algorithms for numerical implementation of the energy approach to cohesive fracture. Two computational algorithms have been proposed in a discontinuous Galerkin finite element framework, including a continuation algorithm which entails successive smooth approximations of the nondifferentiable functional and a block coordinate descent algorithm which uses generalized differential calculus for the treatment of nondifferentiability. These methods allow for a seamless transition from the uncracked to the cracked state, making possible the use of iterative solvers with implicit time-stepping, and completely sidestepping robustness issues of previous computational frameworks. A critical component of this work is validation of the robustness of the proposed numerical methods. Various numerical simulations are presented including time step and mesh size convergence studies and qualitative and quantitative comparison of simulations with experimental observations and theoretical findings. In addition, an energy-based hydro-mechanical model and computational algorithm is presented for hydraulic fracturing in impermeable media, which shows the crucial importance of continuity in multi-physics modelling. A search algorithm is developed on the basis of graph theory to identify the set of fluid-pressurized cracks among cracks in naturally fractured domains

Near-Linear-Time Deterministic Plane Steiner Spanners and TSP Approximation for Well-Spaced Point Sets

We describe an algorithm that takes as input n points in the plane and a parameter {\epsilon}, and produces as output an embedded planar graph having the given points as a subset of its vertices in which the graph distances are a (1 + {\epsilon})-approximation to the geometric distances between the given points. For point sets in which the Delaunay triangulation has bounded sharpest angle, our algorithm's output has O(n) vertices, its weight is O(1) times the minimum spanning tree weight, and the algorithm's running time is bounded by O(n \sqrt{log log n}). We use this result in a similarly fast deterministic approximation scheme for the traveling salesperson problem.Comment: Appear at the 24th Canadian Conference on Computational Geometry. To appear in CGT

Topological changes in 2D simplicial meshes for the simulation of fractures

A method for the introduction of strong discontinuities into a mesh will be developed. This method, applicable to a number of eXtended Finite Element Methods (XFEM) with intra-element strong discontinuities will be demonstrated with one specific method: the Generalized Cohesive Element (GCE) method. The algorithm utilizes a subgraph mesh representation which may insert the GCE either adaptively during the course of the analysis or a priori. Using this subgraphing algorithm, the insertion time is O(n) to the number of insertions. Numerical examples are presented demonstrating the advantages of the subgraph insertion method

Engineering photonic and plasmonic light emission enhancement

Thesis (Ph.D.)--Boston UniversitySemiconductor photonic devices are a rapidly maturing technology which currently occupy multi-billion dollar markets in the areas of LED lighting and optical data communication. LEDs currently demonstrate the highest luminous efficiency of any light source for general lighting. Long-haul optical data communication currently forms the backbone of the global communication network. Proper design of light management is required for photonic devices, which can increase the overall efficiency or add new device functionality. In this thesis, novel methods for the control of light propagation and confinement are developed for the use in integrated photonic devices. The first part of this work focuses on the engineering of field confinement within deep subwavelength plasmonic resonators for the enhancement of light-matter interaction. In this section, plasmonic ring nanocavities are shown to form gap plasmon modes confined to the dielectric region between two metal layers. The scattering properties, near-field enhancement and photonic density of states of nanocavity devices are studied using analytic theory and 3D finite difference time domain simulations. Plasmonic ring nanocavities are fabricated and characterized using photoluminescence intensity and decay rate measurements. A 25 times increase in the radiative decay rate of Er:Si02 is demonstrated in nanocavities where light is confined to volumes as small as 0.01(λ/n)^3 . The potential to achieve lasing, due to the enhancement of stimulated emission rate in ring nanocavities, is studied as a route to Si-compatible plasmon-enhanced nanolasers. The second part of this work focuses on the manipulation of light generated in planar semiconductor devices using arrays of dielectric nanopillars. In particular, aperiodic arrays of nanopillars are engineered for omnidirectional light extraction enhancement. Arrays of Er:SiNx nanopillars are fabricated and a ten times increase in light extraction is experimentally demonstrated, while simultaneously controlling far-field radiation patterns in ways not possible with periodic arrays. Additionally, analytical scalar diffraction theory is used to study light propagation from Vogel spiral arrays and demonstrate generation of OAM. Using phase shifting interferometry, the presence of OAM is experimentally verified. The use of Vogel spirals presents a new method for the generation of OAM with applications for secure optical communications

Radial projection statistics: a different angle on tilings

Jakobi T. Radial projection statistics: a different angle on tilings. Bielefeld: Universität Bielefeld; 2017

Engineering aperiodic spiral order for photonic-plasmonic device applications

Thesis (Ph.D.)--Boston UniversityDeterministic arrays of metal (i.e., Au) nanoparticles and dielectric nanopillars (i.e., Si and SiN) arranged in aperiodic spiral geometries (Vogel's spirals) are proposed as a novel platform for engineering enhanced photonic-plasmonic coupling and increased light-matter interaction over broad frequency and angular spectra for planar optical devices. Vogel's spirals lack both translational and orientational symmetry in real space, while displaying continuous circular symmetry (i.e., rotational symmetry of infinite order) in reciprocal Fourier space. The novel regime of "circular multiple light scattering" in finite-size deterministic structures will be investigated. The distinctive geometrical structure of Vogel spirals will be studied by a multifractal analysis, Fourier-Bessel decomposition, and Delaunay tessellation methods, leading to spiral structure optimization for novel localized optical states with broadband fluctuations in their photonic mode density. Experimentally, a number of designed passive and active spiral structures will be fabricated and characterized using dark-field optical spectroscopy, ellipsometry, and Fourier space imaging. Polarization-insensitive planar omnidirectional diffraction will be demonstrated and engineered over a large and controllable range of frequencies. Device applications to enhanced LEDs, novel lasers, and thin-film solar cells with enhanced absorption will be specifically targeted. Additionally, using Vogel spirals we investigate the direct (i.e. free space) generation of optical vortices, with well-defined and controllable values of orbital angular momentum, paving the way to the engineering and control of novel types of phase discontinuities (i.e., phase dislocation loops) in compact, chip-scale optical devices. Finally, we report on the design, modeling, and experimental demonstration of array-enhanced nanoantennas for polarization-controlled multispectral nanofocusing, nanoantennas for resonant near-field optical concentration of radiation to individual nanowires, and aperiodic double resonance surface enhanced Raman scattering substrates

Near-Linear-Time Deterministic Plane Steiner Spanners for Well-Spaced Point Sets

See article for Abstract.This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/computational-geometry/Keywords: Sparse, Euclidean TSP, Spanne

MAGNETIZATION DYNAMICS IN KAGOME ARTIFICIAL SPIN ICE CONSIDERING THE EFFECT OF VERTEX AND GEOMETRICAL LATTICE DISTORTION

Artificial spin ices (ASI) have been shown to exhibit dynamic magnetic responses that are dramatically different from plane magnetic thin films. A number of magnetic ASI have been fabricated and measured in recent years. However, some important effects including influence of vertex and geometrical distortion on their dynamic response have not been addressed. This dissertation adopts Ferromagnetic Resonance (FMR) spectroscopy to study magnetization dynamics in fabricated artificial spin ices with a contentiously distorted Honeycomb geometry with the specific goal of exploring how the vertex and lattice distortion affect the dynamic magnetic response. Samples were patterned using electron beam lithography techniques. FMR spectroscopy was developed by designing a new printed microwave transmission line. The transmission line was fabricated using photolithography technique. A Vector Network Analyzer (VNA) and an electromagnet were used to measure the FMR spectrum of the samples. Object Oriented Micromagnetic Framework (OOMMF) and Fast Fourier Transform (FFT) were used to simulate the magnetization texture and FMR spectrum of the samples. The experimental and simulation findings demonstrated the importance of the vertex in the dynamic response of artificial spin ices such that previous research findings were modified. Moreover, Fibonacci-distortion was applied on the geometrical lattice (Honeycomb) of Kagome ASI that offers a mathematical algorithm to design artificial spin ice. The FMR spectra for different degrees of distortion severity were measured. We found that the geometrical distortion is a suitable algorithm to achieve the desired FMR spectrum and multi-step magnetization reversal process for engineering applications. Furthermore, FMR data aided with simulations in the magnetization reversal regime suggest that the distortion can cause the artificial spin ice to relax in partially long-range magnetic order