Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric

International audienceThis article presents a new method to compute a self-intersection free high-dimensional Euclidean embedding (SIFHDE) for surfaces and volumes equipped with an arbitrary Riemannian metric. It is already known that given a high-dimensional (high-d) embedding, one can easily compute an anisotropic Voronoi diagram by back-mapping it to 3D space. We show here how to solve the inverse problem, i.e., given an input metric, compute a smooth intersection-free high-d embedding of the input such that the pullback metric of the embedding matches the input metric. Our numerical solution mechanism matches the deformation gradient of the 3D → higher-d mapping with the given Riemannian metric. We demonstrate applications of the method, by being used to construct anisotropic Restricted Voronoi Diagram (RVD) and anisotropic meshing, that are otherwise extremely difficult to compute. In the SIFHDE-space constructed by our algorithm, difficult 3D anisotropic computations are replaced with simple Euclidean computations, resulting in an isotropic RVD and its dual mesh on this high-d embedding. The results are compared with the state-ofthe-art in anisotropic surface and volume meshings using several examples and evaluation metrics

The dual half-edge-a topological primal/dual data structure and construction operators for modelling and manipulating cell complexes

© 2016 by the authors. There is an increasing need for building models that permit interior navigation, e.g., for escape route analysis. This paper presents a non-manifold Computer-Aided Design (CAD) data structure, the dual half-edge based on the Poincaré duality that expresses both the geometric representations of individual rooms and their topological relationships. Volumes and faces are expressed as vertices and edges respectively in the dual space, permitting a model just based on the storage of primal and dual vertices and edges. Attributes may be attached to all of these entities permitting, for example, shortest path queries between specified rooms, or to the exterior. Storage costs are shown to be comparable to other non-manifold models, and construction with local Euler-type operators is demonstrated with two large university buildings. This is intended to enhance current developments in 3D Geographic Information Systems for interior and exterior city modelling

TCP-Carson: A loss-event based Adaptive AIMD algorithm for Long-lived Flows

The diversity of network applications over the Internet has propelled researchers to rethink the strategies in the transport layer protocols. Current applications either use UDP without end-to-end congestion control mechanisms or, more commonly, use TCP. TCP continuously probes for bandwidth even at network steady state and thereby causes variation in the transmission rate and losses. This thesis proposes TCP Carson, a modification of the window-scaling approach of TCP Reno to suit long-lived flows using loss-events as indicators of congestion. We analyzed and evaluated TCP Carson using NS-2 over a wide range of test conditions. We show that TCP Carson reduces loss, improves throughput and reduces window-size variance. We believe that this adaptive approach will improve both network and application performance

Development of an Unstructured 3-D Direct Simulation Monte Carlo/Particle-in-Cell Code and the Simulation of Microthruster Flows

This work is part of an effort to develop an unstructured, three-dimensional, direct simulation Monte Carlo/particle-in-cell (DSMC/PIC) code for the simulation of non-ionized, fully ionized and partially-ionized flows in micropropulsion devices. Flows in microthrusters are often in the transitional to rarefied regimes, requiring numerical techniques based on the kinetic description of the gaseous or plasma propellants. The code is implemented on unstructured tetrahedral grids to allow discretization of arbitrary surface geometries and includes an adaptation capability. In this study, an existing 3D DSMC code for rarefied gasdynamics is improved with the addition of the variable hard sphere model for elastic collisions and a vibrational relaxation model based on discrete harmonic oscillators. In addition the existing unstructured grid generation module of the code is enhanced with grid-quality algorithms. The unstructured DSMC code is validated with simulation of several gaseous micronozzles and comparisons with previous experimental and numerical results. Rothe s 5-mm diameter micronozzle operating at 80 Pa is simulated and results are compared favorably with the experiments. The Gravity Probe-B micronozzle is simulated in a domain that includes the injection chamber and plume region. Stagnation conditions include a pressure of 7 Pa and mass flow rate of 0.012 mg/s. The simulation examines the role of injection conditions in micronozzle simulations and results are compared with previous Monte Carlo simulations. The code is also applied to the simulation of a parabolic planar micronozzle with a 15.4-micron throat and results are compared with previous 2D Monte Carlo simulations. Finally, the code is applied to the simulation of a 34-micron throat MEMS-fabricated micronozzle. The micronozzle is planar in profile with sidewalls binding the upper and lower surfaces. The stagnation pressure is set at 3.447 kPa and represents an order of magnitude lower pressure than used in previous experiments. The simulation demonstrates the formation of large viscous boundary layers in the sidewalls. A particle-in-cell model for the simulation of electrostatic plasmas is added to the DSMC code. Solution to Poisson\u27s equation on unstructured grids is obtained with a finite volume implementation. The Poisson solver is validated by comparing results with analytic solutions. The integration of the ionized particle equations of motion is performed via the leapfrog method. Particle gather and scatter operations use volume weighting with linear Lagrange polynomial to obtain an acceptable level of accuracy. Several methods are investigated and implemented to calculate the electric field on unstructured meshes. Boundary conditions are discussed and include a formulation of plasma in bounded domains with external circuits. The unstructured PIC code is validated with the simulation of a high voltage sheath formation

A 3D Unstructured Mesh FDTD Scheme for EM Modelling

The Yee finite difference time domain (FDTD) algorithm is widely used in computational electromagnetics because of its simplicity, low computational costs and divergence free nature. The standard method uses a pair of staggered orthogonal cartesian meshes. However, accuracy losses result when it is used for modelling electromagnetic interactions with objects of arbitrary shape, because of the staircased representation of curved interfaces. For the solution of such problems, we generalise the approach and adopt an unstructured mesh FDTD method. This co-volume method is based upon the use of a Delaunay primal mesh and its high quality Voronoi dual. Computational efficiency is improved by employing a hybrid primal mesh, consisting of tetrahedral elements in the vicinity of curved interfaces and hexahedral elements elsewhere. Difficulties associated with ensuring the necessary quality of the generated meshes will be discussed. The power of the proposed solution approach is demonstrated by considering a range of scattering and/or transmission problems involving perfect electric conductors and isotropic lossy, anisotropic lossy and isotropic frequency dependent chiral materials

Diskrete Spin-Geometrie für Flächen

This thesis proposes a discrete framework for spin geometry of surfaces. Specifically, we discretize the basic notions in spin geometry, such as the spin structure, spin connection and Dirac operator. In this framework, two types of Dirac operators are closely related as in smooth case. Moreover, they both induce the discrete conformal immersion with prescribed mean curvature half-density.In dieser Arbeit wird ein diskreter Zugang zur Spin-Geometrie vorgestellt. Insbesondere diskretisieren wir die grundlegende Begriffe, wie zum Beispiel die Spin-Struktur, den Spin-Zusammenhang und den Dirac Operator. In diesem Rahmen sind zwei Varianten für den Dirac Operator eng verwandt wie in der glatten Theorie. Darüber hinaus induzieren beide die diskret-konforme Immersion mit vorgeschriebener Halbdichte der mittleren Krümmung

Visual Techniques for Geological Fieldwork Using Mobile Devices

Visual techniques in general and 3D visualisation in particular have seen considerable adoption within the last 30 years in the geosciences and geology. Techniques such as volume visualisation, for analysing subsurface processes, and photo-coloured LiDAR point-based rendering, to digitally explore rock exposures at the earth’s surface, were applied within geology as one of the first adopting branches of science. A large amount of digital, geological surface- and volume data is nowadays available to desktop-based workflows for geological applications such as hydrocarbon reservoir exploration, groundwater modelling, CO2 sequestration and, in the future, geothermal energy planning. On the other hand, the analysis and data collection during fieldwork has yet to embrace this ”digital revolution”: sedimentary logs, geological maps and stratigraphic sketches are still captured in each geologist’s individual fieldbook, and physical rocks samples are still transported to the lab for subsequent analysis. Is this still necessary, or are there extended digital means of data collection and exploration in the field ? Are modern digital interpretation techniques accurate and intuitive enough to relevantly support fieldwork in geology and other geoscience disciplines ? This dissertation aims to address these questions and, by doing so, close the technological gap between geological fieldwork and office workflows in geology. The emergence of mobile devices and their vast array of physical sensors, combined with touch-based user interfaces, high-resolution screens and digital cameras provide a possible digital platform that can be used by field geologists. Their ubiquitous availability increases the chances to adopt digital workflows in the field without additional, expensive equipment. The use of 3D data on mobile devices in the field is furthered by the availability of 3D digital outcrop models and the increasing ease of their acquisition. This dissertation assesses the prospects of adopting 3D visual techniques and mobile devices within field geology. The research of this dissertation uses previously acquired and processed digital outcrop models in the form of textured surfaces from optical remote sensing and photogrammetry. The scientific papers in this thesis present visual techniques and algorithms to map outcrop photographs in the field directly onto the surface models. Automatic mapping allows the projection of photo interpretations of stratigraphy and sedimentary facies on the 3D textured surface while providing the domain expert with simple-touse, intuitive tools for the photo interpretation itself. The developed visual approach, combining insight from all across the computer sciences dealing with visual information, merits into the mobile device Geological Registration and Interpretation Toolset (GRIT) app, which is assessed on an outcrop analogue study of the Saltwick Formation exposed at Whitby, North Yorkshire, UK. Although being applicable to a diversity of study scenarios within petroleum geology and the geosciences, the particular target application of the visual techniques is to easily provide field-based outcrop interpretations for subsequent construction of training images for multiple point statistics reservoir modelling, as envisaged within the VOM2MPS project. Despite the success and applicability of the visual approach, numerous drawbacks and probable future extensions are discussed in the thesis based on the conducted studies. Apart from elaborating on more obvious limitations originating from the use of mobile devices and their limited computing capabilities and sensor accuracies, a major contribution of this thesis is the careful analysis of conceptual drawbacks of established procedures in modelling, representing, constructing and disseminating the available surface geometry. A more mathematically-accurate geometric description of the underlying algebraic surfaces yields improvements and future applications unaddressed within the literature of geology and the computational geosciences to this date. Also, future extensions to the visual techniques proposed in this thesis allow for expanded analysis, 3D exploration and improved geological subsurface modelling in general.publishedVersio

Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios

Physically-based simulation of ice formation

The geometric and optical complexity of ice has been a constant source of wonder and inspiration for scientists and artists. It is a defining seasonal characteristic, so modeling it convincingly is a crucial component of any synthetic winter scene. Like wind and fire, it is also considered elemental, so it has found considerable use as a dramatic tool in visual effects. However, its complex appearance makes it difficult for an artist to model by hand, so physically-based simulation methods are necessary. In this dissertation, I present several methods for visually simulating ice formation. A general description of ice formation has been known for over a hundred years and is referred to as the Stefan Problem. There is no known general solution to the Stefan Problem, but several numerical methods have successfully simulated many of its features. I will focus on three such methods in this dissertation: phase field methods, diffusion limited aggregation, and level set methods. Many different variants of the Stefan problem exist, and each presents unique challenges. Phase field methods excel at simulating the Stefan problem with surface tension anisotropy. Surface tension gives snowflakes their characteristic six arms, so phase field methods provide a way of simulating medium scale detail such as frost and snowflakes. However, phase field methods track the ice as an implicit surface, so it tends to smear away small-scale detail. In order to restore this detail, I present a hybrid method that combines phase fields with diffusion limited aggregation (DLA). DLA is a fractal growth algorithm that simulates the quasi-steady state, zero surface tension Stefan problem, and does not suffer from smearing problems. I demonstrate that combining these two algorithms can produce visual features that neither method could capture alone. Finally, I present a method of simulating icicle formation. Icicle formation corresponds to the thin-film, quasi-steady state Stefan problem, and neither phase fields nor DLA are directly applicable. I instead use level set methods, an alternate implicit front tracking strategy. I derive the necessary velocity equations for level set simulation, and also propose an efficient method of simulating ripple formation across the surface of the icicles

Framework for The Generation and Design of Naturally Functionally Graded Lattice Structures

Functionally Graded Lattice (FGL) Structures have shown improved performance over uniform lattice structures in different fields. Another form of functional grading can be seen in materials in nature, where the cellular structure can vary in both cell porosity and size. To distinguish between lattice structures that vary in porosity only and lattice structures that vary in both, we will refer to the latter in this research as Naturally Functionally Graded Lattice (NFGL) structures. Research into NFGL structures' performance against FGL structures in the literature is lacking. Furthermore, the current methods in the literature to generate these structures are severely limited and suffer from multiple drawbacks. This research aims to develop a framework, namely the NFGL Framework, to generate NFGL structures without the drawbacks that exist in current methods and to improve the performance of the generated structures using the NFGL Framework against existing FGL structures. The NFGL Framework uses a novel method to generate nodes for NFGL structures from using a developed simplified sphere packing algorithm to generate conformal NFGL structures in a deterministic and computationally efficient manner. Furthermore, the NFGL Framework can perform a similarity analysis using a modified Mean Structural Similarity (MSSIM) index to improve the performance of the generated NFGL structure. The generated structures using the NFGL Framework were tested against the existing methods and showed to overcome the drawbacks of these methods with improved performance and computational time. Furthermore, the generated NFGL structures were tested against FGL structures and the results showed a performance gain from the use of NFGL structures over FGL structures with a reduced computational cost.Ph.D