Digital objects in rhombic dodecahedron grid

Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system

Modelling of radio wave propagation using Finite Element Analysis.

Fourth generation (4G) wireless communication systems are intended to support high data rates which requires careful and accurate modelling of the radio environment. In this thesis, for the first time finite clement based accurate and computationally efficient models of wave propagation in different outdoor and indoor environments has been developed. Three different environments were considered: the troposphere, vegetation and tunnels and wave propagation in these environments were modelled using finite element analysis. Use of finite elements in wave propagation modelling is a novel idea although many propagation models and approaches were used in past. Coverage diagrams, path loss contours and power levels were calculated using developed models in the troposphere, vegetation and tunnels. Results obtained were compared with commercially available software Advanced Refractive Effects Prediction Software (AREPS) to validate the accuracy of the developed approach and it is shown that results were accurate with an accuracy of 3dB. The developed models were very flexible in handling complex geometries and similar analysis can be easily extended to other environments. A fully vectored finite element base propagation model was developed for straight and curved tunnels. An optimum range of values of different electrical parameters for tunnels of different shapes has been derived. The thesis delivered a novel approach to modelling radio channels that provided a fast and accurate solution of radio wave propagation in realistic environments. The results of this thesis will have a great impact in modelling and characterisation of future wireless communication systems

Development of a new method for the wave optical propagation of ultrashort pulses through linear optical systems

The design and simulation of ultrashort pulse shaping systems require pulse propagation methods that take the combined effects of dispersion, diffraction, and system aberrations into account. In the conventional pulse propagation methods based on the spectrum of plane wave, usually large number of sampling points are needed for the correct Fourier transform operations due to the fast oscillating phase of the complex pulse field. In this work, I have developed an alternative pulse propagation method, based on the Gaussian pulsed beam decomposition, as an extension of the monochromatic Gaussian beam propagation method. Methods for the decomposition of an input pulse, with arbitrary spatial and temporal (spectral) profiles, into a set of elementary Gaussian pulsed beams are proposed. Algorithms for computing the spatio-temporal and spatio-spectral profiles of the propagated pulse as the phase correct superposition of individual Gaussian pulsed beams are developed. The proposed decomposition method allows the elementary Gaussian pulsed beams to have different parameters depending on the local spatial and spectral phase of the given input pulse which reduces the number of Gaussian pulsed beams required to decompose an input pulsed beam with a given accuracy. Furthermore, a new kind of beam called the truncated Gaussian beam, is introduced and combined with the conventional Gaussian beam decomposition method to enable decomposition of fields after hard apertures. The analytical propagation equation of the truncated Gaussian beam through a paraxial optical system is derived. Additionally, the application of the Gaussian beam decomposition method is extended to handle the propagation of vectorial fields. Several example calculations are presented to validate the proposed methods and show their application in propagating fields through optical systems which are rather complicated to model using the conventional methods

Conformal electromagnetic wave propagation using primal mimetic finite elements

Elektromagnetische Wellenausbreitung bildet die physikalische Grundlage für unzählige Anwendungen in verschiedenen Bereichen der heutigen Welt. Um räumliche Szenarien zu modellieren, muss der kontinuierliche Raum in geeigneter Weise in ein Rechengebiet umgewandelt werden. Üblich diskretisierte Modelle – welche auf verschiedenen Größen beruhen – berücksichtigen die Beziehungen zwischen Feldvariablen mittels Relationen, welche durch partielle Differentialgleichungen repräsentiert werden. Um mathematische Beziehungen zwischen abhängigen Variablen in zweckdienlicher Art nachzubilden, schaffen hyperkomplexe Zahlensysteme ein passendes alternatives Rahmenwerk. Dieser Ansatz bezweckt das Einbinden bestimmter Systemeigenschaften und umfasst zusätzlich zur Modellierung von Feldproblemen, bei denen alle Variablen vorkommen, auch vereinfachte Modelle. Um eine wettbewerbsfähige Alternative zur üblichen numerischen Behandlung elektromagnetischer Felder in beobachtungsorientierter Weise darzubieten, wird das elektrische und magnetische Feld elektromagnetischer Wellenfelder als eine zusammengefasste Feldgröße, eingebettet im Funktionenraum, verstanden. Dieses Vorgehen ist intuitiv, da beide Felder in der Elektrodynamik gemeinsam auftreten und direkt messbar sind. Der Schwerpunkt dieser Arbeit ist in zwei Ziele untergliedert. Auf der einen Seite wird ein umformuliertes Maxwell-System in einer metrikfreien Umgebung mittels dem sogenannten „bikomplexen Ansatz“ umfassend untersucht. Auf der anderen Seite wird eine mögliche numerische Implementierung hinsichtlich der Finite-Elemente-Methode auf modernem Wege durch Nutzung der diskreten äußeren Analysis mit Fokus auf Genauigkeitsbelange bewertet. Hinsichtlich der numerischen Genauigkeitsbewertung wird demonstriert, dass der vorgelegte Ansatz grundsätzlich eine höhere Exaktheit zeigt, wenn man ihn mit Formulierungen vergleicht, welche auf der Helmholtz-Gleichung beruhen. Diese Dissertation trägt eine generalisierte hyperkomplexe alternative Darstellung von gewöhnlichen elektrodynamischen Ausdrucksweisen zum Themengebiet der Wellenausbreitung bei. Durch die Nutzung einer direkten Formulierung des elektrischen Feldes in Verbindung mit dem magnetischen Feld wird die Rechengenauigkeit von Randwertproblemen erhöht. Um diese Genauigkeitserhöhung zu erreichen, wird eine geeignete Erweiterung der de Rham-Kohomologie unterbreitet.Electromagnetic wave propagation provides the physical basis for countless applications in various subjects of today’s world. In order to model spatial scenarios, the continuous space must be converted to an appropriate computational domain. Ordinarily discretized models – which are based on distinct quantities – consider the connection between field variables by relations which are represented by partial differential equations. To reproduce mathematical relationships between dependent variables in a convenient manner, hypercomplex number systems build a suitable alternative framework. This approach aims to incorporate certain system properties and covers, in addition to the modeling of field problems where all variables are present, also simplified models. To provide a competitive alternative to the ordinary numerical handling of electromagnetic fields in an observation-based way, the electric and magnetic field of electromagnetic wave fields is understood as only one combined field variable embedded in the function space. This procedure is intuitive since both fields occur together in electrodynamics and are directly measureable. The focus of this thesis is twofold. On the one side, a reformulated Maxwell system is broadly investigated in a metric-free environment by the use of the so-called ”bicomplex approach”. On the other side, a possible numerical implementation concerning the Finite Element Method is evaluated in a modern way by the use of discrete exterior calculus with focus on accuracy matters. Regarding the numerical accuracy evaluation, it is demonstrated that the presented approach yields a higher exactness in general when comparing it to formulations which are based on the Helmholtz equation. This thesis contributes generalized hypercomplex alternative representations of ordinary electrodynamic expressions to the topic of wave propagation. By the use of a direct formulation of the electric field in conjunction with the magnetic field, the computational accuracy of boundary value problems is improved. In order to achieve this increase of accuracy, an appropriate enhancement of the de Rham cohomology is proposed

Holistic simulation of optical systems

For many years, the design of optical systems mainly comprised a linear arrangement of plane or spherical components, such as lenses, mirrors or prisms, and a geometric-optical description by ray tracing lead to an accurate and satisfactory result. Today, many modern optical systems found in a variety of different industrial and scientific applications, deviate from this structure. Polarization, diffraction and coherence, or material interactions, such as volume or surface scattering, need to be included when reasonable performance predictions are required. Furthermore, manufacturing and alignment aspects must be considered in the design and simulation of optical systems to ensure that their impact is not damaging to the overall purpose of the corresponding setup. Another important part is the growing field of digital optics. Signal processing algorithms have become an indispensable part of many systems, whereby an almost unlimited number of current and potential applications exists. Since these algorithms are an essential part of the system, their compatibility and impact on the completed system is an important aspect to con- sider. In principle, this list of relevant topics and examples can be further expanded to an almost unlimited extend. However, the simulation and optimization of the single sub-aspects do often not lead to a satisfactory result. The goal of this thesis is to demonstrate that the performance prediction of modern optical systems benefits significantly from an aggregation of the individual models and technological aspects. Present concepts are further enhanced by the development and analysis of new approaches and algorithms, leading to a more holistic description and simulation of complex setups as a whole. The long-term objective of this work is a comprehensive virtual and rapid prototyping. From an industrial perspective, this would reduce the risk, time and costs associated with the development of an optical system

Development of freeform optical systems

One of the essential properties of the freeform surface is that its asymmetric and locally variant surface profile breaks the symmetry of an optical system, thus provoking unique issues that have never been considered in rotationally symmetric optical systems. This thesis focuses on developing an optimization algorithm that automatically eliminates the obscuration in the non-rotationally symmetric reflective optical system, as well as defining and computing the generalized chromatic aberrations in the non-rotationally symmetric refractive optical system. Furthermore, a comprehensive model for the tolerancing of freeform surface is put forward. When optimizing a three-dimensional (3D) reflective optical system by tilting the mirrors, the mirrors can block the ray path and, in consequence, reduce the image brightness and contrast. To take the degree of obscuration into consideration, an error function that mathematically describes all the obscuration cases in 3D reflective systems is proposed. In order to analyze the generalized chromatic aberrations in 3D refractive systems, the reference axis and reference plane are clarified to figure out the precise definition of the generalized chromatic aberrations. Both ray-based and wavefront-based methods are proposed to calculate the generalized chromatic aberrations surface-by-surface. In addition, the influence of pupil aberration is discussed to improve calculation accuracy. The manufacturing error of the freeform surface can be transferred into the frequency domain by Fourier transform. The autocorrelation function (ACF) of the phase pattern is computed in different frequency ranges. By characterizing the width of ACF, the boundary frequency between the deterministic and statistic errors can be found. A comprehensive model representing different types of surface errors is proposed to construct a synthetic freeform surface. By performing the Monte-Carlo simulation, tolerancing of the freeform system can be realized

Visuelle Analyse großer Partikeldaten

Partikelsimulationen sind eine bewährte und weit verbreitete numerische Methode in der Forschung und Technik. Beispielsweise werden Partikelsimulationen zur Erforschung der Kraftstoffzerstäubung in Flugzeugturbinen eingesetzt. Auch die Entstehung des Universums wird durch die Simulation von dunkler Materiepartikeln untersucht. Die hierbei produzierten Datenmengen sind immens. So enthalten aktuelle Simulationen Billionen von Partikeln, die sich über die Zeit bewegen und miteinander interagieren. Die Visualisierung bietet ein großes Potenzial zur Exploration, Validation und Analyse wissenschaftlicher Datensätze sowie der zugrundeliegenden Modelle. Allerdings liegt der Fokus meist auf strukturierten Daten mit einer regulären Topologie. Im Gegensatz hierzu bewegen sich Partikel frei durch Raum und Zeit. Diese Betrachtungsweise ist aus der Physik als das lagrange Bezugssystem bekannt. Zwar können Partikel aus dem lagrangen in ein reguläres eulersches Bezugssystem, wie beispielsweise in ein uniformes Gitter, konvertiert werden. Dies ist bei einer großen Menge an Partikeln jedoch mit einem erheblichen Aufwand verbunden. Darüber hinaus führt diese Konversion meist zu einem Verlust der Präzision bei gleichzeitig erhöhtem Speicherverbrauch. Im Rahmen dieser Dissertation werde ich neue Visualisierungstechniken erforschen, welche speziell auf der lagrangen Sichtweise basieren. Diese ermöglichen eine effiziente und effektive visuelle Analyse großer Partikeldaten