Delaunay Tessellations and Voronoi Diagrams in CGAL

The Cgal library provides a rich variety of Voronoi diagrams and Delaunay triangulations. This variety covers several aspects: generators, dimensions and metrics, which we describe in Section 2. One aim of this paper is to present the main paradigms used in CGAL: Generic programming, separation between predicates/constructions and combinatorics, and exact geometric computation (not to be confused with exact arithmetic!). The first two paradigms translate into software design choices, described in Section 4, while the last covers both robustness and efficiency issues, respectively described in Sec- tion 6 and 7. Other important aspects of the Cgal library are the interface issues, be they for traversing a tessellation, or for interoperability with other libraries or languages, see Section 5. We present in Section 8 some tessellations at work in the context of surface reconstruction and mesh generation. Section 9 is devoted to some on-going and future work on periodic triangulations (triangulations in periodic spaces), and on high-quality mesh generation with optimized tessellations. Section 10 provides typical numbers in terms of efficiency and scalability for constructing tessellations, and lists the remaining weaknesses. We conclude by listing some of our directions for the future

Structure formation and identification in geometrically driven soft matter systems

Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces. Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system’s constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness. In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system “forced” to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate. We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase. The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy

Über ausgewählte numerische Zugänge zu Zellgewebe

Different numerical approaches and algorithms arising in the context of modelling of cellular tissue evolution are discussed in this thesis. Being suited in particular to off-lattice agent-based models, the numerical tool of three-dimensional weighted kinetic and dynamic Delaunay triangulations is introduced and discussed for its applicability to adjacency detection. As there exists no implementation of a code that incorporates all necessary features for tissue modelling, algorithms for incremental insertion or deletion of points in Delaunay triangulations and the restoration of the Delaunay property for triangulations of moving point sets are introduced. In addition, the numerical solution of reaction-diffusion equations and their connection to agent-based cell tissue simulations is discussed. In order to demonstrate the applicability of the numerical algorithms, biological problems are studied for different model systems: For multicellular tumour spheroids, the weighted Delaunay triangulation provides a great advantage for adjacency detection, but due to the large cell numbers the model used for the cell-cell interaction has to be simplified to allow for a numerical solution. The agent-based model reproduces macroscopic experimental signatures, but some parameters cannot be fixed with the data available. A much simpler, but in key properties analogous, continuum model based on reaction-diffusion equations is likewise capable of reproducing the experimental data. Both modelling approaches make differing predictions on non-quantified experimental signatures. In the case of the epidermis, a smaller system is considered which enables a more complete treatment of the equations of motion. In particular, a control mechanism of cell proliferation is analysed. Simple assumptions suffice to explain the flow equilibrium observed in the epidermis. In addition, the effect of adhesion on the survival chances of cancerous cells is studied. For some regions in parameter space, stochastic effects may completely alter the outcome. The findings stress the need of establishing a defined experimental model to fix the unknown model parameters and to rule out further models.Diese Arbeit behandelt verschiedene numerische Verfahren zur Modellierung der Entwicklung von Zellgewebe. Das numerische Hilfsmittel der dreidimensionalen gewichteten, kinetischen und dynamischen, Delaunay-Triangulierung, welches insbesondere für gitterfreie agentenbasierte Modelle geeignet ist, wird eingeführt und auf seine Anwendbarkeit in der Nachbarschaftserkennung diskutiert. Da keine numerische Implementierung existiert, welche alle notwendigen Eigenschaften für die Gewebemodellierung beinhaltet, werden Algorithmen für das inkrementelle Einfügen und Löschen von Punkten in Delaunay-Triangulierungen und das Wiederherstellen der Delaunay-Eigenschaft für Mengen sich bewegender Punkte eingeführt. Weiterhin wird die numerische Lösung von Reaktions-Diffusions-Gleichungen und ihre Verbindung zu agentenbasierten Zellgewebesimulationen diskutiert. Um die Anwendbarkeit der numerischen Algorithmen zu demonstrieren, werden für verschiedene Modellsysteme biologische Probleme studiert: Für multizelluläre Tumorsphäroide stellt die gewichtete Delaunay-Triangulierung einen großen Vorteil für die Nachbarschaftserkennung dar, jedoch muss wegen der großen Zellzahlen das Modell für die Zell-Zell-Wechselwirkung deutlich vereinfacht werden, um eine numerische Lösung zu erlauben. Das agentenbasierte Modell reproduziert makroskopische experimentelle Signaturen, jedoch können nicht alle Parameter mit den verfügbaren Daten bestimmt werden. Ein deutlich einfacheres, aber in Schlüsseleigenschaften analoges Kontinuumsmodell, welches auf Reaktions-Diffusions-Gleichungen basiert, kann gleichfalls die experimentellen Daten reproduzieren. Beide Modellansätze machen jedoch verschiedene Aussagen über nicht-quantifizierte experimentelle Signaturen. Im Falle der Epidermis wird ein kleineres System behandelt, was eine vollständigere Behandlung der Bewegungsgleichungen ermöglicht. Insbesondere wird ein Kontrollmechanismus der Zellproliferation analysiert. Einfache Annahmen reichen aus, um das Fließgleichgewicht zu erklären, welches in der Epidermis beobachtet wird. Zusätzlich wird der Effekt der Adhäsion auf die Überlebenschancen von Krebszellen studiert. Für einige Regionen im Parameterraum können stochastische Effekte den Ausgang komplett verändern. Die Resultate unterstreichen die Notwendigkeit der Etablierung eines definierten experimentellen Modellsystems, um unbekannte Modellparameter zu fixieren und Modelle zu falsifizieren

Influence of microstructure on mechanical properties and damage characteristics of Al-Si alloys

Nowadays Al-Si alloys have an increasing share in automobile parts and are seen as a promising material for new structural applications that require an improved strength and fracture resistance. In this respect, the functionality of the alloys is mostly defined by their mechanical performance. The latter, in turn, strongly depends on the microstructure. With respect to the most relevant mechanical properties, tailoring the microstructure more closely to each specific application needs requires a deep understanding of the relations between morphological and mechanical properties of a structure. In this study, an extensive analysis of morphological properties, mechanical behavior and damage characteristics of Al-Si alloys under different loading conditions is conducted and corresponding structure-properties relations are investigated. The limits of 2D characterization of Si morphology and damage in the eutectic structure are discussed. Furthermore, a model for the simulation of a system of dimples on the fracture surface of the eutectic phase is proposed and its application to studying the relation between Si morphology and eutectic fracture toughness is described. The work thus contributes to better understanding of structure-properties relations and developing quantitative methods enabling a microstructure-based prediction of properties without their direct measurements.Heutzutage werden Al-Si-Legierungen zunehmend in der Automobilindustrie eingesetzt und gelten als vielsprechender Strukturwerkstoff für Anwendungen, die erhöhte Festigkeit und Bruchfestigkeit erfordern. Somit wird die Funktionalität der Legierungen meist über ihre mechanischen Eigenschaften definiert, welche wiederum vom Gefüge abhängen. In Bezug auf die wichtigsten mechanischen Eigenschaften erfordert die Einstellung des Gefüges für spezifische Anwendungen ein tiefes Verständnis des Zusammenhangs zwischen den morphologischen und den mechanischen Eigenschaften einer Struktur. In dieser Arbeit wurde eine grundlegende Analyse der morphologischen Eigenschaften, des mechanischen Verhaltens und des Schädigungsverhaltens von Al-Si-Legierungen in verschiedenen Lastsituationen durchgeführt und die entsprechenden Struktur-Eigenschafts-Beziehungen untersucht. Die Grenzen der 2D-Charakterisierung der Silizium-Morphologie und der Schädigung im eutektischen Gefüge werden diskutiert. Darüber hinaus wird ein Modell für die Simulation der Bruchoberfläche der eutektischen Phase vorgeschlagen und seine Anwendung auf die Untersuchung des Zusammenhangs zwischen der Siliziummorphologie und der Bruchzähigkeit des Eutektikums beschrieben. Damit trägt diese Arbeit zu einem besseren Verständnis der Struktur-Eigenschafts-Beziehung bei und zur Entwicklung gefügebasierter Methoden zur quantitativen Vorhersage von Eigenschaften ohne diese direkt messen zu müssen

Self Assembly Problems of Anisotropic Particles in Soft Matter.

Anisotropic building blocks assembled from colloidal particles are attractive building blocks for self-assembled materials because their complex interactions can be exploited to drive self-assembly. In this dissertation we address the self-assembly of anisotropic particles from multiple novel computational and mathematical angles. First, we accelerate algorithms for modeling systems of anisotropic particles via massively parallel GPUs. We provide a scheme for generating statistically robust pseudo-random numbers that enables GPU acceleration of Brownian and dissipative particle dynamics. We also show how rigid body integration can be accelerated on a GPU. Integrating these two algorithms into a GPU-accelerated molecular dynamics code (HOOMD-blue), make a single GPU the ideal computing environment for modeling the self-assembly of anisotropic nanoparticles. Second, we introduce a new mathematical optimization problem, filling, a hybrid of the familiar shape packing and covering problem, which can be used to model shaped particles. We study the rich mathematical structures of the solution space and provide computational methods for finding optimal solutions for polygons and convex polyhedra. We present a sequence of isosymmetric optimal filling solutions for the Platonic solids. We then consider the filling of a hyper-cone in dimensions two to eight and show the solution remains scale-invariant but dependent on dimension. Third, we study the impact of size variation, polydispersity, on the self-assembly of an anisotropic particle, the polymer-tethered nanosphere, into ordered phases. We show that the local nanoparticle packing motif, icosahedral or crystalline, determines the impact of polydispersity on energy of the system and phase transitions. We show how extensions of the Voronoi tessellation can be calculated and applied to characterize such micro-segregated phases. By applying a Voronoi tessellation, we show that properties of the individual domains can be studied as a function of system properties such as temperature and concentration. Last, we consider the thermodynamically driven self-assembly of terminal clusters of particles. We predict that clusters related to spherical codes, a mathematical sequence of points, can be synthesized via self-assembly. These anisotropic clusters can be tuned to different anisotropies via the ratio of sphere diameters and temperature. The method suggests a rich new way for assembling anisotropic building blocks.Ph.D.Applied Physics and Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91576/1/phillicl_1.pd