Local cardinal interpolation by C^2 cubic B2-splines with a tunable shape parameter

A $C^2$ cubic local interpolating B2-spline, controllable by a shape parameter, is introduced and its properties analyzed. An algorithm for the automatic selection of the free parameter is developed and tested on several examples. Finally, a two-phase subdivision scheme for its efficient evaluation at dyadic points is presented

New strategies for curve and arbitrary-topology surface constructions for design

This dissertation presents some novel constructions for curves and surfaces with arbitrary topology in the context of geometric modeling. In particular, it deals mainly with three intimately connected topics that are of interest in both theoretical and applied research: subdivision surfaces, non-uniform local interpolation (in both univariate and bivariate cases), and spaces of generalized splines. Specifically, we describe a strategy for the integration of subdivision surfaces in computer-aided design systems and provide examples to show the effectiveness of its implementation. Moreover, we present a construction of locally supported, non-uniform, piecewise polynomial univariate interpolants of minimum degree with respect to other prescribed design parameters (such as support width, order of continuity and order of approximation). Still in the setting of non-uniform local interpolation, but in the case of surfaces, we devise a novel parameterization strategy that, together with a suitable patching technique, allows us to define composite surfaces that interpolate given arbitrary-topology meshes or curve networks and satisfy both requirements of regularity and aesthetic shape quality usually needed in the CAD modeling framework. Finally, in the context of generalized splines, we propose an approach for the construction of the optimal normalized totally positive (B-spline) basis, acknowledged as the best basis of representation for design purposes, as well as a numerical procedure for checking the existence of such a basis in a given generalized spline space. All the constructions presented here have been devised keeping in mind also the importance of application and implementation, and of the related requirements that numerical procedures must satisfy, in particular in the CAD context

Arbitrary topology meshes in geometric design and vector graphics

Meshes are a powerful means to represent objects and shapes both in 2D and 3D, but the techniques based on meshes can only be used in certain regular settings and restrict their usage. Meshes with an arbitrary topology have many interesting applications in geometric design and (vector) graphics, and can give designers more freedom in designing complex objects. In the first part of the thesis we look at how these meshes can be used in computer aided design to represent objects that consist of multiple regular meshes that are constructed together. Then we extend the B-spline surface technique from the regular setting to work on extraordinary regions in meshes so that multisided B-spline patches are created. In addition, we show how to render multisided objects efficiently, through using the GPU and tessellation. In the second part of the thesis we look at how the gradient mesh vector graphics primitives can be combined with procedural noise functions to create expressive but sparsely defined vector graphic images. We also look at how the gradient mesh can be extended to arbitrary topology variants. Here, we compare existing work with two new formulations of a polygonal gradient mesh. Finally we show how we can turn any image into a vector graphics image in an efficient manner. This vectorisation process automatically extracts important image features and constructs a mesh around it. This automatic pipeline is very efficient and even facilitates interactive image vectorisation

Conversion of B-rep CAD models into globally G¹ triangular splines

Existing techniques that convert B-rep (boundary representation) patches into Clough-Tocher splines guarantee watertight, that is C0, conversion results across B-rep edges. In contrast, our approach ensures global tangent-plane, that is G1, continuity of the converted B-rep CAD models. We achieve this by careful boundary curve and normal vector management, and by converting the input models into Shirman-Séquin macro-elements near their (trimmed) B-rep edges. We propose several different variants and compare them with respect to their locality, visual quality, and difference with the input B-rep CAD model. Although the same global G1 continuity can also be achieved by conversion techniques based on subdivision surfaces, our approach uses triangular splines and thus enjoys full compatibility with CAD

Controlling the interpolation of NURBS curves and surfaces

The primary focus of this thesis is to determine the best methods for controlling the interpolation of NURBS curves and surfaces. The various factors that affect the quality of the interpolant are described, and existing methods for controlling them are reviewed. Improved methods are presented for calculating the parameter values, derivative magnitudes, data point spacing and twist vectors, with the aim of producing high quality interpolants with minimal data requirements. A new technique for obtaining the parameter values and derivative magnitudes is evaluated, which constructs a C$^1$ cubic spline with orthogonal first and second derivatives at specified parametric locations. When this data is used to create a C$^2$ spline, the resulting interpolant is superior to those constructed using existing parameterisation and derivative magnitude estimation methods. Consideration is given to the spacing of data points, which has a significant impact on the quality of the interpolant. Existing methods are shown to produce poor results with curves that are not circles. Three new methods are proposed that significantly reduce the positional error between the interpolant and original geometry. For constrained surface interpolation, twist vectors must be estimated. A method is proposed that builds on the Adini method, and is shown to have improved error characteristics. In numerical tests, the new method consistently outperforms Adini. Interpolated surfaces are often required to join together smoothly along their boundaries. The constraints for joining surfaces with parametric and geometric continuity are discussed, and the problem of joining $N$ patches to form an $N$-sided region is considered. It is shown that regions with odd $N$ can be joined with G$^1$ continuity, but those with even $N$ or requiring G$^2$ continuity can only be obtained for specific geometries

Analysis and new constructions of generalized barycentric coordinates in 2D

Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer graphics and computational mechanics. There exist many coordinate functions that satisfy all the basic properties of barycentric coordinates, but they differ by a number of other properties. We start by providing an extensive comparison of all existing coordinate functions and pointing out which important properties of generalized barycentric coordinates are not satisfied by these functions. This comparison shows that not all of existing coordinates have fully investigated properties, and we complete such a theoretical analysis for a particular one-parameter family of generalized barycentric coordinates for strictly convex polygons. We also perform numerical analysis of this family and show how to avoid computational instabilities near the polygon’s boundary when computing these coordinates in practice. We conclude this analysis by implementing some members of this family in the Computational Geometry Algorithm Library. In the second half of this dissertation, we present a few novel constructions of non-negative and smooth generalized barycentric coordinates defined over any simple polygon. In this context, we show that new coordinates with improved properties can be obtained by taking convex combinations of already existing coordinate functions and we give two examples of how to use such convex combinations for polygons without and with interior points. These new constructions have many attractive properties and perform better than other coordinates in interpolation and image deformation applications

Doctor of Philosophy

dissertationVolumetric parameterization is an emerging field in computer graphics, where volumetric representations that have a semi-regular tensor-product structure are desired in applications such as three-dimensional (3D) texture mapping and physically-based simulation. At the same time, volumetric parameterization is also needed in the Isogeometric Analysis (IA) paradigm, which uses the same parametric space for representing geometry, simulation attributes and solutions. One of the main advantages of the IA framework is that the user gets feedback directly as attributes of the NURBS model representation, which can represent geometry exactly, avoiding both the need to generate a finite element mesh and the need to reverse engineer the simulation results from the finite element mesh back into the model. Research in this area has largely been concerned with issues of the quality of the analysis and simulation results assuming the existence of a high quality volumetric NURBS model that is appropriate for simulation. However, there are currently no generally applicable approaches to generating such a model or visualizing the higher order smooth isosurfaces of the simulation attributes, either as a part of current Computer Aided Design or Reverse Engineering systems and methodologies. Furthermore, even though the mesh generation pipeline is circumvented in the concept of IA, the quality of the model still significantly influences the analysis result. This work presents a pipeline to create, analyze and visualize NURBS geometries. Based on the concept of analysis-aware modeling, this work focusses in particular on methodologies to decompose a volumetric domain into simpler pieces based on appropriate midstructures by respecting other relevant interior material attributes. The domain is decomposed such that a tensor-product style parameterization can be established on the subvolumes, where the parameterization matches along subvolume boundaries. The volumetric parameterization is optimized using gradient-based nonlinear optimization algorithms and datafitting methods are introduced to fit trivariate B-splines to the parameterized subvolumes with guaranteed order of accuracy. Then, a visualization method is proposed allowing to directly inspect isosurfaces of attributes, such as the results of analysis, embedded in the NURBS geometry. Finally, the various methodologies proposed in this work are demonstrated on complex representations arising in practice and research

AutoGraff: towards a computational understanding of graffiti writing and related art forms.

The aim of this thesis is to develop a system that generates letters and pictures with a style that is immediately recognizable as graffiti art or calligraphy. The proposed system can be used similarly to, and in tight integration with, conventional computer-aided geometric design tools and can be used to generate synthetic graffiti content for urban environments in games and in movies, and to guide robotic or fabrication systems that can materialise the output of the system with physical drawing media. The thesis is divided into two main parts. The first part describes a set of stroke primitives, building blocks that can be combined to generate different designs that resemble graffiti or calligraphy. These primitives mimic the process typically used to design graffiti letters and exploit well known principles of motor control to model the way in which an artist moves when incrementally tracing stylised letter forms. The second part demonstrates how these stroke primitives can be automatically recovered from input geometry defined in vector form, such as the digitised traces of writing made by a user, or the glyph outlines in a font. This procedure converts the input geometry into a seed that can be transformed into a variety of calligraphic and graffiti stylisations, which depend on parametric variations of the strokes

Advanced Numerical Modelling of Discontinuities in Coupled Boundary ValueProblems

Industrial development processes as well as research in physics, materials and engineering science rely on computer modelling and simulation techniques today. With increasing computer power, computations are carried out on multiple scales and involve the analysis of coupled problems. In this work, continuum modelling is therefore applied at different scales in order to facilitate a prediction of the effective material or structural behaviour based on the local morphology and the properties of the individual constituents. This provides valueable insight into the structure-property relations which are of interest for any design process. In order to obtain reasonable predictions for the effective behaviour, numerical models which capture the essential fine scale features are required. In this context, the efficient representation of discontinuities as they arise at, e.g. material interfaces or cracks, becomes more important than in purely phenomenological macroscopic approaches. In this work, two different approaches to the modelling of discontinuities are discussed: (i) a sharp interface representation which requires the localisation of interfaces by the mesh topology. Since many interesting macroscopic phenomena are related to the temporal evolution of certain microscopic features, (ii) diffuse interface models which regularise the interface in terms of an additional field variable and therefore avoid topological mesh updates are considered as an alternative. With the two combinations (i) Extended Finite Elemente Method (XFEM) + sharp interface model, and (ii) Isogeometric Analysis (IGA) + diffuse interface model, two fundamentally different approaches to the modelling of discontinuities are investigated in this work. XFEM reduces the continuity of the approximation by introducing suitable enrichment functions according to the discontinuity to be modelled. Instead, diffuse models regularise the interface which in many cases requires even an increased continuity that is provided by the spline-based approximation. To further increase the efficiency of isogeometric discretisations of diffuse interfaces, adaptive mesh refinement and coarsening techniques based on hierarchical splines are presented. The adaptive meshes are found to reduce the number of degrees of freedom required for a certain accuracy of the approximation significantly. Selected discretisation techniques are applied to solve a coupled magneto-mechanical problem for particulate microstructures of Magnetorheological Elastomers (MRE). In combination with a computational homogenisation approach, these microscopic models allow for the prediction of the effective coupled magneto-mechanical response of MRE. Moreover, finite element models of generic MRE microstructures are coupled with a BEM domain that represents the surrounding free space in order to take into account finite sample geometries. The macroscopic behaviour is analysed in terms of actuation stresses, magnetostrictive deformations, and magnetorheological effects. The results obtained for different microstructures and various loadings have been found to be in qualitative agreement with experiments on MRE as well as analytical results.Industrielle Entwicklungsprozesse und die Forschung in Physik, Material- und Ingenieurwissenschaft greifen in einem immer stärkeren Umfang auf rechnergestützte Modellierungs- und Simulationsverfahren zurück. Die ständig steigende Rechenleistung ermöglicht dabei auch die Analyse mehrskaliger und gekoppelter Probleme. In dieser Arbeit kommt daher ein kontinuumsmechanischer Modellierungsansatz auf verschiedenen Skalen zum Einsatz. Das Ziel der Berechnungen ist dabei die Vorhersage des effektiven Material- bzw. Strukturverhaltens auf der Grundlage der lokalen Werkstoffstruktur und der Eigenschafen der konstitutiven Bestandteile. Derartige Simulationen liefern interessante Aussagen zu den Struktur-Eigenschaftsbeziehungen, deren Verständnis entscheidend für das Material- und Strukturdesign ist. Um aussagekräftige Vorhersagen des effektiven Verhaltens zu erhalten, sind numerische Modelle erforderlich, die wesentliche Eigenschaften der lokalen Materialstruktur abbilden. Dabei kommt der effizienten Modellierung von Diskontinuitäten, beispielsweise Materialgrenzen oder Rissen, eine deutlich größere Bedeutung zu als bei einer makroskopischen Betrachtung. In der vorliegenden Arbeit werden zwei unterschiedliche Modellierungsansätze für Unstetigkeiten diskutiert: (i) eine scharfe Abbildung, die üblicherweise konforme Berechnungsnetze erfordert. Da eine Evolution der Mikrostruktur bei einer derartigen Modellierung eine Topologieänderung bzw. eine aufwendige Neuvernetzung nach sich zieht, werden alternativ (ii) diffuse Modelle, die eine zusätzliche Feldvariable zur Regularisierung der Grenzfläche verwenden, betrachtet. Mit der Kombination von (i) Erweiterter Finite-Elemente-Methode (XFEM) + scharfem Grenzflächenmodell sowie (ii) Isogeometrischer Analyse (IGA) + diffuser Grenzflächenmodellierung werden in der vorliegenden Arbeit zwei fundamental verschiedene Zugänge zur Modellierung von Unstetigkeiten betrachtet. Bei der Diskretisierung mit XFEM wird die Kontinuität der Approximation durch eine Anreicherung der Ansatzfunktionen gemäß der abzubildenden Unstetigkeit reduziert. Demgegenüber erfolgt bei einer diffusen Grenzflächenmodellierung eine Regularisierung. Die dazu erforderliche zusätzliche Feldvariable führt oft zu Feldgleichungen mit partiellen Ableitungen höherer Ordnung und weist in ihrem Verlauf starke Gradienten auf. Die daraus resultierenden Anforderungen an den Ansatz werden durch eine Spline-basierte Approximation erfüllt. Um die Effizienz dieser isogeometrischen Diskretisierung weiter zu erhöhen, werden auf der Grundlage hierarchischer Splines adaptive Verfeinerungs- und Vergröberungstechniken entwickelt. Ausgewählte Diskretisierungsverfahren werden zur mehrskaligen Modellierung des gekoppelten magnetomechanischen Verhaltens von Magnetorheologischen Elastomeren (MRE) angewendet. In Kombination mit numerischen Homogenisierungsverfahren, ermöglichen die Mikrostrukturmodelle eine Vorhersage des effektiven magnetomechanischen Verhaltens von MRE. Außerderm wurden Verfahren zur Kopplung von FE-Modellen der MRE-Mikrostruktur mit einem Randelement-Modell der Umgebung vorgestellt. Mit Hilfe der entwickelten Verfahren kann das Verhalten von MRE in Form von Aktuatorspannungen, magnetostriktiven Deformationen und magnetischen Steifigkeitsänderungen vorhergesagt werden. Im Gegensatz zu zahlreichen anderen Modellierungsansätzen, stimmen die mit den hier vorgestellten Methoden für unterschiedliche Mikrostrukturen erzielten Vorhersagen sowohl mit analytischen als auch experimentellen Ergebnissen überein