Smooth Subdivision Surfaces: Mesh Blending and Local Interpolation

Subdivision surfaces are widely used in computer graphics and animation. Catmull-Clark subdivision (CCS) is one of the most popular subdivision schemes. It is capable of modeling and representing complex shape of arbitrary topology. Polar surface, working on a triangle-quad mixed mesh structure, is proposed to solve the inherent ripple problem of Catmull-Clark subdivision surface (CCSS). CCSS is known to be C1 continuous at extraordinary points. In this work, we present a G2 scheme at CCS extraordinary points. The work is done by revising CCS subdivision step with Extraordinary-Points-Avoidance model together with mesh blending technique which selects guiding control points from a set of regular sub-meshes (named dominative control meshes) iteratively at each subdivision level. A similar mesh blending technique is applied to Polar extraordinary faces of Polar surface as well. Both CCS and Polar subdivision schemes are approximating. Traditionally, one can obtain a CCS limit surface to interpolate given data mesh by iteratively solving a global linear system. In this work, we present a universal interpolating scheme for all quad subdivision surfaces, called Bezier Crust. Bezier Crust is a specially selected bi-quintic Bezier surface patch. With Bezier Crust, one can obtain a high quality interpolating surface on CCSS by parametrically adding CCSS and Bezier Crust. We also show that with a triangle/quad conversion process one can apply Bezier Crust on Polar surfaces as well. We further show that Bezier Crust can be used to generate hollowed 3D objects for applications in rapid prototyping. An alternative interpolating approach specifically designed for CCSS is developed. This new scheme, called One-Step Bi-cubic Interpolation, uses bicubic patches only. With lower degree polynomial, this scheme is appropriate for interpolating large-scale data sets. In sum, this work presents our research on improving surface smoothness at extraordinary points of both CCS and Polar surfaces and present two local interpolating approaches on approximating subdivision schemes. All examples included in this work show that the results of our research works on subdivision surfaces are of high quality and appropriate for high precision engineering and graphics usage

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

Structured meshes: composition and remeshing guided by the Curve-Skeleton

Virtual sculpting is currently a broadly used modeling metaphor with rising popularity especially in the entertainment industry. While this approach unleashes the artists' inspiration and creativity and leads to wonderfully detailed and artistic 3D models, it has the side effect, purely technical, of producing highly irregular meshes that are not optimal for subsequent processing. Converting an unstructured mesh into a more regular and struc- tured model in an automatic way is a challenging task and still open prob- lem. Since structured meshes are useful in different applications, it is of in- terest to be able to guarantee such property also in scenarios of part based modeling, which aim to build digital objects by composition, instead of modeling them from a scratch. This thesis will present methods for obtaining structured meshes in two different ways. First is presented a coarse quad layout computation method which starts from a triangle mesh and the curve-skeleton of the shape. The second approach allows to build complex shapes by procedural composition of PAM's. Since both quad layouts and PAMs exploit their global struc- ture, similarities between the two will be discussed, especially how their structure has correspondences to the curve-skeleton describing the topology of the shape being represented. Since both the presented methods rely on the information provided by the skeleton, the difficulties of using automat- ically extracted curve-skeletons without processing are discussed, and an interactive tool for user-assisted processing is presented

Structured meshes: composition and remeshing guided by the Curve-Skeleton

Virtual sculpting is currently a broadly used modeling metaphor with rising popularity especially in the entertainment industry. While this approach unleashes the artists' inspiration and creativity and leads to wonderfully detailed and artistic 3D models, it has the side effect, purely technical, of producing highly irregular meshes that are not optimal for subsequent processing. Converting an unstructured mesh into a more regular and struc- tured model in an automatic way is a challenging task and still open prob- lem. Since structured meshes are useful in different applications, it is of in- terest to be able to guarantee such property also in scenarios of part based modeling, which aim to build digital objects by composition, instead of modeling them from a scratch. This thesis will present methods for obtaining structured meshes in two different ways. First is presented a coarse quad layout computation method which starts from a triangle mesh and the curve-skeleton of the shape. The second approach allows to build complex shapes by procedural composition of PAM's. Since both quad layouts and PAMs exploit their global struc- ture, similarities between the two will be discussed, especially how their structure has correspondences to the curve-skeleton describing the topology of the shape being represented. Since both the presented methods rely on the information provided by the skeleton, the difficulties of using automat- ically extracted curve-skeletons without processing are discussed, and an interactive tool for user-assisted processing is presented

Simulation of electric field-assisted nanowire growth from aqueous solutions

The present work is aimed at investigating the mechanisms of nanowire growth from aqueous solutions through a physical and chemical modeling. Based on this modeling, deriving an optimized process control is intended. The work considers two methods of nanowire growth. The first is the dielectrophoretic nanowire assembly from neutral molecules or metal clusters. Secondly, in the directed electrochemical nanowire assembly metal-containing ions are reduced in an AC electric field in the vicinity of the nanowire tip and afterwards deposited at the nanowire surface. To describe the transport and growth processes, continuum models are employed. Furthermore, it has been necessary to consider electro-kinetic fluid flows to match the experimental observations. The occurring partial differential equations are solved numerically by means of finite element method (FEM). The effect of the process parameters on the nanowire growth are analyzed by comparing experimental results to a parameter study. The evaluation has yielded that an AC electro-osmotic fluid flow has a major influence on the dielectrophoretic nanowire assembly regarding the growth velocity and morphology. In the case of directed electrochemical nanowire assembly, the nanowire morphology can be controlled by the applied AC signal shape. Based on the nanowire growth model, an optimized AC signal has been designed, whose parametrization allows to adjust to the chemical precursor and the desired nanowire diameter.Ziel der vorliegenden Arbeit ist es, mittels physikalischer und chemischer Modelle die Mechanismen des Nanodrahtwachstums aus wässrigen Lösungen zu erforschen und daraus eine optimierte Prozesskontrolle abzuleiten. Dabei werden zwei Verfahren des Nanodrahtwachstums näher betrachtet: Dies sind die dielektrophoretische Assemblierung von neutralen Molekülen oder Metallclustern sowie die gerichtete elektrochemische Nanodrahtabscheidung (engl. directed electrochemical nanowire assembly), bei der metallhaltige Ionen im elektrischen Wechselfeld an der Nanodrahtspitze zunächst reduziert und anschließend als Metallatome abgeschieden werden. Zur Beschreibung der Transport- und Wachstumsprozesse werden Kontinuumsmodelle eingesetzt. Darüber hinaus hat es sich als notwendig erwiesen, elektrokinetische Fluidströmungen zu berücksichtigen, um die experimentellen Beobachtungen zu reproduzieren. Die auftretenden partiellen Differenzialgleichungen werden mittels der Finiten Elemente Methode (FEM) numerisch gelöst. Die Auswirkungen der Prozessparameter auf das Nanodrahtwachstum werden durch den Vergleich von experimentellen Ergebnissen mit Parameterstudien analysiert. Die Auswertung hat ergeben, dass für das dielektrophoretische Wachstum ein durch Wechselfeldelektroosmose (engl. AC electro-osmosis) angetriebener Fluidstrom die Drahtwachstumsgeschwindigkeit und -morphologie maßgeblich beeinflusst. Im Falle der gerichteten elektrochemischen Nanodrahtabscheidung lässt sich die Drahtmorphologie über das angelegte elektrische Wechselsignal steuern. Unter Verwendung des Wachstumsmodells ist ein optimiertes Signal generiert worden, dessen Parametrisierung eine gezielte Anpassung auf den chemischen Ausgangsstoff und den gewünschten Drahtdurchmesser erlaubt

Generative Mesh Modeling

Generative Modeling is an alternative approach for the description of three-dimensional shape. The basic idea is to represent a model not as usual by an agglomeration of geometric primitives (triangles, point clouds, NURBS patches), but by functions. The paradigm change from objects to operations allows for a procedural representation of procedural shapes, such as most man-made objects. Instead of storing only the result of a 3D construction, the construction process itself is stored in a model file. The generative approach opens truly new perspectives in many ways, among others also for 3D knowledge management. It permits for instance to resort to a repository of already solved modeling problems, in order to re-use this knowledge also in different, slightly varied situations. The construction knowledge can be collected in digital libraries containing domain-specific parametric modeling tools. A concrete realization of this approach is a new general description language for 3D models, the "Generative Modeling Language" GML. As a Turing-complete "shape programming language" it is a basis of existing, primitv based 3D model formats. Together with its Runtime engine the GML permits - to store highly complex 3D models in a compact form, - to evaluate the description within fractions of a second, - to adaptively tesselate and to interactively display the model, - and even to change the models high-level parameters at runtime.Die generative Modellierung ist ein alternativer Ansatz zur Beschreibung von dreidimensionaler Form. Zugrunde liegt die Idee, ein Modell nicht wie üblich durch eine Ansammlung geometrischer Primitive (Dreiecke, Punkte, NURBS-Patches) zu beschreiben, sondern durch Funktionen. Der Paradigmenwechsel von Objekten zu Geometrie-erzeugenden Operationen ermöglicht es, prozedurale Modelle auch prozedural zu repräsentieren. Statt das Resultat eines 3D-Konstruktionsprozesses zu speichern, kann so der Konstruktionsprozess selber repräsentiert werden. Der generative Ansatz eröffnet unter anderem gänzlich neue Perspektiven für das Wissensmanagement im 3D-Bereich. Er ermöglicht etwa, auf einen Fundus bereits gelöster Konstruktions-Aufgaben zurückzugreifen, um sie in ähnlichen, aber leicht variierten Situationen wiederverwenden zu können. Das Konstruktions-Wissen kann dazu in Form von Bibliotheken parametrisierter, Domänen-spezifischer Modellier-Werkzeuge gesammelt werden. Konkret wird dazu eine neue allgemeine Modell-Beschreibungs-Sprache vorgeschlagen, die "Generative Modeling Language" GML. Als Turing-mächtige "Programmiersprache für Form" stellt sie eine echte Verallgemeinerung existierender Primitiv-basierter 3D-Modellformate dar. Zusammen mit ihrer Runtime-Engine erlaubt die GML, - hochkomplexe 3D-Objekte extrem kompakt zu beschreiben, - die Beschreibung innerhalb von Sekundenbruchteilen auszuwerten, - das Modell adaptiv darzustellen und interaktiv zu betrachten, - und die Modell-Parameter interaktiv zu verändern

Mixed-Cell Methods for Diffusion Problems in Multiphase Systems.

We simulate diffusion in multimaterial systems with a cell-centered Eulerian mesh in two dimensions. A system with immiscible fluids contains sharp interfaces. An Eulerian mesh is fixed in space and does not move with the material. Therefore, cells with an interface contain multiple fluids; these are known as mixed cells. The treatment of mixed cells can vary in computational cost and accuracy. In some cases, the primary source of inaccuracy can be attributed to approximations made in modeling the mixed cells. This thesis focuses on the treatment of mixed cells based on the diffusion approximation of the transport equation. We introduce five subgrid, mixed-cell models. Two models have a single temperature for each cell, while the other three allow a separate temperature for each phase. The single-temperature models are implemented using the Support-Operators Method, which is derived herein. The first single-temperature model utilizes an effective tensor diffusivity that distinguishes diffusion tangent and normal to the interface. The second single-temperature model specifies a unique diffusivity in each corner of a mixed cell, which is effectively a mesh refinement of the mixed cell. The three multi-temperature models have increasingly accurate levels of approximation of the flux: (i) flux is calculated between cell-centers for each phase, (ii) flux is calculated between the centroid of each phase, and (iii) flux normal to an interface is calculated between centroids of each phase. The physical interpretations of these models are: (i) each phase occupies the entire cell, (ii) oblique flux is continuous, (iii) only normal flux is continuous. The standard approximation, using the harmonic mean of the diffusivities present in a mixed cell as an effective diffusivity, is also tested for comparison. We also derive two time-dependent analytical solutions for diffusion in a two-phase system, in both one and two dimensions. With the standard model as a reference point, the accuracy of the new models is quantified, and the convergence rates of the error are determined between pairs of spatial resolutions for the two problems with analytical solutions. Simulations of multiphysics and multimaterial phenomenon may benefit from increased mixed-cell fidelity achieved in this dissertation.PhDApplied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/107150/1/leftynm_1.pd