Isotopic Equivalence from Bezier Curve Subdivision

We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035

Finding antipodal point grasps on irregularly shaped objects

Two-finger antipodal point grasping of arbitrarily shaped smooth 2-D and 3-D objects is considered. An object function is introduced that maps a finger contact space to the object surface. Conditions are developed to identify the feasible grasping region, F, in the finger contact space. A “grasping energy function”, E , is introduced which is proportional to the distance between two grasping points. The antipodal points correspond to critical points of E in F. Optimization and/or continuation techniques are used to find these critical points. In particular, global optimization techniques are applied to find the “maximal” or “minimal” grasp. Further, modeling techniques are introduced for representing 2-D and 3-D objects using B-spline curves and spherical product surfaces

A sharp interface isogeometric strategy for moving boundary problems

The proposed methodology is first utilized to model stationary and propagating cracks. The crack face is enriched with the Heaviside function which captures the displacement discontinuity. Meanwhile, the crack tips are enriched with asymptotic displacement functions to reproduce the tip singularity. The enriching degrees of freedom associated with the crack tips are chosen as stress intensity factors (SIFs) such that these quantities can be directly extracted from the solution without a-posteriori integral calculation. As a second application, the Stefan problem is modeled with a hybrid function/derivative enriched interface. Since the interface geometry is explicitly defined, normals and curvatures can be analytically obtained at any point on the interface, allowing for complex boundary conditions dependent on curvature or normal to be naturally imposed. Thus, the enriched approximation naturally captures the interfacial discontinuity in temperature gradient and enables the imposition of Gibbs-Thomson condition during solidification simulation. The shape optimization through configuration of finite-sized heterogeneities is lastly studied. The optimization relies on the recently derived configurational derivative that describes the sensitivity of an arbitrary objective with respect to arbitrary design modifications of a heterogeneity inserted into a domain. The THB-splines, which serve as the underlying approximation, produce sufficiently smooth solution near the boundaries of the heterogeneity for accurate calculation of the configurational derivatives. (Abstract shortened by ProQuest.

The Bernstein basis in set-theoretic geometric modelling

SIGLEAvailable from British Library Document Supply Centre-DSC:DXN037062 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Tissue thickness measurement tool for craniofacial reconstruction

Craniofacial Reconstruction is a method of recreating the appearance of the face on the skull of a deceased individual for identification purposes. Older clay methods of reconstruction are inaccurate, time consuming and inflexible. The tremendous increase in the processing power of the computers and rapid strides in visualization can be used to perform the reconstruction, saving time and providing greater accuracy and flexibility, without the necessity for a skillful modeler.;This thesis introduces our approach to computerized 3D craniofacial reconstruction. Three phases have been identified. The first phase of the project is to generate a facial tissue thickness database. In the second phase this database along with a 3D facial components database is to be used to generate a generic facial mask which is draped over the skull to recreate the facial appearance. This face is to be identified from a database of images in the third phase.;Tissue thickness measurements are necessary to generate the facial model over the skull. The thesis emphasis is on the first phase of the project. An automated facial tissue thickness measurement tool (TTMT) has been developed to populate this database

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

Automatic mesh generation

The objective of this thesis project is a study of Pre-Processors and development of an Automatic Mesh Generator for Finite Element Analysis. The Mesh Generator developed in this thesis project can create triangular finite elements from the geometric database of Macintosh Applications. The user is required to give the density parameter to the program for mesh generation. The research is limited to Mesh Generators of planar surfaces. Delauny Triangulation method maximizes the minimum angles of a triangle. Watson\u27s Delauny Triangulation method can mesh only the \u27convex hull\u27 of a set of nodes. This algorithm has been modified to create triangular elements in convex and non-convex surfaces. The surfaces can have holes also. A node generation algorithm to place nodes on and inside a geometry has been developed in this thesis project. The mesh generation is very efficient and flexible. Geometric modeling methods have been studied to understand and integrate the Geometric Modeler with the Finite Element Mesh Generator. Expert Systems can be integrated with Finite Element Analysis. This will make Finite Element Method fully automatic. In this thesis project, Expert Systems in Finite Element Analysis are reviewed. Proposals are made for future approach for the integration of the two fields

Ισογεωμετρική Στατική Ανάλυση με T-SPLines

Σκοπός αυτής της διπλωματικής είναι η διερεύνηση της ισογεωμετρικής στατικής ανάλυσης χρησιμοποιώντας ΄ενα νέο έιδος συναρτήσεων σχήματος , τις T-SPLines. Τόσο οι T-SPLines όσο και η ανάλυση πεπερασμένων στοιχείων εετάστηκαν ξεχωριστά αφού αποτελούν τις δύο συνιστώσες της ισογεωμετρικής μεθόδου. Τα θέματα που εξετάστηκαν είναι οι T-SPLines και οι ιδιότητές τους, οι τεχνικές πύκνωσης του δικτύου , η μόρφωση του μητρώου στιβαρότητας, η επεξεργασία των αποτελεσμάτων της ανάλυσης (πεδίο μετατοπίσεων, τάσεων και παραμορφώσεων) και εφαρμογές 2Δ για τη διερεύνηση διαφόρων φορέων.The scope of this thesis if the investigation of static isogeometric analysis unsing a new type of shape functions T-SPLines. T-SPLines and finite elements have been examined separately, as the two components of the isogeometric method. The topics considered are T-SPLine formulation and properties, refinement techniques, stiffness matrix formulation , result post-processing (displacement, stress and strain field) and linear 2D applications investigating models of various representations.Δημήτριος Γ. Τσαπέτη

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest