Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Isogeometric analysis : applications for torque and drag models, drillstring and bottom-hole assembly design

The drilling industry today relies on torque and drag models to analyze and ensure success during all phases of well construction and operations, including planning, drilling, and completion. Analytical models are based on equations that are undergoing constant development and improvement. The finite element method is an alternative to complex analytical calculations that is used often to determine torque and drag forces that are present when a drillstring is lowered, raised, and rotated in a wellbore. Traditional finite element analysis (FEA), however, is not time efficient or computationally able to simulate the complexities of a real wellbore. Thus, we introduce an alternative to the traditional finite element approach: isogeometric analysis. Isogeometric analysis is similar to finite element analysis except that it uses NURBS (Non-Uniform Rational B-Splines), as opposed to interpolatory polynomials used in traditional FEA, as the basis functions. NURBS functions are the same as those used in CAD programs, and they are able to construct exact conic shapes, such as circles and ellipses. Adopting NURBS basis functions allows finite element analysis to be performed directly on the exact geometrical surface - not on an approximate geometric surface mesh, as in traditional FEA. IGA yields a significantly faster and more accurate simulation. This thesis presents a real-world application of IGA to a drag force model to determine the resultant surface hook load during run-in-hole (RIH) operations. Real well data is used, and IGA results are compared to a similar FEA analysis. The outcome shows that IGA is indeed a superior finite element method that has immense potential for further application in the industryPetroleum and Geosystems Engineerin

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

Finite Element Analysis for Linear Elastic Solids Based on Subdivision Schemes

Finite element methods are used in various areas ranging from mechanical engineering to computer graphics and bio-medical applications. In engineering, a critical point is the gap between CAD and CAE. This gap results from different representations used for geometric design and physical simulation. We present two different approaches for using subdivision solids as the only representation for modeling, simulation and visualization. This has the advantage that no data must be converted between the CAD and CAE phases. The first approach is based on an adaptive and feature-preserving tetrahedral subdivision scheme. The second approach is based on Catmull-Clark subdivision solids

Smoothness of Nonlinear and Non-Separable Subdivision Schemes

We study in this paper nonlinear subdivision schemes in a multivariate setting allowing arbitrary dilation matrix. We investigate the convergence of such iterative process to some limit function. Our analysis is based on some conditions on the contractivity of the associated scheme for the differences. In particular, we show the regularity of the limit function, in $L^p$ and Sobolev spaces

Recursive subdivision algorithms for curve and surface design

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.The Chinese Educational Commission and The British Council (SBFSS/1987