Optimal Order Convergence Implies Numerical Smoothness

It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability

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

Error bounded approximate reparametrization of NURBS curves

Journal ArticleThis paper reports research on solutions to the following reparametrization problem: approximate c(r(t)) by a NURBS where c is a NURBS curve and r may, or may not, be a NURBS function. There are many practical applications of this problem including establishing and exploring correspondence in geometry, creating related speed profiles along motion curves for animation, specifying speeds along tool paths, and identifying geometrically equivalent, or nearly equivalent, curve mappings. A framework for the approximation problem is described using two related algorithmic schemes. One constrains the shape of the approximation to be identical to the original curve c. The other relaxes this constraint. New algorithms for important cases of curve reparametrization are developed from within this framework. They produce results with bounded error and address approximate arc length parametrizations of curves, approximate inverses of NURBS functions, and reparametrizations that establish user specified tolerances as bounds on the Frechet distance between parametric curves

On multi-degree splines

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to B\'ezier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean\u2013hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Quasi-Interpolation in a Space of C 2 Sextic Splines over Powell–Sabin Triangulations

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell– Sabin triangulations. These spline functions are of class C 2 on the whole domain but fourth-order regularity is required at vertices and C 3 regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators.Erasmus+ International Dimension programme, European CommissionPAIDI programme, Junta de Andalucía, Spai

Numerics and Fractals

Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions themselves are known to be the fixed points of an associated Read-Bajactarevi\'c operator. This paper establishes existence and properties of local fractal functions and discusses how they are computed. In particular, it is shown that piecewise polynomials are a special case of local fractal functions. Finally, we develop a method to compute the components of a local IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note: substantial text overlap with arXiv:1309.0243. text overlap with arXiv:1309.024

Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT

We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients $\sigma(x)$ in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions $s(x)$ over the domain $\Omega$. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of $\Omega$ when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we obtain an optimally robust homogenization algorithm for arbitrary rough coefficients. Next, we consider inverse homogenization and show how to decompose it into a linear ill-posed problem and a well-posed non-linear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT). It is known that the EIT problem admits at most one isotropic solution. If an isotropic solution exists, we show how to compute it from any conductivity having the same boundary Dirichlet-to-Neumann map. It is known that the EIT problem admits a unique (stable with respect to $G$-convergence) solution in the space of divergence-free matrices. As such we suggest that the space of convex functions is the natural space in which to parameterize solutions of the EIT problem