Invariant-geometry conditions for the rational bi-quadratic Bézier surfaces

A generalization of Patterson’s work (Patterson, 1985), on the invariants of the rational Bézier curves, to the case of surfaces is presented. An equation for the determination of the invariants for surfaces of degree (n, n) is derived and solved for the bi-quadratics – for which it is shown that seven independent, invariant functions exist. Explicit forms of the invariants are derived and a number of applications are presented

Parametric Interpolation To Scattered Data [QA281. A995 2008 f rb].

Dua skema interpolasi berparameter yang mengandungi interpolasi global untuk data tersebar am dan interpolasi pengekalan-kepositifan setempat data tersebar positif dibincangkan. Two schemes of parametric interpolation consisting of a global scheme to interpolate general scattered data and a local positivity-preserving scheme to interpolate positive scattered data are described

Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

Multisided generalisations of Gregory patches

We propose two generalisations of Gregory patches to faces of any valency by using generalised barycentric coordinates in combination with two kinds of multisided Bézier patches. Our first construction builds on S-patches to generalise triangular Gregory patches. The local construction of Chiyokura and Kimura providing G1 continuity between adjoining Bézier patches is generalised so that the novel Gregory S-patches of any valency can be smoothly joined to one another. Our second construction makes a minor adjustment to the generalised Bézier patch structure to allow for cross-boundary derivatives to be defined independently per side. We show that the corresponding blending functions have the inherent ability to blend ribbon data much like the rational blending functions of Gregory patches. Both constructions take as input a polygonal mesh with vertex normals and provide G1 surfaces interpolating the input vertices and normals. Due to the full locality of the methods, they are well suited for geometric modelling as well as computer graphics applications relying on hardware tessellation

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

Almost-C1C^1 splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems

Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-$C^1$ are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are $C^1$ smooth almost everywhere, that is, at all vertices and across most edges, and in addition almost (i.e. approximately) $C^1$ smooth across all other edges. Thus, the splines form $H^2$-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-$C^1$ splines are described in an explicit B\'ezier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behavior

The Construction of Optimized High-Order Surface Meshes by Energy-Minimization

Despite the increasing popularity of high-order methods in computational fluid dynamics, their application to practical problems still remains challenging. In order to exploit the advantages of high-order methods with geometrically complex computational domains, coarse curved meshes are necessary, i.e. high-order representations of the geometry. This dissertation presents a strategy for the generation of curved high-order surface meshes. The mesh generation method combines least-squares fitting with energy functionals, which approximate physical bending and stretching energies, in an incremental energy-minimizing fitting strategy. Since the energy weighting is reduced in each increment, the resulting surface representation features high accuracy. Nevertheless, the beneficial influence of the energy-minimization is retained. The presented method aims at enabling the utilization of the superior convergence properties of high-order methods by facilitating the construction of coarser meshes, while ensuring accuracy by allowing an arbitrary choice of geometric approximation order. Results show surface meshes of remarkable quality, even for very coarse meshes representing complex domains, e.g. blood vessels