Ambient Isotopic Meshing of Implicit Algebraic Surface with Singularities

A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a symbolic-numeric method to compute a certified meshing for the surface inside a box containing singularities and use a modified Plantinga-Vegter marching cube method to compute a certified meshing for the surface inside a box without singularities. Nontrivial examples are given to show the effectiveness of the algorithm. To our knowledge, this is the first method to compute a certified meshing for surfaces with singularities.Comment: 34 pages, 17 Postscript figure

Two triangulations methods based on edge refinement

In this paper two curvature adaptive methods of surface triangulation are presented. Both methods are based on edge refinement to obtain a triangulation compatible with the curvature requirements. The first method applies an incremental and constrained Delaunay triangulation and uses curvature bounds to determine if an edge of the triangulation is admissible. The second method uses this function also in the edge refinement process, i.e. in the computation of the location of a refining point, and in the re-triangulation needed after the insertion of this refining point. Results are presented, comparing both approachesPostprint (published version

A scalable, efficient scheme for evaluation of stencil computations over unstructured meshes

pre-printStencil computations are a common class of operations that appear in many computational scientific and engineering applications. Stencil computations often benefit from compile-time analysis, exploiting data-locality, and parallelism. Post-processing of discontinuous Galerkin (dG) simulation solutions with B-spline kernels is an example of a numerical method which requires evaluating computationally intensive stencil operations over a mesh. Previous work on stencil computations has focused on structured meshes, while giving little attention to unstructured meshes. Performing stencil operations over an unstructured mesh requires sampling of heterogeneous elements which often leads to inefficient memory access patterns and limits data locality/reuse. In this paper, we present an efficient method for performing stencil computations over unstructured meshes which increases data-locality and cache efficiency, and a scalable approach for stencil tiling and concurrent execution. We provide experimental results in the context of post-processing of dG solutions that demonstrate the effectiveness of our approach

Subdivision surface fitting to a dense mesh using ridges and umbilics

Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach

Free-form deformation of solid models in CSR.

Lai Chi-fai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.Includes bibliographical references (leaves 98-99).Abstracts in English and Chinese.Chapter 1. --- INTRODUCTION --- p.7Chapter 1.1 --- Motivations and objectives --- p.7Chapter 1.2 --- Thesis Organization --- p.10Chapter 2. --- related works --- p.11Chapter 2.1 --- Deformation Techniques --- p.11Chapter 2.1.1 --- Deformation techniques requiring a deformation tool --- p.11Chapter 2.1.2 --- Directly specified deformation techniques --- p.14Chapter 2.1.3 --- Comparison on Different Deformation Technique --- p.15Chapter 2.2 --- Application of Deformation --- p.16Chapter 2.2.1 --- Deforming superquadrics --- p.16Chapter 2.2.2 --- Volume wraping --- p.16Chapter 2.2.3 --- Deforming linear object --- p.17Chapter 2.2.4 --- FFD for animation synthesis --- p.17Chapter 2.2.5 --- Using FFD on feature-based Surface --- p.18Chapter 2.2.6 --- NURBS-BASED Free-Form Deformation (NFFD) --- p.18Chapter 2.3 --- Algebraic Patch Techniques --- p.20Chapter 2.3.1 --- Dahmen's scheme --- p.20Chapter 2.3.2 --- Lodha and Warren's technique --- p.20Chapter 2.3.3 --- Guo's method --- p.21Chapter 3. --- BACKGROUND THEORIES --- p.22Chapter 3.1 --- Algebraic Patches --- p.22Chapter 3.1.1 --- Bernstein-Bezier representation of a single patch --- p.22Chapter 3.1.2 --- Constructing free-form objects --- p.29Chapter 3.1.2.1 --- Bounding volumes for quadric patches --- p.29Chapter 3.1.2.2 --- Filling two-sided gaps --- p.31Chapter 3.2 --- Constructive Shell Representation --- p.35Chapter 3.2.1 --- Properties of quadric patches and its construction tetrahedron and trunctets --- p.38Chapter 3.3 --- Free-Form Deformation --- p.40Chapter 3.3.1 --- Formulating free-form deformation --- p.40Chapter 4. --- FREE-FORM DEFORMATION OF CSR SOLID MODELS --- p.43Chapter 4.1 --- Determination of Lattice Structure --- p.43Chapter 4.2 --- "Relation between weights, normals and shape of a trunctet" --- p.46Chapter 4.3 --- Applying FFD on CSR solid models --- p.49Chapter 4.3.1 --- Deforming normal at vertices --- p.52Chapter 4.3.2 --- Using vertices' neighborhoods --- p.54Chapter 4.4 --- Free-Form Deformation of CSR objects by Surface Fitting --- p.57Chapter 4.4.1 --- Deforming a single surface patch --- p.57Chapter 4.4.1.1 --- Locating surface points --- p.59Chapter 4.4.1.2 --- Conversion between barycentric and Cartesian coordinates --- p.61Chapter 4.4.1.3 --- Evaluating the deformed surface patch --- p.62Chapter 4.4.1.4 --- Saddle shape trunctet --- p.64Chapter 4.4.1.5 --- Using double tetrahedrons --- p.66Chapter 4.4.1.6 --- Surface subdivision --- p.69Chapter 4.4.2 --- Deforming Entire Solid Model --- p.72Chapter 4.4.3 --- Comparison on different approaches --- p.75Chapter 4.5 --- Conversion of CSG solid Models into CSR --- p.76Chapter 4.5.1 --- Converting halfspaces into CSR objects --- p.77Chapter 5. --- IMPLEMENTATION AND RESULTS --- p.82Chapter 5.1 --- Implementation --- p.82Chapter 5.2 --- Experimental Results --- p.84Chapter 6. --- CONCLUSION AND SUGGESTIONS FOR FURTHER WORK --- p.93Chapter 6.1 --- Conclusion --- p.93Chapter 6.2 --- Suggestions for further work --- p.9

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation

This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g.\ the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. We describe the method both for domains meshed by mapped quadrilaterals and triangles, introducing for each case (i) well-conditioned methods for the production of certain requisite source polynomial interpolants and (ii) efficient translation formulae for polynomial particular solutions. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology