Isogeometric Segmentation: Construction of auxiliary curves

We present a novel technique for segmenting a three-dimensional solid with a 3-vertex-connected edge graph consisting of only convex edges into a collection of topological hexa-hedra. Our method is based on the edge graph, which is defined by the sharp edges between the boundary surfaces of the solid. We repeatedly decompose the solid into smaller solids until all of them belong to a certain class of predefined base solids. The splitting step of the algorithm is based on simple combinatorial and geometric criteria. The segmentation technique described in the paper is part of a process pipeline for solving the isogeometric segmentation problem that we outline in the paper.

Motivated by the discretization problem in isogeometric analysis, we consider the challenge of segmenting a contractible boundary-represented solid into a small number of topological hexahedra. A satisfactory segmentation of a solid must eliminate non-convex edges because they prevent regular parameterizations. Our method works by searching a sufficiently connected edge graph of the solid for a cycle of vertices, called a cutting loop, which can be used to decompose the solid into two new solids with fewer non-convex edges. This can require the addition of auxiliary vertices to the edge graph. We provide theoretical justification for our approach by characterizing the cutting loops that can be used to segment the solid, and proving that the algorithm terminates. We select the cutting loop using a cost function. For this cost function we propose terms which help to select geometrically and combinatorially favorable cutting loops. We demonstrate the effects of these terms using a suite of examples.

We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analysis (IGA). The input into our pipeline is a boundary represented 3D model, either as a triangulation or as a collection of trimmed non-uniform rational B-spline (NURBS) surfaces. The pipeline consists of three stages: computer aided design (CAD) model reconstruction from a triangulation (if necessary); segmentation of the boundary-represented solid into topological hexahe-dra; and volume parameterization. The result is a collection of volumetric NURBS patches. In this paper we discuss our methods for the three stages, and demonstrate the suitability of the result for IGA by performing stress simulations with examples of the output.

Computer Methods in Applied Mechanics and Engineering, 2009

We propose a new isogeometric method using Toric surface patches for trimmed CAD planar surfaces. This method converts each trimmed spline element into a Toric surface patch with conforming boundary representation and converts each non-trimmed spline element into a Bézier element. Because the Toric surface patches are a multi-sided generalization of classical Bézier surface patches, all trimmed and non-trimmed elements of a trimmed CAD surface have a unified geometric representation using Toric surface patches. Toric surface patches share the advantages of isogeometric continuum elements in that they can exactly model the geometry and can be easily implemented in standard finite-element code architectures. Several numerical examples are used to demonstrate the reliability of the proposed method.

Computers & Graphics, 1987

calculated of the interactive ve was generated (1 S) (this implies r n) and then inin monotonicity re the conditions urv, 1 a spline 317 (l':lo6). sciola, Analysis of of"Basic L-splines." sciola, Using intercattered data. IEEE 4(7), 43-45 (1984). \onotone piecewise • Anal. 17(2), 238n bv some Hermite 1ppi. Math. 4, 7-9 ,ch verkoppelte un-1e interpolation. El.

The Visual Computer, 2002

In a landmark paper, Catmull and Clark described a simple generalization of the subdivision rules for bi-cubic B-splines to arbitrary quadrilateral surface meshes. This subdivision scheme has become a mainstay of surface modeling systems. Joy and Mac-Cracken described a generalization of this surface scheme to volume meshes. Unfortunately, little is known about the smoothness and regularity of this scheme due to the complexity of the subdivision rules. This paper presents an alternative subdivision scheme for hexahedral volume meshes that consist of a simple split and average algorithm. Along extraordinary edges of the volume mesh, the scheme provably converges to a smooth limit volume. At extraordinary vertices, the authors supply strong experimental evidence that the scheme also converges to a smooth limit volume. The scheme automatically produces reasonable rules for non-manifold topology and can easily be extended to incorporate boundaries and embedded creases expressed as Catmull-Clark surfaces and B-spline curves.

Computer Methods in Applied Mechanics and Engineering, 2011

We develop optimal approximation estimates for T-splines in the case of geometries obtained by gluing two standard tensor product patches. We derive results both for the T-spline space in the parametric domain and the mapped T-NURBS in the physical one. A set of numerical tests in complete accordance with the theoretical developments is also presented. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c m a the first theoretical result on error analysis of the Isogeometric method based on T-splines.

Journal of Theoretical and Applied Mechanics, 2017

Isogeometric Analysis (IGA) involves unification of modelling and analysis by adopting the same basis functions (splines), for both. Hence, spline based parametric model is the starting step for IGA. Representing a complex domain, using parametric geometric model is a challenging task. Parameterization problem can be defined as, finding an optimal set of control points of a B-spline model for exact domain modelling. Also, the quality of parameterization, too has significant effect on IGA. Finding the B-spline control points for any given domain, which gives accurate results is still an open issue. In this paper, a new planar B-spline parameterization technique, based on domain mapping method is proposed. First step of the methodology is to map an input (non-convex) domain onto a unit circle (convex) with the use of harmonic functions. The unique properties of harmonic functions: global minima and mean value property, ensures the mapping is bi-jective and with no self-intersections. ...

Volume 1: 21st Computers and Information in Engineering Conference, 2001

A computer code for the generation of unstructured two-dimensional triangular meshes around arbitrary complex geometries has been developed. The code is based on Delaunay triangulation with an automatic point insertion scheme and a smoothing technique. The geometrical definition of the domain to be meshed is prescribed by means of B-spline curves obtained from two approaches of interest in Computer-Aided Geometric Design named inverse design and interpolation problems. The presented scheme is based on an interpolation procedure along a B-spline curve proposed by the author in a recent paper. This technique prevents that the resulting grid may overlap convex portions of the boundaries. The main goal is to study the possibility of extend the methodology of unstructured grid generation beginning with boundaries described by polylines to other in which they are prescribed by piecewise polynomials curves capable to drive more realistic problems. Several figures and examples from Computat...

Applied Mathematics and Computation, 1998

Many applications deal with the rendering of trimmed surfaces and the generation of grids for trimmed surfaces. Usually, a structured or unstructured grid must be constructed in the parameter space of the trimmed surface. Trimmed surfaces not only cause problems in the context of grid generation but also when exchanging data between different CAD systems. This paper describes a new approach for decomposing the valid part of the parameter space of a trimmed surface into a set of four-sided surfaces. The boundaries of these four-sided surfaces are line segments, segments of the trimming curves themselves, and segments of bisecting curves that are defined by a generalized Vorondi diagram implied by the trimming curves in parameter space. We use a triangular background mesh for the computation of the bisecting curves of the generalized Vorono'i diagram.