Final Triangulation Research Articles

Terminal Triangles Centroid Algorithms for Quality Delaunay Triangulation

Two Lepp algorithms for quality Delaunay triangulation are discussed. Firstly a terminal triangles centroid Delaunay algorithm is studied. For each bad quality triangle t, the algorithm uses the longest edge propagating path (Lepp(t)) to find a couple of Delaunay terminal triangles (with largest angles less than or equal to 120 degrees) sharing a common longest (terminal) edge. Then the centroid of the terminal quadrilateral is Delaunay inserted in the mesh. Insertion of the midpoints of some constrained edges are also performed to assure convergence close to the constrained edges. We prove algorithm termination and that a graded, optimal size, 30 degrees triangulation is obtained, for any planar straight line graph (PSLG) geometry with constrained angles greater than or equal to 30 degrees. We also prove that the size of the final triangulation is optimal and that this size is independent of the processing order of the bad triangles in the mesh. Next, by introducing the concept of non-improvable triangles (with constrained angle < 30 degrees), we generalize the algorithm to deal with PSLG geometries with N small constrained angles. Thus given a triangle size parameter δ for non-improvable triangles, the generalized algorithm constructs a quality triangulation with non constrained angles ≥ 30 degrees and at most N non-improvable triangles of size δ (longest edge ≤δ). In practice the algorithms behave as predicted by the theory.

Efficiently computing feature-aligned and high-quality polygonal offset surfaces

3D surface offsetting is a fundamental geometric operation in CAD/CAE/CAM. In this paper, we propose a super-linear convergent algorithm to generate a well-triangulated and feature-aligned offset surface based on particle system. The key idea is to distribute a set of moveable sites as uniformly as possible while keeping these sites at a specified distance away from the base surface throughout the optimization process. In order to make the final triangulation align with geometric feature lines, we use the moveable sites to predict the potential feature regions, which in turn guide the distribution of moveable sites. Our algorithm supports multiple kinds of input surfaces, e.g., triangle meshes, implicit functions, parametric surfaces and even point clouds. Compared with existing algorithms on surface offsetting, our algorithm has significant advantages in terms of meshing quality, computational performance, topological correctness and feature alignment.

GPU Local Triangulation: an interpolating surface reconstruction algorithm

AbstractA GPU capable method for surface reconstruction from unorganized point clouds without additional information, called GLT (GPU Local Triangulation), is presented. The main objective of this research is the generation of a GPU interpolating reconstruction based on local Delaunay triangulations, inspired by a pre‐existing reconstruction algorithm. Current graphics hardware accelerated algorithms are approximating approaches, where the final triangulation is usually performed through either marching cubes or marching tetrahedras. GPU‐compatible methods and data structures to perform normal estimation and the local triangulation have been developed, plus a variation of the Bitonic Merge Sort algorithm to work with multi‐lists. Our method shows an average gain of one order of magnitude over previous research.

Triangular mesh generation employing a boundary expansion technique

This paper presents a triangulation method for modelling very large sets of cloud data. The three-dimensional (3D) data sets are produced by a machine vision system and/or coordinate measuring machine (CMM). The algorithm is suitable for processing the data collected from objects composed of free form surface patches especially with interior holes. This is accomplished from the 3D data sets in two steps. Firstly, the original cloud data is reduced into a simplified data set employing a data reduction technique (voxel binning method), in which the error between the cloud data and the meshed surface is used to control the data reduction. Secondly, the triangulation process starts with a randomly selected seed triangle. The triangular mesh extends outward by continuously linking suitable external points to it along the boundary edges of the meshed area. A complex free form surface with interior holes can be triangulated in one computing session without manually dividing it into several simple patches. The error-based data reduction parameters are extracted from the cloud data set, by a series of local surface patches, and the required spatial error between the final triangulation and the cloud data. Experimental results are given to illustrate the efficacy of the technique for rapidly constructing a geometric model from 3D digitised cloud data.

Adaptive multi‐resolution triangulations based on physical compression

AbstractThis paper presents a method for generating multi‐resolution adaptive triangulations of non‐manifold 3D objects composed of several regions with arbitrary geometry. The process first immerses the input boundary elements inside a semi‐regular adaptive tetrahedral mesh known as a BMT triangulation. The mesh elements are then pushed towards the boundary by means of a physically based compression scheme, more specifically, a mass‐spring system. The final triangulation has no degenerated tetrahedra and provides an approximation of the boundary based on a chosen resolution. Copyright © 2005 John Wiley &amp; Sons, Ltd.

Merging of intersecting triangulations for finite element modeling

Surface mesh generation over intersecting triangulations is a problem common to many branches of biomechanics. A new strategy for merging intersecting triangulations is described. The basis of the method is that object surfaces are represented as the zero-level iso-surface of the distance-to-surface function defined on a background grid. Thus, the triangulation of intersecting objects reduces to the extraction of an iso-surface from an unstructured grid. In a first step, a regular background mesh is constructed. For each point of the background grid, the closest distance to the surface of each object is computed. Background points are then classified as external or internal by checking the direction of the surface normal at the closest location and assigned a positive or negative distance, respectively. Finally, the zero-level iso-surface is constructed. This is the final triangulation of the intersecting objects. The overall accuracy is enhanced by adaptive refinement of the background grid elements. The resulting surface models are used as support surfaces to generate three-dimensional grids for finite element analysis. The algorithms are demonstrated by merging arterial branches independently reconstructed from contrast-enhanced magnetic resonance images and by adding extra features such as vascular stents. Although the methodology is presented in the context of finite element analysis of blood flow, the algorithms are general and can be applied in other areas as well.

Cloud data modelling employing a unified, non-redundant triangular mesh

This paper describes an application of error-based triangulation to very large sets of three-dimensional (3D) data. The algorithm is suitable for processing data collected by machine vision systems, co-ordinate measuring machines or laser-based range sensors. The algorithm models the large data sets, termed cloud data, using a unified, non-redundant triangular mesh. This is accomplished from the 3D data points in two steps. Firstly, an initial data thinning is performed, to reduce the copious data set size, employing 3D spatial filtering. Secondly, the triangulation commences utilising a set of heuristic rules, from a user defined seed point. The triangulation algorithm interrogates the local geometric and topological information inherent in the cloud data points. The spatial filtering parameters are extracted from the cloud data set, by a series of local surface patches, and the required spatial error between the final triangulation and the cloud data. Two procedures are subsequently employed to enhance the mesh: (i) the edges of mesh triangles are adjusted to produce a mesh containing approximately equilateral triangles; and (ii) mesh edges are aligned with the boundaries present on the object to minimise smoothing of naturally occurring features. Case studies are presented that illustrate the efficacy of the technique for rapidly constructing a geometric model from 3D digitised data.

A Robust Implementation for Three-Dimensional Delaunay Triangulations

This paper presents an implementation for Delaunay triangulations of three-dimensional point sets. The code uses a variant of the randomized incremental flip algorithm and employs symbolic perturbation to achieve robustness. The algorithm's theoretical time complexity is quadratic in n, the number of input points, and this is optimal in the worst case. However, empirical running times are proportional to the number of triangles in the final triangulation, which is typically linear in n. Even though the symbolic perturbation scheme relies on exact arithmetic, the resulting code is efficient in practice. This is due to a careful implementation of the geometric primitives and the arithmetic module. The source code is available on the Internet.

Linear-size nonobtuse triangulation of polygons

We give an algorithm for triangulatingn-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than ?/2. The number of triangles in the triangulation is onlyO(n), improving a previous bound ofO(n2), and the running time isO(n log2n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.

A fast algorithm for generating constrained delaunay triangulations

A fast algorithm for generating constrained two-dimensional Delaunay triangulations is described. The scheme permits certain edges to be specified in the final triangulation, such as those that correspond to domain boundaries or natural interfaces, and is suitable for mesh generation and contour plotting applications. Detailed timing statistics indicate that its CPU time requirement is roughly proportional to the number of points in the data set. Subject to the conditions imposed by the edge constraints, the Delaunay scheme automatically avoids the formation of long thin triangles and thus gives high quality grids. A major advantage of the method is that it does not require extra points to be added to the data set in order to ensure that the specified edges are present.

Two simple algorithms for constructing a two-dimensional constrained Delaunay triangulation

Often when triangulating a set of node points in the plane it is necessary to fix the position of certain edges because, for example, they define boundaries between regions of different physical properties. The triangulation of the points may not contain some of these fixed edges. Two simple algorithms are presented which perform retriangulation of the nodes such that two given nodes are joined by an edge. Further, since iterative retriangulation of the region lying between these two nodes is used the algorithms ensure that the fixed edges meet other edges only at node points. The first algorithm achieves this by the use of an intermediate point lying on the fixed line between its two end points while the second joins them directly. The algorithms are simple in concept and, due to their iterative nature, are convenient for keeping track of the structure of the triangulation. The final triangulation forms a Delaunay triangulation except perhaps in the region of the constrained edges. The use of these algorithms in building constrained polygons within a convex hull is discussed and simple examples are presented. An iterative triangulation algorithm is briefly described for triangulating the region within the convex hull when it contains fixed polygonal boundaries. The application of this algorithm to the meshing of a magnetic recording head is discussed briefly.

A grid generator based on 4‐triangles conforming mesh‐refinement algorithms

AbstractA procedure to generate 2D meshes of triangles in a quite general fashion allowing for local and selective mesh‐refinement is presented and discussed. The grid generator is based on the iterative application of 4‐triangles conforming mesh‐refinement algorithms for triangulations, which are also introduced in this paper. These algorithms are modified versions of those proposed in Int. j. numer. methods eng., 20, 745–756 (1984), and they can be used for global refinement of a grid, as well as for local refinement. The grid generator works in the following way: given any initial coarse triangulation which properly defines the geometry of the problem, a set of user‐defined refinement subregions Ri, i = 1,2,…, N, and an associated set of tolerance parameters hi, an irregular and conforming final triangulation is automatically constructed in such a way that the diameter of all the triangles contained in Ri is smaller than hi, i= 1,2,…, N. Moreover, all angles in the final triangulation are greater than or equal to half the smallest angle in the initial, coarse one. The refinement is propagated only to assure the conformity and smoothness of the grid, and consequently, the number of involved nodes will be minimized. The refinement of the final mesh will be determined by the subregions Ri and the parameters Ni and will be essentially independent of the initial coarse grid. The 4‐triangles conforming mesh‐refinement algorithms are presented and their properties discussed. The implementation of these techniques is discussed and examples of the application of the grid generator are given.