I discuss algorithms for constructing constrained Delaunay triangulations (CDTs) in dimensions higher than two. If the CDT of a set of vertices and constraining simplices exists, it can be constructed in O(n v n s ) time, where n v is the number of input vertices and n s is the number of output d-simplices. The CDT of a starshaped polytope can be constructed in O(n s log n v ) time, yielding an efficient way to delete a vertex from a CDT. 1 Introduction Mesh generation and interpolation can benefit from triangulations that have properties similar to Delaunay triangulations, but are constrained to contain specified faces. These constraints may arise because a mesh must conform to the shape of an object, or because of the desire to interpolate a discontinuous function. The constrained Delaunay triangulation (CDT) [5, 1, 11] is a Delaunay-like triangulation that conforms to constraints. In two dimensions, the input is a planar straight line graph (PSLG) X , which is a set of vertices a...

We present new algorithms for the robust transmission of geometric data sets, i.e. transmission which allows the receiver to recover (an approximation of) the original geometric object even if parts of the data get lost on the way. These algorithms can be considered as hinted point cloud triangulation schemes since the general manifold reconstruction problem is simplified by adding tags to the vertices and by providing a coarse base--mesh which determines the global surface topology. Robust transmission techniques exploit the geometric coherence of the data and do not require redundant transmission protocols on lower software layers. As an example application scenario we describe the teletext--like broadcasting of 3D models.

In this paper we summarize our experiences with 3D constrained Delaunay triangulation algorithms for industrial applications. In addition, we report a robust implementation process for constructing 3D constrained triangulations from initial unconstrained triangulations, based on a minimalist approach, in which we minimize the use of geometrical operations such as intersections. This is achieved by inserting Steiner points on missing constraining edges and faces in the initial unconstrained triangulations. This approach allowed the generation of tetrahedral meshes for arbitrarily complex 3D domains.

Given the importance of parallel mesh generation in large-scale scientific applications and the proliferation of multilevel SMTbased architectures, it is imperative to obtain insight on the interaction between meshing algorithms and these systems. We focus on Parallel Constrained Delaunay Mesh (PCDM) generation. We exploit coarse-grain parallelism at the subdomain level and fine-grain at the element level. This multigrain data parallel approach targets clusters built from low-end, commercially available SMTs. Our experimental evaluation shows that current SMTs are not capable of executing fine-grain parallelism in PCDM. However, experiments on a simulated SMT indicate that with modest hardware support it is possible to exploit fine-grain parallelism opportunities. The exploitation of fine-grain parallelism results to higher performance than a pure MPI implementation and closes the gap between the performance of PCDM and the state-of-the-art sequential mesher on a single physical processor. Our findings extend to other adaptive and irregular multigrain, parallel algorithms. 1

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The finite element method has grown, in the past 40 years, to be a popular method for the simulation of physical systems in science and engineering. The finite element method is used in a wide array of industries. In fact just about any enterprise that makes a physical product can, and probably does, use finite element technology. The success of the finite element method is due in large part to its ability to allow the use of accurate formulation of partial differential equations (PDEs), on arbitrarily general physical domains with complex boundary conditions. Additionally, the rapid growth in the computational power available in todays computers - for an ever more affordable price - has made finite element technology...

Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions. The main contributions are rigorous, theory-tested definitions of constrained Delaunay triangulations and piecewise linear complexes (geometric domains that incorporate nonconvex faces with “internal ” boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

Algorithms in Computational Geometry and Computer Aided Design are often developed for the Real RAM model of computation, which assumes exactness of all the input arguments and operations. In practice, however, the exactness imposes tremendous limitations on the algorithms – even the basic operations become uncomputable, or prohibitively slow. In some important cases, however, the computations of interest are limited to determining the sign of polynomial expressions. In such circumstances, a faster approach is available: one can evaluate the polynomial in floating point first, together with some estimate of the rounding error, and fall back to exact arithmetic only if this error is too big to determine the sign reliably. A particularly efficient variation on this approach has been used by Shewchuk in his robust implementations of Orient and InSphere geometric predicates. We extend Shewchuk’s method to arbitrary polynomial expressions. The expressions are given as programs in a suitable source language featuring basic arithmetic operations of addition, subtraction, multiplication and squaring, which are to be perceived by the programmer as exact. The source language also allows for anonymous

We present algorithms to produce Delaunay meshes from arbitrary triangle meshes by edge flipping and geometry-preserving refinement and prove their correctness. In particular we show that edge flipping serves to reduce mesh surface area, and that a poorly sampled input mesh may yield unflippable edges necessitating refinement to ensure a Delaunay mesh output. Multiresolution Delaunay meshes can be obtained via constrained mesh decimation. We further examine the usefulness of trading off the geometry-preserving feature of our algorithm with the ability to create fewer triangles. We demonstrate the performance of our algorithms through several experiments.

to appear March 2008.