A Fast Algorithm for Well-Spaced Points and Approximate Delaunay Graphs

We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time $O(2^{O(d)}(n\log n + m))$, where $n$ is the input size, $m$ is the output point set size, and $d$ is the ambient dimension. The constants only depend on the desired element quality bounds. To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on $d$ is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size $2^{O(d)}m$ graph in $2^{O(d)}(n\log n + m)$ expected time. If $m$ is superlinear in $n$, then we can produce a hierarchically well-spaced superset of size $2^{O(d)}n$ in $2^{O(d)}n\log n$ expected time.Comment: Full versio

Conforming restricted Delaunay mesh generation for piecewise smooth complexes

A Frontal-Delaunay refinement algorithm for mesh generation in piecewise smooth domains is described. Built using a restricted Delaunay framework, this new algorithm combines a number of novel features, including: (i) an unweighted, conforming restricted Delaunay representation for domains specified as a (non-manifold) collection of piecewise smooth surface patches and curve segments, (ii) a protection strategy for domains containing curve segments that subtend sharply acute angles, and (iii) a new class of off-centre refinement rules designed to achieve high-quality point-placement along embedded curve features. Experimental comparisons show that the new Frontal-Delaunay algorithm outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.Comment: To appear at the 25th International Meshing Roundtabl

Unstructured and adaptive mesh generation for high Reynolds number viscous flows

A method for generating and adaptively refining a highly stretched unstructured mesh suitable for the computation of high-Reynolds-number viscous flows about arbitrary two-dimensional geometries was developed. The method is based on the Delaunay triangulation of a predetermined set of points and employs a local mapping in order to achieve the high stretching rates required in the boundary-layer and wake regions. The initial mesh-point distribution is determined in a geometry-adaptive manner which clusters points in regions of high curvature and sharp corners. Adaptive mesh refinement is achieved by adding new points in regions of large flow gradients, and locally retriangulating; thus, obviating the need for global mesh regeneration. Initial and adapted meshes about complex multi-element airfoil geometries are shown and compressible flow solutions are computed on these meshes

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains

Unstructured mesh generation and adaptivity

An overview of current unstructured mesh generation and adaptivity techniques is given. Basic building blocks taken from the field of computational geometry are first described. Various practical mesh generation techniques based on these algorithms are then constructed and illustrated with examples. Issues of adaptive meshing and stretched mesh generation for anisotropic problems are treated in subsequent sections. The presentation is organized in an education manner, for readers familiar with computational fluid dynamics, wishing to learn more about current unstructured mesh techniques

TetGen, towards a quality tetrahedral mesh generator

TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation

3D boundary recovery by constrained Delaunay tetrahedralization

Three-dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a {\it constrained Delaunay tetrahedralization} (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so-called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples

An advancing front Delaunay triangulation algorithm designed for robustness

A new algorithm is described for generating an unstructured mesh about an arbitrary two-dimensional configuration. Mesh points are generated automatically by the algorithm in a manner which ensures a smooth variation of elements, and the resulting triangulation constitutes the Delaunay triangulation of these points. The algorithm combines the mathematical elegance and efficiency of Delaunay triangulation algorithms with the desirable point placement features, boundary integrity, and robustness traditionally associated with advancing-front-type mesh generation strategies. The method offers increased robustness over previous algorithms in that it cannot fail regardless of the initial boundary point distribution and the prescribed cell size distribution throughout the flow-field

Parallel generalized Delaunay mesh refinement

The modeling of physical phenomena in computational fracture mechanics, computational fluid dynamics and other fields is based on solving systems of partial differential equations (PDEs). When PDEs are defined over geometrically complex domains, they often do not admit closed form solutions. In such cases, they are solved approximately using discretizations of domains into simple elements like triangles and quadrilaterals in two dimensions (2D), and tetrahedra and hexahedra in three dimensions (3D). These discretizations are called finite element meshes. Many applications, for example, real-time computer assisted surgery, or crack propagation from fracture mechanics, impose time and/or mesh size constraints that cannot be met on a single sequential machine. as a result, the development of parallel mesh generation algorithms is required.;In this dissertation, we describe a complete solution for both sequential and parallel construction of guaranteed quality Delaunay meshes for 2D and 3D geometries. First, we generalize the existing 2D and 3D Delaunay refinement algorithms along with theoretical proofs of mesh quality in terms of element shape and mesh gradation. Existing algorithms are constrained by just one or two specific positions for the insertion of a Steiner point inside a circumscribed disk of a poorly shaped element. We derive an entire 2D or 3D region for the selection of a Steiner point (i.e., infinitely many choices) inside the circumscribed disk. Second, we develop a novel theory which extends both the 2D and the 3D Generalized Delaunay Refinement methods for the concurrent and mathematically guaranteed independent insertion of Steiner points. Previous parallel algorithms are either reactive relying on implementation heuristics to resolve dependencies in parallel mesh generation computations or require the solution of a very difficult geometric optimization problem (the domain decomposition problem) which is still open for general 3D geometries. Our theory solves both of these drawbacks. Third, using our generalization of both the sequential and the parallel algorithms we implemented prototypes of practical and efficient parallel generalized guaranteed quality Delaunay refinement codes for both 2D and 3D geometries using existing state-of-the-art sequential codes for traditional Delaunay refinement methods. On a heterogeneous cluster of more than 100 processors our implementation can generate a uniform mesh with about a billion elements in less than 5 minutes. Even on a workstation with a few cores, we achieve a significant performance improvement over the corresponding state-of-the-art sequential 3D code, for graded meshes