Delaunay Mesh Research Articles

A novel terrain rendering algorithm based on quasi Delaunay triangulation

Terrain rendering has long been an interesting research topic, and Delaunay triangulation has been one of the methods frequently employed. The problem of smooth morphing between successive Delaunay meshes in a dynamic setting does not have a satisfactory solution however. In this paper, we address this issue by temporarily relieving the mesh from the strict constraints of Delaunay triangulation (DT). The proposed algorithm uses an off-line process to compute a relative importance for each sampling point in a Digital Elevation Model (DEM). It then constructs a mesh model in real-time from a set of points selected according to their viewpoint distances and relative importances. The mesh model is initialized to be a genuine DT. As the viewpoint moves, some points are added, and some are removed. We use simple methods for point insertion and removal that allow smooth morphing between successive frames. While the simple methods do not ensure Delaunay properties, we eliminate the slivery triangles gradually by collecting and flipping illegal edges incident to them. Point insertions, removals, edge flips, and their animations are organized by queueing and carried out over time. In this way, we amortize the burst computations to successive frames, so that a balanced workload and a high frame rate are achieved. The proposed algorithm produces a concise and well-composed mesh adaptive to both viewpoint and the terrain’s local geometry, and, most importantly, it supports smooth morphing.

Boundary conforming Delaunay mesh generation

A boundary conforming Delaunay mesh is a partitioning of a polyhedral domain into Delaunay simplices such that all boundary simplices satisfy the generalized Gabriel property. It’s dual is a Voronoi partition of the same domain which is preferable for Voronoi-box based finite volume schemes. For arbitrary 2D polygonal regions, such meshes can be generated in optimal time and size. For arbitrary 3D polyhedral domains, however, this problem remains a challenge. The main contribution of this paper is to show that boundary conforming Delaunay meshes for 3D polyhedral domains can be generated efficiently when the smallest input angle of the domain is bounded by arccos 1/3 ≈ 70.53°. In addition, well-shaped tetrahedra and an appropriate mesh size can be obtained. Our new results are achieved by reanalyzing a classical Delaunay refinement algorithm. Note that our theoretical guarantee on the input angle (70.53°) is still too strong for many practical situations. We further discuss variants of the algorithm to relax the input angle restriction and to improve the mesh quality.

Fully Generalized Two-Dimensional Constrained Delaunay Mesh Refinement

Traditional refinement algorithms insert a Steiner point from a few possible choices at each step. Our algorithm, on the contrary, defines regions from where a Steiner point can be selected and thus inserts a Steiner point among an infinite number of choices. Our algorithm significantly extends existing generalized algorithms by increasing the number and the size of these regions. The lower bound for newly created angles can be arbitrarily close to $30^{\circ}$. Both termination and good grading are guaranteed. It is the first Delaunay refinement algorithm with a $30^{\circ}$ angle bound and with grading guarantees. Experimental evaluation of our algorithm corroborates the theory.

Delaunay Meshing of Piecewise Smooth Complexes without Expensive Predicates

Recently a Delaunay refinement algorithm has been proposed that can mesh piecewise smooth complexes which include polyhedra, smooth and piecewise smooth surfaces, and non-manifolds. However, this algorithm employs domain dependent numerical predicates, some of which could be computationally expensive and hard to implement. In this paper we develop a refinement strategy that eliminates these complicated domain dependent predicates. As a result we obtain a meshing algorithm that is practical and implementation-friendly.

A Second‐Order Accurate, Finite Volume–Based, Integrated Hydrologic Modeling (FIHM) Framework for Simulation of Surface and Subsurface Flow

Surface water, the vadose zone, and groundwater are linked components of a hydrologic continuum. In order to capture the interaction between different components of a hydrologic continuum and to use this understanding in water management situations, an accurate numerical model is needed. The quality of model results depends on accurate representation of the physical processes and the data describing the area of interest, as well as performance of the numerical formulation implemented. Here we present a physics‐based, distributed, fully coupled, second‐order accurate, upwind cell‐centered, constrained unstructured mesh based finite‐volume modeling framework (FIHM) that simultaneously solves two‐dimensional unsteady overland flow and three‐dimensional variably saturated subsurface flow in heterogeneous, anisotropic domains. A multidimensional linear reconstruction of the hydraulic gradients (surface and subsurface) is used to achieve second‐order accuracy. Accuracy and efficiency in raster data and vector‐boundary representations are facilitated through the use of constrained Delaunay meshes in domain discretization. The experiments presented here (i) explore the influence of initial moisture conditions, soil properties, anisotropy, and heterogeneity in determining the pressure head distributions in the vadose and saturated zones, (ii) show the existence of localized “flux rotation” phenomenon due to heterogeneous anisotropy, leading to the creation of convergence–divergence zones, (iii) show the influence of vertical drainage from unsaturated zone on the response of an unconfined aquifer to pumping, and (iv) show the effects of capillarity, saturation excess, infiltration excess, and initial water table location on determining the overland flow generation.

Automatic LEFM crack propagation method based on local Lepp–Delaunay mesh refinement

A numerical method for 2D LEFM crack propagation simulation is presented. This uses a Lepp–Delaunay based mesh refinement algorithm for triangular meshes which allows both the generation of the initial mesh and the local modification of the current mesh as the crack propagates. For any triangle t, Lepp(t) (Longest Edge Propagation Path of t) is a finite, ordered list of increasing longest edge neighbor triangles, that allows to find a pair of triangles over which mesh refinement operations are easily and locally performed. This is particularly useful for fracture mechanics analysis, where high gradients of element size are needed. The crack propagation is simulated by using a finite element model for each crack propagation step, then the mesh near the crack tip is modified to take into account the crack advance. Stress intensify factors are calculated using the displacement extrapolation technique while the crack propagation angle is calculated using the maximum circumferential stress method. Empirical testing shows that the behavior of the method is in complete agreement with experimental results reported in the literature. Good results are obtained in terms of accuracy and mesh element size across the geometry during the process.

Two-dimensional Delaunay-based anisotropic mesh adaptation

Science and engineering applications often have anisotropic physics and therefore require anisotropic mesh adaptation. In common with previous researchers on this topic, we use metrics to specify the desired mesh. Where previous approaches are typically heuristic and sometimes require expensive optimization steps, our approach is an extension of isotropic Delaunay meshing methods and requires only occasional, relatively inexpensive optimization operations. We use a discrete metric formulation, with the metric defined at vertices. To map a local sub-mesh to the metric space, we compute metric lengths for edges, and use those lengths to construct a triangulation in the metric space. Based on the metric edge lengths, we define a quality measure in the metric space similar to the well-known shortest-edge to circumradius ratio for isotropic meshes. We extend the common mesh swapping, Delaunay insertion, and vertex removal primitives for use in the metric space. We give examples demonstrating our scheme’s ability to produce a mesh consistent with a discontinuous, anisotropic mesh metric and the use of our scheme in solution adaptive refinement.

Optimistic parallelism requires abstractions

The problem of writing software for multicore processors is greatly simplified if we could automatically parallelize sequential programs. Although auto-parallelization has been studied for many decades, it has succeeded only in a few application areas such as dense matrix computations. In particular, auto-parallelization of irregular programs, which are organized around large, pointer-based data structures like graphs, has seemed intractable. The Galois project is taking a fresh look at autoparallelization. Rather than attempt to parallelize all programs no matter how obscurely they are written, we are designing programming abstractions that permit programmers to highlight opportunities for exploiting parallelism in sequential programs, and building a runtime system that uses these hints to execute the program in parallel. In this paper, we describe the design and implementation of a system based on these ideas. Experimental results for two real-world irregular applications, a Delaunay mesh refinement application and a graphics application that performs agglomerative clustering, demonstrate that this approach is promising.

A template for developing next generation parallel Delaunay refinement methods

We describe a complete solution for both sequential and parallel construction of guaranteed quality Delaunay meshes for general two-dimensional geometries. We generalize the existing sequential point placement strategies for guaranteed quality mesh refinement: instead of a specific position for a new point, we derive two types of two-dimensional regions which we call selection disks. Both types of selection disks are inside the circumdisk of a poor quality triangle, with the Type I disk always inside the Type II disk. We prove that any point placement algorithm which inserts Steiner points inside selection disks of Type I terminates, and any algorithm which inserts Steiner points inside selection disks of Type II produces an asymptotically size-optimal mesh. In the area of parallel Delaunay mesh refinement, we develop a new theoretical framework for the construction of graded meshes on parallel architectures, i.e., for parallel mesh generation with element size controlled by a user-defined criterion. Our sufficient conditions of point Delaunay-independence allow to select points for concurrent insertion in such a way that the standard sequential guaranteed quality Delaunay refinement procedures can be applied in parallel to attain the required element quality constraints. Finally, we present a novel parallel algorithm which can be used in conjunction with any sequential point placement strategy that chooses points within the selection disks. We implemented our algorithm for shared memory multi-core architectures and present the experimental results. Our data show that even on workstations with a few cores, which are now in common use, our implementation is significantly faster than the best sequential counterpart.

Feature preserving Delaunay mesh generation from 3D multi‐material images

AbstractGenerating realistic geometric models from 3D segmented images is an important task in many biomedical applications. Segmented 3D images impose particular challenges for meshing algorithms because they contain multi‐material junctions forming features such as surface patches, edges and corners. The resulting meshes should preserve these features to ensure the visual quality and the mechanical soundness of the models. We present a feature preserving Delaunay refinement algorithm which can be used to generate high‐quality tetrahedral meshes from segmented images. The idea is to explicitly sample corners and edges from the input image and to constrain the Delaunay refinement algorithm to preserve these features in addition to the surface patches. Our experimental results on segmented medical images have shown that, within a few seconds, the algorithm outputs a tetrahedral mesh in which each material is represented as a consistent submesh without gaps and overlaps. The optimization property of the Delaunay triangulation makes these meshes suitable for the purpose of realistic visualization or finite element simulations.

A multigrain Delaunay mesh generation method for multicore SMT-based architectures

Given the proliferation of layered, multicore- and SMT-based architectures, it is imperative to deploy and evaluate important, multi-level, scientific computing codes, such as meshing algorithms, on these systems. We focus on Parallel Constrained Delaunay Mesh (PCDM) generation. We exploit coarse-grain parallelism at the subdomain level, medium-grain at the cavity level and fine-grain at the element level. This multi-grain data parallel approach targets clusters built from commercially available SMTs and multicore processors. The exploitation of the coarser degree of granularity facilitates scalability both in terms of execution time and problem size on loosely-coupled clusters. The exploitation of medium-grain parallelism allows performance improvement at the single node level. Our experimental evaluation shows that the first generation of SMT cores is not capable of taking advantage of fine-grain parallelism in PCDM. Many of our experimental findings with PCDM extend to other adaptive and irregular multigrain parallel algorithms as well.

Algorithm, software, and hardware optimizations for Delaunay mesh generation on simultaneous multithreaded architectures

This article focuses on the optimization of PCDM, a parallel, two-dimensional (2D) Delaunay mesh generation application, and its interaction with parallel architectures based on simultaneous multithreading (SMT) processors. We first present the step-by-step effect of a series of optimizations on performance. These optimizations improve the performance of PCDM by up to a factor of six. They target issues that very often limit the performance of scientific computing codes. We then evaluate the interaction of PCDM with a real SMT-based SMP system, using both high-level metrics, such as execution time, and low-level information from hardware performance counters.

Uma estrutura de dados dinâmica para geração de malhas de Delaunay bidimensionais e tridimensionais

Neste trabalho, é apresentada a implementação de um algoritmo para geração de malhas de Delaunay triangulares e tetraédricas, por meio de um conjunto de estruturas de dados e métodos genéricos que permitem redimensionar dinamicamente o tamanho de cada uma dessas estruturas, produzindo classes e métodos tanto de ordem 2 quanto de ordem 3, conforme a dimensão do problema em estudo. A idéia principal desse algoritmo é proporcionar a geração de malhas bidimensionais e tridimensionais, sem a necessidade de dois algoritmos ou, ainda, de estruturas de armazenamentos complexas, eliminando, assim, a redundância de código. A metodologia apresentada foi desenvolvida com o auxílio dos contêineres da Standard Template Library (STL), os quais apresentam as ferramentas necessárias para o desenvolvimento desse algoritmo.

A 2-D Delaunay Refinement Algorithm Using an Initial Prerefinement From the Boundary Mesh

An improvement to the quality bidimensional Delaunay mesh generation algorithm, which combines the mesh refinement algorithms strategy of Ruppert and Shewchuk is proposed in this research. The developed technique uses diametral lenses criterion, introduced by L. P. Chew, with the purpose of eliminating the extremely obtuse triangles in the boundary mesh. This method splits the boundary segment and obtains an initial prerefinement, and thus reducing the number of necessary iterations to generate a high quality sequential triangulation. Moreover, it decreases the intensity of the communication and synchronization between subdomains in parallel mesh refinement.

Algorithm 872

Delaunay refinement is a widely used method for the construction of guaranteed quality triangular and tetrahedral meshes. We present an algorithm and a software for the parallel constrained Delaunay mesh generation in two dimensions. Our approach is based on the decomposition of the original mesh generation problem into N smaller subproblems which are meshed in parallel. The parallel algorithm is asynchronous with small messages which can be aggregated and exhibits low communication costs. On a heterogeneous cluster of more than 100 processors our implementation can generate over one billion triangles in less than 3 minutes, while the single-node performance is comparable to that of the fastest to our knowledge sequential guaranteed quality Delaunay meshing library (the Triangle).

Combining improvement and refinement techniques: 2D Delaunay mesh adaptation under domain changes

We propose a framework that combines improvement and Delaunay refinement techniques for incrementally adapting a refined mesh by interactively inserting and removing domain elements. Our algorithms achieve quality mesh by deleting, moving or inserting Steiner points from or into the mesh. The modifications applied to the mesh are local and the number of Steiner points added during the mesh adaptation process remains low. Moreover, since a mesh generation process can be viewed as an adaptation mesh process when domain elements are inserted one by one, our approach can also be applied to the generation of refined Delaunay quality meshes by incorporating our framework in the main body of Delaunay refinement mesh generation algorithms.

Optimistic parallelism requires abstractions

Irregular applications, which manipulate large, pointer-based data structures like graphs, are difficult to parallelize manually. Automatic tools and techniques such as restructuring compilers and run-time speculative execution have failed to uncover much parallelism in these applications, in spite of a lot of effort by the research community. These difficulties have even led some researchers to wonder if there is any coarse-grain parallelism worth exploiting in irregular applications. In this paper, we describe two real-world irregular applications: a Delaunay mesh refinement application and a graphics application thatperforms agglomerative clustering. By studying the algorithms and data structures used in theseapplications, we show that there is substantial coarse-grain, data parallelism in these applications, but that this parallelism is very dependent on the input data and therefore cannot be uncoveredby compiler analysis. In principle, optimistic techniques such asthread-level speculation can be used to uncover this parallelism, but we argue that current implementations cannot accomplish thisbecause they do not use the proper abstractions for the data structuresin these programs. These insights have informed our design of the Galois system , an object-based optimistic parallelization system for irregular applications. There are three main aspects to Galois: (1) a small number of syntactic constructs for packaging optimistic parallelism as iteration over ordered and unordered sets, (2)assertions about methods in class libraries, and (3) a runtime scheme for detecting and recovering from potentially unsafe accesses to shared memory made by an optimistic computation. We show that Delaunay mesh generation and agglomerative clustering can be parallelized in a straight-forward way using the Galois approach, and we present experimental measurements to show that this approach is practical. These results suggest that Galois is a practical approach to exploiting data parallelismin irregular programs.

Sliver-free perturbation for the Delaunay tetrahedrization

This paper studies symbolic perturbation schemes in the context of Delaunay meshing in the three-dimensional space. Symbolic perturbation is a general and powerful technique for removing geometric degeneracy. However, a straightforward application of this technique to Delaunay meshing does not work well, because the perturbation generates volume-zero tetrahedra, called slivers, which should not appear in meshes for the finite element method. First we characterize the set of directions in which a point can be perturbed without generating slivers. Next, as an application of this characterization, we construct a graph-theoretic method for finding a sliver-free perturbation. We also show that an ordinary symbolic perturbation cannot avoid slivers for integer-grid points, and point out that there is a generalized type of perturbation that can avoid slivers completely.

Generation of Unstructured Meshes for Process and Device Simulation by Means of Partial Differential Equations

For process and device simulation, very high mesh densities are often required to obtain accurate simulation results. Unfortunately, the required mesh densities depend often on a direction. Conventional mesh-refinement strategies generate isotropic meshes with a high amount of mesh points, reaching the memory and time limits in particular for three-dimensional simulations. For a better resolution of the carrier concentrations, for instance, a boundary-conforming mesh-generation method with tunable mesh spacings in almost orthogonal directions was developed. Similar to elliptic mesh generation, the mesh points are placed inside the simulation regions based on the solution of partial differential equations. The method used can produce highly anisotropic mesh densities in the regions of particular interest. In contrast to elliptic grid generation, which produces structured grids, the method used generates triangular or tetrahedral (unstructured) Delaunay meshes in two or three dimensions, respectively, which are very well suitable for the process and device simulators

Three-Dimensional Delaunay Mesh Generation

We propose an algorithm to compute a conforming Delaunay mesh of a bounded domain in ${\Bbb R}^3$ specified by a piecewise linear complex. Arbitrarily small input angles are allowed, and the input complex is not required to be a manifold. Our algorithm encloses the input edges with a small buffer zone, a union of balls whose sizes are proportional to the local feature sizes at their centers. In the output mesh, the radius-edge ratio of the tetrahedra outside the buffer zone is bounded by a constant independent of the domain, while that of the tetrahedra inside the buffer zone is bounded by a constant depending on the smallest input angle. Furthermore, the output mesh is graded. Our work is the first that provides quality guarantees for Delaunay meshes in the presence of small input angles.