One machine, one minute, three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh

3D Delaunay triangulation of 1 billion points on a PC

Of course, there is not enough memory on a PC with 16 GB RAM, and tetrahedra constructed have to be output to leave rooms for the creation of new tetrahedra in the next round of point insertion. A segmental zonal insertion scheme is developed, in which large data sets of more than 100 million points are partitioned into zones, each of which is triangulated in turn by the parallel zonal insertion module. An overlapping zone between two steps of insertion has to be allowed to ensure Delaunay tetrahedra formed at the boundary between two insertion zones. Tetrahedra between zones can be easily eliminated by the minimum vertex allocation method. The collection of all the tetrahedra from each insertion zone/step will produce the required triangulation for the point set. As the work of each typical step for the insertion of an equal number of points is very much similar, the processing time bears roughly a linear relationship with the number of points in the set, at a construction rate of more than 5 million Delaunay tetrahedra per second for the triangulation of 1 billion randomly generated points.postprin

A Level Set Approach for Denoising and Adaptively Smoothing Complex Geometry Stereolithography Files

abstract: Stereolithography files (STL) are widely used in diverse fields as a means of describing complex geometries through surface triangulations. The resulting stereolithography output is a result of either experimental measurements, or computer-aided design. Often times stereolithography outputs from experimental means are prone to noise, surface irregularities and holes in an otherwise closed surface. A general method for denoising and adaptively smoothing these dirty stereolithography files is proposed. Unlike existing means, this approach aims to smoothen the dirty surface representation by utilizing the well established levelset method. The level of smoothing and denoising can be set depending on a per-requirement basis by means of input parameters. Once the surface representation is smoothened as desired, it can be extracted as a standard levelset scalar isosurface. The approach presented in this thesis is also coupled to a fully unstructured Cartesian mesh generation library with built-in localized adaptive mesh refinement (AMR) capabilities, thereby ensuring lower computational cost while also providing sufficient resolution. Future work will focus on implementing tetrahedral cuts to the base hexahedral mesh structure in order to extract a fully unstructured hexahedra-dominant mesh describing the STL geometry, which can be used for fluid flow simulations.Dissertation/ThesisMasters Thesis Aerospace Engineering 201

Scalable Parallel Delaunay Image-to-Mesh Conversion for Shared and Distributed Memory Architectures

Mesh generation is an essential component for many engineering applications. The ability to generate meshes in parallel is critical for the scalability of the entire Finite Element Method (FEM) pipeline. However, parallel mesh generation applications belong to the broader class of adaptive and irregular problems, and are among the most complex, challenging, and labor intensive to develop and maintain. In this thesis, we summarize several years of the progress that we made in a novel framework for highly scalable and guaranteed quality mesh generation for finite element analysis in three dimensions. We studied and developed parallel mesh generation algorithms on both shared and distributed memory architectures. In this thesis we present a novel two-level parallel tetrahedral mesh generation framework capable of delivering and sustaining close to 6000 of concurrent work units (cores). We achieve this by leveraging concurrency at two different granularity levels by using a hybrid message passing and multi-threaded execution model which is suitable to the hierarchy of the hardware architecture of the distributed memory clusters. An end-user productivity and scalability study was performed on up to 6000 cores, and indicated very good end-user productivity with about 300 million tets per second and about 3600 weak scaling speedup. Both of these results suggest that: compared to the best previous algorithm, we have seen an improvement of more than 7000 times in performance, measured in terms of speed (elements per second) by using about 180 times more CPUs, for geometries that are by many orders of magnitude more complex

Efficient Generating And Processing Of Large-Scale Unstructured Meshes

Unstructured meshes are used in a variety of disciplines to represent simulations and experimental data. Scientists who want to increase accuracy of simulations by increasing resolution must also increase the size of the resulting dataset. However, generating and processing a extremely large unstructured meshes remains a barrier. Researchers have published many parallel Delaunay triangulation (DT) algorithms, often focusing on partitioning the initial mesh domain, so that each rectangular partition can be triangulated in parallel. However, the comproblems for this method is how to merge all triangulated partitions into a single domain-wide mesh or the significant cost for communication the sub-region borders. We devised a novel algorithm --Triangulation of Independent Partitions in Parallel (TIPP) to deal with very large DT problems without requiring inter-processor communication while still guaranteeing the Delaunay criteria. The core of the algorithm is to find a set of independent} partitions such that the circumcircles of triangles in one partition do not enclose any vertex in other partitions. For this reason, this set of independent partitions can be triangulated in parallel without affecting each other. The results of mesh generation is the large unstructured meshes including vertex index and vertex coordinate files which introduce a new challenge \-- locality. Partitioning unstructured meshes to improve locality is a key part of our own approach. Elements that were widely scattered in the original dataset are grouped together, speeding data access. For further improve unstructured mesh partitioning, we also described our new approach. Direct Load which mitigates the challenges of unstructured meshes by maximizing the proportion of useful data retrieved during each read from disk, which in turn reduces the total number of read operations, boosting performance

Delaunay-kolmioinnin hyödyntäminen infrastruktuurin suunnitteluohjelmistoissa

In Finland, irregular triangulation has traditionally been used in infrastructural design software, such as road, railroad, bridge, tunnel and environmental design software, to model ground surfaces. Elsewhere, methods like regular square and triangle network, approximating surface without a surface presentation, and algebraic surfaces, have been used for the same task. Approximating the ground surface is necessary for tasks such as determining the height of a point on the ground, interpolating 2D polylines onto the ground, calculating height lines, calculating volumes and visualization. In most of these cases, a continuous surface representation, a digital terrain model is needed. Delaunay triangulation is a way of forming an irregular triangulation out of a 2D point set, in such a way that the triangles are well-formed. Well-formed triangles are essential for the accuracy of the surface representation. This Master's Thesis studies how much time and memory it takes to form a Delaunay triangulation for large point sets, and how Delaunay triangulation compares to other methods of forming a surface representation. In addition, the run-time and accuracy of the resulting surface representations is studied in different interpolation and volume calculation tasks.Infrastruktuurin suunnitteluohjelmistoissa, kuten tien-, rautatien-, sillan-, tunnelin-, ja ympäristönsuunnitteluohjelmistoissa, on Suomessa perinteisesti käytetty maaston pinnan mallintamiseen mittapisteistä muodostettua epäsäännöllistä kolmioverkkoa. Muualla maailmassa ovat käytössä olleet säännölliset neliö- ja kolmioverkot, maaston approksimointi ilman pintaesitystä, sekä joissain tapauksissa algebralliset pintaesitykset. Pinnan approksimaatiota tarvitaan em. sovelluksissa mm. pisteen korkeuden arviointiin, 2-ulotteisten murtoviivojen interpolointiin maaston pinnalle, korkeuskäyrien laskemiseen ja massan (tilavuuden) laskentaan annetuilta alueilta sekä visualisointiin. Delaunay-kolmiointi on tapa muodosta 2-ulotteisesta pistejoukosta epäsäännöllinen kolmioverkko, jonka kolmiot hyvin tasamuotoisia. Kolmioiden tasamuotoisuus on oleellisesta pintamallin tarkkuudelle. Tässä työssä tutkitaan Delaunay-kolmioinnin käytettävyyttä maaston mallintamiseen suurilla pistejoukoilla, sekä epäsäännöllisen kolmioinnin käytettävyyttä em. tehtäviin. Työssä vertaillaan Delaunay-kolmioinnin muodostamisen ajan ja muistin kulutusta pintaesityksen muodostamiseen muilla menetelmillä. Lisäksi tutkitaan näin muodostettujen pintamallien tilavuuslaskennan ja interpolaation nopeutta ja tarkkuutta

Parallel Geometric Algorithms for Multi-Core Computers

International audienceComputers with multiple processor cores using shared memory are now ubiquitous. In this paper, we present several parallel geometric algorithms that specifically target this environment, with the goal of exploiting the additional computing power. The d-dimensional algorithms we describe are (a) spatial sorting of points, as is typically used for preprocessing before using incremental algorithms, (b) kd-tree construction, (c) axis-aligned box intersection computation, and finally (d) bulk insertion of points in Delaunay triangulations for mesh generation algorithms or simply computing Delaunay triangulations. We show experimental results for these algorithms in 3D, using our implementations based on the Computational Geometry Algorithms Library (CGAL, http://www.cgal.org/). This work is a step towards what we hope will become a parallel mode for CGAL, where algorithms automatically use the available parallel resources without requiring significant user intervention