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A new parallel algorithm for constructing Voronoi tessellations from distributed input data
Abstract
We present a new parallel algorithm for generating consistent Voronoi diagrams from distributed input data for the purposes of simulation and visualization. The algorithm functions by building upon any serial Voronoi tessellation algorithm. The output of such a serial tessellator is used to determine the connectivity of the distributed domains without any assumptions about how points are distributed across those domains, and then in turn to build the portion of the global tessellation local to each domain using information from that domains neighbors. The result is a generalized methodology for adding distributed capabilities to serial tessellation packages. Results from several two-dimensional tests are presented, including strong and weak scaling of its current implementation.
... The mesoscale components of the internal structure of concrete (cement stone, aggregates, micropores) are specified explicitly ( Figure 1). Spatial placement and determination of the geometry of particular aggregates in the model samples were carried out using the Voronoi tessellation algorithm [71]. The volume fraction of basalt aggregates was assumed to be 10% in the present study. ...
... Each micropore was introduced by removing the corresponding area of the material from the initial (non-porous) matrix. We Spatial placement and determination of the geometry of particular aggregates in the model samples were carried out using the Voronoi tessellation algorithm [71]. The volume fraction of basalt aggregates was assumed to be 10% in the present study. ...
It is well-known that the effect of interstitial fluid on the fracture pattern and strength of saturated high-strength concrete is determined by qualitatively different mechanisms at quasi-static and high strain rate loading. This paper shows that the intermediate range of strain rates (10−4 s−1 < ε˙ < 100 s−1) is also characterized by the presence of a peculiar mechanism of interstitial water effect on the concrete fracture and compressive strength. Using computer simulations, we have shown that such a mechanism is the competition of two oppositely directed processes: deformation of the pore space, which leads to an increase in pore pressure; and pore fluid flow. The balance of these processes can be effectively characterized by the Darcy number, which generalizes the notion of strain rate to fluid-saturated material. We have found that the dependence of the compressive strength of high-strength concrete on the Darcy number is a decreasing sigmoid function. The parameters of this function are determined by both low-scale (capillary) and large-scale (microscopic) pore subsystems in a concrete matrix. The capillary pore network determines the phenomenon of strain-rate sensitivity of fluid-saturated concrete and logistic form of the dependence of compressive strength on strain rate. Microporosity controls the actual boundary of the quasi-static loading regime for fluid-saturated samples and determines localized fracture patterns. The results of the study are relevant to the design of special-purpose concretes, as well as the assessment of the limits of safe impacts on concrete structural elements.
... Delaunay triangulation is arguably known as the most common discrete representation tool for complex 3D shape models. The problem of computing the Delaunay Triangulation (DT) of large point sets has been considered in different scientific and engineering fields from computer graphics [Fuetterling et al., 2014], scientific visualization [Antaki et al., 2000], fluid simulation [Ando et al., 2013], computer vision [Hiep et al., 2009], multimedia [Tekalp, Ostermann, 2000], pattern recognition [Xiao, Yan, 2003], to geology [Kaufmann, Martin, 2009,Wang et al., 2017 and astrophysics [Sousbie, 2011] [Starinshak et al., 2014] [Zhao et al., 2016]. ...
... A popular framework from HPC computing on high end clusters is MPI, which can take advantage of low level synchronizations and efficient message passing to design very efficient algorithms [Remacle et al., 2015], even reaching the milestone of three billion tetrahedra produced in a minute on a single (very high end) machine [Marot et al., 2018]. In contrast, distributed approaches consider separate memory segments for each PE and thus require explicit communications [Lee, Lam, 2008, Starinshak et al., 2014. These approach tend to be less efficient in processing speed but scale out to larger datasets at reasonnable hardware costs. ...
This paper deals with the distributed computation of Delaunay triangulations of massive point sets, mainly motivated by the needs of a scalable out-of-core surface reconstruction workflow from massive urban LIDAR datasets. Such a data often corresponds to a huge point cloud represented through a set of tiles of relatively homogeneous point sizes. This will be the input of our algorithm which will naturally partition this data across multiple processing elements. The distributed computation and communication between processing elements is orchestrated efficiently through an uncentralized model to represent, manage and locally construct the triangulation corresponding to each tile. Initially inspired by the star splaying approach, we review the Tile& Merge algorithm for computing Distributed Delaunay Triangulations on the cloud, provide a theoretical proof of correctness of this algorithm, and analyse the performance of our Spark implementation in terms of speedup and strong scaling in both synthetic and real use case datasets. A HPC implementation (e.g. using MPI), left for future work, would benefit from its more efficient message passing paradigm but lose the robustness and failure resilience of our Spark approach.
... , otherwise s min /(s min + s max ) ∆u ij ≥ 0 and s i ≥ s j , or ∆u ij < 0 and s i < s j s max /(s min + s max ) ∆u ij ≥ 0 and s i < s j , or ∆u ij < 0 and s i ≥ s j (67) s min = min(|s i |, |s j |), s max = max(|s i |, |s j |) 67)). Our standard set of parameters for the viscosity used throughout this paper are C l = 2, C q = 1, 2 = 10 −2 , η crit = 1/n h (where n h is the expected number of nodes per smoothing scale in one dimension), and η fold = 0.2. ...
... In our example, we include the results of a ReALE calculation for comparison. The presented ReALE methodology is based on the original study of [31]; the details of this ReALE implementation can be found in [53,67,65,66]. ...
We present a formulation of smoothed particle hydrodynamics (SPH) that employs a first-order consistent reproducing kernel function, exactly interpolating linear fields with particle tracers. Previous formulations using reproducing kernel (RK) interpolation have had difficulties maintaining conservation of momentum due to the fact the RK kernels are not, in general, spatially symmetric. Here, we utilize a reformulation of the fluid equations such that mass, momentum, and energy are all manifestly conserved without any assumption about kernel symmetries. Additionally, by exploiting the increased accuracy of the RK method's gradient, we formulate a simple limiter for the artificial viscosity that reduces the excess diffusion normally incurred by the ordinary SPH artificial viscosity. Collectively, we call our suite of modifications to the traditional SPH scheme Conservative Reproducing Kernel SPH, or CRKSPH. CRKSPH retains the benefits of traditional SPH methods (such as preserving Galilean invariance and manifest conservation of mass, momentum, and energy) while improving many of the well-known shortcomings of SPH, particularly the overly aggressive artificial viscosity and zeroth-order inaccuracy. We compare CRKSPH to two different modern SPH formulations (pressure based SPH and compatibly differenced SPH), demonstrating the advantages of our new formulation when modeling fluid mixing, strong shock, and adiabatic phenomena.
... Okabe et al. (2000) show an extensive review on VT computation methods. There are also parallel algorithms for VT computation such as Starinshak et al. (2014), however none of them are open source and freely available to the community. Another parallel VT implementation is tess2 4 (Peterka et al., 2014) , which does not take into account buffer zones and is poorly documented. ...
We present a new open source code for massive parallel computation of Voronoi tessellations(VT hereafter) in large data sets. The code is focused for astrophysical purposes where VT densities and neighbors are widely used. There are several serial Voronoi tessellation codes, however no open source and parallel implementations are available to handle the large number of particles/galaxies in current N-body simulations and sky surveys. Parallelization is implemented under MPI and VT using Qhull library. Domain decomposition takes into account consistent boundary computation between tasks, and includes periodic conditions. In addition, the code computes neighbors list, Voronoi density, Voronoi cell volume, density gradient for each particle, and densities on a regular grid.
... Finally, the magnitude of the area or volume containing the molecules can be determined by methods based on the molecular density. They include Delaunay 28,29 (Figure 1c, left) and Voronoi 30,31 ( Figure 1c, middle) tessellations and kernel surface density function, ks-density 32,33 (Figure 1c, right). These are described within the Methods (Supporting Information). ...
Current methods for determining equilibriumconstants often operate in three-dimensional environments, whichmay not accurately reflect interactions with membrane-boundproteins. With our technique, based on single-molecule localizationmicroscopy (SMLM), we directly determine protein−proteinassociation (Ka) and dissociation (Kd) constants in cellularenvironments by quantifying associated and isolated moleculesand their interaction area. We introduce Kernel Surface Density(ks-density,) a novel method for determining the accessible area forinteracting molecules, eliminating the need for user-definedparameters. Simulation studies validate our method’s accuracyacross various density and affinity conditions. Applying thistechnique to T cell signaling proteins, we determine the 2Dassociation constant of T cell receptors (TCRs) in resting cells and the pseudo-3D dissociation constant of pZAP70 molecules fromphosphorylated intracellular tyrosine-based activation motifs on the TCR-CD3 complex. We address challenges of multiple detectionand molecular labeling efficiency. This method enhances our understanding of protein interactions in cellular environments,advancing our knowledge of complex biological processes.
KEYWORDS: equilibrium constant, single-molecule localization microscopy, protein−protein interactions, DNA-PAINT, T cells
... However, algorithms have been proposed for standard CVTs on 2D planes and surfaces which exploit GPU parallelism [42], and in 3D parallel clipped CVTs [52]. A general data distributed parallelism for CVTs was intruduced by Starinshak et al. [45], which also handles restrictions only by clipping. However, clipping will leave randomly fragmented regions along the boundaries depending on the data. ...
Spatial statistical analysis of multivariate volumetric data can be challenging due to scale, complexity, and occlusion. Advances in topological segmentation, feature extraction, and statistical summarization have helped overcome the challenges. This work introduces a new spatial statistical decomposition method based on level sets, connected components, and a novel variation of the restricted centroidal Voronoi tessellation that is better suited for spatial statistical decomposition and parallel efficiency. The resulting data structures organize features into a coherent nested hierarchy to support flexible and efficient out-of-core region-of-interest extraction. Next, we provide an efficient parallel implementation. Finally, an interactive visualization system based on this approach is designed and then applied to turbulent combustion data. The combined approach enables an interactive spatial statistical analysis workflow for large-scale data with a top-down approach through multiple-levels-of-detail that links phase space statistics with spatial features.
... Thus there is a need to compute the Voronoi tessellation at a large scale and in parallel. Currently, there are some parallel approaches in the literature for computing Voronoi cells in a distributed-memory model [73,74]. Here, we consider parallelizing Voro++ using a shared-memory model with multithreading. ...
VORO++ is a software library written in C++ for computing the Voronoi tessellation, a technique in computational geometry that is widely used for analyzing systems of particles. VORO++ was released in 2009 and is based on computing the Voronoi cell for each particle individually. Here, we take advantage of modern computer hardware, and extend the original serial version to allow for multithreaded computation of Voronoi cells via the OpenMP application programming interface. We test the performance of the code, and demonstrate that we can achieve parallel efficiencies greater than 95% in many cases. The multithreaded extension follows standard OpenMP programming paradigms, allowing it to be incorporated into other programs. We provide an example of this using the VoroTop software library, performing a multithreaded Voronoi cell topology analysis of up to 102.4 million particles.
... However, algorithms have been proposed for standard CVTs on 2D planes and surfaces which exploit GPU parallelism [42], and in 3D parallel clipped CVTs [52]. A general data distributed parallelism for CVTs was intruduced by Starinshak et al. [45], which also handles restrictions only by clipping. However, clipping will leave randomly fragmented regions along the boundaries depending on the data. ...
Spatial statistical analysis of multivariate volumetric data can be challenging due to scale, complexity, and occlusion. Advances in topological segmentation, feature extraction, and statistical summarization have helped overcome the challenges. This work introduces a new spatial statistical decomposition method based on level sets, connected components, and a novel variation of the restricted centroidal Voronoi tessellation that is better suited for spatial statistical decomposition and parallel efficiency. The resulting data structures organize features into a coherent nested hierarchy to support flexible and efficient out-of-core region-of-interest extraction. Next, we provide an efficient parallel implementation. Finally, an interactive visualization system based on this approach is designed and then applied to turbulent combustion data. The combined approach enables an interactive spatial statistical analysis workflow for large-scale data with a top-down approach through multiple-levels-of-detail that links phase space statistics with spatial features.
... 27-29 Several parallel implementations are also available. Starinshak et al. 30 use a parallel divide-and-conquer approach, where each processor computes Voronoi cells in a subdomain, after which they are joined together. Other examples include PARAVT, 31 and a GPU implementation. ...
Many physical systems can be studied as collections of particles embedded in space, often evolving in time. Natural questions arise concerning how to characterize these arrangements—are they ordered or disordered? If they are ordered, how are they ordered and what kinds of defects do they possess? Voronoi tessellations, originally introduced to study problems in pure mathematics, have become a powerful and versatile tool for analyzing countless problems in pure and applied physics. We explain the basics of Voronoi tessellations and the shapes that they produce and describe how they can be used to characterize many physical systems.
... 27,28 Several parallel implementations are also available. Starinshak et al. 29 introduce a parallel divide-and-conquer approach, where each processor computes Voronoi cells in a subdomain, after which they are joined together. Other examples include PARAVT, 30 and a GPU implementation. ...
Many physical systems can be studied as collections of particles embedded in space, evolving through deterministic evolution equations. Natural questions arise concerning how to characterize these arrangements - are they ordered or disordered? If they are ordered, how are they ordered and what kinds of defects do they possess? Originally introduced to study problems in pure mathematics, Voronoi tessellations have become a powerful and versatile tool for analyzing countless problems in pure and applied physics. In this paper we explain the basics of Voronoi tessellations and the shapes they produce, and describe how they can be used to study many physical systems.
... We assumed that all the porosity of the refractory is concentrated in the matrix. The spatial placement and determination of the large grains' geometry in the model were executed out using the Voronoi tessellation algorithm [43]. Representing by the model of the grain size distribution as during the pressing of the bricks ignores the changes of the grain size and shape to possibly occur during the sintering of the bricks. ...
Computer modelling is a key tool in the optimisation and development of ceramic refractories utilised as insulation in high-temperature industrial furnaces and reactors. The paper is devoted to the mesoscale computer modelling of silica refractories using the method of homogeneously deformable discrete elements. Approaches to determine the local mechanical properties of the constituents from the global experimental failure parameters and respective crack trajectories are considered. Simulations of the uniaxial compressive and tensile failure in a wide range of quasi-static and dynamic loading rates (102 s−1) are performed. The upper limit of the dynamic loading rates corresponds to the most severe loading rates during the scrap loading on the refractory lining. The dependence of the strength, fracture energy, and brittleness at failure on the loading rate is analysed. The model illustrates that an increase in the loading rate is accompanied by a significant change in the mechanical response of the refractory, including a decrease in the brittleness at failure, a more dispersed failure process, and a higher fraction of the large grain failure. The variation of the grain–matrix interface’s strength has a higher impact on the static compressive than on the static tensile properties of the material, while the material’s dynamic tensile properties are more sensitive to the interface strength than the dynamic compressive properties.
... The regularity of Voronoi diagrams depends on the distribution of seed points. The whole process of generating the random polygon using Voronoi tessellation is described in literature (Sukumar and Tabarraei, 2004;Talischi et al., 2012;Starinshak et al., 2014). By changing the density and distribution of seeds, the desired polygon can be obtained which may represent the size and shape of grain. ...
The scatter in the fatigue behavior of duplex microstructure titanium alloys occurs due to microstructural randomness. The presence of two phases in titanium alloy microstructure also contributes to the scatter in fatigue life data. In the literature, several mathematical frameworks have been developed to consider the effect of microstructural randomness on the prediction of fatigue scatter. However, no numerical study has been found that considers the effect of presence of two phases on the fatigue life scatter. Thus, in the present work, a mathematical framework based on continuum damage mechanics (CDM) and polygonal finite element method (PFEM) is developed to predict the fatigue life scatter in α+β titanium alloys. In this framework, PFEM is applied to perform the numerical simulations and CDM is used to explain the fatigue behavior of material. In PFEM, the Voronoi tessellation approach is used to generate the idealized microstructure models. The Waschpress shape functions are adopted to compute the shape functions over the arbitrary convex polygonal elements. In CDM, a fatigue damage model is adopted to consider the crack initiation and propagation under the cyclic loading conditions. The advantage of combined CDM-PFEM framework is that the numerical simulation can be performed at relatively coarser mesh which decreases the computational cost and increases the efficiency of the computational model.
While predicting the fatigue life, it has been observed that micro cracks may initiate and propagate at several locations inside the microstructure. Thus, to track the nucleation and growth of micro cracks effectively, a machine learning based technique ‘cluster analysis’ is employed in the developed framework. The “cluster analysis” identifies the location and dimensions of the micro and dominant crack patterns in the domain. Furthermore, for an idealized representation of duplex microstructures, both phases of titanium alloy (i.e., primary α and β) are modeled using random distribution function. The effect of different sources of randomness in microstructures i.e., topology of microstructure, grain size, volume fraction of alpha phase, inhomogeneity in elastic modulus and internal voids, on the fatigue life is investigated. Several simulations are performed on the numerically generated microstructures at various loads. The statistical analysis of the predicted fatigue lives data is performed through normal and Weibull distribution fits. The simulation results highlight that the combined CDM-PFEM based framework is simple and an effective tool to predict the fatigue life scatter and can effectively analyze the mean fatigue life in titanium alloys under cyclic loading conditions.
... In this scenario, the post-processing step can be accelerated utilizing modern parallelization techniques. There are several existing methods that can be employed directly in this regard [49,50] and will be considered in the future. ...
In this paper, we propose a consistent parallel unstructured mesh generator based on a multi-phase SPH method. A set of physics-motivated modeling equations are developed to achieve the targets of domain decomposition, communication volume optimization and high-quality unstructured mesh generation simultaneously. A unified density field is defined as the target function for both partitioning the geometry and distributing the mesh-vertexes. A multi-phase Smoothing Particle Hydrodynamics (SPH) method is employed to solve the governing equations. All the optimization targets are achieved implicitly and consistently by the particle relaxation procedure without constructing triangulation/tetrahedralization explicitly. The target of communication reduction is achieved by introducing a surface tension model between distinct partitioning sub-domains, which are characterized by colored SPH particles. The resulting partitioning diagram features physically localized sub-domains and optimized interface communication. The target of optimizing the mesh quality is achieved by introducing a tailored equation-of-state (EOS) and a smooth isotropic kernel function. The mesh quality near the interface of neighboring sub-domains is improved by gradually removing the surface-tension force once a steady state is achieved. The proposed method is developed basing on a new parallel environment for multi-resolution SPH to exploit both coarse- and fine-grained parallelism. A set of benchmarks are conducted to verify that all the optimization targets are achieved consistently within the current framework.
... This work has been initiated thanks to a collaboration with M. Shashkov and R. Loubère from CNRS Toulouse. I also had the opportunity to visit M. Owen from the Lawrence Livermore National Laboratory (LLNL) to work on the generation of polygonal grids using the Polytope library developed by LLNL [55]. Finally, the physical variables are conservatively interpolated from the Lagrangian grid onto the new rezoned one during the remapping phase. ...
This document presents a part of the works done at CELIA Laboratory (CEA, CNRS, Univer-
sité Bordeaux) in the field of High Energy Density Physics (HEDP). The CELIA plasma physics group is particularly involved in HEDP, Inertial Confinement Fusion (ICF) and Laboratory astrophysics. Flows encountered in HEDP are multi-material and characterized by strong shock waves and large changes in the domain shape due to rarefaction waves. Numerical schemes based on the Lagrangian formalism are good candidates to model this kind of flows since the computational grid follows the fluid motion. This provides accurate results around the shocks as well as a natural tracking of multi-material interfaces. The document is structured as follows. First, we recall the main features of the compatible cell-centered Lagrangian discretization. ALE and ReALE strategy are presented by describing the different phases of our multi-material ALE algorithm. Diffusion scheme used to couple all the physical models is presented with the last extension of the CCLADNS scheme for non-conformal and non-orthogonal meshes in 2D and 3D. Finally CHIC code is introduced and some theoretical studies and experiments are presented.
... In this scenario, the post-processing step can be accelerated utilizing modern parallelization techniques. There are several existing methods that can be employed directly in this regard [49,50] and will be considered in the future. ...
In this paper, we propose a consistent parallel unstructured mesh generator based on a multi-phase SPH method. A set of physics-motivated modeling equations are developed to achieve the targets of domain decomposition, communication volume optimization and high-quality unstructured mesh generation simultaneously. A unified density field is defined as the target function for both partitioning the geometry and distributing the mesh-vertexes. A multi-phase Smoothing Particle Hydrodynamics (SPH) method is employed to solve the governing equations. All the optimization targets are achieved implicitly and consistently by the particle relaxation procedure without constructing triangulation/tetrahedralization explicitly. The target of communication reduction is achieved by introducing a surface tension model between distinct partitioning sub-domains, which are characterized by colored SPH particles. The resulting partitioning diagram features physically localized sub-domains and optimized interface communication. The target of optimizing the mesh quality is achieved by introducing a tailored equation-of-state (EOS) and a smooth isotropic kernel function. The mesh quality near the interface of neighboring sub-domains is improved by gradually removing the surface-tension force once a steady state is achieved. The proposed method is developed basing on a new parallel environment for multi-resolution SPH to exploit both coarse- and fine-grained parallelism. A set of benchmarks are conducted to verify that all the optimization targets are achieved consistently within the current framework.
... The Voronoi analysis is a spatial proximity analysis method. Parallel computing for constructing Voronoi diagram (Boltcheva & Lévy, 2017;Starinshak, Owen, & Johnson, 2014) is to speed up data analysis as well as visualization. In addition to the above spatial analysis algorithms, a CG_Hadoop framework (Eldawy, Li, Mokbel, & Janardan, 2013) presents a set of fundamental computational geometry operators, namely, polygon union, skyline, convex hull, farthest pair, and closest pair for other geometric algorithms. ...
Spatial vector data with high-precision and wide-coverage has exploded globally, such as land cover, social media, and other data-sets, which provides a good opportunity to enhance the national macroscopic decision-making, social supervision, public services, and emergency capabilities. Simultaneously, it also brings great challenges in management technology for big spatial vector data (BSVD). In recent years, a large number of new concepts, parallel algorithms, processing tools, platforms, and applications have been proposed and developed to improve the value of BSVD from both academia and industry. To better understand BSVD and take advantage of its value effectively, this paper presents a review that surveys recent studies and research work in the data management field for BSVD. In this paper, we discuss and itemize this topic from three aspects according to different information technical levels of big spatial vector data management. It aims to help interested readers to learn about the latest research advances and choose the most suitable big data technologies and approaches depending on their system architectures. To support them more fully, firstly, we identify new concepts and ideas from numerous scholars about geographic information system to focus on BSVD scope in the big data era. Then, we conclude systematically not only the most recent published literatures but also a global view of main spatial technologies of BSVD, including data storage and organization, spatial index, processing methods, and spatial analysis. Finally, based on the above commentary and related work, several opportunities and challenges are listed as the future research interests and directions for reference.
... Description of the techniques we use to generate Voronoi tessellation in 2D and 3D using the dual Delaunay triangulation computation are presented -see Figure 1. We also explain the algorithm we are using to compute the tessellation in a MPI parallel context based on [2] and establish the global mesh connectivity. ...
... Description of the techniques we use to generate Voronoi tessellation in 2D and 3D using the dual Delaunay triangulation computation are presented -see Figure 1. We also explain the algorithm we are using to compute the tessellation in a MPI parallel context based on [2] and establish the global mesh connectivity. ...
SHAPO • Cross-platform library developed in C++ - based on VTK to build Voronoi meshes from a set of generators/seeds in a closed domain • Developed at Kitware (France) for LANL since 2013 • Binding layers to use the tool in C, Fortran and Python • Works in serial or in parallel (MPI) • Use Shewchuck’s robust predicates • Self-diagnostic (of I/O) & error managemen
... There are several serial open source VT implementations such as Qhull 1 (Barber et al., 1996), or Voro++ 2 (Rycroft, 2009). There are also parallel algorithms for VT computation such as Starinshak et al. (2014), however none of them are open source and freely available to the community. ...
We present a new open source code for massive parallel computation of Voronoi tessellations(VT hereafter) in large data sets. The code is focused for astrophysical purposes where VT densities and neighbors are widely used. There are several serial Voronoi tessellation codes, however no open source and parallel implementations are available to handle the large number of particles/galaxies in current N-body simulations and sky surveys. Parallelization is implemented under MPI and VT using Qhull library. Domain decomposition take into account consistent boundary computation between tasks, and support periodic conditions. In addition, the code compute neighbors lists, Voronoi density and Voronoi cell volumes for each particle, and can compute density on a regular grid.
... Therefore, mapping particle-based simulations onto a mesh using a Voronoi mesh 9 is beneficial for our proposes. Note, Mapping SPH simulations onto a 9 The Voronoi tessellation has been recently implemented into Voronoi mesh has also been carried out by Hubber et al. (2016), Barcarolo et al. (2014), Starinshak et al. (2014) and Zhou et al. (2007). ...
Through synthetic observations of a hydrodynamical simulation of an evolving star-forming region, we assess how the choice of observational techniques affects the measurements of properties which trace star formation. Testing and calibrating observational measurements requires synthetic observations which are as realistic as possible. In this part of the paper series (Paper I), we explore different techniques for how to map the distributions of densities and temperatures from the particle-based simulations onto a Voronoi mesh suitable for radiative transfer and consequently explore their accuracy. We further test different ways to set up the radiative transfer in order to produce realistic synthetic observations. We give a detailed description of all methods and ultimately recommend techniques. We have found that the flux around 20 microns is strongly overestimated when blindly coupling the dust radiative transfer temperature with the hydrodynamical gas temperature. We find that when instead assuming a constant background dust temperature in addition to the radiative transfer heating, the recovered flux is consistent with actual observations. We present around 5800 realistic synthetic observations for Spitzer and Herschel bands, at different evolutionary time-steps, distances and orientations. In the upcoming papers of this series (Paper II, Paper III and Paper IV), we will test and calibrate measurements of the star-formation rate (SFR), gas mass and the star-formation efficiency (SFE) using our realistic synthetic observations.
... In this talk we present ShaPo, a cross platform C++ software library accessible from different programming languages (C, Fortran and Python) that integrates algorithms to generate, in serial or in parallel, 2D Voronoi meshes from a set of generators into a domain defined by non-convex multi-connected boundary polygons. ShaPo allows to generate either Constrained Voronoi or Clipped Voronoi tessellations respectively inspired from [2,3] and [4], and provides the full cells, points and edges connectivity. ...
Voronoi meshes are involved in numerous applications from astrophysics to biology or computer graphics.
Their properties are especially widely used in geophysical flows and computational fluid dynamics.
Our main interest in Voronoi meshes related to Reconnection-Based ALE family of methods introduced
in [1].
In this talk we present ShaPo, a cross platform C++ software library accessible from different programming
languages (C, Fortran and Python) that integrates algorithms to generate, in serial or in parallel,
2D Voronoi meshes from a set of generators into a domain defined by non-convex multi-connected
boundary polygons. ShaPo allows to generate either Constrained Voronoi or Clipped Voronoi tessellations
respectively inspired from [2, 3] and [4], and provides the full cells, points and edges connectivity.
We describe the two different robust Voronoi construction algorithms implemented in ShaPo both based
on the dual Delaunay triangulation computation. Proposed optimizations in regard to the original algorithms
are also described. As ShaPo can also be used in a data-parallelism context where generators
and boundaries are distributed across processors, a quick overview of the parallel algorithm based on
an MPI communication layer is presented. Finally we describe the remeshing functionality of ShaPo
which enables the generation of a Voronoi mesh in the sub-domain of an existing unstructured mesh.
... The generators that are inside the computational domain will be called internal generators. The set of internal generators are denoted by The VT on can be defined as (see, for example, [44,45,50,47]): ...
eW present a new adaptive Arbitrary Lagrangian Eulerian (ALE) method. This method is based on the reconnection-based ALE (ReALE) methodology of Refs. [35,34,6]. The main elements in a standard ReALE method are: an explicit Lagrangian phase on an arbitrary polygonal (in 2D) mesh in which the solution and positions of grid nodes are updated; a rezoning phase in which a new grid is defined by changing the connectivity (using Voronoi tessellation) but not the number of cells; and a remapping phase in which the Lagrangian solution is transferred onto the new grid. In the standard ReALE method, the rezoned mesh is smoothed by using one or several steps toward centroidal Voronoi tessellation, but it is not adapted to the solution in any way.
In the current paper we present a new adaptive ReALE method, A-ReALE, that is based on the following design principles. First, a monitor function (or error indicator) based on the Hessian of some flow parameter(s) is utilized. Second, an equi-distribution principle for the monitor function is used as a criterion for adapting the mesh. Third, a centroidal Voronoi tessellation is used to adapt the mesh. Fourth, we scale the monitor function to avoid very small and large cells and then smooth it to permit the use of theoretical results related to weighted centroidal Voronoi tessellation.
In the A-ReALE method, both number of cells and their locations are allowed to change at the rezone stage on each time step. The number of generators at each time step is chosen to guarantee the required spatial resolution in regions where monitor function reaches its maximum value.
We present all details required for implementation of new adaptive A-ReALE method and demonstrate its performance in comparison with standard ReALE method on series of numerical examples.
... The VT on Ω can be defined as (see, for example, [44,45,50,47]): ...
This cumulative thesis presents several novel numerical methods and parallel techniques relating to Smoothed Particle Hydrodynamics (SPH). More specifically, a new multi-resolution parallel framework for SPH, a Lagrangian Inertial Centroidal Voronoi Particle (LICVP) method for dynamic load balancing of particle-based method and a consistent multi-phase SPH formulation for parallel unstructured isotropic mesh generation are proposed.
Hardware transactional memory is a new parallel programming paradigm supported by current commercial multiprocessors. This paradigm provides optimistic concurrency and overcomes some of the problems associated with classical lock-based synchronization, such as deadlock and serialization. Certain algorithms of computational geometry are found to be good candidates for parallelization with this paradigm. However, hardware transactional approaches to these algorithms lead to poor performance as the resulting transactions are too large for the underlying hardware to deal with. Large transactions overflow hardware resources serializing the execution.
In this paper, we propose using privatizing transactions to parallelize two computational geometry algorithms: Lee’s algorithm, which solves the shortest-route problem, and Ruppert’s algorithm for Delaunay/Voronoi mesh refinement. Privatizing transactions are based on commercial hardware transactional memory extensions, and their goal is to reduce transaction footprint by means of a non-transactional private execution section. This results in effective smaller transactions. Our implementation is able to further reduce the transaction size as we propose a reduced validation set for privatizing transactions. Programming complexity of these implementations is discussed.
Results show that our privatizing transaction implementations indeed enhance performance comparing with existing hardware transactional memory versions. Experiments with Intel’s transactional memory extensions yield speedups ranging from 2× to 3.5× with four threads.
A new algorithm, featuring overlapping domain decompositions, for the
parallel construction of Delaunay and Voronoi tessellations is
developed. Overlapping allows for the seamless stitching of the partial
Delaunay tessellations constructed by individual processors. The
algorithm is then modified, by the addition of stereographic
projections, to handle the parallel construction of spherical Delaunay
and Voronoi tessellations. The algorithms are then embedded into
algorithms for the parallel construction of planar and spherical
centroidal Voronoi tessellations that require multiple constructions of
Delaunay tessellations. Computational tests are used to demonstrate the
efficiency and scalability of the algorithms for spherical Delaunay and
centroidal Voronoi tessellations. Compared to serial versions of the
algorithm and to the STRIPACK-based approaches, the new parallel
algorithm results in significant speedups for the construction of
spherical centroidal Voronoi tessellations.
Load balance is critical for performance in large parallel applications. An imbalance on today's fastest supercomputers can force hundreds of thousands of cores to idle, and on future exascale machines this cost will increase by over a factor of a thousand. Improving load balance requires a detailed understanding of the amount of computational load per process and an application's simulated domain, but no existing metrics sufficiently account for both factors. Current load balance mechanisms are often integrated into applications and make implicit assumptions about the load. Some strategies place the burden of providing accurate load information, including the decision on when to balance, on the application. Existing application-independent mechanisms simply measure the application load without any knowledge of application elements, which limits them to identifying imbalance without correcting it.
Our novel load model couples abstract application information with scalable measurements to derive accurate and actionable load metrics. Using these metrics, we develop a cost model for correcting load imbalance. Our model enables comparisons of the effectiveness of load balancing algorithms in any specific imbalance scenario. Our model correctly selects the algorithm that achieves the lowest runtime in up to 96% of the cases, and can achieve a 19% gain over selecting a single balancing algorithm for all cases.
A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applica- tions of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods for computing these tessellations, provide some analyses concerning both the tessellations and the methods for their determination, and, finally, present the results of some numerical experiments.
We describe ann-processor,O(log(n) log log(n))-time CRCW algorithm to construct the Voronoi diagram for a set ofn point-sites in the plane.
This paper describes the design and implementation of a practical parallel algorithm for Delaunay triangulation that works well on general distributions. Although there have been many theoretical parallel algorithms for the problem, and some implementations based on bucketing that work well for uniform distributions, there has been little work on implementations for general distributions. We use the well known reduction of 2D Delaunay triangulation to find the 3D convex hull of points on a paraboloid. Based on this reduction we developed a variant of the Edelsbrunner and Shi 3D convex hull algorithm, specialized for the case when the point set lies on a paraboloid. This simplification reduces the work required by the algorithm (number of operations) from O(n log 2
n) to O(n log n) . The depth (parallel time) is O( log 3
n) on a CREW PRAM. The algorithm is simpler than previous O(n log n) work parallel algorithms leading to smaller constants.
Initial experiments using a variety of distributions showed that our parallel algorithm was within a factor of 2 in work from the best sequential algorithm. Based on these promising results, the algorithm was implemented using C and an MPI-based toolkit. Compared with previous work, the resulting implementation achieves significantly better speedups over good sequential code, does not assume a uniform distribution of points, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for the IBM SP2, Cray T3D, SGI Power Challenge, and DEC AlphaCluster.
A procedure has been developed to improve polygonal surface mesh quality while maintaining the essential characteristics of the discrete surface. The sur- face characteristics are preserved by repositioning mesh vertices so that they remain on the original discrete surface. The repositioning is performed in a series of triangular-facet-based local parametric spaces. The movement of the mesh vertices is driven by a nonlinear numerical optimization process. Two optimization ap- proaches are described, one which improves the quality of elements as much as possible and the other which improves element quality but also keeps the new mesh as close as possible to the original mesh.
Fields as found in the geosciences have properties that are not usually found in other disciplines: the phenomena studied are often three-dimensional, they tend to change continuously over time, and the collection of samples to study the phenomena is problematic, which often results in highly anisotropic distributions of samples. In the GIS community, raster structures (voxels or octrees) are the most popular solutions, but, as we show in this paper, they have shortcomings for modelling and analysing 3D geoscientific fields. As an alternative to using rasters, we propose a new spatial model based on the Voronoi diagram (VD) and its dual the Delaunay tetrahedralization (DT), and argue that they have many advantages over other tessellations. We discuss the main properties of the 3D VD/DT, present some GIS operations that are greatly simplified when the VD/DT is used, and, to analyse two or more fields, we also present a variant of the map algebra framework where all the operations are performed directly on VDs. The usefulness of this Voronoi-based spatial model is demonstrated with a series of potential applications.
The performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation (PDE) models. It is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix. In this paper, for typical second-order elliptic problems, some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes. The derivation of such relations for general high-order elements is based on a new trace formula for the element stiffness matrix. It is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces. These results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing.
Delaunay triangulation has been much used in such applications as volume rendering, shape representation, terrain modeling and so on. The main disadvantage of Delaunay triangulation is large computation time required to obtain the triangulation on an input points set. This time can be reduced by using more than one processor, and several parallel algorithms for Delaunay triangulation have been proposed. In this paper, we propose an improved parallel algorithm for Delaunay triangulation, which partitions the bounding convex region of the input points set into a number of regions by using Delaunay edges and generates Delaunay triangles in each region by applying an incremental construction approach. Partitioning by Delaunay edges makes it possible to eliminate merging step required for integrating subresults. It is shown from the experiments that the proposed algorithm has good load balance and is more efficient than Cignoni et al.'s algorithm (1993) and our previous algorithm (1996)
Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.
We present the results of an evaluation study on the re-structuring of a latency-bound mesh generation algorithm into a latency-tolerant parallel kernel. We use concurrency at a fine-grain level to tolerate long, variable, and unpredictable latencies of remote data gather operations required for parallel guaranteed quality Delaunay triangulations. Our performance data from a 16 node SP2 and 32 node Cluster of Sparc Workstations suggest that more than 90% of the latency from remote data gather operations can be masked effectively at the cost of increasing communication overhead between 2 and 20% of the total run time. Despite the increase in the communication overhead the latency-tolerant mesh generation kernel we present in this paper can generate tetrahedral meshes for parallel field solvers eight to nine times faster than the traditional approach. Copyright © 2003 John Wiley & Sons, Ltd.
We introduce a geometric transformation that allows Voronoi diagrams to be computed using a sweepline technique. The transformation
is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted
point sites. All algorithms haveO(n logn) worst-case running time and useO(n) space.
Here we report our recent results for the fast synthesis of Bi2Sr2Ca2Cu3O10+x phase in powder and single crystal forms using modified alkali chlorides flux technique. For many factors such as flux, crucibles materials, chemical and phase composition of the precursor, temperature and duration of heat treatment, evaporation rate of the flux, their effects on the formation and growth of 2223 have been studied. By optimization of these parameters, the single-phase samples of Pb-doped and Pb-free 2223 phase were obtained in the temperature range of 850–870°C and heat treatment for <15 h, that is much faster, compared with a solid-state reaction technique. The single crystals with the size of 0.5 mm were isolated and characterized using XRD, SEM/EDX, susceptibility and resistivity measurements. The effect of post-annealing in different atmospheres on the oxygen content and superconductivity of the samples was studied by TG and chemical analyses.
We present a new reconnection-based arbitrary-Lagrangian–Eulerian (ALE) method. The main elements in a standard ALE simulation are an explicit Lagrangian phase in which the solution and grid are updated, a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred (conservatively interpolated) onto the new grid. In standard ALE methods the new mesh from the rezone phase is obtained by moving grid nodes without changing connectivity of the mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In our new method we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and allows to follow Lagrangian features of the mesh much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tessellation. We demonstrate performance of our new method on series of numerical examples and show it superiority in comparison with standard ALE methods without reconnection.
We describe an implementation of a compact parallel algo- rithm for 3D Delaunay tetrahedralization on a 64-processor shared-memory machine. Our algorithm uses a concurrent version of the Bowyer-Watson incremental insertion, and a thread-safe space-efficient structure for representing the mesh. Using the implementation we are able to generate sig- nificantly larger Delaunay meshes than have previously been generated—10 billion tetrahedra on a 64 processor SMP us- ing 200GB of RAM. The implementation makes use of a locality based rela- beling of the vertices that serves three purposes—it is used as part of the space efficient representation, it improves the memory locality, and it reduces the overhead necessary for locks. The implementation also makes use of a caching tech- nique to avoid excessive decoding of vertex information, a technique for backing out of insertions that collide, and a shared work queue for maintaining points that have yet to be inserted. The sequential algorithm is based on incremental inser- tion using the well-known Bowyer-Watson kernel (12, 43). During the course of the algorithm a Delaunay triangula- tion of the current pointset is maintained. An incremental step inserts a new vertex into the mesh by determining the tetrahedra that violate the Delaunay condition. We use the idea of Clarkson and Shor (18) and maintain an association between uninserted vertices and the tetrahedra containing those vertices. We keep a work queue of tetrahedra whose
It has long been realized that in pulse-code modulation (PCM), with a given ensemble of signals to handle, the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall. It has been shown by Panter and Dite that, in the limit as the number of quanta becomes infinite, the asymptotic fractional density of quanta per unit voltage should vary as the one-third power of the probability density per unit voltage of signal amplitudes. In this paper the corresponding result for any finite number of quanta is derived; that is, necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy. The optimization criterion used is that the average quantization noise power be a minimum. It is shown that the result obtained here goes over into the Panter and Dite result as the number of quanta become large. The optimum quautization schemes for
2^{b}
quanta,
b=1,2, \cdots, 7
, are given numerically for Gaussian and for Laplacian distribution of signal amplitudes.
We propose a new isotropic remeshing method, based on Centroidal Voronoi Tessellation (CVT). Constructing CVT requires to repeatedly compute Restricted Voronoi Diagram (RVD), defined as the intersection between a 3D Voronoi diagram and an input mesh surface. Existing methods use some approximations of RVD. In this paper, we introduce an efficient algorithm that computes RVD exactly and robustly. As a consequence, we achieve better remeshing quality than approximation-based approaches, without sacrificing efficiency. Our method for RVD computation uses a simple procedure and a kd-tree to quickly identify and compute the intersection of each triangle face with its incident Voronoi cells. Its time complexity is O(mlogn), where n is the number of seed points and m is the number of triangles of the input mesh. Fast convergence of CVT is achieved using a quasi-Newton method, which proved much faster than Lloyd's iteration. Examples are presented to demonstrate the better quality of remeshing results with our method than with the state-of-art approaches.
We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities---short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number.
Voro++ is a free software library for the computation of three dimensional Voronoi cells. It is primarily designed for applications in physics and materials science, where the Voronoi tessellation can be a useful tool in the analysis of densely-packed particle systems, such as granular materials or glasses. The software comprises of several C++ classes that can be modified and incorporated into other programs. A command-line utility is also provided that can use most features of the code. Voro++ makes use of a direct cell-by-cell construction, which is particularly suited to handling special boundary conditions and walls. It employs algorithms which are tolerant for numerical precision errors, and it has been successfully employed on very large particle systems.
A new algorithm for molecular dynamics simulations of biological macromolecules on parallel computers, point-centered domain decomposition, is introduced. The molecular system is divided into clusters that are assigned to individual processors. Each cluster is characterized by a center point and comprises all atoms that are closer to its center point than to the center point of any other cluster. The point-centered domain decomposition algorithm is implemented in the new program OPALpusinga standard message passing library, so that it runs on both shared memory and massively parallel distributed memory computers. Benchmarks show that the program makes efficient use of up to 100 and more processors for realistic systems of a protein in water comprising 10000 to 20000 atoms. 2000 Elsevier Science B.V. All rights reserved. Keywords: Molecular dynamics; Parallel computing; Message passing; Domain decomposition; MPI; OPALp 1. Introduction Molecular dynamics (MD) simulations [1] have be...
Polygon Voronoi Library (Version 1.0)
- A Sydorchuk
- Boost
- Lee