Poisson Voronoi Tessellation Research Articles

Generalised Voronoi tessellation for generating microstructural finite element models with controllable grain-size distributions and grain aspect ratios

Summary A generalised Voronoi tessellation is proposed to create three-dimensional microstructural finite element model, which can effectively reproduce the grain size distribution and grain aspect ratio obtained from experiments. This new approach consists of two steps. The first step generates the desired lognormal grain size distribution with a given average grain volume and standard deviation. The second step requires grouping meshed elements to create a specific grain aspect ratio, using the Voronoi generators from the first step. A new concept is introduced to describe the transition from the Poisson–Voronoi tessellation to the centroidal Voronoi tessellation. More importantly, instead of using the conventional way where the Voronoi cells are first generated and then meshed into finite elements, this new approach discretises the pre-meshed specimen with the Voronoi generators. This new technique prevents the presence of high density mesh at the vertices of Voronoi cells, and can tessellate irregular geometry much more easily. Examples of microstructures with different size distributions, non-equiaxed grains and complicated specimen geometries further demonstrate that the proposed approach can offer great flexibility to model various specimen geometries while keeping the process simple and efficient. Copyright © 2015 John Wiley & Sons, Ltd.

Spatial Spectrum and Energy Efficiency of Random Cellular Networks

It is a great challenge to evaluate the network performance of cellular mobile communication systems. In this paper, we propose new spatial spectrum and energy efficiency models for Poisson-Voronoi tessellation (PVT) random cellular networks. To evaluate the user access the network, a Markov chain based wireless channel access model is first proposed for PVT random cellular networks. On that basis, the outage probability and blocking probability of PVT random cellular networks are derived, which can be computed numerically. Furthermore, taking into account the call arrival rate, the path loss exponent and the base station (BS) density in random cellular networks, spatial spectrum and energy efficiency models are proposed and analyzed for PVT random cellular networks. Numerical simulations are conducted to evaluate the network spectrum and energy efficiency in PVT random cellular networks.

Percolation on stationary tessellations: models, mean values, and second-order structure

We consider a stationary face-to-face tessellation X of Rd and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We are interested in geometric properties of the union Z of black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes of Z ∩ W, where the observation window W is assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes of X. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.

Percolation on stationary tessellations: models, mean values, and second-order structure

We consider a stationary face-to-face tessellationXofRdand introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probabilitypand white otherwise. We are interested in geometric properties of the unionZof black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes ofZ∩W, where the observation windowWis assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes ofX. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.

Many-faced cells and many-edged faces in 3D Poisson–Voronoi tessellations

Motivated by recent new Monte Carlo data we investigate a heuristic asymptotic theory that applies to n-faced 3D Poisson–Voronoi cells in the limit of large n. We show how this theory may be extended to n-edged cell faces. It predicts the leading order large-n behavior of the average volume and surface area of the n-faced cell, and of the average area and perimeter of the n-edged face. Such a face is shown to be surrounded by a toroidal region of volume n/λ (with λ the seed density) that is void of seeds. Two neighboring cells sharing an n-edged face are found to have their seeds at a typical distance that scales as n−1/6 and whose probability law we determine. We present a new data set of 4 × 109 Monte Carlo generated 3D Poisson–Voronoi cells, larger than any before. Full compatibility is found between the Monte Carlo data and the theory. Deviations from the asymptotic predictions are explained in terms of subleading corrections whose powers in n we estimate from the data.

Fractal random series generated by Poisson-Voronoi tessellations

In this paper, we construct a new family of random series defined on R D \mathbb {R}^D , indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of R D \mathbb {R}^D are not the usual dyadic meshes but random Voronoi tessellations generated by Poisson point processes. This approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and Hausdorff dimensions. The proof of this main result is based on several new distributional properties of the Poisson-Voronoi tessellation on the one hand, and an estimate of the oscillations of the function coupled with an application of a Frostman-type lemma on the other hand. Finally, we introduce two related models and provide in particular a box-dimension calculation for a derived deterministic Takagi-Knopp series with hexagonal bases.

Spatial cluster point processes related to Poisson–Voronoi tessellations

We discuss how to construct models for cluster point processes within ‘territories’ modelled by $$d$$ -dimensional Voronoi cells whose nuclei are generated by a latent Poisson process (where the planar case $$d=2$$ is of our primary interest). Conditional on the territories/cells, the clusters are independent Poisson processes whose points may be aggregated around or away from the nuclei and along or away from the boundaries of the cells. Observing the superposition of clusters within a bounded region, we discuss how to account for edge effects. Bayesian inference for a particular flexible model is discussed in connection to a botanical example.

Rate of Poisson approximation for nearest neighbor counts in large-dimensional Poisson point processes

Consider two independent homogeneous Poisson point processes Π of intensity λ and Π′ of intensity λ′ in d-dimensional Euclidean space. Let qk,d, k=0,1,…, be the fraction of Π-points which are the nearest Π-neighbor of precisely k Π′-points. It is known that as d→∞, the qk,d converge to the Poisson probabilities e−λ′/λ(λ′/λ)k/k!, k=0,1,…. We derive the (sharp) rate of convergence d−1/2(4/33)d, which is related to the asymptotic behavior of the variance of the volume of the typical cell of the Poisson–Voronoi tessellation generated by Π. An extension to the case involving more than two independent Poisson point processes is also considered.

Extreme values for characteristic radii of a Poisson-Voronoi Tessellation

A homogeneous Poisson-Voronoi tessellation of intensity γ is observed in a convex body W. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in W. We prove that when $\gamma \rightarrow \infty $ , these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between W and its so-called Poisson-Voronoi approximation.

Large-n conditional facedness mn of 3D Poisson–Voronoi cells

We consider the three-dimensional Poisson–Voronoi tessellation and study the average facedness mn of a cell known to neighbor an n-faced cell. Whereas Aboav’s law states that mn = A + Bn−1, theoretical arguments indicate an asymptotic expansion mn = 8 + k1n−1/6 + ⋯ . Recent new Monte Carlo data due to Lazar et al based on a very large data set now clearly rule out Aboav’s law. In this work we determine the numerical value of k1 and compare the expansion to the Monte Carlo data. The calculation of k1 involves an auxiliary planar cellular structure composed of circular arcs, which we will call the Poisson–Möbius diagram. It is a special case of more general Möbius diagrams (or multiplicatively weighted power diagrams) and is of interest for its own sake. We obtain exact results for the total edge length per unit area, which is a prerequisite for the coefficient k1, and a few other quantities in this diagram.

Scaling properties of planar discrete Poisson-Voronoi tessellations with von Neumann neighborhoods constructed according to the nucleation and growth mechanism

In contrast to the conventional continual case, discrete Poisson-Voronoi tessellations resulting from the growth to impingement of random nuclei differ from tessellations constructed from the nearest tile loci. Previously studied tessellations were based directly on the notion of locus [A. Korobov, Phys. Rev. E 79, 031607 (2009); A. Korobov, Phys. Rev. E 87, 014401 (2013)]. This paper presents results for tessellations constructed by the growth of random nuclei. Their boundaries have a different structure and scaling properties are comparably more robust. One more scalable characteristic may be introduced for them, the perimeter distribution function, which is well approximated by the normal distribution function with the unit mean and the standard deviation equal to 0.25.

Mathematical algorithm development and parametric studies with the GEOFRAC three-dimensional stochastic model of natural rock fracture systems

This paper presents results from research conducted at MIT during 2010–2012 on modeling of natural rock fracture systems with the GEOFRAC three-dimensional stochastic model. Following a background summary of discrete fracture network models and a brief introduction of GEOFRAC, the paper provides a thorough description of the newly developed mathematical and computer algorithms for fracture intensity, aperture, and intersection representation, which have been implemented in MATLAB. The new methods optimize, in particular, the representation of fracture intensity in terms of cumulative fracture area per unit volume, P32, via the Poisson-Voronoi Tessellation of planes into polygonal fracture shapes. In addition, fracture apertures now can be represented probabilistically or deterministically whereas the newly implemented intersection algorithms allow for computing discrete pathways of interconnected fractures. In conclusion, results from a statistical parametric study, which was conducted with the enhanced GEOFRAC model and the new MATLAB-based Monte Carlo simulation program FRACSIM, demonstrate how fracture intensity, size, and orientations influence fracture connectivity.

Second-order comparison of three fundamental tessellation models

We consider three random tessellation models which, due to their analytical tractability, play an important role as reference models for random tessellations: Poisson Voronoi tessellations, Poisson hyperplane tessellations, and STIT tessellations. We present a systematic comparison of second-order properties of these three tessellation models in ℝ2 and ℝ3. If possible, we make use of analytical formulae for the considered characteristics. In cases where explicit formulae are still unknown, the characteristics are studied by simulation.

Energy Efficiency Evaluation of Cellular Networks Based on Spatial Distributions of Traffic Load and Power Consumption

Energy efficiency has gained its significance when service providers' operational costs burden with the rapidly growing data traffic demand in cellular networks. In this paper, we propose an energy efficiency model for Poisson-Voronoi tessellation (PVT) cellular networks considering spatial distributions of traffic load and power consumption. The spatial distributions of traffic load and power consumption are derived for a typical PVT cell, and can be directly extended to the whole PVT cellular network based on the Palm theory. Furthermore, the energy efficiency of PVT cellular networks is evaluated by taking into account traffic load characteristics, wireless channel effects and interference. Both numerical and Monte Carlo simulations are conducted to evaluate the performance of the energy efficiency model in PVT cellular networks. These simulation results demonstrate that there exist maximal limits for energy efficiency in PVT cellular networks for given wireless channel conditions and user intensity in a cell.

Publisher’s Note: Scaling properties of the area distribution functions and kinetic curves of dense plane discrete Poisson-Voronoi tessellations [Phys. Rev. E87, 014401 (2013)

Received 1 February 2013DOI:https://doi.org/10.1103/PhysRevE.87.029901©2013 American Physical Society

Scaling properties of the area distribution functions and kinetic curves of dense plane discrete Poisson-Voronoi tessellations

This brief report supplements and refines previous publications [Korobov, Phys. Rev. B 76, 085430 (2007); Korobov, Phys. Rev. E 79, 031607 (2009); Korobov, Phys. Rev. E 84, 021602 (2011)] to show that the scaling properties of the area distribution functions and kinetic curves actually hold for dense plane discrete Poisson-Voronoi tessellations. The previously noted apparent violations are due to the involved structures of boundaries of random domains.

Revisiting the Cosmological Principle in a Cellular Framework

The cosmological principle in its various versions states that: (i) the galaxy does not occupy a particular position, (ii) the Universe is homogeneous and isotropic. This statement does not agree with the recent astronomical observations in the range z lower than 0.05 which are in agreement with a cellular structure of the Universe. Here we present a local analysis of the inhomogeneity of the Universe. When z is greater than 0.05 our analysis cannot be applied because the astronomical sample of galaxies here processed is not complete. The two tools of the Poisson Voronoi Tessellation (PVT) and the luminosity function for galaxies allow building a new version of the local cosmological principle.

Complete Topology of Cells, Grains, and Bubbles in Three-Dimensional Microstructures

We introduce a general, efficient method to completely describe the topology of individual grains, bubbles, and cells in three-dimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of three-dimensional microstructures that are often regarded as close analogues in the literature: one resulting from normal grain growth (mean curvature flow) and another resulting from a random Poisson-Voronoi tessellation of space. Grain growth strongly favors particular grain topologies, compared with the Poisson-Voronoi model. Moreover, the frequencies of highly symmetric grains are orders of magnitude higher in the grain growth microstructure than they are in the Poisson-Voronoi one. Grain topology statistics provide a strong, robust differentiator of different cellular microstructures and provide hints to the processes that drive different classes of microstructure evolution.

Three-dimensional virtual grain structure generation with grain size control

A three-dimensional (3D) controlled Poisson Voronoi tessellation (CPVT) model has been developed for generating 3D polycrystalline grain structures for micromechanics simulations. A virtual grain structure generated using the CPVT model has the property that its grain size distribution is statistically equivalent to the actual grain structure in term of the specified physical parameters: the mean grain size, a small grain size, a large grain size, and the percentage of grains within that range. Development of the CPVT model requires three steps: (1) Defining the regularity that specifies the uniformity of a tessellation, and deriving the control parameter based on the regularity, (2) establishing the mapping from the regularity to the distribution parameter of a one-parameter gamma distribution, (3) defining the mapping from the set of physical parameters to the distribution parameter. Relations between the regularity and distribution parameter, for a range of regularity values, are determined by a comprehensive set of statistical experiments, in which data fitting for the grain size distribution model is in each case obtained by an evolutionary optimisation algorithm. A software system (VGRAIN) has been developed for implementing the proposed three-dimensional CPVT model to generate the grain structure for crystal plasticity finite element (CPFE) analysis. To demonstrate the proposed scheme and the VGRAIN system, CPFE analyses of compression of micro-pillars are performed, and the effects of both regularity and grain size on the deformation are studied.

Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

The heat transfer problem in a polycrystal with nonlinear jump conditions on the grain boundaries will be homogenized using the method of stochastic two-scale convergence developed by Zhikov and Pyatnitskii [V.V. Zhikov and A.L. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math. 70(1) (2006), pp. 19–67] and recently extended by the author [M. Heida, An extension of stochastic two-scale convergence and application, Asympt. Anal. (2010) (in press)]. It will be shown that for monotone Lipschitz jump conditions differentiable in 0, the nonlinearity vanishes in the limit. Additionally, existing Poincaré inequalities will be extended to more general geometric settings with the only restriction of local C 1-interfaces with finite intensity. In particular, the result can now be applied to the Poisson–Voronoi tessellation.