Poisson Voronoi Research Articles

Energy Harvesting and Cell Zooming in $K-$ Tier Heterogeneous Random Cellular Networks

In this paper, we study the performance of ${K}$ –tier heterogeneous random cellular networks (HetNets) in terms of energy efficiency (EE) and spectral efficiency (SE). In order to improve efficiency, two cell zooming (CZ) techniques, namely telescopic and ON/OFF schemes, are applied to these networks. Accordingly, we investigate two scenarios in this paper. First, we focus on a single tier scenario and derive a new expression for the ergodic capacity of a Poisson–Voronoi tessellation random cellular network using stochastic geometry. Besides, the EE and SE of this model are depicted for several user densities and CZ parameters. In the second scenario, we investigate multitier HetNets in which the base stations gather their required energy from the environment. Therefore, two important concepts in green cellular networks, namely CZ and energy harvesting, are addressed in this paper. Subsequently, numerical results show the performance improvement in terms of the coverage and blocking probabilities in addition to EE and SE.

Noise sensitivity and Voronoi percolation

In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $\mathbb{R} ^2$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

Sharpness of the phase transition for continuum percolation in $$\mathbb {R}^2$$ R 2

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $$\lambda _c$$ . Moreover, The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.

Achieving Ultra Reliable Communication in 5G Networks: A Dependability Perspective Availability Analysis in the Space Domain

As part of the 5G communication paradigm, ultra reliable communication (URC) is envisaged as an important technology pillar for providing anywhere and anytime services to end users. While most existing studies on reliable communication do not investigate this problem from a dependability theory perspective, those dependability-based studies tend to define reliability merely in the time domain. In this letter, we advocate extending the concept of URC from the dependability perspective also in the space domain . We initiate definitions on cell availability and system availability. The availability and the probability of providing a guaranteed level of availability in a network are analyzed both/either cell-wise and/or system-wise. Poisson point process and Voronoi tessellation are adopted to model the spatial characteristics of cell deployment in both homogeneous and heterogeneous networks.

Planar discrete birth-growth Poisson–Voronoi tessellations with the von Neumann neighbourhood

Poisson–Voronoi tessellations are widely used as the generic model for studying various birth-growth processes and resulting morphologies in physics, chemistry, materials science, and related fields. This paper studies planar discrete Poisson–Voronoi tessellations constructed directly by the growth to impingement of random square germs. They materially differ from similar tessellations constructed of the nearest tile loci according to the basic definition. The boundary structure is described in detail. Its peculiarities are used to extend the concept of Gabriel edges to the considered discrete case and also to quantify this concept. The averaged percentage of Gabriel edges appears to be practically independent of the germs density, = 70%. The studied densities range from 0.01 to 0.000 01. Statistical results are presented for the whole tessellation and also for subsets of random domains with the given number of edges ν. Two sets of results are compared: for edges of each random domain arranged from the longest to the shortest and for edges arranged from the nearest to the most distant. Averaged distances to neighbours in the metric determined by the growth mode of islands are compared with that in the Euclidean metric. Also, the cyclic sequences of edge lengths of random domains are examined. The linearity with respect to ν is revealed for four scaling-related characteristics: the area of random domains, the perimeter length of random domains, the area of complete concentric belts, and the coordinates of maxima of kinetic curves.

Towards efficient spall generation simulation in rolling element bearing

AbstractRolling element bearing prognosis is the process of forecasting the remaining operational life, future condition or probability of failure of the bearing. While operational, bearings are subjected to rolling contact fatigue (RCF), and, as a result, a spall is generated on the raceway of the bearing. Complete understanding of the fatigue process is critical for predictive modelling to estimate bearing remaining useful life, which allows improved scheduling of maintenance actions. This work presents an RCF model that was implemented using abaqus finite element software. The RCF model is based on a damage mechanics approach that relates the accumulated microscopic failure mechanisms to a damage state variable and includes representation of material grain structure by a Poisson–Voronoi tessellation. Different microstructures, with a variety of material properties and grain topologies, were constructed for simulation purposes. The geometry of the simulated spalls and the Weibull slopes of the fatigue lives are in good agreement with published theoretical and experimental data. It can be concluded that the assumptions and the simplifications of the current, convenient to use, RCF model yield a sufficiently accurate tool on the basis of previous publications and experimental data.

Development of physically based atomistic microstructures: The effect on the mechanical response of polycrystals

A method for the generation of atomistic realizations of polycrystalline aggregates from a phase field grain growth model is presented. Topologies of computational microstructures constructed from the proposed method as well as conventional Poisson Voronoi tessellation are quantitatively compared. While little difference is exhibited in the macroscale mechanical response, substantial differences in the resolved contribution to strain of deformation mechanisms of the structures under uniaxial tension are uncovered using post-processing kinematic metrics. These differences in the fundamental strain accommodation processes suggest that grain topology and grain boundary character significantly affect local responses of polycrystals in molecular dynamics simulations and that significant attention should be paid to the chosen starting microstructure.

Asymptotic theory for statistics of the Poisson–Voronoi approximation

This paper establishes expectation and variance asymptotics for statistics of the Poisson--Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approximation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formula applicable in the setting of sums of stabilizing functionals.

Plasticity in Materials with Heterogeneous Microstructures

The heterogeneous microstructure has a predominant effect on the mechanical behavior of polycrystalline material. In most instances, a homogenized parameter such as mean grain size is used to describe and to represent the microstructure. However, these models do not account for a measure of heterogeneity in the grain size and grain shape distributions. In this work, we introduce the grain size distribution into a multi-scale stress–strain-gradient model using a controlled Poisson Voronoi tessellation. The correlation between grain size distribution and strength is studied with various cases of grain size distribution with a fixed grid area and mean grain size. In addition, the effect of the spatial distribution of second phases and grain size on the material strength and ductility is also investigated. The results show that introducing heterogeneity into the microstructure can enhance the strength and ductility of the material compared with its equivalent homogeneous microstructure. In addition, different spatial distributions of phases can also lead to different mechanical responses.

Discrete dislocation and crystal plasticity analyses of load shedding in polycrystalline titanium alloys

The focus of this paper is the mechanistic basis of the load shedding phenomenon that occurs under the dwell fatigue loading scenario. A systematic study was carried out using a discrete dislocation plasticity (DDP) model to investigate the effect of crystallographic orientations, localised dislocation behaviour and grain combinations on the phenomenon. Rate sensitivity in the model arises from a thermal activation process at low strain rates, which is accounted for by associating a stress- and temperature-dependent release time with obstacles; the activation energy was determined by calibrating an equivalent crystal plasticity model to experimental data. First, the application of Stroh's dislocation pile-up model of crack nucleation to facet fracture was quantitatively assessed using the DDP model. Then a polycrystalline model with grains generated using a controlled Poisson Voronoi tessellation was used to investigate the soft-hard-soft rogue grain combination commonly associated with load shedding. Dislocation density and peak stress at the soft/hard grain boundary increased significantly during the stress dwell period, effects that were enhanced by dislocations escaping from pile-ups at obstacles. The residual stress after dwell fatigue loading was also found to be much higher compared to standard fatigue loading. Taylor (uniform strain) and Sachs (uniform stress) type assumptions in a soft-hard grain combination have been assessed with a simple bicrystal DDP model. Basal slip nucleation in the hard grain was found to be initiated by high stresses generated by strong pile ups in the soft grain, and both basal and pyramidal slip nucleation was observed in the hard grain when the grain boundary orientation aligned with that of an active slip system in the soft grain. The findings of this study give new insight into the mechanisms of load shedding and faceting associated with cold dwell fatigue in Ti alloys used in aircraft engines.

Study of Metal-Gate Work-Function Variation Using Voronoi Cells: Comparison of Rayleigh and Gamma Distributions

We have demonstrated, via validation to experimental data for TiN and Ru, that the grains that appear in the metal gate-stacks of nanoscale CMOS devices can be characterized via a two-parameter Gamma distribution ( $p$ -values 0.17 and 0.42 for TiN and Ru). Conversely, a previously presented fit that used Rayleigh distribution does not reproduce the experimental data ( $p$ -values $3\times 10^{-14}$ and 0.0029 for TiN and Ru). Poisson Voronoi diagrams (PVDs) are shown as a suitable algorithm to generate grains with Gamma distribution, via fitting of the distribution of 10 000 grains. We have also compared the PVD variability against the Rayleigh model for both TiN and TaN metal gates, and concluded that Rayleigh approach overestimates the device variability (by 11.9% for the TiN and by 7.14% for the TaN).

Interplay between Kolmogorov–Johnson–Mehl–Avrami kinetics and Poisson–Voronoi tessellation

In this paper we investigate the connection between Voronoi tessellation and the KJMA approach of space filling. In particular, we study how nuclei, in their growth, cover a given Voronoi cell. This approach leads to an integral equation for the cell-size distribution function. Starting from the 1D case, that is solved exactly, we extend the results to the dD case. The analysis allows to find a rationale to the phenomenological parameter entering the Gamma distribution function and to improve the description of the transformation through the knowledge of the kinetics of grain formation. Moreover, the nucleus size distribution function has been calculated as a function of the transformed fraction.

A multilevel approach to the evolutionary generation of polycrystalline structures

The Poisson–Voronoi tessellation is commonly used as an approximation to the microstructure of polycrystalline material. Although simple, this approximation fails to respect basic physical properties observed empirically, including the generally lognormal distribution of grain sizes. Stochastic approximations such as genetic algorithms can be used to adjust a Poisson–Voronoi tessellation to better reflect such a distribution. We apply techniques from multilevel optimisation to give a new approach to the evolutionary generation of polycrystalline structures, in a way that allows approximation of a target lognormal grain size distribution with unit mean and arbitrary variance. Results obtained through this method indicate almost perfect distribution fitting, show up to two orders of magnitude improvement in the number of evolutionary steps required for an acceptable fit, and suggest a reduction of the overall problem complexity from Θ(N3) to Θ(N2).

Computational homogenisation from a 3D finite element model of asphalt concrete—linear elastic computations

Asphalt concrete (AC) is a composite material consisting of bituminous binders, mineral aggregate, and voids. Experimental and computational efforts to assess the bulk mechanical properties of AC ubiquitously raise the question of representative sample size. It is well known that the dimension of a sample has to be larger than the largest morphological entity. However, it has been shown that the required size is a function of the morphological and physical properties under consideration, e. g. the difference between the properties of the constituents at the microscale, and their volume fractions. In the present contribution, representative sample sizes are determined numerically for a variation of parameters in terms of a tolerable amount of statistical scatter by conducting virtual experiments. If the scatter between distinct heterogeneous samples falls below a certain threshold, the corresponding volume is called a representative volume element (RVE). Finite element (FE) discretisations of AC volume samples are generated by means of a shrunk Poisson Voronoi tessellation. In order to provide estimates for the evolution of the RVE size under changing material properties of the mortar, the stiffness ratio between rocks and mortar is varied by two orders of magnitude. Furthermore, the influence of a variation in volume fraction is investigated.

Asymptotic properties of Euclidean shortest-path trees in random geometric graphs

We consider asymptotic properties of two functionals on Euclidean shortest-path trees appearing in random geometric graphs in R2 which can be used, for example, as models for fixed-access telecommunication networks. First, we determine the asymptotic bivariate distribution of the two backbone lengths inside a certain class of typical Cox–Voronoi cells as the size of this cell grows unboundedly. The corresponding Voronoi tessellation is generated by a stationary Cox process which is concentrated on the edges of the random geometric graph and whose intensity tends to 0. The limiting random vector can be represented as a simple geometric functional of a decomposition of a typical Poisson–Voronoi cell induced by an independent random sector. Using similar methods, we consider the asymptotic bivariate distribution of the total lengths of the two subtrees inside the Cox–Voronoi cell.

Enhancing Coverage and Rate of Cell Edge User in Multi-Antenna Poisson Voronoi Cells

This paper analyzes the cell edge mobile user performance in the downlink cellular system. We develop frame-work for coverage probability and spectral efficiency. In particular, we analyzed the performance of multi-antenna mobile users under multi-antenna base stations (BSs). The expressions of coverage probability and spectral efficiency are derived for cell edge user using stochastic geometry. We investigate how much the performance of cell edge user is improved when distances connecting BSs and cell edge users are modeled with cell edge null probability distribution. The probability of coverage and spectral efficiency is studied using zero-forcing beam-forming and the performance metrics are compared between coordinated scheduling (CS) and without coordinated scheduling (w/o CS). The interesting observation from our results is that the edge user coverage and rate is closely approaching towards the inner cell typical mobile user's rate and coverage, and the performance is verified with relative probability of coverage gain analysis.

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.

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.

Surface order scaling in stochastic geometry

Let ${{ \mathcal{P}}}_{{\lambda}}:={{ \mathcal{P}}}_{{\lambda}{\kappa}}$ denote a Poisson point process of intensity ${\lambda}{\kappa}$ on $[0,1]^{d}$, $d\geq2$, with ${\kappa}$ a bounded density on $[0,1]^{d}$ and ${\lambda}\in(0,\infty)$. Given a closed subset ${ \mathcal{M}}\subset[0,1]^{d}$ of Hausdorff dimension $(d-1)$, we consider general statistics $\sum_{x\in{{ \mathcal{P}}}_{{\lambda}}}\xi(x,{{ \mathcal{P}}}_{{\lambda}},{ \mathcal{M}})$, where the score function $\xi$ vanishes unless the input $x$ is close to ${ \mathcal{M}}$ and where $\xi$ satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics $\sum_{x\in{{ \mathcal{P}}}_{{\lambda}}}\xi({\lambda}^{1/d}x,{\lambda}^{1/d}{{ \mathcal{P}}}_{{\lambda}},{\lambda}^{1/d}{ \mathcal{M}})$ as ${\lambda}\to\infty$. When ${ \mathcal{M}}$ is of class $C^{2}$, we obtain weak laws of large numbers and variance asymptotics for these statistics, showing that growth is surface order, that is, of order $\mathrm{Vol} ({\lambda}^{1/d}{ \mathcal{M}})$. We use the general results to deduce variance asymptotics and central limit theorems for statistics arising in stochastic geometry, including Poisson–Voronoi volume and surface area estimators, answering questions in Heveling and Reitzner [Ann. Appl. Probab. 19 (2009) 719–736] and Reitzner, Spodarev and Zaporozhets [Adv. in Appl. Probab. 44 (2012) 938–953]. The general results also yield the limit theory for the number of maximal points in a sample.

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.