Poisson Voronoi Tessellation Research Articles

A numerical method for the generation of hierarchical Poisson Voronoi microstructures applied in micromechanical finite element simulations\u2014part I: method

Poisson Voronoi (PV) tessellations as artificial microstructures are widely used in investigations of material deformation behaviors. However, a PV structure usually describes a relative homogeneous field. This work presents a simple numerical method for generating 2D/3D artificial microstructures based on hierarchical PV tessellations. If grains/particles of a phase cover a large size span, the concept of “artificial phases” can be used to create a more realistic size distribution. From case to case, detailed microstructural features cannot be directly achieved by commercial or free softwares, but they are necessary for a deep or thorough study of the material deformation behavior. PV tessellations created in our process can fulfill individual requirements from material designs. Another reason to use PV tessellations is due to the limited experimental data. Concerning the application of PV microstructures, four examples are given. The FE models and results will be presented in consecutive works, i.e. “part II: applications”.

Poisson–Voronoi Tessellation on a Riemannian Manifold

Abstract In this paper, we consider a Riemannian manifold $M$ and the Poisson–Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda $ on $M$. We obtain a two-term asymptotic expansion, when $\lambda $ goes to infinity, of the mean number of vertices of the Voronoi cell associated with $x_0$. The 1st term of the estimate is equal to the mean number of vertices in the Euclidean setting, while the 2nd term involves the scalar curvature of $M$ at $x_0$. This settles with the proper and rigorous frame the former 2D statement from [ 19] and extends it to higher dimension. The key tool for proving this result is a new change of variables formula of Blaschke–Petkantschin type in the Riemannian setting, which brings out the Ricci curvatures of the manifold.

Distribution of Cell Area in Bounded Poisson Voronoi Tessellations with Application to Secure Local Connectivity

Poisson Voronoi tessellations have been used in modeling many types of systems across different sciences, from geography and astronomy to telecommunications. The existing literature on the statistical properties of Poisson Voronoi cells is vast, however, little is known about the properties of Voronoi cells located close to the boundaries of a compact domain. In a domain with boundaries, some Voronoi cells would be naturally clipped by the boundary, and the cell area falling inside the deployment domain would have different statistical properties as compared to those of non-clipped Voronoi cells located in the bulk of the domain. In this paper, we consider the planar Voronoi tessellation induced by a homogeneous Poisson point process of intensity $\lambda\!>\!0$ in a quadrant, where the two half-axes represent boundaries. We show that the mean cell area is less than $\lambda^{-1}$ when the seed is located exactly at the boundary, and it can be larger than $\lambda^{-1}$ when the seed lies close to the boundary. In addition, we calculate the second moment of cell area at two locations for the seed: (i) at the corner of a quadrant, and (ii) at the boundary of the half-plane. We illustrate that the two-parameter Gamma distribution, with location-dependent parameters calculated using the method of moments, can be of use in approximating the distribution of cell area. As a potential application, we use the Gamma approximations to study the degree distribution for secure connectivity in wireless sensor networks deployed over a domain with boundaries.

Investigation of microstructure effect on fretting fatigue crack initiation using crystal plasticity

AbstractThe onset of fretting fatigue is characterized by material microstructural changes in which the extent of the damage is comparable to grain size, and hence, the microstructure characteristics could have a significant effect on fatigue crack initiation. In this paper, a three‐dimensional finite element crystal plasticity framework is presented for simulation of the fretting fatigue. Controlled Poisson Voronoi tessellation (CPVT) method is employed to generate the polycrystalline region. In the CPVT method, regularity parameter controls the shape of grains. In this study, the impact of grain size and regularity parameter on crack initiation life and initiation site has been investigated. Cumulative plastic slip was used as a parameter of microstructure‐sensitive fatigue indicator. This parameter could effectively predict the location of crack initiation and its life. The results show that regularity parameter has a significant effect on the location of crack initiation. Furthermore, the effect of grain size on the fretting fatigue life of 316L stainless steel was investigated experimentally through testing different specimens with different grain sizes, to validate the simulation results.

Anchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaces

We show that a random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson–Voronoi tessellation and the hyperbolic Poisson–Delaunay triangulation, have $1$-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive hyperbolic speed. Finally, we include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson–Voronoi tessellation.

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.

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.

Modeling structures of open cell foams

This work proposes an original geometrical model based on randomly packed spheres using Laguerre-Voronoi tessellations to simulate geometrical and topological characteristics in the microstructure of open cell foams. The model can be used to analyze the effect of coefficient of variation on the pores distribution in real foams. The distribution of foam-cell volumes in foam structures generated in this work is dependent on the log-normal distribution of sphere volumes in corresponding randomly packed spheres. The statistical data of modeled foam structures, including distribution of the cell volume, face and edge number is very close to the characteristics of real materials. The results also show that a higher coefficient of variation in the sphere diameter would decrease the average number of faces per cell. The average number of faces varies from 13.56 to 14.43 for different coefficients of variation of sphere diameter, while the average number of faces in the Poisson-Voronoi tessellation structures is approximately 15.5. Furthermore, the porosity of foam structures, ε, decreases with the ratio of strut diameter to the average diameter of randomly packed spheres, ds/E(d), while the specific surface area of foams, SV, increases with ds/E(d).

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.

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.

Exact asymptotic statistics of the n-edged face in a 3D Poisson-Voronoi tessellation

This work considers the 3D Poisson-Voronoi tessellation. It investigates the joint probability distribution for an arbitrarily selected cell face to be n-edged and for the distance between the seeds of the two adjacent cells to be equal to 2L. For this quantity an exact expression is derived, valid in the limit with n1/6L fixed. The leading order correction term is determined. Good agreement with earlier Monte Carlo data is obtained. The cell face is shown to be surrounded by a three-dimensional domain that is empty of seeds and is the union of n balls; it is pumpkin-shaped and analogous to the flower of the 2D Voronoi cell. For this domain tends towards a torus of equal major and minor radii. The radii scale as n1/3, in agreement with earlier heuristic work. A detailed understanding is achieved of several other statistical properties of the n-edged cell face.

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.

Some remarks on second-order characteristics of planar Poisson-Voronoi tessellations

Some remarks on second-order characteristics of planar Poisson-Voronoi tessellations

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).

On Non-Poissonian Voronoi Tessellations

<p>The Voronoi tessellation is the partition of space for a given seeds pattern and the result of the partition depends completely on the type of given pattern ”random”, Poisson-Voronoi tessellations (PVT), or ”non-random”, Non Poisson-Voronoi tessellations. In this note we shall consider properties of Voronoi tessellations with centers gener-ated by Sobol quasi random sequences which produce a more ordered disposition of the centers with respect to the PVT case. A probability density function for volumes of these Sobol Voronoi tessellations (SVT) will be proposed and compared with results of numerical simulations. An application will be presented concerning the local struc-ture of gas (CO<sub>2</sub>) in the liquid-gas coexistence phase. Furthermore a probability distribution will be computed for the length of chords resulting from the intersections of random lines with a three-dimensional SVT. The agreement of the analytical formula with the results from a computer simulation will be also investigated. Finally a new type of Voronoi tessellation based on adjustable positions of seeds has been introduced which generalizes both PVT and SVT cases.</p>

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.

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.