Poisson Voronoi Tessellation Research Articles

Effects of random ferroelectric and dielectric phase distributions on junctionless ferroelectric field effect transistors

In this study, we comprehensively investigated the effects of random ferroelectric (FE) and dielectric (DE) phase distributions on junctionless ferroelectric field-effect transistors (JL-FeFETs). The Poisson–Voronoi tessellation (PVT) algorithm, which corresponds to the physical growth mechanism, was used to obtain grain nucleation in the ferroelectric layer. The simulation results demonstrated that as the probability of FE phase decreased from 80% to 40%, the standard deviation of the memory window (σMW) increased from 62.4 to 99.5 mV, and the possibility of forming a blocking current path from the source to the drain increased, which degraded the memory window (MW). The simulation results indicated that decreasing the gate length and width increased device variations. Furthermore, σMW decreased from 84.5 to 58.9 mV as the grain size decreased from 5 to 3 nm.

Effective electrical conductivity of random resistor networks generated using a Poisson–Voronoi tessellation

We studied the effective electrical conductivity of dense random resistor networks (RRNs) produced using a Voronoi tessellation when its seeds are generated by means of a homogeneous Poisson point process in the two-dimensional Euclidean space. Such RRNs are isotropic and in average homogeneous; however, local fluctuations of the number of edges per unit area are inevitable. These RRNs may mimic, e.g., crack-template-based transparent conductive films. The RRNs were treated within a mean-field approach. We found an analytical dependency of the effective electrical conductivity on the number of conductive edges (resistors) per unit area, nE. The effective electrical conductivity is proportional to nE when nE≫1.

Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks.

We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a random bipartite associator graph in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an independent PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in R2 as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of kth contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its load. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed.

On impact loading of Voronoi functional graded porous structure

In the realm of developing novel lightweight forms with enhanced mechanical properties, structural innovation introduces a fresh perspective, with a key focus on porous structures that must be addressed. Different from previous gradient control methods, this paper focuses on porosity (relative density) control criteria and establishes porous structures with different porosity gradients. This paper aims to evaluate the potential application of Voronoi functional graded porous structures (FGPS). The study employs Poisson-Voronoi tessellation to generate FGPS models, creating four Voronoi porous structures with diverse porosity gradients. Model verification involves comparing results with the Gibson-Ashby porous structure theory. Subsequent analyses rely on numerical modeling and finite element analysis (FEA) using Abaqus. The research explores the impact of varying impact velocities and directions as parameters for analysis. Parametric studies reveal that the most influential factor affecting the deformation behavior of Voronoi FGPS is the impact velocity. As impact velocity increases, the influence of gradient magnitude and direction becomes relatively less pronounced. Notably, the stress-strain relationship on the impact side shows greater sensitivity to impact velocity compared to the fixed end. Furthermore, the energy absorption of Voronoi FGPS demonstrates varying advantageous ranges under different operational conditions, including impact velocity and final strain levels. In summary, the size, direction, and impact velocity of the porosity gradient in Voronoi FGPS distinctly influence deformation behavior, stress-strain relationships, densification strain, plateau stress, and energy absorption at different strain levels. The future of FGPS lies in the development of advanced materials with increasingly sophisticated gradients, allowing for superior performance optimization in demanding applications, such as the lightweight and crashworthiness design of vehicles and aircrafts, biomimetic porosity designs and protective layers for civil engineering. These findings provide valuable insights for the practical design of porosity-controlled FGPS.

Fundamentals of Vehicular Communication Networks With Vehicle Platoons

Vehicular platooning is a promising way to facilitate efficient movement of vehicles with a shared route. Despite its relevance, the interplay of platooning and the communication performance in the resulting vehicular network (VN) is largely unexplored. Inspired by this, we develop a comprehensive approach to statistical modeling and system-level analysis of VNs with platooned traffic. Modeling the network of roads using the by-now well-accepted Poisson line process (PLP), we place vehicles on each road according to an independent Matérn cluster process (MCP) that jointly captures randomness in the locations of platoons on the roads and vehicles within each platoon. The resulting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">triply-stochastic</i> point process is a PLP-driven-Cox process, which we term the PLP-MCP. We first present this new point process’s distribution and derive several fundamental properties essential for the resulting VN’s analysis. Assuming that the cellular base-stations (BSs) are distributed as a Poisson point process (PPP), we derive the distribution of the loads served by the typical BS and the BS associated with the typical user. In deriving the latter, we also present a new approach to deriving the length distribution of a tagged chord in a Poisson Voronoi tessellation. Using the derived results, we present the rate coverage of the typical user while considering partial loading of the BSs. We also provide a comparative analysis of VNs with and without platooning of traffic.

Fatigue crack growth in polysilicon microstructures subjected to multi-axial loading: A Poisson–Voronoi-based finite element analysis

Polycrystalline silicon (polysilicon) is the most commonly used structural material of microscopic electromechanical devices, such as sensors and actuators. Almost all of these miniaturized devices contain mechanical elements experiencing high-frequency loading cycles. Due to the rapidly accumulating loading cycles and very small thicknesses of these microstructures, basic understanding of fatigue crack growth in polysilicon at the microscale is critical to the design of durable microdevices that satisfy application requirements. In this investigation, fatigue crack growth in a typical polysilicon microstructure subjected to multi-axial loading was analyzed with the finite element method. To account for the inherent heterogeneity and anisotropy of polysilicon at the microscale, a Poisson-Voronoi tessellation was incorporated in the highly stressed region of the resonating microdevice to model a polycrystalline microstructure. Simulation results illuminated the effect of local texture on the direction and rate of crack growth. Transgranular or intergranular crack growth were predicted, depending on the angle between the crack-path direction and the grain boundary and the fracture resistance of the grain and the grain boundary. From a fundamental fracture mechanics perspective, the computational approach developed in this study provides a capability for examining the effect of local texture anisotropy on cracking in polycrystalline microstructures.

Beta-star polytopes and hyperbolic stochastic geometry

Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in Rd with density proportional to (‖x‖2−1)−β, where ‖x‖>1 and β>d/2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k-dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected f-vector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry.

Criteria for Poisson process convergence with applications to inhomogeneous Poisson–Voronoi tessellations

This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables, point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson–Voronoi tessellations and large nearest neighbor distances in a Boolean model of disks.

Palm theory, random measures and Stein couplings

We establish a general Berry–Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in four examples: completely random measures, excursion random measure of a locally dependent random process, and the total edge length of Ginibre–Voronoi tessellations and of Poisson–Voronoi tessellations. Moreover, we apply the general bound to Stein couplings and discuss the special cases of local dependence and additive functionals in occupancy problems.

On point processes defined by angular conditions on Delaunay neighbors in the Poisson–Voronoi Tessellation

AbstractConsider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. The present note discusses the properties of two stationary point processes associated with the latter and depending on a parameter $\theta$ . The first is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and such that the angle with which they see the two nuclei defining the facet is $\theta$ . The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight line having a random orientation. Its intensity is well known. The intersection points almost surely belong to one-dimensional facets. The main question here concerns the Palm distribution of the angle with which the points of this second point process see the two nuclei associated with the facet. We will give answers to these two questions and briefly discuss their practical motivations. We also discuss natural extensions to three dimensions.

Variability Analysis in a 3-D Multigranular Ferroelectric Capacitor

A simulation-based study of variability of remnant polarization $\left (P_r \right)$ in a multi-granular 3-D ultra-thin ferroelectric (FE) capacitor is presented in this paper. The Poisson Voronoi Tessellation Diagram (PVD) is used for the nucleation of grains in the FE region, which corresponds to the physical growth mechanism. The PVD algorithm implemented in MATLAB is coupled with TCAD simulations, to trace the ferroelectric hysteresis loop. It is found that the grains which have linear profile of $P_r$ show larger variability in the FE hysteresis loop, compared to the grains, which follow the Gaussian distribution of $P_r$. Additionally, the impact of dielectric content in the FE grains is analyzed. It is seen that the dielectric grains cause very large amount of variability in the FE hysteresis loop. An increase in the dielectric grains also leads to a loss in the retentivity of the hysteresis loop.

Continuum line-of-sight percolation on Poisson–Voronoi tessellations

AbstractIn this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

3D Network Modeling for THz-Enabled Ultra-Fast Dense Networks: A 6G Perspective

Terahertz (THz) communications are envisioned as a critical technology for 6G and enable ultra-fast dense networks with tiny cells. Implementing THz-enabled ultra-fast cells requires new techniques due to its 3D nature. Stochastic geometry is a tool that is widely used for network deployment, which provides tractability and feasibility. In this article, we provide a summary of 3D network modeling for the THz system. We discuss the new 6G ultra-cell scenario along with a potential 3D network architecture. Several recent developments in network modeling applying a stochastic geometry-based clustered process are also presented. We also discuss several potential network modeling techniques such as Poisson Voronoi (PV) tessellation, Thomas cluster process (TCP), and Matérn cluster process (MCP). Subsequently, a 3D network model for THz-band ultra-cell is demonstrated. Finally, we discuss major KPIs for 3D stochastic geometry and conclude with research challenges and open issues for 3D THz architectures.

Joint Spatial-Propagation Modeling of Cellular Networks Based on the Directional Radii of Poisson Voronoi Cells

In coverage-oriented networks, base stations (BSs) are deployed in a way such that users at the cell boundaries achieve sufficient signal strength. The shape and size of cells vary from BS to BS, since the large-scale signal propagation conditions differ in different geographical regions. This work proposes and studies a joint spatial-propagation (JSP) model, which considers the correlation between cell radii and the large-scale signal propagation (captured by shadowing). We first introduce the notion of the directional radius of Voronoi cells, which has applications in cellular networks and beyond. The directional radius of a cell is defined as the distance from the nucleus to the cell boundary at an angle relative to the direction of a uniformly random location in the cell. We study the distribution of the radii in two types of cells in the Poisson Voronoi tessellations: the zero-cell, which contains the origin, and the typical cell. The results are applied to analyze the JSP model. We show that, even though the Poisson point process (PPP) is often considered as a pessimistic spatial model for BS locations, the JSP model with the PPP achieves coverage performance close to the most optimistic one-the standard triangular lattice model. Further, we show that the network performance depends critically on the variance of the large-scale path loss along the cell boundary.

Contact Dynamics modeling of viscoelastic granular materials using irregular polyhedral particles

Viscoelastic granular materials are present in several disciplines. One example is asphalt mixture employed in road construction. In the last three decades, discrete element modeling has been positioned as a valid tool for the analysis of this multiphase material at the grain-scale. All this despite the simplification of the shape of the particles used in these studies. In this work, it is proposed a simplified procedure for the generation of viscoelastic granular samples composed of irregular polyhedra. The numerical aggregates were generated by a Poisson-Voronoi tessellation based on the particle size distribution (PSD) and statistic data of aggregates, without using complex imaging technics. This procedure set the porosity of the packing, while controlling the PSD. Using this procedure implies a significant computational-time reduction by skipping several preparation stages for polyhedral samples, such as deposition by gravity and compaction. This approach can be used for the study granular materials as inclusion in a solid matrix as concrete or asphalt mixtures, particle breaking, and fatigue damage of viscoelastic materials.

Poisson approximation with applications to stochastic geometry

This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference between the cumulative distribution functions. Special attention is dedicated to estimating the difference when the cumulative distribution functions are evaluated at 0. This permits to approximate the minimum (or maximum) of a collection of random variables by a suitable random variable in the Kolmogorov distance. The main theoretical results are obtained by combining the Chen-Stein method with size-bias coupling and a generalization of size-bias coupling for integer-valued random variables developed herein. A wide variety of applications are then discussed with a focus on stochastic geometry. In particular, transforms of the minimal circumscribed radius and the maximal inradius of Poisson-Voronoi tessellations as well as the minimal inter-point distance of the points of a Poisson process are considered and bounds for their Kolmogorov distances to extreme value distributions are derived.

Statistical properties of Poisson-Voronoi tessellation cells in bounded regions

Many spatial statistics methods require neighbourhood structures such as the one determined by a Voronoi tessellation, so understanding statistical properties of Voronoi cells is crucial. While distributions of cell properties when data locations follow an unbounded homogeneous Poisson process have been studied, little attention has been given to how these properties change when a boundary is imposed. This is important when geographical data are gathered within a restricted study area, such as a national boundary or a coastline. We study the effects of imposing a boundary on the cell properties of a Poisson Voronoi tessellation. The area, perimeter and number of edges of individual cells with and without boundary conditions are investigated by simulation. Distributions of these properties differ substantially when boundaries are imposed, and these differences are affected by proximity to the boundary. We also investigate how changes in such properties when boundaries are imposed vary over two-dimensional space.

Distance from the Nucleus to a Uniformly Random Point in the 0-Cell and the Typical Cell of the Poisson–Voronoi Tessellation

Consider the distances $\tilde{R}_o$ and $R_o$ from the nucleus to a uniformly random point in the 0-cell and the typical cell, respectively, of the $d$-dimensional Poisson-Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of $\tilde{R}_o$ and $R_o$. First, using the well-known relationship between the 0-cell and the typical cell, we show that the random variable $\tilde{R}_o$ is equivalent in distribution to the contact distance of the Poisson point process. Next, we derive a multi-integral expression for the exact distribution of $R_o$. Further, we derive a closed-form approximate expression for the distribution of $R_o$, which is the contact distribution with a mean corrected by a factor equal to the ratio of the mean volumes of the 0-cell and the typical cell. An additional outcome of our analysis is a direct proof of the well-known spherical property of the PV cells having a large inball.

Almost sure central limit theorems in stochastic geometry

AbstractWe prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals arising in stochastic geometry. As a consequence, we provide almost sure central limit theorems for (i) the total edge length of thek-nearest neighbors random graph, (ii) the clique count in random geometric graphs, and (iii) the volume of the set approximation via the Poisson–Voronoi tessellation.

Beta polytopes and Poisson polyhedra: f-vectors and angles

We study random polytopes of the form [X1,…,Xn] defined as convex hulls of independent and identically distributed random points X1,…,Xn in Rd with one of the following densities:fd,β(x)=cd,β(1−‖x‖2)β,‖x‖<1,(beta distribution, β>−1) orf˜d,β(x)=c˜d,β(1+‖x‖2)−β,x∈Rd,(beta' distribution, β>d/2). Here, cd,β and c˜d,β are normalizing constants. This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of k-faces of [X1,…,Xn] for arbitrary k∈{0,1,…,d−1}. We prove that for any such k this expected number is strictly monotonically increasing with n. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large n limit in the beta' case, we compute the expected f-vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of k-faces of the zero cell for a class of isotropic Poisson hyperplane tessellations in Rd. This family includes the zero cell of a classical stationary and isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson–Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this f-vector in the high-dimensional regime, as d→∞. Finally, we relate the d-dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory.