Laguerre Tessellation Research Articles

An analytical representation of the 2d generalized balanced power diagram

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and the Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.

Persistent homology based goodness-of-fit tests for spatial tessellations

Motivated by the rapidly increasing relevance of virtual material design in the domain of materials science, it has become essential to assess whether topological properties of stochastic models for a spatial tessellation are in accordance with a given dataset. Recently, tools from topological data analysis such as the persistence diagram have allowed to reach profound insights in a variety of application contexts. In this work, we establish the asymptotic normality of a variety of test statistics derived from a tessellation-adapted refinement of the persistence diagram. Since in applications, it is common to work with tessellation data subject to interactions, we establish our main results for Voronoi and Laguerre tessellations whose generators form a Gibbs point process. We elucidate how these conceptual results can be used to derive goodness of fit tests, and then investigate their power in a simulation study. Finally, we apply our testing methodology to a tessellation describing real foam data.

A computational framework based on 3D microstructure modelling to predict the mechanical behaviour of polycrystalline materials

In this work, a computational framework is proposed to predict the microstructure sensitive mechanical behaviour of polycrystalline materials i.e., fatigue crack initiation life and anisotropic deformation behaviour. A methodology is developed to generate 3D microstructural RVEs using Laguerre tessellation technique in Abaqus software at a particular location of the component. In any commercial software (Abaqus), the modelling of morphology of 3D polycrystalline microstructures is quite challenging due to the complex geometries of grains, and large number of interfaces or grain boundaries, however it can be achieved efficiently and automatically with the developed methodology. The continuum damage mechanics based elasto-plastic constitutive model is used to predict the microstructure sensitive fatigue crack initiation life of titanium alloy notched specimens. To capture an accurate fatigue response, damage coupled elasto-plastic constitutive equations are implemented via user defined material subroutine UMAT. The predicted fatigue crack initiation lives show good agreement with the experimental data. To further showcase the application of the developed framework, an anisotropic analysis is performed by generating 3D polycrystalline microstructure of β-titanium alloy. The developed methodology enhances the capabilities of commercial software in the field of microstructure modelling of complex problems.

Fitting the grain orientation distribution of a polycrystalline material conditioned on a Laguerre tessellation

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel–titanium shape memory alloy.

Dynamic grain models via fast heuristics for diagram representations

ABSTRACT The present paper introduces a mathematical model for analysing dynamic grain growth. In particular, we show how the characteristic measurements grain volumes, centroids, and central second-order moments at discrete moments in time can be quickly turned into a continuous description of the grain growth process in terms of geometric diagrams (which largely generalize the well-known Voronoi and Laguerre tessellations). We give a theoretical analysis of common optimization-free heuristics in terms of discriminant analysis and evaluate the computational behaviour of our algorithm on real-world data.

Numerical simulation of the moisture transport in unsaturated concrete

The capillary absorption of water in unsaturated concrete plays a critical role in the ingress of aggressive agents when the concrete structure is built in corrosive environment. This paper proposes a three-dimensional (3D) pore network model to simulate the moisture transport mechanism in concrete under the action of capillary forces. The Fuller particle distribution is used to generate the aggregate particles in the geometrical domain of concrete and the principle of Laguerre tessellation is implemented using the Voro++ library to develop the network model, where the interfaces between the polyhedron cells are assumed as the equivalent capillary pores in concrete through which the capillary absorption takes place. Then the diffusion equation is solved on the interfaces by the finite element method. To consider the effect of gravity and change in pore structure, the time-dependent hydraulic diffusivity is assumed and solved by using appropriate subroutine. The cumulative water absorption determined by the 3D pore network model shows good agreement with the experimental results. This model is useful to predict the moisture content distribution within the concrete specimen after any elapsed time.

Geometric modelling of polycrystalline materials: Laguerre tessellations and periodic semi-discrete optimal transport

In this paper we describe a fast algorithm for generating periodic RVEs of polycrystalline materials. In particular, we use the damped Newton method from semi-discrete optimal transport theory to generate 3D periodic Laguerre tessellations (or power diagrams) with cells of given volumes. Complex, polydisperse RVEs with up to 100,000 grains of prescribed volumes can be created in a few minutes on a standard laptop. The damped Newton method relies on the Hessian of the objective function, which we derive by extending recent results in semi-discrete optimal transport theory to the periodic setting.

Prediction of bulk mechanical properties of PVC foam based on microscopic model:Part I-Microstructure characterization and generation algorithm

The mechanical properties of porous materials greatly depend on their microstructures, so the generation of realistic microstructures is the premise for a sound numerical simulation and analytical prediction. This paper presented the microstructure generation of three kinds of transversely isotropic closed-cell polyvinyl chloride (PVC) foams with different densities based on the geometric characteristics obtained from the X-ray computed tomography (CT) technology. Advancing front method was used to densely pack a set of spheres whose volume distribution is proportion to the cell volume distribution and the microstructure was formed by Laguerre tessellation. Then the geometric characteristics of the numerical models were compared with those from the CT. The good consistence proves that the proposed method is accurate and efficient to produce the representative microstructure of the PVC foams. Finally, a method was proposed to optimize the microstructure of small size numerical specimens, by which the small specimens could give similar geometric characteristics to those of the large specimens.

Influence of tessellation morphology on ultrasonic scattering.

Material properties, such as hardness, yield strength, and ductility, depend on the microstructure of the material. If the microstructural organization can be quantified nondestructively, for example, with ultrasonic scattering techniques, then it may be possible to predict the mechanical performance of a component. Three-dimensional digital microstructures have been increasingly used to investigate the scattering of mechanical waves within a numerical framework. These synthetic microstructures can be generated using different tessellation algorithms that result in different grain shapes. In this study, the variation of ultrasonic scattering is calculated for microstructures of different morphologies for a nickel polycrystal. The ultrasonic properties are calculated for the Voronoi, Laguerre tessellations, and voxel-based synthetic microstructures created by DREAM.3D. The results show that the differences in the two-point statistics and ultrasonic attenuation for different morphologies become more significant at wider size distributions and higher frequencies.

A sophisticated periodic micro-model for closed-cell foam based on centroidal constraint and capacity constraint

A deeper understanding of the uniaxial compression damage behavior and its microscopic mechanisms in closed-cell Polyurethane foam (PUF) at higher strains would be the first step to comprehend its energy absorption behavior in application. A physical-based micro-structure of the PUF model, therefore, is a crucial precondition in the prediction of the material property. A highly efficient physical-based model with adapted periodic boundary conditions (PBCs) is developed to simulate the mechanical behavior of the rigid closed-cell foam using 100 cells, much less than that mentioned in the literature (800 cells). The microstructure characteristics of PUF have been obtained by scanning electron microscopy (SEM), including the cell size distribution, the wall thickness, the cell aspect ratio, etc. Using these parameters, a periodic representative volume elements (RVEs) model based on the Laguerre tessellation with centroidal constraint and capacity constraint (LTCCC) has been proposed. The centroidal constraint and capacity constraint on the LTCCC has a sound physical meaning that the volume distribution of the cells in the RVEs model obtained by the iterations of the weights of the seed points can be precisely matched to that measured using the SEM. Meanwhile, the features of the RVEs model, including the number of the faces per cell and the number of the edges per face, also match those of the real foam. The anisotropic compressive behavior of the PUF is predicted and well validated using the experimental results. The quasi-static crushing progress of the cells as well as the struts is detailed discussed using the proposed model to reproduce the layer-wise collapse phenomena and reveal the reasons that cause the different crushing behavior in axial direction and perpendicular to the axis direction.

Elastic properties of Laguerre approximations of random foams

Stochastic models are valuable tools to study the mechanical behaviour of foams whose characteristic features include random cellular morphology and cells of different size (polydispersity). Random Laguerre tessellations, additively weighted generalisations of Voronoi tessellations, possess these features which makes them natural models for foams. This work studies how well a random foam can be represented by a Laguerre tessellation. For this means the Surface Evolver is used to simulate random, polydisperse soap froths which are subsequently reconstructed by Laguerre tessellations. By design, the resulting Laguerre approximations resemble the cellular morphology of the underlying froth. The Young’s modulus of low-density solid foams with open cells is then calculated for random, polydisperse foams based on Laguerre approximations and simulated soap froths. For the special case of monodisperse foams, we compare Laguerre approximated foams with the well known Kelvin and Weaire–Phelan foams.

The microstructure characterization and elastic properties of closed-cell foams

The present study investigated the microstructure and compressive response of ALPORAS closed-cell foam. The micro-CT technique was adopted to study the interior structure of the foam. Image processing algorithms were then incorporated to measure the geometrical characteristics of the foam cellular microstructure, including the size, surface area, geometrical shape, aspect ratio, and sphericity of cells. The cells were irregular polyhedra with approximately 14–15 faces and 5 edges per face; they had longer dimensions in the gravitational direction than in the other two mutually orthogonal directions. Three statistical measurements were employed in the development of random 3D numerical models based on the Laguerre tessellation of random close packed spheres along with three probability density functions for sphere size distributions. The effects of parameters such as probability density functions, sphere volume fraction, and coefficient of variation in sphere volume on the geometrical characteristics of the developed numerical models were examined. The difference in geometric characteristics of the numerical models generated using different probability functions was nonsignificant. Numerical simulations were subsequently conducted using the developed numerical models. The comparison between numerical and experimental results revealed that numerical models overestimated the experimental value by a factor of approximately 3. This discrepancy can be attributed to the difference in detailed microstructural features between the generated numerical models and real foam.

Polydisperse solid foams: Multiscale modeling and simulations of elasto-acoustic properties including thin membrane effects

This work is concerned with the prediction of the elasto-acoustic properties of polydisperse solid foam structures. A highly polydisperse foam sample is first characterized using microtomography and scanning electron microscopy. Relevant geometrical properties are then determined by image processing and utilized to model the partially closed cell system with random Laguerre tessellations. The macroscopic visco-thermal transport properties of the solid foams are next calculated by numerical techniques, using either finite element computations or pore-network simulations. The permeability and sound absorption coefficient at normal incidence are also measured and a good agreement is obtained with the calculations when the elasto-acoustic coupling is modeled from the Biot’s equations (including characterized visco-elastic parameters). Our results demonstrate that stochastic geometry provides a robust framework to understand the structure–property relationships of polydisperse foam.

Random Tessellations and Gibbsian Solutions of Hamilton–Jacobi Equations

We pursue two goals in this article. As our first goal, we construct a family $\mathcal{M}_G$ of Gibbs like measures on the set of piecewise linear convex functions $g:\mathbb{R}^2\to\mathbb{R}$. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and $\textit{Laguerre tessellations}$. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector $\rho\in\mathbb{R}^2$. Each measure $\nu^f\in\mathcal{M}_G$ in our family is uniquely characterized by a kernel $f(x,\rho^-,\rho^+)$, which represents the rate at which a line separating two cells associated with marks $\rho^-$ and $\rho^+$ passes through $x$. To construct our measures, we give a precise recipe for the law of the restriction of our tessellation to a box. This recipe involves a boundary condition, and a dynamical description of our random tessellation inside the box. As we enlarge the box, the consistency of these random tessellations requires that the kernel satisfies a suitable kinetic like PDE. As our second goal, we study the invariance of the set $\mathcal{M}_G$ with respect to the dynamics of such Hamilton-Jacobi PDEs. In particular we $\textit{conjecture}$ the invariance of a suitable subfamily $\widehat{\mathcal{M}_G}$ of $\mathcal{M}_G$. More precisely, we expect that if the initial slope $u_x(\cdot,0)$ is selected according to a measure $\nu^{f}\in \widehat{\mathcal{M}_G}$, then at a later time the law of $u_x(\cdot, t)$ is given by a measure $\nu^{\Theta_t(f)}\in\widehat{\mathcal{M}_G}$, for a suitable kernel $\Theta_t(f)$. As we vary $t$, the kernel $\Theta_t(f)$ must satisfy a suitable kinetic equation. We remark that the function $u$ is also piecewise linear convex function in $(x,t)$, and its law is an example of a Gibbs-like measure on the set of Laguerre tessellations of certain convex subsets of $\mathbb{R}^3$.

Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points, we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points, we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step, parameter estimation based on maximum pseudolikelihood methods is tractable. For model selection, we consider maximized log pseudolikelihood functions for models of the radii conditioned on the points. Model checking is performed using global envelopes and corresponding tests in both steps and moreover by comparing observed and simulated tessellation characteristics in the second step. We apply our methodology for a 3D Laguerre tessellation data set representing the microstructure of a polycrystalline metallic material, where simulations under a fitted model may substitute expensive laboratory experiments.

Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures

Ultrasonic testing of polycrystalline media relies heavily on simulation of the expected signals in order to detect and correctly interpret deviations due to defects. Many effects disturb ultrasonic waves propagating in polycrystalline media. One of them is scattering due to the granular microstructure of the polycrystal. The thus arising so-called microstructural noise changes with grain size distribution and testing frequency. Here, a method for simulating this noise is introduced. We geometrically model the granular microstructure to determine its influence on the backscattered ultrasonic signal. To this end, we utilize Laguerre tessellations generated by random sphere packings dividing space into convex polytopes—the cells. The cells represent grains in a real polycrystal. Cells are characterized by their volume and act as single scatterers. We compute scattering coefficients cellwise by the Born approximation. We then combine the Generalized Point Source Superposition technique with the backscattered contributions resulting from the cell structure to compute the backscattered ultrasonic signal. Applying this new methodology, we compute the backscattered signals in a pulse-echo experiment for a coarse grain cubic crystallized Inconel-617 and a fine grain hexagonal crystallized titanium. Fitting random Laguerre tessellations to the observed grain structure allows for simulating within multiple realizations of the proposed model and thus to study the variation of the backscattered signal due to microstructural variation.

Semi-discrete optimal transport methods for the semi-geostrophic equations

We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

Permeability of polydisperse solid foams.

The effect of polydispersity on foam permeability is investigated by numerical simulations. Foam structures are first generated by Laguerre tessellations via the Neper software and relaxed to minimize the surface energy via the Surface Evolver software. The fluid flow and permeability are then calculated by means of pore-network simulations, by considering either fully open-cell foams or foams with randomly selected closed windows. Different configurations of window aperture are used, including identical window aperture size, identical window aperture ratio, or random window aperture ratio. The main results are obtained for the case of foams having identical and uniform window aperture ratios. For such foams and at constant mean pore size, foam permeability is found to strongly increase with the polydispersity degree. The numerical results are employed to discuss the validity of the mean pressure field assumption used to calculate the foam permeability, the effect of small pores, and the definition of an equivalent Kelvin foam size. We show that as long as the fluctuations of the window aperture ratio remain low, foam permeability can be estimated by using the mean pressure field hypothesis. The weak effect of small pores on permeability is related to their small contribution to the overall fluid volume fraction. Finally, various estimations of the equivalent Kelvin foam size based on pore-size distribution are proposed.

A new method for design and calculating the mechanical properties and energy absorption behavior of cellular structures using foam microstructure modeling based on Laguerre tessellation

In this paper, the elastic modulus and energy absorption behavior of cellular structures such as aluminum foams have been investigated. The main purpose of this study is to develop microstructure modeling of cellular structures and to study their elastic–plastic behavior through finite element method. In this research, first a unit cell is prepared to reduce numerical calculations, which has the characteristics of the main foam. The behavior of the unit cell under impact is then simulated and analyzed by the finite element method. Using simulated experiments, a model using the response surface method to obtain the densification strain, elastic modulus, mean plastic stress, yield stress as well as foam compaction behavior will be presented. Finally, the results of finite element and response surface methods for cellular structures with experimental results is validated. The results of this study showed that the possibility of modeling and analysis of energy absorption (impact) of foams is possible using the response surface model presented in this study and finally a suitable model can be achieved to analyze the microstructure of different foams. As a consequence, the hybrid cellular structures are simulated in view of applications of foams as filler in the thin-walled structures about energy absorption subject.

Efficient Fitting of 3D Tessellations to Curved Polycrystalline Grain Boundaries

The curvature of grain boundaries in polycrystalline materials is an important characteristic, since it plays a key role in phenomena like grain growth. However, most traditional tessellation models that are used for modeling the microstructure morphology of these materials, e.g., Voronoi or Laguerre tessellations, have flat faces and thus fail to incorporate the curvature of the latter. For this reason, we consider generalizations of Laguerre tessellations—variations of so-called generalized balanced power diagrams (GBPDs)—that exhibit non-convex cells. With as many as ten parameters for each cell, it is computationally demanding to fit GBPDs to three-dimensional image data containing hundreds of grains. We therefore propose a modification of the traditional definition of GBDPs that allows gradient-based optimization methods to be employed. The resulting reduction in runtime makes it feasible to find approximations to real experimental datasets. We demonstrate this on a three-dimensional x-ray diffraction (3DXRD) mapping of an AlCu alloy, but we also evaluate the modeling errors for simulated data. Furthermore, we investigate the effect of noisy image data and whether the smoothing of image data prior to the fitting step is advantageous.