Laguerre Tessellation Research Articles

Angles in Laguerre tessellation models for solid foams

Geometric stochastic models proved to be very helpful to improve the understanding of how physical properties of materials depend on their microstructures. The microstructures of cellular materials are well modeled by random Laguerre tessellations generated by systems of non-overlapping spheres. We enrich the geometric characterization of Laguerre tessellations by computing angles between facets and angles between edges. Their distributions, which depend on the model parameters, are investigated in detail. To include angles in the model fit, we developed a method for estimating angles in 3D images of foams, e. g. obtained by micro-computed tomography. Furthermore, we analyzed the influence of different characteristics on the model’s fit to samples of closed and open foams.

VLDP web server: a powerful geometric tool for analysing protein structures in their environment

Protein structures are an ensemble of atoms determined experimentally mostly by X-ray crystallography or Nuclear Magnetic Resonance. Studying 3D protein structures is a key point for better understanding protein function at a molecular level. We propose a set of accurate tools, for analysing protein structures, based on the reliable method of Voronoi–Laguerre tessellations. The Voronoi Laguerre Delaunay Protein web server (VLDPws) computes the Laguerre tessellation on a whole given system first embedded in solvent. Through this fine description, VLDPws gives the following data: (i) Amino acid volumes evaluated with high precision, as confirmed by good correlations with experimental data. (ii) A novel definition of inter-residue contacts within the given protein. (iii) A measure of the residue exposure to solvent that significantly improves the standard notion of accessibility in some cases. At present, no equivalent web server is available. VLDPws provides output in two complementary forms: direct visualization of the Laguerre tessellation, mostly its polygonal molecular surfaces; files of volumes; and areas, contacts and similar data for each residue and each atom. These files are available for download for further analysis. VLDPws can be accessed at http://www.dsimb.inserm.fr/dsimb_tools/vldp.

The Influence of Varying Cell Sizes on the Tensile Behaviour of Open-Cell Foams

Abstract The aim of this paper is to study the mechanical tensile properties of open-cell foam structures on the cell size variation by numerical simulations. For this purpose, random Laguerre tessellations were used, which allow the creation of foam structures with strongly varying cell sizes. First, a model was fitted to real open-cell foams based on x-ray computed tomography (XCT) scans by parameter optimization. Two more virtual foam models with different cell size variations were produced according to the first fitted one. Tensile properties of the model realizations were computed by finite-element analysis using beam elements. The numerical results were presented and discussed.

Urea in aqueous solution studied by quantum mechanical charge field-molecular dynamics (QMCF-MD)

This work presents a quantum mechanical charge field-molecular dynamics (QMCF-MD) simulation of urea in dilute aqueous solution. Detailed data for structure and dynamics are provided and compared to previous works of other groups. Radial and angular distributions are employed, as well as higher degree spatial investigations, two-dimensional particle mapping, volume maps and the previously proposed SLICE formalism. Information on dynamical properties are presented in the form of hydrogen bond correlation functions and mean lifetime analysis based on weighted Voronoi decomposition. Dihedral and tilt/theta angle distributions substantiate the previous findings of other groups, that urea is far from being planar within aqueous solution. In addition to the analysis of the complete hydration shell, several specific regions of hydration have been identified, for which individual analysis has been performed in terms of hydrogen bond lifetime correlation functions and re-orientational times. A decomposition study based on Laguerre tessellation further investigates the structure and dynamics of the individual hydration layers. It is found that urea does not show properties found in the case of typical structure breaking agents, such as Rb(+) or Cs(+), which is in accordance with spectroscopic data of Rezus and Bakker.

Stochastic Multiscale Modeling of Metal Foams

A procedure for the computation of eigenfrequencies for structures made of metal foam is proposed. The heterogeneity of the foam geometry has an influence on these macroscopic properties and has to be taken into account. This is done by fitting a model of the microstructure based on Laguerre tessellations by means of statistical information obtained from CT image analysis. As the length scale of the representative volume element is nearly of the same order as the length scale of the structures under consideration, classical homogenization techniques for the computation of effective properties can not be applied. A stochastic homogenization method that necessitates the computation of empirical marginal distributions and correlation functions for apparent properties defined on the mesoscale is proposed. This information allows to define random fields for elastic properties and the density on the structure level. Statistical properties of the eigenfrequencies can then be inferred.

Elastic foam compression in a finite element (FE) context

A procedure to generate statistical virtual representative volume elements of foam in a finite element context is described. This technique, based on Laguerre tessellations and advancing front method, level-set description of interfaces and anisotropic meshing adaptation, is detailed. The capability of the procedure to respect statistical data could be insured by the advancing front method. To simulate biaxial foam compression, a uniform velocity is imposed on the domain’s upper and lower boundaries. By considering the presence of air inside the foam’s cells which are bounded by the elastic solid skeleton, a fluid–structure interaction problem occurs between a compressible elastic solid and a compressible fluid. A monolithic formulation is used for solving this problem. Such strategy gives rise to an extra stress tensor in the Navier–Stokes equations coming from the presence of the structure in the fluid.

Precise generation of complex statistical Representative Volume Elements (RVEs) in a finite element context

A precise procedure to generate complex 2D and 3D statistical virtual microstructures in a finite element (FE) context is described. This technique, based on Laguerre tessellations and advancing front method, level-set description of interfaces and anisotropic meshing adaptation, is detailed for equiaxial polycrystal and powder RVEs defined by a size distribution law. The capability of the procedure to respect statistical data could be insured by the advancing front method coupled with an optimization procedure depending on the nature of the considered RVE. Moreover, by applying a graph coloration technique, the number of level-set functions used to represent the cells and also in anisotropic mesh adaptation is greatly reduced.

Laguerre tessellations for elastic stiffness simulations of closed foams with strongly varying cell sizes

The dependency of the elastic stiffness, i.e., Young’s modulus, of isotropic closed-cell foams on the cell size variation is studied by microstructural simulation. For this purpose, we use random Laguerre tessellations which, unlike classical Voronoi models, allow to generate model foams with strongly varying cell sizes. The elastic stiffness of the model realizations is computed by micro finite element analysis using shell elements. The main result is a moderate decrease of the effective elastic stiffness for increasing cell size variations if the solid volume fraction is assumed to be constant.

On the Use of Laguerre Tessellations for Representations of 3D Grain Structures

AbstractAccurate descriptions of 3D grain structures in polycrystalline materials are of key interest as the grain structure is closely correlated to the macroscopic properties of the material. In the present study, we investigate the accuracy of using Laguerre tessellations to represent 3D grain structures from only the spatial center of mass location and the volume of the grains. The ability of Laguerre tessellations to describe accurate grain shapes and topologies of real 3D grain structures are revealed by direct comparison to 3D reconstructions of an un‐deformed meta‐stable β ‐titanium alloy obtained by phase‐contrast micro‐tomography. This study reveals that (volume weighted) Laguerre tessellations are superior to classical Voronoi tessellations when it comes to providing accurate representations of real 3D grain structures. Furthermore, although the Laguerre tessellations were only able to correctly describe the local arrangements of grains (i.e., the grain neighbors and number of grain facets) for 31.8% of the investigated grains, the Laguerre tessellations were able to accurately describe statistical grain characteristics such as grain size distributions and grain neighbor distributions.

Modeling of Ceramic Foams for Filtration Simulation

AbstractWe present a stochastic model for the microstructure of a ceramic foam filter which is produced by coating an open polymer foam with liquid ceramic. First, we fit a random Laguerre tessellation to the polymer foam skeleton. Then, the coating is modeled using locally adaptable morphology which allows for the reproduction of the ceramic foam's locally varying strut thickness. Furthermore, we introduce methods for the estimation of the intensity and orientation distribution of the closed facets which are formed during the coating process. By computation of the permeability of the foam in different directions we show that the anisostropic behavior of the material is reproduced correctly in the model.

A Novel Evaluation of Residue and Protein Volumes by Means of Laguerre Tessellation

Amino acids control the protein folding process and maintain its functional fold. This study underlines the interest of the Laguerre tessellation to determine relevant amino acid volumes in proteins. Previous studies used a limited number of proteins and only buried residues. The present computations improve the method and results on three main points: (i) a large, high-quality updated and refined data bank of proteins is used; (ii) all residues are taken into account, including those at the protein surface, thanks to (iii) the addition of a realistic solvent. The new values of the average and standard deviation of amino acid volumes show significant corrections with respect to previous studies. Another issue of the method is the polyhedral protein/water interface area (PIA) which quantifies the exposure of atoms or residues to the solvent. We propose this PIA as a new, parameter-free, alternative for measuring accessibility. The comparison with NACCESS is satisfactory; however, the methods disagree in pointing out buried residues: where NACCESS evaluates to zero, the exposure given by PIA ranges from 0 to 20%. Variations of average residue volumes have been analyzed under several conditions, e.g., how they depend on protein size and on secondary structure environments. As it is based on strong mathematical grounds and on numerous high-quality protein structures, our work gives a reliable methodology and up-to-date values of amino acid volumes and surface accessibility.

Microstructure models for cellular materials

Laguerre tessellations generated by random sphere packings are promising models for the microstructure of cellular or polycrystalline materials. In this paper, the case of hard sphere packings with lognormal or gamma distributed volumes is investigated. The dependence of the geometric characteristics of the Laguerre cells on the volume fraction of the sphere packing and the coefficient of variation of the volume distribution is studied in detail. The moments of certain cell characteristics are described by polynomials, which allows to fit tessellation models to real materials without further simulations. The procedure is demonstrated by the examples of open polymer and aluminium foams.

Random Laguerre tessellations

A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

Random Laguerre tessellations

A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

Fitting three-dimensional Laguerre tessellations to foam structures

Foam models, especially random tessellations, are powerful tools to study the relations between the geometric structure of foams and their physical properties. In this paper, we propose the use of random Laguerre tessellations, weighted versions of the well-known Voronoi tessellations, as models for the microstructure of foams. Based on geometric characteristics estimated from a tomographic image of a closed-cell polymer foam, we fit a Laguerre tessellation model to the material. It is shown that this model allows for a better fit of the geometric structure of the foam than some classical Voronoi tessellation models.

Modelling a ceramic foam using locally adaptable morphology

In this paper, a ceramic open foam is modelled using a two-step procedure. First, a random Laguerre tessellation is fitted to the edge system of the polyurethane foam forming the core of the ceramic foam. In a second phase, a model of the ceramic foam is obtained using dilations with balls of locally varying size. The model fitting is based on geometric characteristics of both polyurethane and ceramic foam estimated from reconstructed tomographic images of these structures.

Effective conductive properties of open-cell foams

The work is devoted to the problem of simulation of the effective electro and thermoconductive properties of open-cell foam materials with slim highly conductive ligaments. The solution of this problem depends on two small parameters: the ratio of the typical length and the cross-section size of the ligaments, and the ratio of the conductivities of the host medium and the ligaments. The principal terms of the asymptotic solutions (with respect to these parameters) for the fields inside the ligaments are obtained and used in the framework of a finite element method for the numerical simulation of the fields inside the representative volume element (RVE) of the foam material. The Laguerre tessellation procedure is applied for simulation of foam microstructures with various distributions of cell-sizes inside the RVE. Effective conductivity constants of the foams are obtained by averaging the detailed fields over the RVE. The number of cells inside the RVE for reliable calculations of the effective conductivities is indicated. Dependences of the effective conductive properties of the foams on the shapes of the ligaments and the distribution law of the cell-sizes are obtained and analyzed.

On the quasi-stationary grain size distribution from two Gamma size distributions in three-dimensional grain growth

Two different grain structures of which the grain size distribution could be well described by the Gamma distribution were generated by the combination of the Laguerre tessellation and Monte Carlo technique. Monte Carlo simulation shows that the three-dimensional grain growth process can evolve to the quasi-stationary state. The quasi-stationary grain size distribution could be well fit by the Gamma distribution with different parameters. The results in literature from various simulation methods support this point.

Mechanical Properties of Open Cell Foams: Simulations by Laguerre Tesselation Procedure

The Laguerre tessellation procedure is used for simulation of microstructures of open-cell foams. In contrast with the conventional Voronoii tessellation, the Laguerre one permits to simulate the foam microstructures with a given law of distribution of cell diameters. An original finite element method is developed for calculating the elastic properties: the ligaments are modelled as Timoshenko beams and each ligament is treated as one finite element. The size of the representative volume element for reliable calculations of the effective elastic properties is evaluated by computational experiments. Dependence of the properties on the cell size distributions and ligament shapes are analyzed.

A model of binary assemblies of spheres

We propose a theoretical model of random binary assemblies of spheres at any packing fraction. We use the notion of geometrical neighborhood between grains that is defined through two generalizations of the Voronoi tessellation: the radical (or Laguerre) tessellation and the navigation map. The model is tested on different numerical packings. We find a weak local segregation for high packing fraction. We also find that the higher the size ratio of the particles, the more important the segregation.