Application PaperRadical Voronoï tessellation from random pack of polydisperse spheres: Prediction of the cells’ size distribution☆
Introduction
Polycrystalline materials, such as metals or minerals, are usually composed of numerous individual grains, connected to each other by their grain boundaries. As a result, their thermomechanical properties are strongly influenced by the morphology and the crystalline orientation of each grain. The numerical generation of random geometries, representative of real microstructures, is still challenging because it requires to statistically reproduce the physical properties of the modelled grains in order to model the behaviour of polycrystalline aggregates [[1], [2], [3]].
The Poisson–Voronoï (PV) tessellation [4] has been extensively used over the last decades to numerically generate microstructure similar to real ones [[1], [2], [5], [6], [7], [8], [9]]. Consider a set of separate points (called seeds) in a given volume . In PV geometry, the th cell is defined such that: where denotes the euclidean distance. The PV tessellation results in a series of convex and non-overlapping cells bounded by planar surfaces. Nevertheless, considering the definition given in (1), size and shape of each Voronoï cell are only driven by the distance from its centre to those of its neighbours. As a result, cell sizes cannot easily be constrained. For instance, Groeber et al. [7] have developed the Constrained Voronoi Tessellation Method (CVTM), consisting in recursively adding new seeds in order to result in realistic cell morphologies.
The Radical Voronoï (RV) tessellation, also known as Laguerre–Voronoï [10], helps to impose the size of each cell through the introduction of a local radius, denoted . Consider a set of seeds and a set of radii . For the RV tessellation, (1) becomes: where denotes the euclidean distance in Laguerre geometry, so that: Consider two cells and ; according to (2), if then the boundary between the two cells is located at the distance (resp. ) from (resp. ). Eq. (4) is satisfied if and are the centres of two spheres, in contact with each other, with respective radii and . As a result, the dimensions of the cells resulting from the RV tessellation computed from Random Close Packing (RCP) of spheres are strongly inherited from their radii.
It is worth mentioning that the numerical generation of RCP of spheres is not straightforward. The two main methods used in the literature are the collective rearrangement method [[11], [12], [13], [14], [15]] and the sequential generation. The latter usually consists in numerically pouring a closed volume with the spheres, by simulating a gravitational effect [[16], [17], [18]]. Yang et al. [11] have investigated the evolution of the packing factor of RCPs computed from sphere radii following a lognormal distribution. They have shown that the packing factor was an increasing function of the so-called shape parameter of the distribution (denoted below, see Section 2.1) starting from 0.637. Falco et al. [19] have adapted the “3D-clew” technique [20] to generate the RCPs of sphere. This method consists in adding the spheres one by one into the pack, ensuring the tangency with other spheres. Rodrigues et al. [9] use a virtual container with moving walls instead of gravity. Fan et al. [14] have generated RCPs of spheres by collective rearrangement, in order to investigate the geometry of the RV tessellation thanks to the Qhull software package [21]. They have shown that if the volumes of the spheres follow a lognormal distribution, then those of the resulting RV cells follow a lognormal distribution too. In addition, they have focused on the geometrical characterization of each RV cell. In particular, they have shown that the mean number of faces per RV cell was a decreasing function of the Coefficient of Variation (CV) of sphere volumes, ranging from 13.04 to 14.11. Finally, Fan et al. [14] have investigated the relationship between the CV of the sphere volumes with that of the RV cell volumes.
It is well known that sequential algorithms usually result in anisotropic RCPs [[16], [22], [23]]. For instance, Tory et al. [16] have reported that the length of projection along the vertical direction (i.e. direction of applied gravitational force) of monodisperse spheres was 11 % greater than those computed along the horizontal directions.
The geometry of a real polycrystalline structure is usually described by the morphology and size of its grains. In order to numerically reproduce those structures using a RV tessellation, it is necessary to predict the adequate distribution for the RCP spheres. Grain sizes can be measured by 3D imaging techniques (such as X-ray tomography or neutron diffraction) but they are more frequently evaluated from 2D cross sections, for instance by microscopy or Electron Backscatter Diffraction (EBSD) mapping. Those 2D observations induce a bias in grain sizes, for they only provide information about the intersection of the 3D grains and the cutting plane. For instance, it can be shown that the mean apparent radius of a sphere with unit radius cut at random latitudes is [24]. For polydisperse particles, deriving their 3D sizes from 2D observations is called stereology problem. It is usually solved using an iterative routine applied on finite histograms [[25], [26], [27]]. Lopez-Sanchez and Llana-Fúnez [27] have proposed a routine, namely the two-step method, for the resolution of stereology problems. It consists in:
deriving the 3D histogram from 2D sections, thanks to the so-called Saltykov method [25];
fitting a lognormal distribution on the 3D histograms.
The Saltykov method (also known as Scheil–Schwartz–Saltykov method) is an iterative procedure, consisting in evaluating the probability of cutting each grain (considered as spherical) at a certain latitude, resulting in a reduced apparent size. This method is based on the finite histogram (finite number of classes) of apparent sizes. The fundamental assumption is that the upper class (i.e. larger apparent sizes) is given by the largest grains cut near their equatorial planes. When cutting a particle of radius at a random latitude, the probability of finding an apparent radius comprised in between and is given by: Eq. (5) is first used for evaluating the number of grains with radius , and being the limits of the upper class. Then, (5) can be used for each other class to subtract the contribution of grains with size in the overall histogram. Finally, the previous steps are reiterated, starting from the (remaining) second upper class, and so on.
The aim of this work is to establish continuous inverse functions for predicting the radius distribution parameters (denoted and below) based on 3D or 2D observations. Fig. 1 schematically illustrates the method used in this paper for this purpose: RCPs have been computed with different distributions for the sphere radii, then RV tessellations have been carried out on each RCP. Finally, the geometries of the RV cells have been statistically evaluated. The procedure used in this work for generating the geometries is detailed in Section 2. Section 3 investigates the geometry of each RCP. A continuous relationship between the radius distribution of the spheres and that of the cell volumes is proposed in this section. In Section 4, the bias introduced when one attempts to characterize the grain size from 2D sections is studied and methods for evaluating the 3D distribution based on 2D sections are proposed. In order to assess those methods, a numerical microstructure, representative of a real one, has been generated; those microstructures are compared in Section 5.
Section snippets
Geometry generation
This section describes the procedure used in this work for generating RCPs of spheres with radii following lognormal distribution, then computing the RV tessellations.
Analysis of the resulting 3D geometries
In this section, the analysis of the geometries resulting from RV tessellation of each RCP of spheres has led to the following statements:
the RCPs of spheres generated in this work lead to the same packing factor of that obtained by Fan et al. [14];
the mean number of faces per RV cell is in good agreement with real grains, found in various granular material;
the geometry resulting from RV tessellation can be considered as isotropic;
the equivalent radii of RV cells follows a lognormal
Stereology problem
In this section, 2D slicing has been performed on each RCP in order to study the bias introduced when one tries to evaluate the grain size based on 2D sections. The following statements are made:
the apparent radii distributions can be characterized using both the mean value () and the Pearson’s second skewness coefficient ();
relationships between both and versus can be established;
the accuracy of the two-step method [27] has been investigated. It has been shown that this method
Application on a real polycrystalline microstructure
In this section, the methods proposed in Section 4 to numerically generate a 3D polycrystal, based on a 2D section of a real material, are tested. The following statements are made:
both the - and the three-step methods give valuable results;
the three-step method appears to be more accurate in terms of equivalent radii distribution.
Conclusion
RCPs of spheres have been generated in LAMMPS, with radii following a lognormal distribution for a shape parameter () ranging from 0 to 0.9 and unit expectation (). From each pack, the RV tessellation has been performed. Then, efforts have been made to find an analytical continuous correlation between the resulting geometry and the input parameters (shape parameter and expectation).
If one tries to numerically generate a polycrystalline aggregate knowing the grain volume distribution of a
References (38)
- et al.
Intergranular and intragranular behavior of polycrystalline aggregates. Part 1: F.E. model
Int J Plast
(2001) - et al.
Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing
Comput Methods Appl Mech Engrg
(2011) The grain boundary impedance of random microstructures: numerical simulations and implications for the analysis of experimental data
Solid State Ion
(2002)- et al.
Simulation of microplasticity-induced deformation in uniaxially strained ceramics by 3-D Voronoi polycrystal modeling
Int J Plast
(2005) - et al.
A framework for automated analysis and simulation of 3D polycrystalline microstructures. Part 2: Synthetic structure generation
Acta Mater
(2008) - et al.
Integrated approach for geometric modeling and interactive visual analysis of grain structures
Comput Aided Des
(2018) - et al.
Dynamic simulation of the centripetal packing of mono-sized spheres
Physica A
(1999) - et al.
Simulation of polycrystalline structure with Voronoi diagram in Laguerre geometry based on random closed packing of spheres
Comput Mater Sci
(2004) - et al.
Effect of rectangular container’s sides on porosity for equal-sized sphere packing
Powder Technol
(2012) - et al.
Generation of 3D polycrystalline microstructures with a conditioned Laguerre–Voronoi tessellation technique
Comput Mater Sci
(2017)
3D particle size distributions from 2D observations: stereology for natural applications
J Volcanol Geotherm Res
An extension of the Saltykov method to quantify 3D grain size distributions in mylonites
J Struct Geol
Fast parallel algorithms for short-range molecular dynamics
J Comput Phys
A framework for automated analysis and simulation of 3D polycrystalline microstructures.: Part 1: Statistical characterization
Acta Mater
DREAM.3D: A digital representation environment for the analysis of microstructure in 3D
Integr Mater Manuf Innov
Spatial tessellations
Periodic three-dimensional mesh generation for crystalline aggregates based on Voronoi tessellations
Comput Mech
Power diagrams: properties, algorithms and applications
SIAM J Comput
Simulation of correlated and uncorrelated packing of random size spheres
Phys Rev E
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2021, European Journal of Operational ResearchCitation Excerpt :Techniques in the first group are based on analytical representations of regular shapes used to compose irregular figures and the main modeling problem is to state analytically non-overlapping and containment conditions (see, e.g. Chernov, Stoyan, and Romanova (2010), Stoyan et al. (2015); Fasano and Pintér (2016); Ma et al. (2018); Pankratov, Romanova, and Litvinchev (2020), Zhao et al. (2020); Romanova et al. (2018); Stoyan et al. (2019b)). The second approach is based on tessellating the container/objects with a grid and then representing approximately complex shapes by corresponding collections of the grid nodes (see, e.g., Depriester and Kubler (2019); Edelkamp and Wichern (2015); Lucarini (2009); Mollon and Zhao (2014) and the references therein). This way, detecting the overlap is reduced to verify if two shapes share the same node.
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This paper has been recommended for acceptance by M. Sitharam.