Modeling polycrystalline materials with elongated grains

Abstract

A novel algorithm to reproduce the arrangement of grains in polycrystalline materials was recently published by the authors. In this original approach, a dense package of circles (or spheres) with the same distribution as the grains is generated to produce a set of Voronoi cells that are later modified to Laguerre cells representing the original structure. This algorithm was successfully applied to materials with somewhat equidimensional grains; however, it fails for long-shaped grains. In this paper, modifications are provided in order to overcome these drawbacks. This is accomplished by moving each vertex of the Voronoi cells in such a way that the vertex should be equidistant from the particles with respect to the Euclidean distance. The algorithm is applied to packages of ellipses and spherocylinders in 2D. An example for a package of spheres is also provided to illustrate the application for a simple 3D case. The adherence between the generated packages and the corresponding tessellations is verified by means of the Jaccard coefficient (J). Several packages are generated randomly and the distribution of J coefficients is investigated. The obtained values satisfy the theoretical restraints and the quality of the proposed algorithm is statistically validated.

Laguerre tessellations and polycrystalline microstructures: a fast algorithm for generating grains of given volumes

“…Sphere-packing methods are popular for fitting Laguerre diagrams to volume distributions [4,8,10,18]. For n non-overlapping spheres S 1 , .…”

“…Voronoi diagrams and their generalisations are often used to represent the microstructure of polycrystalline metals and foams, e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14], with individual Voronoi cells representing grains in metals and pores or bubbles in foams. They can be used to generate complex microstructures quickly using a relatively small number of parameters, and they are often used as representative volume elements (RVEs) for computational homogenisation, e.g.…”

“…One popular method for approximately controlling the grain size distribution of Laguerre diagrams is using random close packing of spheres and ellipsoids [4,8,10,18]. This method is inexact, however, since it is impossible to tessellate Euclidean space with spheres or ellipsoids.…”

“…In other words, (w init ) i = r 2 i where r i is the radius of a ball of area m i in 2D or volume m i in 3D. This is motivated by sphere-packing methods [4,8,10,18].…”

“…The relative error of 79% of the grain centroids is less than 1 and the relative error of 93% of the grain centroids is less than 2. The run time for this example was 376 s on an Intel Xeon E5-1620V3 (3.5 GHz, 4 cores, 8 threads) with the initial guess (w init ) i = (3m i /(4p)) 2/3 , which was inspired by sphere-packing methods [4,8,10,18].…”