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

Abstract

Generation of realistic artificial 3D grain structures for use in modeling has gained increasing attention during the last two decades due to significant enhancements in the capabilities of large-scale 3D computer simulations. One commonly chosen model is the Laguerre tessellation (also known as weighted Voronoi tessellations or power diagrams). [1] As in the case of the classical Voronoi tessellation, Laguerre tessellations also partition space into convex polyhedra with planar faces. The advantage of the Laguerre tessellations, compared to the classical Voronoi tessellation, is that it allows for a wider range of grain structures by means of weighting factors.

Optimal polyhedral description of 3D polycrystals: Method and application to statistical and synchrotron X-ray diffraction data

“…Compared to the direct approach consisting of computing the seed attributes from the grain centroids and volumes (which was used as initial solution), the errors on the centroids and volumes are significantly lower: 0.7% on the centroids, 3.2% on the volumes (0.7% on the diameters) while, for the direct approach: 10.1% on the centroids and 14.4% on the volumes (3.2% on the diameters). The present method also supersedes the one of Lyckegaard and co-authors [40], which provides errors of 6.2% on the centroids and 13.5% on the volumes (9.6% on the diameters). This is an important improvement since such changes can lead to appreciably different grain morphologies and neighbours, as was seen in Section 4.…”

Fitting Laguerre tessellation approximations to tomographic image data

“…However, it is not able to reproduce the neighbor structure as successfully as our approach, as evidenced by rows 2 through 5 of Table 1. Our method is slightly less successful at approximating the PLT data but, in this case, considerably outperforms the heuristics of [6].…”

“…As with the artificial data, we compare our results with the heuristic approach proposed in [6] and the orthogonal regression approach introduced in [48]. As mentioned above, the orthogonal regression approach reconstructs the individual cells quite well but does not result in a tessellation.…”

“…The parameters used by the heuristic approach of [6] are very close to the initial conditions of the CE algorithm. Thus, from the difference in performance, it seems that the initial conditions are quite far from the optimal parameters.…”

“…Namely, by manually increasing the number of variance injections to 20 (instead of the 8 used in the Table 1. Evaluation of the artificial data approximations: H denotes the heuristic approach of [6]; CE denotes the CE method considered in the present paper; OR denotes orthogonal regression proposed in [48]. standard approach), we obtained an approximation which correctly labeled 97.3% of the voxels (instead of 96.2%).…”