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.

Inverting Laguerre Tessellations

“…Pertinent solutions possess extra structure that is imposed by the modeler. Typical requirements arising in materials science (see, e.g., [14]), geometry (e.g., sphere packing [17]), molecular biology and biochemistry (see, e.g., [18]), include the following:…”

“…However, the inversion problem for Laguerre tessellations seems to be much less studied. Recently, an approximate inversion method was given in [14], although with a different motivation, i.e., the possibility of reconstructing the complete tessellation using only cells' centers of mass and cell volumes.…”

Measuring the stress field around an evolving crack in tensile deformed Mg AZ31 using three-dimensional X-ray diffraction

“…The 2D projections were produced on a 10×10 µm 2 grid by Laguerre tessellation [37], thus for each grid point the contribution was taken from the grain for which the distance from grid point to the projected grain centre divided by the grain radius building up in front of the notch, while axial stresses (σ 33 ) build up in a butterfly-shaped contour around the tip. In the plastic regime the simulated stresses tend to be higher than the measured stresses in accordance with the stress relaxation observed during the diffraction experiment which results in reduced stresses relative to the hardening curve used in the simulation (Fig.…”

Description of the 3d Morphology of Grain Boundaries in Aluminum Alloys Using Tessellation Models Generated by Ellipsoids

“…Many parametric tessellation models have been invented to describe cellular structures in various fields of research (Okabe et al, 2010). Concerning materials science, great interest has been devoted to the application of convex tessellation models, in which the grains are convex polyhedra; see, e.g., Lyckegaard et al (2011). The simplicity of these models allows a simple evaluation of size and shape characteristics of the grains (Lautensack and Zuyev, 2008) and relatively fast and accurate fitting to empirical data (Spettl et al, 2016).…”

“…Voronoi and Laguerre tessellations have extensively been used in literature to model grain microstructures; see, e.g., Kumar and Kurtz (1994) and Lyckegaard et al (2011). However, modern methods of microscopic research have reliably proven that the grain boundaries in real microstructures are not perfectly planar.…”