Efficient Representation of Laguerre Mosaics with an Application to Microstructure Simulation of Complex Ore

Abstract

Laguerre mosaics have been an important modeling approach in astronomy, physics, crystallography, geology and mathematics for several decades. In materials science, they are used as models for cellular and polycrystalline materials, networks and cell foams. In this study, Laguerre mosaics are used to model the three-dimensional internal mineral microstructure of complex ores. Here, the difficulties arise in representing and simulating these microstructure mosaics for dimensions larger than two. Therefore, this manuscript introduces a general workflow for the representation in arbitrary dimensions and presents a realization of this workflow using generalized maps for representation in two and three dimensions. With this approach, lower-dimensional components such as cells, facets, edges and vertices can be accessed directly, which enables us to efficiently create the mosaics and derive statistics, plane sections and new mosaic models by intersection. Furthermore, it allows for easy deduction of the dual mosaic and efficient storage. The mineral microstructure of complex ores can be very complicated and often shows a highly fractal structure. Therefore, numerical modeling and representation of these microstructures is challenging. The proposed approach for Laguerre mosaic creation and representation is successfully applied to the modeling of mineral microstructures and particles. These microstructure models are used for mineral processing simulations in order to determine optimal processing strategies to conserve valuable resources.

Algorithm for Identification of Individual Cells as Intersection Candidates

Fig. 12

Collection algorithm for dimension $n=2$

In practice, microstructure mosaic realizations can contain a very high number of individual n-cells. For particle sampling based on fracture mosaics, each cell has to be identified as either completely outside, inside or on an intersecting n-cell. Naively, this would mean testing each of the N cells from the microstructure mosaic with all M cells from the fracture mosaic for intersection. This can be avoided by applying a so-called collection algorithm that defines i-cells as intersection candidates. The general n-dimensional method leads recursively back to the two-dimensional case as described in what follows.

Let $\mathcal {C}_1 = \{\mathcal {E}_1, \mathcal {V}_1\}$ be a polygon and $M_2 = \{\mathcal {C}_2, \mathcal {E}_2, \mathcal {V}_2\}$ be a mosaic in $\mathbb {R}^2$ with cells $\mathcal {C}$ (2-cells), edges $\mathcal {E}$ (1-cells) and vertices $\mathcal {V}$ (0-cells). First obtain the so-called connectivity function. It is defined to be the mapping $g : \mathcal {V}_1 \longrightarrow \mathcal {C}_2$ such that for $V \in \mathcal {V}_1$, $g(V) \in \mathcal {C}_2$ is the cell with $V \in g(V)$. The next step is called Collection two-dimensional (Algorithm 1; Fig. 12). Here, all cells of mosaic $M_2$ which are crossed by an edge of $\mathcal {E}_1$ are stored in the collection list. A simple example is shown in Fig. 13.

Fig. 13

Intersection polygon $\mathcal {C}_1$ (blue) and mosaic $M_2$ (red). After intersection, the result is the cell complex (green)

Fig. 14

Collecting algorithm in any dimension

This list contains cells of $M_2$ which need to be processed and handled in the so-called construction step. Here, the list entries are collected for each cell that occurs giving new edges of the sub-mosaic. A third step adds all enclosed cells to the sub-mosaic. Therefore, all neighboring cells of the cells contained in the collection list with vertices inside $\mathcal {C}_1$ and their unvisited neighboring cells are copied into the sub-mosaic structure (see Sect. 2.2).

Generally, let $\mathcal {C}$ be a polytope (n-cell) and M be a mosaic in $\mathbb {R}^n$. Application of collect ($\mathcal {C}$, M) from Algorithm 2 (Fig. 14) derives the collection list, which is handled afterwards as in the two-dimensional case.