Tessellation Models Research Articles

An analytical representation of the 2d generalized balanced power diagram

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and the Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.

Continuum-based heterogenous models for capturing brittle damage and failure of hard rocks under loading and unloading conditions

Grain-scale heterogeneities play a crucial role in the failure process of hard brittle rocks. Previous numerical investigations have often overlooked certain sources of heterogeneity and the inherent variability in laboratory test data during model calibration. This study aims to assess the impact of grain-scale heterogeneities on macro-property variability, internal stress variations, and unloading-induced brittle damage using continuum-based heterogeneous models. For this purpose, ten realizations of homogeneous (consisting of one mineral type) and heterogeneous (consisting of four mineral types) Voronoi Tessellated Models (VTMs) were generated. These models were calibrated to match the mean strength of undamaged Lac du Bonnet (LdB) granite. The heterogeneous VTM demonstrated a stronger agreement with LdB granite in terms of the shape of the strength envelope and peak strength variability compared to the homogeneous VTM. Analysis of internal elastic stresses revealed higher induced tensile stresses within the heterogeneous VTM than the homogeneous VTM during compressive loading. Furthermore, it was found that, in contrast to the homogeneous VTM, unloading-induced brittle damage in the heterogeneous VTM led to a reduction in its peak strength and Young’s modulus, aligning with laboratory test results. Therefore, the heterogeneous VTM is deemed a more representative model for hard rocks than the homogeneous VTM.

The β-Delaunay tessellation IV: Mixing properties and central limit theorems

Various mixing properties of [Formula: see text]-, [Formula: see text]- and Gaussian-Delaunay tessellations in [Formula: see text] are studied. It is shown that these tessellation models are absolutely regular, or [Formula: see text]-mixing. In the [Formula: see text]- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the [Formula: see text]-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of [Formula: see text]- and Gaussian-Delaunay tessellations are established. This includes the number of [Formula: see text]-dimensional faces and the [Formula: see text]-volume of the [Formula: see text]-skeleton for [Formula: see text].

Simulation of hard rock pillar failure using 2D continuum-based Voronoi tessellated models: The case of Quirke Mine, Canada

The Quirke Mine, ON, Canada, is a typical example of a mine that experienced a chain reaction of pillar failure. In this mine, the rib pillars were initially laid out with their long axis parallel to the dip direction of the orebody. In the central part of the mine, the pillars were re-aligned at 45° to the orebody true dip. This realignment resulted in adverse shear loading conditions that led to their failure. In this study, a two-dimensional finite element program was used to simulate the failure of Quirke Mine pillars under compressive and shear loading conditions. The pillars were simulated as a heterogeneous material, consisting of randomly generated Voronoi blocks. The calibration of this model, referred to as a Voronoi Tessellated Model (VTM), was conducted against the rock mass strength estimated based on a tri-linear, brittle failure criterion. The calibrated VTM was then employed to investigate the pillar behavior during drift development and stope excavation. It was found that the pillar under shear loading experiences higher confinement loss and tensile stress at its core and, therefore, fails at a lower stress level than that under compression. The simulated pillar failure process was found to agree with documented field observations.

Efficient Fitting of 3D Tessellations to Curved Polycrystalline Grain Boundaries

The curvature of grain boundaries in polycrystalline materials is an important characteristic, since it plays a key role in phenomena like grain growth. However, most traditional tessellation models that are used for modeling the microstructure morphology of these materials, e.g., Voronoi or Laguerre tessellations, have flat faces and thus fail to incorporate the curvature of the latter. For this reason, we consider generalizations of Laguerre tessellations—variations of so-called generalized balanced power diagrams (GBPDs)—that exhibit non-convex cells. With as many as ten parameters for each cell, it is computationally demanding to fit GBPDs to three-dimensional image data containing hundreds of grains. We therefore propose a modification of the traditional definition of GBDPs that allows gradient-based optimization methods to be employed. The resulting reduction in runtime makes it feasible to find approximations to real experimental datasets. We demonstrate this on a three-dimensional x-ray diffraction (3DXRD) mapping of an AlCu alloy, but we also evaluate the modeling errors for simulated data. Furthermore, we investigate the effect of noisy image data and whether the smoothing of image data prior to the fitting step is advantageous.

Comparing cost and print time estimates for six commercially-available 3D printers obtained through slicing software for clinically relevant anatomical models

Background3D printed patient-specific anatomical models have been applied clinically to orthopaedic care for surgical planning and patient education. The estimated cost and print time per model for 3D printers have not yet been compared with clinically representative models across multiple printing technologies. This study investigates six commercially-available 3D printers: Prusa i3 MK3S, Formlabs Form 2, Formlabs Form 3, LulzBot TAZ 6, Stratasys F370, and Stratasys J750 Digital Anatomy.MethodsSeven representative orthopaedic standard tessellation models derived from CT scans were imported into the respective slicing software for each 3D printer. For each printer and corresponding print setting, the slicing software provides a print time and material use estimate. Material quantity was used to calculate estimated model cost. Print settings investigated were infill percentage, layer height, and model orientation on the print bed. The slicing software investigated are Cura LulzBot Edition 3.6.20, GrabCAD Print 1.43, PreForm 3.4.6, and PrusaSlicer 2.2.0.ResultsThe effect of changing infill between 15% and 20% on estimated print time and material use was negligible. Orientation of the model has considerable impact on time and cost with worst-case differences being as much as 39.30% added print time and 34.56% added costs. Averaged across all investigated settings, horizontal model orientation on the print bed minimizes estimated print time for all 3D printers, while vertical model orientation minimizes cost with the exception of Stratasys J750 Digital Anatomy, in which horizontal orientation also minimized cost. Decreasing layer height for all investigated printers increased estimated print time and decreased estimated cost with the exception of Stratasys F370, in which cost increased. The difference in material cost was two orders of magnitude between the least and most-expensive printers. The difference in build rate (cm3/min) was one order of magnitude between the fastest and slowest printers.ConclusionsAll investigated 3D printers in this study have the potential for clinical utility. Print time and print cost are dependent on orientation of anatomy and the printers and settings selected. Cost-effective clinical 3D printing of anatomic models should consider an appropriate printer for the complexity of the anatomy and the experience of the printer technicians.

Data-Driven Selection of Tessellation Models Describing Polycrystalline Microstructures

Tessellation models have proven to be useful for the geometric description of grain microstructures in polycrystalline materials. With the use of a suitable tessellation model, the complex morphology of grains can be represented by a small number of parameters assigned to each grain, which not only entails a significant reduction in complexity, but also facilitates the investigation of certain geometric features of the microstructure. However, for a given set of microstructural data, the choice of a particular geometric model is traditionally based on researcher intuition. The model should provide a sufficiently good approximation to the data, while keeping the number of parameters small. In this paper, we discuss general aspects of the process of model selection and suggest several criteria for selecting an appropriate candidate from a certain set of tessellation models. The choice of candidate represents a trade-off between accuracy and complexity of the model. Here, the selected model is used solely to approximate given data samples, but it also provides guidance for developing stochastic tessellation models and generating virtual microstructures. Model fitting is carried out by simulated annealing, applied in a consistent manner to twelve different tessellation models.

Line segments which are unions of tessellation edges

Planar tessellation structures occur in material science, geology (in rock formations), physics (of foams, for example), biology (especially in epithelial studies) and in other sciences. Their mathematical and statistical study has many aspects to consider. In this paper, line-segments which are either a tessellation edge or a finite union of edges are studied. Our focus is on a sub-class of such line-segments – those we call M-segments – that are not contained in a longer union of edges. These encompass the so-called I-segments that have arisen in many recent tessellation models. We study the expected numbers of edges and cell-sides contained in these M-segments, and the prevalence of these entities. Many examples and figures, including some based on tessellation nesting and superposition, illustrate the theory. M-segments are much more prevalent when a tessellation is not side-to-side, so our paper has theoretical connections with the recent IAS paper by Cowan and Thäle (2014); that paper characterised non side-to-side tessellations.

Effects of cell size and cell wall thickness variations on the strength of closed-cell foams

In this work, the effects of cell size and cell wall thickness variations on the compressive and shear strengths of closed-cell foams were investigated using Laguerre tessellation models. It is found that the compressive and shear strengths of closed-cell foams decrease as cell size and cell wall thickness variations increase, and the compressive strength reduces more significantly. At a given level of variation, the effect of cell size variation on strength reduction is comparable to that of cell wall thickness variation on them. In the foam studied (M130 foam), cell wall thickness has a larger dispersion than cell size, and therefore is the main contributor to the strength reduction of the foam. In comparison to compressive and shear stiffnesses, cell size and cell wall thickness variations reduce compressive and shear strengths more significantly. Cell size variation results in strain concentration at the end of foam samples, thereby reducing the compressive strength of closed-cell foams. Cell wall thickness variation leads to cell wall buckling and thus reduction in compressive and shear strengths of closed-cell foams. As cell size and cell wall thickness in foams become less uniform, the relationships of compressive and shear strengths to relative density become less linear.

DESCRIPTION OF THE 3D MORPHOLOGY OF GRAIN BOUNDARIES IN ALUMINUM ALLOYS USING TESSELLATION MODELS GENERATED BY ELLIPSOIDS

Parametric tessellation models are often used to approximate complex grain morphologies of polycrystalline microstructures. A big advantage of such models is the substantial reduction in disk space required to store large, three-dimensional data sets, especially when compared with voxel-based alternatives. By selection of an appropriate tessellation model, a reasonably small loss of information on the real grain shapes can usually be achieved. Special attention has recently been devoted to models based on ellipsoidal approximations fitted to each grain. Faces of these tessellations are portions of quadric surfaces whose parameters can be derived easily. In this paper, we deal with geometric features of the structure, notably curvatures and dihedral angles, which are closely related to the kinetics of grain growth. These characteristics are computed for ellipsoidbased tessellations fitted to two different aluminum alloys with nominal composition Al-3 wt% Mg-0.2 wt% Sc and Al-1 wt% Mg. The results are then compared with estimations based on meshed empirical data. We observe that the model offers more consistent estimations of grain shape characteristics than do the meshed empirical data. Precise description of grain boundaries by the model is also promising with respect to possible applications of these tessellations in stochastic space-time modeling of grain growth.

Determination of the thickness of the embedding phase in 0D nanocomposites

0D nanocomposites formed by small nanoparticles embedded in a second phase are very interesting systems which may show properties that are beyond those observed in the original constituents alone. One of the main parameters to understand the behavior of such nanocomposites is the determination of the separation between two adjacent nanoparticles, in other words, the thickness of the embedding phase. However, its experimental measurement is extremely complicated. Therefore, its evaluation is performed by an indirect approach using geometrical models. The ones typically used represent the nanoparticles by cubes or spheres.In this paper the used geometrical models are revised, and additional geometrical models based in other parallelohedra (hexagonal prism, rhombic and elongated dodecahedron and truncated octahedron) are presented. Additionally, a hybrid model that shows a transition between the spherical and tessellated models is proposed. Finally, the different approaches are tested on a set of titanium carbide/amorphous carbon (TiC/a-C) nanocomposite films to estimate the thickness of the a-C phase and explain the observed hardness properties.

Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids

Control over the deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials, in particular those with negative Poisson’s ratio. This article sheds light on the old question of the relationship between geometric microstructure and mechanical response, by comparison of the deformation properties of bar-and-joint-frameworks with those of their realisation as a cellular solid made from linear-elastic material. For ordered planar tessellation models, we find a classification in terms of the number of degrees of freedom of the framework model: first, in cases where the geometry uniquely prescribes a single deformation mode of the framework model, the mechanical deformation and Poisson’s ratio of the linearly-elastic cellular solid closely follow those of the unique deformation mode; the result is a bending-dominated deformation with negligible dependence of the effective Poisson’s ratio on the underlying material’s Poisson’s ratio and small values of the effective Young’s modulus. Second, in the case of rigid structures or when geometric degeneracy prevents the bending-dominated deformation mode, the effective Poisson’s ratio is material-dependent and the Young’s modulus E˜cs large. All analysed structures of this type have positive values of the Poisson’s ratio and large values of E˜cs. Third, in the case, where the framework has multiple deformation modes, geometry alone does not suffice to determine the mechanical deformation. These results clarify the relationship between mechanical properties of a linear-elastic cellular solid and its corresponding bar-and-joint framework abstraction. They also raise the question if, in essence, auxetic behaviour is restricted to the geometry-guided class of bending-dominated structures corresponding to unique mechanisms, with inherently low values of the Young’s modulus.

3D reconstruction of grains in polycrystalline materials using a tessellation model with curved grain boundaries

A compact and tractable representation of the grain structure of a material is an extremely valuable tool when carrying out an empirical analysis of the material’s microstructure. Tessellations have proven to be very good choices for such representations. Most widely used tessellation models have convex cells with planar boundaries. Recently, however, a new tessellation model — called the generalised balanced power diagram (GBPD) — has been developed that is very flexible and can incorporate features such as curved boundaries and non-convexity of cells. In order to use a GBPD to describe the grain structure observed in empirical image data, the parameters of the model must be chosen appropriately. This typically involves solving a difficult optimisation problem. In this paper, we describe a method for fitting GBPDs to tomographic image data. This method uses simulated annealing to solve a suitably chosen optimisation problem. We then apply this method to both artificial data and experimental 3D electron backscatter diffraction (3D EBSD) data obtained in order to study the properties of fine-grained materials with superplastic behaviour. The 3D EBSD data required new alignment and segmentation procedures, which we also briefly describe. Our numerical experiments demonstrate the effectiveness of the simulated annealing approach (compared to heuristic fitting methods) and show that GBPDs are able to describe the structures of polycrystalline materials very well.

Tessellations in GIS: Part II–making changes

We attempt to describe the role of tessellated models of space within the discipline of Geographic Information Systems (GIS) – a speciality coming largely out of Geography and Land Surveying, where there was a strong need to represent information about the land’s surface within a computer system rather than on the original paper maps. We look at some of the basic operations in GIS, including dynamic and kinetic applications. We examine issues of topology and data structures, and produced a tessellation model that may be widely applied both to traditional “object” and “field” data types. The Part I of this study examined object and field spatial models, the Voronoi extension of objects, and the graphs that express the resulting adjacencies. The required data structures were also briefly described, along with 2D and 3D structures and hierarchical indexing. The importance of graph duality was emphasized. Here, this second paper builds on the structures described in the first, and examines how these may be modified: change may often be associated with either viewpoint or time. Incremental algorithms permit additional point insertion, and applications involving the addition of skeleton points, for map scanning, contour enrichment or watershed delineation and simulation. Dynamic algorithms permit skeleton smoothing, and higher order Voronoi diagram applications, including Sibson interpolation. Kinetic algorithms allow collision detection applications, free-Lagrange flow modeling, and pen movement simulation for map drawing. If desired these methods may be extended to 3D. Based on this framework, it can be argued that tessellation models are fundamental to our understanding and processing of geographical space, and provide a coherent framework for understanding the “space” in which we exist.

Tessellations in GIS: Part I—putting it all together

This article attempts to describe the role of tessellated models of space within the discipline of geographic information systems (GIS)—a speciality coming largely out of geography and land surveying, where there was a strong need to represent information about the land’s surface within a computer system rather than on the original paper maps. We look at some of the basic operations in GIS, including dynamic and kinetic applications. We examine issues of topology and data structures and produce a tessellation model that may be widely applied both to traditional “object” and “field” data types. Based on this framework, it can be argued that tessellation models are fundamental to our understanding and processing of geographical space, and provide a coherent framework for understanding the “space” in which we exist. This first article examines static structures, and a subsequent article looks at “change”—what happens when things move.

Tessellation growth models for polycrystalline microstructures

Abstract This work proposes novel models to represent and parametrize random morphology of polycrystalline microstructures. The reliability of high-fidelity mechanical analysis of polycrystalline microstructures depends upon the morphological representation of the virtual model. Two models addressed in this work are spherical growth and ellipsoidal growth tessellations in which grains grow as spheres (or ellipsoids) with random velocities which initiate from random nucleation sites represented by a spatial point process. All of the stochastic parameters can be represented by a marked point process random field model, for which simulation algorithms exist. Probability distributions of the model parameters are estimated by obtaining best-fit realizations of the models to a data set of a reconstructed microstructure specimen. The accuracy to which these tessellation models can represent real microstructures is evaluated using two example data sets by computing numerous microstructure features as well as the mismatch volume between the best-fit realizations and the data. The spherical growth and ellipsoidal growth tessellations demonstrate very significant improvements over the Voronoi tessellation, while remaining low dimensional representations of the microstructure. Realizations generated from a marked point process random field model show very good agreement in grain size, aspect ratio, and nearest neighbor distributions compared to an example data set. Thus, subsequent realistic instantiations of microstructures having the same statistical characteristics of the data can be trivially obtained, which are necessary to propagate the uncertainty associated with morphological randomness on response quantities of interest in mechanics-based applications.

Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams

This paper concerns with the micromechanical modelling of closed-cell polymeric foams (M130) using Laguerre tessellation models incorporated with realistic foam cell size and cell wall thickness distributions. The cell size and cell wall thickness distributions of the foam were measured from microscope images. The Young’s modulus of cell wall material of the foam was characterised by nanoindentation tests. It is found that when the cell size and cell wall thickness are assumed to be uniform in the models, the Kelvin, Weaire–Phelan and Laguerre models overpredict the stiffness of the foam. However, the Young’s modulus and shear modulus predicted by the Laguerre models incorporating measured foam cell size and cell wall thickness distributions agree well with the experimental data. This emphasizes the fact that the integration of realistic cell wall and cell size variations is vital for foam modelling. Subsequently the effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams were investigated using Laguerre models. It is found that the Young’s modulus and shear modulus decrease with increasing cell size and cell wall thickness variations. The degree of stiffness variation of closed-cell foams resulting from the cell size dispersion and cell wall thickness dispersion are comparable. There is little interaction between the cell size variation and cell wall thickness variation as far as their effects on foam moduli are concerned. Based on the simulation results, expressions incorporating cell size and cell wall thickness variations were formulated for predicting the stiffness of closed-cell foams. Lastly, a simple spring system model was proposed to explain the effects of cell size and cell wall thickness variations on the stiffness of cellular structures.

Can plasticity make spatial structure irrelevant in individual-tree models?

Distance-dependent individual-tree models have commonly been found to add little predictive power to that of distance-independent ones. One possible reason is plasticity, the ability of trees to lean and to alter crown and root development to better occupy available growing space. Being able to redeploy foliage (and roots) into canopy gaps and less contested areas can diminish the importance of stem ground locations. Plasticity was simulated for 3 intensively measured forest stands, to see to what extent and under what conditions the allocation of resources (e.g., light) to the individual trees depended on their ground coordinates. The data came from 50 × 60 m stem-mapped plots in natural monospecific stands of jack pine, trembling aspen and black spruce from central Canada. Explicit perfect-plasticity equations were derived for tessellation-type models. Qualitatively similar simulation results were obtained under a variety of modelling assumptions. The effects of plasticity varied somewhat with stand uniformity and with assumed plasticity limits and other factors. Stand-level implications for canopy depth, distribution modelling and total productivity were examined. Generally, under what seem like conservative maximum plasticity constraints, spatial structure accounted for less than 10% of the variance in resource allocation. The perfect-plasticity equations approximated well the simulation results from tessellation models, but not those from models with less extreme competition asymmetry. Whole-stand perfect plasticity approximations seem an attractive alternative to individual-tree models.

Development of teaching materials to learn crystallographic symmetry

Authors have developed proper polyhedron models to enable people to learn the concept of three-dimensional symmetry. Touching and operating the symmetry elements of the proper polyhedron enables people to understand symmetry. In this study, authors made three-dimensional tessellation models. Certain polyhedra can be stacked in a regular periodic pattern to fill three-dimensional space completely. Figures show our models. The cube (Fig. (a)) is the only regular polyhedron to fill three-dimensional space completely. The cube is a Voronoi region of the simple cubic lattice (sc). The truncated octahedron (Fig. (b)) is the only Archimedean solid to fill three-dimensional space completely. The truncated octahedron is a Voronoi region of the body-centered cubic lattice (bcc). The rhombic dodecahedron (Fig. (c)) is the only Catalan solid (or Archimedean dual) to fill three-dimensional space completely. The rhombic dodecahedron is a Voronoi region of the face-centered cubic lattice (fcc). Figs. (a), (b), and (c) show three kinds of their aggregate respectively. In each of left-hand aggregate, there is a two-fold rotational axis along a vertical direction. In each of central aggregate, there is a three-fold rotational axis along a vertical direction. In each of right-hand aggregate, there is a four-fold rotational axis along a vertical direction. Fig. (d) is a nontrivial polyhedron to fill three-dimensional space completely. The external shape of the polyhedron was designed as a tree shape. We call such a model three-dimensional Escher shape (3DES) [1]. This can be stacked in a regular periodic pattern too.

3D RECONSTRUCTION OF A MULTISCALE MICROSTRUCTURE BY ANISOTROPIC TESSELLATION MODELS

In the area of tessellation models, there is an intense activity to fully understand the classical models of Voronoi, Laguerre and Johnson-Mehl. Still, these models are all simulations of isotropic growth and are therefore limited to very simple and partly convex cell shapes. The here considered microstructure of martensitic steel has a much more complex and highly non convex cell shape, requiring new tessellation models. This paper presents a new approach for anisotropic tessellation models that resolve to the well-studied cases of Laguerre and Johnson-Mehl for spherical germs. Much better reconstructions can be achieved with these models and thus more realistic microstructure simulations can be produced for materials widely used in industry like martensitic and bainitic steels.