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Robert D. MacPherson

Publications and source records attributed to Robert D. MacPherson.

7 recordsLinked to original sources
arXiv2020

Distribution of Topological Types in Grain-Growth Microstructures

An open question in studying normal grain growth concerns the asymptotic state to which microstructures converge. In particular, the distribution of grain topologies is unknown. We introduce a thermodynamic-like theory to explain these distributions in two- and three-dimensional systems. In particular, a bending-like energy $E_i$ is associated to each grain topology $t_i$, and the probability of observing that particular topology is proportional to $\frac{1}{s(t_i)}e^{-βE_i}$, where $s(t_i)$ is the order of an associated symmetry group and $β$ is a thermodynamic-like constant. We explain the physical origins of this approach, and provide numerical evidence in support.

cond-mat.mtrl-sci↗
arXiv2017

Roundness of grains in cellular microstructures

Many physical systems are composed of polyhedral cells of varying sizes and shapes. These structures are simple in the sense that no more than three faces meet at an edge and no more than four edges meet at a vertex. This means that individual cells can usually be considered as simple, three-dimensional polyhedra. This paper is concerned with determining the distribution of combinatorial types of such polyhedral cells. We introduce the terms \emph{fundamental} and \emph{vertex-truncated} types and apply these concepts to the grain growth microstructure as a testing ground. For these microstructures we demonstrate that most grains are of particular fundamental types, whereas the frequency of vertex-truncated types decreases exponentially with the number of truncations. This can be explained by the evolutionary process through which grain growth structures are formed, and in which energetically unfavorable surfaces are quickly eliminated. Furthermore, we observe that these grain types are `round' in a combinatorial sense: there are no `short' separating cycles that partition the polyhedra into two parts of similar sizes. A particular microstructure derived from the Poisson--Voronoi initial condition is identified as containing an unusually large proportion of round grains. This Round microstructure has an average of $14.036$ faces per grain, and is conjectured to be more resistant to topological change than the steady-state grain growth microstructure.

cond-mat.mtrl-sci↗
arXiv2015

Geometric and topological properties of the canonical grain growth microstructure

Many physical systems can be modeled as large sets of domains "glued" together along boundaries - biological cells meet along cell membranes, soap bubbles meet along thin films, countries meet along geopolitical boundaries, and metallic crystals meet along grain interfaces. Each class of microstructures results from a complex interplay of initial conditions and particular evolutionary dynamics. The statistical steady-state microstructure resulting from isotropic grain growth of a polycrystalline material is canonical in that it is the simplest example of a cellular microstructure resulting from a gradient flow of a simple energy, directly proportional to the total length or area of all cell boundaries. As many properties of polycrystalline materials depend on their underlying microstructure, a more complete understanding of the grain growth steady-state can provide insight into the physics of a broad range of everyday materials. In this paper we report geometric and topological features of these canonical two- and three-dimensional steady-state microstructures obtained through large, accurate simulations of isotropic grain growth.

cond-mat.mtrl-sci↗
arXiv2014

Statistical Topology of Three-Dimensional Poisson-Voronoi Cells and Cell Boundary Networks

Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simulated Poisson-Voronoi structures containing a total of 250,000,000 cells to provide topological and geometrical statistics of this important class of networks. We also report correlations between some of these topological and geometrical measures. Using these results, we are able to corroborate several conjectures regarding the properties of three-dimensional Poisson-Voronoi networks and refute others. In many cases, we provide accurate fits to these data to aid further analysis. We also demonstrate that topological measures represent powerful tools for describing cellular networks and for distinguishing among different types of networks.

physics.comp-ph↗
arXiv2012

Complete topology of cells, grains, and bubbles in three-dimensional microstructures

We introduce a general, efficient method to completely describe the topology of individual grains, bubbles, and cells in three-dimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of three-dimensional microstructures that are often regarded as close analogues in the literature: one resulting from normal grain growth (mean curvature flow) and another resulting from a random Poisson-Voronoi tessellation of space. Grain growth strongly favors particular grain topologies, compared with the Poisson-Voronoi model. Moreover, the frequencies of highly symmetric grains are orders of magnitude higher in the the grain growth microstructure than they are in the Poisson-Voronoi one. Grain topology statistics provide a strong, robust differentiator of different cellular microstructures and provide hints to the processes that drive different classes of microstructure evolution.

cond-mat.mtrl-sci↗
arXiv1997

Intersection Homology of Toric Varieties and a Conjecture of Kalai

We prove that the intersection homology Poincare' polynomial P(X) of an affine toric variety X is bounded below by the product P(Y)P(X/Y), where Y is the closure of any orbit in X and X/Y is a slice transverse to the orbit. This proves a combinatorial conjecture of Kalai for rational polytopes.

alg-geom↗