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A cornerstone of materials science is that material properties are determined by their microstructure. While the community has already developed a wide variety of approaches to describe microstructure, most of these are tailored to specific material systems or classes. This work proposes a way to quantitatively measure the similarity of microstructures based on the geometry of the grain boundary network, a feature which is fundamental to and characteristic of all polycrystalline materials. Specifically, a distance on all single-phase polycrystalline microstructures is proposed such that two microstructures that are close with regard to the distance have grain boundary networks that are statistically similar in all geometric respects below a user-specified length scale. Given a pair of micrographs, the distance is approximated by sampling windows from the micrographs, defining a distance between pairs of windows, and finding a window matching that minimizes the sum of the pairwise window distances. The approach is used to compare a variety of synthetic microstructures and to develop a procedure to query a proof-of-concept database suitable for general single-phase polycrystalline microstructures. 

Abstract The Eden cell growth model is a simple discrete stochastic process which produces a “blob” (aggregation of cells) in $$\mathbb {R}^d$$ $R^d$: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers at timetgrow at a rate between$$t^{(d-1)/d}$$ $t^{(d-1)/d}$and$$P_d(t)$$ $P_d\left(t\right)$, where$$P_d(t)$$ $P_d\left(t\right)$is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes.