Insights from the persistent homology analysis of porous and granular materials

Persistent homology is an algebraic topological tool developed for data analysis that measures changes in topology of a filtration: a growing sequence of spaces indexed by a single real parameter. It produces invariants called the barcodes or persistence diagrams that are sets of intervals recording the birth and death parameter values of each homology class in the filtration. When the filtration parameter is a length-scale, persistence diagrams provide a comprehensive description of geometric structure over the given parameter range. The physical properties of porous and granular materials critically depend on the topological and geometric details of the material micro-structure. For example, the way water flows through sandstone depends on the connectivity and diameters of its pores, and the balance of forces in a grain silo on the contacts between individual grains. These materials are therefore a natural application area for persistent homology. My work with the x-ray micro-CT group at ANU has produced topologically valid and efficient algorithms for studying and quantifying the intricate structure of complex porous materials. Our code package, diamorse, for computing skeletons, partitions, and persistence diagrams from 2D and 3D images is available on GitHub. The code contains several optimisations that allow it to process images with up to 2000^3 voxels on a high-end desktop PC. This software is enabling us to explore the connections between topology, geometry and physical properties of sandstone rock cores and granular packings. We have shown that persistence diagrams display a clear signal of crystallisation in bead packings, the degree of consolidation in sandstones, percolating length scales in porous media, and the trapping of non-wetting phase in two-phase fluid flow experiments.

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