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Packing3D: An Open-Source Analytical Framework for Computing Packing Density and Mixing Indices Using Partial Spherical Volumes
Freddie J. Barter, Christopher R. K. Windows-Yule
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Accurate quantification of local packing density and mixing in simulations of particulate systems is essential for many industrial applications. Traditional methods which simply count the number of particle centres within a given volume of space (cell) introduce discontinuities at cell boundaries, leading to unreliable measurements of packing density. We introduce Packing3D.jl, an open-source Julia package providing analytic partial-volume calculations for spheres intersecting Cartesian and cylindrical meshes. We derive closed-form solutions for single, double and triple spherical-cap intersections, plus sphere-cylinder overlaps. We implement efficient mesh-generation routines, principal-cell indexing, and data-splitting functions for time-series analyses. Performance and accuracy were validated against simple cubic and face-centred cubic lattices and via boundary-shift continuity tests. Packing3D.jl converges exactly to theoretical lattice densities, eliminates discontinuities at sub-particle resolution, and scales linearly with particle count. Memory usage remains modest (40 B per particle, 48 B per cell). Packing3D.jl provides researchers with continuous, reproducible volume-fraction fields and robust mixing indices at high performance, facilitating sensitivity analyses and optimisation in granular process engineering.
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