Packing3D.jl: An open-source analytical framework for computing packing density and mixing indices using partial spherical volumes

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.在颗粒系统的模拟工作中，精准量化局域堆积密度与混合程度，对诸多工业应用而言均具有关键意义。传统方法仅统计给定空间单元（cell）内的粒子质心数量，这类方法会在单元边界处引入不连续性，进而导致堆积密度的测量结果缺乏可靠性。我们研发了开源Julia程序包Packing3D.jl，可对与笛卡尔网格、圆柱网格相交的球体开展解析偏体积计算。我们推导了单、双、三球形帽相交场景以及球-圆柱重叠场景的闭合形式解。本工具还实现了高效的网格生成例程、主单元索引以及用于时序分析的数据拆分功能。我们通过简单立方、面心立方晶格，以及边界位移连续性测试，验证了Packing3D.jl的性能与计算精度。Packing3D.jl可精准收敛至理论晶格密度，消除了亚粒子分辨率下的边界不连续性，且计算耗时随粒子总数呈线性缩放。其内存占用始终维持在较低水平：单粒子内存占用为40字节（B），单单元内存占用为48字节（B）。Packing3D.jl可为研究人员提供高性能的连续、可复现体积分数场与可靠的混合指数，助力颗粒过程工程中的敏感性分析与优化工作。