Voronoi Tessellations in Polycrystalline Microstructures

Voronoi tessellations have become a cornerstone in the quantitative modelling of polycrystalline microstructures, offering a mathematically rigorous yet computationally efficient way to represent grains by partitioning a domain into regions around seed points. In their simplest form, these tessellations capture key statistical features of grain size distributions and topological arrangements, enabling rapid generation of representative volume elements for finite‐element simulations and homogenisation studies. Extensions such as anisotropic Voronoi and Laguerre (power) tessellations introduce control over grain shape, orientation and volume, closely mimicking growth kinetics observed in solidification and annealing processes. These methods underpin the design of materials with bespoke mechanical, thermal and functional properties, feeding into applications ranging from additive manufacturing and aerospace alloys to ceramic membranes and energy storage materials. Advances in computational geometry and optimal transport theory have improved the fidelity of these models, allowing the incorporation of curved boundaries, non‐convex cells and spatially varying growth fields. Together, these developments have forged a versatile toolkit for scientists and engineers to probe the interplay between microstructural morphology and macroscopic behaviour.

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Voronoi Tessellations in Polycrystalline Microstructures publication trend

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Technical terms

Voronoi tessellation: A partition of space into convex cells surrounding seed points such that each location is assigned to its nearest seed.

Laguerre tessellation (power diagram): A generalisation of Voronoi tessellation where cells are weighted by seed‐specific values to control volumes or shapes.

Representative volume element (RVE): A small, statistically representative sample of a heterogeneous material used to predict bulk properties via simulation.

Grain boundary: The interface between crystallographically distinct regions (grains) in a polycrystalline material, influencing mechanical and transport properties.

Anisotropy: Directional dependence of material behaviour or growth, here realised by non‐spherical expansion fields in tessellation models.

References

1. Modeling of Polycrystalline Material Microstructure with 3D Grain Boundary Based on Laguerre–Voronoi Tessellation. Materials (2022).
2. Geometric modelling of polycrystalline materials: Laguerre tessellations and periodic semi-discrete optimal transport. Mechanics Research Communications (2023).
3. Anisotropic power diagrams for polycrystal modelling: Efficient generation of curved grains via optimal transport. Computational Materials Science (2024).