Laguerre tessellations and polycrystalline microstructures: a fast algorithm for generating grains of given volumes

ABSTRACT

We present a fast algorithm for generating Laguerre diagrams with cells of given volumes, which can be used for creating RVEs of polycrystalline materials for computational homogenisation, or for fitting Laguerre diagrams to EBSD or XRD measurements of metals. Given a list of desired cell volumes, we solve a convex optimisation problem to find a Laguerre diagram with cells of these volumes, up to any prescribed tolerance. The algorithm is built on tools from computational geometry and optimal transport theory which, as far as we are aware, have not been applied to microstructure modelling before. We illustrate the speed and accuracy of the algorithm by generating RVEs with user-defined volume distributions with up to 20,000 grains in 3D. We can achieve volume percentage errors of less than 1% in the order of minutes on a standard desktop PC. We also give examples of polydisperse microstructures with bands, clusters and size gradients, and of fitting a Laguerre diagram to 3D EBSD measurements of an IF steel.

KEYWORDS:

• Laguerre diagrams
• power diagrams
• polycrystalline materials
• grains
• foams
• volume distribution

Acknowledgments

The authors would like to thank Carola Celada-Casero for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Technical remark: It can be shown that evaluating the optimal transport (Wasserstein) distance $W_2({\chi }_{\mathrm{\Omega }},\sum _i{m}_i{\delta }_{x_i})$ between the Lebesgue measure and a discrete measure generates a partition of $\mathrm{\Omega }$ into Laguerre cells of size $m_1,\dots ,m_n$.

Additional information

Funding

D.P.B. would like to thank the Engineering and Physical Sciences Research Council (EPSRC) for financial support via the grant EP/R013527/1, EP/R013527/2 Designer Microstructure via Optimal Transport Theory. Some of the work of D.P.B. was carried out at Durham University. The work on generating 3D EBSD data has received funding from the European Union's Horizon 2020 research and innovation programme Euratom research and training programme 2014–2018 under grant agreement No 709418 MuSTMeF.