Laguerre approximation of random foams

Abstract

Stochastic models for the microstructure of foams are valuable tools to study the relations between microstructure characteristics and macroscopic properties. Owing to the physical laws behind the formation of foams, Laguerre tessellations have turned out to be suitable models for foams. Laguerre tessellations are weighted generalizations of Voronoi tessellations, where polyhedral cells are formed through the interaction of weighted generator points. While both share the same topology, the cell curvature of foams allows only an approximation by Laguerre tessellations. This makes the model fitting a challenging task, especially when the preservation of the local topology is required. In this work, we propose an inversion-based approach to fit a Laguerre tessellation model to a foam. The idea is to find a set of generator points whose tessellation best fits the foam’s cell system. For this purpose, we transform the model fitting into a minimization problem that can be solved by gradient descent-based optimization. The proposed algorithm restores the generators of a tessellation if it is known to be Laguerre. If, as in the case of foams, no exact solution is possible, an approximative solution is obtained that maintains the local topology.

Keywords:

• Laguerre tessellation
• power diagram
• optimization
• modelling
• cellular structures
• foams

Acknowledgements

The author would like to thank A.M. Kraynik and T. Dahlheimer for supporting the implementation of the foam simulation procedures. Additional thanks goes to C. Redenbach and M. Godehardt for proof reading and valuable comments.

Notes

No potential conflict of interest was reported by the author.

This work is part of the author’s PhD thesis that was defended on 23 June 2014.

Additional information

Funding

This work was supported by the German Research Foundation (DFG) [grant number RE 3002/2-1].