Multi-scale models of random sets

Complex microstructures often involve multi-scale heterogeneous textures, that can be modelled by random closed sets derived from Mathematical Morphology [1]. Starting from 2D or 3D images, a complete morphological characterization is performed, and used for the identification of a probabilistic model of random structure.

This presentation briefly reviews some random models and their probabilistic properties, illustrated by examples of application and by simulations. Extensions of the Boolean random closed sets model provide multi-scale models:

Cox Boolean models, long range random sets generated by Boolean varieties, iteration of Poisson varieties, sequential Cox Boolean models.

Simulations of realistic microstructures generated by these models can be introduced in a numerical solver to compute appropriate fields (electric, elastic, velocity,...) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields.

[1] Jeulin D. (2021) Morphological Models of Random Structures, Springer.