Introduction to multi-scale probabilistic models and simulations of random structures
Résumé
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 by image analysis, and used for the identification of a probabilistic model of random structure. This approach gives access to the synthesis of random textures mimicking as closely as possible real images, and to virtual textures useful to explore a wide range of microstructures. 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, iterated 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. Combining multi-scale models with fracture criteria enables us to provide probabilistic models of fracture. Examples are given about crack initiation and growth in fatigue of materials. New numerical approaches for fatigue are based on the implementation of phase field models accounting for the degradation of toughness during cracks propagation. Finally, simulations of probabilistic models can feed machine learning or deep learning tools like CNN (Convolutional Neural Networks) for further tasks like texture recognition or prediction of physical properties of random media. [1] Jeulin D. (2021) Morphological Models of Random Structures, Springer.
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