Energetic upscaling strategy for grain growth. II: Probabilistic macroscopic model identified by Bayesian techniques

This paper is the second part of an energetic upscaling strategy to simulate grain growth at the macroscopic scale with state variables that contain statistical descriptors of the grain structure. The first part was dedicated to the derivation of a fast mesoscopic model of grain growth based on orientated tessellation updating method, which consists in a succession of Voronoi-Laguerre tessellations obtained by establishing an evolution law directly on the parameters defining the tessellations. In this contribution, the final step of the upscaling strategy is detailed by deriving macroscopic evolutions laws of the state variables representing statistical distributions of the grain structure. The approach relies on macroscopic free energy and dissipation potentials that are identified not axiomatically, but using a large database of mesoscopic computations. The macroscopic energy is found to be purely deterministic, although the dissipation necessitates to introduce a probabilistic framework. Indeed, an epistemic uncertainty arises due to the loss of information in the reduction of the amount of data between the detailed mesoscopic state and the statistical macroscopic state (i.e., several mesoscopic states can share the same macroscopic state). Classical Bayesian inference has been used to identify the probability density functions associated to the epistemic uncertainty. From the computational point of view, this work can be used to simulate grain growth at very large scale with short computation time, while processing rich statistical information about the grain structure, such as statistical indicators of the grain boundary character distribution. The resulting stochastic macroscopic model has been compared to several particular mesoscopic evolutions, and good agreement is observed. However, the macroscopic model still requires an experimental validation.