Bayesian multi-scale analysis

Multi-scale structural modelling of microstructures requires upscaling of constitutive behavior from lower to higher scales. The upscaling process for linear material behavior can be addressed with a homogenization approach in which averaged stress and strain of a lower micro- or meso-scale is imposed on the higher macro-scale. However, the classical homogenization is incapable of capturing nonlinear effects, or uncertainties present in the material description. In this paper we propose a Bayesian learning method that couples the respective scales in terms of energy observables. The probabilistic setting enables nonlinear propagation of stochastic information from the lower scale material description to the macro-scale. The method is tested on the coupling of the intricate microstructure of porous trabecular bone discretized by the finite element method to the macro-scale homogeneous representative volume element. The Bayesian upscaling procedure is implemented using a Markov Chain Monte Carlo sampling algorithm and a custom likelihood function that incorporates existing finite element software.