Bayesian stochastic multi-scale analysis via energy considerations

Multi-scale processes governed on each scale by separate principles for evolution or equilibrium are coupled by matching the stored energy and dissipation in line with the Hill-Mandel principle. We are interested in cementitious materials, and consider here the macro- and meso-scale behaviour of such a material. The accurate representations of stored energy and dissipation are essential for the depiction of irreversible material behaviour, and here a Bayesian approach is used to match these quantities on different scales. This is a probabilistic upscaling and as such allows to capture, among other things, the loss of resolution due to scale coarsening, possible model errors, localisation effects, and the geometric and material randomness of the meso-scale constituents in the upscaling. On the coarser (macro) scale, optimal material parameters are estimated probabilistically for certain possible behaviours from the class of generalised standard material models by employing a nonlinear approximation of Bayes’s rule. To reduce the overall computational cost, a model reduction of the meso-scale simulation is achieved by combining unsupervised learning techniques based on a Bayesian copula variational inference with functional approximation forms.