  Concurrent multiscale finite element analysis (FE2) is a powerful approach
for high-fidelity modeling of materials for which a suitable macroscopic
constitutive model is not available. However, the extreme computational effort
associated with computing a nested micromodel at every macroscopic integration
point makes FE2 prohibitive for most practical applications. Constructing
surrogate models able to efficiently compute the microscopic constitutive
response is therefore a promising approach in enabling concurrent multiscale
modeling. This work presents a reduction framework for adaptively constructing
surrogate models based on statistical learning. The nested micromodels are
replaced by a machine learning surrogate model based on Gaussian Processes
(GP). The need for offline data collection is bypassed by training the GP
models online based on data coming from a small set of fully-solved anchor
micromodels that undergo the same strain history as their associated macro
integration points. The Bayesian formalism inherent to GP models provides a
natural tool for uncertainty estimation through which new observations or
inclusion of new anchors are triggered. The surrogate constitutive manifold is
constructed with as few micromechanical evaluations as possible by enhancing
the GP models with gradient information and the solution scheme is made robust
through a greedy data selection approach embedded within the conventional
finite element solution loop for nonlinear analysis. The sensitivity to model
parameters is studied with a tapered bar example with plasticity, while the
applicability of the model to more complex cases is demonstrated with the
elastoplastic analysis of a plate with multiple cutouts and a crack growth
example for mixed-mode bending. Significant efficiency gains are obtained
without resorting to offline training.
