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Concurrent multiscale finite element analysis (FE2) is a powerful approachfor high-fidelity modeling of materials for which a suitable macroscopicconstitutive model is not available. However, the extreme computational effortassociated with computing a nested micromodel at every macroscopic integrationpoint makes FE2 prohibitive for most practical applications. Constructingsurrogate models able to efficiently compute the microscopic constitutiveresponse is therefore a promising approach in enabling concurrent multiscalemodeling. This work presents a reduction framework for adaptively constructingsurrogate models based on statistical learning. The nested micromodels arereplaced by a machine learning surrogate model based on Gaussian Processes(GP). The need for offline data collection is bypassed by training the GPmodels online based on data coming from a small set of fully-solved anchormicromodels that undergo the same strain history as their associated macrointegration points. The Bayesian formalism inherent to GP models provides anatural tool for uncertainty estimation through which new observations orinclusion of new anchors are triggered. The surrogate constitutive manifold isconstructed with as few micromechanical evaluations as possible by enhancingthe GP models with gradient information and the solution scheme is made robustthrough a greedy data selection approach embedded within the conventionalfinite element solution loop for nonlinear analysis. The sensitivity to modelparameters is studied with a tapered bar example with plasticity, while theapplicability of the model to more complex cases is demonstrated with theelastoplastic analysis of a plate with multiple cutouts and a crack growthexample for mixed-mode bending. Significant efficiency gains are obtainedwithout resorting to offline training.

On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning