[en] When developing stochastic models or performing uncertainty quantification in the context of multi-scale models, considering direct numerical simulations at the different scales is unreachable because of the overwhelming computational cost. Surrogate models of the micro-scale boundary value problems (BVP), typically Stochastic Volume Elements (SVE), are then developed and can be constructed or trained using off-line simulations. In such a data-driven approach, different kinds of surrogate models exist including in the context of non-linear behaviours, but difficulties arise when irreversible or history-dependent responses have to be accounted for as in the context of elasto-plastic composites. In this paper we investigate three kinds of surrogate models that can handle elasto-plasticity.
Once trained using a synthetic database, neural-networks (NNWs) can substitute the micro-scale BVP resolution while reducing the computation time by more than 5 orders of magnitude. In the context of reversible behaviours or proportional loading, feed-forward NNWs can predict a homogenised response, possibly for different parametrised micro-structures. In order to introduce the history dependency, recurrent neural networks (RNNs) were shown to be efficient and accurate in approximating the history-dependent homogenised stress-strain relationships.
The limitations of NNWs are mainly two-fold. On the one hand they are unable to extrapolate responses (they can only interpolate), and on the other hand they require a large synthetic database to be trained. A physics informed alternative is the deep material network (DMN) approach which consists in a network of mechanistic building blocks. During the training process, the DMN “learns” the weight ratio and interactions of the building blocks. Once trained, the DMN is able to predict nonlinear responses, including for unseen material responses and loading conditions, in a thermodynamically consistent way, although they are less computationally efficient than the NNWs in their online stage.
A last approach is to identify the parameters of a semi-analytical mean-field-homogenization (MFH) model from the resolutions of different micro-scale BVP or SVEs: a set of MFH parameters is associated to each SVE. Since the surrogate is purely micro-mechanistic, it can handle damage-enhanced elasto-plasticity including strain-softening by considering objective quantities such as the critical energy release rate.
The different surrogates are applied in two different contexts: On the one hand the Bayesian inference of multi-scale model parameters and on the other hand, the stochastic multi-scale simulation of composite coupons.