Multi-layer perceptron-based data-driven multiscale modelling of granular materials with a novel Frobenius norm-based internal variable

One objective of developing machine learning (ML)-based material models is to integrate them with well-established numerical methods to solve boundary value problems (BVPs). In the family of ML models, recurrent neural networks (RNNs) have been extensively applied to capture history-dependent constitutive responses of granular materials, but these multiple-step-based neural networks are neither sufficiently efficient nor aligned with the standard finite element method (FEM). Single-step-based neural networks like the multi-layer perceptron (MLP) are an alternative to bypass the above issues but have to introduce some internal variables to encode complex loading histories. In this work, one novel Frobenius norm-based internal variable, together with the Fourier layer and residual architecture-enhanced MLP model, is crafted to replicate the history-dependent constitutive features of representative volume element (RVE) for granular materials. The obtained ML models are then seamlessly embedded into the FEM to solve the BVP of a biaxial compression case and a rigid strip footing case. The obtained solutions are comparable to results from the FEM-DEM multiscale modelling but achieve significantly improved efficiency. The results demonstrate the applicability of the proposed internal variable in enabling MLP to capture highly nonlinear constitutive responses of granular materials.