Machine-learning-based asymptotic homogenisation and localisation of spatially varying multiscale configurations made of materials with nonlinear stress-strain relationships

This article is aimed to propose a general method in support of efficient and reliable predictions of both the global and local behaviours of spatially-varying multiscale configurations made of materials bearing general nonlinear history-independent stress-strain relationships. The framework is developed based on a complementary approach that integrates asymptotic analysis with machine learning. The use of asymptotic analysis is to identify the homogenised constitutive relationship and the implicit relationships that link the local quantities of interest, say, the site where the maximum Von Mises stress lies, with other onsite mean-field quantities. As for the implementation of the proposed asymptotic formulation, the aforementioned relationships of interest are represented by neural networks using training data generated following a guideline resulting from asymptotic analysis. With the trained neural networks, the desired local behaviours can be quickly accessed at a homogenised level without explicitly resolving the microstructural configurations. The efficiency and accuracy of the proposed scheme are further demonstrated with numerical examples, and it is shown that even for fairly complex multiscale configurations, the predicting error can be maintained at a satisfactory level. Implication from the present study to speed up classical computational homogenisation schemes is also discussed.