On-the-fly adaptive sampling strategy for data-driven computational mechanics: Applications to computational homogenisation

To overcome well-known drawbacks of classical phenomenological constitutive modelling of non-linear heterogeneous materials, a popular approach is the so-called computational homogenisation (CH), which relies on a combined description of constitutive behaviours and spatial morphology of smaller scales constituents. Their computational implementation usually encompasses two-level finite element numerical models (FE${}^2$), which quickly leads to prohibitive computational costs, even for problems with a modest number of degrees of freedom. A popular strategy in the literature to alleviate such computational burden is to use machine learning-based surrogate constitutive models, although fundamental drawbacks such as the absence of interpretability, limited extrapolation capability, and limited mathematical analysis, are still present. On the other hand, the so-called (model-free) data-driven computational mechanics (DDCM) paradigm Kirchdoerfer and Ortiz (2016) proposes the direct integration of “experimental data”, completely bypassing the need for explicit constitutive laws. The main goal of this work is to show how the DDCM approach can be used in synergy with CH to bypass fully-coupled multiscale computations. A naive approach to using multiscale constitutive behaviour along with DDCM is the offline construction of a database via CH by assuming some sampling of the strain-space. This approach has little interest since the region of the strain-space covered during a simulation is problem-dependent. During the iterations of the DDCM solver, a finite set of strain–stress pairs is used as input, while another set, comprising mechanically admissible states, is dynamically generated. Such a scenario naturally unveils the most relevant goal-oriented phase-space instances to guide a material dataset enrichment iterative procedure, bridging DDCM and CH towards computationally effective two-scale simulations. On the other hand, the performance is limited if the quality of the mechanically admissible states is low, particularly when the material dataset is not insufficiently dense. To address these challenges, we propose meaningful ranking scores alongside numerical techniques to enhance the DDCM solver in sparse data scenarios. As result, we show through meaningful numerical examples that the proposed framework results in significant computational savings if compared to standard FE${}^2$.