At the close of the 20th century, the influx of vast amounts of data across various scientific fields spurred the rapid growth of machine learning techniques. This shift has sparked a scientific debate: on one side are those who advocate for material constitutive descriptions rooted in mechanistic modeling approaches, while on the other are those exploring the potential of machine learning to identify correlations and describe constitutive relationships. In response, our group is committed to developing novel hybrid approaches that leverage both decades of advancements in mechanistic modeling and the latest breakthroughs in data-driven strategies.
A model-data-driven approach for multiscale hyperelastic descriptions
The accuracy of concurrent methods, the efficiency of staggered onesOn one hand, fully concurrent computational multiscale approaches, in which microscale mechanical properties are obtained using numerical homogenization techniques and consistently mapped to integration points in macroscale numerical simulations, often incur prohibitive computational costs. These high costs significantly limit their applicability in real-world engineering applications.
On the other hand, existing machine learning approaches for material modeling in finite-strain applications require vast amounts of data, which are often either experimentally inaccessible or numerically impractical to obtain. Moreover, surrogate models generated from data-driven approaches can encounter significant challenges when applied within a numerical finite element environment.
Our research activity addresses some challenges in the field. Firstly, we developed the first-ever framework based on local approximate Gaussian process (laGPR) regression for data-driven constitutive modeling under finite-strain, three-dimensional hyperelasticity. The results demonstrated that laGPR outperformed traditional artificial neural networks. To extend the use of laGPR in structural finite element problems, we also introduced a modified Newton-Raphson procedure specifically tailored for laGPR. Secondly, we conceived and implemented a hybrid model-data-driven approach for introducing high-fidelity constitutive information in macroscale finite element simulations under finite-strain elasticity. The rationale is to combine the response of well-known constitutive equations (model-driven component) with a correction obtained from machine learning techniques (data-driven component).
Overall, the presented model-data-driven methodology proves to be more versatile and accurate than methods based on classical model-driven, as well as pure data-driven techniques. In particular, a lower number of training samples is required and robustness is higher than for simulations which solely rely on data. Model-data-driven simulations inherit the level of information of fully concurrent schemes, such as FE2, at the computational cost of purely model-driven simulations. Accordingly, this strategy potentially makes computational multiscale approaches more widely applicable.
A model-data-driven approach for elasto-plastic problems
Efficient learning of complex yield responsesMachine learning strategies are typically rooted in a big-data mindset, where large datasets from computer simulations or full-field measurements are assumed to be readily available. As a result, these approaches cannot be directly applied to reproduce yield responses with complex behaviors—such as highly orthotropic or tension/compression asymmetries—since experimental data on the yield locus are often limited to uniaxial and biaxial tests.
To address these challenges, we proposed a hybrid model-data-driven approach for correcting phenomenological yield functions when supplementary data on the yield surface is available. The correction is performed using surrogate models built via machine learning techniques, with a focus on data-scarce scenarios. While any machine learning technique can be applied within this hybrid framework, special attention is given to ensuring the convexity of the resulting model-data-driven yield function with respect to its stress-dependent arguments. Therefore, convex extensions of three commonly used machine learning methods are investigated to enhance the phenomenological model.
