Data-Driven nonlocal mechanics: Discovering the internal length scales of materials

Manifold embedding data-driven mechanics

“…The distance-minimization paradigm has also been extended for different applications such as elastodynamics (Kirchdoerfer and Ortiz, 2018), finite-strain elasticity (Nguyen and Keip, 2018), plasticity (Eggersmann et al, 2019), fracture mechanics (Carrara et al, 2020), geometrically exact beam theory (Gebhardt et al, 2020), poroelasticity (Bahmani and Sun, 2021a), and micro-polar continuum (Karapiperis et al, 2021). Its scalability issue with respect to the data size is addressed in Bahmani and Sun (2021a) where an isometric projection is introduced to efficiently organize the database via the k-d tree data structure that provides a logarithmic time complexity (in average) for nearest neighbor search.…”

“…On-the-fly multiscale calculations, such as hierarchical upscaling methods (e.g., FEM 2 ) Matouš et al, 2017) or concurrent multiscale domain coupling methods (Hughes et al, 1998;Sun and Mota, 2014;Sun et al, 2017;Badia et al, 2008) may eliminate the need to compose constitutive laws that captures the responses of the effective medium through computational homogenization. However, the computational cost of these multiscale techniques remains a major technical barrier despite years of research progress (Wang and Sun, 2018;Karapiperis et al, 2021).…”

“…In data-driven mechanics research, such as Ortiz (2016, 2018), the independent components of the input-output pairs (e.g., the strain-stress pair) of the constitutive laws are often used to constitute a product space, and the measure of distance relies on the equipped weighted energy norms. This idea is utilized not only for elasticity problems but also for multiphysics (e.g., Bahmani and Sun (2021a) and higher-order continua (e.g., Karapiperis et al (2021)). Noticeably, Leygue et al (2018) compare simulation results obtained from different choices of distance measure and demonstrate a dependence between the energy-minimized solutions and the choice of the measure (see Fig.…”

Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties

“…In contrast, the model-free data-driven approach (Kirchdoerfer and Ortiz, 2016;Ibañez et al, 2017;Kirchdoerfer and Ortiz, 2017;Conti et al, 2018;Nguyen and Keip, 2018;Eggersmann et al, 2019;Carrara et al, 2020;Karapiperis et al, 2021) bypasses constitutive relations by mapping a material point's deformation to an appropriate stress state (subject to compatibility constraints) directly from a large dataset of stress-strain pairs. Recent approaches (Ibáñez et al, 2019;González et al, 2019) also proposed adding data-driven corrections to the existing constitutive models.…”

NN-EUCLID: deep-learning hyperelasticity without stress data

“…Modern data-driven methods aim to take a further step to overcome the limited expressive power of classical material models. In the (material) model-free paradigm proposed by Kirchdoerfer and Ortiz (2016) and first formulated for elasticity, a material's state of deformation is directly mapped to a stress state closest to the stress-strain pair available in the dataset (subject to physical compatibility constraints) (Kirchdoerfer and Ortiz, 2016;Ibañez et al, 2017;Kirchdoerfer and Ortiz, 2017;Conti et al, 2018;Nguyen and Keip, 2018;Eggersmann et al, 2019;Carrara et al, 2020;Karapiperis et al, 2021). Alternative approaches keep the concept of a material model and surrogate it by learning an approximate mapping between the strains and stresses (referring again to the easiest case of elasticity) using, e.g., sparse regression with feature engineering (Flaschel et al, 2021(Flaschel et al, , 2022Joshi et al, 2022;Wang et al, 2021), manifold learning methods and polynomial approximations (Ibañez et al, 2018(Ibañez et al, , 2017González et al, 2019b), Gaussian process regression (Rocha et al, 2021;Fuhg et al, 2022), and artificial neural networks (NNs) (Ghaboussi et al, 1991;Fernández et al, 2021;Klein et al, 2022;Vlassis and Sun, 2021;Kumar et al, 2020;Bastek et al, 2022;Zheng et al, 2021;Mozaffar et al, 2019;Bonatti and Mohr, 2021;Vlassis et al, 2020;Kumar and Kochmann, 2021;As'ad et al, 2022;Liang et al, 2022), with the list being roughly in order of decreasing physical interpretability and increasing approximation power.…”