COMPLAS 2023

A k-means Clustering FE2 Method Based on Unsupervised Machine Learning for Nonlinear Multiscale Calculations

  • Chaouch, Souhail (University Gustave Eiffel, MSME, CNRS UMR 820)
  • Yvonnet, Julien (University Gustave Eiffel, MSME, CNRS UMR 820)

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Classical multi-scale methods for the modeling of nonlinear, heterogeneous materials, with time-dependent behaviour (e.g FE2 [1]) suffer from the involvement of huge amount of computational time and memory storage. Several developments have provided ways to tackle these drawbacks, including reduced order models (ROM) and machine learning. One of the most common developped strategies is to create a surrogate model for the microscale problem during an offline stage by means of regression. However, these methods are problem-dependent and require a large number of pre-computations to be accurate. A new and different approach based on k-means clustering was introduced in [2]. This approach relies on unsupervised learning avoiding the need for an offline stage. The main idea is to reduce the number of nonlinear micro problems to be solved at each Newton iteration at the macro mesh level. In other words, for a set of macro scale Gauss points that are mechanically close to each other, only one representative nonlinear problem is solved. We propose several novelties including: (a) the use of non-constant macro stress into the clusters; (b) the use of appropriate vectors used for the classification including anelastic strains and (c) treatments for improved convergence at the macro scale. Two-dimensional applications using different behaviours and loading conditions are presented. The method is first applied to heterogeneous material with Neo-Hookean hypereleastic properties. Then, it is extended to take into account internal variables, and is tested for a problem including viscoelastic properties with small strains. The number of RVEs to be solved is reduced up to a factor of (1/60) as compared with FE2 solutions. Finally, the method is applied to a heterogenous problem with elasto-plastic micro phases to illustrate its efficiency for a cyclic loading-unloading path.