A k‐means clustering machine learning‐based multiscale method for anelastic heterogeneous structures with internal variables

Abstract

A new machine-learning based multiscale method, called k-means FE 2 , is introduced to solve general nonlinear multiscale problems with internal variables and loading history-dependent behaviors, without use of surrogate models. The macro scale problem is reduced by constructing clusters of Gauss points in a structure which are estimated to be in the same mechanical state. A k-means clustering-machine learning technique is employed to select the Gauss points based on their strain state and sets of internal variables. Then, for all Gauss points in a cluster, only one micro nonlinear problem is solved, and its response is transferred to all integration points of the cluster in terms of mechanical properties. The solution converges with respect to the number of clusters, which is weakly depends on the number of macro mesh elements. Accelerations of FE 2 calculations up to a factor 50 are observed in typical applications. Arbitrary nonlinear behaviors including internal variables can be considered at the micro level. The method is applied to heterogeneous structures with local quasi-brittle and elastoplastic behaviors and, in particular, to a nuclear waste package structure subject to internal expansions.

A multiscale FEM-MD coupling method for investigation into atomistic-scale deformation mechanisms of nanocrystalline metals under continuum-scale deformation

“…One of the solutions to this problem is to utilize k-means clustering on integration points with similar deformation patterns, leading to a reduction in computational cost. The employment of k-means clustering has already been demonstrated in some studies on the homogenization method [58][59][60]. Nevertheless, the present method can evaluate the dislocation dynamics of nanocrystalline models that contain multiple grains based on realistic macroscale deformation history.…”

Accelerating the distance-minimizing method for data-driven elasticity with adaptive hyperparameters

“…which specifies for a number of N data discrete strain states j the associated stress states σ j . In a classical boundary value problem, we seek the solution to (3) that preserves both the compatibility condition (1) and a given material law (4). By contrast, in a data-driven framework for constitutive modeling, we seek the data points in D that minimize the violation of ( 1) and (3).…”

Reduced order modeling based inexact FETI‐DP solver for lattice structures

“…The goal of these approaches is to compute a homogenized macro‐scale behavior from its underlying heterogeneous microstructure by different interchange of information between scales. Among others, let us mention the multilevel FEM (FE$$$ {}^2 $$$ 6–9 ) which computes homogenized fields from the micro‐scale and assigns them at each integration point of the macro‐scale model; the Multiscale FEM (MsFEM 10,11 ) where numerically computed basis functions encode the micro‐scale heterogeneity; global/local coupling 12 that allows to replace the homogenized global model by the fine high‐fidelity model in a certain local region; or direct numerical homogenization 13–16 where macro‐scale material parameters are identified. However, these methods mostly rely on a true separation of scales to alleviate size effects 17,18 .…”