Accelerating hierarchical FEM–MD simulations using clustering

In this paper, a method is proposed to reduce computational complexity in multiscale Finite Element Method (FEM) – Molecular Dynamics (MD) simulations. In FEM–MD simulations as proposed by Murashima et al. (2019), a representative unit cell (in the paper (Murashima et al., 2019) it was referred to as the r-cell) is associated to each Gauss point of the FEM mesh describing the macro model. In this FEM–MD approach, information is transferred strictly in a one-way manner from the macroscopic FEM to the microscopic MD. Since the microscopic MD results are not fed back into the macroscopic FEM calculations, atomistic-scale responses do not influence or perturb the macroscopic field. However, this independent treatment of each Gauss point induces dramatic computational costs, especially in 3D models of nano-metallic polycrystals where the r-cell contains a large number of atoms. In the present work, Gauss points are grouped into clusters using k-mean++ clustering. A single r-cell calculation is performed for each cluster, and the corresponding mechanical response is re-distributed to all Gauss points in the cluster. The proposed technique offers a simple and robust way of reducing the computational costs in such time-dependent nonlinear multiscale simulations. It has the advantage of being convergent to the FEM–MD solution with respect to the number of clusters, and does not require neither off-line calculations nor database construction. The concept is illustrated on simple structures made of nano polycrystalline metallic models, showing significant time reductions as compared to direct FEM–MD simulations.