The bridging scale for two-dimensional atomistic/continumn coupling

In this paper, we present all necessary generalisations to extend the bridging scale, a finite-temperature multiple scale method which couples molecular dynamics (MD) and finite element (FE) simulations, to two dimensions. The crucial development is a numerical treatment of the boundary condition acting upon the reduced atomistic system, as such boundary conditions are analytically intractable beyond simple one-dimension systems. The approach presented in this paper offers distinct advantages compared to previous works, specifically the compact size of the resulting time history kernel, and the fact that the time history kernel can be calculated using an automated numerical procedure for arbitrary multi-dimensional lattice structures and interatomic potentials. We demonstrate the truly two-way nature of the Coupled FE and reduced MD equations of motion via two example problems, wave propagation and dynamic crack propagation. Finally, we compare both problems to benchmark full MID Simulations to validate the accuracy and efficiency of the proposed method.