Consistent atomic-scale finite element: A unified framework for the multiscale modeling of carbon nanostructures

A new multiscale modeling technique called the Consistent Atomic Finite Element (CAFÉ) is introduced. Unlike traditional approaches for linking the atomic structure to its equivalent continuum, this method directly connects the atomic degrees of freedom to a reduced set of finite element degrees of freedom without passing through an intermediate homogenized continuum. As a result, there is no need to introduce stress and strain measures at the atomic level. The Tersoff-Brenner interatomic potential is used to calculate the consistent tangent stiffness matrix of the structure. In this finite element formulation, all local and non-local interactions between carbon atoms are taken into account using overlapping finite elements. In addition, a consistent hierarchical finite element modeling technique is developed for adaptively coarsening and refining the mesh over different parts of the model. This process is consistent with the underlying atomic structure and, by refining the mesh, molecular dynamic results will be recovered. This method is valid across the scales and can be used to concurrently model atomistic and continuum phenomena. Applicability of the method is shown with several examples of deformation of a graphite sheet under different loadings and boundary conditions.