A new hierarchical technique for the multiscale modeling of carbon nanostructures

We present a new hierarchical modeling technique called the Consistent Atomic-scale Finite Element (CAFÉ) method [1]. Unlike traditional approaches for linking the atomic structure to its equivalent continuum [2-7], 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. This technique partitions atoms to masters and salves and reduces the total number of degrees of freedom by establishing kinematic constraints between them [5-6]. The Tersoff-Brenner interatomic potential [8] 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 (Figure 1b). In addition, a consistent hierarchical finite element modeling technique is developed for adaptively coarsening and refining the mesh over different parts of the model (Figure 2a, 2b). The stiffness of higher-rank elements is approximated using the stiffness of lower-rank elements and kinematic constraints. 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 so, in contrast with most other multiscale methods [4-7], there is no need to introduce artificial boundaries for coupling atomistic and continuum regions. Effect of the length scale of the nanostructure is also included in the model by building the hierarchy of elements from bottom up using a finite size atom cluster as the building block (Figures 2a, 2b). In this method by introducing two independent field variables, the so-called inner displacement is taken into account (Fig. 3b). Applicability of the method is shown with several examples of deformation of carbon nanostructures such as graphene sheet, nanotube, and nanocone, subjected to different loads and boundary conditions.