# Meshing highly regular structures: the case of super carbon nanotubes of arbitrary order

C. Schröppel, J. Wackerfuß

Super carbon nanotubes (SCNT) can be thought of as an extension of regular carbon nanotubes (CNT), which are graphene-based three-dimensional molecular carbon structures. Whereas in CNTs, each carbon atom is bonded to three neighbor atoms, in an SCNT, the bonds are replaced by carbon nanotubes, while the atoms themselves are replaced by CNT junctions (see figure 1). The resulting structures are complex, yet highly ordered. In particular, they exhibit geometric as well as structural symmetries at each hierarchy level, and display self-similarity across levels (see figure 2).

In order to perform mechanical simulations on SCNTs, it is necessary to generate a mathematical representation of their structure. Furthermore, obtaining meaningful data with regard to the symmetry and self-similarity of the structure, the mathematical representation does not only need to provide a “mesh” of the structure at an elementary level, but needs to preserve the information related to these characteristics.

As part of the work of the MISMO research group, the “Hierarchical Graph Meshing” (HGM) method, a graph-algebraic approach to the problem has been developed, which meets the requirements described above. Due to its generality, it may be used not only for generating representations of SCNTs, but also other hierarchical and/or symmetric structures, both in atomistic and conventional finite element analysis. The main feature of the HGM approach is that nodes (or atoms, in the specific case of SCNTs) are not represented by a list of consecutive integers, but by tuples consisting of multiple integer values. On this basis, a number of well-defines basic algebraic functions is declares, which provide the foundation for the more complex steps in the construction of actual representations of structures.

The HGM method is computationally efficient and exhibits good scaling characteristics. In particular, it scales linearly for super carbon nanotube structures and is working much faster than geometry-based methods employing neighborhood search algorithms. Its modular character makes it conducive to automatization. Figure 3 shows a comparison of the computational efficiency of the HGM method versus an advanced divide-and-conquer neighborhood search algorithm, indicating that the HGM method is much faster for larger structures such as SCNTs of order 2.

The intrinsically hierarchic description of the resulting mesh greatly reduces the effort of determining mesh hierarchies for multigrid and multiscale applications and helps to exploit symmetry-related methods in the mechanical analysis of complex structures.

Publications:

Schröppel, C. and Wackerfuß, J.: “Meshing Highly Regular Structures: The Case of Super Carbon Nanotubes of Arbitrary Order”, J. of Nanomaterials, Special Issue “Hierarchically Structured Materials”. In press.

Schröppel, C. and Wackerfuß, J.: “Algebraic Graph theory and its applications for mesh generation”, Proc. in Applied Mathematics and Mechanics 12(1), 2012, p. 663–664.