Abstract
Mesh generation is an important step in many numerical methods. We present the “Hierarchical Graph Meshing” (HGM) method as a novel approach to mesh generation, based on algebraic graph theory. The HGM method can be used to systematically construct configurations exhibiting multiple hierarchies and complex symmetry characteristics. The hierarchical description of structures provided by the HGM method can be exploited to increase the efficiency of multiscale and multigrid methods. In this paper, the HGM method is employed for the systematic construction of super carbon nanotubes of arbitrary order, which present a pertinent example of structurally and geometrically complex, yet highly regular, structures. The HGM algorithm 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. For the generation of a mesh, the information about the geometry of the structure in a given configuration is added in a way that relates geometric symmetries to structural symmetries. 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.
Highlights
In recent years, mesh generation, which is an important part of most numerical analyses, has emerged as a research subject in its own right [1]
carbon nanotubes (CNTs) and tube-like connections in super carbon nanotubes (SCNTs) have been modeled as cylinders as well as structures composed of Euler and Timoshenko beams [32]
The Hierarchical Graph Meshing (HGM) method proposed in this paper provides a description of configurations such as SCNTs that help to integrate such approaches into a multiscale concept for analyzing large-scale molecular structures
Summary
Mesh generation, which is an important part of most numerical analyses, has emerged as a research subject in its own right [1]. If not all, structures that may be generated by the Hierarchical Graph Meshing (HGM) method presented here can as well be produced on an ad hoc basis, applying elementary transformations on indices representing the nodes and interdependencies of the mesh Such ad hoc methods lead to descriptions of the resulting configurations that are highly dependent on the specific operations chosen to generate the mesh. CNTs and tube-like connections in SCNTs have been modeled as cylinders as well as structures composed of Euler and Timoshenko beams [32] In this context, the HGM method proposed in this paper provides a description of configurations such as SCNTs that help to integrate such approaches into a multiscale concept for analyzing large-scale molecular structures. In SCNTs, multiple hierarchy and symmetry properties are intertwined in a complex way Such structures are excellent examples that illustrate the characteristics of the HGM method presented in this paper. Some proofs related to algebraic graph operations are given in Appendix
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