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

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.

[1]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[2]  C. Bayreuther,et al.  Mehrskalenmodelle in der Festkörpermechanik und Kopplung von Mehrgittermethoden mit Homogenisierungsverfahren , 2005 .

[3]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[4]  Robert D. Falgout,et al.  Coarse-Grid Selection for Parallel Algebraic Multigrid , 1998, IRREGULAR.

[5]  K. Koohestani,et al.  Exploitation of symmetry in graphs with applications to finite and boundary elements analysis , 2012 .

[6]  Alphose Zingoni,et al.  Group‐theoretic exploitations of symmetry in computational solid and structural mechanics , 2009 .

[7]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[8]  Bin Liu,et al.  The atomic-scale finite element method , 2004 .

[9]  Jeffrey Butler,et al.  Improving Coarsening and Interpolation for Algebraic Multigrid , 2006 .

[10]  S. Iijima.  Helical microtubules of graphitic carbon , 1991, Nature.

[11]  J. C. Wohlever.  Some computational aspects of a group theoretic finite element approach to the buckling and postbuckling analyses of plates and shells-of-revolution , 1999 .

[12]  E Weinan,et al.  Heterogeneous multiscale methods: A review , 2007 .

[13]  Christian Wagner,et al.  Multilevel ILU decomposition , 1999, Numerische Mathematik.

[14]  K. Stüben.  Algebraic multigrid (AMG): experiences and comparisons , 1983 .

[15]  Van Emden Henson,et al.  Robustness and Scalability of Algebraic Multigrid , 1999, SIAM J. Sci. Comput..

[16]  X. Qiu,et al.  A comprehensive study on the mechanical properties of super carbon nanotubes , 2008 .

[17]  Jens Wackerfuß,et al.  Molecular mechanics in the context of the finite element method , 2009 .

[18]  Krishnan Suresh,et al.  Automated symmetry exploitation in engineering analysis , 2006, Engineering with Computers.

[19]  Gebräuchliche Fertigarzneimittel,et al.  V , 1893, Therapielexikon Neurologie.

[20]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[21]  A. Brandt.  Algebraic multigrid theory: The symmetric case , 1986 .

[22]  G. Hahn,et al.  Graph homomorphisms: structure and symmetry , 1997 .

[23]  A. Kaveh,et al.  Block diagonalization of Laplacian matrices of symmetric graphs via group theory , 2007 .

[24]  Donald J. Miller.  The Categorical Product of Graphs , 1968, Canadian Journal of Mathematics.

[25]  Peter Wriggers,et al.  An Introduction to Computational Micromechanics , 2004 .

[26]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[27]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[28]  S. Pellegrino,et al.  AN INTRODUCTION TO THE ANALYSIS OF SYMMETRIC STRUCTURES , 1999 .

[29]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[30]  W. Imrich,et al.  Handbook of Product Graphs, Second Edition , 2011 .

[31]  Ali Kaveh,et al.  An efficient analysis of repetitive structures generated by graph products , 2010 .

[32]  Ado Jorio,et al.  Geometric and electronic structure of carbon nanotube networks: ‘super’-carbon nanotubes , 2006 .

[33]  Neil Genzlinger.  A. and Q , 2006 .

[34]  V. E. Henson,et al.  BoomerAMG: a parallel algebraic multigrid solver and preconditioner , 2002 .

[35]  E. Vanden-Eijnden,et al.  The Heterogeneous Multiscale Method: A Review , 2007 .

[36]  Xiong Zhang,et al.  Equivalent parameter study of the mechanical properties of super carbon nanotubes , 2007 .

[37]  Christian Miehe,et al.  Multilevel FEM for Heterogeneous Structures: From Homogenization to Multigrid Solvers , 2006 .

[38]  Ali Kaveh,et al.  Graph products for configuration processing of space structures , 2008 .

[39]  Wolfgang A. Wall,et al.  Towards a taxonomy for multiscale methods in computational mechanics: building blocks of existing methods , 2007 .