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 geometrybased 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 symmetryrelated methods in the mechanical analysis of complex structures.
In recent years, mesh generation, which is an important part of most numerical analyses, has emerged as a research subject in its own right [
We propose a new method, the “Hierarchical Graph Meshing” (HGM) method, to generate meshes of complex, yet highly regular, structures. We apply this method to the construction of super carbon nanotubes (SCNTs) of arbitrary order. SCNTs are derived from carbon nanotubes (CNTs). CNTs are graphitic microtubules, first described in greater detail in [
Hierarchy of a super carbon nanotube. Each hierarchy level is composed of elements of the preceding hierarchy level. Starting with the carboncarbon bond (i), it is thus possible to obtain super carbon nanotubes (SCNTs) of arbitrary order. The image is based on data obtained by the graphbased HGM method.
Symmetry characteristics. A super carbon nanotube exhibits multiple symmetries at each hierarchy level.
Combining symmetry characteristics in a hierarchical structure.
Translational symmetry
Reflectional symmetry
Combined translational symmetry (vertically) and reflectional symmetry (horizontally)
The geometric aspect of obtaining a monolithic mesh of a super carbon nanotube of higher order is a nontrivial task, involving nonlinear transformations of the constituent elements at each hierarchy order. We would like to stress, however, that this text focuses almost exclusively on the structural aspect of the construction of super carbon nanotubes and does not offer any detailed description of their geometry.
The HGM method is applicable to all situations in which a mesh characterized by regularities such as hierarchy and symmetry either reflects the actual physical characteristics of a structure or emerges as part of the modeling process. It can be used both to build a structure based on certain regularities and to describe a given hierarchically symmetric structure. In particular, as an example of the latter case, geometrically irregular structures may be represented by meshes that exhibit certain structural regularities. Hierarchy and symmetry characteristics may also emerge as a result of design decisions. For example, frameworks of trusses are often characterized by multiple symmetries (see Figure
Regular meshes are often used for academic examples and for the analysis of structures that exhibit simple forms of symmetry. In contrast, more complex structures that, in principle, would lend themselves to a description by meshes that preserve their hierarchy and symmetry characteristics are often described by irregular meshes. As a consequence, important hierarchy and symmetry information is lost and can no longer be exploited in the subsequent analysis. In contrast, the HGM method proposed in this paper results in a tuplebased indexing system which preserves such regularities and provides a systematic and formalized way to describe and to generate meshes (see Figure
Commonly used sequential indexing and tuplebased indexing of the HGM method.
Sequential indexing: the information about the reflection symmetry cannot be retrieved from the indices
Tuplebased indexing: the leading index identifies the symmetric parts of the graphs as well as the invariant nodes. The trailing index identifies mutually symmetric nodes
Many, 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. However, such ad hoc methods lead to descriptions of the resulting configurations that are highly dependent on the specific operations chosen to generate the mesh.
In this situation, it is difficult to communicate the actual meaning of the description of the resulting structure, as one has to refer to elementary operations rather than abstract, but exactly defined, steps of the generation process. Such abstract steps include translational, reflectional, and rotational symmetries and hierarchy characteristics, including selfsimilarity. A systematic and modularized method of generating hierarchical and symmetric meshes, such as the graphtheoretic method presented here, is also a precondition for the development of computer algorithms and programs that can be used for the production of large classes of meshes, providing defined interfaces for both input and output of meshrelated information.
Seen from this perspective, one of the major advantages of the HGM method is precisely the fact that it enables generic program modules to do most of the basic work associated with the meshing process. As a result, researchers and practitioners can focus on the specific characteristics of their respective problems and task instead of coping with the often tedious work associated with elementary indexing systems and operations.
Multigrid methods are usually based on coarse meshes that are derived from the fine mesh often by heuristic methods. With the HGM method, coarse meshes, which can be based on the leading indices of the tuples identifying the nodes, can be built in parallel with the generation of the fine mesh. With regard to complex, highly regular structures, multigrid and multiscale methods can therefore benefit greatly from the application of the HGM method.
Both multiscale and multigrid methods have become important tools in the analysis of a wide range of physical phenomena, including problems in solid and fluid mechanics [
Generic multigrid methods, such as the algebraic multigrid method (AMG), can be applied to a wide range of problems and lead to algorithms generally characterized by good accuracy, robustness, and convergence. For unstructured grids, such generic methods, in combination with heuristic algorithms for coarsening and interpolation, are often the methods of choice. Coarsening methods include the RugeStüben (RS) algorithm [
Meshes may result from the discretization of a continuous configuration, or they may be essentially determined by the underlying physical model as in atomistic simulations. Regularities may be given by exact or approximate geometric periodicity, symmetry, or hierarchy of the structure itself, or they may be introduced by the meshing process as a result of the specific discretization chosen to model an otherwise irregular structure. In the latter case, irregular configurations may be discretized by regular meshes, or topological as well as approximate geometric regularities may be described by meshes that are regular from a structural, or graphtheoretic, viewpoint.
In the presence of such regularities, the HGM method proposed in this paper preserves these characteristics as an integral part of the description of the mesh, thus enabling the systematic construction of mesh hierarchies for multigrid and multiscale methods. While regularities at the finer grid levels may be most interesting in the context of the variational multiscale method [
Algebraic multigrid (AMG) methods [
Thus, HGM and AMG, as well as other multigrid algorithms, can be combined to obtain efficient, modular, and widely applicable strategies for the solution of engineering problems.
Such hierarchically symmetric graphs may exhibit more than one hierarchy level and more than one symmetry. In particular, symmetries may be of different kind, that is, reflectional as well as translational and rotational symmetries, and may have one or more invariant node sets. The modularized nature of the HGM method allows setting up the parameters for each hierarchy and/or symmetry characteristic separately. Thus, all information of a structure, such as information related to the geometry, the topology, or the connectivity, as well as the symmetries at different hierarchy levels of the structure, is being introduced in welldefined steps of the algorithm, avoiding a patchwork of ad hoc solutions that are often being applied in the generation of actual meshes for finite element analyses or other purposes.
Retaining the information related to the symmetry characteristics opens new and efficient methods for the mechanical analysis of hierarchically symmetric structures. For example, the stiffness matrix of a geometrically symmetrical configuration can be blockdiagonalized using symmetry group operations [
Algebraic graph theory is a powerful tool for the systematic generation of meshes, and the elementary steps of the construction algorithm rely on algebraic operations of directed and undirected graphs. In this paper, a particular formulation of graphs, based on directed graphs, is being used. Reference [
The graphalgebraic approach allows for a computationally highly efficient implementation of the algorithm. As complex steps can be described as a welldefined sequence of elementary operations, and operations can be arranged in a way that minimizes computational cost, code optimizations can focus on a small subset of the overall program. In addition, as a result of the identification of arcs as pairs of tuples of indices, basic operations such as intersections, unions, and compositions of graphs can be calculated in a very efficient way, especially for highly hierarchical and symmetric graphs. Generally, the exploitation of hierarchy and symmetry characteristics, if present, significantly reduces the computational cost. For carbon nanotubes (CNTs) and super carbon nanotubes (SCNTs) [
The atomicscale finite element method can be used to explore the mechanical properties of molecular structures [
Section
Section
The main results are summarized in the concluding section. Some proofs related to algebraic graph operations are given in Appendix.
The structure of super carbon nanotubes (SCNTs) can be thought of as a network of carbon nanotubes (CNTs) connected by Yshaped structural elements, that is, CNT junctions. Conceptually, a SCNT can be generated by replacing the carboncarbon bonds in a singlewalled carbon nanotube by CNTs and the carbon atoms by Yshaped junctions. A number of recent publications on SCNTs, such as [
This process can be recursively applied to generate SCNT structures of arbitrary order of hierarchy. Following [
Figure
The plane
The planes
Symmetries of a CNT junction.
Axonometric projection
Projection on the horizontal plane
Projection along an axis
In order to retain the multiple symmetries of SCNTs in mesh generation, it is helpful to abandon the traditional interpretation of SCNTs as collections of tube elements and junction elements. Instead, we interpret a SCNT as a CNT in which the carbon atoms are being replaced by Yshaped junctions that comprise both the “traditional” junction elements and half of the adjacent carbon nanotube elements. These Yshaped junctions are being connected at their ends by carboncarbon bonds, in the case of the zigzag orientation, or, alternatively, share a “ring” of carbon atoms, in the case of the armchair orientation.
Super carbon nanotubes, as well as CNTs, are graphenebased structures. A graphene sheet can be viewed as a hexagonal lattice of carbon atoms or as a trigonal lattice of hexagons that are being formed by six carbon atoms, respectively. Following the latter interpretation, a graphene sheet in the Euclidean space can be thought of as a 2dimensional surface embedded in a 3dimensional space.
Similarly, a CNT, as a graphenebased structure, can be thought of as a 2dimensional surface embedded in a 3dimensional space. Therefore, the theory of hypersurfaces, a branch of differential geometry, can be used to explore its geometric properties. The following exposition of such basic properties draws on concepts and terms originating from differential geometry.
In the case of the simple CNT, the surface, when interpreted as the surface of a cylinder, has zero Gaussian curvature and can thus be unrolled onto a plane. Therefore, a CNT may consist entirely of hexagons. The respective surface of a SCNT, however, is a surface of higher genus (i.e., it necessarily has “holes”) with a negative Gaussian curvature. This implies that this surface cannot be unrolled onto a plane and that it is not possible to construct such a surface based exclusively on hexagons.
Differential geometric observations show that the number of different types of polygons in each junction element is not arbitrary. While different combinations of nonhexagonal shapes, for example, pentagons, heptagons, or octagons, are possible, and these “defects” may be located at different positions on the junction element, the resulting structure’s Euler characteristic, defined as the sum of atoms and surface elements minus the number of bonds, must match the total curvature of its surface, divided by
The elementary operations employed in the HGM method are based on graphtheoretic concepts. Therefore, in order to lay the groundwork for the subsequent presentation of hierarchically symmetric graph and the description of the construction of super carbon nanotubes, we develop a graph algebra based on elementary unary and binary graph operations.
The following exposition, in part, draws on, and in some cases expands on, the algebraic approach to graph theory as presented in [
Graphs are an abstract concept of relations between elements of a set. The elements of such a set may be actual physical items, or they may be abstract entities, such as the nodes derived from the discretization of a structure. Hierarchies and symmetries can be defined on graphs, and such properties of a graph can be used to characterize the structure that it represents. Generally, graphbased hierarchies and symmetries are not dependent on geometric hierarchies and symmetries, although structures that exhibit regularities from a graphbased perspective often show related regularities from a geometric viewpoint.
In many practical applications, graphs are primarily interpreted as a systematic collection of data. In graph theory, the structure of graphs and of the operations that can be performed on graphs is more formalized. However, different concepts exist to describe graphs.
In textbooks on graph theory, graphs are generally introduced by defining
Various definitions and notations of graphs are being used in the literature. In order to build a more comprehensive algebraic structure, we identify
A
In order to simplify the following exposition, we introduce two basic operations for tuples, the natural projection, denoted by
For tuples of sets, we define set operations based on the representation of such tuples as disjoint unions. Thus, for tuples of sets given by
A nonempty node corridor
The
The graph
The arcs of a graph can be understood as directed connections between source nodes and target nodes; that is, an arc connects a node
We introduce two functions that can be used to identify the source nodes and the target nodes, respectively, of an arc, a set of arcs, or a node corridor. The
If
As a unary operation in
The union and the intersection of two sets of arcs,
We can thus define two binary operations in the graph space
Graph operations.
Operation  Notation  Remark 

Opposite 

See Section 
Union 

See Section 
Intersection 

See Section 
Composition 

See Section 
Conjugation 

See Section 
Categorical product 

See Section 
Binary operations.
Union
Intersection
Composition
While an arc is a directed connection between two nodes, an
We introduce a function
With regard to a nonempty node corridor
When applied to graphs, the function
In addition to the union and the intersection, we introduce the
We define the composition of two graphs of different dimensions as follows: for graphs
We denote the
The empty graph without nodes, which we will also call the
With regard to the opposite graph, we observe that
For two graphs
Composition and conjugation of undirected graphs.
The composition of two undirected graphs generally results in a graph that is not undirected
The conjugation of an undirected graph results in an undirected graph
For brevity, we generally omit the composition symbol “” and the juxtaposition of two graphs indicates the composition. Often, we assume that
If, for a node corridor
For two graphs
We introduce a
The composition of a graph
Conjugation of an undirected graph with a transfer graph
We denote a directed graph of
The categorical product allows combining graphs in a way that models the hierarchical aspects of the underlying structure. From two graphs,
The categorical product of two arcs is given by
The
Figure
The categorical product.
If a structure exhibits both symmetries and hierarchies, then these properties and their relationship should be preserved when describing such a structure as a graph. Often, different symmetry characteristics present in a given structure are hierarchically related, making the hierarchical aspect of the graphbased description the precondition for the precise description of the symmetry characteristics.
Therefore, the concept of the hierarchically symmetric graph is central to an efficient description of such structures. Hierarchically symmetric graphs of order
The super carbon nanotubes described in the preceding sections are pertinent examples of hierarchically symmetric graphs.
In graphs, symmetries can be characterized by graph automorphisms. We therefore recall the formal definition of a graph automorphism, which is based on the more general concept of a graph homomorphism.
Minors play an important role in graph theory, as they help to describe basic properties of graphs. In the following, we will combine the concept of a graph minor with the concept of a hierarchical graph, introducing the notion of a
A
Graph homomorphisms.
For the graphs
The graph
A
A graph is said to have an
We recall that a hierarchical graph of order
Hierarchically symmetric graphs.
A 3fold hierarchical symmetry, given by the automorphism
The 3fold symmetry axis (perpendicular to the plane) of the geometric representation of
A 2fold hierarchical symmetry, given by the automorphism
One of the 2fold symmetry axes (dashed line) of the geometric representation of
The construction of a natural minor can also be thought of as a natural projection of a hierarchical graph of order
A graph automorphism can be said to be induced by an automorphism of its minor, or be described as a
We say that a hierarchical graph has an
A natural minor of a hierarchical graph can be understood as its natural projection on its outermost hierarchy levels. However, hierarchical graphs may also be projected on arbitrary hierarchy levels. Let
For a projectioninduced automorphism
Hierarchically symmetric graphs can be constructed by combining multiple instances of a given graph in a way that preserves graph isomorphisms between these instances. In the simplest case, a graph is duplicated by taking the categorical product:
Symmetries of hierarchically symmetric graphs. Isolated notes are omitted in the figures.
Reflectional symmetry. The node branches of the hierarchically symmetric graph and the subgraphs of the constituent graph
Translational symmetry. The node branches of the hierarchically symmetric graph and the subgraphs of the constituent graph. Border branch sets may be distinguished into interior and boundary border branch sets
Rotational symmetry. The node branches of the hierarchically symmetric graph and the subgraphs of the constituent graph
In most cases, the interior and the border subgraphs of a constituent graph will not be natural branch sets of that graph. In order to modularize the overall construction, it is useful to add another dimension to the graph that separates the subgraphs into natural branch set. In this case, it is much easier to define transfer graphs that can be used to actually construct the hierarchically symmetric graph. Figure
Construction of a hierarchically symmetric graph. Isolated notes are omitted in the figures.
The constituent graph
The transfer graph
The symmetry graph
The hierarchically symmetric graph resulting from the operation
Note the composition of the graph
The concepts introduced in the preceding sections enable us to model the structure of carbon nanotubes (CNTs) of higher order in a systematic way. The result, a hierarchically symmetric graph, allows one to readily identify hierarchies and symmetries at all hierarchy levels, as well as similarities between the different levels of the hierarchy. In addition, by evaluating the geometryrelated functions set up in conjunction with the graph, a mesh in the traditional sense, that is, based on a onedimensional, sequentially ordered set of nodes, can be derived from the graph at low computational cost.
Meshes are generally understood as a collection of information of (a) a list of nodes together with their geometric position and (b) a list of elements (which may reduce to simple edges) together with a function that associates each element (or edge) to a list of nodes. The latter information may be provided by a connectivity list, a common approach in FE models.
A graph, however, does not contain any information about the geometry. While geometric representations of graphs are commonly used to illustrate graphs, all operations on graphs are independent of the geometric position assigned to its nodes, and graphs indeed do not necessarily represent objects that do have a reasonable geometric representation.
Therefore, if a graph created by the Hierarchical Graph Meshing method presented in this paper is to be used to investigate the mechanical properties of a structure, it must be combined with geometric information about the position of the nodes.
In the most general case, a mesh
For a
For a
The SCNTs that will be constructed in the following sections can be described by hierarchically symmetric meshes. A graphene sheet, before being “rolled” into a tube, can be described as a strongly hierarchically symmetric mesh.
In particular, the construction of each hierarchical order starts with the result of the construction of the preceding order, and the steps of the construction process of each hierarchical order are structurally identical, providing a generalized description of the construction process of the configurations of CNTs of arbitrary hierarchical order. The sequence of hierarchical orders is shown in Table
Hierarchical structures based on carbon nanotubes.
Order 0  Order 1  ⋯  

(Initial structure)  Single atom 
Junction  ⋯ 
Step 
Graphene sheet  Sheet of CNTs  ⋯ 
Step 
Tube  Tube of CNTs  ⋯ 
Step 
Junction  Junction of CNTs  ⋯ 
Sequence of the iterative construction of a SCNT of arbitrary order.
Figure
Iterative construction of a hierarchically structured SCNT of arbitrary order. For arbitrarily large orders, the table can be thought of as extending to an infinite number of columns to the right. For any order
However, as the initial structure for order
The following description will therefore generally refer to the situation for order 1 and above and provide some additional remarks concerning the special case of the construction of the CNT of order 0.
We write
The junction
(b) The second part of the first step again duplicates the result of Step 1(a) and adds arcs between the respective node sets of the junctions, identified by the leading indices
(c) Subsequently, the junctions along the circumference of the resulting CNT (or super CNT) are created by replicating the result of the previous step. Choosing this direction with regard to the geometry of the structure created in the previous step results in the zigzag configuration, while choosing the orthogonal direction would result in the armchair configuration. This operation can be expressed as
(d) The construction of a rectangular graphene (or CNT) sheet is completed by replicating the graphene (or CNT) ribbons resulting from the previous step along the length axes of the resulting CNT (or SCNT), resulting in
Construction of
(b) Again, two configurations of the preceding step, that is,
(c) The construction of a junction of order
Transfer graph
The authors have implemented directed graphs and their functions, following the definitions of the graph algebra presented in this paper, as a set of MATLAB classes. The unary and binary operations for graphs are thus available as methods of a class of graphs. The code of a number of elementary functions associated with the basic graphalgebraic method has been optimized, resulting in a computationally efficient implementation.
Some additional functions, such as the creation of symmetrical graphs, have been implemented as highlevel methods, allowing the user to focus on the characteristics of the structure rather than on identifying the relationships of indices. The methods implemented in the MATLAB classes do not presuppose any particular structure of the objects apart from those given by the graph algebra itself. In particular, no assumptions are made with regard to the sequence of elements in an array that stores the elements of a set.
The identification of each node by a tuple of indices, as opposed to a sequential numbering of the nodes, is essential for the efficient implementation of the graph algebra. In particular, the composition of graphs, which includes the identification of matching nodes in different graphs, is much faster if the comparison operation can be subdivided into comparison operations at each hierarchical level, that is, each position in the tuples identifying the nodes.
The speed and scaling characteristics of the method depend on both the size of the structure and the complexity of its hierarchy and symmetry characteristics. As the matching of nodes in the composition of graphs will be the computationally most expensive part of the process in most cases, the construction of configurations with a low relative number of nodes that form invariant subsets with regard to the graph homomorphisms underlying the graph’s symmetry characteristics will generally be faster than the construction of configurations that have a large relative number of such nodes.
Hierarchically symmetric graphs form a large class of very diverse items. In addition to differences in the number of nodes, arcs, or edges, graphs may consist of different numbers of hierarchy levels, and they may exhibit few symmetries of large subgraphs or a large number of symmetries between smaller subgraphs. A description of the different characteristics and their influence on the efficiency and scaling of the implementation is thus outside the scope of this paper. In addition, while the MATLABbased implementation has been optimized to some degree, so that it will probably not constitute the bottleneck in any workflow including mesh construction and numerical analysis, the elementary algorithms can be further optimized, and the functions can be adapted to the specific processor and memory characteristics. Programming languages such as FORTRAN or C would bring more control over elementary steps in the process, resulting in further optimization possibilities.
Super carbon nanotube structures, due to their rather complex hierarchical symmetries, can be used as a reference case to illustrate the basic characteristics of the implementation of the HGM method. We recall the basic sequence in the construction of a SCNT structure (see also Figure
Start with the zerodimensional graph.
Repeat the following steps for each hierarchy order:
Construct a rectangular graphene (supergraphene) sheet from carbon atoms (CNT junctions) and cut out a trapezoid area, resulting in the
Construct a CNT (SCNT) junction by applying symmetry operations to the trapezoid part of the graphene (supergraphene) sheet.
Construct a rectangular graphene (supergraphene) sheet from carbon atoms (CNT junctions) and roll it into a SCNT.
Sequence of the construction of a SCNT.
Of these steps, step
The following results have been obtained on Xeon E5620 CPU at 2.40 GHz, with 48 GB RAM installed on a Supermicro X8DA3 mainboard. Each data point reflects the median of the results of six runs.
The graph generated by the HGM method contains all information that is needed to describe the structure. However, in order to use the data in commonly used solvers and finite element calculation programs, the hierarchical description must be converted into a format that such programs can process. In most cases, this involves the calculation of an explicit description of the structure’s geometry, that is, the coordinates of the nodes, the linear ordering of the nodes, that is, the conversion of their index tuples to a strictly totally ordered set, usually identified with sequential numbers, and the creation of a connectivity list (in this case, up to neighbors of third degree). Figure
Sequence of the generation of a mesh with the HGM method.
Figure
Construction of a SCNT of order 2. The outer level circumference is set to six hexagons and the outer length is set to eight zigzag rings (a) or two zigzag rings (b). The inner circumference is also held constant within each series. The inner length varies between two and seven (a) and four to nine (b) zigzag rings, respectively, resulting in different numbers of arcs in the respective structures.
Inner circumference: 6 hexagons
Inner circumference: 18 hexagons
Figure
Construction of the mesh of SCNTs of order 2. See the caption of Figure
Inner circumference: 6 hexagons
Inner circumference: 18 hexagons
As illustrated in Figure
To the knowledge of the authors, there are no benchmark structures for the construction of graphs that could be used for the purpose of comparisons between algorithms and implementations. The large variety of possible structures, which would include different sizes, differences in overall complexity, extent of the hierarchy, and symmetry characteristic, makes it unlikely that a single concept or algorithm would be more efficient than others in all possible cases. The Hierarchical Graph Meshing (HGM) method presented here is capable of exploiting various forms of regularities to the extent that they exist in a given structure, while nonregular structures, or nonregular parts of a structure, may still be described by the common singlenumber index system, which would constitute a special case in the context of the HGM method.
One particular advantage of the HGM method over other approaches, such as eliminating duplicate points, edges, or surfaces after creating a symmetric mesh by replicating the mesh of a smaller part of the structure, is the fact that all relevant information with regard to the connectivity of the structure is preserved at each step of the process. This obviates the need to use special methods that reconstruct such information when processing the resulting mesh. While such methods can be optimized by specific designs, such as divideandconquer algorithms, achieving good scaling characteristics is difficult without accessible knowledge about the structure, and better scaling often involves the use of complicated algorithms and data management features that in turn slow down the computation.
Concepts related to symmetry group operations have not been incorporated into generalpurpose finite element programmes in a way that allows for an automatized processing of symmetric configurations [
In the process of the mechanical analysis of SCNT structures, the authors, prior to the development of the HGM method, have also created a program that treated the symmetry operations, as well as the operations related to the hierarchy characteristics, as separate from the actual creation of the mesh of the structure. Although a powerful neighborhood search algorithm has been employed, this program was far slower compared to the HGM method. Figure
The HGM method (circles) in comparison with a geometrybased algorithm (squares). The triangles indicate the different scaling characteristics of both algorithms. Data connected by straight lines share the same order and circumference parameter; that is, they differ only with regard to their respective length parameters.
Due to overhead cost, the underlying scaling function for the HGM algorithm is visible only for the largest tubes. For these configurations, the HGM method achieves approximately linear scaling. This characteristic is a result of the fact that the number of atoms located on the border of symmetric elements of the configuration, at the respective hierarchical level, is increasing much slower than the total number of atoms. As most of the computation is related to these border nodes, a near linear scaling is obtained for large configurations.
For the geometrybased algorithm, the situation is more complex, as each of the parameters of a SCNT, that is, the hierarchical order, the circumference, and the length, has a different impact on the neighborhood search algorithm (i.e., the part of the program that reconstructs the information that is lost in the geometrybased algorithm while being preserved in the graphbased approach) and thus on the overall scaling characteristics. With regard to the volume
Compared to a geometrybased neighborhood search algorithm, the HGM method offers a significant improvement in computational efficiency both with regard to the scaling characteristics and with regard to the absolute time spent for practically relevant computations. The linear scaling of the HGM algorithm is a significant advantage with regard to the construction of large hierarchically symmetric structures.
The Hierarchical Graph Meshing (HGM) method presented in this paper provides an efficient and accurate method for the systematic generation of meshes that represent hierarchical and/or symmetric structures. The resulting graph preserves the information about the hierarchy and symmetry characteristics of such structures. This information, in turn, can be used to apply multigrid approaches that exploit such information, for example, in mesh coarsening or domain decomposition methods.
Hierarchically symmetric graphs, due to their use of multiindexed node labels, can describe structures exhibiting hierarchy and/or symmetry characteristics in a way that is much more accessible than a linear numbering scheme, the standard approach in mesh generation. This allows identifying nodes with particular properties (e.g., nodes that are located on a symmetry axis of the structure) by their respective combinations of indices. Symmetries are reflected by welldefined graph automorphisms, and, for structures that exhibit geometric symmetries in addition to structural symmetries, such automorphisms correspond to welldefined geometric symmetry groups.
The construction algorithm is based on a concise set of elementary algebraic graph operations, which can easily be organized into a modular process. Each characteristic of a hierarchical or symmetric structure corresponds to a specific step in the algorithm. Therefore, any single characteristic may be changed by modifying the parameters or the operations contained in a specific module. The modular character of the HGM method also makes it conducive to automatization.
The HGM method has been implemented as a MATLAB code. Test runs of the implementation show that the method is computationally efficient and scales clearly better than a geometrybased method employing a neighborhood search algorithm. For super carbon nanotube structures, the code scales linearly with the problem size, that is, the number of nodes and edges contained in the structure, and the speedup reaches a factor of
The conversion from a multiindex graphbased mesh to a mesh based on a linear ordering of nodes is computationally inexpensive, allowing the integration of the HGM method with common meshbased programs and workflows.
The distributive law of the graph algebra applied in the HGM method allows for the description of graphs as the disjoint union of smaller graphs, which in turn can be written as categorical products. In this way, it is possible to obtain a standardized notation of graphs. As the factorization of the subgraphs allows for the elimination of redundant data in the description of the graph, the graph algebra employed in the HGM method can also be used to significantly reduce memory requirements. The details of this approach, however, must be left for further research at this point.
With
Let
The proposition
Let
The proposition
The proposition
The authors declare that there is no conflict of interests regarding the publication of this paper.
Financial support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG) via the Emmy Noether Program under Grant no. Wa 2516/31. This support is gratefully acknowledged.
In contrast to [
The common definition of directed graphs assumes that nodes are elements of a onedimensional node space; that is, they are not represented by tuples, and it also assumes that the set of source nodes is equal to the set of target nodes; that is,
For the respective proofs, see Appendix
In general, however, the composition of a graph and its opposite,
For the respective proofs, see Appendix
We use the symbol
The notation
In addition to the geometry, a connectivity list can be created from the graph by composing it with itself, as the composition
For translational symmetries, the isomorphism must exist between
The construction of a tube is quite similar to the construction of a base element and does not introduce any new elements to the construction method. Therefore, the elementary steps of this part of the construction are not described here.
Note that