Graph Products and Its Applications in Mathematical Formulation of Structures
The first step in the analysis of a structure is to generate its configuration. Different means are available for this purpose. The use of graph products is an example of such tools. In this paper, the use of product graphs is extended for the formation of different types of structural models. Here weighted graphs are used as the generators and the connectivity properties of different models are expressed in terms of the properties of their generators through simple algebraic relationships. In this paper by using graph product concepts and spatial structured matrices, a new algebraic closed form is proposed for mathematical formulation and presentation of structures. For clarification some examples are included.
Data generation is the first step in the analysis of every structure. Configuration processing of large scale problems without automatic approaches can be erroneous and occasionally impossible. Formex configuration processing is one such a means introduced by Nooshin  and further developed by Nooshin et al.  and Nooshin and Disney . Similar methods are developed based on set theory by Behravesh et al. . Kaveh applied graph theory for this formation  (see also Kaveh et al. ). The use of product graphs in structural mechanics is suggested in [7, 8] and application of the corresponding concepts utilizing the directed and looped generators is due to Kaveh and Koohestani , weighted graph products by Kaveh and Nouri  and weighted triangular and circular graph products employed by Kaveh and Beheshti .
There are many other references in the field of data generation; however, most of them are prepared for specific classes of a problem. For example, many algorithms have been developed and successfully implemented on mesh or grid generation; a complete review of which may be found in a paper by Thacker  and in the books by Thompson et al. , Liseikin , and Topping et al. .
In this paper the configuration processing of regular structures is considered. A structure is called regular if it can be considered as the product of two or three subgraphs (generators) . The weighted graph products developed in  and their application are extended. Weighted paths and cycles are considered as the generators, and it is shown that many such product graphs can algebraically be expressed by simple relationships and a new algebraic closed form proposed for mathematical formulation and presentation of structures. Once this is done, then the existing methods can be applied to eigensolution and analysis of such structures [17–19]. However, this paper is limited to the generalization of product graphs for configuration processing of space structures. The methods of this paper can easily be adopted in the mesh generation of the finite element models.
2. Definitions from Graph Theory
A graph consists of a set of elements, , called nodes and a set of elements, , called members, together with a relation of incidence which associates two distinct nodes with each member, known as its ends. If weights are assigned to the members and nodes of a graph, then it becomes a weighted graph, (Figure 1). Two nodes of a graph are called adjacent if these nodes are the end nodes of a member. A member is called incident with a node if that node is an end node of the member. The degree of a node is the number of members incident with that node. A subgraph of a graph is a graph for which and , and each member of has the same ends as in . A path graph is a simple connected graph with that can be drawn in a way that all of its nodes and members lie on a single straight line. A path graph with nodes is denoted by , and a weighted path is shown by . A cycle graph is a simple connected graph with identical number of nodes and members that can be drawn so that all of its nodes and members lie on a circle. A cycle graph with nodes is shown by , and a weighted cycle is denoted by . Examples of these graphs are shown in Figure 1. For further definitions the reader may refer to Kaveh [7, 20].
3. Algebraic Representation of Path and Cycles
Most of the space structures can be viewed as the product of some weighted paths and cycles. Therefore in this section some simple mathematical relationships are presented for defining such generators.
3.1. Weighted Path
The adjacency matrix of a path in general can be expressed as where the weights are divided into 3 groups, , , and U.
Using this definition a weighted path, in general, can be expressed as
3.2. Weighted Cycle
The adjacency matrix of a weighted cycle can similarly be expressed as where the weights are also divided into 3 groups , , and U.
Considering these, a weighted cycle, in general, can be shown as
3.3. Unit and Zero Vectors
The unit vector is defined as the following.
is an by 1 vector with all entries being 1. In addition is a vector of the same dimension with all entries as 1 except the entry at th row which is zero: The zero vector is defined as the following.
is an by 1 vector with all entries being 0. In addition is a vector of the same dimension with all entries as 0 except the entry at the th row which is 1:
3.4. Extension of the Zero and Unit Vectors
In this section the zero and unit vectors are extended to represent in an efficient manner.
If we want to create a vector with some entries as 1 and the remaining also as 0, we use the following expression:
If we want to create a vector with the , , as 1 and the remaining entries as 0, we use the following expression:
For creating a vector with the , , as 1 and the remaining entries as 0, we use the following expression:
For creating a vector with the , , , and as 1 and the remaining entries as 0, we use the following expression: In general case the following relation exists between the zero and unit vectors:
As an example the weighted graphs shown in Figure 2 are expressed in algebraic form.
In compact algebraic representation the difference between a simple and a weighted graph is illustrated. As an example, for Figures 2(a) and 2(a1), which are both simple paths, a is weighted and a1 is simple, the algebraic representations are as follows: For the weighted case , and are , respectively.
4. Graph Products
In this section, weighted graph products which are introduced in  are formulated and generalized for configuration processing of structural models. These products are formulated in the algebraic form defined in Section 3. The operators used for each product are provided in Table 1.
Graph products of simple and weighted graphs are fully explained in . For weighted case the first step is the formation of the coordinates of the nodes using the nodes of the generators. As an example, for two paths these nodes are generated in Figure 3.
4.1. Weighted Cartesian Product
In this product after the formation of the nodes according to the nodes of the generators (Figure 3), a member is added between two typical nodes and , (Figure 4), if the following conditions are fulfilled.
We use the weights −1, 0, and +1 to assign to the nodes and elements in order to control the generation of the members and nodes:
As an example, Figure 5 illustrates some weighted Cartesian products and their compact representation.
4.2. Weighted Strong Cartesian Product
In this product after the formation of the nodes, according to the nodes of the generator, (Figure 3), a member is added between two typical nodes and , (Figure 4), if the following conditions are fulfilled:
Examples of strong Cartesian products of weighted graphs and their compact presentations are provided in Figure 6. As it can be observed, the compact products of paths and/or cycles are a powerful means for configuration processing and can be employed similarly to Formex configuration processing of Nooshin .
4.3. Weighted Direct Product
In this product after the formation of the nodes according to the nodes of the generator (Figure 3), a member is added between two typical nodes and , (Figure 4), if the following conditions are fulfilled:
Some examples of these weighted products and their compact representations are illustrated in Figure 7.
5. Geometrical Transformation of Graph Products
In this section using simple transformations, the weighted graph products of the previous section are employed for configuration processing of different types of space structures.
5.1. Transformation between Cartesian Coordinate System and Oblique System
In Cartesian coordinate systems (or rectangular coordinates), the “address’’ of a point is given by two real numbers indicating the positions of the perpendicular projections from the point to two fixed perpendicular lines, known as the -axis and the -axis, and we write , (Figure 8).
In this figure , , , , and . The axes divide the plane into four quadrants: is in the first quadrant, in the second, in the third, and in the fourth. is on the positive -axis.
The following generalization of Cartesian coordinates is useful for configuration processing of space structures. Consider two axes, intersecting at the origin but not necessarily perpendicularly. Let the angle between these axes be . In this system of oblique coordinates, a point is given by two real numbers indicating the positions of the projections from the point to each axis, in the direction of the other axis (Figure 9). The first axis (-axis) is generally drawn horizontally. The case = 90° yields a Cartesian coordinate system.
In this coordinate system we have , , , , and . Compare to Figure 8.
Connectivity and topological properties of a graph do not depend on its view in a coordinate system. One can present a graph with the same connectivity and different shapes in a different coordinate system.
We use Cartesian and oblique coordinate systems and the transformation between these systems for configuration processing of the space structures, as illustrated in Figure 10.
5.2. Coordinate Conditions
Additing or restricting the conditions on the domains of the weighted graph products result in different configurations. As an example, additing of the condition on and transforming the coordinate system, one can obtain the configuration shown in Figure 11.
5.3. Stretching of Nodal Point
6. Generalized Weighted Graph Products
In this section using the previously defined products, transforming the coordinate systems, moving the nodes, adding new conditions to the conditions of different graph products, and also using generalized coordinate systems, the domain of the applications of graph products in configuration processing of space structures is extended.
For configuration processing using the graph products, we extend the forms by defining the coordinate systems shown in Figure 13.
Product of adjacent axes of each coordinate system’s new weighted graph products can be produced. As an example some products of this kind are illustrated in Figure 14. The algebraic form of each configuration is shown in Table 2.
Figure 14(a) is obtained by the multiplication of axis 1 and axis 2, where the characteristics of the axes are shown in algebraic form in Table 2. Figure 14(b) is formed by multiplication of axis 1 by 2, axis 2 by 3, and axis 3 by 1. The remaining configurations of Figure 14 are obtained similarly by multiplication of each pair of adjacent axes. The properties of the axes are provided in Table 2.
The mathematical formulations of the configurations in Figure 14 are provided in Table 2. In these relationships the type of the coordinate system, the generators, the type of the products, and the imposed conditions are provided.
In this section, the generalized weighted graph products examples of different configurations are formulated. First the configuration is formed and then appropriate geometric transformations are imposed to generate the final configuration of the models.
Examples of Cartesian, strong Cartesian, and direct products are illustrated in Figure 15. For each case, the compact formulation is provided underneath the corresponding figure.
In this paper the graph products and their applications in configuration processing are extended. Topology of a structure is viewed as the product of two weighted subgraphs like paths and/or cycles as its generators. The paths and cycles are formulated in a mathematical form, and the configuration of a space structure is expressed as different products of these weighted subgraphs as one expression. In the presented method the topological information of space structures can be stored as simple algebraic relationships. More complex configurations can be formulated using different graph theory operators and new conditions can be added to the domains of the products. The application of the introduced products of weighted graphs can also be extended to the mesh generation of finite element models.
M. Nouri is grateful for the support of the Shabestar Branch, Islamic Azad University.
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