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 [
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 [
In this paper the configuration processing of regular structures is considered. A structure is called
A
Examples of simple and weighted graphs.
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.
The adjacency matrix of a path in general can be expressed as
Using this definition a weighted path, in general, can be expressed as
The adjacency matrix of a weighted cycle can similarly be expressed as
Considering these, a weighted cycle, in general, can be shown as
The
In this section the zero and unit vectors are extended to represent
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
For creating a vector with the
For creating a vector with the
As an example the weighted graphs shown in Figure
Some weighted graphs.
In compact algebraic representation the difference between a simple and a weighted graph is illustrated. As an example, for Figures
In Figure
In this section, weighted graph products which are introduced in [
Operators of graph products.
Product  Operator 

Cartesian 

Strong Cartesian 

Direct  ◯ 
Graph products of simple and weighted graphs are fully explained in [
Nodal coordinate system of the Boolian product of
In this product after the formation of the nodes according to the nodes of the generators (Figure
Two random nodes selected from a product domain.
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
Examples of two weighted Cartesian products.
In this product after the formation of the nodes, according to the nodes of the generator, (Figure
Examples of strong Cartesian products of weighted graphs and their compact presentations are provided in Figure
Different weighted strong Cartesian products of two simple weighted graphs.
In this product after the formation of the nodes according to the nodes of the generator (Figure
Some examples of these weighted products and their compact representations are illustrated in Figure
Examples of weighted direct products: product of
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.
In Cartesian coordinate systems (or rectangular coordinates), the “address’’ of a point
Points in the Cartesian coordinate system.
In this figure
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
Points in an oblique coordinate system.
In this coordinate system we have
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
Transformation between Cartesian and oblique coordinate systems.
Additing or restricting the conditions on the domains of the weighted graph products result in different configurations. As an example, additing of the condition
Geometrical conditions and transformation between Cartesian and oblique coordinate systems applied to a weighted graph product.
Moving certain nodes in a graph model can produce different suitable configurations. Examples of such operations from [
Suitable transformations of nodal coordinates.
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
Generalized coordinate systems.
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
Compact algebraic representation of graph products presented in Figure
(a) 

(b) 

(c) 

(d) 

(e) 

(f) 

(g) 

(h) 

Weighted graph products in the shown coordinate system.
Figure
The mathematical formulations of the configurations in Figure
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
Different configurations in generalized graph products.
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.