For a large system, configuration processing is one of the most tedious and time-consuming parts of the analysis. Different methods have been proposed for configuration processing and data generation. Nooshin developed a mathematical tool, so-called Formex algebra, for configuration process. Behravesh et al. employed set theory and showed that some concepts of set algebra can be used to build up a general method for describing the interconnection patterns of structural systems, and a graph theoretical method has been developed by Kaveh. In all these methods a sub-model is expressed in algebraic forms and then functions are used to produce the entire model. The basic functions include translation, rotation, reflection, and projection, or combination of these functions.
In all the above mentioned methods neither the sub-models nor the operations to be selected are unique. A model can be generated using translation function. The same model may be obtained by pure rotation or pure reflection. One can also use combination of these operations in a hybrid form to generate a model.
Now the question is whether there is a general operation which can be used to generate any required configurations. The answer seems to be positive, and rotation may be considered as the right choice. This problem and similar question are open and will be investigated in future research.
On the other hand many structural models can be viewed as the graph products of two or three subgraphs, known as their generators. Many properties of structural models can be obtained by considering the properties of their generators. This simplifies many complicated calculations, particularly in relation with eigensolution of regular structures, as shown by Kaveh and Rahami.
In our recent research, weights are assigned to nodes and members of the generators to generate configurations which can not be formed using the four existing graph products Kaveh and Koohestani, and Kaveh and Nouri. The new products are especially suitable for the formation of the models of space structures. This type of approaches can be used in the formation of other systems, and can easily be extended to the formation of finite element models.
Rigidity of Structures
The Rigidity of structures has been studied by pioneering structural engineers such as Henneberg and Müller-Breslau. The methods they developed for examining the rigidity of skeletal structures are useful for the structures either with a small number of joints and members or possessing special connectivity properties. Rigid-jointed structures (frames), when supported in an appropriate form containing no releases, are always rigid. Therefore, only truss structures require to be studied for rigidity.
Various types of method have been employed for the study of rigidity; however, the main approaches are either algebraic or combinatorial. The first combinatorial approach to the study of rigidity is due to Laman who found the necessary and sufficient conditions for a graph to be rigid, when its member and nodes correspond to rigid rods (bars) and rotatable pin-joints of a planar truss. Also, there are some special methods for recognizing the rigidity of planar trusses.
On the other hand, rigidity of trusses can be studied in different levels. The first level is combinatorial – is the graph of joints and members (bars) correct? The second level is geometrical – is the placement of joints appropriate? The third level is mechanical – are the selected materials and methods of construction are suitable? In this paper, we focus on the most famous methods of the first level that are based on Laman’s theorem.
Laman’s conditions for the topological rigidity of planar trusses are explained by Kaveh and Ehsani. Two main types of methods are discussed for recognizing the generically independence of graphs, which use complete matching and decomposition approaches. The modified version of the Edmonds' algorithm is studied in detail and it is found to be highly efficient. Examples are included to illustrate the performance of this approach. All these methods are applicable to planar trusses, however, methods for checking the rigidity of space trusses have many problem which will be investigated in future research of the CEFFSS.
Symmetry and Regularity of Structures
In spite of considerable advances in capability of computers in recent years, efficient methods for more time saving solutions of structures are of great interest. Large problems arise in many scientific and engineering problems. While the basic mathematical ideas are independent of the size of the matrices, the numerical determination of the displacement and internal forces become more complicated as the dimensions of matrices increase and their sparsity decrease. The use of prefabrication in industrialized building construction, often results in structures with regular patterns of elements exhibiting symmetry of various types, and special methods are beneficial for efficient solution of such problems.
Well established techniques exist for the eigensolution of bilateral symmetry in the work of Kaveh and Sayarinejad. Other eigensolution methods are also available for cyclically symmetric structures in Thomas, Williams and Hasan and Hasan, Aghayere, Kaveh, and Rahami and Kaveh and Nemati, among many others. Group theoretical methods can be found in the work of Zloković, Zingoni, and Kaveh and Nikbakht. Methods based on graph products are developed by Kaveh and Rahami. The history of the developments in symmetry and application of different mathematical tools can be found in the excellent review paper of Kangwai et al..
Recently a new combined graph-group method is developed for eigensolution of special graphs by Kaveh and Nikbakht. Their study of symmetric graphs with regularity is the main objective of this study. Many structural models can be viewed as the product of simple graphs. Such models are called regular, and usually have symmetric configurations. The word regular in here is used in its literal sense and should not be taken as its graph theoretical definition, where such a graph has nodes of identical valency (degree). The developed method operates symmetry analysis of the entire structure via symmetry properties of its simple generators. The model of a structure is considered as a product graph, and Laplacian matrix, as one of the most important matrices associated with a graph, is studied. Characteristic problem of this matrix is investigated using symmetry analysis via group theory enriched by graph theory. The decomposition of Laplacian matrix of such graphs is performed in a step by step manner, based on the proposed method. This method focuses on simple paths which generate large networks, and finds the eigenvalues of the network via the analysis of the simple generators. Group theory is utilized as the main tool, improved by some concepts of graphs products. Here, as an application of the method, a benchmark problem of group theory from structural mechanics is studied. Vibration of cable nets is analyzed and the frequencies of the networks are calculated using a hybrid graph-group method. Though considerable advances are made for regular structures however, most of them deal with circulant regular structures and much is left to be done for general regular structures.
There are two general methods to optimize a function, namely, mathematical programming and meta-heuristic methods. Various mathematical programming methods such as linear programming, homogenous linear programming, integer programming, dynamic programming, and nonlinear programming have been applied for solving the optimization problems. These methods use gradient information to search the solution space near an initial starting point. In general, gradient-based methods converge faster and can obtain solutions with higher accuracy compared to stochastic approaches in fulfilling the local search task. However, for effective implementation of these methods, the variables and cost function of the generators need to be continuous. Furthermore, a good starting point is vital for these methods to be executed successfully. In many optimization problems, prohibited zones, side limits and non-smooth or non-convex cost functions need to be considered. As a result, these non-convex optimization problems cannot be solved by the traditional mathematical programming methods. Although, dynamic programming or mixed integer nonlinear programming, and their modifications offer some facility in solving non-convex problems, these methods, in general, require considerable computational effort.
As an alternative to the conventional mathematical approaches, the meta-heuristic optimization techniques have been used to obtain global or near global optimum solutions. Due to their capability of exploring and finding promising regions in the search space at an affordable time, these methods are quite suitable for global searches, and furthermore alleviate the need for continuous cost functions and variables used for the mathematical optimization methods. Though these are approximate methods, i.e. their solution are good, but not necessarily optimal, they do not require the derivatives of the objective function and constraints and employ probabilistic transition rules instead of deterministic ones.
Nature has always been a major source of inspiration to engineers and natural philosophers and many meta-heuristic approaches are inspired by solutions that nature herself seems to have chosen for hard problems. The Evolutionary Algorithm (EA) proposed by Fogel et al., De Jong and Koza, and the Genetic Algorithm (GA) proposed by Holland and Goldberg are inspired from the biological evolutionary process. Studies on animal behavior led to the method of Tabu Search presented by Glover. Ant Colony Optimization proposed by Dorigo et al. and Particle Swarm Optimizer formulated by Eberhart and Kennedy . Also, Simulated Annealing proposed by Kirkpatrick et al., Big Bang–Big Crunch algorithm proposed by Erol and Eksin and improved by Kaveh and Talatahari , Gravitational Search Algorithm presented by Rashedi et al. are introduced using physical phenomena.
Recently a new optimization algorithm based on principles from physics and mechanics, which will be called Charged System Search (CSS) by Kaveh and Talatahari. They utilize the governing Coulomb law from physics and the governing motion from Newtonian mechanics. Applications of this method are currently performed and extended to multi-objective optimization. Further research is needed for increasing the capability of meta-heuristic algorithms.
Optimal Analysis of Structures
Recent advances in structural technology require greater accuracy, efficiency and speed in the analysis of structural systems, referred to as Optimal Structural Analysis. It is therefore not surprising that new methods have been developed for the analysis of the structures with complex configurations.
The requirement of accuracy in analysis has been brought about by need for demonstrating structural safety. Consequently, accurate methods of analysis had to be developed since conventional methods, although perfectly satisfactory, when used on simple structures, have been found inadequate when applied to complex and large-scale structures. Another reason why greater accuracy is required results from the need to achieve efficient and optimal use of the material, i.e. optimal design.
The methods of analysis that meet the requirements mentioned above, employ matrix algebra and graph theory, which are ideally suited for modern computational mechanics. Although this paper deals primarily with analysis of structural engineering systems, it should be recognized that these methods are also applicable to other types of structures. The concepts presented in here are not only applicable to skeletal structures, but can equally be used for the analysis of other systems such as hydraulic and electrical networks. These concepts can easily be extended to finite element methods.
Analysis of systems and in particular structures can be decomposed into three phases:
1. Approximation, followed by choosing an appropriate model.
2. Specifying topological properties followed by a topological analysis.
3. Assigning algebraic variables, followed by an algebraic analysis.
Such a decomposition results in a considerable simplification in the analysis and leads to a clear understanding of the problems involved in studying the structural behaviour.
For the optimal analysis of structures, three conditions need to be fulfilled, Kaveh. The structural matrices (stiffness or flexibility) should be sparse, properly structured (e.g. banded) and well-conditioned.
Pattern equivalence of structural matrices and matrices associated with graph theory simplifies structural problems and allows advances made in this field to be transferred to structural mechanics. As an example, for rigid-jointed frames the sparsity of flexibility matrices can be provided by construction of sparse cycle adjacency matrices. Similarly, using sparse cut set bases, the formation of sparse stiffness matrices become feasible. Proper structuring of the flexibility and stiffness matrices of a structure can also be accomplished by structuring the pattern of cycle and cut set adjacency matrices of its model, respectively.
Kaveh and students have presented methods involved in optimal structural analysis. Topological transformations are used to achieve some of the goals of the optimal analysis. For some problems embedding on higher or lower dimensional spaces is needed. For some other it is beneficial to associate a new graph to the original model more simple connectivity properties. These transformations are further developed and a book is under preparation containing many such transformations.
Topological properties of structures
In the analysis of skeletal structures, three different properties are encountered, which can be classified as topological, geometrical and material. Separate study of these properties results in a considerable simplification in the analysis and leads to a clear understanding of the structural behavior. The topological properties of skeletal structures, are important since many structural properties can be investigated using these properties. Many such transformations are developed by Kaveh and applied to may problems in structural mechanics. These studies are recently continued and made the finite element analysis by force method feasible. Further study is under investigation.
The Force Method of Structural Analysis
The force method of structural analysis, in which the member forces are used as unknowns, is appealing to engineers, since the properties of members of a structure most often depend on the member forces rather than joint displacements. This method was used extensively until 1960. After this, the advent of the digital computer and the amenability of the displacement method for computation attracted most researchers. As a result, the force method and some of the advantages it offers in non-linear analysis and optimization has been neglected.
Five different approaches are adopted for the force method of structural analysis, which will be classified as:
1. Topological force methods, 2. Graph theoretical force method
3. Algebraic force methods, 4. Mixed algebraic-combinatorial force methods,
5. Integrated force method.
Topological methods have been developed by Henderson and Maunder for rigid-jointed skeletal structures using manual selection of the cycle bases of their graph models. Methods suitable for computer programming using graph theory are due to Kaveh. These methods are generalized to cover all types of skeletal structures, such as rigid-jointed frames, pin-jointed planar trusses and ball-jointed space trusses.
Algebraic methods have been developed by Denke, Robinson, Topçu, Kaneko et al., Soyer and Topçu and mixed algebraic-topological methods have been used by Gilbert et al., Coleman and Pothen, and Pothen.
The integrated force method has been developed by Patnaik, in which the equilibrium equations and the compatibility conditions are satisfied simultaneously in terms of the force variables.
Methods using graph-theoretical concepts are applied to finite element analysis by Kaveh and Koohestani and further developments are under intensive research at the centre.
Matroids for Structural Mechanics
The theory of matroids, which was introduced by Whitney in his pioneering paper as early as 1935, is concerned with the abstract properties of independence. He conceived a "matroid" as an abstract generalization of a matrix; hence some of the language, including the name of the theory, is based on that of linear algebra. At the same time he refers to matroids as generalized graphs, and uses some terms from graph theory.
Matroids have received a great deal of attention from both the theoretical and the application points of view. Contributions have been made to its extension by Tutte, Rado, Welsh, Mirsky, Edmonds and many others; an excellent introduction is Welsh. Matroids have been applied to various fields of engineering such as electrical networks by Minty, structural analysis by Kaveh and rigidity of structures by Crapo, Recski, and Whiteley among many other fields of science and engineering. Our interest in matroids has been motivated by the active presence of both matrices and graphs in the matrix analysis of structures, and the need for generalizing some of the existing concepts of graph theory.
Further applications of the theory of matroids to other problems in structural mechanics is under intensive study at our centre.