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The purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization and structural mechanics. Some important applications of graph products consist of nodal ordering and graph partitioning for structuring the structural matrices and finite element subdomaining, respectively.

In the existing methods for the eigensolution of Laplacian matrices, members have been added to the model of a graph product such that for its Laplacian matrix an algebraic relation between blocks become possible. These methods are categorized as topological approaches. Here, using concepts of linear algebra a general algebraic method is developed.

A new algebraic method is introduced for calculating the eigenvalues of Laplacian matrices in graph products.

The present method provides a simple tool for calculating the eigenvalues of the Laplacian matrices without using the configurational model and merely by using the Laplacian matrices. The developed formula for calculating the eigenvalues contains approximate terms which can be managed by the analyst.

An improvement is presented for the existing minimal cycle basis selection algorithms increasing their efficiency. This consists of reducing the number of cycles to be considered as candidates for being the elements of a minimal cycle basis and makes practical use of the Greedy algorithm feasible. A modification is also included to form suboptimal‐minimal cycle bases in place of minimal bases. An efficient algorithm is developed to form suboptimal cycle bases of graphs, in which the Greedy algorithm is applied twice. First a suboptimal minimal cycle basis is formed, and then ignoring the minimality, a basis with elements having smaller overlaps is selected.

This paper aims to present a new multi-community meta-heuristic optimization algorithm, which is called shuffled shepherd optimization algorithm (SSOA). In this algorithm.

The agents are first separated into multi-communities and the optimization process is then performed mimicking the behavior of a shepherd in nature operating on each community.

A new multi-community meta-heuristic optimization algorithm called a shuffled shepherd optimization algorithm is developed in this paper and applied to some attractive examples.

A new metaheuristic is presented and tested with some classic benchmark problems and some attractive structures are optimized.

For the solution of equations with sparse matrices, the problem of bandwidth reduction is an important issue. Though graph theoretical algorithms are available, the purpose of this paper is to examine the feasibility of ant systems (AS).

For band optimization an ant colony algorithm based on AS is utilized. In this algorithm a local search procedure is also included to improve the solution.

AS algorithms are found to be suitable for bandwidth optimization.

Application of AS to the bandwidth reduction is the main purpose of this paper, which is successfully performed. The results are compared to those of a graph theoretical bandwidth optimization algorithm.

The generality of the genetic search in the light of proper coding schemes, together with its non‐gradient‐based search, has made it popular for many discrete problems including structural optimization. However, the required computational effort increases as the cardinality of the search space and the number of design variables increase. Memetic algorithms are formal attempts to reduce such a drawback for real‐world problems incorporating some kind of problem‐specific information. This paper aims to address this issue.

In this paper both Lamarckian and Baldwinian approaches for meme evolution are implemented using the power of graph theory in topology assessment. For this purpose, the concept of load path connectivity in frame bracing layouts is introduced and utilized by the proposed graph theoretical algorithms. As an additional search refinement tool, a dynamic mutation band control is recommended. In each case, the results are studied via a set of ultimate design family rather than one pseudo optimum. The method is further tested using a number of steel frame examples and its efficiency is compared with conventional genetic search.

Here, the problem of bracing layout optimization in steel frames is studied utilizing a number of topological guidelines.

The method of this paper attempts to reduce the computational effort for optimal design of real‐world problems incorporating some kind of problem‐specific information.

In the analysis of some near‐regular structures one can solve the regular part independently and then superimpose the effect of the additional part. For such models, the matrices corresponding to regular part have canonical forms and their eigensolution or inversion can easily be performed. The effect of member changing the regular to a near‐regular structure can then be added. The purpose of this paper is to analyze near‐regular structures using the force method.

The paper uses the force method, and instead of selecting a statically determinate basic structure (standard method), the paper employs the regular part of the structure as the basic structure.

A new algebraic method is introduced for the force method of analysis for efficient analysis of large near‐regular structures.

In this paper, the force method is used, however, instead of selecting a statically determinate basic structure, the regular part of the structure is employed as the basic structure. Those additional elements are considered as redundant elements. This method is applied to truss and frame structures. In the present approach it is possible to have missing elements instead of additional elements.

Although genetic algorithm (GA) has already been extended to various types of engineering problems, tuning its parameters is still an interesting field of interest. Some recent works have addressed attempts requiring several GA runs, while more interesting approaches aim to obtain proper estimate of a tuned parameter during any run of genetic search. This paper seeks to address this issue.

In this paper, a competitive frequency‐based methodology is proposed to explore the least proper population size as a major affecting control parameter of GAs. In the tuning stage, the indirect shared memory in ant strategies is borrowed in a discrete manner to generate a dynamic colony of the most successive recent solutions to be added into each new population. An adaptive variable band mutation based on direct index coding for structural problems is also employed to increase the convergence rate as well as to prevent premature convergence especially after determining a proper population size. As an important field of engineering problems, the method is then applied to a number of structural size and layout optimization examples in order to illustrate and validate its capability in capturing the problem optimum with reduced computational effort.

It was shown that improper fixed size population can lead to premature convergence. Applying the proposed method could result in a more efficient convergence to the global optimum compared with the fixed size population methods.

A novel combination of genetic and ant colony approaches is proposed to provide a dynamic short‐term memory of the sampled representatives which can enrich the current population, avoiding unnecessary increase in its size and the corresponding computational effort in the genetic search. In addition, a dynamic band mutation is introduced and matched with such a search, to make it more efficient for structural purposes.

The purpose of this paper is to present an efficient method for dynamic analysis of structures utilizing a modal analysis with the main purpose of decreasing the computational complexity of the problem. In traditional methods, the solution of initial-value problems (IVPs) using numerical methods like finite difference method leads to step by step and time-consuming recursive solutions.

The present method is based on converting the IVP into boundary-value problems (BVPs) and utilizing the features of the latter problems in efficient solution of the former ones. Finite difference formulation of BVPs leads to matrices with repetitive tri-diagonal and block tri-diagonal patterns wherein the eigensolution and matrix inversion are obtained using graph products rules. To get advantage of these efficient solutions for IVPs like the dynamic analysis of single DOF systems, IVPs are converted to boundary-value ones using mathematical manipulations. The obtained formulation is then generalized to the multi DOF systems by utilizing modal analysis.

Applying the method to the modal analysis leads to a simple and efficient formulation. The laborious matrix inversion and eigensolution operations, of computational complexities of O(n2.373) and O(n3), respectively, are converted to a closed-form formulation with summation operations.

No limitation.

Swift analysis has become possible.

Suitability of solving IVPs and modal analysis using conversion and graph product rules is presented and applied to efficient seismic optimal analysis and preliminary design.

There are many structures that have a repetitive pattern. If a relationship can be established between a repetitive structure and a circulant structure, then the repetitive structure can be analyzed by using the properties of the corresponding circulant structure. The purpose of this paper is to develop such a transformation.

A circulant matrix has certain properties that can be used to reduce the complexity of the analysis. In this paper, repetitive and near-repetitive structures are transformed to circulant structures by adding and/or eliminating some elements of the structure. Numerical examples are provided to show the efficiency of the present method.

A transformation is established between a repetitive structure and a circulant structure, and the analysis of the repetitive structure is performed by using the properties of the corresponding circulant structure.

Repetitive and near-repetitive structures are transformed to circulant structures, and the complexity of the analysis of the former structures is reduced by analyzing the latter structures.

The purpose of this paper is to describe how the method recently developed for mass‐spring systems and frame structures is modified to include the free vibration of trusses.

Here, two methods are presented for calculating the eigenfrequencies of structures. The first approach is graph theoretical and uses graph symmetry. The graph models are decomposed into submodels and healing processes are employed such that the union of the eigenvalues of the healed submodels contain the eigenvalues of the entire model. The second method has an algebraic nature and uses special canonical forms. The present method is illustrated through three simple examples with odd and even number of bays.

The inter‐relation for the mechanical properties of elements is established using new weighted graphs, enabling easy calculation of the eigenvalues involved. Two methods are presented for calculating the eigenfrequencies of the truss structures.

Symmetry is used for easy calculation of the eigenfrequencies of structures.