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Meta-heuristics have already received considerable attention in various engineering optimization fields. As one of the most rewarding tasks, eigenvalue optimization of truss structures is concerned in this study. In the proposed problem formulation the fundamental eigenvalue is to be maximized for a constant structural weight. The optimum is searched using Particle Swarm Optimization, PSO and its variant PSOPC with Passive Congregation as a recent meta-heuristic. In order to make further improvement an additional hybrid PSO with genetic algorithm is also proposed as PSOGA with the idea of taking benefit of various movement types in the search space. A number of benchmark examples are then treated by the algorithms. Consequently, PSOGA stood superior to the others in effectiveness giving the best results while PSOPC had more efficiency and the least fit ones belonged to the Standard PSO.

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In this paper simple formulae are derived for calculating the number of spanning trees of different product graphs. The products considered in here consists of Cartesian, strong Cartesian, direct, Lexicographic and double graph. For this purpose, the Laplacian matrices of these product graphs are used. Form some of these products simple formulae are derived and whenever direct formulation was not possible, first their Laplacian matrices are transformed into single block diagonal forms and then using the concept of determinant, the calculations are performed.

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Keeping the eigenfrequencies of a structure away from the external excitation frequencies is one of the main goals in the design of vibrating structures in order to avoid risk of resonance. This paper is devoted to the topological design of freely vibrating continuum structures with the aim of maximizing the fundamental eigenfrequency. Since in the process of topology optimization some areas of domain can potentially be removed, it is quite possible to encounter the problem of localized modes. Hence, the modified Solid Isotropic Material with Penalization (SIMP) model is here used to avoid artificial modes in low density areas. As during the optimization process, the first natural frequency increases, it may become close to the second natural frequency. Due to lack of the usual differentiability of the multiple eigenfrequencies, their sensitivity are calculated by the mathematical perturbation analysis. The optimization problem is formulated by a variable bound formulation and it is solved by the Method of Moving Asymptotes (MMA). Two dimensional plane elasticity problems with different sets of boundary conditions and attachment of a concentrated nonstructural mass are considered. Numerical results show the validity and supremacy of this approach.

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The mass matrix formulation is very important to achieve a high-convergent model in structural dynamics. This study calculates the optimum mass matrix for in-plane free vibrations of the plane problems. In fact, the parameterized mass and stiffness for a rectangular element are formulated by the template approach. By using perturbation theory and sensitivity analysis, the changes of the natural frequencies are obtained as a function of the free parameter variations. Based on the natural frequencies, the objective function is established. Through an optimization process, the optimum values for template-free parameters are determined. Findings are used to calculate the plane problems’ natural frequencies. Some structural analyses and comparative studies with the other schemes are performed. Base on the obtained results, the efficiencies and high-convergence properties of the optimal element are demonstrated by numerical examples.

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In this study, the support vector machine and Monte Carlo simulation are applied to predict natural frequencies of truss structures with uncertainties. Material and geometrical properties (e.g., elasticity modulus and cross-section area) of the structure are assumed to be random variables. Thus, the effects of multiple random variables on natural frequencies are investigated. Monte Carlo simulation is used for probabilistic eigenvalue analysis of the structure. In order to reduce the computational cost of Monte Carlo simulation, a support vector machine model is trained to predict the required natural frequencies of the structure computed in the simulations. The provided examples demonstrate the computational efficiency and accuracy of the proposed method compared to the direct Monte Carlo simulation in the computation of the natural frequencies for trusses with random parameters.