Keywords: particle swarm optimization, damage identification, inverse problems, frequency response function, truss structure, nonlinear oscillator.

In this study, a new particle swarm optimization (PSO) model based damage identification procedure (PSO-dip) using frequency domain data is proposed. The formulation of the objective function, for the minimization problem, is based on the frequency response functions (FRFs) of the system. PSO-dip avoids the drawbacks of conventional damage detection methods because it does not need modal expansion or model reduction techniques (normally used in modal-based damage detection methods) and additionally, PSO-dip is not restricted to linear systems. The proposed technique was applied successfully in two examples: first, in a damaged linear 10-bar truss structure and second, in a nonlinear oscillator.

A critical issue on using a heuristic approach to solve an optimization problem is the selection of an extreme excess of parameters that controls its performance [1]. Here we propose a novel strategy for the proper choice of the PSO parameters based on the simplex method [2]. Thus, the convergence of PSO-dip becomes independent of the heuristic constants.

In order to manipulate the PSO parameters, Parsopoulus and Vrahatis [1] used the differential evolution algorithm (DEA) [3] but the DEA itself contains user defined heuristics constants and, therefore, the convergence of the combined algorithm is not totally independent of the parameter selection. To avoid this problem, we propose the use of the simplex method [2] to select the proper PSO constants. Despite all the known failures and inefficiencies of the Simplex method, we used it here to find parameters values that make the PSO independent of the initial parameter selection and to produce improvement with only few iterations of the modified PSO (m-PSO). Additionally, the use of the simplex method has one advantage: it works only with functions values (is a derivative free algorithm).

As was explained, the m-PSO algorithm is a combination of the Simplex method with the basic PSO. The issue here is to determine appropriate values for the PSO algorithm in order to control the swarm behaviour. The principal task of the simplex method is to search improved values of these parameters. In this work, we used the original version of the Nealder and Mead simplex algorithm [2]. Consequently, each point of the simplex and any reflection, contraction, expansion and shrinkage is evaluated with an independent swarm characterized by the vector of heuristic parameters, X. This fact enhances the capability search of the m-PSO. Additionally, we include, in our m-PSO, the variable N (swarm size) in order to find the best swarm size.

When a damage event occurs, the stiffness matrix will change and consequently, the complex receptance matrix will change. FRFs have a large amount of extra information since there are many points in a FRF than damage parameters in the model. Is therefore necessary to decide which points of the FRF should be used. The general rule for selecting FRF points to be used in the SADR problem is that, the chosen points should be nearby the natural frequencies. The optimization problem is formulated as determining the vector of variables aⁱ that minimizes a natural cost function representing the divergence between the measured FRF data and the analytical FRF.

In all mathematical examples presented, m-PSO never failed to find the global minimum. Additionally, it outperforms the Simulated Annealing Algorithm accuracy. As expected, the total number of functions evaluations needed to reach the minimum was larger than for the others methods. But the relevant aspect is that the m-PSO convergence does not dependent on initial or user's defined parameters. The mean values of w, C₁, C₂ and H are in accordance with the values related in the literature [4].

The modified PSO algorithm has been introduced. The approach is based on the combination of the basic PSO [5] and the simplex algorithm [2]. The principal advantages of this approach are: first, the algorithm's high reliability and second, its independence on initial estimative of heuristic parameters. The m-PSO algorithm was able to locate the global optimum with great accuracy but using a higher number of function evaluations (when compared to Simulated Annealing and EPSO).

Based on the m-PSO, the PSO-dip methodology was developed. This procedure showed high accuracy in noise-polluted scenarios and allowed the proper identification of damaged elements in the truss studied.

1

K.E. Parsopoulos, M.N. Vrahatis, "Recent approaches to global optimization problems through particle swarm optimization", Natural Computing, 1. 235-306, 2002. doi:10.1023/A:1016568309421

2

J.A. Nelder, R. Mead, "A Simplex method for function minimization", Computer Journal 7, 308-313, 1965.

3

R. Storm, K. Price, "Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces", Journal of Global Optimization, 11, 341-359, 1997. doi:10.1023/A:1008202821328

4

M.A. Abido, "Optimal design of power-systems stabilizers using particle swarm optimization", IEEE Transactions on Energy Conversion, 17(3), 406-413, 2002. doi:10.1109/TEC.2002.801992

5

J. Kennedy, R. Eberhart, "Particle swarm optimization", Proc. IEEE Int. Conf. Neural Networks, 4, 1942-1948, 1995. doi:10.1109/ICNN.1995.488968

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