Particle swarm approach for structural design optimization

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Abstract

This paper presents in detail the background and implementation of a particle swarm optimization algorithm suitable for constraint structural optimization tasks. Improvements, effect of the different setting parameters, and functionality of the algorithm are shown in the scope of classical structural optimization problems. The effectiveness of the approach is illustrated by three benchmark structural optimization tasks. Results show the ability of the proposed methodology to find better optimal solutions for structural optimization tasks than other optimization algorithms.

Introduction

Over the past decade a number of optimization algorithms have been used extensively in structural optimization task, from gradient-based algorithms using convex and continuous design spaces, to non-gradient probabilistic-based search algorithms widely applied for global and non-convex design exploration. In the latter category many of these algorithms have been developed by mimicking natural phenomena like simulated annealing [1], genetic algorithms [2], and bacterial foraging [3] among others. Recently, a family of optimization algorithms has been developed based on the simulation of social interactions among members of a specific species looking for food sources. From this family, the two most promising algorithms are ant colony optimization, which is based on the pheromone pathways used by ants to guide other ants in a colonies towards food sources [4], and particle swarm optimization or PSO, which is based on the social behaviour reflected in flock of birds, bees, and fish that adjust their physical movements to avoid predators, and to seek the best food sources [5].

The PSO algorithm was first proposed by Kennedy and Eberhart [6]. It is based on the premise that social sharing of information among members of a species offers and evolutionary advantage. A number of advantages with respect to other algorithms make PSO an ideal candidate to be used in optimization tasks. The algorithm is robust, well suited to handle non-linear, non-convex design spaces with discontinuities. It can handle continuous, discrete and integer variable types with ease. As compared to other robust design optimization methods PSO is more efficient, requiring fewer number of function evaluations, while leading to better or the same quality of results [7], [8]. In addition, it easiness of implementation makes it more attractive as it does not required specific domain knowledge information, internal transformation of variables or other manipulations to handle constraints. Furthermore, it is a population-based algorithm, so it can be efficiently parallelized to reduce the total computational effort.

Recently, the PSO has been proved useful on diverse engineering design applications such as logic circuit design [9], control design [10], [11], [12], and power systems design [13], [14], [15] among others. Applications in structures had been done in the area of structural shape optimization [16], [17], and in topology optimization [18], with promising results in such structural design applications. This paper focuses on the implementation and application of PSO for structural optimization. The general PSO methodology as well as the enhancement of the basic algorithm is introduced. Application of the algorithm to classical constraint structural optimization problems is shown. The development of the paper is as follows: Section 2 presents the general formulation of the particle swarm optimization approach. In Section 3 analytical properties of the algorithm are discussed and improvement of the basic algorithms are presented. Section 4 presents different structural optimization case studies. They are used to analyze the behaviour and sensitivity of the PSO parameters, and demonstrate the effectiveness of the approach in finding optimal structural optimization solutions. Finally, the paper closes with some concluding remarks.

Section snippets

Mathematical formulation

The particle swarm process is stochastic in nature; it makes use of a velocity vector to update the current position of each particle in the swarm. The velocity vector is updated based on the “memory” gained by each particle, conceptually resembling an autobiographical memory, as well as the knowledge gained by the swarm as a whole. Thus, the position of each particle in the swarm is updated based on the social behaviour of the swarm which adapts to its environment by returning to promising

Inertia weight update

Due to the importance of the inertia weight in controlling the global/local search behaviour of the PSO algorithm, a dynamic improvement has proven useful by forcing an initial global search with a high inertia weight (w  1) and subsequently narrowing down the algorithm exploration to feasible areas of the design space by decreasing its value towards local search values (w < 0.5). Two main approaches had been used to deal with such dynamic update. In Shi and Eberhart [19], a dynamic variation of

Numerical examples

The effectiveness of the implemented PSO algorithm on structural optimization is shown through the use of four classical truss optimization examples.

Conclusions

PSO is a population-based optimization algorithm, which mimics the social behaviour of animals in a flock. It makes use of individual and group memory to update each particle position allowing global and local search optimization. Analytical properties of the PSO were explored, showing that the general PSO formulation can be expressed in terms of a traditional optimization problem with a defined stochastic step length and search direction. Furthermore, the PSO can be represented in a

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