Keywords: PSO algorithm, optimal design, finite element method, engineering structures, mixed variables, constraints.

In the past decades, many optimization algorithms have been applied to solve structural design optimization problems. Among them, evolutionary algorithms (EAs) such as genetic algorithms (GAs), evolutionary programming (EP) and evolution strategies (ES) are attractive because they do not apply mathematical assumptions to the optimization problems and have better global search abilities than conventional optimization algorithms [1]. Most structural optimal design problems are hard to solve for both conventional optimization algorithms and EAs, because they involve problem-specific constraints. To handle these constrains, various approaches have been proposed. Normally, the constrained problems are solved as unconstrained. The most common approach of them uses a penalty function [2]. However the major drawback of using penalty functions is that they require additional tuning parameters. In particular, the penalty coefficients have to be fine tuned in order to balance the objective and penalty functions. Inappropriately tuned penalty coefficients will make the optimization problem intractable [3]. Another difficulty for solving structural optimization problems is that structural optimal design problems may contain mixed-variables. Recently a particle swarm optimization (PSO) has been extended to handle mixed-variable nonlinear optimization problems more effectively [4].

This paper presents an approach to integrate the finite element method (FEM) with the particle swarm optimization (PSO) algorithm to deal with structural optimization problems. The proposed methodology is concerned with two main aspects. First, the problem definition must be established, expressing an explicit relationship between design variables and objective functions as well as constraints. The second aspect is to resolve the minimization problem of structure design using the PSO technique. In this paper, an improved particle swarm optimizer is extended to solve structural design optimization problems which involve problem-specific constraints and mixed variables such as integer, binary, discrete and continuous variables. Especially the improved PSO is combined with the finite element method to deal with the constraints related to the boundary conditions of structures controlled by stresses or displacements. The proposed algorithm has been successfully applied to solve two structure design problems. The calculation results show that the proposed algorithm is able to achieve better convergence performance and higher accuracy in comparison with other conventional optimization methods used in civil engineering.

A structural design optimization problem can be formulated as a nonlinear programming problem. In contrast to generic nonlinear problems which only contain continuous or integer variables, structural design optimizations usually involve mixed variables. The binary variables are usually involved in the selection of alternative loads. The discrete variables are used to represent the standardization constraints such as the diameters of bars. The integer variables are referred as to the numbers of objects which are design variables, such as the number of bars.

This paper's objective is to offer a practical methodology for optimization of structures. It takes advantages of the FEM and PSO minimization strategy. The constraint handling technique proposed in this paper requires a feasible initial population to guarantee that the solutions of successive generations are feasible. For the structure optimization design problem, a feasible initial population can be easily obtained since their feasible search spaces are usually large and feasible particles can be easily generated by the FEM. Small sized populations are preferred to reduce the time to find a feasible initial population. The updated particles are also used by the FEM to adjust particle positions so that the stress and deformation constraints are not violated.The constraints handling is also improved to fit the structural optimization problem solving. If any variables in iteration violate the constraints, then this variable is put back to the randomly chosen previous particle position. This will accelerate the convergence speed and prevent premature convergence.

Two truss structure design problems have been investigated, through comprehensive simulation studies. The simulation results show that the application of improved PSO to structural optimization problems is feasible and effective.

1

C.A. Coello Coello, "Theoretical and Numerical Constraint-handling Techniques Used with Evolutionary Algorithms: A Survey of the State of the Art", Computer Methods in Applied Mechanics and Engineering, 191(11-12), 1245-1287, 2002. doi:10.1016/S0045-7825(01)00323-1

2

T. Abdo, R. Rackwitz, "A New Beta Point Algorithm for Large Time Invariant Reliability Problems. In: Reliability and Optimization of Structures", Proceedings of the third WG 7.5 IFIP conference, 1-11, 1990.

3

R.G. Le Riche, C. Knopf-Lenoir, R.T. Haftka, "A Segregated Genetic Algorithm for Constrained Structural Optimization", Sixth International Conference on Genetic Algorithms, University of Pittsburgh, 558-565, 1995.

4

S. He, E. Prempain, Q.H. Wu, "An Improved Particle Swarm Optimizer for Mechanical Design Optimization Problems", Engineering Optimization. 36(5), 585-605, 2004. doi:10.1080/03052150410001704854

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