Structural optimization for post-buckling behavior using particle swarms

The aim of this paper is to develop a new algorithm based on the particle swarm optimization (PSO) concept and then to apply it in the solution of some new structural optimization problems for post-buckling behavior. Proposed modifications of the algorithm regard both the PSO kernel and the constraints handling. The "controlled reflection" technique is proposed for dealing with inequality constraints. The values of the objective are calculated for some control points chosen along a move vector. The position for which the objective is the smallest one and the constraints are not violated is selected. For the case of equality constraints, the "particle trap" strategy is proposed. First, equalities are transformed into inequalities forming constraint "zone of influence." If a particle from a swarm drops into this "zone" it remains trapped there and can move further only inside this subspace. Simultaneously, a penalty term is added to the objective function to force particles to be "captured" and constraints to become active at the optimum. The new PSO algorithm has been successfully applied to problems of structural optimization against instability. The standard maximization of the critical load is performed both for single and double buckling loads. The modified optimization for post-buckling behavior is also performed. A new problem of reconstruction of a predicted post-buckling path is formulated. The sum of squared distances between the control points of a given equilibrium path and the reconstructed one is minimized. Another new problem regards the modification of the slope of nonlinear equilibrium curve. This is obtained by adding a set of post-buckling constraints imposed on derivative values calculated for selected control points at the equilibrium curve.