Keywords: global optimization, non-deterministic method, truss and grillage structures.

A non-deterministic method of optimization (PGSL - Probabilistic Global Search Lausanne, developed by Raphael [1] at the Swiss Federal Institute of Technology of Lausanne) has been interfaced with a finite element code, then applied and tested on several structures such as trusses and grillages. The objective of this work is to test the ability of the PGSL method to find a global optimum for the problems of optimal sizing of truss and grillage structures. The mechanical problem is concerned with the minimization of the volume or the compliance of a structure with constraints on displacements or stresses.

The optimization method uses a probability distribution function to sample a search space and select the best structure. The probability distribution function represents the probability of finding a good solution at any given point in the search domain. Each solution (structure) is evaluated by the objective function. The better solutions are more likely to be found in the neighbourhood of good solutions; hence, during the iterations, the probabilities are increased in regions containing good solutions and decreased in regions with less attractive solutions. This implies that more potential solutions are generated in regions with higher probabilities. The method contains four nested cycles. During the cycles, the research space is progressively narrowed by selecting a subdomain of smaller size centred on the best solution. This method permits a uniform and exhaustive search over the entire search space and is well suited to search a global optimum. The tests realized by Raphael on nonlinear benchmark mathematical functions indicated that PGSL performs better than genetic algorithms and simulated annealing for most problems. Furthermore, its relative performance increases as the number of variables is increased. This suggests that it is very suitable for large-scale structural optimization problems.

The purpose of this paper is to describe the use of the PGSL algorithm together with a finite element code in determining the global optimal sizing designs of two-dimensional truss and grillage structures for stress and displacement constraints. The grillage optimization problems have a highly non-convex design space and numerous relative optima, and can not be efficiently solved by the traditional methods of nonlinear programming based on the sensitivity analysis. Several optimal solutions of truss and grillage optimization problems are presented in the paper, and compared with the results given by other codes.

1

B. Raphael, I.F.C. Smith, "A direct stochastic algorithm for global search", Journal of Applied Mathematics and Computation, 146(2-3):729-758, 2003. doi:10.1016/S0096-3003(02)00629-X

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