Aim of the article is to suggest technology for optimization of pile positions in a grillage-type foundations seeking for the minimum possible pile quantity. The objective function to be minimized is the largest reactive force that arises in any pile under the action of statical loading. When piles of the grillage have different characteristics, the alternative form of objective function may be employed: the largest difference between vertical reaction and allowable reaction at any pile. Several different allowable reactions with a given number of such piles may be intended for a grillage. The design parameters for the problem are positions of the piles. The feasible space of design parameters is determined by two constraints. First, during the optimization process piles can move only along the connecting beams. Therefore, the two-dimensional grillage is “unfolded” to a one-dimensional construct, and the supports are allowed to range through this space freely. Second, the minimum allowable distance between two adjacent piles is introduced due to the specific capacities of pile driver.The initial data for the problem are the following: the geometrical scheme of the grillage, the cross-section and material data of connecting beams, minimum possible distance between adjacent supports, characteristics of piles, and the loading data given in the form of concentrated loads or trapezoidal distributed loadings. The results of solution are the required number of piles and their positions.The entire optimization problem is solved in two steps. First, the grillage is transformed to a one-dimensional construct, and the optimizer decides about a routine solution (i.e. the positions of piles in this construct). Second, the backward transformation returns the pile positions into the two-dimensional grillage, and the “black-box” finite element program returns the corresponding objective function value. On the basis of this value the optimizer predicts the new positions of piles, etc. The finite element program idealizes the connecting beams as the beam elements and the piles – as the finite element mesh nodes with a given boundary conditions in form of vertical and rotational stiffnesses. The optimizing program is an elitist genetic algorithm or a random local search algorithm. At the beginning of problem solution the genetic algorithm is employed. In the optimization problems under consideration, the genetic algorithms usually demonstrate very fast convergence at the beginning of solution and slow non-monotonic convergence to a certain local solution point after some number of generations. When the further solution with a genetic algorithm refuses to improve the achieved answer, i.e. a certain local solution is obtained; the specific random search algorithm is used. The moment, at which the transition from genetic algorithm to the local search is optimal, is sought in the paper analyzing the experimental data. Thus, the hybrid genetic algorithm that combines the genetic algorithm itself and the local search is suggested for the optimization of grillages.
A Two-Dimensional Genetic Algorithm and Its Application to Aircraft Scheduling Problem
