Keywords: structural topology design, multi-objective optimization, genetic algorithm, evolutionary algorithms.

The topology optimization problem of structures can be viewed as the process of identifying solid or void regions within an allowed domain, in order to obtain the optimal structure topology for given criterion and a given amount of material.

This problem, as stated above, is an integer programming problem (material/no-material) difficult to solve by classical programming methods. To overcome this difficulty a relaxation of the problem is possible, by introducing a material volume fraction parameter that has a continuous variation from zero to one (see e.g. Bendsoe [1]).

Alternatively, the binary chromosome design storage and global search capabilities of the Genetic Algorithm (GA) make it a powerful tool for solving topology design problem. Due to the equality constraint on volume, here considered, all the chromosomes used in the GA lead to the same volume value, hence all of them have the same number of ones (implicitly, the same number of zeros) along the evolutionary process.

For a typical chromosome defined by genes, we have a total of possible solution, in the representation scheme. Considering the volume equality constraint, a significant reduction in this number can be achieved. If a genes chromosome is considere, with genes being one (), we will only have

possible variations. The present analysis explores this fact leading to a reduction in computational effort. To proceed in this way, it is necessary to define these chromosomes and to create new operators of crossover and mutation, that preserve volume.

In contrast to single-objective optimization, where objective and fitness functions are often identical, both fitness assignment and selection must allow for several objectives when multi-objective optimization problems are considered.

Hence, instead of a single optima, multi-objective optimization problems solution is often a family of points, known as Pareto optimal set, where each objective component of any point along the Pareto-front can only be improved by degrading at least one of its other objective components. In the total absence of information regarding the preferences of objectives, a ranking scheme based upon the Pareto optimality is regarded as an appropriate approach to represent the strength of each individual in an evolutionary algorithm for multi-objective optimization (Fonseca and Fleming [2], Srinivas and Deb [3]).

Since a great diversity of solutions exists, additional information is introduced in the algorithm. This action leads to a reduction in the computational effort needed. The information supplied to the algorithm is obtained a priori from classical methods, regarding single-objective optimization, and the initial population individuals are derived from this information.

A very powerful tool the resolution of multi-objective problems is developed by considering the present alliance between genetic and classical methods. The computation model is tested in numerical applications, some results are shown in Figure 150.1.

1

M.P. Bendsoe, "Optimization of Structural Topology, Shape, and Material", Springer-Verlag, 1995.

2

C.M. Fonseca and P.J. Fleming, "Genetic algorithms for multi-objective optimization: Formulation, discussion, and generalization.", Proceedings of the fifth international conferance on Genetic Algorithms, 416-423, 1993.

3

N. Srinivas and K. Deb, "Muiltiobjective Optimization Using Nondominated Sorting in Genetic Algorithms", Evolutionary Computation, 2(3), pp.221-248, 1994. doi:10.1162/evco.1994.2.3.221

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