The continuum problem, with its applied loads and boundary conditions, is usually discretized by the finite element method (FEM). The topology optimization will consist then on determining which elements of the original FE mesh will be present in the structure and which will be removed, originating voids. The process of removal or insertion of elements can lead to infeasible structures that are physically unacceptable and moreover may delay the GA convergence. This topic will be addressed below.

In [1] a GA for multi-objective topology optimization of elastic structures was developed. The genotype employed was a characteristic vector of "ones" and "zeros" to indicate the presence or absence of material on a FE mesh. As an equality constraint on volume has to be enforced, all chromosomes generated individuals with the same volume value, i.e., the same number of "ones". This means that to have feasible individuals the number of "ones" and, implicitly, the number of "zeros", has to be kept the same for all individuals along the evolutionary process. It was thus necessary to: (i) define chromosomes satisfying this propriety; (ii) create the corresponding crossover and mutation operators to preserve this number. However, as these operators do not guarantee individuals representing connected structures and the selection pressure proved to be insufficient for the elimination of these infeasible individuals, in the course of the evolutionary process, with the subsequent waste of computer time, a repair mechanism was created to increase the relative number of feasible individuals in the population [2].

In this paper a different strategy is adopted by developing a representation method based on trees to generate initial feasible individuals that remain feasible upon crossover and mutation and as such no repairing is required.

The technique of tree encodings has received some attention in the GA literature: RK (random key encoding) seems to have been first introduced for encoding order and scheduling problems; typical applications of RKs in optimization of constrained facility layout problems can also be seen; LNB (link and node biased encoding), where the graph vertices where assigned random weights and a minimum spanning tree was constructed; NetKey (network random key encoding); and determinant encoding.

As an application example we study the topology optimization of structures where the objective function is the maximization of the first and the second eigenfrequencies of a plate. All cases have a prescribed material volume constraint.

1

Aguilar Madeira J.F., Rodrigues H.C., Pina H.L., "Multi-objective optimization of structures topology by genetic algorithms", Advances in Engineering Software, 36, Issue 1, pp. 21-28, 2005. doi:10.1016/j.advengsoft.2003.07.001

2

Aguilar Madeira J.F., Rodrigues H.C., Pina H.L., "Multiobjective topology optimization of structures using genetic algorithms with chromosome repairing", Structural and Multidisciplinary Optimization, 32, Number 1, pp. 31-39, 2006. doi:10.1007/s00158-006-0007-0

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