Keywords: dynamic, composites, reliability, optimisation, genetic algorithm, hierarchical.

Uncertainties of physical properties lead to a probabilistic failure analysis of the composite materials. This feature explains the great interest in the reliability analysis of composite structures [1]. On other hand, the reliability based optimal design under dynamic response is an emerging research area due to the difficulties associated with coupling optimisation and reliability analysis. In the present work it is intended to develop a simple model of optimisation based on structural reliability maximisation of composite structures under dynamic loading conditions.

A very important problem in structural reliability analysis of composites is the existence of multiple Most Probable failure Points (MPPs) of the limit state functions. Multiple MPPs are similar to the local minima in structural optimisation. Many problems in structural optimisation are stopped once a local minimum is reached. This is an unacceptable procedure in reliability analysis since the local MPP may not represent the worst failure and the actual failure may occur below the predicted level. Only the global MPP represents the actual structural reliability [2].

In order to obtain every reliability index it is necessary to implement the global MPP search [3,4]. In the proposed approach this represents the inner optimisation problem. The optimal design aiming the maximisation of structural reliability is defined as the external problem. Thus a Hierarchical Genetic Algorithm is proposed with an appropriate topology where two kinds of populations are identified: a master population and a net of small size sub-populations. The master population performs the evolutionary process associated with the external optimisation problem while the sub-populations evolve linked to the MPP search problems. A Micro Genetic Algorithm is used to solve these inner problems associated with MPP search. Sampling space reduction variance techniques together with weak convergence conditions are used to accelerate the MPP search in the proposed approach.

In the implemented genetic code each chromosome has two segments activated alternatively when the evolutionary process passes from the master population to the slave sub-populations. In both cases a binary code format is used to manipulate the exchange data.

The Micro Genetic Algorithm and the Genetic Algorithm applied to the master population are based on four principal operators: selection, crossover, implicit mutation and replacement of similar individuals. These operators are supported by an elitist strategy that always preserves a core of best individuals of the population that is transferred into the next generations.

The offspring group formed by the crossover operator will take part of the population of the next generation. To avoid the rising of local minima a chromosome set which genes are generated randomly is introduced into the population. This operation is called mutation and is quite different from classic techniques where a reduced number of genes is changed. The mutation operator guarantees the diversity of the population in each generation [4]. To control the genetic diversity, a scheme that detects individuals belonging to the same neighbourhood has been implemented. The analysis is made from the genetic point of view. The best individual is kept in the population and others generated randomly replace the similar ones.

An example using a displacement limit state function is presented and the results for the reliability analysis and the optimisation procedure show the robustness of the proposed approach. The genetic search history for the solution of the optimisation problem shows that the solution is matured after a few generations what is considered a good trial considering the number of design and random variables intervening in the global search. The worst fitted individual of the elite group continues to be improved and at the end of the optimisation procedure is close to the best one. The efficiency of the presented evolutionary process is measured by the success rate of individuals coming from crossover or mutation, actually enter the elite group and the worst individuals of this group are eliminated.

1

C.A. Conceição António, A.T. Marques, and J.F. Gonçalves, "Reliability based design with a degradation model of laminated composite structures", Structural Optimization, 12, 16-28, 1996. doi:10.1007/BF01270440

2

L. Wang and R.V. Grandhi, "Intervening variables and constraint approximations in safety index and failure probability calculations", Structural Optimization, 10, 2-8, 1995. doi:10.1007/BF01743688

3

C.A. Conceição António, "An approach to reliability based design of composite structures under dynamic response", ECCOMAS 2000 - European Congress on Computational Methods in Applied Sciences and Engineering, incorporating the VI International Conference on Computational Plasticity (COMPLAS VI), 11-14 Setembro, Barcelona, Spain, 2000.

4

C.A. Conceição António, "A hierarchical genetic algorithm for reliability based design of geometrically non-linear composite structures", Composite Structures, Vol. 54, pp. 37-47, 2001. doi:10.1016/S0263-8223(01)00068-X

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