Keywords: structural optimization, robust design, multi-objective optimization, uncertainty, frames, genetic algorithms, Monte-Carlo simulation.

The objective of robust design is to optimize simultaneously the mean and typical deviation of the performance of the system, which is subjected to uncertain variables or parameters (noise factors), minimizing the disadvantages introduced by these imprecise values. Optimum structural bar designs are mainly conditioned by the requirements of the multiple types of necessary constraints: stresses, displacements and buckling related, being often the accurate fitting of these constraints that limits the optimum design. An additional consideration is the sensibility of these obtained optimum designs to load variations. An optimum design with a highly accurate accomplishment of constraints could violate them easily with a minimum increase in some loads. However, a design with more constraint margin (robustness) will not suffer this consequence. In general, a load variation distribution in a particular structure produces a constraint violation distribution. So, it is desirable the minimization of the obtained distribution (in terms of mean and variance). Therefore, it is reasonable to strive after a null constraint violation, even under this possible load variation, simultaneously with the minimization of structural weight.

In this work Monte-Carlo simulations are performed assuming normal distributions in the external loads (considering simulations per structural design in order to construct its constraint violation distribution, being the number of different load distributions) and a multicriteria optimization with evolutionary algorithms [1] for structural robust design with simultaneous minimization of the constrained mass of the frame structure and the variance of the constraints violation distribution (including stresses, buckling and displacement constraints), both conflicting objectives and therefore suitable for a multicriteria optimization.

The variation condition in loads is in real structures frequent, and it is considered in the design codes. The variation of the load actions which act over a structure from the viewpoint of the probabilistic or semi-probabilistic safety criteria, is associated with considering the loads as stochastic variables and to the existence of some limit ultimate states that guide to the total or partial failure of the structure and limit service states that when achieved result in its malfunctioning. In order to define the actions, it is assumed that their variation follows a Gaussian probability density function. The characteristic value of an action is defined as the value that belongs to the 95% percentile, that is, a probability of 0.05 to be surpassed.

It has been considered a well known four-sized bar plane frame structure reference test case [2]. The loading of the deterministic optimum design has been considered as a characteristic load in the robust design with a real discrete cross-section type optimization for comparison.

Ten independent executions were performed using two evolutionary multiobjective algorithm (EMO): NSGAII, non-dominated sorting genetic algorithm II and SPEA2, strength pareto evolutionary approach 2. A population size of 200 individuals, uniform crossover, uniform mutation rate of 0.06, and a stop criterion of 100 generations were considered in all cases.

A total of eighteen different frame structural designs make up the obtained Pareto optimal front. They are detailed graphically and numerically in the paper. Both EMOs have similar behaviour, being capable to locate the extreme frame structural design solutions. The number of final Pareto front designs located is similar, although the NSGAII has a slightly higher variability in the produced front. Both algorithms also locate all the eighteen design solutions, although the constraint variations caused by the Monte-Carlo simulation produces the effect that some of this points are dominated and not shown in the final front results.

The deterministic optimum design has no constraint violations with its fixed loads, having a mass of 3324.3 kg. When the robust design is considered including the load variations, it is observed, that this design violation distribution has a typical deviation of 71 kg. and a mean of 18 kg. Therefore, the engineer or decision-maker, should select an individual among this deterministic optimum design and the most right solution of the front, which has no constraint violations at all, even in the stochastic case (corresponding to a design mass of 3618.8 kg), despite the uncertainty of the external loads. In this test case, the consideration of the five percent excess over the characteristic load has guided to an increment of 9% in structural mass. It is not a negligible value and it is an indicator of the failure probability allowed in the limit state theory used.

Robust optimum design of frame structures with real discrete cross-section types including all the possible constraints violations as variables has been handled successfully in this paper, having considered the modelling of uncertain loads by Monte-Carlo simulation and the multiobjective optimization using two efficient EMOs, both with similar performance.

1

C.A. Coello Coello, "Evolutionary Multi-Objective Optimization: A Historical View of the Field", IEEE Computational Intelligence Magazine, 1, 28-36, 2006. doi:10.1109/MCI.2006.1597059

2

D. Greiner, G. Winter, J.M. Emperador. "Optimising Frame Structures by different strategies of genetic algorithms", Finite Elements in Analysis and Design, 37(5), 381-402, 2001. doi:10.1016/S0168-874X(00)00054-8

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