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The ESO method revisited

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Abstract

This paper examines the evolutionary structural optimisation (ESO) method and its shortcomings. By proposing a problem statement for ESO followed by an accurate sensitivity analysis a framework is presented in which ESO is mathematically justifiable. It is shown that when using a sufficiently accurate sensitivity analysis, ESO method is not prone to the problem proposed by Zhou and Rozvany (Struct Multidiscip Optim 21(1):80–83, 2001). A complementary discussion on previous communications about ESO and strategies to overcome the Zhou-Rozvany problem is also presented. Finally it is shown that even the proposed rigorous ESO approach can result in highly inefficient local optima. It is discussed that the reason behind this shortcoming is ESO’s inherent unidirectional approach. It is thus concluded that the ESO method should only be used on a very limited class of optimisation problems where the problem’s constraints demand a unidirectional approach to final solutions. It is also discussed that the Bidirectional ESO (BESO) method is not prone to this shortcoming and it is suggested that the two methods be considered as completely separate optimisation techniques.

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Notes

  1. It should be noted that K is only invertible when the components corresponding to the restrained degrees of freedom are eliminated from it. Hereafter, by K we are actually referring to this reduced matrix.

  2. 2 In general we have δ m c≥δ m+n c, but the equal sign is only applicable when the whole structure is removed resulting in \({\Delta } c \rightarrow \infty \).

  3. A hyper-ball in (N−1)-dimensional space

  4. Due to the binary nature of design variables, the only feasible non-zero values in Δx are ±1. Keeping the volume constant requires the sum of the components in Δx to vanish. Thus the smallest feasible positive value of ||Δx|| is \(\sqrt {2}\).

  5. The difference between this number and the value of c=1121 reported for this design by Zhou and Rozvany (2001) is due to using analytically integrated stiffness matrices for finite elements in this paper. Also see Stolpe and Bendsøe (2011).

  6. 6 In contrast to what seems to be generally believed, it can be shown that the same can be true even for hard-kill BESO. This is however beyond the scope of this work and the author wishes to address this matter in a separate communication. For the current discussion, it is enough to accept that soft-kill BESO will not suffer from this shortcoming of ESO.

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  1. School of Civil Engineering and Surveying, University of Southern Queensland, Toowoomba, QLD, 4350, Australia

    Kazem Ghabraie

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  1. Kazem Ghabraie
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Ghabraie, K. The ESO method revisited. Struct Multidisc Optim 51, 1211–1222 (2015). https://doi.org/10.1007/s00158-014-1208-6

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