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On the equivalent static loads approach for dynamic response structural optimization

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Abstract

The equivalent static loads algorithm is an increasingly popular approach to solve dynamic response structural optimization problems. The algorithm is based on solving a sequence of related static response structural optimization problems with the same objective and constraint functions as the original problem. The optimization theoretical foundation of the algorithm is mainly developed in Park and Kang (J Optim Theory Appl 118(1):191–200, 2003). In that article it is shown, for a certain class of problems, that if the equivalent static loads algorithm terminates then the KKT conditions of the original problem and the final sub-problem are identical. The proof of this important theoretical result is unfortunately both incomplete and incorrect and the result is generally not valid. The missing parts of the proof are herein identified and corrected and the critical mistake in the proof is located and explained. We suggest a modified method in the same spirit for which the requested result is proved.

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Notes

  1. Since the static structural optimization problem (4) in general is non-convex solving should be interpreted as finding a point satisfying first-order necessary optimality conditions.

  2. A univariate example to illustrate this argument is f(x)= sin(x) which is zero for all x = ± k π for k=0,1,2,…. The derivative f ′(x)= cos(x) ≠ 0 at all those points.

  3. A univariate example to illustrate this argument is given by the two functions f 1(x)=x and f 2(x)=x 2. These are both zero at x=0 but f 1 ′(x)=1 and f 2 ′(x)=2x which is zero at x=0.

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Acknowledgements

Sincere thanks to my colleagues José Blasques and Susana Rojas Labanda for constructive and insightful comments and suggestions on a draft version of this note. I would also like to thank two reviewers for many valuable comments, questions, and suggestions.

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Authors and Affiliations

  1. Department of Wind Energy, Technical University of Denmark, Frederiksborgvej 399, 4000, Roskilde, Denmark

    Mathias Stolpe

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  1. Mathias Stolpe
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Correspondence to Mathias Stolpe.

Additional information

This work was funded by The Danish Council for Independent Research | Technology and Production Sciences through the research project Optimal Design of Composite Structures under Manufacturing Constraints.

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Stolpe, M. On the equivalent static loads approach for dynamic response structural optimization. Struct Multidisc Optim 50, 921–926 (2014). https://doi.org/10.1007/s00158-014-1101-3

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