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Symmetry and asymmetry of solutions in discrete variable structural optimization

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Abstract

In this paper symmetry and asymmetry of optimal solutions in symmetric structural optimization problems is investigated, based on the choice of variables. A group theory approach is used to define the symmetry of the structural problems in a general way. This approach allows the set of symmetric structures to be described and related to the entire search space of the problem. A relationship between the design variables and the likelihood of finding symmetric or asymmetric solutions to problems is established. It is shown that an optimal symmetric solution (if any) does not necessarily exist in the case of discrete variable problems, regardless of the size of the discrete, countable set from which variables can be chosen. Finally a number of examples illustrate these principles on simple truss structures with discrete topology and sizing variables.

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Notes

  1. The notation \(\mathbf {x}_{\iota \kappa }\) refers to the characteristics of the vector under the symmetry transformations \(\iota \) and \(\kappa \).

  2. The vertical gaps in the graph are a result of groups of binary numbers whose equivalent structures share common mechanical instabilities. These instabilities are considered non-feasible and therefore not plotted.

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Acknowledgement

The first author would like to thank the Fonds National de la Recherche Scientifique (FNRS, Belgium) for financial support of this research.

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Correspondence to James N. Richardson.

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Richardson, J.N., Adriaenssens, S., Bouillard, P. et al. Symmetry and asymmetry of solutions in discrete variable structural optimization. Struct Multidisc Optim 47, 631–643 (2013). https://doi.org/10.1007/s00158-012-0871-8

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  • DOI: https://doi.org/10.1007/s00158-012-0871-8

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