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On some fundamental properties of structural topology optimization problems

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Abstract

We study some fundamental mathematical properties of discretized structural topology optimization problems. Either compliance is minimized with an upper bound on the volume of the structure, or volume is minimized with an upper bound on the compliance. The design variables are either continuous or 0–1. We show, by examples which can be solved by hand calculations, that the optimal solutions to the problems in general are not unique and that the discrete problems possibly have inactive volume or compliance constraints. These observations have immediate consequences on the theoretical convergence properties of penalization approaches based on material interpolation models. Furthermore, we illustrate that the optimal solutions to the considered problems in general are not symmetric even if the design domain, the external loads, and the boundary conditions are symmetric around an axis. The presented examples can be used as teaching material in graduate and undergraduate courses on structural topology optimization.

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Acknowledgements

The presented examples came up after, sometimes lively, discussions about symmetry in the optimal design group at DTU Mathematics. I would like to thank my colleagues Oded Amir, Anders Astrup Larsen, Eduardo Munoz, and Nam Nguyen Canh for interesting discussions on this topic. I would also like to thank Ole Sigmund and two anonymous referees for their feedback on the manuscript which helped improve the presentation.

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  1. Department of Mathematics, Technical University of Denmark (DTU), Matematiktorvet, Building 303 S, 2800, Kgs. Lyngby, Denmark

    Mathias Stolpe

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  1. Mathias Stolpe
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Correspondence to Mathias Stolpe.

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Stolpe, M. On some fundamental properties of structural topology optimization problems. Struct Multidisc Optim 41, 661–670 (2010). https://doi.org/10.1007/s00158-009-0476-z

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  • DOI: https://doi.org/10.1007/s00158-009-0476-z

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