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Benchmarking optimization solvers for structural topology optimization

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Abstract

The purpose of this article is to benchmark different optimization solvers when applied to various finite element based structural topology optimization problems. An extensive and representative library of minimum compliance, minimum volume, and mechanism design problem instances for different sizes is developed for this benchmarking. The problems are based on a material interpolation scheme combined with a density filter. Different optimization solvers including Optimality Criteria (OC), the Method of Moving Asymptotes (MMA) and its globally convergent version GCMMA, the interior point solvers in IPOPT and FMINCON, and the sequential quadratic programming method in SNOPT, are benchmarked on the library using performance profiles. Whenever possible the methods are applied to both the nested and the Simultaneous Analysis and Design (SAND) formulations of the problem. The performance profiles conclude that general solvers are as efficient and reliable as classical structural topology optimization solvers. Moreover, the use of the exact Hessians in SAND formulations, generally produce designs with better objective function values. However, with the benchmarked implementations solving SAND formulations consumes more computational time than solving the corresponding nested formulations.

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Notes

  1. There are implementations of these methods which do allow equality constraints see e.g. (Zhang et al. 1996; Wang et al. 2011). However, to the best of our knowledge there are no numerical results suggesting that these methods can be used to solve SAND formulation of topology optimization problems.

  2. MMA and GCMMA are considered as first-order methods. IPOPT, SNOPT and FMINCON are considered as second-order methods even though for certain problem formulations, limited memory BFGS (Broyden-Fletcher-Goldfarb-Shanno) is used to approximate the Hessian of the Lagrangian.

  3. A design, which is deemed incorrect by the optimality conditions, can indeed be a capable design and visually describe the correct topology. However, we experience that tight optimality conditions lead to better objective function values.

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Acknowledgments

We would like to thank Professor Krister Svanberg at KTH in Stockholm for providing the implementations of both MMA and GCMMA.

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Authors and Affiliations

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

    Susana Rojas-Labanda & Mathias Stolpe

Authors
  1. Susana Rojas-Labanda
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  2. Mathias Stolpe
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Corresponding author

Correspondence to Susana Rojas-Labanda.

Additional information

This research is funded by the Villum Foundation through the research project Topology Optimization – the Next Generation (NextTop).

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Rojas-Labanda, S., Stolpe, M. Benchmarking optimization solvers for structural topology optimization. Struct Multidisc Optim 52, 527–547 (2015). https://doi.org/10.1007/s00158-015-1250-z

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