Topology optimization with worst-case handling of material uncertainties
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RESEARCH PAPER
Topology optimization with worst-case handling of material uncertainties Jannis Greifenstein1 · Michael Stingl1 Received: 26 January 2019 / Revised: 13 August 2019 / Accepted: 11 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this article, a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worst-case sense, i.e., the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative concave relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov-type and a barrier regularization scheme are developed, which render the resulting minimization problem continuously differentiable. The barrier regularization scheme turns out to be more suitable for the practical solution of the problem, as it can be closely linked to a highly efficient interior point approach used for the evaluation of the optimal value function and its gradient. Based on this, the outer minimization problem can be approached by a gradient-based optimization solver like the method of moving asymptotes. Examples from additive manufacturing as well as material degradation are examined, demonstrating the efficiency of the suggested method. Finally, the impact of the concave relaxation of the inner problem is investigated. In order to test the conservatism of the latter, a RAMP-type continuation scheme providing a lower bound for the inner problem is suggested and numerically tested. Keywords Topology optimization · Robust optimization · Interior point methods · Additive manufacturing
1 Introduction In the last decade, structural optimization has seen a vast increase in publications considering uncertainties in the optimization problem. Uncertainties in structural optimization can arise from a multitude of different sources, such as modelling errors, inaccurate loading scenarios, manufacturing precision, or defective material. In the literature, models and solution approaches are discussed for load uncertainties, see e.g., Elishakoff et al. (1994), Ben-Tal and Nemirovski (1997),
Responsible Editor: Xu Guo Jannis Greifenstein
[email protected] Michael Stingl [email protected] 1
Applied Mathematics (Continuous Optimization), Department of Mathematics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg (FAU), Cauerstr. 11, 91058 Erlangen, Germany
Lombardi and Haftka (1998), Jiang et al. (2007), Kogiso et al. (2008), de Gournay et al. (2008), Guo et al. (2009a), Dunning et al. (2011), Takezawa et al. (2011), Brittain et al. (2012), and Guo et al. (2015) or Thore et al. (2017). For geometric uncertainties including topological uncertainties at the boundary of a design body, see e.g., Guest and Igusa (2008), Sigmund (2009), Wang et al. (2011), Chen and Chen (2011), Scheve
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