Structural and Multidisciplinary Optimization https://doi.org/10.1007/s00158-018-2122-0 RESEARCH PAPER Non-probabilistic robust continuum topology optimization with stress constraints Gustavo Assis da Silva 1 · Eduardo Lenz Cardoso 2 · Andr ´ e Te ´ ofilo Beck 1 Received: 6 May 2018 / Revised: 21 September 2018 / Accepted: 9 October 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract This paper proposes a non-probabilistic robust design approach, based on optimization with anti-optimization, to handle unknown-but-bounded loading uncertainties in stress-constrained topology optimization. The objective of the proposed topology optimization problem is to find the lightest structure that respects the worst possible scenario of local stress constraints, given predefined bounds on magnitudes and directions of applied loads. A solution procedure based on the augmented Lagrangian method is proposed, where worst-case local stress constraints are handled without employing aggregation techniques. Results are post-processed, demonstrating that maximum stress of robust solutions is almost insensitive with respect to changes in loading scenarios. Numerical examples also demonstrate that obtained robust solutions satisfy the stress failure criterion for any load condition inside the predefined range of unknown-but-bounded uncertainties in applied loads. Keywords Topology optimization · Stress constraints · Uncertainties · Non-probabilistic · Robust · Worst case 1 Introduction Topology optimization of continuum structures with local stress constraints was introduced in the literature by Duysinx and Bendsøe (1998) considering the density approach. Since then, several papers were developed aiming at overcoming well-known difficulties associated with this formulation: the local nature of stress failure criterion (Pereira et al. 2004), the singularity phenomenon (Cheng and Guo 1997; Duysinx and Bendsøe 1998), and the artificial stress concentration on jagged boundaries (Sv¨ ard 2015). While these difficulties were not completely overcome in the scope of density-based approach, research advanced over parallel topics, such as the consideration of uncertain- Responsible Editor: Xu Guo  Gustavo Assis da Silva gustavoassisdasilva@gmail.com 1 Department of Structural Engineering, S˜ ao Carlos School of Engineering, University of S˜ ao Paulo, S˜ ao Carlos, S˜ ao Paulo, 13.566-590, Brazil 2 Department of Mechanical Engineering, State University of Santa Catarina, Joinville, Santa Catarina, 89.219-710, Brazil ties in the stress-based formulations. These considerations naturally increase problem nonlinearity and, hence, the dif- ficulty of obtaining topology optimization solutions. In a comprehensive literature review, we identified few papers addressing topology optimization problems of continuum structures with stress constraints under uncertainty (Luo et al. 2014; Holmberg et al. 2017; da Silva and Cardoso 2017; Thore et al. 2017; da Silva et al. 2018; da Silva and Beck 2018). These works can be classified into two categories: probabilistic and non-probabilistic. In probabilistic approaches, uncertainties are quantified in terms of probabilities, and uncertain variables are represented as random variables. These approaches can be divided in two categories: reliability-based topology optimization (RBTO) and robust topology optimization (RTO). In the RBTO approach, stress constraints are imposed in terms of an allowable probability of failure for each point of stress computation (Luo et al. 2014; da Silva and Beck 2018). In the RTO approach, stress constraints are rewritten as a weighted sum between their expected value and standard deviation (da Silva and Cardoso 2017; da Silva et al. 2018). Both approaches are suitable for handling uncertainties described as random variables. RBTO approaches may be used when marginal probability density functions of random variables are available, whereas RTO approaches require